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EC : ELECTRONICS AND COMMUNICATIONS ENGINEERING
Compendium of Important Results and Formulae
CONTENTS
Chapter
Topic Page No.
No.
1. Networks 1
2. Electronic Devices 9
3. Digital Circuits 12
4. Analog Integrated Circuits 20
5. Signals and Systems 28
6. Control Systems 38
7. Communication Systems 45
8. Electromagnetics 54
Compendium of Important Results and Formulae
1. NETWORKS
Network Elements & Graph
Formulae
dq
x i= Ampere
dt
…..[ i = current in Ampere, q = charge in coulomb, t = time in seconds]
w(joules)
x v= volts
q(coulomb)
x P = vi watts
x R=
VS
…..[ = Length of conductor, V = conductivity, S = CrossSectional Area]
x Rt = Ro (1 + D't)
…..[Rt = resistance at tqC, Ro = resistance at reference temperature, D = temperature
coefficient of resistance]
dv
x i=C Amp
dt
1 1 1 1
x (a) ........ farad1
Cser C1 C2 Cn
(b) Cpar = C1 + C2 + ………Cn farad
HA
x C= farad
d
…..[C = parallel plate capacitance, H = permittivity of the medium, A = surface Area of
plate, d = distance between plates]
PN2 A
x L= Henry
…..[P = permeability of medium, N = Number of terms, = length of coil ]
x \ = Li Weber
1113/GATE/EC/Compendium/Pg.1
(2) Vidyalankar GATE EC
d\
x v= volts
dt
di
=L
dt
x Energy
1 2 1
EL = Li , EC = Cv 2 joules
2 2
Network graph
x No. of twigs = N1
x No. of link = BN+1
x No. of KVL equation = BN+1
x No. of KCL equation = N1
x No. of Trees = NN2 (N > 2)
^ `
= det [A][A T ] ..…[A] = reduced incidence matrix
Network Theorems
x Ohm’s Law
V(t) = i(t).R
x KVL
N
∑ v (t)
j 1
j 0 …..[in a loop/mesh]
x KCL
N
∑ I (t)
j 1
j 0 …..[at a given node]
x Voltage divider rule
⎛ R1 ⎞ ⎛ R2 ⎞
V1 ⎜ ⎟V , V2 ⎜ ⎟V
⎝ R1 R2 ⎠ ⎝ R1 R2 ⎠
x Current divider rule
⎛ R2 ⎞ ⎛ R1 ⎞
I1 = ⎜ ⎟I , I2 ⎜ ⎟I
⎝ R1 R2 ⎠ ⎝ R1 R2 ⎠
1113/GATE/EC/Compendium/Pg.2
Compendium of Important Results and Formulae (3)
x Maximum power transfer
z zR*
E2
P=
4zR
x Millman’s Theorem
n
∑
i 1
Ei yi
E n
…..[for voltage sources]
∑
i 1
yi
N
Ii
∑y
i 1 i
I N
…..[for current sources]
1
∑
i 1 yi
x Star to ' & ' to star Transformation:
A A
RA
R1 R2
RB RC
B C
B C R3
Y to ' transformation
R R RBRC R ARC
R1 = A B
RC
R ARB RBRC R ARC
R2 =
RB
R ARB RBRC R ARC
R3 =
RA
' to Y transformation
R1R2 R1R3 R2R3
RA = , RB = , RC =
R1 R2 R3 R1 R2 R3 R1 R2 R3
1113/GATE/EC/Compendium/Pg.3
(4) Vidyalankar GATE EC
x Coefficient of coupling (K)
M
K=
L1L 2
x For complex impedance V and I relationship is given as :
V = Vm cos Zt
I = Im cos (Zt + T)
Average power is given as
Pav = Irms . Vrms cos T
Reactive power is given as
Pr = Irms . Vrms sin T
Apparent power is given as
Papp = Irms . Vrms
Power factor is given as
Paverage Pav
cosT
Papparent Papp
Linear Differential Equations
dx
x Px Q
dt
Then,
x = ept ∫ Qept dt Kept
if P & Q both are constant
x(t) = x(f) [x(f) x(o)] e t/T …..[T = Time constant]
x Element with initial conditions Equivalent circuit at t = O+
R R
oIo Io
+
qo Vo
Vo=
C
1113/GATE/EC/Compendium/Pg.4
Compendium of Important Results and Formulae (5)
x Element with initial Conditions Element equivalent at t = f
R R
oIo
Io
+
qo Vo
Vo=
C
x Nature of roots Description Form of solution Graph of response
(i) Negative real overdamped x(t) = K1 es1t K 2 es2 t
& unequal
o t
(ii) Negative real critically damped x(t) = (K1 + K2t) es1t
& equal
t
o
(iii) Complex Underdamped x(t) = eVt ( K1 cos w1t K 2 sin w1t )
Conjugate S1,2 = V rjw1
(Real past ve)
o t
1113/GATE/EC/Compendium/Pg.5
(6) Vidyalankar GATE EC
(iv) imaginary undamped x(t) = K1 cos w1t K 2 sin w1t
S1,S2 = rjw
o t
x To find particular solution
Form of excitation function Form of particular solution
Ko A
Kot A + Bt
Koebt (b z characteristic root) Aebt
Koebt (b z characteristic root) Atebt
Kosin bt Asin bt + Bcos bt
or
Kocos bt C cos(bt + I)
Frequency Response
Series resonance (RLC) Parallel resonance (RLC)
Resonant 1 1
fr = fr =
Frequency 2S LC 2S LC
WnL 1 1 R 1
Selectivity(Q) Q= Q = WnRC =
R 2[ WnRC WnL 2[
Bandwidth Wr R Wr 1
Bandwidth = rad/sec. Bandwidth = rad/sec.
