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EE All Formula

The document is a formula sheet for the Electrical Engineering Department, covering essential topics such as Mathematics, Electromagnetic Fields, and Control Systems, with page references for each section. It also outlines online and offline services for exam aspirants, detailing access levels based on the type of coaching or materials purchased. Additionally, it includes mathematical concepts, theorems, and properties relevant for GATE, IES, and PSU examinations.

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0% found this document useful (0 votes)

247 views244 pages

EE All Formula

Uploaded by

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FOR UPCOMING EXAMS

 GATE …..
 PSU …..
 IES ….
Formula Sheet
(EE Department)

Index Page. No.

1. Mathematics --- --- 1 to 9

2. Electromagnetic Fields --- 10 to 16
3. Signals & Systems --- --- 17 to 22
4. Electrical Machines --- --- 23 to 58
5. Power Systems --- --- 59 to 89
6. Control Systems --- --- 90 to 92
7. Measurements --- --- 93 to 114
8. Analog Electronics --- --- 115 to 123
9. Digital Electronics --- --- 124 to 126
10. Microprocessors --- --- 127 to 130
11. Power Electronics --- --- 131 to 143
Online & Offline Services for Aspirants

1. Aspirants, who joined for Classroom Coaching in our Institute of

Engineering Studies, get Diamond user access in our portal along
with Classes.
2. Aspirants who take Postal series get Diamond access along with
books of our Study materials, Question Banks, Practice problems
books.
3. Aspirants who take Practice problems books (Without Study
Materials) get Shift access in our portal.
4. Aspirants who want to take Online Tests Conducted for GATE, IES
Objective and for Different PSU/Govt sector jobs will get Gold access
in our portal.
5. Those who don’t pay will get Silver access in our portal.

Silver Gold Shift Diamond

Few Online Tests Premium Online Tests Premium Online Tests Premium Online Tests
(for both GATE/PSU) (for both GATE/PSU) (for both GATE/PSU)
Practice Qns. In online Premium Downloads Premium Downloads Premium Downloads
without Solutions
Answering to Practice Questions in Practice Questions in Practice Questions in
reported Questions online with Solutions online with Solutions online with Solutions
with Least Priority
-NA- Answering to Answering to Answering to
reported Qns with reported Qns with reported Qns with
Highest Priority Highest Priority Highest Priority
-NA- -NA- All Practice problems All Practice problems
books to your & Study Materials
doorstep books to your
doorstep
-NA- Check Soln of any Qn Check Soln of any Qn Check Soln of any Qn
from Practice books & from Practice books & from Practice books &
Report Report Report
Institute of Engineering Studies (IES,Bangalore) Mathematics Formula Sheet
MatheMatics

Matrix :-
• If |A| = 0 → Singular matrix ; |A| ≠ 0 Non singular matrix
• Scalar Matrix is a Diagonal matrix with all diagonal elements are equal
• Unitary Matrix is a scalar matrix with Diagonal element as ‘1’ (AQ = (A∗ )T = A−1 )
• If the product of 2 matrices are zero matrix then at least one of the matrix has det zero
• Orthogonal Matrix if AAT = AT .A = I ⇒ AT = A−1
• A = AT → Symmetric
A = - AT → Skew symmetric

Properties :- (if A & B are symmetrical )

• A + B symmetric
• KA is symmetric
• AB + BA symmetric
• AB is symmetric iff AB = BA
• For any ‘A’ → A + AT symmetric ; A - AT skew symmetric.
• Diagonal elements of skew symmetric matrix are zero
• If A skew symmetric A2n → symmetric matrix ; A2n−1 → skew symmetric
• If ‘A’ is null matrix then Rank of A = 0.

Consistency of Equations :-
• r(A, B) ≠ r(A) is consistent
• r(A, B) = r(A) consistent &
if r(A) = no. of unknowns then unique solution
r(A) < no. of unknowns then ∞ solutions .

Hermition , Skew Hermition , Unitary & Orthogonal Matrices :-

• AT = A∗ → then Hermition
• AT = −A∗ → then Hermition
• Diagonal elements of Skew Hermition Matrix must be purely imaginary or zero
• Diagonal elements of Hermition matrix always real .
• A real Hermition matrix is a symmetric matrix.
• |KA| = K n |A|

Eigen Values & Vectors :-

• Char. Equation |A – λI| = 0.
Roots of characteristic equation are called eigen values . Each eigen value corresponds to non zero
solution X such that (A – λI)X = 0 . X is called Eigen vector .
• Sum of Eigen values is sum of Diagonal elements (trace)
• Product of Eigen values equal to Determinent of Matrix .
• Eigen values of AT & A are same
|A|
• λ is Eigen value of A then 1/ λ → A−1 & is Eigen value of adj A.
λ
• λ1 , λ2 …… λn are Eigen values of A then

KA → K λ1 , K λ2 ……..K λn

Am → λ1m , λm m
2 ………….. λn .

A + KI → λ1 + k , λ2 + k , …….. λn + k
(A − KI)2 → (λ1 − k)2 , ……… (λn − k)2

• Eigen values of orthogonal matrix have absolute value of ‘1’ .

• Eigen values of symmetric matrix also purely real .
• Eigen values of skew symmetric matrix are purely imaginary or zero .
• λ1 , λ2 , …… λn distinct eigen values of A then corresponding eigen vectors X1 , X 2, .. … X n for
linearly independent set .
2
• adj (adj A) = |A|n−2 ; | adj (adj A) | = |A|(n−1)

Complex Algebra :-

• Cauchy Rieman equations

𝜕𝜕𝑢𝑢 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
= ; =−
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 1 𝜕𝜕𝜕𝜕
=
� Neccessary & Sufficient Conditions for f(z) to be analytic
𝜕𝜕𝜕𝜕 r 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 1 𝜕𝜕𝜕𝜕
=− r
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

2πi
• ∫c f(z)/(Z − a)n+1 dz = n!
[ f n (a) ] if f(z) is analytic in region ‘C’ & Z =a is single point
(z−z0 ) (z−z0 )2 (z−z )n
• f(z) = f(z0 ) + f ′ (z0 ) + f ′′ (z0)
+ …… + f n (z0 ) n!0 + ………. Taylor Series
1! 2!
⇓
fn (z0 )
if z0 = 0 then it is called Mclauren Series f(z) = ∑∞ n
0 a n (z − z0 ) ; when a n = n !
• If f(z) analytic in closed curve ‘C’ except @ finite no. of poles then

∫c f(z)dz = 2πi (sum of Residues @ singular points within ‘C’ )

Res f(a) = lim(𝑍𝑍 − 𝑎𝑎 𝑓𝑓(𝑧𝑧)

𝑧𝑧→𝑎𝑎

= Φ(a) / φ′ (a)

1 𝑑𝑑 𝑛𝑛−1
= lim (𝑛𝑛−1)! 𝑑𝑑𝑧𝑧 𝑛𝑛−1
((Z − a)n f(z) )
𝑍𝑍→𝑎𝑎

Calculus :-

Rolle’s theorem :-

If f(x) is

(a) Continuous in [a, b]

Langrange’s Mean Value Theorem :-

If f(x) is continuous in [a, b] and differentiable in (a, b) then there exists atleast one value ‘C’ in (a, b)
f(b)−f(a)
such that f ′ (c) =
b−a

Cauchy’s Mean value theorem :-

If f(x) & g(x) are two function such that

(a) f(x) & g(x) continuous in [a, b]

(b) f(x) & g(x) differentiable in (a, b)

(c) g ′ (x) ≠ 0 ∀ x in (a, b)

Then there exist atleast one value C in (a, b) such that

f(b)−f(a)
f ′ (c) / g ′ (c) =
g(b)−g(a)

Properties of Definite integrals :-

b c b
• a < c < b ∫a f(x). dx = ∫a f(x). dx + ∫c f(x). dx
a a
• ∫0 f(x)dx = ∫0 f(a − x)dx

a a
• ∫−a f(x). dx = 2 ∫0 f(x)dx f(x) is even

= 0 f(x) is odd
a a
• ∫0 f(x). dx = 2 ∫0 f(x)dx if f(x) = f(2a- x)

• = 0 if f(x) = - f(2a – x)
na a
• ∫0 f(x). dx = n ∫0 f(x)dx if f(x) = f(x + a)

b b
• ∫a f(x). dx = ∫a f(a + b − x). dx

π/2 π/2 (n−1)(n−3)(n−5)………2

• ∫0 sinn x = ∫0 cosn x = if ‘n’ odd
n (n−2)(n−4)……….3

(n−1)(n−3)……1 𝜋𝜋
= . � � if ‘n’ even
n (n−2)(n−4)……….2 2

π/2 {(m−1)(m−3)….(m−5)……(2 or 1)} {(n−1)(n−3)…….(2 or 1)}.K

• ∫0 sinm x . cosn x . dx = (m+n) (m+n−2)(m+n−4)………2 or 1

Where K = π / 2 when both m & n are even otherwise k = 1

Maxima & Minima :-

A function f(x) has maximum @ x = a if f ′ (a) = 0 and f ′′ (a) < 0

A function f(x) has minimum @ x = a if f ′ (a) = 0 and f ′′ (a) > 0

Constrained Maximum or Minimum :-

To find maximum or minimum of u = f(x, y, z) where x, y, z are connected by Φ (x, y, z) = 0

Working Rule :-

(i) Write F(x, y, z) = f(x, y, z) + λ ϕ(x, y, z)

(ii) Obtain Fx = 0, Fy = 0 , Fz = 0

(ii) Solve above equations along with ϕ = 0 to get stationary point .

Laplace Transform :-

𝑑𝑑 𝑛𝑛
• L �𝑑𝑑𝑑𝑑 𝑛𝑛 𝑓𝑓(𝑠𝑠)� = s n f(s) - s n−1 f(0) - s n−2 f ′ (0) …… f n−1 (0)

dn
• L { t n f(t) } = (−1)n dsn
f(s)

f(t) ∞
• ⇔ ∫s f(s) ds
t

t
• ∫0 f(u) du ⇔ f(s) / s .

Inverse Transforms :-

s2 1
• (s2 +a2 )2
= [ sin at + at cos at]
2a

1 1
• = [ sin at - at cos at]
(s2 +a2 )2 2a3

s
• = Cos hat
s 2 − a2

a
• = Sin hat
s 2 − a2

T
∫0 e−st f(t)dt
Laplace Transform of periodic function : L { f(t) } = 1−e−sT

Numerical Methods :-

Bisection Method :-
x1 +x2
(1) Take two values of x1 & x2 such that f(x1 ) is +ve & f(x2 ) is –ve then x3 = find f(x3 ) if f(x3 )
2
+ve then root lies between x3 & x2 otherwise it lies between x1 & x3 .

Regular falsi method :-

x −x
Same as bisection except x2 = x0 - f(x 1)−f(x
0
f(x0 )
1 0)

Newton Raphson Method :-

f(xn )
xn+1 = xn –
f′ (xn )

Pi cards Method :-
x dy
yn+1 = y0 + ∫x f(x, yn ) ← = f(x, y)
0 dx

Taylor Series method :-

dy (x− x0 )2 (x− x0 )n
= f(x, y) y = y0 + (x- x0 ) (y ′ )0 + (y)′′
0 + …………. (y)n0
dx 2! n!

Euler’s method :-
dy
y1 = y0 + h f(x0 , y0 ) ← = f(x, y
dx

(1) h
y1 = y0 + [f(x0 , y0 ) + f(x0 + h, y1 )
2

Calculate till two consecutive value of ‘y’ agree

y2 = y1 + h f(x0 + h, y1 )

(1) h
y2 = y0 + [f(x0 + h, y1 ) + f(x0 + 2h, y2 )
2

………………

Runge’s Method :-
k1 = h f(x0 , y0 )
h k1 1
k 2 = h f( x0 + , y0 + ) finally compute K = (K1 + 4K 2 + K 3 )
2 2 6

k ′ = h f(x0 +h , y0 + k1 )

k 3 = h ( f (x0 +h , y0 + k ′ ))

Runge Kutta Method :-

k1 = h f(x0 , y0 )
h k1 1
k 2 = h f( x0 + , y0 + ) finally compute K = (K1 + 2K 2 + 2K 3 + K 4 )
2 2 6

h k2
k 3 = h f(x0 + , y0 + ) ∴ approximation vale y1 = y0 + K .
2 2

k 3 = h f (x0 +h , y0 + k 3 )

Trapezoidal Rule :-
x +nh h
∫x 0 f(x). dx =
2
[ ( y0 + yn ) + 2 (y1 + y2 + ……. yn−1 )]
0

f(x) takes values y0 , y1 …..

@ x0 , x1 , x2 ……..

Simpson’s one third rule :-

Simpson three eighth rule :-

x0 +nh 3h
∫x f(x). dx = 8
[ ( y0 + yn ) + 3 (y1 + y2 + y4 + y5 + ……. yn−1 )+ 2 (y3 + y6 + ⋯ … . + yn−3 ) ]
0

Differential Equations :-

Variable & Seperable :-

General form is f(y) dy = ϕ(x) dx

Sol: ∫ f(y) dy = ∫ ϕ(x) dx + C .

Homo generous equations :-

dy f(x,y)
General form = f(x, y) & ϕ(x, y) Homogenous of same degree
dx ϕ(x,y)

dy dv
Sol : Put y = Vx ⇒ =V+x & solve
dx dx

Reducible to Homogeneous :-
dy ax+by+c
General form =
dx a′ x+b′ y+c′

a b
(i) ≠
a′ b′

Sol : Put x=X+h y=Y+k

dy ax+by+(ah+bk+c) dy
⇒ = Choose h, k such that becomes homogenous then solve by Y = VX
dx a′ x+b′ y+(a′ h+b′ k+c′ ) dx

a b
(ii) =
a′ b′

a b 1
Sol : Let = =
a′ b′ m

dy ax+by+c
=
dx m(ax+by)+c

dy 𝑑𝑑𝑑𝑑
Put ax + by = t ⇒ =� − 𝑎𝑎�/b
dx 𝑑𝑑𝑑𝑑

Libnetz Linear equation :-

dy
General form +py = Q where P & Q are functions of “x”
dx

I.F = e∫ p.dx

Sol : y(I.F) = ∫ Q. (I. F) dx + C .

Exact Differential Equations :-

General form M dx + N dy = 0 M → f (x, y)

N → f(x, y)

∂M ∂N
If ∂y
= ∂x
then

Sol : ∫ M. dx + ∫(terms of N containing x ) dy = C

( y constant )

Rules for finding Particular Integral :-

1 1
eax = eax
f(D) f(a)

1
= x eax if f (a) = 0
f′ (a)

1
= x2 eax if f ′ (a) = 0
f′′ (a)
1 1
sin (ax + b) = sin (ax + b) f(- a2 ) ≠ 0
f(b2 ) f(−a2 )

1
= x sin (ax + b) f(- a2 ) = 0 Same applicable for cos (ax + b)
f′ (−a2 )

1
= x2 sin (ax + b)
f′′ (−a2 )

1
x m = [f(D)]y x m
f(D)

1 1
eax f(x) = eax f(x)
f(D) f(D+a)

Vector Calculus :-

∂Ψ ∂ϕ
∫C (ϕ dx + φ dy) = ∫ ∫ � ∂x − ∂y � dx dy

This theorem converts a line integral around a closed curve into Double integral which is special case of
Stokes theorem .

Series expansion :-

Taylor Series :-

f′ (a) f′′ (a) fn (a)

f(x) = f(a) + (x-a) + (x − a)2 + …………+ (x − a)n
1! 2! n!

f′ (0) f′′ (0) fn (0)

f(x) = f(0) + x + x 2 + …………+ x n + ……. (mc lower series )
1! 2! n!

n(n−1)
(1 + x)n = 1+ nx + x 2 + …… | nx| < 1
2

x2
ex = 1 + x + 2 ! + ……..

x3 x5
Sin x = x - 3!
+ 5!
- ……..

x2 x4
Cos x = 1 - 2!
+ 4 ! - ……..

Vector Calculus:-
→ A. (B × C) = C. (A × B) = B. (C × A)
→ A×(B×C) = B(A.C) – C(A.B) → Bac – Cab rule
(A.B)
→ Scalar component of A along B is AB = A Cos θAB = A . aB =
|B|
(A.B) B
� B = A Cos θAB . aB =
→ Vector component of A along B is A
|B|2

Laplacian of scalars :-
• ∮ A. ds = 𝑣𝑣 ∫(∇.𝐴𝐴)𝑑𝑑𝑑𝑑 → Divergence theorem
• L∮ A.dI = 𝑠𝑠 ∫(∇× 𝐴𝐴)𝑑𝑑𝑑𝑑 → Stokes theorem
• ∇2 A = ∇ (∇ .A) - ∇ × ∇ × A
• ∇ .A = 0 → solenoidal / Divergence loss ; ∇ .A > 0 → source ; ∇ .A < 0 ⇒ sink
• ∇ × A = 0 → irrotational / conservative/potential.
• ∇2 A = 0 → Harmonic .

Electrostatics :-
Q Qk (r−rk ) Q Q
• Force on charge ‘Q’ located @ r F = ∑N
k=1
� 12
; F12 = 4πε1 R23 . R
4πε0 |r−rk |3 0
1 (r−rk )
• E @ point ‘r’ due to charge located @ 𝑟𝑟 𝑠𝑠 ′
𝐸𝐸� = ∑N Q
4πε0 K=1 |r−rk 3 k
ρL
• E due to ∞ line charge @ distance ‘ ρ ‘ E = 2πε ρ
. aρ (depends on distance)
0
ρ
• E due to surface charge ρs is E = 2εs an . an → unit normal to surface (independent of distance)
0
• For parallel plate capacitor @ point ‘P’ b/w 2 plates of 2 opposite charges is
ρ ρ
E = s an - � s � (−𝑎𝑎𝑛𝑛 )
2ε0 2ε0
Q
• ‘E’ due to volume charge E = ar .
4πε0 R2
→ Electric flux density D = ε0 E D → independent of medium
Flux Ψ = s ∫ D .ds

Gauss Law :-
→ Total flux coming out of any closed surface is equal to total charge enclosed by surface .
Ψ = Q enclosed ⇒ ∫ D . ds = Q enclosed = ∫ ρv . dv
ρv = ∇. D
w B
→ Electric potential VAB = = - ∫A E. dI (independent of path)
Q
B Q
VAB = - ∫A ar . dr ar = VB - VA (for point charge )
4πε0 r2

• Potential @ any point (distance = r), where Q is located same where , whose position is vector @ r ′
Q
V= |r−r′ |
4πε0
Q
→ V(r) = + C . [ if ‘C’ taken as ref potential ]
4πε0 r
→ ∇ × E = 0, E = - ∇V

Boundary Conditions :- Et1 = Et2

• Tangential component of ‘E’ are continuous across dielectric-dielectric Boundary .
• Tangential Components of ‘D’ are dis continues across Boundary .
D
• Et1 = Et2 ; D1t = ε1 / ε2 .
2t
• Normal components are of ‘D’ are continues , where as ‘E’ are dis continues.
ε tan θ ε ε
• D1n - D2n = ρs ; E1n = 2 E2n ; tan θ1 = ε1 = εr1
ε1 2 2 r2
µ1
• H1t = H2t B12 = µ2
B2 t

µ2
B1n = B2n H1n = H2n
µ1

Maxwell’s Equations :-
d
→ faraday law Vemf = ∮ E. dI = - ∫ B. ds
dt

∂B ∂B
→ Transformer emf = ∮ E. dI = - ∫ ds ⇒ ∇ × E = -
∂t ∂t
s

→ Motional emf = ∇ × Em = ∇ × (μ × B).

∂D
→∇×H=J+
∂t

Electromagnetic wave propagation :-

• ∇ × H = J + 𝐷𝐷̇ D = εE ∇2 E = με𝐸𝐸̈
∇ × E = - 𝐵𝐵̇ B = μH ∇2 H = με𝐻𝐻̈
∇. D = ρv J = σE
∇.B = 0
Ey Ez
• Hz
=- Hy
= �µ/ε ; E.H = 0 E ⊥ H in UPW

For loss less medium ∇2 E - ρ2 E = 0 ρ = �jωµ(σ + jωϵ) = α + jβ.

µϵ σ 2
α = ω � ��1 + � � − 1�
2 ωϵ

• E(z, t) = E0 e−αz cos(ωt – βz) ; H0 = E0 / η .

jωµ
• η= � |η | < θη
σ+jωϵ
�µ/ε
• |η| = 1/4 tan 2θη = σ/ωε.
σ 2
�1+� � �
ωϵ
• η= α + jβ α → attenuation constant → Neper /m . | Np | = 20 log10 𝑒𝑒 = 8.686 dB
• For loss less medium σ = 0; α = 0.
• β → phase shift/length ; μ = ω / β ; λ = 2π/β .
Js 𝜎𝜎𝜎𝜎
• Jd
=� � = σ / ωϵ = tan θ → loss tanjent θ = 2θη
𝑗𝑗ωϵE
• If tan θ is very small (σ < < ωϵ) → good (lossless) dielectric
• If tan θ is very large (σ >> ωϵ) → good conductor
𝑗𝑗𝑗𝑗
• Complex permittivity ϵC = ϵ �1 − � = ε′ - j ε′′ .
𝜔𝜔𝜔𝜔
ε′′ σ
• Tan θ = ε′
= .
ωϵ

Plane wave in loss less dielectric :- ( σ ≈ 0)

1
• α = 0 ; β = ω√µϵ ; ω = ; λ = 2π/β ; η = �µr /εr ∠0.
√µϵ
• E & H are in phase in lossless dielectric

Free space :- (σ = 0, μ = µ0 , ε = ε0 )
• α = 0 , β = ω �µ0 ε0 ; u = 1/ �µ0 ε0 , λ = 2π/β ; η = �µ0 /ε0 < 0 = 120π ∠0
Here also E & H in phase .

Good Conductor :-
σ > > ωϵ σ/ωϵ → ∞ ⇒ σ = ∞ ε = ε0 ; μ = µ0 µr

Wµ
• α = β = �πfµσ ; u = �2ω/µσ ; λ = 2π / β ; η = � ∠450
σ
• Skin depth δ = 1/α
1 1+j
• η = √2 ejπ/4 =
σδ σδ
1 πfµ
• Skin resistance R s = =�
σδ σ
Rs .l
• R ac =
w
l
• R dc = .
σs

Poynting Vector :-
∂ 1
• ∫ (E × H) ds = - ∫ [ εE 2 + μH 2 ] dv – ∫ σ E 2 dv
dt 2
S v

Direction of propagation :- (𝐚𝐚𝐤𝐤 )

ak × aE = aH

aE × aH = ak
→ Both E & H are normal to direction of propagation
→ Means they form EM wave that has no E or H component along direction of propagation .

Reflection of plane wave :-

(a) Normal incidence
E η −η
Reflection coefficient Γ = Er0 = η2 + η1
i0 2 1
E 2η2
Txn coefficient Τ = Et0 = η
i0 2 + η1

Medium-I Dielectric , Medium-2 Conductor :-

𝛈𝛈𝟐𝟐 > 𝛈𝛈𝟏𝟏 :-
Γ>0 , there is a standing wave in medium & Txed wave in medium ‘2’.
Max values of | E1 | occurs
−nλ
Zmax = - nπ/β1 = 2 1 ; n = 0, 1, 2….
−(2n+1)π −(2n+1)λ1
Zmin = =
2β1 4

−(2n+1)π −(2n+1)π −(2n+1)λ1

𝛈𝛈𝟐𝟐 < 𝛈𝛈𝟏𝟏 :- E1max occurs @ β1 Zmax = ⇒ Zmax = =
2 2β1 4

−nπ −nλ1
β1 Zmin = nπ ⇒ Zmin = =
β1 2
H1 min occurs when there is |t1 |max
|E | |H | 1+|Γ| s−1
S = |E1 |max = |H1 |max = ;|Γ|=
1 min 1 min 1−|Γ| s+1
Since |Γ| < 1 ⇒ 1 ≤ δ ≤ ∞

Transmission Lines :-
• Supports only TEM mode
• LC = με ; G/C = σ /ε .
d2 Vs d2 Is
• dz2
- r 2 Vs = 0 ; dz2
- r 2 Is = 0
• Γ = �(R + jωL)(G + jωC) = α + jβ
• V(z, t) = V0+ e−αz cos (ωt- βz) + V0− eαz cos (ωt + βz)
V− R+jωL γ R+jωL
• Z0 = − I−0 = = =�
0 γ G+jωC G+jωC

Lossless Line : (R = 0 =G; σ = 0)

Distortion less :(R/L = G/C)

G R
→ α = √RG ; β = ωL� = ωC� = ω √LC
R G
R L 1
→ Z0 = � = � ; λ = 1/f √LC ; u = = Vp ; uz0 = 1/C , u /z0 = 1/L
G C √LC

i/p impedance :-
𝑍𝑍 + Z tan hl
Zin = Z0 � 𝐿𝐿+ Z 0 tan hl� for lossless line γ = jβ ⇒ tan hjβl = j tan βl
𝑍𝑍0 L
𝑍𝑍𝐿𝐿 + 𝑗𝑗Z0 tan βl
Zin = Z0 � + Z tan βl �
𝑍𝑍0 L

Z −Z
• VSWR = ΓL = ZL+Z0
L 0
• CSWR = - ΓL
• Transmission coefficient S = 1 + Γ
V I 1+| ΓL | Z Z0
• SWR = Vmax = Imax = = ZL =
min min | 1−|ΓL 0 ZL
(ZL > Z0 ) (ZL < Z0 )

Vmax
• |Zin |max = Imin
= SZ0
Vmin
• |Zin |min = = Z0 /S
Imax

Shorted line :- ΓL = -1 , S = ∞ Zin = Zsc = jZ0 tan βl

• ΓL = -1 , S = ∞ Zin = Zsc = j Z0 tan βl.

• Zin may be inductive or capacitive based on length ‘0’

If l < λ / 4 → inductive (Zin +ve)

λ
< l < λ/2 → capacitive (Zin -ve)
4

Open circuited line :-

Zin = Zoc = -jZ0 cot βl
Γl = 1 s = ∞ l < λ / 4 capacitive
λ
< l < λ/2 inductive
4
Zsc Zoc = Z02

Matched line : (ZL = Z0 )

Zin = Z0 Γ = 0 ; s =1
No reflection . Total wave Txed . So, max power transfer possible .

𝑍𝑍𝑠𝑠𝑠𝑠 =∞
l = λ /4 → 𝑍𝑍𝑜𝑜𝑜𝑜 =0
� → impedance inverter @ l = λ /4

𝑍𝑍𝑠𝑠𝑠𝑠 =0
l = λ /2 : Zin = Z0 ⇒ 𝑍𝑍𝑜𝑜𝑜𝑜 =∞
� impedance reflector @ l = λ /2

Wave Guides :-
TM modes : (Hz = 0)
𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛
Ez = E0 sin � �x sin � � y e−nz
𝑎𝑎 𝑏𝑏

mπ 2 nπ 2
h2 = k 2x + k 2y ∴ γ = �� + � � − ω2 µε where k = ω √µϵ
a b
m→ no. of half cycle variation in X-direction
n→ no. of half cycle variation in Y- direction .
1 2 2
Cut off frequency ωC = ��mπ� + �nπ� γ = 0; α = 0 = β
√µϵ a b
mπ 2 nπ 2
• k2 < � � + � � → Evanscent mode ; γ=α; β=0
a b
mπ 2 nπ 2
• k2 > � � + � � → Propegation mode γ = jβ α = 0
a b

mπ 2 nπ 2
β = �k 2 − � � −� �
a b

u′p 2 2
• fc = ��m� + �n� u′p = phase velocity = 1
is lossless dielectric medium
2 a b √µϵ

2
• λc = u′ /fc = m n
�( a )2 +(b)2

f 2
• β = β′ �1 − � c � β′ = ω/ W β′ = phase constant in dielectric medium.
f

• up = ω/β λ = 2π/β = up /f → phase velocity & wave length in side wave guide

E E β µ f 2
• ηTM = Hx = - Hy = = � �1 − � fc �
y x ωϵ ε

f 2
ηTM = η′ �1 − � fc � η′ → impedance of UPW in medium

TE Modes :- (𝐄𝐄𝐳𝐳 = 0)

𝑚𝑚𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛𝑛𝑛
→ Hz = H0 cos � � cos � � e−nz
𝑎𝑎 𝑏𝑏

wµ f 2
→ ηTE = = η′ / �1 − � fc �
β

→ ηTE > ηTM

→ TE10 Dominant mode

Antennas :-
jI β dl
Hertzian Dipole :- HΦs = 0 sin θ e−jβγ Eθs = ηHΦs
4πr

Half wave Dipole :-

π
jI0 e−jβγ cos� cos θ�
2
Hϕs = 2πγ sin θ
; Eθs = ηHΦs

Signals & Systems

∞
→ Energy of a signal ∫−∞ |x(t)|2 dt = ∑∞
𝑛𝑛=−∞ |𝑥𝑥[𝑛𝑛]|
2

1 T 1
→ Power of a signal P = lim ∫−T |x(t)|2 dt = lim ∑N
n=−N |x[n]|
2
𝑇𝑇→∞ 2𝑇𝑇 𝑁𝑁→∞ 2𝑁𝑁+1

→ x1 (t) → P1 ; x2 (t) → P2
x1 (t) + x2 (t) → P1 + P2 iff x1 (t) & x2 (t) orthogonal

→ Shifting & Time scaling won’t effect power . Frequency content doesn’t effect power.

→ if power = ∞ → neither energy nor power signal

Power = 0 ⇒ Energy signal
Power = K ⇒ power signal

→ Energy of power signal = ∞ ; Power of energy signal = 0

→ Generally Periodic & random signals → Power signals

Aperiodic & deterministic → Energy signals

Precedence rule for scaling & Shifting :

x(at + b) → (1) shift x(t) by ‘b’ → x(t + b)

(2) Scale x(t + b) by ‘a’ → x(at + b)

x( a ( t + b/a)) → (1) scale x(t) by a → x(at)

(2) shift x(at) by b/a → x (a (t+b/a)).

𝑡𝑡−𝑏𝑏
→ x(at +b) = y(t) ⇒ x(t) = y � �
𝑎𝑎

t
• Step response s(t) = h(t) * u(t) = ∫−∞ h(t)dt S’ (t) = h(t)
𝑛𝑛
S[n] = ∑𝑘𝑘=0 ℎ[𝑛𝑛] h[n] = s[n] – s[n-1]
1
• e−at u(t) * e−bt u(t) = [ e−at - e−bt ] u(t) .
b−a
• A1 Rect (t / 2T1 ) * A2 Rect(t / 2T2 ) = 2 A1 A2 min (T1 , T2 ) trapezoid (T1 , T2 )

• Rect (t / 2T) * Rect (t / 2T) = 2T tri(t / T)

Hilbert Transform Pairs :

∞ 2 /2σ2 ∞ 2 /2σ2
∫−∞ e−x dx = σ √2π ; ∫−∞ x 2 e−x dx = σ3 √2π σ > 0

1 σ+j∞
x(t) = ∫ X(s) est ds
2πj σ−j∞

∞
X(s) = ∫−∞ x(t) e−st ds

Initial & Final value Theorems :

x(t) = 0 for t < 0 ; x(t) doesn’t contain any impulses /higher order singularities @ t =0 then

x( 0+ ) = lim 𝑠𝑠 𝑋𝑋(𝑠𝑠)
𝑠𝑠→∞

x(∞) = lim 𝑠𝑠 𝑋𝑋(𝑠𝑠)

𝑠𝑠→0

Properties of ROC :-

1. X(s) ROC has strips parallel to jω axis

2. For rational laplace transform ROC has no poles

3. x(t) → finite duration & absolutely integrable then ROC entire s-plane

4. x(t) → Right sided then ROC right side of right most pole excluding pole s = ∞

5. x(t) → left sided ROC left side of left most pole excluding s= - ∞

6. x(t) → two sided ROC is a strip

7. if x(t) causal ROC is right side of right most pole including s = ∞

8. if x(t) stable ROC includes jω-axis

Z-transform :-
1
x[n] =
2πj
∮ x(z) z n−1 dz

X(z) = ∑∞
n=−∞ x[n] z
−n

Initial Value theorem :

If x[n] = 0 for n < 0 then x[0] = lim 𝑋𝑋(𝑧𝑧)
𝑧𝑧→∞

Final Value theorem :-

lim 𝑥𝑥[𝑛𝑛] = lim(𝑧𝑧 − 1) X(z)
→∞ 𝑧𝑧→1

Properties of ROC :-

1.ROC is a ring or disc centered @ origin

2. DTFT of x[n] converter if and only if ROC includes unit circle
3. ROC cannot contain any poles
4. if x[n] is of finite duration then ROC is enter Z-plane except possibly 0 or ∞
5. if x[n] right sided then ROC → outside of outermost pole excluding z = 0
6. if x[n] left sided then ROC → inside of innermost pole including z =0
7. if x[n] & sided then ROC is ring
8. ROC must be connected region
9.For causal LTI system ROC is outside of outer most pole including ∞
10.For Anti Causal system ROC is inside of inner most pole including ‘0’
11. System said to be stable if ROC includes unit circle .
12. Stable & Causal if all poles inside unit circle
13. Stable & Anti causal if all poles outside unit circle.

Phase Delay & Group Delay :-

When a modulated signal is fixed through a communication channel , there are two different delays to be
considered.

(i) Phase delay:

Signal fixed @ o/p lags the fixed signal by ∅(ωc ) phase
∅(ωc )
τP = - where ∅(ωc ) = K H(jω)
ωc
↓
Frequency response of channel
d∅(ω)
Group delay τg = − � for narrow Band signal
dω ω= ωc
↓
Signal delay / Envelope delay

Probability & Random Process:-

P(A∩B)
→ P (A/B) =
P(B)
→ Two events A & B said to be mutually exclusive /Disjoint if P(A ∩ B) =0
→ Two events A & B said to be independent if P (A/B) = P(A) ⇒ P(A ∩ B) = P(A) P(B)
B
P(Ai ∩B) P� � P(Ai)
Ai
→ P(Ai / B) = = B
P(B) ∑n
i=1 P� � P(Ai)
Ai

CDF :-
Cumulative Distribution function Fx (x) = P { X ≤ x }

Properties of CDF :

• Fx (∞) = P { X ≤ ∞ } = 1
• Fx (- ∞) = 0
• Fx (x1 ≤ X ≤ x2 ) = Fx (x2 ) - Fx (x1 )

PDF :-
d
Pdf = fx (x) = Fx (x)
dx

Pmf = fx (x) = ∑∞
i=−∞ P{X = xi } δ(x = xi )

Properties:-
• fx (x) ≥ 0
x
• Fx (x) = fx (x) * u(x) = ∫−∞ fx (x) dx

∞
• Fx (∞) = ∫−∞ fx (x) dx =1 so, area under PDF = 1

x
• P { x1 < X ≤ x2 } = ∫x 2 fx (x)dx
1

Mean & Variance :-

∞
Mean µx = E {x} = ∫−∞ x fx (x) dx

Variance σ2 = E { (X − µx )2 } = E {x 2 } - µ2x
∞
→ E{g(x)} = ∫−∞ g(x) fx (x) dx

Uniform Random Variables :

Random variable X ~ u(a, b) if its pdf of form as shown below

1
; 𝑎𝑎 < 𝑥𝑥 ≤ 𝑏𝑏
fx (x) = �𝑏𝑏−𝑎𝑎
0 , 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

1 ; 𝑥𝑥 > 𝑏𝑏
𝑥𝑥−𝑎𝑎
Fx (x) = �𝑏𝑏−𝑎𝑎 ; 𝑎𝑎 < 𝑥𝑥 < 𝑏𝑏
0 ; 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
a+b
Mean =
2

a2 + ab+ b2
Variance = (b − a)2 / 12 E{ x 2 } = 3

Gaussian Random Variable :-

X ~ N (µ1 σ2)
∞ 1 2 /2σ2
Mean = ∫−∞ x e−(x−µ) dx = μ
√2πσ2

1 ∞ 2 /2σ2
Variance =
√2πσ2
∫−∞ x 2 e−(x−µ) dx = σ2

Exponential Distribution :-

fx (x) = λ e−λx u(x)

Fx (x) = ( 1- e−λx ) u(x)

Laplacian Distribution :-
λ
fx (x) = e−λ |x|
2

Multiple Random Variables :-

• FXY (x , y) = P { X ≤ x , Y ≤ y }
• FXY (x , ∞) = P { X ≤ x } = Fx (x) ; Fxy (∞ , y) = P { Y < y } = FY (y)
• Fxy (-∞, y) = Fxy (x, - ∞) = Fxy (-∞, -∞) = 0
∞ ∞
• fx (x) = ∫−∞ fxy (x, y) dy ; fY (y) = ∫−∞ fxy (x, y) dx

Y P{Y ≤y, X ≤x} FXY (x,y)

• FY/X � ≤ x� = =
X P{X ≤x} FX (x)

fxy (x,y)
• fY/X (y/x) = fx (x)

Independence :-
• X & Y are said to be independent if FXY (x , y) = FX (x) FY (y)
⇒ fXY (x, y) = fX (x) . fX (y) P { X ≤ x, Y ≤ y} = P { X ≤ x} . P{Y ≤ y}

Correlation:
∞ ∞
Corr{ XY} = E {XY} = ∫−∞ ∫−∞ fxy (x, y). xy. dx dy
If E { XY} = 0 then X & Y are orthogonal .

Uncorrelated :-
Covariance = Cov {XY} = E { (X - µx ) (Y- µy }
= E {xy} – E {x} E{y}.
If covariance = 0 ⇒ E{xy} = E{x} E{y}

Random Process:-
Take 2 random process X(t) & Y(t) and sampled @ t1 , t 2

X(t1 ) , X(t 2 ) , Y(t1 ) , Y (t 2 ) → random variables

→ Auto correlation R x (t1 , t 2 ) = E {X(t1 ) X(t 2 ) }

→ Auto covariance Cx (t1 , t 2 ) = E { X(t1 ) - µx (t1 )) (X(t 2 ) - µx (t 2 ) } = R x (t1 , t 2 ) - µx (t1 ) µx (t 2 )
→ cross correlation R xy(t1 , t 2 ) = E { X(t1 ) Y(t 2 ) }
→ cross covariance Cxy(t1 , t 2 ) = E{ X(t1 ) - µx (t1 )) (Y(t 2 ) - µy (t 2 ) } = R xy(t1 , t 2 ) - µx (t1 ) µy (t 2 )

→ CXY (t1 , t 2 ) = 0 ⇒ R xy(t1 , t 2 ) = µx (t1 ) µy (t 2 ) → Un correlated

→ R XY (t1 , t 2 ) = 0 ⇒ Orthogonal cross correlation = 0
→ FXY (x, y ! t1 , t 2 ) = Fx (x! t1 ) Fy (y ! t 2 ) → independent

Properties of Auto correlation :-

• R x (0) = E { x 2 }
• R x (τ) = R x (-τ) → even
• | R x (τ) | ≤ R x (0)

Cross Correlation

• R xy(τ) = R yx(-τ)
• R2xy(τ) ≤ R x (0) . R y (0)
• 2 | R xy(τ)| ≤ R x (0) + R y (0)

Power spectral Density :-

∞
• P.S.D Sx (jω) = ∫−∞ R x (τ) e−jωτ dτ

1 ∞
R x (τ) = ∫ 𝑆𝑆 (jω)ejωτ dω
2π −∞ 𝑥𝑥
2
• Sy (jω) = Sx (jω) |H(jω)|
1 ∞
• Power = R x (0) = ∫ 𝑆𝑆 (jω) dω
2π −∞ 𝑥𝑥
• R x (τ) = k δ(τ) → white process

Properties :
• Sx (jω) even
• Sx (jω) ≥ 0

Lap Winding Wave Winding

(1) Coil Span : S S
Ycs = P Ycs = P
(2) Back Pitch Yb = U Ycs Yb = U Ycs
(3) Commutator Pitch Yc = 1 2 (c+1)
Yc = p
for Progressive winding
for Progressive winding
Yc = -1 2 (c+1)
for Retrogressive winding Yc = - p
for Restrogressive winding
(Yc Must be integer)
(4) Front Pitch Yf =Yb +2 Yf =2Yc - Yb
for Progressive winding
Yf =Yb -2
for Retrogressive winding

(5) Parallel Paths A=P A=2

(6) Conductor Current I I
Ic = Aa Ic = 2a
(7) No of brushes No of brushes = A = P No of brushes = 2

• S = No of commutator segments
• P = No of poles
2C
• U = No of coil sides / No of poles = S
• C = No of coils on the rotor
• A = No of armature parallel paths
• Ia = Armature current

phasor sum coil emf chord 2

→ Distribution factor (K d ) = arthematic sum of coil emf = arc
=π
elecrrical angle of coil
→ Pitch factor ( K p ) = 1800
*100%
P
→ θ0electrical = 2 θ0mechanical
ZI
→ Armature mmf/Pole (Peak) , ATa = 2AP
a

ZI pole arc
→ AT (Compensating Winding) = 2AP
a
* pole pitch
B
→ AT(Inter pole) = ATa + µ i lgi
0
Where Bi = Flux density in inter pole airgap

AT(Inter pole)
→ No of turns in each interpole , Ninterpole = Ia
Z pole arc
→ The no of compensating conductor per pole, Ncw /pole = 2 A P (pole pitch )
→ The Mechanical power that is converted is given by Pconv = Tind ωm
Where T = Induced torque

ωm = Angular speed of the machines rotor

→ The resulting electric power produced Pconv = EA IA

→ The power balance equation of the DC Machine is Tind ωm = EA IA
∅ZNP
→ The induced emf in the armature is Ea = 60A
PZ
→ Torque developed in Dc machine , Te = 2πA ∅ Ia

Where ∅ = Flux\pole , Z = No of armature conductors , P = No of poles , N = Speed in rpm ,

A = No of armature parallel paths, Ia = Armature current

→ The terminal voltage of the DC generator is given by Vt = Ea - Ia R a

→ The terminal voltage of the DC motor is given by Vt = Ea + Ia R a
ωnl− ωfl Nnl− Nfl
→ Speed regulation of dc machine is given by ,SR = * 100 % = * 100 %
ωfl Nfl
Vnl− Vfl
→ Voltage regulation , VR = Vfl
* 100 %

Shunt Generator:

→ For a shunt generator with armature induced voltage Ea, armature current Ia and
armature resistance Ra, the terminal voltage V is:
V = Ea - IaRa
→ The field current I f for a field resistance R f is:
If = V / Rf
→ The armature induced voltage Ea and torque T with magnetic flux Φ at angular
speed ω are:
Ea = k fΦω = kmω
T = k fΦIa = kmIa
where k f and km are design coefficients of the machine.
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Institute of Engineering Studies (IES,Bangalore) Electrical Machines Formula Sheet

Note that for a shunt generator:

- induced voltage is proportional to speed,
- torque is proportional to armature current.

→ The airgap power Pe for a shunt generator is:

Pe = ωT = EaIa = kmω Ia

Series Generator:

→ For a series generator with armature induced voltage Ea, armature current Ia,
armature resistance Ra and field resistance R f, the terminal voltage V is:
V = Ea - ( IaRa + IaR f )= Ea - Ia(Ra + R f)
The field current is equal to the armature current.
→ The armature induced voltage Ea and torque T with magnetic flux Φ at angular
speed ω are:
Ea = k fΦω Ia = kmω Ia
T = k fΦIa2 = kmIa2
where k f and km are design coefficients of the machine.

Note that for a series generator:

- induced voltage is proportional to both speed and armature current,
- torque is proportional to the square of armature current,
- armature current is inversely proportional to speed for a constant Ea

→ The airgap power Pe for a series generator is:

Pe = ωT = EaIa = kmω Ia2
→ Cumulatively compounded DC generator : - ( long shunt)
(a) Ia = If + IL
(b) Vt = Ea - Ia (R a + R s )
V
(c) Isf = Rx = shunt field current
f
(d) The equivalent effective shunt field current for this machine is given by
Nse Armature reaction MMF
Isf =Isf + Ia - ( )
Nf Nf

Where Ns e = No of series field turns

Nf = = No of shunt field turns

Where Ns e = No of series field turns

Nf = = No of shunt field turns

Shunt Motor:

→ For a shunt motor with armature induced voltage Ea, armature current Ia and
armature resistance Ra, the terminal voltage V is:
V = Ea + IaRa
The field current I f for a field resistance R f is:
If = V / Rf
→ The armature induced voltage Ea and torque T with magnetic flux Φ at angular
speed ω are:
Ea = k fΦω = kmω
T = k fΦIa = kmIa
where k f and km are design coefficients of the machine.

Note that for a shunt motor:

- induced voltage is proportional to speed,
- torque is proportional to armature current.

→ The airgap power Pe for a shunt motor is:

Pe = ωT = EaIa = kmω Ia
V Ra
→ The speed of the shunt motor , ω = K∅ - (K∅)2T

PZ
Where K = 2πA

Series Motor :

→ For a series motor with armature induced voltage Ea, armature current Ia,
armature resistance Ra and field resistance R f, the terminal voltage V is:

Note that for a series motor:

→ The airgap power Pe for a series motor is:

Pe = ωT = EaIa = kmω Ia2

Losses:

→ constant losses (P k) = Pw f + Pi o

Where, Pio = No of load core loss

→ Pwf = Windage & friction loss

→ Variable losses (Pv ) = Pc + Ps t + Pb
2
where Pc = Copper losses = Ia R a

Ps t = Stray load loss = α I2

Pb = Brush Contact drop = Vb Ia , Where Vb = Brush voltage drop

→ The total machine losses , PL = Pk +Vb Ia + K v Ia 2

Efficiency

→ The per-unit efficiency η of an electrical machine with input power Pin, output
power Pout and power loss Ploss is:

η = Pout / Pin = Pout / (Pout + Ploss) = (Pin - Ploss) / Pin

→ Rearranging the efficiency equations:

Pin = Pout + Ploss = Pout / η = Ploss / (1 - η)

Pout = Pin - Ploss = ηPin = ηPloss / (1 - η)

Ploss = Pin - Pout = (1 - η)Pin = (1 - η)Pout / η

Temperature Rise:

→ The resistance of copper and aluminium windings increases with temperature,

and the relationship is quite linear over the normal range of operating
temperatures. For a linear relationship, if the winding resistance is R1 at
temperature θ1 and R2 at temperature θ2, then:

R1 / (θ1 - θ0) = R2 / (θ2 - θ0) = (R2 - R1) / (θ2 - θ1)

where θ0 is the extrapolated temperature for zero resistance.

→ The ratio of resistances R2 and R1 is:

R2 / R1 = (θ2 - θ0) / (θ1 - θ0)

→ The average temperature rise ∆θ of a winding under load may be estimated from
measured values of the cold winding resistance R1 at temperature θ1 (usually
ambient temperature) and the hot winding resistance R2 at temperature θ2, using:
∆θ = θ2 - θ1 = (θ1 - θ0) (R2 - R1) / R1

→ Rearranging for per-unit change in resistance ∆Rpu relative to R1:

∆Rpu = (R2 - R1) / R1 = (θ2 - θ1) / (θ1 - θ0) = ∆θ / (θ1 - θ0)

.Copper Windings:

→ The value of θ0 for copper is - 234.5 °C, so that:

∆θ = θ2 - θ1 = (θ1 + 234.5) (R2 - R1) / R1

→ If θ1 is 20 °C and ∆θ is 1 degC:
∆Rpu = (R2 - R1) / R1 = ∆θ / (θ1 - θ0) = 1 / 254.5 = 0.00393

→ The temperature coefficient of resistance of copper at 20 °C is 0.00393 per

degC.

Aluminium Windings:

→ If θ1 is 20 °C and ∆θ is 1 degC:
∆Rpu = (R2 - R1) / R1 = ∆θ / (θ1 - θ0) = 1 / 248 = 0.00403

→ The temperature coefficient of resistance of aluminium at 20 °C is 0.00403 per

degC.

Dielectric Dissipation Factor:

→ If an alternating voltage V of frequency f is applied across an insulation system

comprising capacitance C and equivalent series loss resistance RS, then the
voltage VR across RS and the voltage VC across C due to the resulting
current I are:
VR = IRS
VC = IXC
V = (VR2 + VC2)½

→ The dielectric dissipation factor of the insulation system is the tangent of the
dielectric loss angle δ between VC and V:
tanδ = VR / VC = RS / XC = 2πfCRS
RS = XCtanδ = tanδ / 2πfC

→ The dielectric power loss P is related to the capacitive reactive power QC by:
P = I2RS = I2XCtanδ = QCtanδ

→ The power factor of the insulation system is the cosine of the phase
angle φ between VR and V:
cosφ = VR / V
so that δ and φ are related by:
δ + φ = 90°

→ tanδ and cosφ are related by:

tanδ = 1 / tanφ = cosφ / sinφ = cosφ / (1 - cos2φ)½
so that when cosφ is close to zero, tanδ ≈ cosφ

TRANSFORMERS:

→ Gross cross sectional area = Area occupied by magnetic material + Insulation

material.

t
Weight of
→ Specific weight of t/f = f
t
KVA rating of
f

Net Cross Sectional area

→ Stacking/iron factor :- (k s ) = Gross Cross Sectional area

→ k s is always less than 1

→ Gross c.s Area = AG = length × breadth
→ Net c.s Area = An = k s × AG
Effective C.S.Area
→ Utilization factor of transformer core = Total C.S Area
U.F of cruciform core = 0.8 to
0.85
mmF
→ Flux = Reluctance = = ϕm sin ωt
dϕ d
→ According to faradays second law e1 = −N1 dt = −N1 dt �ϕm sin ωt�
Instantaneous value e1 = N1 ϕm ω sin�ωt − π�2�
of emf in primary

→ Transformer emf equations :-

E1 = 4.44 N1 Bmax An f  (1)
E2 = 4.44 N2 Bmax An f  (2)

E
→ Emf per turn in Iry = N1 = 4.44 Bmax An f
1
ry E2
→ Emf per turn in II = = 4.44 Bmax An f
N2

⟹ Emf per turn on both sides of the transformer is same

E1 E2
=
N1 N2
E1 N1 1
⟹ = =
E2 N2 k
E2
Transformation ratio = K = =
E1

→ For a single-phase transformer with rated primary voltage V1, rated primary
current I1, rated secondary voltage V2 and rated secondary current I2, the voltampere
rating S is:
S = V1I1 = V2I2

→ For a balanced m-phase transformer with rated primary phase voltage V1, rated
primary current I1, rated secondary phase voltage V2 and rated secondary current I2,
the voltampere rating S is:
S = mV1I1 = mV2I2

→ The primary circuit impedance Z1 referred to the secondary circuit for an ideal
transformer with N1 primary turns and N2 secondary turns is:
Z12 = Z1(N2 / N1)2

→ During operation of transformer :-

E1 V1
Bm ∝ ∝
f f
V1
Bmax = constant ⟹ = constant
f

Equivalent ckt of t/f under N.L condition :-

Iw Iµ
V1 R0 X0 E1 E2

No load /shunt branch.

→ No load current = I0 = Iµ + Iw = I0 �−ϕ0
Iw = I0 cos ϕ0
Iµ = I0 sin ϕ0
→ No load power = v1 I0 cos ϕ0 = v1 Iw = Iron losses.
v1 v1 No load power
R0 = Iw1
; X0 = Iµ
⟹ Iw =
V1
𝐫𝐫𝐫𝐫
Transferring from 𝐈𝐈𝐈𝐈 to 𝐈𝐈 𝐫𝐫𝐫𝐫:-
R1 R 21

I22 R 2 = I12 R 21
I 2
R 21 = R 2 �I2 �
1
R
= 22
K
∴ 1 R
R 2 = K22

From 𝐈𝐈 𝐫𝐫𝐫𝐫 to 𝐈𝐈𝐈𝐈 𝐫𝐫𝐫𝐫 :-

I12 R1 = I22 . R11
I12
R11 = I22
. R1
R11 = R1 . K 2

→ Total resistance ref to primary = R1 + R 21

R 01 = R1 + R 2 /k 2
→ Total resistance ref to secondary = R 2 + R11
R 02 = R 2 + k 2 R1
→ Total Cu loss = I12 R 01
Or
2
I2 R 02
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Institute of Engineering Studies (IES,Bangalore) Electrical Machines Formula Sheet

Per unit resistance drops :-

I1 R1
→ P.U primary resistance drop = E1
I R
→ P.U secondary resistance drop = 2E 2
2
I1 R01
→ Total P.U resistance drop ref to I ry = E1
ry I2 R02
→ Total P.U resistance drop ref to II = E
2
→ The P.U resistance drops on both sides of the t/f is same
I1 R01 I2 R02
E1
= E2

Losses present in transformer :-

t/f windings
1. Copper losses
major losses
2. Iron losses t/f core
cu parts
3. Stray load losses Iron parts
4. Dielectric losses minor losses
insulating materials.
1. Cu losses in t/f:

Total Cu loss = I12 R1 + I22 R1

= I12 R 01
= I22 R 02
VA rating of t/f
→ Rated current on I ry =
E1
ry VA rating of t/f
Similarly current on II =
E2
→ Cu losses ∝ I12 or I22. Hence there are called as variable losses.
FL Cu loss in watts
→ P.U Full load Cu loss =
VA rating of t/f
I12 R01
= E1 I1
→ If VA rating of t/f is taken as base then P.U Cu loss ∝ I12 as remaining terms are constant.

→ P.U Cu loss at x of FL = x 2 × PU FL Cu loss

ry
→ P. U resistance drop ref to I � =
I1 R01 I1
×
or E1 I1
P. U resistance ref to Iry
I12 R01
= E1 I1

% FL Cu loss = % R = % Resistance drop.

Iron (or) Core losses in t/f :-

1. Hysteresis loss :

Steinmetz formula :-

x
Wh = η Bmax . f . v Area under one hysteresis loop.

Where
η = stienmetz coefficient
Bmax = max. flux density in transformer core.
f = frequency of magnetic reversal = supply freq.
v = volume of core material
x = Hysteresis coeff (or) stienmetz exponent
= 1.6 (Si or CRGo steel)
2. Eddycurrent loss:
Eddy current loss ,(We ) ∝ R ce × Ie2
As area decreases in laminated core resistance increases as a result conductivity decreases.
2
We = K. Bmax f 2. t2
thickness of laminations.
Supply freq
Constant
(it is a function of σ )
During operation of transformer :-
V1
Bm ∝ f
V1
Case (i) :- f
= constant, Bmax = const.
we ∝ f 2

we = B f 2

Const.

∴ wi = wh + we When Bmax = const.

2
wi = Af + Bf

P.U iron loss :-

Iron loss in watts
→ P.U iron loss =
VA rating of t/f

→ As VA rating is choosen as base then the P.U iron loss are also constant at all load conditions.

To find out constant losses :-

 W0 = Losses in t/f under no load condition

= Iron losses + Dielectric loss + no load primary loss (I02 R1)

 Constant losses = W0 − I02 R1

Where , R1 = LV winding resistance.

To find out variable losses :-

 Wsc = Loss in t/f under S.C condition

= F.L Cu loss + stray load losses (Cu and Iron) + Iron losses in both wdgs
 Variable losses = WSC − Iron losses corresponding to VCC

O.C test :-
V1 rated → Wi
S.C test :-

VSC → (Wi )S.C

Wi ∝ V1 2
Wi V1 rated 2
(Wi )SC
= � �
VSC
VSC 2
(Wi )S.C = Wi × � �
V1 rated
VSC 2
∴ Variable losses = WSC − (Wi )SC �V �
1 rated

Efficiency :-

output power
 Efficiency of transformer is given by η =
input power
output power
=
output power+losses
E2 I2 cos ϕ2
=E
2 I2 cos ϕ2 + F.L cu losses+Iron losses
E2 I2 cos ϕ2
ηF.L =
E2 I2 cos ϕ2 + I22 R02 + Wi

x (E2 I2 ) cos ϕ2
ηx of F.L =
x (E2 I2 ) cos ϕ2 + x 2 (I2 2 R 02 ) + Wi

S.C test O.C test

KVA × cos ϕ
→ Transformer efficiency =
KVA × cos ϕ + wi + Cu losses

x × MVA × Pf
→ Efficiency = η =
x × MVA × Pf + wcu × x2 + wi

1
→ Total losses in transformer = � − 1� output
η

→ Condition for maximum effieciency is, Cu losses = Iron losses

→ Total losses at ηmax = 2Wi
W Iron loss
→ %load at which maximum efficiency occurs % x = �I 2 Ri *100 %= � *100 %
2 02 F.L.cu loss

Iron loss
→ KVA corresponding to ηmax = F.L KVA �
F.L culoss

→ Voltage drop in t/f at a Specific load p.f = I2 R 02 cos ϕ2 ± I2 X02 sin ϕ2

I2 R02 cos ϕ2 ± I2 X02 sin ϕ2

→ % Voltage regulation = ×100
V1′

I R02 I X02
=�2 � cos ϕ2 ± � 2 � sin ϕ2
V1′ V1′
↓ ↓
P.U resistance P.U reactance
% Regulation = �(P. U R) cos ϕ2 + (P. U X) sin ϕ2 � × 100

Condition for max. regulation :-

% regulation = (% R) cos ϕ2 ± (% X) sin ϕ2

d regn
=0
d ϕ2
%X X02
Tan ϕ2 = =
%R R02
X02
ϕ2 = Tan−1 � � lagging
R02

X
At maximum regulation ϕ2 = Tan−1 �RL �
L
X
= Tan−1 �R02 �
02
XL X02
RL
= R02

Value of maximum regulation :-

% Regulation = (% R) cos ϕ2 + (% X) sin ϕ2
%R
At max. regulation cos ϕ2 =
%Z
%X
Sin ϕ2 =
%Z
%R (% X)
max. % regulation = (% R) + (% X)
%Z %Z
(% R) + (% X)2
2
=
%Z
(% Z)2
= %Z
max. % regn = % Z
= % of rated voltage required to produce rated short ckt current
.

X R02
→ At zero regulation condition : ϕ2 = Tan−1 � C � = Tan−1 � �
RL X02
XC R02
RL
= X02

→ Regulation at x of FL = x [% R cosϕ2 ± % X sin ϕ2 ]

= x × F.L regn

Regulation at U.P.F:-
Regulation at UPF = % R

= % F.L Cu loss

Scott Connection:
ia
Ia 86.6%
A 0.866
� �
√3
V1 0.866
0.577 V1 2 Va
N :
1
0.289 IA
IA�
IB 2 M 2
B
Vb
V1 IBC ib
IC
C
VAM = 0.866 V1
V
VAN = � 31 � = 0.577 V1
√
VMN = 0.866 V1 − 0.577 V1 = 0.289 V1

Teaser t/f :-

Ia N2
ia
= 0.866 N1

N2
Ia = × ia
0.866 N1

Let N1 ∶ N2 = 1 ∶ 1
IA = 1.15 ia
Main t/f
IBC N
ib
= N2
1
N2
IBC = × ib
N1

Let N1 : N2 = 1 ∶ 1
IBC = ib
IB = IBC − IA� IC = −IBC − IA�
2 2
→ Capacity of Scott Connection :-
(KVA)Scott = √3 VL IL
V2 = V1 IL = I1
↙ ↓
→ Vol. rating of 1 – ϕ t/f Current rating of 1 – ϕ t/f
(KVA)Scotf = √3 V1 I1
(KVA)Scott = √3 (KVA)1− ϕ
(KVA)
→ Utilization factor = availableScott
KVA
√3 V1 I1
= 2V1 I1
= 0.866
→ Utilization factor of Scott connection with 2 identical 1 – ϕ t/f’s is 86.6%

AUTO TRANSFORMER:

LV
→ K of auto transformer =
HV
(KVA)induction = (V1 − V2 ) I1

I/P KVA = V1 I1

(KVA)induction (V1 − V2 ) I1
i/p KVA
= V1 I1
LV
=1–
HV
=1–K
∴ (KVA) induction = (1 – K) i/p KVA

(KVA) conduction = I/p KVA – (KVA)ind

(KVA)conduction = K × I/p KVA

→ Wt. of conductor in section AB of auto t/f ∝ (N1 − N2 ) I1

→ Wt of conductor in section BC of auto t/f ∝ (I2 − I1 )N2

∴ Total wt. of conductor in auto t/f is
∝ I1 (N1 − N2 ) + (I2 − I1 )N2
∝ 2 (N1 − N2 ) I1
→ Total wt. of conductor in 2 wdg transformer
∝ I1 N1 + I2 N2
∝ 2 I1 N1
wt.of conductor in an auto t/f 2(N1 −N2 )I1
→ =
wt.of conductor in 2 wdg t/f 2N1 I1
N
= 1– 2
N1
=1–K
Wt. of conductor in auto t/f = (1 – K) (wt. of conductor in 2 wdg t/f)
→ Thus saving of conductor material if auto – t/f is used} = K × {conductor wt in 2 wdg transformer.
→ (% FL losses)Auto t/f = (1 − K)(% FL losses)2 wdg t/f
→ (% Z)AT = (1 − K) (% Z)2 wdg t/f
1
→ (KVA)AT = (KVA)2 wdg t/f .
1−K

→ Principle of operation :-

Whenever a conductor cuts the magnetic flux, an emf is induced in that conductor”

 Faraday’s law of electromagnetic induction.

→ Coil span (β) :- It is the distance between two sides of the coil. It is expressed in terms of
degrees, pole pitch, no. of slots / pole etc
→ Pole pitch :- It is the distance between two identical points on two adjacent poles.
Pole pitch is always 180° e = slots / pole.
P
→ θelec = θmech
2
→ Slot pitch or slot angle :- (T)Slot angle is the angle for each slot.
P(180°)
→ For a machine with ‘P’ poles and ‘s’ no. of slots, the slot angle = γ = d
180°
γ= s
� �
p

→ Pitch factor or coil span factor or chording factor :- (K P)

The emf induced | coil in short pitched winding

KP =
The emf induced |coil in full pitched winding

The vector sum of induced emf | coil

=
Arithmetic sumof induced emf | coil

2E cos∝/2
KP =
2E

K p = cos ∝/2

→ Pitch factor for nth harmonic i.e,K p = cos n ∝

n 2

180°
→ chording angle to eliminate nth harmonics (α)=
n
n−1
→ coil spam to eliminate nth harmonics ,(β) = 180 � �
n
→ Distribution factor | spread factor | belt factor | breadth factor(kd) :-

The emf induced when the winding is distributed

Kd =
The emf induced when the winding is concentrated

Vector sum of emf induced

Kd =
Arithmetic sum of emf induced

mnγ
sin
For nth harmonic, kdn = 2
nγ
m sin
2

360°
→ To eliminate nth harmonics ,phase spread (mγ) =
n
→ Generally, KVA rating, power output ∝ kd and Eph (induce emf) ∝ k d . Tph .
60
KVA60 (3− ϕ) Pout60 (3 ϕ) kd60 sin m120 sin 30° 120
∴ = = = 2
120 × = × = 1.15
KVA120 (3− ϕ) Pout120 (3ϕ) kd120 sin m60 sin 60° 60
2

60
KVA60 (3ϕ) Pout60 (3ϕ) kd60 sin 90
KVA90 (2ϕ)
= Pout90° (2ϕ)
= kd90
= 2
90 × = 1.06
sin 60
2

60
KVA60 (3ϕ) Pout60 (3ϕ) kd60 sin 180
= = = 2
180 × = 1.5
KVA180 (1ϕ) Pout180 (1− ϕ) kd180 sin 60
2

KVA90 (2− ϕ) Pout90 kd90 sin90⁄2 180

KVA180 (1− ϕ)
= = = sin180⁄2
× = 1.414
Pout180 kd180 90

1
→ Speed of space harmonics of order (6k ± 1) is (6k . Ns
±1)

120 f
where Ns = synchronous speed =
p

2S
The order of slot harmonics is � ± 1�
P

where S = no. of slots , P = no. of poles

→ Slot harmonics can be eliminated by skewing the armature slots and fractional slot winding.

The angle of skew = θs = γ (slot angle)

= 2 harmonic pole pitches

= 1 slot pitch.
2s
→ Distribution factor for slot harmonics, k d � ± 1�
p

→ The synchronous speed Ns and synchronous angular speed s of a machine with p pole
pairs running on a supply of frequency fs are:

ωs = 2πfs / p
NS − N
→ Slip S =
NS

120 f
Where NS = = synchronous speed
p
→ The magnitude of voltage induced in a given stator phase is Ea = √2 π Nc ∅ f = K∅ω
Where K = constant

→ The output power Pm for a load torque Tm is:

Pm = ωsTm

→ The rated load torque TM for a rated output power PM is:

TM = PM / ωs = PM p/ 2πfs = 120PM / 2πNs

Synchronous Generator:

→ For a synchronous generator with stator induced voltage Es, stator current Is and
synchronous impedance Zs, the terminal voltage V is:

V = E - IsZs = Es - Is(Rs + jXs)

where Rs is the stator resistance and Xs is the synchronous reactance

E = �(V cos ϕ + Ia R a )2 + (V sin ϕ ± Ia Xs )2

+ ⇒ lag p.f
− ⇒ leading p.f.

→ For a synchronous motor with stator induced voltage Es, stator current Is and synchronous
impedance Zs, the terminal voltage V is:

V = Es + IsZs = Es + Is(Rs + jXs)

where Rs is the stator resistance and Xs is the synchronous reactance

Voltage regulation :

|E|− |V|
→ % regulation = |V|
×100
E – V = Ia Zs
E−V
∴ % regulation =
V
Ia Zs
= × 100
V
∴ regulation ∝ Zs
∴ As Zs increases, voltages regulation increases.

Ia Zs
→ Condition for zero | min. voltage regulation is, Cos (θ + ϕ) = − 2V
→ Condition for max. Voltage regulation is, ϕ = θ
Ifm 1 1
→ Short circuit ratio (SCR) = = =
Ifa Zs (adjusted)|unit Xs (adjusted)|unit

1 1
SCR ∝ ∝
Xa Armature reaction

Voltage regulation ∝ Armature reaction

1
∴ SCR ∝
Voltage regulation

∴ Small value of SCR represent poor regulation.

armature mmf
ϕa =
reluctance

But reluctance ∝ Air gap

armature mmf
∴ ϕa =
airgap

1
ϕa ∝
Air gap length
1
Armature reaction ∝ ϕa ∝
Airgap length

1
∴ SCR ∝ ∝ Airgap length
Armature reaction

Air gap length ∝ SCR

∴ machine size ∝ SCR.

Cost ∝ SCR
EV
Power = sin δ
Xs

1
⇒P∝ ∝ SCR
Xs

Power ∝ SCR
∴ Large value of SCR represent more power output.
→ Synchronizing power coefficient or stability factor Psy is given as
dp d EV
Psy = = � sin δ�
dδ dδ Xs
EV
= cos δ
Xs
Psy is a measure of stability
∴ stability ∝ Psy
1
But Psy ∝ ∝ SCR
Xs

∴ Stability ∝ SCR
Stability ∝ SCR ∝ Air gap length

∴ Stability ∝ Air gap length

→ When the stator mmf is aligned with the d – axis of field poles then flux ϕd perpole is set up
and the effective reactance offered by the alternator is X d .

maximum Voltage (Vt )line (at min. Ia )

Xd = = = Direct axis reactance
minimum current √3 Ia (min )

→ When the stator mmf is aligned with the q – axis of field poles then flux ϕq per pole is set up
and the effective reactance offered by the alternator is X q.

minimum voltge Vt line (at maximum Ia )

Xq = = = Quadrature axis reactance
maximum voltage √3 Ia (max )

→ Salient pole synchronous machine ,

The per phase power delivered to the infinite bus is given by

Ef Vt Vt2 1 1
P= Xd
sin δ + 2
�X − � sin 2δ
q Xd

Condition for max. power:-

→ For cylindrical rotor machine :-

dp
At constant Vt and Ef , the condition for max. power is obtained by putting =0
dδ
dp Ef Vt
∴ = cos δ =0
dδ Xs
Cos δ = 0
δ = 90°
Hence maximum power occurs at δ = 90°

dp
→ For salient – pole synchronous machine :- =0
dδ
Vt Ef 1 1
⇒ cos δ + Vt 2 � − � cos 2δ =0
Xd Xq Xd

2
Ef Xq 1 E Xq
± � + �4V
f
Cos δ = − 4V �
t �Xd − Xq � 2 �X t d − Xq �

The value of load angle is seed to be less than 90°.

∴ max. power occurs at δ < 90°
→ Synchronizing power = Psy. ∆ δ.
EV
cos 𝛿𝛿 . ∆ 𝛿𝛿 .
=
Xs
Synchronizing power
→ Synchronizing torque = .
ω

Power flow in Alternator :-

→ Complex power = S = P + jQ = VIa∗

EV V2
Where Active power flow (P) = cos(θ − δ) − cos θ ;
Zs Zs

If R a = 0; θ = δ = 90° ; then max power is given by

EV V2
Pmax = Zs
− Z cos θ
s

SYNCHRONOUS MOTORS:

NN.L − NF.L
→ Speed regulation = × 100
NF.L
N −N
= S S × 100 = 0%
NS
NS − N NS − NS
⇒ Slip S = = N = 0%
NS S
120 f
NS =
p
→ The speed can be controlled by varying the frequency
V↑
↓ϕ ∝
f↑
v
ratio control is preferred for rated torque operation
F

Power flow in synchronous motor is given by

complex power i/p s = p + jQ = V Ia∗
where P = Resl power flow , Q = Reactive power flow
V2 EV V2 EV
P = Z cos θ − ZS
cos(θ + δ) : Q = Z sin θ − ZS
sin(θ + δ)
2 2

:
⇒ If R a = 0 ; ZS = X S ; θ 90°
EV V
Pin = X sin δ Q = X [V − E cos δ]
S S

→ Condition for max power :-

δ = 180° − 0

V2 EV
Pmax = ZS
cos θ + ZS

Expression for mechanical power developed :-

→ Mechanical power developed = Pm = active component + [E Ia∗ ]
EV E2
⇒ Pm = cos(θ − δ) + cos θ
ZS ZS

→ Condition for max. mechanical power developed :-

d pm EV
dδ
= sin(θ − δ) = 0
ZS
Sin (θ – δ) = 0 = sin 0
δ=θ
EV E2
Pm max = ZS
− ZS
cos θ

This is the expression for the mechanical power developed interms of load angle and the
internal machine angle θ, for constant voltage Vph and constant E i.e., excitation
Pm
→ Gross Torque =
w
Pm
= 2π Ns
60
Ns = synchronous speed in r.p.m
60 Pm
∴ Tg = .
2π NS

9.55 Pm
Tg = Ns

→ Condition for excitation when motor develops Pmax :-

For max power developed is
dPm
=0
dE
d EV E2
� cos(θ −
dt ZS
δ) − ZS
cos θ� = 0
VZS
→ Condition for excitation when motor develops Pmax is , Eb =
2Ra

Power flow in synchronous motors :-

Tg = Pin
2π Ns
60

Pin Stator Mechanical power Friction

√3 VL IL cos ϕ copper loss developed in armature and Iron
2 Pm = 3Ebph . Iaph cos
(input) 3 Iaph Ra losses
(ϕ ± δ)
Tsh =
+ ve lead
Pout
–ve for lag 2 π Ns
60. Pout
output

→ For leading p.f

Ia �Xq cos ϕ+ Ra sin ϕ�

tan δ = V
t + Ia �Xq sin ϕ− Ra cos ϕ�

→ The mechanical power developed per phase is given by,

s
Eb Vph Vph2 1 1
Pm = sin δ + �X − � sin 2δ
Xd 2 q Xd

INDUCTION MACHINES:

→ The power flow diagram of 3 – ϕ induction motor is

Power of
rotor shaft

Stator Stator Rotor Rotor core loss Friction loss Windage

I2R core I2R (negligible for at bearings loss
loss loss loss small slips) and sliprings
of (if any)

ns − nr
The slip of induction machine is (S) = ns
Ns − Nr
= Ns
Where Ns is synchronous speed in rpm
ns is synchronous speed in rps
⇒ Nr = Ns (1 − s)
⇒ Ns − Nr = SNs
P . SNs PNs
∴ Rotor frequency, f2 = =S = Sf1
120 120

For an induction machine with rotor resistance Rr and locked rotor leakage reactance Xr, the
rotor impedance Zr at slip s is:Zr = Rr + jsXr
The stator circuit equivalent impedance Zrf for a rotor / stator frequency ratio s is:
Zrf = Rrs / s + jXrs

For an induction motor with synchronous angular speed ωs running at angular speed ωm and
slip s, the airgap transfer power Pt, rotor copper loss Pr and gross output power Pm for a
gross output torque Tm are related by:
Pt = ωsTm = Pr / s = Pm / (1 - s)
Pr = sPt = sPm / (1 - s)
Pm = ωmTm = (1 - s)Pt
The power ratios are:
Pt : Pr : Pm = 1 : s : (1 - s)
The gross motor efficiency ηm (neglecting stator and mechanical losses) is:
ηm = Pm / Pt = 1 - s

Rotor emf, Current Power :-

At stand still, the relative speed between rotating magnetic field and rotor conductors is
synchronous speed Ns ; under this condition let the per phase generated emf in rotor circuit
be E2 .