(B.W.) Q L Q RC
2 2
Lower cuttoff R ⎛R ⎞ 1 1 ⎛ 1 ⎞ 1
= ⎜ ⎟ rad/sec. = ⎜ ⎟ rad/sec.
frequency 2L ⎝ 2L ⎠ LC 2RC ⎝ 2RC ⎠ LC
2
2 1 ⎛ 1 ⎞ 1
higher cuttoff R ⎛R ⎞ 1 = ⎜ ⎟ rad/sec
= ⎜ ⎟ rad/sec 2RC ⎝ 2RC ⎠ LC
frequency 2L ⎝ 2L ⎠ LC
At resonance VL = VC = Q u applied voltage IL = IC = Q u applied current
1113/GATE/EC/Compendium/Pg.6
Compendium of Important Results and Formulae (7)
Note: For Series resonance (RLC)
(a) Frequency at which max. voltage across Inductor.
1 1
fL
2S LC R2 C
1
2L
(b) Frequency at which max. voltage across capacitor.
1 1 R2
fC
2S LC 2L2
1
x For series RLC and parallel RLC circuits Bandwidth =
Timeconstant
L
? Series RLC has time constant = and parallel RLC has time constant = RC
R
Network functions & two part network
x InterRelationship between Parameters
Parameters [z] [y] [T] [h]
⎡ z11 z12 ⎤ 1 ⎡ y 22 y12 ⎤ 1 ⎡ A 'T ⎤ 1 ⎡ 'h h12 ⎤
[z] ⎢z ⎥ ⎢ ⎥ ⎢ h ⎥
⎣ 21 z22 ⎦ 'y ⎣ y 21 y11 ⎦ C ⎢⎣ 1 D ⎥⎦ h22 ⎣ 21 1 ⎦
1 ⎡ z22 z12 ⎤ ⎡ y11 y12 ⎤ 1 ⎡ D 'T ⎤ 1 ⎡ 1 h12 ⎤
[y] ⎢ ⎥ ⎢y ⎥ ⎢ ⎥
'z ⎣ z21 z11 ⎦ ⎣ 21 y 22 ⎦ B ⎢⎣ 1 A ⎥⎦ h11 ⎣h21 'h ⎦
1 ⎡z11 'z ⎤ 1 ⎡ y 22 1⎤ ⎡A B ⎤ 1 ⎡ 'h h11 ⎤
[T] ⎢ ⎥ ⎢
z21 ⎣ 1 z22 ⎦ y 21 ⎣ 'y y11 ⎥⎦ ⎢C D ⎥
⎣ ⎦
⎢
h21 ⎣h22 1 ⎦
⎥
1 ⎡ 'z z12 ⎤ 1 ⎡ 1 y12 ⎤ 1 ⎡ B 'T ⎤ ⎡h11 h12 ⎤
[h] ⎢ z ⎥ ⎢ ⎥ ⎢h ⎥
z22 ⎣ 21 1 ⎦ y11 ⎣ y 21 'y ⎦ D ⎢⎣ 1 C ⎥⎦ ⎣ 21 h22 ⎦
'X = X11X22 X12X21
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(8) Vidyalankar GATE EC
x Condition for Symmetry & Reciprocity
Name Equation Condition for Condition for
Reciprocity Symmetry
(i) Open circuit impedance V1 = z11I1 + z12I2
z12 = z21 z11 = z22
[z] V2 = z21I1 + z22I2
(ii) Short circuit admittance I1 = y11v1 + y12v2
y12 = y21 y11 = y22
[y] I2 = y21v1 + y22v2
(iii) Transmission or chain V1 = AV2 BI2
'T = 1 A=D
[T] I1 = CV2 DI2
(iv) Inverse Transmission V2 = A1V1 B1I1
'T1 = 1 A1 = D1
[T1] I2 = C1V1 D1I1
(v) Hybrid parameters V1 = h11I1 + h12V2
h12 = h21 'h = 1
[h] I2 = h21I1 + h22V2
(vi) Inverse hybrid I1 = g11V1 + g12I2
g12 = g21 'g = 1
[g] V2 = g21V1 + g22I2
1113/GATE/EC/Compendium/Pg.8