∴ E2 /ph = 4.44 Nphr ϕ1 f1 K dr K pr

E2 /ph = 4.44 Nphr ϕ1 f1 K wr

→ But during running conditions the frequency of the rotor becomes, running with speed Nr

P(Ns − Nr ) P SNs
120
= 120
= Sf1
∴ fr = Sf1
∴ Emf under running conditions is
E = √2 π fr Kw2 Nphr ϕ1
= SE2

→ Rotor leakage reactance = 2π (Rotor frequency)

(Rotor leakage Inductance)
∴ Rotor leakage reactance at stand still = 2π f1 𝑙𝑙2
= x2 Ω
→ Rotor leakage reactance at any slips = 2π f2 𝑙𝑙2

= sx2 Ω

→ Rotor leakage impedance at stand still

= �r22 + x22

→ At any slip s, rotor

= �r22 + (sx2 )2

→ Per phase rotor current at stand still

E
=
�r22 + x22

→ Per phase rotor current at any slip s is given by

SE2 E2
I2 = =
2
�r2 + (sx2 ) 2 �(r1 /s)2 + x22

→ The rotor current I2 lags the rotor voltage E2 by rotor power factor angle θ2 given by
sx
θ2 = tan−1 r 2
2

r2 /s
=
�(r2 /s)2 + (x2 )2

r2 /s
∴ Pg = E2 I2 ×
�(r2 /s)2 + (x2 )2
`
E2 r2
= × I2
�(r2 /s)2 + (x2 )2 s

r2
= I22 s
→ Pg is the power transferred from stator to rotor across the air gap. There fore Pg is called air
gap power
r
Pg = I22 s2
1−S
= I22 r2 + I22 r2 � �
S

Pg = (Rotor ohmic loss) + Internal mechanical power developed in rotor (Pm )

= S Pg + (1 − S)Pg

1−S
∴ Pm = (1 − S) Pg = I22 r2 � �
S

S
Rotor ohmic loss = � � Pm = SPg
1−S

→ Internal (or gross) torque developed per phase is given by

Internal mechanical power developed inrotor

Te =
Rotor speed in mechanical radian per sec

Pm (1−S)Pg Pg
Te = ωr
= (1−S)ωs
= ωs

→ Electromagnetic torque Te can also be expressed as

Pg 1 I22 r2 Rotor ohmic loss

Te = = × =
ωs ωs S (ωs ) slip
Rotor ohmic loss
∴ Te = (ωs ) slip

→ Power available at the shaft can be obtained from Pg as follows.

Output or shaft power, Psh = Pm − Mechanical losses

→ Mechanical losses implies frication and windage losses

Psh = Pg − Rotor ohmic loss – Friction and windage losses

= Net mechanical power output or net power output
P Psh
Output or shaft torque Tsh = Rotorsh
speed
= (1−s) ωs

→ If the stator input is known. Then air gap power Pg is given by

Pg = stator power input – stator I 2 R loss – stator core loss.

→ Ratio of Rotor input power, rotor copper losses and gross mechanical output is
1
Ir2 R 2 /s : Ir2 R 2 : Ir2 R 2 � − 1�
s
⇒ 1 : S : (1 – S)
∴ Rotor copper losses = S × Rotor input
Gross mechanical output =(1 – S) × Rotor input.
S
Rotor copper losses = (Gross Mechanical output) ×
1−S
Efficiency of the rotor is approximately
Gross mechanical power output
Equal to ηrotor =
Rotor input
(1−S) Rotor input
=
Rotor input
=1–S
NS − N
=1−
NS
N
=
Ns
N
ηrotor ≃ Ns

Total torque is
m Ve2 r2
Te = × r2 2
× Nm
ωs �Re + � + (x2 + Xe )2 s
s
m is the number of stator phases.
Torque equation can be written as
m r
Te = × I22 × s2
ωs
m
Te = × rotor input per phase.
ωs

Substituting the value of maximum slip in the torque equation, gives maximum torque
m Ve2
Tem = ωs
×
2�Re + �Re2 + X22 �

r2
If stator parameters are neglected then applying maximum transfer theorem to r2 /s then s
=x2

Slip corresponding to maximum torque is

r2
Sm = (Breakdown slip)
x2

Nm = Ns (1 − Sm )
⇒ Nm = Ns (1 − R 2 /x2 )

Nm is the stalling speed at the maximum torque

Starting torque:-
At starting, slip S = 1.00, starting torque is given by
m Ve2 r2
Test = × (Re + r2 )2 + X2
ωs

Motor torque in terms of 𝐓𝐓𝐞𝐞𝐞𝐞 :

→ The torque expression of an induction motor can also be expressed in terms of maximum
S
torque Tem and dimension less ratio . In order to get a simple and approximate
Sm,T
expression, stator resistance r1 , or the stator equivalent resistance R e , is neglected.
2�Re + �Re2 + X2 �
Te r2
∴ Tem
= r 2
× s
�Re + 2� + X2
s

→ Since r1 or Re is neglected
Te 2X r2
Tem
= r 2
× s
� 2� + X2
s

→ The slip at which maximum torque occurs is

r
SmT = X2 ∴ r2 = SmT X
Te 2X SmT X
∴ Tem
= X 2
×
S
� mT � + X2 s
s
Te 2
⇒ = SmT S
Tem +
S SmT

Power slip characteristics :-

→ The total internal mechanical power developed is
1−s
Pm = m(1 − S)Pg = m I22 r2 � �w
s
mVe2 1−s
Pm = 2 × r2 � �
r (1−s) s
�Re + r2 + 2 � + X2
s
Maximum power transfer theorem is invoked again to obtain maximum value of internal
1−s
mechanical power developed. Since Pm per phase is the power delivered to r2 � s
�, internal
mechanical power developed is maximum, when
rs �1−Smp �
SmP
= �(R e + r2 )2 + X 2
r2
SmP =
�(Re + r2 )2 + X2 + r2
r2 �1− Smp � r2 (1−s)
In order to get maximum power Pm ,substitute Smp
, in place of s
in power equation
mve2
Pmm =
2[Re + r2 + �(Re + r2 )2 + X2

In order to get maximum power output from an induction generator, the rotor must be deiven
at a speed given by
r2
ns �1 + �
�(R e + r2 )2 + X 2 + r2

Losses and efficiency :-

There are three cases in iron losses.
Case (i) : If the ratio of voltage to frequency is constant and flux is also constant then
Iron loss = Hysteresis loss + eddy current loss
Ph = K h + Bm1.6 Pe = K e f 2 Bm2
V V
Given is constant. As Bm ∝
f f
⇒ Bm is constant

∴ Ph ∝ f and Pe ∝ f 2

Case (ii) : If the ratio of voltage to frequency is not constant and flux is also not constant
v
⇒ ≠ const ϕ ≠ const
f

∴ Ph ∝ v1.6 f −0.6 Pe ∝ v 2

Case (iii) : If frequency is constant and voltage is variable then

1.6
Ph = K h f Bm Pe = K h f 2 Bm
2

v 1.6
= Kh f � �
f

Ph ∝ v1.6 Pe ∝ v12

→ Short circuit current with normal voltage applied to stator is

V
I = Ibr ×
Vbr

I = short circuit current with normal voltage

Ibr = short circuit current with voltage Vbr .
→ Power factor on short circuit is found from
Pbr = √3 Vbr Ibr cos ϕbr
Pbr
⇒ Cos ϕbr =
√ br Ibr
3 V

→ As Pbr is approximately equal to full load copper losses

Pbr
R br = Ibr2

The blocked rotor impedance is

Vbr
Zbr =
Ibr

∴ Blocked rotor reactance = X br = �Zbr 2 − R br2

Efficiency of Induction machines :-
output power
Generally efficiency =
input power
Net mechanical output
∴ Efficiency of Induction motor =
Electrical power input

Net electrical output

∴ Efficiency of Induction generator =
mechanical power input

Squirrel cage rotor:

Stator Cu loss = 3 Isc2 r1 = 3 Isc2 r1
∴ Rotor Cu loss = Psc − 3 Isc2 r1

Wound rotor
GD I22 r2 r2 I 2
= I12 r1
= �I2 �
GF r1 1

Direct – on line (across the line) starting :-

→ The relation between starting torque and full load torque is
1 r2
Te = × I22
ωs s
2 r2
Te.st I2.st
∴ Te.fl
= 2
1
r2
I2.ft
Sfl

I 2
= � I2 st � × Sfl
2 fl
The above equation valids of rotor resistance remains constant.
Te.st I 2
Te.fl
= � Ist� × Sfl
fl

Ist (Effective rotor to stator turns ratio) I2 st

Where = (Effective rotor to stator turns ratio) I2 fl
Ifl
→ Per phase short – circuit current at stand still (or at starting) is,
V1
Isc = Zsc

Where Zsc = (r1 + r2 ) + j(x1 + x2 )

Here shunt branch parameters of equivalent circuit are neglected.
→ Therefore, for direct switching,
V1
Ist = Isc =
Zsc

T I 2
∴ Test = � Isc � Sfl .
esf fl

Stator resistor (or reactor) starting :-

Since per phase voltage is reduced to xv, the per phase starting current Ist is given by
xv
Ist = Z 1 = xIsc
sc
Te.st I 2
As be fore Te.fl
= � Ist� × Sfl
fl
xI 2
= � I sc � × Sfl
fl

→ In an induction motor, torque ∝ (voltage)2

starting torque with reactor starting xv 2

∴ starting torque with direct switching
= � v 1� = x2
1

Auto transformer starting :-

→ Per phase starting current from the supply mains is Ist = x 2 Isc

Te.st Per phase starting current in motor winding

Te fl
= × Sfl
Per−phase motor full load current

Te.st Ist Isc

Te.fl
= Ifl2
× Sfl

Test with an autotransformer xv 2

= � 1� = x2
Test with direct switching v1

Star – delta method of starting :

starting torque with star delta starter
→
starting torque with direct switching in delta
V 2
� L� 1
√3
= [V 2 =
L] 3
∴ star delta starter also reduces the starting torque to one – third of that produced by direct switching
in delta.
→ With star – delta starter, a motor behaves as if it were started by an auto transformer starter with x =
1
= 0.58 i.e with 58% tapping.
√3
1 2r 1 2
Starting torque with star delta starter Te .st �I
ws st .y
� 12 � I � 1 I d 2
√3 st .d
→ = 1 r = (Ifl .d )2
× Sfl = �I sc . d� × Sfl
Full load torque with startor winding in delta,Tefld (I )2 2 3 fl .
ws fl d Sfl

Thermal Power Station :-

Heat equivalent of mech−energy Transmitted toTurbine shat
 Thermal efficiency, ηThermal =
Heat of coal combustion
 Thermal efficiency = ηboiler × ηturbine
Heat equvivalent of electrical o/p
 Overall efficiency, ηoverall =
Heat of combustion of coal
 Overall efficiency = Thermal efficiency × Electrical efficiency.
 Energy output = coal consumption × calorific value.
= coal consumption × 6500 k.cal
Output in k.cal
η=
Input in k.cal

Water Power equation:-

 Water Head : The difference of water level is called the water head.
 Gross Head : The total head between the water level at inlet and tail race is called as gross
head
 Rated Head : Head utilized in doing work on the turbine
 Net Head : It is the sum of the Rated Head and the loss of head in guide passage and
entrance
H = Head of water in metre
Q = Quantity of water in m3 /sec or lit/sec.
W = specific gravity of water
= 1 kg/lit when ‘Q’ represented in lit/sec.
= 100 kg/m3 when ‘Q’’ represented in m3 /sec.
η = efficiency of the system.
Effective work done
or � = WQH × η kg- m/sec.
Output of system

WQH × η
 Metric output = (H.P)
75

1H.P = 75 kg-m/sec

WQH × η
 Metric output in watt = × 735.5
75

WQH
 Output = × η kw
102

 Volume of water available per annum = catchment area × Annual Rainfall

→ Electric energy generated = weight × head × overall η.
GAS TURBINE POWER PLANT :

→ The thermal efficiency of gas turbine plant is about 22% to 25%

→ The air fuel ratio may be of the order of 60 : 1 in this ase.

ηoverall
 Engine efficiency ηengines = ηalt
ηengine
 Thermal efficiency ηthey = mech.η of englnd
 Heat produced by fuel per day = coal consumption / day × caloritic value

Terms and Definitions :-

1. Connected load :-
It is the sum of ratings in kilo watts of equipment installed in the consumer’s premises
2. Demand :-
It is the load or power drawn from the source of supply at the receiving end averaged over a
specified period.
3. Maximum Demand :-
Maximum demand (M.D) of a power station is the maximum load on the power station in a given
period.
4. Average load:-
If the number of KWH supplied by a station in one day is divided by 24 hours, then the value so
obtained is known as daily average load.
KWH deliverd in one day
Daily average load =
24
KWH delivered in one month
Monthly average load =
30 ×24
KWH delivered in one year
Yearly average load =
365 ×24
5. Plant capacity :-
It is the capacity or power for which a plant or station is designed. It should be slightly more than
M.D. it is equal to sum of the ratings of all the generators in a power station.
6. Firm Power :-
It is the power which should be always be available even under emergency
7. Prime Power :-
It is the maximum power (may be thermal or hydraulic or mechanical) continuously available for
conversion into electrical power.
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Institute of Engineering Studies (IES,Bangalore) Power Systems Formula Sheet
8. Dump power :-
This is the term usually used in hydro electric plants and it represents the power in excess of the
load requirements. It is made available by surplus water.
9. Spill Power :-
Is that power which is produced during floods in a hydro power station.
10. Cold reserve :-
Is that reserve generating capacity which is not in operation but can be made available for service.
11. Hot reserve :-
It that reserve generating capacity which is in operation but not in service
12. Spinning reserve :-
Is that reserve generating capacity which is connected to bus-bars and is ready to take the load.

Load factor :-
It is defined as the ratio of number of units actually generated in a given period to the number
of units that could have been generated with maximum demand.
Average load or Average Demand
 Load factor =
Maximum Demand.

Energy generated in a given period

= (Maximum Demand) ×(Hours of operaation in the given period)
.
→ The load factor will be always less than one (< 1)
Demand factor:-
It is defined as the ratio of maximum demand on the station to the total connected load to
the station.
Maximum Demand on the station
 ∴ Demand factor =
Total connected load to the station
 Its value also will be always less than one (< 1)

Diversity Factor :-
Diversity factor may be defined as “the sum of individual maximum demand to the station
to the maximum demand on the power station”.
Sum of individual consumers maximum demand
 Diversity factor =
Maximum demand on the station.
 Its value will be always greater than one (> 1)

Plant Factor or Plant Use Factor :-

station output in kwh

 Plant factor =
Σ (KW1 ) H1 + (KW2 ) H2 + (KW3 )H3 + ……
Where KW1, KW2, KW3 etc. are the kilowatt ratings of each generator and H1 , H2 , H3 etc.
are the number of hours for which they have been worked.

→ It is defined as the ratio of average demand on the station to the maximum installed capacity.

Average demand on the station

i.e. capacity factor =
Max.installed capacity of the station

→ Coincidence factor:-

It is the reciprocal of diversity factor and is always less than 1

Maximum demand
→ Utilization factor =
Plant capacity
Service hours
→ Operation factor =
Total duration
Actual energy produced
→ Use factor =
Plant capacity ×Time (hrs)the plant has been in operation

D.C. Distribution calculations

Uniformly loaded Distributor fed at one end.

→ Fig (a) shows the single lien diagram of a 2 – wire d.c. distributor AB fed at one end A and
loaded uniformly with i amperes per metre length.
𝑙𝑙 𝑙𝑙
A C
B
A B
x dx
i i i i
Fig. (a)
→ Then the current at point c is.
= δ𝑙𝑙 − ix amperes
= i(𝑙𝑙 – x) amperes.
→ Total voltage drop is the distributor upto point C is
x x2
𝑣𝑣 = ∫0 ir (𝑙𝑙 − x)dx = ir �𝑙𝑙x − 2
�
→ Voltage drop over the distributor AB
1
= ir𝑙𝑙 2
2
1
= IR
2
Where i𝑙𝑙 = I, the total current entering at point A

Uniformly loaded distributor fed at both ends.

(i) Distributor fed at both ends with equal voltages
Current supplied from each feeding point
9𝑙𝑙
=
2

𝑙𝑙
x dx
A B
C

V C
i i i i i
ir
→ Voltage drop upto point C = (𝑙𝑙x − x 2 ).
2
1
→ Max. voltage drop = IR
8
IR
→ Min. voltage =V– volts
8
(ii) Distributor fed at both ends with unequal voltages :-
The point of minimum potential C is situated at a distance x meters from the feeding point A.
irx2
Voltage drop in section AC = volts.
2

x 𝑙𝑙 - x
C
A B

VA VB
i i i i i
VA − VB 𝑙𝑙
→ x= ir𝑙𝑙
+
2

Comparison of 3 – wire and 2 – wire D.C. distribution

The area of cross – section of neutral is half the cross – section of outers in 3 – wire system.

volume of w for 3−wire system 5 1 5

→ ∴ = a𝑙𝑙 × =
volume of w for 2−wire system 8 2a𝑙𝑙 16

If the neutral has the same cross – section as the outer, then.

Transmission Lines:

𝑙𝑙 3P
→ The empirical formula for the economical voltage line to line is V = 5.5 � +
1.6 100
where ‘V’ = line pressure in KV, l = distance of transmission in KM,
P = estimated max.KW per phase to be delivered over one pole or tower line

Performance of Lines

→ By performance of lines is meant the determination of efficiency and regulation of

lines.
The efficiency of lines is defined as

Power delivered at the receiving end

→ % efficiency = × 100
Power sent from sending end

Power delivered at the receiving end

→ % efficiency = × 100
Power delivered at the receiving end+losses

Vr′ − Vr
→ % regulation = Vr
× 100

Where Vr ′ is the receiving end voltage under no load condition and Vr the
Receiving end voltage under full load condition.

Effect of Earth on a 3 – 𝛟𝛟 line:-

S. No Line Description R L XL C XC
1. Length Increases Increases Increases Increases Increases Decreases
2. Distance of separation No change Increases Increases Decreases Increases
increases

→ The capacitance C of the conductor with reference to grund

2 πε0
C= F/metre
ln �2h�r�

Where h = distance between earth and conductor

Effect of earth on the capacitance of single – phase transmission line :-

a b

Daa′

a′ b′

Effect of earth on the capacitance of single – phase transmission line is

π ε0
→ C= Dab Daa′
F/metre
ln
r Dab

The effect of earth on capacitance of the system is to increases it.

Capacitance of a 1 – 𝛟𝛟 transmission line

h
1 – 𝛟𝛟 transmission line
π ε0
→ C= 2
F/metre.
h h
+� 2
r r

ρL π ε0
→ C= = F/metre.
V ln h�r

→ The capacitance C of the 3 – ϕ transmission line with reference to grund is

P 2π εD
C = Va = F/metre.
a 3 D D D
� ab bc ca 3 Daa′ Dbb′ Dcc′
ln �D
r D D
ab′ bc′ ca′

→ Capacitance of a 3 – Phase Unsymmetrically Spaced Transmission Line is

P 2πε0
C = Va = GMD F/metre
a ln
r

Vertical Spacing:-
h
a c′ c b′ b a′

d g

b f b′ a a′ c c′

c a′ b c′ a b′

1 2 3

→ For Vertical Spacing conductors, The capacitance C per phase of the system is
4 πε0
C= 2� F/metre/phase.
3 d g 3
ln √2 � �
r f

→ Capacitance of a hexagonal spacing Double Circuit Line is

ρ 2πε0
C = Va = √3D
F/metre/coloumb
a ln
2

Bundled conductors:-

→ For a two conductor bundle, the equation for maximum gradient at the surface of a
sub-conductor is
2r
V�1+ �
s
g= d
2r ln
√rs
where ‘s’ is the seperation between the sub – conductors.

→ Let the equivalent radius or geometric mean radius be P0 then for two conductors
1 1 1
P0 = (rd) �2 = r �2 d �2
→ When there are 3 conductors
1�
1� 1� 2� 3 3
P0 = (r d′ d′ ) 3 = r 3 d 3 � �
4
→ For 4 conductors
1� 1�
d d 4 1� 3� 1 4
P0 = �r . d� = r 4 d 4 � � .
√2 √2 2

5 1�
1� d �6 1� d 5 6
→ For six conductors , P0 = r 6� � 6 6 = �6r � � �
2 2

Inductance of a double 3 – 𝛟𝛟 line :-

a c′ c b′ b a′

d g

b f b′ a a′ c c′

c a′ b c′ a b′

1 2 3
Transposed double circuit line.
→ Inductance per phase =
1� 1�
1� d 2 g 3
2 × 10−7 ln 2 6 � 1� � � H/metre/phase
r f

Inductance of composite conductors :-

1
2
m
3
2
1 4
3
n

A B
A

Inductance of composite conductors – 1 – 𝛟𝛟 transmission line.

mn
��D′11 D′12 ⋯ D′1n � �D′21 D′22 ⋯ D′2n � ⋯�D′m1 D′m2 D′mn �
−7
→ LA = 2 × 10 ln m2
�(R′ D12 D13 ⋯ D1m ) (R′ D21 D23 ⋯ D2m ) ⋯(R′ Dm1 Dm2 ⋯ Dmm )
The mnth root of the product of the mn distances between m strands of conductor A and n strands
of conductor B is called geometric mean distance (GMD) and is denoted as Dm .

The m2 th root of m2 distances i.e., the distance of the various strands from one of the strands
and the radius of the same strand, the distances of such m groupings constitute m2 terms
in the denominator, is called the geometric mean radius (GMR) of self GMD and is denoted
as Ds .
Dm
→ LA = 2 × 10−7 ln Ds
Henry/metre.

Composite Conductors,

→ Inductance L per unit length

µ µ0 D
= �4π0 + π
ln � Henry/metre.
R
−7 D 1�
=4× 10 ln ′ Henry/metre [where R′ = R. e 4]
R

Inductance of 3 – 𝛟𝛟 Unsymmetrically Spaced Transmission Line

b c

Ic a Ib
3 – ϕ transmission with
Unsymmetrical spacing

La + Lb + Lc
→ L=
3
−3
√abc
= 2 × 10−7 ln R′
Henry/metre.

Short Transmission Line

→ The equivalent circuit and vector diagram for a short transmission line are shown in fig.
Vr 2Ir R cos ϕr 2Ir X sin ϕr Ir2
VS = �1 + + + (R2 + X 2 )
Vr Vr Vr2
→ In practice the last term under the square root sign is generally negligible; therefore.
1�
2Ir R 2Ir X 2
VS = Vr �1 + � cos ϕr + sin ϕr ��
Vr Vr

R + JX
vS
Ir jIrX
IS

vS vr
vr IrR
ϕr ϕa
Ir

This means A is the voltage impressed at the sending end per volt at the receiving end when
receiving end is open. It is dimensionless.
Vs
B= � Vr = 0
Ir
B is the voltage impressed at the sending end to have one ampere at the short circuited receiving
end. This is known as transfer impedence in network theory.
I
C = Vs � Ir = 0
r

C is the current in amperes into the sending end per volt on the open – circuited receiving end. It
has the dimension of admittance.
I
D = s � Vr = 0
I r

D is the current at the sending end for one ampere of current at the short circuited receiving end
.
 The constants A, B, C, and D are related for a passive network as follows
AD – BC = 1

→ The sending end voltage and current can be written from the equivalent network as,
Vs = Vr + Ir Z
Is = I r
→ The constants for short transmission lines are,
A=1
B=Z
C=0
D=1
VS� − Vr
→ % regulation = A
Vr
×100

Medium Length Lines:-

→ Transmission lines with lengths between 80 km and 160 km are categorized as medium lines
where the parameters are assumed to be lumped. .
→ The two configurations are known as nominal – T and nominal – π respectively.
R jX R jX
+2 +2 Ir
2 vC 2 R + jX I𝑙𝑙 Ir
IS IC IC1
Y Y vr
vS Y = jwc vr vS
2 2

Nominal – T
vS
X
jIS 2

vc
R
IS 2
vr
jIr X/2
IS
IC

Ir
−j
|VS |� �
Vr ′ = R jX
wc
j
+ −
2 2 wc
Vr′ − Vr
% of regulation = Vr
× 100
P
%η= R × 100
P+3 �I 2 + I52 �
2 r

A, B, C, D constant for nominal – T

YZ
A=1+
2
YZ
B = Z �1 + �
2
C=Y
YZ
D = �1 + �
2
Nominal – 𝛑𝛑

I + ∆I I V + ∆ V1 I + ∆I Z∆X
V1 I

vS V + ∆V V vr Y∆X
C∆X

∆X X ∆X
→ Let Z = series impedence per unit length
Y = shunt admittance per unit length
𝑙𝑙 = length of line
Z = zl = total series impedence
Y = yl = total shunt admittance.
V = Aerx + Be−rx
I
I= (Aerx − Be−rx )
ZC
V +I Z V −I Z
V = r 2 r c erx + r 2r C e−rx
1 V +I Z V −I Z
I = � r 2 r C erx − r 2 r C e−rx �
ZC

→ The propagation constant r = ∝ + jβ ; the real part is known as attenuation constant and the
quadrature component β the phase constant and is measured in radians per unit length.
Vr + Ir ZC Vr −Ir ZC
V= 2
e∝x . ejβx + 2
e−∝x . e−jβx
Vs = Vr cos hrl + Ir Zc sin hrl
sin hrl
IS = Vr + Ir cos hrl
Zc
A = cosh rl
B = Zc sinh rl
sinh rl
C=
Zc
D = cosh rl

The equivalent Circuit Representation of a Long Line equivalent – 𝛑𝛑 Representation

Z sinhr𝑙𝑙
Z1 =
r𝑙𝑙
y1 y1 y Tanhr𝑙𝑙/2
VS =
2 2 2 r𝑙𝑙/2

equivalent – T Representation of Long Line.

Z′ Z Tanhr𝑙𝑙/2
=
2 2 r𝑙𝑙/2
IS

sinh r𝑙𝑙
VS Y1 = Y r𝑙𝑙
Vr

Constants for Two networks in Tandem

A B A B1 A2 B2
equivalent � �=� 1 ��
C D C1 D1 C2 D2

Constants for networks in parallel

IS 1
A1 , B1
C1 , D1
IS Ir

vS vr

IS 2
A2 , B2
C 3 , D2

A1 B2 + A2 B1
A=
B1 + B2

Equivalent B .B
B = B 1+ B2
Single 1 2

Network A1 B2 + A2 B1 D1 B2 + D2 B1
A=D= B1 + B2
= B1 + B2
Parameters
(A1 − A2 )(D2 − D1 )
C = C1 + C2 +
B1 + B2

Critical disruptive voltage :-

 Critical disruptive voltage is defined as the voltage at which complete disruption of

dielectric occurs

 At any other temperature and pressure

g 0′ = g 0 . δ
Where δ is the air density correction factor and is given by
3.92 b
δ=
273+t
where b is the barometric pressure in cm of Hg and t the temperature in ℃.
d
∴ The critical disruptive voltage is given by V ′ = rg 0 δ ln kV
r

Polished wires −1
Roughened or weathered wires − 0.98 to 0.93
Seven strand cable − 0.87 to 0.83
Large cables with more then seven strands − 0.90 approx

The distance between g v and g 0 is called energy distance. According to peek this distance
is equal to (r + 0.301√r) for coaxial conductors.
0.3
→ g v = g 0 δ �1 + � kv/cm for two wires in parallel
√rδ
0.3 d
→ vv = rg 0 δ �1 + � ln kV
√rδ r
→ In case the irregularity factor is taken into account,
0.3 d
Vv = 21.1 mv δr �1 + � ln r kV rms
√rδ
Where r is the radius in cms.
The irregularity factor mv has the following values:
mv = 1.0 for polished wires
= 0.98 to 0.93 for rough conductor exposed to atmospheric severities
= 0.72 for local corona on stranded conductors.

 Peek made a number of experiments to study the effect of various parameters on the
corona loss and he deduced an empirical relation
(f+25) r 2
P = 241 × 10−5 δ
�d �Vp − V0 � kw/km/phase
Where f = frequency of supply
δ = The air density correction factor
Vp = The operating voltage in kV
V0 = the critical disruptive voltage

Sag in Transmission and Distribution Lines

wl2
→ When the Supports are at the Same Level , Sag = s = 2T
→ Where w = weight in kg/m run
𝑙𝑙 = half the span length in metre

→ w = weight of conductor kg/m run

wi = weight of ice kg/m run
ww = wind pressure kg/m run acting horizontally
→ Weight of ice (Wi ) = πR (D + R) × 1 × specific wt of ice
Specific weight of ice = weight of ice/m3 = 1000 kg
→ Wind pressure = Ww = (D + 2R) × 1 × wind pressure kg/m3
∴ W = �(w + wi )2 + ww
2

Wl2
∴ Sag = m
2T

Factor of Safety
Max. Stress
→ factor of safety =
permissible stress
→ Vertical sag = S cos θ

When the Supports are at different level

→ Let l = Span length

h = difference in levets between two supports
x1 = Distance of support at low level from 0
x2 = Distance of support at higher level from 0
T = Tension in the conductor
W = weight of the conductor per unit length, then
Wx12 Wx22
S1 = ; S2 =
2T 2T
𝑙𝑙 Th
x1 = −
2 w𝑙𝑙
𝑙𝑙 Th
x2 = +
2 w𝑙𝑙
x1 + x2 = 𝑙𝑙
S2 − S1 = h
→ The sag is provided in over headlines so that the safe tension is not exceeded.

Most Economical Conductor size in a cable :-

2V
→ g max = D volts/m
d loge
d

Power System Operation and Control:

Incremental change in input

→ incremental fuel rate IFR =
Incremental change in output
∆ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 d (input) dF
→ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = = =
∆ 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 d (output) dp
(output) dp
→ Incremental efficiency = (input)
=
dF

→ If Pi and Q i are scheduled electrical generation, PDi and Q Di are the respective load demands,
Pi − PDi − P𝑙𝑙 = Mi = 0
Q i − Q Di − Q 𝑙𝑙 = Ni = 0 where
Mi and Ni represent the power residuals at bus i and P𝑙𝑙 and Q 𝑙𝑙 the power flow to the neighbouring
system given by
P𝑙𝑙 = ∑Nj=1 Vi Vj Yij cos�δij − θij �
Q 𝑙𝑙 = ∑N
j=1 Vi Vj Yij �δij − θij �
→ For proper operation, each generator should have a minimum and maximum permissible output
and the unit production should be constrained to ensure that
Pimin ≤ Pi ≤ Pimax , i = 1, 2, … … , NP
Q imin ≤ Q i ≤ Q imax , i = 1, 2, … … , Na
Np and NQ being total number of real and reactive sources in the system

→ The condition for optimum operation is

dF1 dF2 dFn
= = ∙∙ = = λ
dP1 dP2 dPn
dF
→ Here dPn = incremental production cost of plant n in RS. per MWhr.
n

→ The incremental production cost of a given plant over a limited range is represented by
dFn
dPn
= Fnn Pn + fn
Where Fnn = slope of incremental production cost curve.
fn = intercept of incremental production cost curve.

→ The optimal load dispatch problem including transmission losses is defined as

 The solution of coordination equation requires the calculation of ∂PL / ∂Pn which
is obtained from equation as

∂ PL d Fn
∂ Pn
= 2 ∑m Bmn Pm also d Pn
= Fnn Pn + fn

 The coordination equations can be written as

Fnn Pn + fn + λ Σ 2Bmn Pm = λ

→ Solving for Pn we obtain

fn
∑ 2 Bmn Pm
1−
Pn = λ m ≠n
Fnn
+ 2 Bnn
λ
Where Pm and Pn are the source loadings, Bmn the transmission loss co – efficient.

→ The loss formula equation is expressed in terms of generations and is

approximately expressed as
PL = � � Pm Bmn Pn
m n
Where Pm and Pn are the source loadings, Bmn the transmission loss co – efficient.

→ For two generated plants are there then Power loss,

PL = B11 P12 + B12 P1 P2 + B22 P22
Where B11 , B12 , B22 are loss coefficients of transmission line

→ 𝑇𝑇𝑇𝑇𝑇𝑇 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑏𝑏𝑏𝑏 𝑎𝑎 𝑡𝑡𝑡𝑡𝑡𝑡 – 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 then the

𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑓𝑓𝑓𝑓𝑓𝑓 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 1 𝑎𝑎𝑎𝑎𝑎𝑎 2 𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
1 1
𝐿𝐿1 = & L2 =
𝜕𝜕 𝑃𝑃𝐿𝐿 ∂ PL
1− 1−
𝜕𝜕 𝑃𝑃1 ∂ P2

V= phase voltage

X= reactance in ohms per phase

→ Alternatively percentage reactance (%X) (an also be expressed in terms of KVA and KV
under

(𝐾𝐾𝐾𝐾𝐾𝐾)𝑋𝑋
%X=
10(𝐾𝐾𝐾𝐾)2

Where X is the reactance in ohms.

→ If X is the only reactance element in the circuit then short circuit currenr is given by
𝑉𝑉 100
Isc= = I × ( )
𝑋𝑋 %𝑋𝑋

i.e short circuit current is obtained by multiplying the full load current by 100/%X
100
Short- circuit KVA=Base KVA ×
%𝑋𝑋

Symmetrical components in terms of phase currents:-

→ The unbalanced phase current in a 3-phase system can be expressed in terms

of symmetrical components as under.

𝐼𝐼��⃗ ��⃗ ��⃗ ��⃗

𝑅𝑅 = 𝐼𝐼𝑅𝑅1 +𝐼𝐼𝑅𝑅2 +𝐼𝐼𝑅𝑅0
𝐼𝐼��⃗𝑌𝑌 = 𝐼𝐼��⃗ ��⃗ ��⃗
𝑌𝑌1 +𝐼𝐼𝑌𝑌2 +𝐼𝐼𝑌𝑌0
��⃗
𝐼𝐼𝐵𝐵 = 𝐼𝐼��⃗ ��⃗ ��⃗
𝐵𝐵1 +𝐼𝐼𝐵𝐵2 +𝐼𝐼𝐵𝐵0

Where The positive phase current (𝐼𝐼��

𝑅𝑅1 , 𝐼𝐼𝑌𝑌1 , &𝐼𝐼𝐵𝐵1 )

Negative phase sequence currents (𝐼𝐼��

𝑅𝑅2 , 𝐼𝐼𝑌𝑌2 , &𝐼𝐼𝐵𝐵2 ) and

→ The operator ‘a’is one, which when multiplied to a vector rotates the vector through 1200
in the anticlockwise direction.
→ A=-0.5+j 0.866 ; a2=-0.5- j 0.866

a3=1

→ Properties of operator ‘a’:

1+a+a2=0
a-a2=j √3
→ Positive sequence current 𝐼𝐼��⃗ ��⃗ ��⃗ ��⃗
𝐵𝐵1 in phase B leads 𝐼𝐼𝑅𝑅1 by 120 and therefore 𝐼𝐼𝐵𝐵1 = 𝑎𝑎 𝐼𝐼𝑅𝑅1
0

similarly, positive sequence current in phase Y is 2400 ahead of 𝐼𝐼��⃗ 2��⃗

𝑌𝑌1 = a 𝐼𝐼𝑅𝑅1
𝐼𝐼��⃗ ��⃗ ��⃗ ��⃗
𝑅𝑅 = 𝐼𝐼𝑅𝑅1 + 𝐼𝐼𝑅𝑅2 + 𝐼𝐼𝑅𝑅0

𝐼𝐼��⃗𝑌𝑌 = 𝐼𝐼��⃗ ��⃗ ��⃗ 2 ��⃗ ��⃗ ��⃗

𝑌𝑌1 + 𝐼𝐼𝑌𝑌2 + 𝐼𝐼𝑌𝑌0 = a 𝐼𝐼𝑅𝑅1 + 𝑎𝑎 𝐼𝐼𝑅𝑅2 + 𝐼𝐼𝑅𝑅0

𝐼𝐼��⃗ ��⃗ 2 ��⃗ ��⃗ ��⃗ ��⃗ ��⃗

𝐵𝐵 = a 𝐼𝐼𝑅𝑅1 + a 𝐼𝐼𝑅𝑅1 + 𝐼𝐼𝑅𝑅0 = 𝐼𝐼𝐵𝐵0 +𝐼𝐼𝐵𝐵1 +𝐼𝐼𝐵𝐵2

→ Zero sequence current:

𝐼𝐼��⃗ ��⃗ ��⃗ ��⃗ 2 ��⃗ 2 ��⃗
𝑅𝑅 + 𝐼𝐼𝑌𝑌 + 𝐼𝐼𝐵𝐵 = 𝐼𝐼𝑅𝑅1 (1+a+a ) + 𝐼𝐼𝑅𝑅2 (1+a+a ) + 3𝐼𝐼𝑅𝑅0
= 3 𝐼𝐼��⃗
𝑅𝑅0
1
∴ ��⃗
𝐼𝐼𝑅𝑅0 = [ 𝐼𝐼��⃗ ��⃗ ��⃗
𝑅𝑅 + 𝐼𝐼𝑅𝑅 + 𝐼𝐼𝑅𝑅 ]
3
→ Positive sequence current :
𝐼𝐼��⃗ ��⃗ 2 ��⃗ ��⃗ 3 3 ��⃗ 2 4 ��⃗
𝑅𝑅 +a 𝐼𝐼𝑌𝑌 +a 𝐼𝐼𝐵𝐵 = 𝐼𝐼𝑅𝑅1 (1+a +a ) + 𝐼𝐼𝑅𝑅2 (1+a +a ) + 𝐼𝐼𝑅𝑅0 (1+a+a )
2

��⃗1
= 3𝐼𝐼𝑅𝑅
1
∴ 𝐼𝐼⃗R1 = = [ 𝐼𝐼��⃗ ��⃗ 2 ��⃗
𝑅𝑅 +a 𝐼𝐼𝑌𝑌 +a 𝐼𝐼𝐵𝐵 ]
3
→ Negative sequence current:-
𝐼𝐼��⃗ 2 ��⃗ ��⃗ ��⃗ ��⃗ ��⃗
𝑅𝑅 +a 𝐼𝐼𝑌𝑌 +a 𝐼𝐼𝐵𝐵 = 𝐼𝐼𝑅𝑅1 (1+a +a ) + 𝐼𝐼𝑅𝑅2 (1+a +a ) + 𝐼𝐼𝑅𝑅0 (1+a +a)
4 2 3 3 2

��⃗2
= 3 𝐼𝐼𝑅𝑅
1
∴ 𝐼𝐼⃗R2 = [ 𝐼𝐼��⃗ 2 ��⃗ ��⃗
𝑅𝑅 +a 𝐼𝐼𝑌𝑌 +a𝐼𝐼𝐵𝐵 ]
3

�⃗ R=0 and 𝐼𝐼⃗B =𝐼𝐼⃗Y=0

→ 𝑉𝑉

The sequence currents in the red phase in terms of line currents shall be :-

3 𝐸𝐸𝑅𝑅 ��⃗
→ Fault current:- Fault current, 𝐼𝐼��⃗ ��⃗
𝑅𝑅 =3 𝐼𝐼0 =��⃗+𝑍𝑍
��⃗
𝑧𝑧0 1 +��
𝑍𝑍2

Phase voltage at fault

Since the generated emf system is of positive sequence only ,the sequence

components of emf in R-phase are:

The sequence voltage at the fault for R-phase are: This is ecpected because R-phase is
shorted to ground.

∴ The phase voltages at fault are:

Line –To-Line fault:-

→
Again taking R-phase as the reference, we have

Expressing in terms of sequence components of red line, we have

Also,

Fault current:

→ Phase voltages: - since the generated emf system is of positive phase sequence only,the
sequence components of emf in R-phase are:

→ The sequence voltages at the fault for R-phase are:

→ Double Line- To – Ground Fault:-

The conditions created by this fault lead to :

Also,

→ Fault current:-

Phase voltages: - the sequence voltages for phase R are:

Now

TRANSIENTS IN SIMPLE CIRCUITS:

𝑁𝑁𝑁𝑁𝑁𝑁 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑏𝑏𝑏𝑏 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 .

b) Inductance only:- when switch s is closed, the current in the circuit will be given by
𝑉𝑉(𝑠𝑠) 𝑉𝑉 1 𝑉𝑉 1
𝐼𝐼(𝑠𝑠) = = . = .
𝑍𝑍(𝑠𝑠) 𝑆𝑆 𝐿𝐿𝐿𝐿 𝐿𝐿 𝑆𝑆 2
𝑉𝑉
𝑖𝑖(𝑡𝑡) = t
𝐿𝐿
c) Capacitance only:- when switch s is closed, the current in the circuit is given
𝑉𝑉(𝑠𝑠) 𝑉𝑉
I(s) = = .CS =VC
𝑍𝑍(𝑠𝑠) 𝑆𝑆
Which is an impulse of strength (magnitude)VC

d) R-L circuit: when switch s is closed, the current in the circuit is given by
𝑉𝑉(𝑠𝑠) 𝑉𝑉 1 𝑉𝑉 1/𝐿𝐿
I(s) = = . = .
𝑍𝑍(𝑠𝑠) 𝑆𝑆 𝑅𝑅+𝐿𝐿𝐿𝐿 𝑆𝑆 𝑆𝑆+𝑅𝑅/𝐿𝐿
𝑉𝑉 1 1 𝐿𝐿
= � − �.
𝐿𝐿 𝑆𝑆 𝑆𝑆+𝑅𝑅/𝐿𝐿 𝑅𝑅
𝑉𝑉 1 1
= � − �
𝑅𝑅 𝑆𝑆 𝑆𝑆+𝑅𝑅/𝐿𝐿
𝑉𝑉 −𝑅𝑅
𝑖𝑖(𝑡𝑡) = �1 − 𝑒𝑒𝑒𝑒𝑒𝑒 � 𝑡𝑡��
𝑅𝑅 𝐿𝐿

e) R-L circuit: After the switch s is closed, current in the circuit is given by
𝑉𝑉(𝑠𝑠) 𝑉𝑉 1
I(s) = = .
𝑍𝑍(𝑠𝑠) 𝑆𝑆 𝑅𝑅+1/𝐶𝐶𝐶𝐶
1
𝑉𝑉 �𝑅𝑅𝑅𝑅�𝐶𝐶𝐶𝐶 𝑉𝑉 1
= = .
𝑆𝑆 𝑆𝑆+1/𝑅𝑅𝑅𝑅 𝑅𝑅 𝑆𝑆+1/𝑅𝑅𝐶𝐶
𝑉𝑉
i (t)= .𝑒𝑒 −𝑡𝑡/𝐶𝐶𝐶𝐶
𝑅𝑅

→ R-L-C circuit: - After the switch S is closed, the current in the circuit is given by
𝑉𝑉 1
Type equation here. I(s) =
𝑆𝑆 𝑅𝑅+𝐿𝐿𝐿𝐿+1𝐶𝐶𝐶𝐶

𝑉𝑉 1
I(s) = .(𝑆𝑆+𝑎𝑎−𝑏𝑏)(𝑆𝑆+𝑎𝑎+𝑏𝑏)
𝐿𝐿

𝑅𝑅 𝑅𝑅2 1
where = a and � − = b; then
2𝐿𝐿 4𝐿𝐿2 𝐿𝐿𝐿𝐿

→ There are three conditions based on the value of to

𝑅𝑅2 1
∗ If 4𝐿𝐿2 > ,b is real
𝐿𝐿𝐿𝐿
𝑅𝑅2 1
∗ If = ,b is zero
4𝐿𝐿2 𝐿𝐿𝐿𝐿
𝑅𝑅2 1
∗ If 4𝐿𝐿2
< ,b is imaginary
𝐿𝐿𝐿𝐿

Case I: when b is real

𝑉𝑉 𝑅𝑅 𝑅𝑅2 1 𝑅𝑅 𝑅𝑅2 1
→ i(t) = �𝑒𝑒𝑒𝑒𝑒𝑒 �− �2𝐿𝐿 + �4𝐿𝐿2 − 𝐿𝐿𝐿𝐿 � +� − 𝑒𝑒𝑒𝑒𝑒𝑒 �− �2𝐿𝐿 − �4𝐿𝐿2 − 𝐿𝐿𝐿𝐿 � 𝑡𝑡� �
𝑅𝑅2 1
2� 2 − .𝐿𝐿
4𝐿𝐿 𝐿𝐿𝐿𝐿

Case II: when b= 0

The expression for current becomes

𝑉𝑉
→ i(t)= {𝑒𝑒 −𝑎𝑎𝑎𝑎 − 𝑒𝑒 −𝑎𝑎𝑎𝑎 } which is indeterminate.
2𝑏𝑏𝑏𝑏

→ Now at b=0
𝑉𝑉 𝑉𝑉𝑉𝑉𝑒𝑒
i (t) = 𝑡𝑡 𝑒𝑒 −𝑎𝑎𝑎𝑎 = − (𝑅𝑅/2𝐿𝐿)𝑡𝑡
𝐿𝐿 𝐿𝐿

Case III. When b is imaginary

𝑉𝑉 𝑉𝑉
→ i (t) =
2𝑏𝑏𝑏𝑏
�𝑒𝑒 −𝑎𝑎𝑎𝑎 . 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 − 𝑒𝑒 −𝑎𝑎𝑎𝑎 . 𝑒𝑒 −𝑗𝑗𝑗𝑗𝑗𝑗 � = 2𝑏𝑏𝑏𝑏 𝑒𝑒 −𝑎𝑎𝑎𝑎 . 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

𝑉𝑉 −𝑅𝑅2 1
= 𝑒𝑒 −𝑎𝑎𝑎𝑎 .2 sin�� 4𝐿𝐿2 + � t
𝑅𝑅2 1 𝐿𝐿𝐿𝐿
2𝐿𝐿� 2 −
4𝐿𝐿 𝐿𝐿𝐿𝐿

A.C source:

→ R-L circuit: when switch is is closed, the current in the circuit is given by

𝑉𝑉(𝑆𝑆) 𝜔𝜔 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1

I (s) = = 𝑉𝑉𝑚𝑚 � 2 + 2 � 𝑅𝑅+𝐿𝐿𝑆𝑆
𝑍𝑍(𝑆𝑆) 𝑆𝑆 2+𝜔𝜔 𝑆𝑆 2+𝜔𝜔

𝑉𝑉𝑚𝑚 𝜔𝜔 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1

= �
𝐿𝐿 𝑆𝑆 2+𝜔𝜔2
+ 2 � 𝑆𝑆+𝑅𝑅/𝐿𝐿
𝑆𝑆 2+𝜔𝜔

→ R-L circuit connected to an ac source

𝑅𝑅
Let = 𝑎𝑎; 𝑡𝑡ℎ𝑒𝑒𝑒𝑒
𝐿𝐿
𝑉𝑉𝑚𝑚 𝜔𝜔 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
I(S) = �
𝐿𝐿 (𝑠𝑠+𝑎𝑎)(𝑆𝑆 2+𝜔𝜔2 )
+ 2 �
(𝑠𝑠+𝑎𝑎)(𝑆𝑆 2+𝜔𝜔 )
𝑉𝑉𝑚𝑚
i(t) = {sin(𝜔𝜔𝜔𝜔 + 𝜑𝜑 − 𝜃𝜃) − sin(𝜑𝜑 − 𝜃𝜃)𝑒𝑒 −𝑎𝑎𝑎𝑎 }
�(𝑅𝑅2 +𝜔𝜔2 𝐿𝐿2 )1�2

𝜔𝜔𝜔𝜔
Where θ= 𝑡𝑡𝑡𝑡𝑡𝑡−1 𝑅𝑅

Circuit Breaker ratings :

→ The value of resistor required to be connected across the breaker contacts which will

𝐿𝐿
give no transient oscillations, is R= 0.5�
𝐶𝐶

Where L,C are the inductance and capacitance upto the circuit breaker
2Vr
→ The average RRRV =
π√Lc

→ Maximum value of RRRV = wn Epeak

→ Where wn = 2πfn ,
1 1
→ Natural frequency of oscillations, fn = �
2π LC

Where L , C are the reactance and capacitance up to the location of circuit breaker

1 1 1
→ Frequency of demand oscillations, f = � − 2 2
2π LC 4R C

→ Symmetrical breaking current = r.m.s value of a.c component

x
= 2
√
→ Asymmetrical breaking current = r.m.s value of total current.
𝑋𝑋 2
= �� 2� + 𝑌𝑌 2
√
Where X = maximum value of a.c component
Y = d.c component

→ Is the rated service line voltage in volts, then for 3-phae circuit? Breaking capacity = √3 ×
V × I × 10−6 MVA

Voltage across the string

→ String efficiency =
n × voltage acrosss the unit near power conductor
Where, n = no. of insulators

Making capacity :-

→ Making capacity = 2.55 × symmetrical breaking capacity.

The Universal Relay Torque Equation:-

→ The universal relay torque equation is given as follows

T = K1 I 2 + K 2 V 2 + K 3 VI(θ − τ) + K
Distance Relays:

impedance relays :

From the universal torque equation putting 𝐾𝐾3 = 0 and giving negative sign to voltage term,
it becomes
→ T = K1 I 2 − K 2 V 2 (Neglecting spring torque)
For the operation of the relay the operating toque should be greater than the
restraining torque i.e
K1 I 2 > K 2 V 2
→ Here V and I are the voltage and current quantities fed to the relay.
V2 K
→ I2 < 1�K
2
K1
→ Z<� �K
2

Reactance Relay:

The directional element is so designed that its maximum torque angle is 900
i.e. in the universal torque equation.
T = K1 I 2 − K 3 VI cos(θ − τ)
= K1 I 2 − K 3 VI cos(θ − 90)
= K1 I 2 − K 3 VI sin θ
For the operation of the relay
KI 2 > K 3 VI sinθ
VI
sinθ < K1 /K 3
I2
K
Z sinθ < 1�K
3
K1
X < �K
3

The mho relay:-

→ In the relay the operating torque is obtained by the V – I element and restraining torque due
tot the voltage elemen

T = K 3 VI cos(θ − τ) − K 2 V 2
→ For relay to operate
K 3 VI cos (θ − τ) > K 2 V 2
V2 K
< 3�K cos(θ − τ)
VI 2
K3
Z < �K cos(θ − τ)
2

Time Response of 2nd order system :-

Step i/P :
e− ζωn t �1−ζ2
• C(t) = 1- (sin ωn �1 − ζ2 t ± tan−1 � � )
�1−ζ2 ζ
e− ζωn t �1−ζ2
• e(t) = �sin 𝜔𝜔𝑑𝑑 𝑡𝑡 ± tan−1 � ��
�1−ζ2 ζ

e− ζωn t �1−ζ2
• ess = lim �sin 𝜔𝜔𝑑𝑑 𝑡𝑡 ± tan−1 � ��
𝑡𝑡→∞ �1−ζ2 ζ

→ ζ → Damping ratio ; ζωn → Damping factor

ζ < 1(Under damped ) :-

e− ζωn t �1−ζ2
C(t) = 1- = Sin �𝜔𝜔𝑑𝑑 𝑡𝑡 ± tan−1 � ��
�1−ζ2 ζ

ζ = 0 (un damped) :-

c(t) = 1- cos ωn t

ζ = 1 (Critically damped ) :-

C(t) = 1 - e−ωn t (1 + ωn t)

ζ > 1 (over damped) :-

−�𝛇𝛇− �𝛇𝛇𝟐𝟐 −𝟏𝟏� 𝛚𝛚𝐧𝐧 𝐭𝐭

e
C(t) = 1 -
2 �𝛇𝛇𝟐𝟐 −𝟏𝟏 �𝛇𝛇− �𝛇𝛇𝟐𝟐 −𝟏𝟏�

1
T=
�𝛇𝛇− �𝛇𝛇𝟐𝟐 −𝟏𝟏�ωn
Tundamped > Toverdamped > Tunderdamped > Tcriticaldamp

Time Domain Specifications :-

π−∅ �1−ζ2
• Rise time t r = ∅ = tan−1 � �
ωn �1−ζ2 ζ
nπ
• Peak time t p =
ωd
2
• Max over shoot % Mp = e−ζωn/�1−ζ × 100
• Settling time t s = 3T 5% tolerance

Static error coefficients :-

𝑆𝑆𝑆𝑆(𝑠𝑠)
• Step i/p : ess = lim 𝑒𝑒(𝑡𝑡) = lim 𝑠𝑠 𝐸𝐸(𝑠𝑠) = lim
t→∞ 𝑠𝑠→0 𝑠𝑠→0 1+𝐺𝐺𝐺𝐺

1
ess = (positional error) K p = lim 𝐺𝐺(𝑠𝑠) 𝐻𝐻(𝑠𝑠)
1+KP 𝑠𝑠→0

1
• Ramp i/p (t) : ess = K v = lim 𝑆𝑆 𝐺𝐺(𝑠𝑠)𝐻𝐻(𝑠𝑠)
Kv 𝑠𝑠→0

• Parabolic i/p (t 2 /2) : ess = 1/ K a K a = lim s 2 𝐺𝐺(𝑠𝑠)𝐻𝐻(𝑠𝑠)

𝑠𝑠→0

Type < i/p → ess = ∞

Type = i/p → ess finite
Type > i/p → ess = 0

∂A/A
• Sensitivity S = sensitivity of A w.r.to K.
∂K/K
• Sensitivity of over all T/F w.r.t forward path T/F G(s) :
Open loop: S =1
1
Closed loop : S=
1+G(s)H(s)

• Minimum ‘S’ value preferable

G(s)H(s)
• Sensitivity of over all T/F w.r.t feedback T/F H(s) : S =
1+G(s)H(s)

Stability
RH Criterion :-

• Take characteristic equation 1+ G(s) H(s) = 0

• All coefficients should have same sign
• There should not be missing ‘s’ term . Term missed means presence of at least one +ve real part root

Derived Units:

Area m2 L2 MMF A I
Volume m3 L3 Frequency Hz T −1
Density Kg/ m3 L−3 M Velocity m/sec LT −1
Angular rad / sec [L]0 T −1 Acceleration m/sec 2 LT −2
Velocity
Angular rad / sec 2 [L]0 T −2 Force Kg m/sec 2 LMT −2
Acceleration
Pressure, Kg/m/sec 2 L−1 MT −2 Luminous flux lm(cd Sr)
stress
Luminance cd/m2 Illumination lm/m2
2 −2
Work , Joule L MT Power Watt L2 MT−3
Energy (Nm) (J/sec)
Charge Coulomb TI EMF Volt (W/A) L2 MT −3 I−1
Electric field V/m LMT −3 I −1 Resistance V/A L2 MT −3 I2
strength
Capacitance (AS/v) L−2 M−1 T 4 I2 Magnetic flux Vs L2 MT −2 I−1
Magnetic Wb / m2 MT −2 I −1 Inductance Vs / A L2 MT −2 I2
flux density

Static error:-

Static error is defined as the difference b/w the measured value and the true value of

the quantity.

 δA = ϵ0 = Am − At
where Am = measured value of quantity

 At = True value of quantity

 δA = Absolute static error
Absolute error
 Relative static error (ϵr ) = True value

∂A ϵ
= = A0
At t

 Percentage static error (% ϵ0 ) = ϵr × 100

Am
 True value At = 1+ϵ
r

 δc = Am − At = - δA

Sensitivity :

It is the ratio of infinitesimal change in output to an infinitesimal change in input.

Infinitesimal change in output

 Static sensitivity = Infinitesimal change in input

Infinitesimal change in input

 Inverse sensitivity (or) deflection factor = Infinitesimal change in output
Linearity:

A measurement system is considered to be linear if the output is linearly proportional to the input .

(maximum deviation of output from the idealized straight line

 Non-linearity = actual reading
× 100

(maximum deviation of output from the idealized straight line)

 Non-linearity = × 100
full scale deflection

 The magnitude of a quantity having a nominal value As and a maximum Aa error or

limiting error of ± δA must have a magnitude Aa between the limits As − δA and
As ± δA

 Actual value of quantity Aa = As + δA

 Relative limiting error , ϵr = ϵ0 / As
A a = A s [ 1 ± ϵr ]

 Magnitude of limiting error = full scale reading × guaranteed accuracy

∑ n
Xi
 Mean : Arithmetic Mean = 𝑋𝑋� = i=1
n
The root mean square value is the standard deviation σ .

2
∑n
i=1 di
 If the number of readings is less than 20, then S.D, σ = � n−1
Variance = σ2

Moving Coil Permanent Magnet Type:

 The deflecting torque Td is given by

Td = NBI l.b = NBA.I

Where N is the number of turns of the coil,

B is air gap flux density , wb/m2 .
I is current in the coil, ampere.
L is active length of conductor , metre and
b is width of coil, metre

 The controlling torque , TC is given by TC = Sθ

Where S is spring constant and θ is the angle of deflection.

 When steady deflection is reached , Td = Tc and

Hence θ ∝ I.

Moving Coil Dynamometer Type :

 The torque developed in a dynamometer type ammeter or voltmeter is given by

Td = I 2 (dM/Dθ)

Where I is the current through fixed coil and moving coil (connected in series) and

M is the mutual inductance between fixed and moving coils.

Moving Iron Instruments :

 The deflecting torque is given by Td = (1/2) I2 (dL/dθ).

Where I is the current through the coil and

L is the inductance.

Linearization of scale:

 θ.(dL/dθ)=K.

 Compensation towards frequency errors can be done by connecting a capacitor

across a part of series resistance in MI voltmeter C = 0.41 × (L/R2 ).

Half wave Rectifier type Instruments:

Vdc
 d.c supply → Im = R
m +RL
 a.c supply → V = Vrms voltage of source.
Vdc = (Vm /π)

 Im = (√2V) / (π(R L + R m ))
= (0.45 V)/(R m + R L )

= (√2 V)/(π)

In half wave rectifier type instrument , the sensitivity of a.c is 0.45 times that of d.c.

Full wave rectifier type instruments:

V
 d.c. supply → Im = R
m +RL

2Vm
 a.c supply → Im = π(R
m +RL )

0.9 V
=
Rm +RL

In full wave rectifier type instrument , the sensitivity of a.c is 0.9 times that of d.c.

Thermal instruments :

The change in the length of the wire due to temperature is given

 ∆L = [(RLα)/(UA)]I2 (or) ∆L = [(Lα)/(UAR)]V 2

Where L in the original length of the wire,

α is the temperature coefficient of linear expansion of wire material,

R is the resistance of wire,

U is the coefficient of heat dissipation,

A is the area of heat dissipating surface.

Thermo couple instruments

 The e.m.f generated in thermocouple is given by

E = a(∆T) + b(∆T)2

Where ∆T is the temperature difference between hot and cold junctions

‘a’ is a constant (40 to 50 μv/0C )

‘b’ is a constant (1 to 2 μv / 0C )

1. Kelvin’s Double Bridge is used for the measurement of low resistance

r
 Vad = Vam +Vmd =I2 R + I2 p+q+r
P
P qr 𝑃𝑃 𝑝𝑝
 R= S.Q p+q+r �𝑄𝑄 − 𝑞𝑞 �
2.Measurement of Medium Resistance:

(a)Voltmeter-Ammeter method:
VR +Va
 Measured value of resistance, R ml = IR
Where R is the true value of the resistance.

 Error = R a % Error = (R a /R)

In this method, always the measured value of resistance is greater than true value

of resistance.

In this method, the measured value of resistance is always less than the true value

of resistance.

This method is suitable for measurement of low resistance among the range.

 The resistance where both the methods give same error is obtained by equating the
two errors.
Ra R
 =R ,
R v
 R = �R a R v
Wheatstone Bridge:
P R
 Balanced condition → Q = S
Q S
=
P R

P R
=
P+Q R+S

θ
 Sensitivity of the galvanometer, Sv = e
Where θ = deflection of the galvanometer

e = emf across galvanometer

= Vb - Vd

= (E - Vab )- (E - Vad )

= Vad - Vab
(R+∆R) E.P
= E−
R+∆R+S P+Q

(R+S)2
 ∴ Sensitivity of galvanometer, Sv = .θ
E.S∆.

E.S∆R
Or θ = Sv .
(R+S)2

θ
 Sensitivity of the Bridge = SB = ∆R
R

E E
SB = Sv . R S = Sv = P Q
+2+ +2+
S R Q P

Sv .E
SBmax =
4

P R
 When Q
=S=1

Bridge sensitivity is useful for

(a) Selecting the galvanometer with which the given unbalance can be observed.
(b) Determining the deflection to be expected for a given unbalance.

Measurement of High Resistance:

(a) Loss of charge method :

 At t, voltage across capacitor, v = Ve−t/RC

−t
R=
2.303C log(V/v)

0.434 t
R=
C log10 (V/v)

 At balance, the Pd across BA = Pd across BC in magnitude and phase

EBA = EBC = I1 Z1 = I2 Z2

Z1 Z4 ∠(Q1+Q 4 ) = Z2 Z3 ∠(Q 2+Q 3 )

Equating the magnitudes and angles

Z1 Z4 =Z2 Z3

∠(Q1+Q 4 ) = ∠(Q 2 +Q 3 )

Measurement of Self Inductance:-

The following bridges are used for measurement of self inductance.

1. Maxwell’s Inductance Bridge

2. Maxwell’s Inductance- Capacitance Bridge
3. Hay’s Bridge
4. Owen’s Bridge
5. Anderson’s Bridge

1.Maxwell’s Inductance Bridge:-

This bridge circuit measures an inductance by comparison with a variable standard

self-inductance.

At balance condition,

 L1 = R 3 L2 /R 4 ; R1 = R 3 (R 2 +r2 )/R 4

Where L1 = unknown inductance of resistance R1

L2 = variable inductance of fixed resistance r2

R 2 = variable resistance connected in series with inductor L2

R 3 , R 4 = known non-inductive resistance

In this bridge, an inductance is measured by comparison with a standard

variable capacitance.

At balance conditions,

 R1 = R 2 R 3 /R 4 ; L1 = R 2 R 3 C4

 Q-factor, Q= wL1 /R1 =wC4 R 4

 This bridge is used for measurement of low Q coils (1< Q<10).

3.Hay’s Bridge:-

This is a modification of Maxwell’s bridge. It is used for measurement of high Q coils.

L1 = unknown inductance having a resistance

R1 , R 2 , R 3 , R 4 = known non-inductive resistance

C4 = standard capacitor

At balance conditions,

R R C ω2 R2 R3 R4 C24
 L1 = 1+W22 +C
3 4
2 +R2 ; R 1 =
4 4 1+ω2 C24 R24

Q=ωL1 /R1 = 1/ωC4 R 4

4.Anderson’s Bridge:-

In this method, the self-inductance is measured in terms of a standard capacitor..

At balance conditions,

5.Owen’s Bridge:-

This bridge may be used for the measurement of an inductance in terms of capacitance.

Under balance conditions,

 L1 = R 2 R 3 C4
C
 R 1 = R 3 C4
2
Measurement of Capacitance:-

The following bridges are used for the measurement of capacitance:

1. De Sauty’s Bridge
2. Modified De Sauty’s Bridge
3. Schering Bridge.

1.De Sauty’s Bridge :- This bridge is the simplest method of comparing two capacitances.

 At balance condition, C1 =C2 . R 4 /R 3

2.Modified De Sauty’s Bridge:-

C R +r R
 At balance condition, C1 = R2 +r2 =R4
2 1 1 3

 Dissipation factor, D1 = tan δ1 = ω c1

 D2 = tan δ2 = ω c2 r2

𝑅𝑅1 𝑅𝑅4
 D1 - D2 = ω c2 � − 𝑅𝑅2 �
𝑅𝑅3
3.Schering’s Bridge :-

 At balance conditions, r1 = R 3 C4 /C2 ;

 Dissipation factor, D1 = tan δ1 = ω c1 r1 = ωc4 r4

Measurement of Mutual Inductance:

The following bridges are used for the measurement of mutual inductance:

(1). Heaviside Mutual Inductance Bridge

(2). Campbell’s modification of Heaviside’s bridge

(3). Heaviside Campbell equal ratio bridge

(4). Carry Foster Bridge/ Heydweiler Bridge →

Ammeter Shunts:

These are small resistances in parallel to basic meter to increase current

measuring capacity

 Im R m = Ish R sh

m R
 = m−1

Where m is the multiplying power of ammeter

Series Multipliers:

These are used for increasing the voltage measuring capacity of basic met
V V
 Rse + Rm
=R
m

 R se = R m (m-1)
Where m is the multiplying power of voltmeter.

Multi Range Ammeter: A range of current settings can be obtained using different shunts.

R I
R sh2 = m m−1 , m2 = I 2
2 m

R I
R sh3 = m m−1 , m3 = I 3
3 m

4.Universal Shunt (or) Ayrton shunt:

Im R m = (I1 - Im ) R1

R I1
R1 = m m−1 , m1 = Im
1

R I2
R 2 = m m−1 , m2 = Im
2

R I3
R 3 = m m−1 , m3 = Im
3

MEASUREMENT OF POWER & ENERGY:

Power in D.C. Circuits

Power measured = Pm1 = VR IR + I 2 R R a

True value = Measured power – power loss in ammeter

Power measured = Pm2 = VR IR + (V 2 R /R v )

True power = Measured power- Power loss in voltmeter

Power in A.C. Circuits:

 Value of power but in its average value over a cycle.

Instantaneous power = VI

Average power = VI cos∅

Where V and I are r.m.s values of voltage and current and cos ∅ is the power factor of the
load.

3.Electro Dynamometer Wattmeter:

 The deflecting torque in electrodynamometer instruments is given by ,

Td = i1 i2 (dM/dθ)
Where i1 and i2 are instantaneous values of currents in two coils.

 Many watt meters are compensated for errors caused by inductance of pressure coil
by means of a capacitor connected in parallel with a portion of multiplier.
Capacitance C = (L/r 2 )

Low Power Factor Wattmeter:

 In present case of wattmeter , i1 is the load current and is the current flowing
through pressure coil.
Td = iP iC (dM/dθ)

 Average deflection torque = IP I cos ∅ (dM/dθ)

= (V/R P ).I cos ∅ (dM/dθ)

Td ∝ VI cos ∅ (dM/dθ)

 At balance condition, Td = TC
K1 VI cos ∅ (dM/dθ) = kθ
θ ∝ VI cos ∅

Errors in Electro Dynamometer Wattmeter:

In general , pressure coil offers inductance, in addition to resistance. Due to this

inductance, wattmeter reads more power on lagging loads and less power in case of leading
loads.

Correction is to be made for wattmeter reading to get true value of power.

cos ∅
 True power for lagging pf loads = cos β ∗cos(∅−β) × actual wattmeter reading

cos ∅
 True power for leading pf loads = cos β ∗cos(∅+β) × actual wattmeter reading

 ERROR = tan ∅ tan β × true power = VI sin ∅ tan β

∅ = pf angle

β = tan−1 (X P/R P )

 ERROR = tan ∅ tan β

 β→ is the angle between PC current and voltage.

Measurement Of Power in Three circuits:

(a)Three watt meter Method:

 In the three wattmeter method to determine the power in 3-∅, 4 wire system.

Sum of the instantaneous readings of watt meters = P= P1 +P2 +P3

P= V1 i1 +V2 i2+V3 i3

 Instantaneous power of load = V1 i1 +V2 i2 +V3 i3

Hence the summation of readings of three watt meters gives the total power of load.

In a 3-∅, three wire system we required 3 elements. But if we make the common
points of pressure coils coincide with one of the lines, then we will require only (n-1
l =2) elements.
 Instantaneous reading of P1 wattmeter (P1 ) = i1 (V1 - V3 )

 Instantaneous reading of P2 wattmeter (P2 ) = i2 (V2 - V3 )

 Sum of instantaneous reading of two watt meters = P1+P2

 From Kirchoff’s current law, i1 +i2 +i3 =0

 P1+P2 = i1 (V1 - V3 ) + i2 (V2 - V3 )

= V1 i1 - V3 i1 + V2 i2 - V3 i2

= V1 i1 + V2 i2 - V3 (i1 +i2 )

= V1 i1 + V2 i2 - V3 (-i3 )

= V1 i1 + V2 i2 + V3 i3

Hence, the sum of two watt meter readings is equal to power consumed by load.

Let the load be balanced, V1 , V2 , V3 be the rms value of phase voltage and I1 , I2 , I3 be the rms
values of phase currents.

 For star connection,

Phase voltages V1 = V2 = V3 = V (say)

Line voltages V13 = V23 = V12 = √3 V

Line currents I1 = I2 = I3 = I

Power factor cos ∅ (lag)

In general , reading of wattmeter is given by

 Wattmeter reading = current through wattmeter current coil * voltage across its
pressure coil * cos( phase angle between this current and voltage).

 ∴ Reading of P1 wattmeter = V13 I1 cos(30- ∅) = √3 VI cos(30-∅)

 Reading of P2 wattmeter = V23 I2 cos(30+ ∅) = √3 VI cos(30+∅)

 Sum of reading of two watt meters = P= P1 +P2

= √3 VI cos (30-∅) + √3 VI cos (30+∅)

= √3 VI cos ∅

This is the total power consumed by load.

 Difference of reading of two watt meters = P1 - P2

= √3 VI [cos (30-∅) - cos (30+∅)]

= √3 VI 2 sin 30 * sin ∅

= √3 VI sin ∅

∴ Reactive power consumed by load = √3 (Difference of two wattmeter readings).

= √3 (P1 - P2 )

√3 (P1 − P2 )
 Power factor cos ∅ = cos tan−1 � P1 +P2
�

7.Effect Of Power Factor On the Readings of Watt Meter:

The readings of two watt meters are equal , each watt meter reads half of the total power.

 When cos ∅ = 0.5, ∅ =60

P1 = (3/2) VI, P2 =0

When power factor is 0.5 , one of the watt meter reads zero and the other reads total power.

 When cos ∅ =0 , ∅= 900

P1 = �(3/2) V I , P2 = - �(3/2) V I
Therefore , width zero power factor, the readings of two meters are equal but of opposite
sign.

8.Measurement of Reactive Power in Three Phase Circuits:

 Reading of wattmeter = v23 i1 cos (angle between i1 and v23 )

= v23 i1 cos (90- ∅)

= √3 VI sin ∅

 Total reactive power of the circuit

= √3 (watt meter reading).

Measurement of Energy:

 Energy is the total power delivered or consumed over a time interval, that is.
Energy = Power × time

 Electrical energy developed as work or dissipated as heat over an interval of time t

may be expressed as :
t
W= ∫0 VI dt

Construction and Principle of Single Phase Induction Energy Meter:

 Net driving torque is given as, Td = k1 ∅1 ∅2 (f/z) sin β cos α

Where β = Phase between fluxes ∅1 and ∅2

α= Phase angle of eddy current paths.

These two fluxes will be produced by two currents which are described earlier.

 At steady speed the driving torque must equal to the breaking torque.
k 4N= k 3 VI sin (∆-∅)

 If ∆ = 900 , sped, N = K VI sin(90- ∅)

= K VI cos ∅

= k x(power)

 Total number of revolutions = ∫ N dt = k ∫ VI sin(∆ − ∅) 𝑑𝑑𝑑𝑑

 If ∆= 900 , total number of revolutions = k ∫ VI cos ∅ dt
= k * (energy)

Single Phase Electrodynamometer Power Factor Meter

 The values of R and L are so adjusted that the two coils carry the same value of
current
at normal frequency (i.e) R = ωL

 Deflecting torque acting on coil is: TA = KVI Mmax cos ∅ sin θ

Where θ= angular deflection from the plane of reference.

Mmax = maximum value of mutual inductance between the two coils.

This torque say acts in the clockwise direction.

 Deflecting torque acting on coil B is: TB = KVI Mmax cos (900 -∅) sin (900 +θ)
= KVI Mmax sin ∅ cos θ

This torque acts in anticlockwise direction.

 At equilibrium , TA = TB ⇒ θ = ∅
Therefore the deflection of the instrument is a measure of phase angle of the circuit.

Q-Meter:

 Q Meters are intended to measure the Q (quality factor) of an inductance or

capacitor.

ωL 1 IX 𝐈𝐈𝐗𝐗 𝐂𝐂 𝐕𝐕𝐂𝐂 𝐨𝐨𝐨𝐨 𝐕𝐕𝐚𝐚

Q= = = IRL = =
R ωCR 𝐈𝐈𝐈𝐈 𝐕𝐕𝐚𝐚

So the voltage across the capacitor or coil is Q times the applied voltage

𝑎𝑎) 𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪:

 In series connection method the unknown value component is connected in series

with the resonant circuit. This method is employed for measurement of low value
resistors, small coils and large capacitors.
1 𝜔𝜔𝜔𝜔 1
𝐴𝐴𝐴𝐴 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟, 𝜔𝜔𝜔𝜔 = 𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄1 = =
𝜔𝜔𝐶𝐶1 𝑅𝑅 𝜔𝜔𝐶𝐶1 𝑅𝑅

𝐶𝐶1 − 𝐶𝐶2
𝑋𝑋𝑠𝑠 =
𝜔𝜔𝐶𝐶1 𝐶𝐶2

𝑄𝑄1 𝐶𝐶1 − 𝑄𝑄2 𝐶𝐶2

𝑅𝑅𝑠𝑠 =
𝜔𝜔𝐶𝐶1 𝐶𝐶2 𝑄𝑄1 𝑄𝑄2

𝑋𝑋𝑠𝑠 (𝐶𝐶1 − 𝐶𝐶2 )𝑄𝑄1 𝑄𝑄2

𝑄𝑄𝑥𝑥 = =
𝑅𝑅𝑠𝑠 𝑄𝑄1 𝐶𝐶1 − 𝑄𝑄2 𝐶𝐶2

b) 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪:

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Institute of Engineering Studies (IES,Bangalore) Measurements Formula Sheet
𝐼𝐼𝐼𝐼 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 , 𝑡𝑡ℎ𝑒𝑒 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑡𝑡𝑡𝑡 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 , 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
.
𝑇𝑇ℎ𝑖𝑖𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑜𝑜𝑜𝑜 𝑖𝑖𝑖𝑖 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑜𝑜𝑜𝑜 ℎ𝑖𝑖𝑖𝑖ℎ 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟, 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

1 ωC1 1
 RP
= = RQ
Q2 1

1
 XP = ω(C
2 −C1 )

Rp (C1 −C2 )Q1 Q2

 Qx = X =
p Q1 C1 −Q2 C2

 The main error in the measurement of Q is due to distributed or stray capacitance of

the circuitry. To check for this, the Q is measured at two frequencies f1 and 2f1 . It
should be same if not,

C1 −C1 C2
Cd = 3

Cathode Ray Tube:

Lld Ed
 For electrostatic deflection, D= 2dEa
Where D-Deflection, m

L – distance from centre of deflection plates to screen, m

ld - effective length of deflection plates, m

Ed - deflection voltage, volts

d- separation between the plates, m

Ea - accelerating voltage, volts

D Ll
 Deflection sensitivity is S = E = 2dEd , m/V
d a

lLB e
 Magnetic Deflection D = �2m , m
�Va

l- width of magnetic coil, m

L- length from centre of 1 to screen, m

B- magnetic flux density, wb/m2

e, m – charge and mass of electron

Va - acceleration potential

D e
 Magnetic Deflection Sensitivity is B = IL �2mV , m
a

Oscilloscope Specifications:

1.Sensitivity:

It means the vertical sensitivity . It refers to smallest deflection factor G = (1/S) and
expressed as mv/div. The alternator of the vertical amplifier is calibrated in mv/div.

2.Band width:

It is the range of frequencies between ± 3 dB of centre frequency.

3.Rise Time:

Rise time is the time taken by the pulse to rise from 10% to 90% of its amplitude.
1
 BW = 2πRC BW = band width in MHz

90% of amplitude is normally reached in 2.2 RC or 2.2 time constants

 Synchronization means the frequency of vertical signal input is an integral multiple

of the sweep frequency.
Fin = n fs

𝐸𝐸G/si =1.21− 3.6 × 10−4 .T ev

• Energy gap 𝐸𝐸G/Ge =0.785− 2.23 × 10−4 .T ev
� Energy gap depending on temperature
𝑁𝑁 𝑁𝑁
• EF = EC - KT ln�𝑁𝑁𝐶𝐶 � = Ev + KT ln �𝑁𝑁𝑣𝑣 �
𝐷𝐷 𝐴𝐴
• No. of electrons n = Nc e−(Ec−Ef)/RT (KT in ev)
• No. of holes p = Nv e−(Ef−Ev )/RT
• Mass action law np = n2i = Nc Nv e−EG/KT
• Drift velocity 𝑣𝑣d = μE (for si 𝑣𝑣d ≤ 107 cm/sec)
B.I
• Hall voltage 𝑣𝑣H = . Hall coefficient R H = 1/ρ . ρ → charge density = qN0 = ne …
we
• Conductivity σ = ρμ ; μ = σR H .
q q
• Max value of electric field @ junction E0 = - Nd . nn0 = - NA . np0 .
ϵsi ϵsi
• Charge storage @ junction Q + = - Q − = qA xn0 ND = qA xp0 NA

EDC

dp dn
• Diffusion current densities Jp = - q Dp Jn = - q Dn
dx dx
• Drift current Densities = q(p µp + nµn )E
• µp , µn decrease with increasing doping concentration .
Dn Dp
• = = KT/q ≈ 25 mv @ 300 K
µn µp
• Carrier concentration in N-type silicon nn0 = ND ; pn0 = n2i / ND
• Carrier concentration in P-type silicon pp0 = NA ; np0 = n2i / NA
𝑁𝑁𝐴𝐴 𝑁𝑁𝐷𝐷
• Junction built in voltage V0 = VT ln � 𝑛𝑛𝑖𝑖2
�
2εs 1 1
• Width of Depletion region Wdep = xp + xn = � � + � (V0 + VR )
q NA ND
2𝜀𝜀𝑓𝑓𝑓𝑓
*� = 12.93𝑚𝑚 𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠�
𝑞𝑞
xn N
• = A
xp ND
q.NA ND
• Charge stored in depletion region qJ = NA +ND
. A . Wdep
εs A εs A
• Depletion capacitance Cj = ; Cj0 =
Wdep Wdep / VR =0

VR m
Cj = Cj0 /�1 + �
V0
Cj = 2Cj0 (for forward Bias)
Dp
• Forward current I = Ip + In ; Ip = Aq n2i L �𝑒𝑒 𝑉𝑉/𝑉𝑉𝑇𝑇 − 1�
p ND
Dn
In = Aq n2i L �𝑒𝑒 𝑉𝑉/𝑉𝑉𝑇𝑇 − 1�
n NA
Dp Dn
• Saturation Current Is = Aq n2i �L +L �
p ND n NA
• Minority carrier life time τp = L2p / Dp ; τn = L2n / Dn

Transistor :-
• IE = IDE + InE
• IC = ICo – α IE → Active region
• IC = – α IE + ICo (1- eVC /VT )

Common Emitter :-
α
• IC = (1+ β) ICo + βIB β=
1−α
I
• Co
ICEO = 1−α → Collector current when base open
• ICBO → Collector current when IE = 0 ICBO > ICo .
0 V
• VBE,sat or VBC,sat → - 2.5 mv / C ; VCE,sat → BE,sat
10
= - 0.25 mv /0 C
IC − ICBo
• Large signal Current gain β = IB + ICBo
IC
• D.C current gain βdc = = hFE
IB
• (βdc = hFE ) ≈ β when IB > ICBo
∂I hFE
• Small signal current gain β′ = ∂IC � = hfe = ∂h
R VCE 1−(ICBo + IB ) FE
∂IC
βactive
• Over drive factor = ∵ IC sat = βforced IB sat
βforced →under saturation

Conversion formula :-
CC ↔ CE
• hic = hie ; hrc = 1 ; hfc = - (1+ hfe ) ; hoc = hoe

CB ↔ CE
h hie hoe −h hoe
• hib = 1+hie ; hib = 1+h - hre ; hfb = 1+hfe ; hob = 1+h
fe fe fe fe

CE parameters in terms of CB can be obtained by interchanging B & E .

Specifications of An amplifier :-

AI ZL hf hr Av .Rs Avs .Rs

AV = Y0 = ho - AIs = =
Zi hi + Rs Zi +Rs ZL

Choice of Transistor Configuration :-

• For intermediate stages CC can’t be used as AV < 1
• CE can be used as intermediate stage
• CC can be used as o/p stage as it has low o/p impedance
• CC/CB can be used as i/p stage because of i/p considerations.

Stability & Biasing :- ( Should be as min as possible)

∆I ∆I ∆IC
• For S = ∆I C � S ′ = ∆V C � S ′′ = ∆β V
�
Co VB0,β BE IC0,β BE,ICo

∆IC = S. ∆ICo + S ′ ∆VBE + S ′′ ∆β

1+β
• For fixed bias S = dI =1+β
1−β B
dIC

1+β 1+β
• Collector to Base bias S = RC 0 < s < 1+ β = RC + RE
1+β 1+β� �
RC +RB RC + RE + RB

1+β Rth
• Self bias S = RE ≈ 1+ βR E > 10 R 2
1+β Re
RE +Rth

Vcc Rth V R
• R1 = Vth
; R 2 = V cc−Vth
cc th

VCC
• For thermal stability [ Vcc - 2Ic (R C + R E )] [ 0.07 Ico . S] < 1/θ ; VCE <
2

Hybrid –pi(π)- Model :-

g m = |IC | / VT

rb′ e = hfe / g m
rb′ b = hie - rb′ e
rb′ c = rb′ e / hre
g ce = hoe - (1+ hfe ) g b′ c

For CE :-
g ′ gm
• fβ = 2π(Cb +e C =
e c) hfe 2π(Ce + Cc )

For CC :-
1+gm RL gm fT Ce gm + gb′e
• fH = ≈ = =
2πCL RL 2πCL CL 2π(CL + Ce )

For CB:-
1+ hfe
• fα = 2πr = (1 + hfe ) fβ = (1 + β) fβ
b′ e (CC + Ce )

β
• fT = f fα > fT > fβ
1+β α

Ebress moll model :-

IC = - αN IE + ICo (1- eV/VT )

IE = - αI IC + IEo (1- eV/VT )

αI ICo = αN IEo

Multistage Amplifiers :-
fL
• fH * = fH √21/n − 1 ; fL∗ =
�21/n −1
0.35 0.35
• Rise time t r = =
fH B.W
• t ∗r = 1.1 2 2
�t r1 + t r2 + ⋯

• fL∗ = 1.1 �fL21 + fL22 + ⋯

1 1 1
• = 1.1 � 2 + +⋯
f∗H f H1 f2H2

Differential Amplifier :-

• Zi = hie + (1 + hfe ) 2R e = 2 hfe R e ≈ 2βR e

α0 |IEE | I
• gm = = 4VC = g m of BJT/4 α0 → DC value of α
4VT T

h R
• CMRR = R fe+he ; R e ↑ , → Zi ↑ , Ad ↑ & CMRR ↑
s ie

Darlington Pair :-

(1+hfe )2 Re2
• Zi = 1+h Ω [ if Q1 & Q 2 have same type ] = AI R e2
fe hoc Re2

R 2h
• R o = (1+hs 2 + 1+hie
fe ) fe

• g m = (1 + β2 ) g m1

Tuned Amplifiers : (Parallel Resonant ckts used ) :

1
• f0 = Q → ‘Q’ factor of resonant ckt which is very high
2π√LC

• B.W = f0 /Q

∆BW
• fL = f 0 -
2
∆BW
• fH = f0 +
2
• For double tuned amplifier 2 tank circuits with same f0 used . f0 = �fL fH .

MOSFET (Enhancement) [ Channel will be induced by applying voltage]

• NMOSFET formed in p-substrate

• If VGS ≥ Vt channel will be induced & iD (Drain → source )
• Vt → +ve for NMOS
• iD ∝ (VGS - Vt ) for small VDS

• VDS ↑ → channel width @ drain reduces .

VDS = VGS - Vt channel width ≈ 0 → pinch off further increase no effect

• For every VGS > Vt there will be VDS,sat

2 1 𝑊𝑊
• iD = K ′n [ (VGS - Vt ) VDS - VDS ] � � → triode region ( VDS < VGS - Vt )
2 𝐿𝐿

K ′n = µn Cox

1 2 𝑊𝑊
• iD = K ′n � � [ VDS ] → saturation
2 𝐿𝐿

1
• rDS = 𝑊𝑊 → Drain to source resistance in triode region
K′n � �(VGS − Vt )
𝐿𝐿

PMOS :-

VGS ≤ Vt → induced channel VDS ≥ VGS - Vt → Continuous channel

𝑊𝑊 2 1
iD = K ′p � � [(VGS − Vt )2 - VDS ] K ′p = µp Cox
𝐿𝐿 2

VDS ≤ VGS - Vt → Pinched off channel .

• NMOS Devices can be made smaller & thus operate faster . Require low power supply .
• Saturation region → Amplifier
• For switching operation Cutoff & triode regions are used

• NMOS PMOS

VGS ≥ Vt VGS ≤ Vt → induced channel

VGS - VDS > Vt VGS - VDS < Vt → Continuous channel(Triode region)

VDS ≥ VGS - Vt VDS ≤ VGS - Vt → Pinchoff (Saturation)

Depletion Type MOSFET :- [ channel is physically implanted . i0 flows with VGS = 0 ]

• For n-channel VGS → +ve → enhances channel .

→ -ve → depletes channel

• iD - VDS characteristics are same except that Vt is –ve for n-channel

• Value of Drain current obtained in saturation when VGS = 0 ⇒ IDSS .

1 𝑊𝑊
∴ IDSS = K ′n � � Vt2 .
2 𝐿𝐿

MOSFET as Amplifier :-

• For saturation VD > VGS - Vt

• To reduce non linear distortion 𝑣𝑣gs < < 2(VGS - Vt )
𝑊𝑊 𝑊𝑊
• id = K ′n � � (VGS − Vt ) 𝑣𝑣gs ⇒ g m = K ′n � � (VGS − Vt )
𝐿𝐿 𝐿𝐿

𝑣𝑣d
• 𝑣𝑣gs
= - gm RD

gm
• Unity gain frequency fT = 2π(C
gs +Cgd )

JFET :-
• VGS ≤ Vp ⇒ iD = 0 → Cut off

• Vp ≤ VGS ≤ 0 , VDS ≥ VGS - Vp

2
𝑉𝑉 I
iD = IDSS �1− 𝐺𝐺𝐺𝐺 � ⇒ VGS = Vp �1−� 𝐷𝐷 �
𝑉𝑉𝑝𝑝 I DSS
2IDSS 𝑉𝑉 2I I
� → Saturation
gm = �1− 𝐺𝐺𝐺𝐺 � = DSS � 𝐷𝐷
|Vp | 𝑉𝑉 𝑝𝑝 |V | I
p DSS

Zener Regulators :-

Vi − Vz
• For satisfactory operation Rs
≥ IZmin + ILmax
Vsmin − Vz0 − IZmin rz
• R Smax = IZmin + ILmax

• Load regulation = - (rz || R s )

rz
• Line Regulation = Rs +rz
.

• For finding min R L take Vs min & Vzk , Izk (knee values (min)) calculate according to that .

Operational Amplifier:- (VCVS)

• Fabricated with VLSI by using epitaxial method

• High i/p impedance , Low o/p impedance , High gain , Bandwidth , slew rate .
• FET is having high i/p impedance compared to op-amp .
• Gain Bandwidth product is constant .
A
• Closed loop voltage gain ACL = 1± βOLA β → feed back factor
OL

−1
• ⇒ V0 = ∫ Vi dt → LPF acts as integrator ;
RC

−R −L dvi
• ⇒ V0 = ∫ 𝑉𝑉i dt ; V0 = (HPF)
L R dt

−1 dvi
• For Op-amp integrator V0 = ∫ 𝑉𝑉i dt ; Differentiator V0 = - τ
τ dt

∆V0 ∆V0 ∆Vi ∆Vi

• Slew rate SR = ∆t
= ∆t
. ∆t
= A. ∆t

slew rate slew rate

• Max operating frequency fmax = = .
2π . ∆V0 2π × ∆Vi ×A

• In voltage follower Voltage series feedback

• In non inverting mode voltage series feedback

• In inverting mode voltage shunt feed back

𝑉𝑉
• V0 = -η VT ln �𝑅𝑅I𝑖𝑖 �
0

• V0 = - VBE

𝑉𝑉
= - η VT ln �𝑅𝑅I 𝑠𝑠 �
𝐶𝐶0

1 𝑉𝑉
• Error in differential % error = � 𝑐𝑐 �× 100 %
CMRR 𝑉𝑉𝑑𝑑

Power Amplifiers :-
B 2 B21
• Fundamental power delivered to load P1 = � 21 � R L = RL
√ 2
𝐵𝐵12 𝐵𝐵22
• Total Harmonic power delivered to load PT = � 2 + 2
+ ⋯ . . � 𝑅𝑅𝐿𝐿

𝐵𝐵 2 𝐵𝐵 2
= P1 �1 + �𝐵𝐵2 � + �𝐵𝐵3 � + … … �
1 1
= [ 1+ D2 ] P1

B
Where D = �+D22 + ⋯ . . +D2n Dn = Bn
1
D = total harmonic Distortion .

Class A operation :-
• o/p IC flows for entire 3600
• ‘Q’ point located @ centre of DC load line i.e., Vce = Vcc / 2 ; η = 25 %
• Min Distortion , min noise interference , eliminates thermal run way
• Lowest power conversion efficiency & introduce power drain
• PT = IC VCE - ic Vce if ic = 0, it will consume more power
• PT is dissipated in single transistors only (single ended)

Class B:-

• IC flows for 1800 ; ‘Q’ located @ cutoff ; η = 78.5% ; eliminates power drain
• Higher Distortion , more noise interference , introduce cross over distortion
• Double ended . i.e ., 2 transistors . IC = 0 [ transistors are connected in that way ] PT = ic Vce
• PT = ic Vce = 0.4 P0 PT → power dissipated by 2 transistors .

Class AB operation :-

• IC flows for more than 1800 & less than 3600

• ‘Q’ located in active region but near to cutoff ; η = 60%
• Distortion & Noise interference less compared to class ‘B’ but more in compared to class ‘A’
• Eliminates cross over Distortion

Class ‘C’ operation :-

• IC flows for < 180 ; ‘Q’ located just below cutoff ; η = 87.5%
• Very rich in Distortion ; noise interference is high .

Oscillators :-
1 29
• For RC-phase shift oscillator f = hfe ≥ 4k + 23 + where k = R c /R
2πRC √6+4K k

1
f= μ > 29
2πRC√6

1
• For op-amp RC oscillator f = | Af | ≥ 29 ⇒ R f ≥ 29 R1
2πRC√6

Wein Bridge Oscillator :-

1
f= hfe ≥ 3
2π√R′ R′′ C′ C′′
μ≥3
A ≥ 3 ⇒ R f ≥ 2 R1

Hartley Oscillator :-
1 L
f= |hfe | ≥ L2
2π�(L1 +L2 )C 1
L
| μ | ≥ L2
1
L2
|A| ≥
L1
↓
Rf
R1

Colpits Oscillator :-
1 C
f= C C
|hfe | ≥ C1
2π�L 1 2 2
C1 +C2
C
| μ | ≥ C1
2

C
| A | ≥ C1
2

Digital Electronics
IOH IOL
• Fan out of a logic gate = IIH
or IIL
• Noise margin : VOH - VIH or VOL - VIL
I +I
• Power Dissipation PD = Vcc Icc = Vcc � 𝐶𝐶𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶𝐶𝐶 � I𝐶𝐶𝐶𝐶𝐶𝐶 → Ic when o/p low
2
I𝐶𝐶𝐶𝐶𝐶𝐶 → Ic when o/p high .
• TTL , ECL & CMOS are used for MSI or SSI
• Logic swing : VOH - VOL
• RTL , DTL , TTL → saturated logic ECL → Un saturated logic
• Advantages of Active pullup ; increased speed of operation , less power consumption .
• For TTL floating i/p considered as logic “1” & for ECL it is logic “0” .
• “MOS” mainly used for LSI & VLSI . fan out is too high
• ECL is fastest gate & consumes more power .
• CMOS is slowest gate & less power consumption
• NMOS is faster than CMOS .
• Gates with open collector o/p can be used for wired AND operation (TTL)
• Gates with open emitter o/p can be used for wired OR operation (ECL)
• ROM is nothing but combination of encoder & decoder . This is non volatile memory .
• SRAM : stores binary information interms of voltage uses FF.
• DRAM : infor stored in terms of charge on capacitor . Used Transistors & Capacitors .
• SRAM consumes more power & faster than DRAM .
• CCD , RAM are volatile memories .
• 1024 × 8 memory can be obtained by using 1024 × 2 memories
• No. of memory ICs of capacity 1k × 4 required to construct memory of capacity 8k × 8 are “16”

DAC ADC
1
• FSV = VR �1 − � * LSB = Voltage range / 2n
2𝑛𝑛
step size VR /2n 1 FSV
• Resolution = = 1 = × 100% * Resolution =
FSV VR �1− n � 2n −1 2n −1
2
1 1 V
• Accuracy = ± LSB = ± n+1 * Quantisation error = 2nR %
2 2
• Analog o/p = K. digital o/p

PROM , PLA & PAL :-

AND OR

Fixed Programmable PROM

Programmable fixed PAL

Programmable Programmable PLA

Fastest ADC :-

• Successive approximation ADC : n clk pulses

• Counter type ADC : 2n - 1 clk pulses
• Dual slope integrating type : 2n+1 clock pulses .

Flip Flops :-

• a(n+1) = S + R′ Q
=D
= JQ′ + K ′ Q
= TQ′ + T ′ Q

Excitation tables :-

S R J K D T
0 0 0 x 0 0 0 x 0 0 0 0 0 0
0 1 1 0 0 1 1 x 0 1 1 0 1 1
1 0 0 1 1 0 x 1 1 0 0 1 0 1
1 1 x 0 1 1 x 0 1 1 1 1 1 0

• For ring counter total no.of states = n

• For twisted Ring counter = “2n” (Johnson counter / switch tail Ring counter ) .
• To eliminate race around condition t pd clock < < t pd FF .
• In Master slave master is level triggered & slave is edge triggered

Combinational Circuits :-

Multiplexer :-

Decoder :-

• n i/p & 2n o/p’s

• used to implement the Boolean function . It will generate required min terms @ o/p & those terms
should be “OR” ed to get the result .
• Suppose it consists of more min terms then connect the max terms to NOR gate then it will give the
same o/p with less no. of gates .
• If you want to Design m × 2m Decoder using n × 2n Decoder . Then no. of n × 2n Decoder
2m
required = 2n .
• In Parallel (“n” bit ) total time delay = 2n t pd .
• For carry look ahead adder delay = 2 t pd .

Microprocessors
1
• Clock frequency = crystal frequency
2
• Hardware interrupts
TRAP (RST 4.5) 0024H both edge level
RST 7.5 → Edge triggered 003CH
RST 6.5 0034 H
RST 5.5 level triggered 002C
INTR Non vectored

• Software interrupts RST 0 0000H

RST 1 0008H
2 0010H Vectored
: 0018H
:
7 0038H
•
S1 S0

0 0 Halt
0 1 write
1 0 Read
1 1 fetch

• HOLD & HLDA used for Direct Memory Access . Which has highest priority over all interrupts .

Flag Registers :-

S Z X AC X P X CY

• Sign flag :- After arthematic operation MSB is resolved for sign flag . S = 1 → -ve result
• If Z = 1 ⇒ Result = 0
• AC : Carry from one stage to other stage is there then AC = 1
• P : P =1 ⇒ even no. of one’s in result .
• CY : if arthematic operation Results in carry then CY = 1
• For INX & DCX no flags effected
• In memory mapped I/O ; I/O Devices are treated as memory locations . You can connect max of
65536 devices in this technique .
• In I/O mapped I/O , I/O devices are identified by separate 8-bit address . same address can be used
to identify i/p & o/p device .
• Max of 256 i/p & 256 o/p devices can be connected .

Programmable Interfacing Devices :-

• 8155 → programmable peripheral Interface with 256 bytes RAM & 16-bit counter
• 8255 → Programmable Interface adaptor
• 8253 → Programmable Interval timer
• 8251 → programmable Communication interfacing Device (USART)
• 8257 → Programmable DMA controller (4 channel)
• 8259 → Programmable Interrupt controller
• 8272 → Programmable floppy Disk controller
• CRT controller
• Key board & Display interfacing Device

RLC :- Each bit shifted to adjacent left position . D7 becomes D0 .

CY flag modified according to D7

RAL :- Each bit shifted to adjacent left position . D7 becomes CY & CY becomes D0 .

ROC :-CY flag modified according D0

RAR :- D0 becomes CY & CY becomes D7

CALL & RET Vs PUSH & POP :-

CALL & RET PUSH & POP

• When CALL executes , μp automatically stores * Programmer use PUSH to save the contents
16 bit address of instruction next to CALL on the rp on stack
Stack
• CALL executed , SP decremented by 2 * PUSH executes “SP” decremented by “2” .
• RET transfers contents of top 2 of SP to PC * same here but to specific “rp” .
• RET executes “SP” incremented by 2 * same here

Some Instruction Set information :-

CALL Instruction

CALL → 18T states SRRWW

CC → Call on carry 9 – 18 states

CM → Call on minus 9-18

CNC → Call on no carry

CP → Call on +ve

CPE → Call on even parity

CPO → Call on odd parity

RET : - 10 T

RC : - 6/ 12 ‘T’ states

Jump Instructions :-

JMP → 10 T

JC → Jump on Carry 7/10 T states

JNC → Jump on no carry

JZ → Jump on zero

JNZ → Jump on non zero

JP → Jump on Positive

JM → Jump on Minus

JPE → Jump on even parity

JPO → Jump on odd parity .

• PCHL : Move HL to PC 6T
• PUSH : 12 T ; POP : 10 T
• SHLD : address : store HL directly to address 16 T
• SPHL : Move HL to SP 6T
• STAX : R p store A in memory 7T
• STC : set carry 4T
• XCHG : exchange DE with HL “4T”

XTHL :- Exchange stack with HL 16 T

• For “AND “ operation “AY” flag will be set & “CY” Reset
• For “CMP” if A < Reg/mem : CY → 1 & Z → 0 (Nothing but A-B)
A > Reg/mem : CY → 0 & Z → 0

A = Reg/mem : Z → 1 & CY → 0 .

1.Turn on time of scr = t d + t r +t s

Where t d = delay time

t r = rise time
t S = settling time

2.Device turn off time, t q = t rr + t gr

Where t rr = Reverse recovery time and

t gr = gate recovery time

3.SERIES OPERATION:

→ SCR,s are connected in series to increase the voltage rating.

𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 V1 + V2

String efficiency ηs = =
𝑁𝑁𝑁𝑁.𝑜𝑜𝑜𝑜 𝑆𝑆𝑆𝑆𝑆𝑆,𝑠𝑠 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑿𝑿 voltage rating of each SCR 2 × V1

Derating factor =1 – string efficiency

C C

T1 T2
𝑏𝑏𝑏𝑏−𝑉𝑉𝑠𝑠𝑛𝑛𝑉𝑉
→ Static equalizing Resistance Rs = (𝑛𝑛−1)∆
𝐼𝐼𝐼𝐼

(n−1)∆ Q
→ Dynamic equalizing capacitance C =
n Vbm−Vs

4. Parallel operation is applied for SCR, s with higher current ratings

𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 I1 + I2

String efficiency ηS = =
𝑁𝑁𝑁𝑁.𝑜𝑜𝑜𝑜 𝑆𝑆𝑆𝑆𝑆𝑆,𝑠𝑠 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑿𝑿 current rating of each SCR 2 I1

1. Single Phase Half wave rectifier R load:

 Circuit turn off time = t c = π/ω

 Average output voltage V0 = Vm /2π[1+ cos α]

 r. m. s output voltage Vor = Vm / (2√π ) [π − α + (Sin 2α) /2]1/2

 R .m .s output current, Ior / R

Power supplied to load

 Power factor of input supply =
Source volt ampere

V2or / R Vor
pf= p.f =
Vs Is Vs

2. Single Phase Half wave rectifier R-L load:

 Circuit turn off time = t c = (2π- β)/ ω

 Average output voltage V0 = (Vm /2π) [cos α – cos β]

1
 r.m.s output voltage Vor = Vm /(2√π ) � β − α + ½ �sin 2α – sin 2β��2

 Average output current, I0 = V0 /R

 R.m.s value of output current, Ior = Vor / R

Vor
 Power factor of supply, pf = Vs

3. Single Phase Half wave rectifier R.L load and free wheeling Diode:

 Circuit turn off time = t c =

π
ω

m V
 Average output voltageV0 = � 2π � [ 1 + 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶]

1
1
 r.m.s output voltageVor = Vm / (2 √π ) �π − α + (sin 2α)�2
2

 Average output current, , I0 = V0 /R

 R.m.s value of output current, , Ior = Vor / R

V
= or
 Power factor of supply, Pf Vs

4. Single Phase Half wave rectifier RLE load:

2π +θ1 −β
 Circuit turn off time, t c = , θ1 = Sin−1 (𝐸𝐸/𝑉𝑉𝑚𝑚 ) and θ2 = 1800 - θ1
ω

 Average Output voltage, V0 =E+I0 R

1
 Average Output currentI0 = � � [𝑉𝑉𝑚𝑚 (𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 − 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶) − 𝐸𝐸(𝛽𝛽 − 𝛼𝛼)]
2𝜋𝜋𝜋𝜋

I2 or R + I0 E
 Supply power factor, pf = Vs Ior

 R.m.s value of output current

1
1 V2S 2
Ior = �2πR2 ��VS2 2
+ E � (β − α) − 2
(sin 2β − 2 sin 2α) − 2Vm E(cosα − cosβ)�

5. Single Phase Full wave rectifier – Mid point converter type:

π−α
 Circuit turn off time t c =
ω

2Vm cos α
 Average output voltage , V0 =
π

6.Single Phase Full wave Bridge type Rectifier:

π−α
 Circuit turn off time , t c =
ω

2Vm cos α
 Average output voltage , V0 =
π

 r.m.s output voltageVor = Vs

 The inductance of source results in an lesser value of voltage

2V ωL
∴ V0 = πm cos α - π s I0

7. Single Phase semi converter:

π−α
 Circuit turn off time t c =
ω

Vm
 Average output voltage V0 = π
[1+cos α]

8.Three Phase controlled half wave rectifier with R load:

3 3√3
 Average output voltage V0 = Vml cos α= Vmp cos α
2π 2π

 Average Output current I0 = V0 /R

1
VML 1 √3 2
 r.m.s output voltageVor = � + (cos 2α)�
R 6 8π

 R.m.s value of output current, Ior = Vor / R

9. Three Phase full converter:

3
 voltage V0 Average output = Vml cos α
π
1
3 π √3 2
 r.m.s output voltageVor = VML � � + 2 (cos 2α)�
2π 3
𝟑𝟑
 𝐢𝐢𝐬𝐬= 𝐢𝐢𝐨𝐨 �𝟐𝟐

10. Three Phase full converter:

3
 voltage V0 Average output = Vml (1+cos α)
2π

1
VML 3 2π √3 2
 r.m.s output voltageVor = � � + (1 + cos 2α)�
2 π 3 2

11. For a 3-∅ converter , The inductance of source results in an lesser value of voltage

3√6 3ωLs
V0 = π
Vph cos α - π
I0

a.STEP UP CHOPPER:

T
 Duty cycle, = � on �(T � = (Ton / T)
on + Toff )

VS
 Average output voltage across load ,V0 = 1−α

(Imin +Imax )
 Energy supplied by inductor = Wout = (𝑉𝑉0 − 𝑉𝑉𝑠𝑠 ) 2
x ToFF

b.STEP DOWN CHOPPER :

T
 Duty cycle,=� on �(T � = (Ton / T)
on + Toff )

 Average output voltage across load ,V0 = VS (Ton / T) = f Ton VS = α Vs

Where V =supply voltage

VS
 Average output current through load,𝐼𝐼0 =𝛂𝛂 R

 Rms value of output voltage = √α Vs

√α Vs
 Rms value of thyristor current = R

R
 Effective input resistence of chopper=
α

The minimum and maximum values of load current is given by

Ton
−
Vs /R [1−e Ta ]
 Imax = T - (E/R)
−
[1−e Ta ]

Ton
−
Vs /R [e Ta −1]
 Imn = T - (E/R)
−
[e Ta −1]

Ton Toff
− −
�1−e Ta ��1−e Ta �
Vs
 Ripple ∆I = R
� T �
−
[1−e Ta ]

 Maximum value of ripple current is given by (∆I)max = Vs / 4fL

A.C.ripple voltage 1
 Ripple factor = = �� − 1
D.C voltage α

Voltage Commutated Chopper:

 Minimum turn on time of chopper → π√LC seconds

 Minimum duty cycle of voltage commutated chopper αmn = πf √LC

Cvs −(−vs )
 The output current , I0 =
2tc

I0 tc V 2
 where C = , L ≥ � S� C
vs I0

t1 = π√LC

Current Commutated Chopper:

 The values of L & C are given by

vs tc
 L= 1
xI0 �π−2 Sin−1 � � �
x
x I0 tc
 C= 1
vs � π−2 sin−1 � � �
x
 Where x = ICp / I0
 t c = [ π-2 sin−1 (1/x) ] √LC
 t c = ICp sin ωt
1
 At ωt = θ1 , ic = I0 = ICp sin θ1 , θ1 = sin−1 [I0 /ICp ] = sin−1 � �
x
 Peak capacitor voltage = vs +I0 �L/C

INVERTERS:

Fourier Analysis Of Single Phase Inverter Output Voltage :

1-phase Half bridge Inverter:

2Vs
 V0 = ∑∞
n=1,3 sin nωt
nπ

2Vs
 i0 = ∑∞
n=1,3 Sin(nωt − ∅n )
nπZn

1-Phase Full Bridge Inverter:

4Vs
V0 = ∑∞
n=1,3 nπ
sin nωt

4Vs
i0 = ∑∞
n=1,3 sin (nωt - ∅n )
nπZn

Zn is the impedance offered to nth harmonic

1 2
 Zn = �R2 + �nωL − �
nωc

 Vor = Vs - full bridge inverter

 Vor = Vs /2 – Half bridge inverte

 Harmonic factor of nth harmonic , H.Fn = Vn / V1

Where Vn = r.m.s value of nth harmonic component

V1 = fundamental component r.m.s value.

→ Total harmonic distortion (T.H.D) → it is a measure of closeness in shape between

a wave form and its fundamental component.
1/2
THD = 1/V1 �∑∞ 2
𝑛𝑛=2,3 𝑉𝑉𝑛𝑛 �

�V2or −V21
=
V1
Vn
 Distortion factor of nth harmonic is defined as V 2
1 .n

Three Phase Bridge Inverter :

4Vs nπ
 Vab = ∑∞
n=1,3 Cos Sin n(ωt + π/6)
nπ 6

4Vs nπ
 Vbc = ∑∞
n=1,3 Cos Sin n(ωt - π/2)
nπ 6

4Vs nπ
 Vca = ∑∞
n=1,3 Cos Sin n(ωt + 5π/6)
nπ 6

All triple n harmonics are absent from the line voltages.

1 2π/3
 Vl = � ∫0
π
Vs2 d(ωt) = Vs �2/3 = 0.8165 Vs

0.8165 Vs
 Vph = = 0.4714 Vs
√3

2Vs nπ π
 Va0 = ∑∞
n=1,3 nπ
Cos Sin n(ωt+ )
6 6

2Vs nπ π
 Vb0 = ∑∞
n=1,3 nπ
Cos Sin n(ωt - )
6 2

2Vs nπ 5π
 Vc0 = ∑∞
n=1,3 nπ
Cos Sin n(ωt+ )
6 6

1 2π/3 Vs 2
�Vph = ∫0 �2� d(ωt) = Vs �1/6
π

 Vph = 0.4082 Vs , Vl = 0.4082 Vs × √3 = 0.7071 Vs

Voltage Control In 1-∅ Inverter:

1.Single Pulse Modulation:

 Vor = Vs �2d/π

4V
 V0 = ∑∞ s
n=1,3( nπ Sin nπ/2. sin nd) sin(nωt)
↓
Maximum value of nth harmonic
 To eliminate nth harmonic , nd = π
 i.e., width of the pulse = 2d = 2π/n

2.Multiple Pulse Modulation:

 Vor = √2 d/π
8Vs nd
 V0 = ∑∞n=1,3 nπ sin nγ sin sin (nωt)
2

Length of half cycle of reference wave form

 Number of pulses per half cycle =
Width of one cycle of triangle carrier wave
1/2fr fc ωc
 N= = =
1/fc 2fr 2ωr
 N= ωc /2ωr per half cycle
Where fc is carrier wave frequency and fr is reference wave frequency

AC VOLTAGE CONTROLLERS:

Single Phase Voltage controller with R Load:

 Circuit turn off time = π/ω sec.

 VC = ∑ω ω
n=1,3,5 An sin nωt + ∑n=1,3,5 Bn cos nωt d(ωt)

Vm sin(𝑛𝑛+1)𝛼𝛼 sin(𝑛𝑛−1)𝛼𝛼
 Where An = π
� (𝑛𝑛+1) − (𝑛𝑛−1) �
Vm cos(𝑛𝑛+1)𝛼𝛼−1 sin(𝑛𝑛+1)𝛼𝛼−1
 Bn = π
� (𝑛𝑛+1) − �
(𝑛𝑛−1)

 Vnm = �A2n + Bn2

B
 ∅n = tan−1 An
n

For n=1, Vor of output voltage is given by

Vm 1 1/2
 Vor = �(𝜋𝜋 − 𝛼𝛼) + 2 sin 2𝛼𝛼�
√2π

Vor
 Ior = R

2 V2or V2
m
 P = Ior R= R
= 2πR (π-α) + ½ sin 2α

V2 1
= πRs �(𝜋𝜋 − 𝛼𝛼) + sin 2𝛼𝛼�
2

Real power V2 /R V
 Power factor = = Vor /R = Vor
Apparent power Vs or s

1/2
1 1
= � �(𝜋𝜋 − 𝛼𝛼) + sin 2𝛼𝛼��
𝜋𝜋 2

PHASE VOLTAGE CONTROLLER WITH RL LOAD →

Vm Vm
 i0 = 2
sin (ωt - ∅) - 2
sin (α - ∅). exp[𝑅𝑅𝑅𝑅 / 𝜔𝜔𝜔𝜔]. e−RT/L

Vm Vm −𝑅𝑅
 i0 = 2
sin (ωt - ∅) - 2
sin (α - ∅). exp � (𝜔𝜔𝜔𝜔 − 𝛼𝛼)�
𝜔𝜔𝜔𝜔

DC & AC DRIVES

DC Motor equations :

Z∅N 𝐏𝐏
 Ea = N → rpm
60 𝐀𝐀

P
= Z∅n n → rps
A

 If ωm = 2πn
ω 𝐏𝐏
 Ea = Z- ∅ 2πm
𝐀𝐀
𝒁𝒁 𝑷𝑷
=� �∅ 𝝎𝝎𝒎𝒎
𝟐𝟐𝟐𝟐 𝑨𝑨

 Ea = k a ∅ωm k a = motor constant = ((z/2π) (P/A)) volts/webers rad/sec

1 P
 Torque T = Z∅ Ia
2π A
Z P
=� � ∅ Ia
2π A

For a dc separately excited motor:

 Flux , ∅ is constant
∴ Ea = k m ωm k m → motor constant volts/rad/sec
 Torque = k a ∅Ia
 Te = k m Ia k m → Newton meter/Ampere
For a dc series motor:

 ∅ ∝ Ia , ∅ = c I a
 Ea = k a c Ia ωm
volt sec
 Ea = k1 Ia ωm k1 → motor constant
rad.amp

 Te = k a ∅Ia
= k a c Ia2
 Te = k1 Ia2 k1 → Nm /Amp2

1-∅ half wave converter drive:

 Ia is assumed to be constant for speed control of the drive.

Vm
 Vt = V0 = 2π (1+cos α1 )

Vm
 Vf = π
(1+cos α2 )
1 π 2
 Is rms = � ∫ I d(ωt)
2π α a
1
= Ia � (π − α)
2π
 Is rms = Ia [(π − α)/2π]1/2
 IF.D.R = Ia [(π + α)/2π]1/2

power delivered to load

 Supply power factor =
Source VA

Ea Ia +I2a ra
= Vs Isr

Ia [Ea +Ia ra ]
=
Vs Isr

 Input supply pf = (Ia Vt )/(Vs Isr )

1-∅ half controlled converter drive →

Vm
 v0 = vt = π
[1+ cos α1 ]

Vm
 Vf = π
[1+cos α2 ]
 Isr = Ia �(π − α)/π

 Ifdr = Ia �2α/2π = Ia �α/π

 ITr = Ia �(π − α)/2π = I0r
VI
 Input supply pf = V tI a
s sr

Single Phase full wave Converter Drive:

2Vm
 V0 = Vt = π
cos α1
2V
 Vf = πm cos α2
 Isr = Ia
 ITr = Ia �π/2π = Ia /√2
 Pf = Vt Ia / Vs Isr

3-∅ half wave converter drive:

3√6
 V0 = Vt = V cos α1
2π ph
 Isr = Ia �120/360 = Ia �1/3
 ITr = Ia �120/360 = Ia �1/3

3-∅ full wave converter drive:

3√6
 V0 = Vt = Vph cos α
π

 Isr = Ia �240/360 = Ia �2/3

 ITr = Ia �120/360 = Ia �1/3

3-∅ semi converter drive:

3√6
 V0 = Vt = V (1 + cos α)
2π ph

 For α1 < 600 , Isr = Ia �2/3 , ITr = Ia �1/3

 For α1 > 600 , Isr = Ia �180 − α1 /180 , ITr = Ia �(180 − α1 )/360

= R × (T-Ton )/T

 R eff = R(1-α) α → duty cycle of the chopper

Introduction
6th Edition 2011 351

Formula
Handbook

including
Engineering
Formulae,
Mathematics,
Statistics
and
Computer Algebra

http://ubuntuone.com/p/ZOF/ - pdf
Name__________________________ http://ubuntuone.com/p/dAn - print
http://ubuntuone.com/p/ZOE/ - OOo (edit)
Course__________________________
Introduction

This handbook was designed to provide engineering students at Aberdeen College

with the formulae required for their courses up to Higher National level (2nd year
university equivalent).

In order to use the interactive graphs you will need to have access to Geogebra
(see 25). If you are using a MS Windows operating system and you already have
Java Runtime Environment loaded then no changes will be required to the registry.
This should mean that no security issues should be encountered. For Mac and Linux
(and for MS Windows if you have problems)
see http://www.geogebra.org/cms/en/portable

It is typed in Open Office.org. Future developments will include more hyperlinks

within the handbook and to other maths sites, with all the illustrations in it
produced with Geogebra (see 25) or OOo.

Any contributions will be gratefully accepted and acknowledged in the handbook.

If you prefer, you can make changes or add to the handbook within the terms of
the Creative Commons licence . Please send me a copy of your work and be
prepared to have it incorporated or adapted for inclusion in my version.
My overriding concern is for the handbook to live on and be continuously improved.
I hope that you find the handbook useful and that you will enjoy using it and that
that you will feel inspired to contribute material and suggest hyperlinks that could
be added.

Many thanks to my colleagues at Aberdeen College for their contributions and help
in editing the handbook. Special thanks are due to Mark Perkins at Bedford College
who adopted the handbook for his students, helped to format the contents and
contributed to the contents. Without Mark's encouragement this project would
have never taken off.

If you find any errors or have suggestions for changes please contact the editor:
Peter K Nicol. (p.nicol@abcol.ac.uk) (peterknicol@gmail.com) Contents

6th Edition XI/MMXI

19/01/12
351
Contents
1 Recommended Books........................................................................................................... 3
1.1 Maths........................................................................................................................... 3
1.2 Mechanical and Electrical Engineering ............................................................................3
2 Useful Web Sites..................................................................................................................4
3 Evaluation............................................................................................................................ 6
3.1.1 Accuracy and Precision................................................................................................................... 6
3.1.2 Units.................................................................................................................................................... 6
3.1.3 Rounding............................................................................................................................................ 6
4 Electrical Formulae and Constants .........................................................................................7
4.1 Basic ........................................................................................................................... 7
4.2 Electrostatics................................................................................................................ 7
4.3 Electromagnetism .........................................................................................................7
4.4 AC Circuits ...................................................................................................................8
5 Mechanical Engineering.........................................................................................................9
5.1.1 Dynamics: Terms and Equations...................................................................................................9
5.1.2 Conversions...................................................................................................................................... 9
5.2 Equations of Motion.......................................................................................................9
5.3 Newton's Second Law..................................................................................................10
5.3.1 Centrifugal Force............................................................................................................................10
5.4 Work done and Power..................................................................................................10
5.5 Energy....................................................................................................................... 10
5.6 Momentum / Angular Impulse........................................................................................11
5.7 Specific force / torque values........................................................................................11
5.8 Stress and Strain......................................................................................................... 11
5.9 Fluid Mechanics.......................................................................................................... 12
5.10 Heat Transfer............................................................................................................ 12
5.11 Thermodynamics.......................................................................................................13
6 Maths for Computing........................................................................................................... 14
6.1.1 Notation for Set Theory and Boolean Laws ..............................................................................14
7 Combinational Logic............................................................................................................ 15
7.1.1 Basic Flowchart Shapes and Symbols........................................................................................15
8 Mathematical Notation – what the symbols mean....................................................................16
8.1.1 Notation for Indices and Logarithms............................................................................................17
8.1.2 Notation for Functions....................................................................................................................17
9 Laws of Mathematics........................................................................................................... 18
9.1 Algebra – sequence of operations.................................................................................19
10 Changing the subject of a Formula (Transposition)................................................................20
11 Simultaneous Equations with 2 variables.............................................................................21
12 Matrices ......................................................................................................................... 22
13 The Straight Line ............................................................................................................. 24
14 Quadratic Equations .......................................................................................................25
15 Areas and Volumes........................................................................................................... 26
16 The Circle......................................................................................................................... 27
16.1.1 Radian Measure........................................................................................................................... 27
17 Trigonometry.................................................................................................................... 28
17.1.1 Notation for Trigonometry...........................................................................................................28
17.2 Pythagoras’ Theorem.................................................................................................28
The Triangle...................................................................................................................... 29
17.2.1 Sine Rule....................................................................................................................................... 29
17.2.2 Cosine Rule................................................................................................................................... 29
17.2.3 Area formula.................................................................................................................................. 29
17.3 Trigonometric Graphs................................................................................................30
17.3.1 Degrees - Radians Conversion..................................................................................................31
17.4 Trigonometric Identities..............................................................................................32
17.5 Multiple / double angles..............................................................................................32
Contents p1 8 Notation 1 26 Computer Input
17.6 Sinusoidal Wave........................................................................................................33
18 Complex Numbers............................................................................................................. 34
19 Vectors............................................................................................................................ 35
20 Co-ordinate Conversion using Scientific Calculators..............................................................36
21 Indices and Logs............................................................................................................... 39
21.1.1 Rules of Indices: ..........................................................................................................................39
21.1.2 Definition of logarithms................................................................................................................39
21.1.3 Rules of logarithms:..................................................................................................................... 39
21.1.4 Infinite Series................................................................................................................................ 40
21.1.5 Hyperbolic Functions .................................................................................................................. 40
21.1.6 Graphs of Common Functions...................................................................................................41
22 Calculus .......................................................................................................................... 42
22.1.1 Notation for Calculus....................................................................................................................42
22.2 Differential Calculus - Derivatives................................................................................43
22.2.1 Maxima and Minima..................................................................................................................... 45
22.2.2 Differentiation Rules..................................................................................................................... 45
22.2.3 Formula for the Newton-Raphson Iterative Process...............................................................46
22.2.4 Partial Differentiation ..................................................................................................................46
22.2.5 Implicit Differentiation..................................................................................................................46
22.2.6 Parametric Differentiation............................................................................................................46
22.3 Integral Calculus - Integrals........................................................................................47
22.3.1 Integration by Substitution..........................................................................................................48
22.3.2 Integration by Parts......................................................................................................................48
22.3.3 Indefinite Integration....................................................................................................................49
22.3.4 Area under a Curve...................................................................................................................... 49
22.3.5 Mean Value................................................................................................................................... 49
22.3.6 Root Mean Square (RMS)..........................................................................................................49
22.3.7 Volume of Revolution ..................................................................................................................50
22.3.8 Centroid.......................................................................................................................................... 50
22.3.9 Partial Fractions............................................................................................................................ 50
22.3.10 Approximation of Definite Integrals..........................................................................................51
22.3.10.1 Simpson's Rule..................................................................................................................51
22.3.10.2 Trapezium Method.............................................................................................................51
22.4 Laplace Transforms ..................................................................................................52
22.5 Approximate numerical solution of differential equations................................................53
22.6 Fourier Series. .........................................................................................................54
22.6.1 Fourier Series - wxMaxima method...........................................................................................55
23 Statistics.......................................................................................................................... 56
23.1.1 Notation for Statistics................................................................................................................... 56
23.2 Statistical Formulae...................................................................................................57
23.2.1 Regression Line ........................................................................................................................... 58
23.2.2 Tables of the Normal Distribution .............................................................................................59
23.2.3 Critical Values of the t Distribution.............................................................................................60
23.2.4 Normal Distribution Curve...........................................................................................................61
23.2.5 Binomial Theorem........................................................................................................................ 61
23.2.6 Permutations and Combinations................................................................................................61
24 Financial Mathematics.......................................................................................................62
25 Recommended Computer Programs...................................................................................63
26 Computer Input ................................................................................................................ 64
26.1 wxMaxima Input........................................................................................................ 65
26.1.1 Newton Raphson..........................................................................................................................65
26.1.2 Differential Equations................................................................................................................... 65
26.1.3 Runge-Kutta.................................................................................................................................. 65
26.2 Mathcad Input .......................................................................................................... 66
27 Using a Spreadsheet to find the ‘best fit’ formula for a set of data. ..........................................67
28 Calibration Error................................................................................................................ 68
29 SI Units - Commonly used prefixes......................................................................................69
30 Electrical Tables................................................................................................................ 69
31 THE GREEK ALPHABET...................................................................................................70

Contents p1 8 Notation 2 26 Computer Input

1 Recommended Books
referred to by author name in this handbook
1.1 Maths

General pre-NC and NC : Countdown to Mathematics; Graham and Sargent

Vol. 1 ISBN 0-201-13730-5, Vol. 2 ISBN 0-201-13731-3

NC Foundation Maths, Croft and Davison

ISBN 0-131-97921-3

NC and HN and Degree: Engineering Mathematics through Applications;

K Singh Kuldeep Singh, ISBN 0-333-92224-7.
www.palgrave.com/science/engineering/singh

Engineering Mathematics, 6th Edition, J Bird

ISBN 1-8561-7767-X

HN and degree: Higher Engineering Mathematics, 4th Edition, J Bird,

J Bird ISBN 0-7506-6266-2

Degree Engineering Mathematics 6th Edition , K A Stroud

ISBN 978-1- 4039-4246-3
-----------------------------------------------------------------------------------------------

1.2 Mechanical and Electrical Engineering

NC Advanced Physics for You, K Johnson, S Hewett et al.

ISBN 0 7487 5296 X

Mechanical Engineering

NC and HN Mechanical Engineering Principles, C Ross, J Bird

ISBN 0750652284

Electrical Engineering

NC and HN Basic Electrical Engineering Science

Ian McKenzie Smith, ISBN 0-582-42429-1

Contents p1 8 Notation 3 26 Computer Input

2 Useful Web Sites
If you use any of the sites below please read the instructions first. When entering
mathematical expressions the syntax MUST be correct. See section 26 of this
book.
Most sites have examples as well as instructions. It is well worth trying the examples
first.
If you find anything really useful in the sites below or any other site please tell us so
that we can pass on the information to other students.

Efunda A US service providing a wealth of engineering

information on materials, processes, Maths,
unit conversion and more. Excellent calculators (like
quickmath). http://www.efunda.com

Freestudy Mechanical engineering notes and exercises and

Maths notes and exercises. http://www.freestudy.co.uk

matek.hu An online calculator which also does calculus and

produces graphs. (Based on Maxima). http://www.matek.hu

Mathcentre Try the Video Tutorials. http://www.mathcentre.ac.uk

MC The other stuff is excellent too.
Also see http://www.mathtutor.ac.uk

QuickMath Links you to a computer running MATHEMATICA

- the most powerful mathematical software.
http://www.quickmath.com

Mathway Try the problem solver for algebra, trig and calculus
and it draws graphs too. See 26 for input syntax.
http://www.mathway.com

Mathsnet Look under Curriculum for Algebra for some excellent

online exercises. http://www.mathsnet.net

BetterExplained It is true – maths and some other topics explained

BE better. http://BetterExplained.com/
how to learn maths how to learn maths

Just the Maths A complete text book – all in pdf format

http://nestor.coventry.ac.uk/jtm/contents.htm

WolframAlpha Almost any maths problem solved!

http://www.wolframalpha.com/

Khan Academy The "free classroom of the World"

Many video lectures using a blackboard
http://www.khanacademy.org

Contents p1 8 Notation 4 26 Computer Input

The Open University There are a lot of excellent courses to study and if you
want to improve your maths I suggest that you start here
http://mathschoices.open.ac.uk/
Read the text very carefully on all the pages and then go to
http://mathschoices.open.ac.uk/routes/p6/index.html and try the
quizzes.

Plus Magazine Plus magazine opens a door to the world of maths, with
all its beauty and applications, by providing articles from
the top mathematicians and science writers on topics as
diverse as art, medicine, cosmology and sport. You can
read the latest mathematical news on the site every week,
browse our blog, listen to our podcasts and keep
up-to-date by subscribing to Plus (on email, RSS,
Facebook, iTunes or Twitter).
http://plus.maths.org/content/

Paul's Online Math Notes Recommended by June Cardno,

Banff and Buchan College
http://tutorial.math.lamar.edu/

Waldomaths Some excellent interactive tools - Equations 1 and 2 in

particular for transposition practice.
http://www.waldomaths.com/

HND Engineer As Alasdair Clapperton says “The aim of this website to

assist, enlighten and inspire Scottish NC/HNC/HND
engineering students within the current Scottish
Government drive towards renewable energy targets”.
http://www.hndengineer.co.uk/

If you come across any Engineering or Mathematics sites that might be useful
to students on your course please tell me (Peter Nicol) - p.nicol@abcol.ac.uk

Contents p1 8 Notation 5 26 Computer Input

3 Evaluation
3.1.1 Accuracy and Precision

Example: Target = 1.234 - 4 possible student answers

Not Accurate, not Precise 1.270, 2.130, 0.835, 1.425

Accurate but not Precise 1.231, 1.235, 1.232, 1.236

Precise but not Accurate 1.276, 1.276, 1.276. 1,276

Precise and Accurate 1.234, 1.234, 1.234, 1.234

-------------------------------------------------------------------------------------------------------
3.1.2 Units
Treat units as algebra -
1 2 m
for example KE = m v where m=5 kg and v=12 .
2 s
2
1
KE = ×5×kg ×
2  12×m

2
s 
2
Standard workshop

1 12 ×m
KE = ×5×kg × tolerance ±0.2 mm
2 s
2

2
1 2 kg×m
KE = ×5×12 × 2
2 s
2
kg m
KE =360 2 KE =360 J
s
-------------------------------------------------------------------------------------------------------
3.1.3 Rounding
Do not round calculations until the last line.
Round to significant figures preferably in engineering form
2
d
Example: A= where d =40
4
3
A=1256.637061 A=1.256637061×10
3
A=1.257×10 rounded to 4 sig fig ( A=1257 )

There should be at least 2 more significant figures in the calculation than in

the answer.
Contents p1 8 Notation 6 26 Computer Input
4 Electrical Formulae and Constants
4.1 Basic
Unit symbol
Series Resistors R T =R1 R 2R 3 … . 

1 1 1 1
Parallel Resistors =   ….  8
R T R1 R2 R 3

Potential Difference V =I R V

2 V2
Power P= I V or P= I R or P= W
R

Energy (work done) W =P t J or kWh

1
Frequency f= Hz
T
-------------------------------------------------------
4.2 Electrostatics
1 1 1 1
Series Capacitors =   …. F
CT C1 C2 C3

Parallel Capacitors C T =C 1C 2C 3 …. F

Charge Q=I t or Q=C V C

A  A 0r
Capacitance C= = F
d d
−12
Absolute Permittivity  0≈8.854×10 F/m
-------------------------------------------------------
4.3 Electromagnetism

Magnetomotive Force F=I N At or A

IN
Magnetisation H= At/m or A/m
ℓ

l l
Reluctance S= = At/Wb or A/Wb
 A  o r A
−7
Absolute Permeability  0=4 ×10 H/m
--------------------------------------------------------
Contents p1 8 Notation 7 26 Computer Input
4.4 AC Circuits
Unit Symbol
Force on a conductor F =B I ℓ N

Electromotive Force E= B ℓ v V

Instantaneous emf e= E sin  V

d di
Induced emf e= N e= L V
dt dt

1
RMS Voltage V rms= ×V V rms≈0.707 V peak V
 2 peak
2 V AV ≈0.637 V peak V
Average Voltage V AV = ×V peak


Angular Velocity =2  f rad/s

17.7

V s Ns I p
Transformation Ratios = =
V p N p Is

Potential Difference V =I Z V

Power Factor pf =cos 

1
Capacitive Reactance X C= 
2 f C

Inductive Reactance X L=2 f L 

1
Admittance Y= S
Z

True Power P=V I cos  W

Reactive Power Q=V I sin  VAr

Apparent Power S=V I * = P j Q VA

Note: I * is the complex conjugate of the phasor current. See 17

_________________________________________________________
Thanks to Iain Smith, Aberdeen College

Contents p1 8 Notation 8 26 Computer Input

5 Mechanical Engineering
[K Singh pp 2 – 98 especially 32 – 40 and 69 - 73]

5.1.1 Dynamics: Terms and Equations

Linear Angular
s= displacement (m)  = angular displacement (rad)
u= initial velocity (m/s) 1= initial velocity (rad/s)
v= final velocity (m/s)  2= final velocity (rad/s)
a= acceleration 2
(m/s )  = acceleration (rad/s2)
t= time (s) t = time (s)
--------------------------------------------------------
5.1.2 Conversions

Displacement s=r 
s 
Velocity v=r  v= =
t t
Acceleration a=r 

o
o
2  radians = 1 revolution = 360 , i.e. 1 rad =
360
2   o
≈57.3 see 17.4.1
2 N
If N = rotational speed in revolutions per minute (rpm), then = rad/s
60
--------------------------------------------------------
5.2 Equations of Motion

Linear Angular

v=ua t  2= 1 t

1 1
s= uvt = 12 t
2 2

1 1
s=ut  a t 2 =1 t t 2
2 2

v 2=u 22 a s  22= 212 

v–u  2− 1
a= =
t t
-------------------------------------------------------------

Contents p1 8 Notation 9 26 Computer Input

5.3 Newton's Second Law

Linear Angular

∑ F =ma ∑T=I 
where T = F r , I =m k 2
and k = radius of gyration
---------------------------------------------------------
5.3.1 Centrifugal Force
m v2
CF=
r
CF=m  2 r
--------------------------------------------------------
5.4 Work done and Power

Linear Angular

Work Done WD= F s WD=T 

Work done
P=
Time taken
Power Fs P=T 
=
t
=F v
--------------------------------------------------------
5.5 Energy

Linear Angular

1 1
Kinetic Energy KE= m v 2 KE= I  2
2 2
1
KE= m k 2  2
2

Potential Energy PE=m g h

KE of a rolling wheel = KE (linear) + KE (angular)

--------------------------------------------------------

Contents p1 8 Notation 10 26 Computer Input

5.6 Momentum / Angular Impulse

Impulse = Change in momentum

Linear Angular

Ft=m2 v – m 1 u Tt= I 2  2− I 1  1

If the mass does not change: Ft=m v−mu

--------------------------------------------------------
5.7 Specific force / torque values

Force to move a load: F = m g cos m g sin m a

Force to hoist a load vertically =90 o  F =m gm a=m ga

Force to move a load

along a horizontal surface =0 o  F = m gm a

Winch drum torque T app =T F  F r  I 

--------------------------------------------------------
5.8 Stress and Strain

F
Stress   = load / area =
A

l x
Strain = change in length / original length = or =
l l


E= Stress / Strain E=


M  E
Bending of Beams = =
I y R

b d3
2nd Moment of Area (rectangle) I=  A h2
12

T  G
Torsion Equation = =
J r L

 D4  d 4
2nd Moment of Area (cylinder) J= −
32 32
--------------------------------------------------------
Thanks to Frank McClean and Scott Smith, Aberdeen College

Contents p1 8 Notation 11 26 Computer Input

5.9 Fluid Mechanics

Mass continuity ṁ= A V , or ṁ= A C

p C2
Bernoulli’s Equation   z = constant
g 2g

p 1 C 21 p 2 C 22
or   z 1=   z 2z F
g 2g g 2g

Volumetric flow rate Q=A v

  
m
2gh –1
f
Actual flow for a venturi-meter Qactual = A1 c d

 
A1
A2
–1
Efunda Calculator

  
m
2gh –1
f
Actual flow for an orifice plate Q= A0 c d 4
D
1– 0
D1 
ρV D VD
Reynold’s number Re= Re= Efunda calculator
v 
4 f l v2 4 f l v2
Darcy formula for head loss h= , h= energy loss
2gd 2d
Efunda Calculator
-----------------------------------------------------------------------------------------------------
5.10 Heat Transfer

k AT 1 – T 2
Through a slab Q̇=
x
T x1 x2 1 1
Through a composite Q̇= where  R=    …
R k 1 k 2 h1 h2
T
Through a cylindrical pipe Q̇=
R

where
 R=
1

R
ln 2
R1    

R
ln 3
R2

1
2 R 1 h 1 2 k 1 2 k 2 2 R 3 h 3
------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 12 26 Computer Input

5.11 Thermodynamics

Boyle’s Law p 1 V 1= p 2 V 2

V1 V2
Charles’s Law =
T1 T2

p1 V 1 p 2 V 2
Combined Gas Law =
T1 T2

Perfect Gas pV =m R T

Mass flow rate ṁ= A C

Polytropic Process pV n = constant

Isentropic Process cP
(reversible adiabatic) pV  = constant where =
cV

Gas constant R=c p −c v

Enthalpy (specific) h=u p v

 
2 2
C C
Steady flow energy equation Q̇=ṁ h 2 – h1 2 – 1  g  z 2 – z 1   Ẇ
2 2

Vapours v x= x v g

u x =u f  x u g −u f 

h x =h f  xh g – h f  or h x =h f  x h f g
___________________________________________________________________
Thanks to Richard Kaczkowski and Scott Smith, Aberdeen College.

Contents p1 8 Notation 13 26 Computer Input

6 Maths for Computing
an a to the base n

a 10 decimal; denary ( a d) a2 binary ( a b)

a 16 hexadecimal ( a h) a8 octal ( a o)

-----------------------------------------------------------------------------------------------

10 3 (1000) kilo 2 10 1024 kilobyte

10 6 Mega 2 20 1024 2  megabyte

but
10 9
Giga 2 30 1024 3 gigabyte

1012 Tera 2 40 1024 4  terabyte

5
1015 Peta 2 50 1024  petabyte
_____________________________________________________________
6.1.1 Notation for Set Theory and Boolean Laws
[J Bird pp 377 - 396]

E universal set E
A B .b .c
A={ a , b , c …} a set A with elements a , b , c etc .a

a∈ A a is a member of A B⊂ A

{ } the empty set ( Ø is also used) E

A B
B⊂ A B is a subset of A

A∪ B A B
Set theory Boolean
E
∪ union ∨  OR A B

∩ intersection ∧ ⋅ AND A∩ B A⋅B

E
A' complement of A A NOT A

A' A
Contents p1 8 Notation 14 26 Computer Input
7 Combinational Logic

A0= A A⋅0=0

A1=1 A⋅1= A

A⋅A= A A A= A

A A=0 A A=1

A= A

A⋅B=B⋅A A B= B A

A⋅ BC = A⋅B A⋅C 

A B⋅C = A B⋅ AC 

A⋅ B⋅C =C⋅ A⋅B A BC=C  AB

A⋅ AB= A A A⋅B= A

De Morgan's Laws

A⋅B⋅C⋅...= ABC... A BC...= A⋅B⋅C⋅...

------------------------------------------------------------------------------------------------------
7.1.1 Basic Flowchart Shapes and Symbols

Start / End Input / Output

Action or Process Connector

Decision Flow Line

______________________________________________________________

Contents p1 8 Notation 15 26 Computer Input

8 Mathematical Notation – what the symbols mean
∈ is a member of. ( x∈ℝ means x is a member of ℝ )

ℕ the set of natural numbers 1, 2, 3, ........

ℤ the set of all integers ....., -2, -1, 0, 1, 2, 3, ......

ℚ the set of rational numbers including ℤ and

p
fractions ; p , q∈ℤ
q

ℝ the set of all real numbers. Numbers represented by

drawing a continuous number line.

ℂ the set of complex numbers. Numbers represented by

drawing vectors.

 therefore

w.r.t. with respect to

∗ used as a multiplication sign ( × ) (in computer algebra)

^ used as “power of” ( x y ) in computer algebra

≠ not equal to

≈ approximately equal to

 greater than. x2 means x is greater than and not equal to 2

≥ greater than or equal to.

 less than. a2 means a is less than and not equal to 2.

≤ less than or equal to.

a≤ x≤b x is greater than or equal to a and less than or equal to b

ab abbreviation for a×b or a∗b or a⋅b

a×10n a number in scientific (or standard) form. ( 3×103=3000 )

use EXP or ×10x key on a calculator

n! “ n factorial” n×n – 1×n – 2×n−3×...×1

Contents p1 8 Notation 16 26 Computer Input
A∝B implies A=k B where k is a constant (direct variation)

∣x∣ the modulus of x . The magnitude of the number x ,

irrespective of the sign. ∣−3∣=3=∣3∣

∞ infinity

⇒ implies

--------------------------------------------------------
8.1.1 Notation for Indices and Logarithms

an abbreviation for a×a×a×a ...×a (n terms). see 21

x ▄ or ^ or x y or y x or a b on a calculator.
1
a the positive square root of the number a .  x= x = x 0.5
2

1
k a k th root of a number a . 3 8=2 k a=a k .

ex exp  x (2.71828.... to the power of x ). See 21.

log e x ln  x on a calculator. The logarithm of x to the base e

log 10 x log  x  on a calculator. The logarithm of x to the base 10

--------------------------------------------------------
8.1.2 Notation for Functions

f x a function of x . Also seen as g  x , h x , y  x

f −1  x the inverse of the function labelled f  x

g° f the composite function - first f then g . or g  f  x .

------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 17 26 Computer Input

9 Laws of Mathematics
Associative laws - for addition and multiplication

abc=abc a b c=a b c
--------------------------------------------------------
Commutative laws - for addition and multiplication

ab=ba but a – b≠b−a

a b
a b=b a but ≠
b a
--------------------------------------------------------
Distributive laws - for multiplication and division

bc b c
a bc=a ba c = 
a a a
--------------------------------------------------------
Arithmetical Identities

x 0= x x ×1=x  x×0=0

--------------------------------------------------------
Algebraic Identities K Singh pp 73 – 75

ab2=abab=a 22 a bb 2 a 2 – b 2= aba−b

ab3 =ab a 22 a bb 2 =a 33 a 2 b3 ab2b3 see 21.1.6
--------------------------------------------------------
Other useful facts

a a 1
a – b=a−b =a÷b= ×
b 1 b

a−−b=a−−b=ab
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ------

a c a d b c a c ac
 = × = see 22.3.8, 4
b d bd b d bd

a c a d
÷ = × MC
b d b c

abcd =acad bcbdFOIL

--------------------------------------------------------
MC

Contents p1 8 Notation 18 26 Computer Input

9.1 Algebra – sequence of operations
[K Singh pp 40 - 43]
Sequence of operations - the same sequence as used by scientific
calculators.

------------------------------------------------------------------
Brackets   come before

-------------------------------------------------------------------
Of x 2,  x , sin x , e x , comes before
“square of x , sine of x
-------------------------------------------------------------------
Multiplication × comes before

Division ÷ comes before

-------------------------------------------------------------------
Addition  comes before

Subtraction −

-------------------------------------------------------------------

3sin a x 2b−5 would be read in this order

left bracket

x squared

times a

plus b

right bracket

sine of the result ( sin a x 2b )

times 3

minus 5

Contents p1 8 Notation 19 26 Computer Input

10 Changing the subject of a Formula (Transposition)
[K Singh pp 53 - 66]

An equation or formula must always be BALANCED -

whatever mathematical operation you do to one side of an equals sign
must be done to other side as well. (to all the terms)

5a - 7 + 7 = 3b + 7

You can’t move a term (or number) from one side of the equals sign to
the other.
You must UNDO it by using the correct MATHEMATICAL operation.

UNDO × with ÷ AND ÷ with ×

UNDO  with − AND − with 
UNDO  with x 2 AND x2 with 
UNDO xn with n AND n with xn
UNDO sin x with sin−1 x AND sin−1 x with sin x
UNDO ex with ln x AND ln x with ex
UNDO 10 x with log 10 x AND log 10 x with 10 x
dy dy
UNDO
dx
with ∫ dx AND ∫ dx with dx

etc

Generally (but not always) start with the terms

FURTHEST AWAY from the new subject FIRST.
Think of the terms in the formula as layers of an onion
- take the layers off one by one.

a  x 2 b

Try http://www.mathsnet.net/algebra/equation.html for getting started.

Contents p1 8 Notation 20 26 Computer Input

11 Simultaneous Equations with 2 variables
[K Singh p 90-98]

General method:

Write down both equations and label (1) and (2).

a 1 xb 1 y=c1 1
a 2 xb 2 y=c 2  2
Multiply every term on both sides of (1) by a 2 and every term on
both sides of (2) by a 1 and re-label as (3) and (4).
a 2 a1 xa 2 b1 y =a 2 c 1 3
a 1 a 2 xa1 b 2 y=a 1 c 2 4
Multiply every term on both sides of (4) by -1 and re-label.
a 2 a1 xa 2 b 1 y=a 2 c 1 3
−a1 a 2 x−a 1 b 2 y=−a1 c 2 5
Add (3) to (5) to eliminate x
Calculate the value of y

Substitute the value of y into equation (1)

Calculate the value of x MC

Check by substituting the values of x and y into (2)

----------------------------------------------------
Graphical Solution

a 1 xb1 y=c1

y y2

a 2 xb 2 y=c2
y1

x x1 x2

If f  x = g  x then f  x – g  x=0 - also see 13 and 14

_________________________________________________________

Contents p1 8 Notation 21 26 Computer Input

12 Matrices
[K Singh pp 507 – 566]

Notation:

[ ]
1 0 0 ..
0 1 0 ..
Identity =
0 0 1 ..
. . . ..

A m×n matrix has m rows and n columns.

a ij an element in the i th row and j th column.

-----------------------------------------------------------------------------------------------

If A=
[ a 11 a 12
a 21 a 22 ]
and B=
b 11 b12
b 21 b 22[ ]
then A B=
[ a 11b11 a 12b12
a 21b 21 a 22b22 ]
and A× B=
[ a 11 b11a 12 b 21 a 11 b 12a 12 b 22
a 21 b11a 22 b 21 a 21 b 12a 22 b 22 ] Columns A=Rows B

-------------------------------------------------------------------------------------------------------
Solution of Equations 2 x 2

If A X = B then X = A−1 B

If A= [ ]
a b
c d

then the inverse matrix,

−1
A =
1
[
d −b
det A −c a ]
, a d −b c≠0

where det A= ∣ ∣a b
c d
=ad −bc
--------------------------------------------------------

Contents p1 8 Notation 22 26 Computer Input

Inverse Matrix, 3 x 3 or larger

[ ∣ ]
a 11 a12 a 13 1 0 0
Start with a 21 a22 a 23 0 1 0 carry out row operations to:
a 31 a32 a 33 0 0 1

[ ∣ ] [ ]
1 0 0 b 11 b12 b13 b 11 b12 b13
−1
0 1 0 b 21 b22 b23 where b 21 b22 b 23 = A
0 0 1 b 31 b32 b33 b 31 b32 b33

1
or for 3x3 A−1= ×transpose of the co-factors of A [place signs!!]
det A

∣ ∣
a 11 a 12 a13
where
a
det A= a 21 a 22 a 23 =a 11 22
a 31 a 32 a33
∣a 23
a32 a 33 ∣ ∣
a a
a 31 a 33
a
∣ ∣
a
−a 12 21 23 a 13 21 22
a 31 a 32 ∣
or use Sarrus' Rule as below _ _ _

∣ ∣[ ]
a 11 a 12 a 13 a 11 a 12 a 13 a 11 a 12
det A= a 21 a 22 a 23 = a 21 a 22 a 23 a 21 a 22
a 31 a 32 a 33 a 31 a 32 a 33 a 31 a 32

+ + +

detA=a 11 a 22 a 33 a 12 a 23 a 31 a 13 a 21 a 32
−a 31 a 22 a 13 −a 32 a 23 a 11 −a 33 a 21 a 12

Thanks to Richard Kaczkowski, Aberdeen College.

Contents p1 8 Notation 23 26 Computer Input

13 The Straight Line
[K Singh pp 100 – 108]
y

y2 ( x2 , y2 )
+ve gradient
dy
y1
( x1 , y1 ) -ve gradient
c
dx
x
x1 x2

The general equation of a straight line of gradient m cutting the y

axis at 0, c is

y =m xc

where the gradient

 y2 – y1  dy  y 2 – y 1
m= or = . See 22.1.1 and 17.3
 x 2− x 1 dx  x 2− x 1

or y 1=m x 1+c (1)

y 2=m x 2+c (2) then (1) – (2) and solve for m (then c )

Also:

A straight line, gradient m passing through a , b has the equation:

 y−b=m x−a

Also see 27, back to 22.2.3, 22.5, 23.2.1, 22.3.10 MC

------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 24 26 Computer Input

14 Quadratic Equations
[K Singh pp 88 - 90 & 109 - 113]
y
−b -ve a
x=
2a
y=a x 2b xc +ve a

x1 x2 x

Geogebra quadratic

a minimum turning point

The solutions (roots) x 1 and x 2 of the equation a x 2b xc=0 are the
value(s) of x where y=a x 2 b xc crosses the x axis.

The solutions (roots) x 1 and x 2 of a x 2b xc=0 are given by the

Quadratic Formula.

x=
−b  b – 4 a c
±
2
or x=
 −b± b2 – 4 a c 
2a  2a 2 a

Definition of a root: The value(s) of x which make y equal to zero.

….........................................................................................................................
Also:

a x 2 b xc=0
2 b c
x  x =0 2

 

a a b
2 where c a see 22.4

  
b d 2= −
a 2
a 2
x d =0
2

If y=k  x  A2 B the turning point is − A , B Geogebra

-------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 25 26 Computer Input

15 Areas and Volumes
Volume of a Prism
b
Rectangle A=l b

Area
length
l
V =A l
1
Triangle A= b h
h 2

r πd2
Circle A= =π r 2
4
d
C=π d =2 π r

π d2
Cylinder Total surface area = π d h2
4
side + 2 ends
h 2 π r h2 π r 2
d
 d2h
V= = r 2 h
4

l πdl
h Cone Curved surface area = =π r l
2
Total surface area =  r l r 2
d
 d2h  r2h
V= =
12 3

Sphere Total surface area = π d 2 =4 π r 2

d
π d 3 4 π r3
V= =
6 3

Contents p1 8 Notation 26 26 Computer Input

16 The Circle

A Minor Sector C Minor Segment

C
A

B D

B Major Sector D Major Segment

--------------------------------------------------------
y

(x,y)

r
The equation  x – a2 y – b2=r 2
b
represents a circle centre a , b
and radius r .

a x
------------------------------------------------------------------------------------------------------
16.1.1 Radian Measure
r
A radian: The angle  subtended (or r
made by) an arc the same
θ
length as the radius of a circle.
Notice that an arc is curved. r
BE.com degrees and radians

Contents p1 8 Notation 27 26 Computer Input

17 Trigonometry
[K Singh pp 168 - 176]
17.1.1 Notation for Trigonometry
A
Labelling of a triangle

c b

B C
a

sin  the value of the sine function of the angle 

cos  the value of the cosine function of the angle 

tan  the value of the tangent function of the angle 

=sin−1 b arcsin b the value of the basic angle  whose sine function
value is b . −90o≤o≤90 o or
−
2
≤≤

2 
=cos−1 b arccos b the value of the basic angle  whose cosine function
value is b .  0o ≤o ≤180 o  or  0≤≤ 

=tan−1 b arctan b the value of the basic angle  whose tangent function
value is b . −90o≤ o≤90o  or
−
2
≤≤

2 
------------------------------------------------------------------------------------------------------
17.2 Pythagoras’ Theorem

In a right angled triangle, with hypotenuse, length R ,

and the other two sides of lengths a and b ,
then
R 2=a 2b 2
R
or R=  a b2 2 b

use of Pythagoras' Theorem BE surprising uses

Pythagorean distance BE pythagorean distance a
Interactive proof http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Py-
thagoras/pythagoras.html
---------------------------------------------------------
Contents p1 8 Notation 28 26 Computer Input
The Triangle

In a right angled triangle, with hypotenuse, (which is the longest side),

of length H ,

SOHCAHTOA
H O
The other two sides have lengths
A (adjacent, or next to angle  )
and O (opposite to angle  ) θ
then
A
MC
O A O
sin = cos = tan = see also 20
H H A
and 13
------------------------------------------------------------------------------------------------------
[K Singh pp 187 - 192] A

c b
In any triangle ABC, where A is the
angle at A, B is the angle at B and C
is the angle at C the following hold:
B C
a
--------------------------------------------------------
17.2.1 Sine Rule
a b c
Sine Rule = =
sin A sin B sin C

sin A sin B sin C

or = =
a b c
http://www.ies.co.jp/math/java/trig/seigen/seigen.html
--------------------------------------------------------
17.2.2 Cosine Rule
b 2c 2 – a 2
cos A=
2b c

or a 2=b 2c 2 – 2 b ccos A

http://www.ies.co.jp/math/java/trig/yogen1/yogen1.html
--------------------------------------------------------
17.2.3 Area formula
b c sin A
Area =
2
-------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 29 26 Computer Input

17.3 Trigonometric Graphs
[K Singh pp 177 - 187]

y=sin x o

Calculator answer
Geogebra Sine wave slider http://www.ies.co.jp/math/java/trig/graphSinX/graphSinX.html

y=cos x o

Calculator answer
Geogebra Cosine wave slider http://www.ies.co.jp/math/java/trig/graphCosX/graphCosX.html

Contents p1 8 Notation 30 26 Computer Input

y=tan x o

Calculator answer
------------------------------------------------------------------------------------------------------
17.3.1 Degrees - Radians Conversion

0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300 315, 330, 360
    2 3 5 7 5  4 3 5  7 11
0  2
6 4 3 2 3 4 6 6 4 3 2 3 4 6

r
Degrees to radians o
x ÷180×= rad r

r
o
Radians to degrees  rad ÷×180=x

=1 radian
Geogebra Radians
BE degrees and radians see 5.1.2
------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 31 26 Computer Input

17.4 Trigonometric Identities
[K Singh pp 203 - 213]

sin A 1 cos A
tan A= cot A= = , (the cotangent of A )
cos A tan A sin A
--------------------------------------------------------
1 1
sec A= , (the secant of A ), cosec A= , (the cosecant of A )
cos A sin A
--------------------------------------------------------
2 2
sin 2 Acos 2 A=1
entered as  sin A   cos A 
--------------------------------------------------------
sin −=−sin  (an ODD function)

cos −=cos  (an EVEN function)

------------------------------------------------------------------------------------------------
17.5 Multiple / double angles

sin  A B=sin A cos Bcos Asin B sin 2 A=2sin A cos B

sin  A – B=sin Acos B – cos Asin B

cos  A B=cos Acos B – sin Asin B cos 2 A=cos 2 A – sin 2 A

=2 cos2 A−1
1
cos 2 A= cos 2 A1
2
cos 2 A=1−2sin 2 A
1
sin 2 A= 1−cos 2 A
2

cos  A – B=cos Acos Bsin Asin B

tan Atan B 2 tan A

tan  AB= tan 2 A=
1 – tan A tan B 1 – tan 2 A

tan A−tan B
tan  A− B=
1tan A tan B

---------------------------------------------------------

Contents p1 8 Notation 32 26 Computer Input

Products to Sums

1
sin A cos B= sin  ABsin  A− B
2

1
cos Asin B= sin  AB−sin  A−B
2

1
cos Acos B= cos  A Bcos  A− B
2

1
sin Asin B= cos A−B−cos  A B
2
---------------------------------------------------------
Sums to Products

sin Asin B=2sin   

A B
2
cos
A– B
2 
sin A−sin B=2 cos   
AB
2
sin
A– B
2 
cos Acos B=2cos   
AB
2
cos
A– B
2 
cos A−cos B=−2 sin   
A B
2
sin
A– B
2 
-----------------------------------------------------------------------------------------------------
17.6 Sinusoidal Wave
[K Singh pp 195 - 202]

R V =R sin  t 

t see 22.6, 4.4




Period =
2
 [Frequency = 2 ]
 = phase angle 
= phase shift

Thanks to Mark Perkins, Bedford College

Contents p1 8 Notation 33 26 Computer Input

18 Complex Numbers
[K Singh pp 464 - 506]
Notation for Complex Numbers
BE - imaginary numbers
j symbol representing −1 . ( i used on most calculators)
a j b a complex number in Cartesian (or Rectangular) form
( x y i on a calculator). a , b∈ℝ , j b imaginary part.

z a complex number z =a j b (or x y i )

r ∠ a complex number in polar form

z complex conjugate of the complex number

If z=a j b then the complex conjugate z=a – j b
or if z=r ∠  then the complex conjugate z=r ∠−

z=a j b=rcos  j sin =r ∠=r e j where j 2=−1

---------------------------------------------------------
Im
Modulus, r =∣z∣= a 2b 2
(or magnitude) see 17.2, 17.2 jb
r

Argument, =arg z =tan−1 

b
a
θ
a
Re
(or angle)
BE - Complex arithmetic - better explained Argand Diagram

Addition a jbc j d =ac j bd 

 a jbc− jd 
Multiplication a jbc jd  Division
c jd c− jd 

Polar Multiplication z 1 z 2=r 1 ∠ 1×r 2 ∠ 2=r 1 r 2 ∠12

z 1 r 1 ∠1 r 1
Polar Division = = ∠1−2 
z 2 r 2 ∠ 2 r 2

r ∠n=r n ∠ n =r n cos n j sin n  r ∠= r ∠ 2 

http://www.justinmullins.com/home.htm
____________________________________________________________
Contents p1 8 Notation 34 26 Computer Input
19 Vectors
Notation for Graphs and Vectors [K Singh pp 568 - 600]

x , y the co-ordinates of a point, where x is the distance

from the y axis and y is the distance from the x axis

v a vector. Always underlined in written work


AB a vector

a i b j a vector in Cartesian form (Rectangular form)

r ∠ a vector in polar form (where r=∣v∣ ) )

 a
b
a vector in Component form (Rectangular Form)

∣v∣ modulus or magnitude of vector v .

---------------------------------------------------------
Vectors y
b x (a,b)
r bj

θ
a ai
A point a , b A vector v=  a
b
or v=r 

Vector Addition  

a
b

c
d
=
ac
bd
Geogebra

Scalar Product a×b=∣a∣∣b∣cos  θ

b
Dot Product a⋅b=a 1 b 1a 2 b 2a 3 b 3 ...

 
a1 b1
a b
where a= 2 and b= 2
a3 b3
. .
-------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 35 26 Computer Input

20 Co-ordinate Conversion using Scientific Calculators

R to P Rectangular to Polar 
x
y
to r ∠ (  x jy to r ∠ )

P to R Polar to Rectangular r ∠ to 
x
y
( r ∠ to  x jy )
see also 17.3

Casio Natural Display Edit keystrokes for your calculator

r
R to P SHIFT Pol( x SHIFT , y ) = out


x
P to R SHIFT Rec( r SHIFT ,  ) = out
y

Casio S-VPAM and new Texet Edit keystrokes for your calculator
R to P SHIFT Pol( x SHIFT , y ) = r out RCL tan  out

P to R SHIFT Rec( r SHIFT ,  ) = x out RCL tan y out

Sharp ADVANCED D.A.L. Edit keystrokes for your calculator

R to P x 2ndF , y 2ndF  r  r out   out

or MATH 1 r out 2ndF ⋅  out

P to R r 2ndF ,  2ndF  x y x out  y out

or MATH 2 x out 2ndF ⋅ y out

Old Casio fx & VPAM

R to P x SHIFT R  P y = r out SHIFT X  Y  out

P to R r SHIFT P  R  = x out SHIFT X  Y y out

Contents p1 8 Notation 36 26 Computer Input

Texet - albert 2
R to P x INV x ↔ y y R  P r out INV x ↔ y  out

P to R r INV x ↔ y  P  R x out INV x ↔ y y out

Casio Graphics (1)

R to P SHIFT Pol( x SHIFT , y ) EXE r out ALPHA J EXE  out

P to R SHIFT Rec( r SHIFT ,  ) EXE x out ALPHA J EXE y out

Casio Graphics (2)

R to P FUNC 4 MATH 4 COORD 1 Pol( x , y ) EXE r ALPHA J EXE 

P to R FUNC 4 MATH 4 COORD 1 Rec( r ,  ) EXE x ALPHA J EXE y

Casio Graphics (7 series)

R to P OPTN ▶ F2 ▶ ▶ Pol( x , y ) EXE r , out

R to P OPTN ▶ F2 ▶ ▶ Rec( r ,  ) EXE x , y out

Old Texet and old Sharp and some £1 calculators

You must be in Complex Number mode.
2ndF CPLX

R to P x a y b 2ndF a r out b  out

P to R r a  b 2ndF b x out b y out

Texas - 36X

R to P x x ↔ y y 3rd R  P r out x ↔ y  out

P to R r x ↔ y  2nd P  R x out x ↔ y y out

Contents p1 8 Notation 37 26 Computer Input

Texas Graphics (TI 83)

R to P 2nd Angle R  Pr ( x , y ) ENTER r out

2nd Angle R  P  ( x , y ) ENTER  out

P to R 2nd Angle P  R x ( r , ) ENTER x out

2nd Angle P  R y ( r , ) ENTER y out

Sharp Graphics

R to P MATH (D)CONV (3) xy  r ( x y ) ENTER r out

MATH (D)CONV (4) xy   ( x y ) ENTER  out

P to R MATH (D)CONV (5) r   x ( r  ) ENTER x out

MATH (D)CONV (6) r   y ( r  ) ENTER y out

Insert the keystrokes for your calculator here (if different from above)
R to P

P to R

------------------------------------------------------------------------------------------------------
Degrees to Radians ÷180× Radians to degrees ÷×180
_____________________________________________________________

Contents p1 8 Notation 38 26 Computer Input

21 Indices and Logs
21.1.1 Rules of Indices:
[K Singh pp 224 - 245] notation 8.1.1
MC
m n mn
1. a ×a =a

am
2. =a m− n
an

3. a m n =a mn

4.
a

m
n = a
n m  1
n
a = a
n

k
5. k a−n =
an
Also,
1
a 0=1
2
 x= x = x 0.5
2 and  a= a

a 1=a
n
 a=b⇔ bn =a
------------------------------------------------------------------------------------------------------
21.1.2 Definition of logarithms

If N =a n then n=log a N

---------------------------------------------------------
21.1.3 Rules of logarithms:
MC
1. log  A× B =log Alog B

2. log A
B
=log A – log B

3. log A n =n log A

log b N
4. log a N =
log b a

------------------------------------------------------------------------------------------------------
exp x≡e x log e x≡ln x log 10 x≡lg x
------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 39 26 Computer Input

21.1.4 Infinite Series
[K Singh pp 338 - 346]

x x x2 x3 x 4 x5 x6 x7
e =1       ... for ∣x∣∞
1! 2! 3 ! 4 ! 5! 6! 7 !
BE exponential functions better explained

 
jx − jx 3 5 7
e −e x x x
sin x= =x−  − ... for ∣x∣∞
j2 3! 5! 7 !

 
jx − jx 2 4 6
e e x x x
cos x= =1−  − ... for ∣x∣∞
2 2! 4! 6!

x−1  x−12  x−13

ln x= –  −... for 0 x≤2
1 2 3
BE- demystifying the natural logarithm
---------------------------------------------------------
21.1.5 Hyperbolic Functions
- definitions [K Singh p 246]

MC pronunciation

 
x −x 3 5 7
e −e x x x
sinh x= =x   ... “shine x”
2 3! 5! 7!

 
x −x 2 4 6
e e x x x
cosh x= =1   ... “cosh x”
2 2! 4! 6!

e x −e −x
tanh x= x −x “thaan x”
e e
______________________________________________________________
y = cosh x
y = ex y=x y = sinh x

y = ln x y = tanh x

y = tanh x

y = sinh x

ax
ke slider k lna x  slider

Contents p1 8 Notation 40 26 Computer Input

21.1.6 Graphs of Common Functions

y=a x 3b x 2c xd y=a x 4b x 3c x 2d x f

a
y= b y=x 2 and y= x
x

y=k 1−e− t y=k e− tb

Contents p1 8 Notation 41 26 Computer Input

22 Calculus
22.1.1 Notation for Calculus
see also section 8
Differentiation

dy
the first derivative of y where y is a function of x (Leibniz)
dx
Also see 13

f ' x the first derivative of f  x . (as above). (Euler)

v̇ the first derivative of v w.r.t. time. (Newtonian mechanics)

D u the first derivative of u

d2 y dy dy
the second derivative of y w.r.t x . The of
dx 2 dx dx

f ''  x the second derivative of f  x . ( f 2  x  is also used)

v̈ the second derivative of v w.r.t. time. (Newtonian mechanics)

∂z
the partial derivative of z w.r.t. x . ( ∂ “partial d”)
∂x

x a small change (increment) in x . (  “delta”)

------------------------------------------------------------------------------------------------------
Integration

∫ the integral sign (Summa)

∫ f  x dx the indefinite integral of f  x  (the anti-differential of f  x )

b
∫ f  x dx the definite integral of f  x from x =a to x=b
a
the area under f  x between x=a and x=b

F  x the primitive of f  x ( ∫ f  x dx without the c )

L[ f t] the Laplace operator (with parameter s )

-------------------------------------------------------------------------------------------------------
BE - gentle introduction to learning Calculus discovring pi - betterexplained.com
Contents p1 8 Notation 42 26 Computer Input
22.2 Differential Calculus - Derivatives
dy
[K Singh pp 258 - 358]
dx
dy
y or f  x  or f '  x
dx
________________________________________________
xn n x n−1
sin x cos x
cos x −sin x
ex ex
1
ln x
x
________________________________________________
k 0
k xn k n x n−1
sin a x a cos a x
cos a x −a sin a x
ea x a ea x
a 1
ln a x =
ax x
________________________________________________
k a xbn k n a a xbn−1
k sin a xb k a cos a xb
k cosa xb −k a sin  a xb
ka
k tan a xb k a sec 2 a xb= 2
cos a xb
k e axb k a e ax b e
x
gradient slider

ka
k ln a xb
a xb
________________________________________________

Contents p1 8 Notation 43 26 Computer Input

Further Standard Derivatives
dy
y or f  x or f '  x
dx
______________________________________________
f '  x
ln [ f  x]
f  x
1
sin−1  x
a  a 2
– x 2
, x 2a 2

cos 
−1 x
a
−1
a – x
2 2
, x 2a 2

tan 
−1 x
a
a
a  x2
2

sinh a xb a cosha xb

cosh a xb a sinh a xb
tanh a xb a sech 2 a xb

sinh
−1

x
a
1
 x2a2
cosh
−1
 x
a
1
 x −a
2 2
, x 2a 2

tanh
−1
 x
a
a
2
a −x
2
, x 2a 2

_____________________________________________________________
Differentiation as a gradient function (tangent to a curve).

y=k x n c dy
=k n x n−1
dx

x
------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 44 26 Computer Input

22.2.1 Maxima and Minima
(Stationary Points) [K Singh pp 308 - 325]
dy
If y= f  x then at any turning point or stationary point = f '  x=0
dx
Determine the nature (max, min or saddle) of the turning points by evaluating
gradients locally (i.e. close to turning point). MC

dy
+ 0 − − 0 + + 0 + − 0 −
dx

d2 y
– + ? ?
dx2
-------------------------------------------------------------------------------------------------------
22.2.2 Differentiation Rules
[K Singh pp 274 – 285]

For D read differentiate D [k f  x]=k f '  x , k a constant

--------------------------------------------------------
Function of a function rule D [ f  g  x]= f '  g  x×g '  x

dy dy du
= × MC
dx du dx
--------------------------------------------------------
If u and v are functions of x then:

du dv
Addition Rule D uv=  =u ' v '
dx dx
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
du dv
Product Rule D uv=v u =v u ' u v ' MC
dx dx
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
du dv
–u v
Quotient Rule
D
u
v
=
dx
v 2
dx vu ' – uv '
=
v2
MC

-------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 45 26 Computer Input

22.2.3 Formula for the Newton-Raphson Iterative Process
[K Singh pp 352 - 356]

Set f  x =0 with guess value x 0 (from graph) see 13

Test for Convergence

∣ f  x 0  f ' '  x 0
[ f '  x 0]2 ∣
1 see 8 - modulus

f  x n
xn f  x n f '  xn  x n1= x n –
f '  x n
(where f '  x n ≠0 )
f  x =0 when x n1=x n to accuracy required.

http://archives.math.utk.edu/visual.calculus/3/newton.5/1.html
-------------------------------------------------------------------------------------------------------
22.2.4 Partial Differentiation
[K Singh pp 695 - 704]

If z= f  x , y then a small change in x , named  x (delta x) and a small

change in y , named  y etc. will cause a small change in z , named  z
∂z ∂z ∂z
such that  z ≃  x  y... where is the partial derivative of z
∂x ∂y ∂x
∂z
w.r.t. x and is the partial derivative of z w.r.t y . see 8
∂y
-------------------------------------------------------------------------------------------------------
22.2.5 Implicit Differentiation

If z = f  x , y then
dy
=
∂z
∂x   Also dy = 1
dx ∂z
∂y   dx
 
dx
dy
-------------------------------------------------------------------------------------------------------
22.2.6 Parametric Differentiation
[K Singh pp 291 - 296]
If x= f t and y=g t
dx dy
= f ' t and =g ' t 
dt dt

 
dy
dy g ' t 
=
dx f ' t 
or
dy
dx
=
dt
 dx
f ' t  , ≠0  MC

 
dx
dt
dt

______________________________________________________________

Contents p1 8 Notation 46 26 Computer Input

22.3 Integral Calculus - Integrals
[K Singh pp 359 - 462] ∫
dy
or f  x y or ∫ f  x dx or F  x + c
dx
____________________________________________________
x n1
xn n≠−1
n1
sin x −cos x
cos x sin x
ex ex
1
= x−1 ln x (when n=−1 )
x
____________________________________________________
k kx
kx n1
k xn n≠−1
n1
−cos a x
sin a x
a
sin a x
cos a x
a
ax
e
e ax
a
k
=k x−1 k ln x (where n=−1 )
x
___________________________________________________
n k a xbn1
k a xb n≠−1
 n1a
−k cos a xb
k sin  a xb
a
k sin  a xb
k cos a xb
a
k tan a xb
k sec2  a xb
a
 a xb
ke
k e a xb 
a
k k ln a xb
n=−1
a xb a
_____________________________________________________

Contents p1 8 Notation 47 26 Computer Input

Further Standard Integrals
dy
or f  x or F  x + c
y or ∫ f  x dx
dx
________________________________________________________


dy
f '  x
dx ln  f  x  ln  y
f  x
y
1
 a 2− x 2
, x 2
a 2
sin−1 
x
a

2
1
a x
2
1
a
tan−1 
x
a
1
sinh a xb cosh a xb
a
1
cosh a xb sinh  a xb
a
1
sech 2  a xb tanh a xb
a
1
 x a
2 2
, x 2a 2 sinh−1 
x
a
or ln  x x 2a 2 

1
 x −a
2 2
, x 2a 2 cosh−1 
a
x
or ln  x x 2−a 2 

2
1
a −x
2
, x 2a 2 1
a
tanh−1  x
a
or
1
ln
2 a a – x ∣
 a x
∣
2
1
x −a
2
, x 2a 2 −1
a 
coth −1
x
a
or
1
ln
 x−a
2 a  xa ∣ ∣
______________________________________________________________
Addition Rule ∫ f  x g  x dx=∫ f  x dx∫ g  x dx
-------------------------------------------------------------------------------------------------------
22.3.1 Integration by Substitution
[K Singh p 368]
∫ f  g  x dx MC
du du
∫ f u du where u=g  x then = g '  x and dx=
dx g '  x
x=b u when x=b

Note change of limits ∫ f g  xdx to ∫ f udu

x=a u when x=a
du is a function of u or du ∈ℝ
--------------------------------------------------------------------------------------------------------
22.3.2 Integration by Parts
[K Singh pp 388 - 395]
see 22.6
∫ u dv=u v−∫ v du MC
------------------------------------------------------------------------------------------------------
Contents p1 8 Notation 48 26 Computer Input
22.3.3 Indefinite Integration
dy
= f  x
dx
dy= f  x  dx
∫ 1dy=∫ f  x dx
y=F  xc
------------------------------------------------------------------------------------------------------
22.3.4 Area under a Curve y
- Definite Integration [K Singh p 442] y = f(x)

∫ f  x dx
a
b
=[ F  x c ]a
F(b) - F(a)
= F bc – F  ac
a b x
Hyperlink to interactive demo of areas by integration
http://surendranath.tripod.com/Applets/Math/IntArea/IntAreaApplet.html

Procedure y
Plot between limits - a and b
Check for roots ( R1 , R 2 .. R n ) and evaluate
+ve +ve
See Newton Raphson 22.2.3
Integrate between left limit, a , and R1 a R1 R2 b x
then between R1 and R 2 and so on to -ve
last root R n and right limit b
Add moduli of areas. (areas all +ve)
-------------------------------------------------------------------------------------------------------
22.3.5 Mean Value
[K Singh p 445]
If y= f  x then y , y y = f(x)
the mean (or average) value of y
over the interval x=a to x=b is
y
b
1
y= ∫ y dx
b−a a
a b x
--------------------------------------------------------
22.3.6 Root Mean Square (RMS)


b
1 2
where y= f  x
y rms= ∫
b−a a
y dx

-------------------------------------------------------------------------------------------------------
Contents p1 8 Notation 49 26 Computer Input
22.3.7 Volume of Revolution
around the x axis [J Bird pp 207-208]

b
2
V = ∫ y dx where y= f  x
a
-------------------------------------------------------------------------------------------------------
22.3.8 Centroid
[J Bird pp 208 - 210]
The centroid of the area of a lamina y
bounded by a curve y= f  x and
y = f(x)
limits x=a and x=b
has co-ordinates  x , y  .

b b
1 2
∫ x y dx ∫
2 a
y dx x Centroid = (x, y)
a
x= b and y= b
y
∫ y dx ∫ y dx
a a
a b x
------------------------------------------------------------------------------------------------------
22.3.9 Partial Fractions
[K Singh pp 397 - 402]

f x A B
≡  see 8
 xa xb  x a  x b

f  x A B C
≡  
 xa2  xb  xa  xa2  xb

f  x Ax B C
2
≡ 2  2 
 x a xb  x a  x a  xb
MC
-----------------------------------------------------------------------------------------------------

Contents p1 8 Notation 50 26 Computer Input

22.3.10 Approximation of Definite Integrals
[K Singh p 434]

22.3.10.1 Simpson's Rule

y
y = f(x)

b−a
w= yn
n
y1 y2 y3 yn-1 yn

x1 x2 x3 xn-1 xn x w
a b
b
w
∫ f  x dx≈ Area≈ 3  y 14 y 22 y 3…2 y n−14 y n y n1 
a
( n is even)
b

∫ f  x dx≈ w3 [ firstlast4 ∑ evens 2 ∑ odds  ]

n xn yn
Multiplier m Product m y n
1 a 1 y1 1× y 1
2 aw 4 y2 4× y 2
3 a2w 2 y3 2× y 3
. . . . .
. . . . .
. . . . .
n−1 . y n−1 2 2× y n−1
n . yn 4 4× y n
n1 b y n1 1 1× y n1
Sum =
×w =
÷3 =
---------------------------------------------------------

22.3.10.2 Trapezium Method

b
w
∫ f  xdx≈  y 12 y 22 y 3......2 y n y n1 
a 2
-------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 51 26 Computer Input

22.4 Laplace Transforms
[J Bird pp 582 – 604] L [ f (t)]
Table of Laplace Transforms
∞
L[ f (t )] is defined by ∫ f t  e−st dt and is written as F  s
0
f t L[ f t ]
1
1 1  L[0]=0
s
1
2 t
s2
n!
3 tn
s n 1
1
4 e−a t
sa
a
5 1−e−a t
ssa
1
6 t e−a t
 sa2
n!
7 t n e−a t
 san1

8 sin (ω t )
s  2
2

s
9 cos (ω t ) 2 2
s 
ω2
10 1−cos(ω t)
s(s 2+ω2)
2 2 s
11 ω t sin (ω t )
 s 22 2
23
12 sin (ω t)−ω t cos (ω t )
 s 22 2

13 e−a t sin(ω t ) 2 2 see 14
 sa 
sa
14 e− a t cos(ω t) 2 2
 sa 
a s
15 e−a t (cos (ω t)− ω sin (ω t )) 2 2
 sa 
ssin  cos 
16 sin t φ 2 2
s 
a a 22
17 e−a t + ω sin (ω t)−cos(ω t)
 sa s 2 2 
Contents p1 8 Notation 52 26 Computer Input
f t L[ f t]

18 sinh (βt )
s − 2
2

s
19 cosh(βt ) 2 2
s −

20 e− a t sinh(β t) 2 2
 sa −
sa
21 e−a t cosh(βt ) 2 2
 sa −

First order differential equation:

L [ ]
dy
dt
=s L[ y ] – y 0 where y 0 is the value of y at t=0

y 1= y 0h y ' 0 13 Range x=ahb

where h is the step size

a and b are limits

x0 y0  y ' 0 y1

Plot the graph of y against x from values in first 2 columns.

Contents p1 8 Notation 53 26 Computer Input

22.6 Fourier Series.
[J Bird pp 611 - 657] and next page and 26 and 26.1

For period T , the smallest period of f t  . (determine from a graph)

2
Fundamental angular frequency =
T

f t =a 0a 1 cos t a 2 cos 2 t a 3 cos3 t …

a n , b n constants
b1 sin  t b 2 sin 2  t b3 sin 3  t …

or
∞
f t =a 0∑ a n cosn  t b n sin n  t 
n=1

where
T
2
1
a 0=
T
∫ f t  dt mean value of f t over period T
−T
2
see 22.3.9
T
2
2
a n=
T
∫ f t cosn t dt n=1, 2, 3…
−T
2

T
2
2
b n=
T
∫ f t sin n  t dt n=1, 2, 3…
−T
2

Alternatively written as:

f t =a 0c1 sin t  1c 2 sin 2 t  2 …c n sin n t  n 

a 0 constant, c n = a 2nb 2n and α n=tan

−1
()
an
bn

f t = constant + first harmonic + second harmonic + ......

See 17.7 of this book.

See Fourier series applet http://www.falstad.com/fourier/index.html

---------------------------------------------------------

Contents p1 8 Notation 54 26 Computer Input

22.6.1 Fourier Series - wxMaxima method.

Close wxMaxima and start again F6 for text

------------------------------------------------------------------------------------------------
T 2 1
Write down the values of T , , , and 
2 T T
------------------------------------------------------------------------------------------------
!! use  (type as w ) in input, not a number.
------------------------------------------------------------------------------------------------
2
an Input f t cosn w t  For piecewise functions
T
−T T −T T
Integrate between and and 0 and 0 and
2 2 2 2
or smaller intervals

Add the parts of a n

---------------------------------------------------
2
bn Input f tsin  n w t For piecewise functions
T
−T T
Integrate between and as above
2 2

Add the parts of b n

------------------------------------------------------------------------------------------------
Make up the sum a n cos n w t bn sin n w t
------------------------------------------------------------------------------------------------
Sum Calculus; Calculate Sum Start with 6 terms ( n from 1 to 6)
but you may need more.
------------------------------------------------------------------------------------------------
Substitute in the value for w
------------------------------------------------------------------------------------------------
Trial plot
------------------------------------------------------------------------------------------------
a 0 By observation OR
1
Input f t  For piecewise functions
T
−T T
Integrate between and as above, but your
2 2
interval may have to be
T
0 to
2
-----------------------------------------------------------------------------------------------
Add a 0 to the Sum
------------------------------------------------------------------------------------------------
Plot You will have to adjust horizontal range to
be able to see the result.
______________________________________________________________
Contents p1 8 Notation 55 26 Computer Input
23 Statistics
[K Singh pp 726 - 796]

23.1.1 Notation for Statistics

n sample size

x a sample statistic (a data value) OR

xi the variate

X a population statistic

x the arithmetic mean point of a sample set of data

s standard deviation of a sample

 the mean value of a population

 standard deviation of a population

∑ the sum of all terms immediately following

f frequency

Q quartile. ( Q 1 lower; Q 2 median; Q 3 upper)

df degrees of freedom n−1 of a sample.

P= X −x the probability that the population statistic equals the sample
statistic

x! x× x−1× x−2× x −3×…×1, x ∈ℕ

Range maximum value – minimum value

Quartiles in a set of ordered data,

Median, Q 2 : the middle value.
Lower, Q 1 : the middle value between minimum and Q 2 .
Upper, Q 3 : the middle value between Q 2 and the maximum.
k 1
Percentile: the k th percentile is in position ×n .
100 2

Mode in a set of data the mode is the most frequently occurring

value.
-----------------------------------------------------------------------------------------------
Contents p1 8 Notation 56 26 Computer Input
23.2 Statistical Formulae

Mean, x =
∑fx or x=
∑ xi where x i is the variate,
∑f n
f is frequency
BE - averages n is the sample size
--------------------------------------------------------

Population Standard Deviation =

 ∑  x i – x2
n

--------------------------------------------------------
=
 ∑ f d2
∑f
d = xi – x

Sample Standard Deviation

where n is the sample size

s=
 ∑  xi – x2
n−1

--------------------------------------------------------
Table for the calculation of Sample Mean and Standard Deviation

xi f f xi x− x f  x−x 
2

. . . . .

∑ f xi = ∑ f  x−x 2=
∑ f xi =
x =
n s=
 ∑ f  x−x 2 =
n−1

--------------------------------------------------------
Coefficient of Variation
s
of a sample (as a %) ×100
x
------------------------------------------------------------------------------------------------------
Q 3−Q 1
Semi-interquartile Range SIR=
2
------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 57 26 Computer Input

23.2.1 Regression Line
- see 13 and 27

For the line y=ab x where b is the gradient and a is the y

intercept and n is the number of pairs of values.

∑ y –b∑ x n ∑ xy – ∑ x ∑ y
a= b= 2 2
n n ∑ x – ∑ x 
--------------------------------------------------------
Product moment coefficient of Correlation (r value)

r=
 n ∑ xy – ∑ x ∑ y 
−1≤r≤1
 n ∑ x 2 – ∑ x 
2
 n ∑ y 2 − ∑ y 
2

------------------------------------------------------------------------------------------------------
x−
Z Scores Z=

-------------------------------------------------------------------------------------------------------
Poisson Distribution - the probability of the occurrence of a rare event

e −  x
Geogbra Poisson slider P  X = x=
x!
-------------------------------------------------------------------------------------------------------
T Test 1 sample

s
Standard Error of the Mean SE  x=
n
x−
T test (1 sample test) t=
SE  x
---------------------------------------------------------
2 sample for n30 ( d f = n 1n 2 – 2 )

Standard Error of Mean SE  x 1−x 2=

s1 s 2

n1 n2
 x − x − 1− 2 

T test (2 sample test) t= 1 2
SE  x 1− x 2
---------------------------------------------------------
2 sample for n30


2 2
n 1 – 1 s 1n2 – 1 s 2
Pooled Standard Deviation sp=
n1n 2−2

Standard Error of Mean SE  x 1−x 2=s p

1 2

n1 n 2
-------------------------------------------------------------------------------------------------------
Contents p1 8 Notation 58 26 Computer Input
23.2.2 Tables of the Normal Distribution

Probability Content from −∞ to Z 0 z

Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9990 0.9990 1

Far Right Tail Probabilities

0 z
Z P{Z to ∞ } Z P{Z to ∞ } Z P{Z to ∞ } Z P{Z to ∞ }
2.0 0.02275 3.0 0.001350 4.0 0.00003167 5.0 2.867E-7
2.1 0.01786 3.1 0.0009676 4.1 0.00002066 5.5 1.899E-8
2.2 0.01390 3.2 0.0006871 4.2 0.00001335 6.0 9.866E-10
2.3 0.01072 3.3 0.0004834 4.3 0.00000854 6.5 4.016E-11
2.4 0.00820 3.4 0.0003369 4.4 0.000005413 7.0 1.280E-12
2.5 0.00621 3.5 0.0002326 4.5 0.000003398 7.5 3.191E-14
2.6 0.004661 3.6 0.0001591 4.6 0.000002112 8.0 6.221E-16
2.7 0.003467 3.7 0.0001078 4.7 0.000001300 8.5 9.480E-18
2.8 0.002555 3.8 0.00007235 4.8 7.933E-7 9.0 1.129E-19
2.9 0.001866 3.9 0.00004810 4.9 4.792E-7 9.5 1.049E-21
These tables are public domain. http://www.math.unb.ca/~knight/utility/NormTble.htm
They are produced by APL programs written by the author, William Knight

Contents p1 8 Notation 59 26 Computer Input

23.2.3 Critical Values of the t Distribution

2-tailed testing 1-tailed testing

df 0.1 0.05 0.01 0.1 0.05 0.01
5 2.015 2.571 4.032 1.476 2.015 3.365
6 1.943 2.447 3.707 1.440 1.943 3.143
7 1.895 2.365 3.499 1.415 1.895 2.998
8 1.860 2.306 3.355 1.397 1.860 2.896
9 1.833 2.262 3.250 1.383 1.833 2.821
10 1.812 2.228 3.169 1.372 1.812 2.764
11 1.796 2.201 3.106 1.363 1.796 2.718
12 1.782 2.179 3.055 1.356 1.782 2.681
13 1.771 2.160 3.012 1.350 1.771 2.650
14 1.761 2.145 2.977 1.345 1.761 2.624
15 1.753 2.131 2.947 1.341 1.753 2.602
16 1.746 2.120 2.921 1.337 1.746 2.583
17 1.740 2.110 2.898 1.333 1.740 2.567
18 1.734 2.101 2.878 1.330 1.734 2.552
19 1.729 2.093 2.861 1.328 1.729 2.539
20 1.725 2.086 2.845 1.325 1.725 2.528
21 1.721 2.080 2.831 1.323 1.721 2.518
22 1.717 2.074 2.819 1.321 1.717 2.508
23 1.714 2.069 2.807 1.319 1.714 2.500
24 1.711 2.064 2.797 1.318 1.711 2.492
25 1.708 2.060 2.787 1.316 1.708 2.485
26 1.706 2.056 2.779 1.315 1.706 2.479
27 1.703 2.052 2.771 1.314 1.703 2.473
28 1.701 2.048 2.763 1.313 1.701 2.467
29 1.699 2.045 2.756 1.311 1.699 2.462
30 1.697 2.042 2.750 1.310 1.697 2.457
40 1.684 2.021 2.704 1.303 1.684 2.423
50 1.676 2.009 2.678 1.299 1.676 2.403
60 1.671 2.000 2.660 1.296 1.671 2.390
80 1.664 1.990 2.639 1.292 1.664 2.374
100 1.660 1.984 2.626 1.290 1.660 2.364
120 1.658 1.980 2.617 1.289 1.658 2.358
140 1.645 1.960 2.576 1.282 1.645 2.327

2 sample test d f = n 1−1n 2−1=n 1n2 – 2

http://facultyweb.berry.edu/vbissonnette/tables/tables.html

Contents p1 8 Notation 60 26 Computer Input

23.2.4 Normal Distribution Curve

 
2
−x−μ
1 2 2
y= e
 2 π

±1sd≈68%
±2sd≈95 %
±3 sd≈99.7 %
Geogebra Normal Dist slider
Geogebra Skewed Dist
----------------------------------------------------------------------------
23.2.5 Binomial Theorem
n
n
 x y = ∑
k=0

n n−k k
k
x y where 
n
=
n!
k k ! n – k !

n! n! n!
 x yn= x n x n−1 y 1 x n−2 y 2... x 1 y n−1 y n
1!n−1! 2 ! n−2! n−1! 1!

-------------------------------------------------------------------------------------------------------
23.2.6 Permutations and Combinations

The number of ways of selecting r objects from a total of n

BE - permutations and combinations

Permutations
n
Repetition allowed P r =nr order does matter

n n!
No repetition Pr= order does matter
 n−r !

Combinations

n n!
No repetition Cr= order doesn’t matter
r !n−r !

n  nr−1!
Repetition allowed Cr= order doesn’t matter
r !r−1!
______________________________________________________________
Thanks to Gillian Cunningham, Aberdeen College.

Contents p1 8 Notation 61 26 Computer Input

24 Financial Mathematics

Notation for Financial Mathematics

i Interest rate (per time period) expressed as a fraction.

(usually written as r )

d Discount rate (per time period) expressed as a fraction.

n Number of time periods (sometimes written as i )

P Principal

A Accrued amount

a Amount

Sn Sum to the n th term (of a geometric progression)

NPV Net Present Value (of an accrued amount)

irr Internal Rate of Return (when NPV =0 )

-------------------------------------------------------------------------------------------------------
Financial Mathematics Formulae

r=1i

A= P 1i n A= P 1 – d n

a r n – 1 a1−r n 
S n= or S n=
r−1 1−r

a 1 – r−n 
(annuities) P=
r−1
BE - visual guide to interest rates

Efunda Calculator
______________________________________________________________

Contents p1 8 Notation 62 26 Computer Input

25 Recommended Computer Programs
wxMaxima free (Open Source) MS Windows and Linux

http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page
Windows: download maxima 5.24.0 (or later version)
http://portableapps.com/node/18166 (portable application)

A open source free download computer algebra system. It is being constantly

updated.
You are not allowed implicit multiplication.

5e 2t3sin  4  typed as 5∗% e ^ 2∗t 3∗sin % pi/ 4

The % sign designates special functions. (numerical values of letters)

Maxima is a system for the manipulation of symbolic and numerical
expressions, including differentiation, integration, Taylor series, Laplace
transforms and ordinary differential equations. Also, Maxima can plot functions
and data in two and three dimensions.
-----------------------------------------------------------------------------------------------
Geogebra
free (Open Source) MS Windows and Linux
http://www.geogebra.org

This program can be accessed over the web i.e. you do not need to download
it although you do usually need to be running Java Runtime Environment (free
download). GeoGebra is a dynamic mathematics software that joins geometry,
algebra and calculus. An expression in the algebra window corresponds to an
object in the geometry window and vice versa.
----------------------------------------------------------------------------------------------------
Mathcad ( £1000 approx.) MS Windows

This is the tool of choice for most engineering mathematics. Notes available.
---------------------------------------------------------------------------------------------------
Graph
free (Open Source) MS Windows

A useful graphing tool which is easy to use. http://www.padowan.dk/graph/

---------------------------------------------------------------------------------------------------
Casio Calculator Manuals (in pdf format)
http://world.casio.com/calc/download/en/manual/
_______________________________________________________________________

Contents p1 8 Notation 63 26 Computer Input

26 Computer Input
wxMaxima and Geogebra are recommended .
Most of this also applies to spreadsheets and online maths sites.
Spreadsheet programs are not recommended (except for statistical
calculations).

Computer (Keyboard) entry

Calculator key
Geogebra (3) Mathcad (2) wxMaxima (5)
× ∗ ∗ (Shift 8) ∗
÷ / / /
^2 ^2
x2 ^2
(Shift 6 then 2) (Shift 6 then 2)
x ▄ or ^ or x y
^ ^ ^
or y x
sqrt() (also on
 drop down list)
\ sqrt( )

n
x 1  1
 ^ ^
x Calculator toolbar x
o o
5sin  x 30 
5sin  x o30o  5*sin(x/180*%pi+
o symbol from 5sin  x deg 30 deg  30/180*%pi)
(1)
drop down list
e from drop %e^( )
x x
e down list then ^ e or
or exp( ) exp( )
ln ln ln log
π pi CTRL g %pi
10××0.7 10 pi *0.7 10 CTRL g*0.7 10*%pi*0.7
sin−1 0.5
means asin(0.5) asin(0.5) asin(0.5)
arcsin(0.5)

(1) As all programs work in radians by default you must change every input
into degrees (if you have to work in degrees).
(2) Also available on toolbars.
(3) Only x allowed as variable
(4) See also 17.5
(5) In wxMaxima typing pi will produce π as a variable NOT 3.1415...
The same is true for e .

Back to 2 Web Sites

------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 64 26 Computer Input

26.1 wxMaxima Input
Note: From version 0.8.1 use Shift+Enter to enter expressions
to change behaviour go to Edit: Configure
See wxMaxima Introduction at http://ubuntuone.com/p/x77
See http://www.math.hawaii.edu/~aaronts/maximatutorial.pdf a simple introduction.
See http://www.neng.usu.edu/cee/faculty/gurro/Maxima.html but put in expression first!
------------------------------------------------------------------------------------------------
Note: Implicit multiplication is NOT allowed. 3 x is always typed as 3∗x
--------------------------------------------------------------------------------------------------------
Insert Text Box F6
------------------------------------------------------------------------------------------------
Zoom in Alt I Zoom out Alt O
------------------------------------------------------------------------------------------------
Copy as an Image to a Edit - Select All
Spreadsheet File Right click – Copy as Image...
Paste onto a worksheet
------------------------------------------------------------------------------------------------------
Assign w:3.7 (means w=3.7 ) f(x):=3*x (means f  x =3x )
-------------------------------------------------------------------------------------------------------
Matrix multiplication
[ ][ ]
⋅ Use . Do not use ∗

-------------------------------------------------------------------------------------------------------
26.1.1 Newton Raphson
load(newton1)
newton  f  x , x , x 0, p . Start with precision p=0.1 and then
p=0.01 etc. until outputs are identical to
significant figures required
---------------------------------------------------------
26.1.2 Differential Equations
see also 22.4 (2nd page)
dy
typed as ‘diff(y,x) note the apostrophe ‘ before diff
dx
d2 y
typed as ‘diff(y,x,2)
d x2
Equations; Solve ODE. Equations; Initial value problem (1) or (2).
---------------------------------------------------------
26.1.3 Runge-Kutta

rk  f  x , y , y , y 0, [ x , x 0, x end , h] See Euler’s Method 22.5

To plot result: wxplot2d([discrete,%o#],[style,points])

you can replace points with line. %o# is a previous output line.
-------------------------------------------------------------------------------------------------------
Contents p1 8 Notation 65 26 Computer Input
26.2 Mathcad Input

Applied Maths
Definition of variables and functions

variable := number and units (:= use colon :)

m
Example: x:3kg will read as x :=3 kg and a:5 m/s^2 as a :=5
s2

Function f  x := function in terms of x

Example: f(x): x*a will be interpreted as f  x:= x⋅a

= gives numerical answer

Example f  x = will produce the answer 15 N ▄

You can type a different unit in place of the box and the number will change to
satisfy the units chosen.
---------------------------------------------------------
Symbolic Maths
f  x = use Boolean (bold) equals

 symbolic units

Implicit multiplication: This is allowed but only with variables that cannot be
confused with units.

For example, 3 x is fine but 3 s must be typed as 3∗s .

When editing expressions use the Ins key to change from editing to the left
to editing to the right of cursor.

Also see Mathcad Notes

_____________________________________________________________

Contents p1 8 Notation 66 26 Computer Input

27 Using a Spreadsheet to find the ‘best fit’ formula for a
set of data.
see 13 and 26 and 23.2.1
Data presented as
x x1 x2 x3 x4 etc
y y1 y2 y3 y4 etc

Basic Procedure:-

Put x data in column A

and y data in column B

Highlight All data

Select Insert (or chart symbol)
Chart
Chart Type XY (Scatter)
Titles Give graph and axes titles
(Chart Location As New Sheet (optional) E)
(Right click on plot Format Plot Area E)
(Area Click to white E)
Right click on data point Add Trendline
Type Choose most appropriate
Options Display equation on chart
Display R 2 value on chart

R 2 value should lie between 0.95 and 1. The closer to 1 the better. Right
click on trendline to change to a better type.

The equation displayed is the formula for the data

All instructions necessary for MS Excel (E). Open Office Calc will provide the
same answers but in a slightly different format.
Mathcad and Maxima can be used but are more complicated mathematically
but will be more accurate. Geogebra can be used to match a line to data.

Note: EXCEL is NOT recommended for any mathematical or

engineering calculation where accuracy or consistency
is vital.
------------------------------------------------------------------------------------------------------

Contents p1 8 Notation 67 26 Computer Input

28 Calibration Error
Output Output

IDEAL Zero Error

Input = Output

Input Input

Output Output

Span Error Linearity Error

First and Last Values
coincide

Input Input

Output Output

Zero-Span Linearity Error

Error First and Last Values
coincide

Input Input

-------------------------------------------------------------------------------------------------------
Thanks to Olaniyi Olaosebikan, Aberdeen College

Contents p1 8 Notation 68 26 Computer Input

29 SI Units - Commonly used prefixes
meaning multiple prefix symbol
×1000000000000000 ×1015 Peta P
×1000000000000 ×1012 Tera T
×1000000000 ×109 Giga G
×1000000 ×106 Mega M
×1000 ×103 kilo k
×1 ×100
÷1000 ×10 −3 milli m
÷1000000 ×10 −6 micro 
÷1000000000 ×10−9 nano n
÷1000000000000 ×10−12 pico p

30 Electrical Tables

Table of Resistivities Relative Static Permittivity

Resistivity  Dielectric
Material Material
 m at 20o C Constant  r
Silver (Ag) 15.9×10−9 Vacuum 1
Copper (Cu) 17.2×10−9 Air 1.00054
Gold (Au) 24.4×10−9 Diamond (C) 5.5 - 10
Tungsten (W) 56.0×10−9 Salt (NaCl) 3 - 15
Nickel (Ni) 69.9×10−9 Graphite (C) 10 - 15
Iron (Fe) 100×10−9 Silicon (Si) 11.68
Lead (Pb) 220×10−9
Carbon (C) 35000×10−9

Permeability Values for some Common Materials

Permeability
Material
 (H/m)
Electrical Steel 5000×10−6
Ferrite (Nickel Zinc) (Ni Zn) 20 – 800×10−6
Ferrite (Manganese Zinc) (Mn Zn) 800×10−6
Steel 875×10−6
Nickel (Ni) 125×10−6
Aluminium (Al) 1.26×10−6
Thanks to Satej Shirodkar, Aberdeen College.
Contents p1 8 Notation 69 26 Computer Input
31 THE GREEK ALPHABET
UPPER lower
CASE case Pronunciation Examples of use
A  Alpha angles, angular acceleration
B  Beta angles
  Gamma shear strain, heat capacity, kinematic viscosity
  Delta DIFFERENTIAL, the change in... (Calculus)
E  Epsilon linear strain, permittivity
Z  Zeta impedance, damping ratio
H  Eta efficiency, viscosity
  Theta angles, temperature, volume strain
I  Iota inertia
K  Kappa compressibility
  Lambda wavelength, thermal conductivity, eigenvalues
M  Mu micro (10-6 ), coefficient of friction
N  Nu velocity
  Xi damping coefficient
O  Omicron
  Pi PRODUCT, 3.141592654...., C= d
P  Rho density, resistivity
  Sigma SUM; standard deviation, normal stress
T  Tau shear stress, torque, time constant
  Upsilon admittance
  Phi angles, flux, potential energy, golden ratio
X  Chi PEARSON’S  2 TEST , angles
  Psi helix angle (gears), phase difference
  Omega RESISTANCE; angular velocity
_____________________________________________________________
This work, to be attributed to Peter K Nicol, Aberdeen College is licensed
under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0
Unported License. To view a copy of this license, visit http://creativecom-
mons.org/licenses/by-nc-sa/3.0/ (click on icon below) or send a letter to
Creative Commons, 171 Second Street, Suite 300, San Francisco, Califor-
nia, 94105, USA.
Contents p1 8 Notation 70 26 Computer Input
Electrical Engineering Formulas
Ohms Law

Recti er E ciency
Ripple Factor

Single Phase AC Power

Two Phase AC Power
Three Phase AC Power
DC Power
Power Factor
Torque to Horsepower (hp)

Horsepower (hp) to Torque

Cylindrical Coil Inductance
Equivalent Resistance - Series & Parallel Circuit
Equivalent Capacitance - Series & Parallel Circuit
Equivalent Inductance - Series & Parallel Circuit
Equivalent Impedance - RLC Series Circuit
Equivalent Impedance - RLC Parallel Circuit
Voltage Drop

Magnetic Field Strength

BASIC ELECTRICAL ENGINEERING FORMULAS

BASIC ELECTRICAL CIRCUIT FORMULAS

IMPEDANCE VOLT-AMP EQUATIONS ENERGY
CIRCUIT
absolute complex instantaneous RMS values for (dissipated on R
ELEMENT
value form values sinusoidal signals or stored in L, C)

RESISTANCE R R v=iR Vrms=IrmsR E=Irms2R×t

INDUCTANCE 2πfL jωL v=L×di/dt Vrms=Irms×2πfL E=Li2/2
CAPACITANCE 1/(2πfC) 1/jωC i=C×dv/dt Vrms=Irms/(2πfC) E=Cv2/2
Notes:
R- electrical resistance in ohms, L- inductance in henrys, C- capacitance in farads, f - frequency
in hertz, t- time in seconds, π≈3.14159;
ω=2πf - angular frequency;
j - imaginary unit ( j2=-1 )

BASIC ELECTRICAL THEOREMS AND CIRCUIT ANALYSIS LAWS

LAW DEFINITION RELATIONSHIP
Ù=Z×Ì, where Ù and Ì - voltage and
Ohm's Law modified Lorentz force law,
current phasors, Z - complex
for AC circuits with Faraday's law and Drude
impedance
sinusoidal signals model
(for resistive circuits: u=R×i )
The sum of electric currents which
Kirchhoff's Current flow into any junction in an electric Conservation of electric
Law (KCL) circuit is equal to the sum of currents charge
which flow out
Kirchhoff's Voltage The sum of the electrical voltages
Conservation of energy
Law (KVL) around a closed circuit must be zero
EQUATIONS FOR SERIES AND PARALLEL CONNECTIONS
CIRCUIT SERIES PARALLEL
ELEMENT CONNECTION CONNECTION

Rparallel=
Rseries=
RESISTANCE 1/
R1+R2+...
(1/R1+1/R2+...)

Lseries= Lparallel=
INDUCTANCE
L1+L2+... 1/(1/L1+1/L2+...)

Cseries=
Cparallel=
CAPACITANCE 1/
C1+C2+...
(1/C1+1/C2+...)

RLC IMPEDANCE FORMULAS

CIRCUIT CONNECTION COMPLEX FORM ABSOLUTE VALUE

Z=R+jωL+1/jωC
Series

Z=
1/(1/R+1/jωL+jωC)

Parallel
MAGNETIC FIELD UNITS AND EQUATIONS

CONVER-
SYM- CGS CGS
QUANTITY SI UNIT SI EQUATION SION
BOL UNIT EQUATION
FACTOR
Magnetic Gauss 1T=
B tesla (T) B=µo(H+M) B = H+4πM
induction (G) 104 G
ampere/ H = NI/lc H = 0.4πNI/lc 1 A/m =
Magnetic Oersted
H meter ( lc - magnetic (lc - magnetic 4 π×10-3
field strength (Oe)
(A/m) path, m) path, cm) Oe
Φ = BAc
weber Maxwell Φ = BAc 1 Wb =
Magnetic flux Φ (Ac - area,
(Wb) (M) (Ac - area, cm2 ) 108 M
m2 )
M=m/V M=m/V
(m- total (m- total
ampere/ 1 A/m =
magnetic 3 magnetic
Magnetization M meter emu/cm 10-3
moment, moment,
(A/m) emu / cm3
V- volume, V- volume,
m3 ) cm3 )
Magnetic
newton/
permeability µo µo= 4π×10-7 1 - 4π×10-7
ampere2
of vacuum
L=
L=µoµN2Ac/lc =0.4πµN2Ac/lc×
(N- turns, ×10-8
Inductance L henry Ac- area, m2, henry (N- turns, 1
lc - magnetic Ac-area, cm2,
path, m) lc - magnetic
path, cm)
V= -10-8×
V= -NdΦ/dt
Emf (voltage) V volt volt ×N×dΦ/dt 1
(N- turns)
(N- turns)
MAXWELL'S EQUATIONS IN FREE SPACE (IN SI UNITS)
LAW DIFFERENTIAL FORM INTEGRAL FORM

Gauss' law for

electricity

Gauss' law for

magnetism

Faraday's law
of induction

Ampere's law

NOTES: E - electric field, ρ - charge density, ε0 ≈ 8.85×10-12 - electric permittivity of free space, π ≈ 3.14159,
k - Boltzmann's constant, q - charge, B - magnetic induction, Φ - magnetic flux, J - current density, i - electric
current,
c ≈ 299 792 458 m/s - the speed of light, µ0 = 4π×10-7 - magnetic permeability of free space, ▼ - del operator (if
V is a vector function, then ▼. V - divergence of V, ▼×V - the curl of V).

For more reference info see EE Information Online

SMPS Power Supply Guide ( http://www.smps.us/ ) - tutorials, electronics reference,
formulas, schematics.
BASIC ELECTRICAL ENGINEERING FORMULAS

BASIC ELECTRICAL CIRCUIT FORMULAS

IMPEDANCE VOLT-AMP EQUATIONS ENERGY
CIRCUIT
absolute complex instantaneous RMS values for (dissipated on R
ELEMENT
value form values sinusoidal signals or stored in L, C)

RESISTANCE R R v=iR Vrms=IrmsR E=Irms2R×t

BASIC ELECTRICAL THEOREMS AND CIRCUIT ANALYSIS LAWS

Rparallel=
Rseries=
1/
RESISTANCE R1+R2+...
(1/R1+1/R2+...)

Lseries= Lparallel=
INDUCTANCE L1+L2+... 1/(1/L1+1/L2+...)

Cseries=
Cparallel=
CAPACITANCE 1/
C1+C2+...
(1/C1+1/C2+...)

RLC IMPEDANCE FORMULAS

CIRCUIT CONNECTION COMPLEX FORM ABSOLUTE VALUE

1 2
Z=R+jωL+1/jωC Z  R 2  ( L  )
Series C

1
Z= Z
1 1 2
1/(1/R+1/jωL+jωC)  ( C  )
R 2
L
Parallel
MAGNETIC FIELD UNITS AND EQUATIONS
CONVER-
SYM- CGS CGS
QUANTITY SI UNIT SI EQUATION SION
BOL UNIT EQUATION
FACTOR
Magnetic Gauss 1T=
B tesla (T) B=µo(H+M) B = H+4πM
induction (G) 104 G
ampere/ H = NI/lc H = 0.4πNI/lc 1 A/m =
Magnetic Oersted
H meter ( lc - magnetic (lc - magnetic 4 π×10-3
field strength (Oe)
(A/m) path, m) path, cm) Oe
Φ = BAc
weber Maxwell Φ = BAc 1 Wb = 108
Magnetic flux Φ (Ac - area,
(Wb) (M) (Ac - area, cm2 ) M
m2 )
M=m/V M=m/V
(m- total (m- total
ampere/ 1 A/m =
magnetic magnetic
Magnetization M meter emu/cm3 10-3
moment, moment,
(A/m) emu / cm3
V- volume, V- volume,
m3 ) cm3 )
Magnetic
newton/
permeability µo µo= 4π×10-7 1 - 4π×10-7
ampere2
of vacuum
L=
2
L=μoμN Ac/lc =0.4πμN2Ac/lc×
(N- turns, ×10-8
Inductance L henry Ac- area, m2, henry (N- turns, 1
lc - magnetic Ac-area, cm2,
path, m) lc - magnetic
path, cm)
V= -10-8×
V=-N×dΦ/dt
Emf (voltage) V volt volt ×N×dΦ/dt 1
(N- turns)
(N- turns)
MAXWELL'S EQUATIONS IN FREE SPACE (IN SI UNITS)
LAW DIFFERENTIAL FORM INTEGRAL FORM

Gauss' law for

electricity

Gauss' law for

magnetism

Faraday's law
of induction

Ampere's law

NOTES: E - electric field, ρ - charge density, ε0 ≈ 8.85×10-12 - electric permittivity of free space, π ≈ 3.14159,
k - Boltzmann's constant, q - charge, B - magnetic induction, Φ - magnetic flux, J - current density, i - electric
current,
c ≈ 299 792 458 m/s - the speed of light, µ0 = 4π×10-7 - magnetic permeability of free space, - del operator (if
.
V is a vector function, then V - divergence of V, ×V - the curl of V).

REFERENCES:

1. EE Information Online

2. Electrical References 2014

SMPS Power Supply Guide ( http://www.smps.us/ ) - tutorials, electronics reference,
formulas, schematics.

Lazar Rozenblat on Google+

www.emersonct.com

800-893-2321

Electrical Formulae AC Three-Phase

(Assuming Balanced Symmetrical Waveform)
Electrical Quantities All quantities r.m.s values:
Quantity Symbol Unit Name Unit Symbol Vl = Line-to-line voltage
Vp = Phase voltage (line-to-neutral)
Electromotive force E, e* Volt V Il = line current (wye)
Potential difference V, v* Volt V Ip = Phase current (delta)
Current I, i* Ampere A In a WYE connected circuit, Vp = Vl 앛 앀3, Vl =앀3Vp, Il = Ip
Magnetic flux Φ Weber Weber In a DELTA connected circuit: Ip = Il 앛 앀3, Il =앀3 Ip Vl =
Vp
Frequency f Hertz Hz
Total of apparent power in VA = 앀3 Vl Il
Flux linkage λ Weber-turns -
Active power in watts,W = 앀3 Vl Il cos ϕ
Resistance R Ohm Ω
Reactive power in VAr = 앀3 Vl Il sin ϕ
Inductance L Henry H Power factor (pf) = cos ϕ
Capacitance C Farad F = Active power / Apparent power
Impedance Z Ohm Ω = W / VAr
Reactance X Ohm Ω
Three-Phase Induction Motors

Symbols & Formulae

Power,dc,or active P Watt W
Volt-ampere All quantities rms values:
Power, reactive Q VAr, var kWmech = horsepower x 0.746
reactive
Power, total or kWelec = 앀3 Vl Il cos ϕ at rated speed and load
S Volt-ampere VA
apparent where Vl = supply voltage Il = rated full load current
Power factor angle ϕ - °, deg. cos ϕ = rated full load power factor
Radians per
Angular velocity ω rads-1
second Efficiency, η = (kWmech 앛 kWelec) x 100 per cent
Revolutions Phase current Ip = Il for wye connection
Rotational velocity n s-1 ,rev s-1
per second Ip = Il 앛 앀3 for delta connection
Revolutions
min-1, rpm
per minute Loads (phase values)
Efficiency η - Resistance R, measured in Ohms (no energy storage)
Number of pairs Inductive reactance, XL = ωL = 2π ƒL Ohms (stores en-
p -
of poles ergy)
* Capital and small letters designate rms and instantaneous value Where ƒ = frequency (Hz), L = Inductance (H)
respectively.
Capacitative reactance, XC = 1/(ωC) = 1/(2πƒC)
Where ƒ = frequency (Hz), C = Capacitance (F)
AC Single-Phase
All quantities r.m.s. values:
Impedance
V=I Z
Impedance is the algebraic sum of the separate load
Total or apparent power in VA = Vl = I2Z = V2앛 Z
values thus:
Active power in watts, W =Vl cos ϕ
Reactive power in VAr = Vl sin ϕ Z = 앀(R2 + XL2) or 앀(R2 + XC2)
If R, XL and XC are present in series in the same circuit
then XL and XC may be summated, treating XC as nega-
tive, thus
Z = 앀(R2 + (XL - XC)2)

399
Phone: 800.894.0412 - Fax: 888.723.4773 - Web: www.clrwtr.com - Email: info@clrwtr.com
www.emersonct.com

800-893-2321

Electrical Formulae Mechanical Variables

Ohms Law
Material Densities
Amperes = Volts Ohms = Volts
or Materials lb/in3 gm/cm3
Ohms Amperes
Aluminum 0.096 2.66
or Volts = Amperes x Ohms Brass 0.299 8.3
Bronze 0.295 8.17
Power in DC Circuits Copper 0.322 8.91
Volts x Amperes Hard Wood 0.029 0.8
Horsepower =
746 Soft Wood 0.018 0.48
Plastic 0.04 1.11
Watts = Volts x Amperes
Glass 0.079-0.090 2.2-2.5
Kilowatts = Volts x Amperes Titanium 0.163 4.51
1,000 Paper 0.025-0.043 0.7-1.2
Volts x Amperes x Hours Polyvinyl chloride 0.047-0.050 1.3-1.4
Kilowatts-Hours =
1,000 Rubber 0.033-0.036 0.92-0.99
Silicone Rubber, without
Power in AC Circuits 0.043 1.2
filler
Kilovolt-Amperes (KVA): Cast Iron, gray 0.274 7.6
Symbols & Formulae

Volts x Amperes Steel 0.28 7.75

kVA (1Ø) =
1,000

kVA (3Ø) = Volts x Amperes x 1.73

Friction Coefficients Ffr=µWL
1,000
Materials µ
Kilowatts (Kw) Steel on Steel (greased) ~0.15
Plastic on Steel ~0.15-0.25
kW (1Ø) = Volts x Amperes x Power Factor Copper on Steel ~0.30
1,000
Brass on Steel ~0.35
kW (3Ø) = Volts x Amperes x Power Factor x 1.73 Aluminum on Steel ~0.45
1,000 Steel on Steel ~0.58
Kilowatts Mechanism µ
Power Factor =
Kilovolts x Amperes Ball Bushings <0.001
Linear Bearings <0.001
Dove-tail Slides ~0.2++
Other Useful Formulae Gibb Ways ~0.5++
Three-Phase (3Ø) Circuits

HP = E x I x 3 x Eff x PF
746 Mechanism Efficiencies
HP x 746 Acme screw with brass nut ~0.35-0.65
Motor Amps =
E x 3 x Eff x PF Acme screw with plastic nut ~0.50-0.85
Ballscrew ~0.85-0.95
Motor Amps = kVA x 1000
Chain and Sprocket ~0.95-0.98
3 xE
Preloaded Ballscrew ~0.75-0.85
Motor Amps = kW x 1000 Spur or Bevel gears ~0.90
3 x E x PF Timing Belts ~0.96-0.98
kW x 1000 Worm Gears ~0.45-0.85
Power Factor =
ExIx 3 Helical Gear (1 reduction) ~0.92

Kilowatt Hours = E x I x Hours x 3 x PF

1000

400 Power (Watts) = E x I x 3 x PF

Phone: 800.894.0412 - Fax: 888.723.4773 - Web: www.clrwtr.com - Email: info@clrwtr.com

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