ALTERNATING CURRENT ROOT MEAN Average value of

SQUARE CURRENT ac is defined for SINGLE COMPONENT CIRCUITS

“If the direction of current in a
resistor or any other element changes positive or negative
alternately, the current is called an half cycle
Irms= I2 = Io 2 = Io
alternating current” 2
2Io 2Vo
2 I= V=
Irms= Io Vrms= Vo
π π
Resistor only Inductor only Capacitor only
2 2 SUMMARY
AVERAGE AND RMS VALUE OF AC R L
i
i i
||
C Z (Impedance) l
O
If the current or voltage is SAWTOOTH FUNCTION
sinusoidal than it can be expressed as Io 1. R only R
IO For half cycle I= 0
i=i0sin(ωt+ ) 2
v=v0sin(ωt+ ) For full cycle I= 0 2. L only XL = ω L
i0 Peak current or current amplitude T/2 T Io 2 -π/2
Mean square current I= 2
3
XC= c_
1
v0 V=V0sin ωt
Peak voltage or voltage amplitude -IO
rms current Irms= Io V=V0sin ωt
V=V0sin ωt
3. C only
2π = 2πf 3 1. V=V0sin ωt ω π/2
ω= T:Time period 1. V=V0sin ωt
T 1. V=V0sin ωt
RECTANGULAR FUNCTION 2. i=i0sin (ωt- π/2 ) 2. i=i0sin (ωt+ π/2 )
f:frequency (Hz or cycle/sec)
I 2. i=i0sin ωt
(ωt+ ): Total phase +Io For half cycle I= Io 3. or current leads to the 3.Current leads the
3. V&i are in phase voltage by π/2 voltage by π/2
For full cycle I= 0
T/2 T

GENERAL GRAPH Mean square current I2= Io2 4. 4.

-Io
V 4. V
) I
rms current Irms= Io I π/2
if i=i0 sinωt
l =0,COS Ol =1
5. O
iO
AVERAGE HEAT PRODUCED DURING A P= εrmsIrms
CYCLE OF AC V
3T/4 T
1
T/4 T/2 t Havg= 2
Io R = Irms R
2 2 V,i )
π/2
I l π/2,COS Ol =0
5. O=
-iO Keep in mind V 6. P=0 (wattless circuits)
V0 l π/2,COS Ol =0
5. O=
⇒ rms value is also called virtual value or effective value I
i=i0 cosωt i0 6. P= 0 (wattless circuits) 7. Inductive reactance
⇒ AC ammeter and voltmeter always measure rms value
iO
t 7. Inductive reactance (XL) XC= c_
1
⇒ Values printed on ac circuits are rms values ω
T/2 XL=L ω Unit-ohm(Ω)
⇒ In houses ac is supplied at 220V which is the rms Unit-ohm(Ω)
T/4 3T/4 T plays role of
of voltage
plays role of resistance resistance
⇒ Peak value is 220√2= 311V V
-iO
6. i0= _0 & irms = Vrms V V
8. i0= _0 & irms = rms V Vrms
⇒ Frequency in general is 50Hz R
XL XL 8. i0= _0 & irms =
⇒ for measuring ac hot wire ⇒ w=2nf=100π rad/sec (314 rad/sec) X c
XC Xc
instruments are used

AVERAGE VALUE OF AC FOR ONE TIME PERIOD PHASOR DIAGRAM SERIES AC CIRCUITS
T T

∫ Idt ∫ I sin ωt dt Diagram representing ac voltage or 1) R-L CIRCUITS

I= 0
T =
0 O
T
= area of I-t graph current as vectors with phase angle 5. Impedance phasor 2. Voltage phasor diagram
∫ dt ∫ dt time v R L XL i0
between them. Z= R2+l2 z VR
0 0
0

V=√VR2+Vc2 )
I0
I= 0 for 0→T for a sinusoidal ac wave.
VR VL XL l
O
The average value of sin or cos function for I=I0sin(ωt)
l _
tan O=
[
one time period or n time periods (n=1,2...) is zero i R Vc [
v=v0sin(ωt+ )
wt+ l _
tan O=
VR
Keep in mind v=vosinwt V
wt 6. i0= _0
Long period is equivalent to one time period Z 3. i=io sin(ωt+ )
1. V=V0sin ωt VR=i0R, VL=i0X L
l
)O
O R VC V
AVERAGE POWER CONSUMPTION
Mean square current for one time Period vo 2. Voltage phasor diagram 2) R-C CIRCUITS 4. Impedance phasor
T
VL R
∫I2dt
I2 = 0 T =
I02 V=√V +VL2 2
V C R
R
Z = R 2 + X C2
||
[
∫ dt 2 Io VL [
l _
tan O= VR XC
0

Remember VR tan =
The average value of square of sin or cosine VC R
1
2
l
3. i=i0sin (ωt-O) XC Z
function for one time period is 2 1
V= Vosinwt Vo
4. _
Vo = V_ =Z=√R2+XL2 l 5. io =
rms
P= VI
∫T sin2kωt = 1 x T Pavg= Vrms Irms cos io irms )O
0
2 VR Z
for cos =1 or =0o i0
∫nTsin2kωt = n x T
Pavg= Vrms Irms 1. V=V0sin ωt VR=i0 R, VC =i0 X C or irms = Vrms
0
2

Z
 3) L-C CIRCUIT RESONANCE IN LCR SERIES CIRCUIT
COMPARISON OF LC OSCILLATION WITH A MASS
QUALITY FACTOR
R1>R2>R3 SPRING SYSTEM
ωr L 1 1 L
i L C In series resonance, impedance of circuit is Q= _ = _ = _ _
R ωr cR R C
minimum & equal to resistance = Z= R, and Mass spring system LC Circuit
[
V = VO sin t
curent is maximum Voltage across C or L [
VL VC or Q= _
~ VL = iO XL , VC = iO XC applied voltage 1.Displacement (x) 1. Charge (q)
Condition for resonance resonance
V=Vosinωt
1 Less sharp the resonance, less is the selectivity 2.Velocity V= _dx 2. Current I= _ dq
dt
Voltage phasor diagram XL = XC L = dt _
C of the circuit.If the Quality factor is large, R dv 3. Rate of change of
V = VL ~ VC [ie, (VL - VC) or (VC - VL)] 1
3. Acceleration a= dt
is low or L is large, the circuit is more selective.
= r = rad / sec
current= (_
dI
4. Mass (m),(inertia)
(
VL XL LC dt
Impedance Phasor Diagram
r resonant frequency (angular) 5. Force constant K 4. Inductance (L), inertia of circuit
Sharpness of Resonance
i
io
1 6. Momentum p = mv 5. Capacitance (C)
io f = fr = ωr 7. Retarding force -m _
dv
Sharpness= Q= _
Hz
Z = XL ~ XC [ (XL - XC) 2 LC ; 2∆ω -bandwidth 6. Magnetic flux O l =LI
2∆ω dt
7. Self induced emf (-L _
VC XC or (XC - XL)] dI (
fr = resonant frequency smaller∆ω, sharper or narrower the 8. Differential equation dt
if XL > Xc , Voltage leads the current by GRAPH
resonance. _
d2x + ω2x=0 8. Differential equation
2
io dt2 d2 q
if XC > XL , current leads the voltage by
2 _ + ω2q=0
if XL = XC , Z = 0, i = POWER IN AC CIRCUIT ω=√ _
k
m
dt2
ω=√ _ 1
w<wr
or, r, XC > XL
i Z w>wr <
ωL = 1 l
Average Power P= Vrms Irms cos O _
1 LC
current leads 9. K.E= 2 mv2 _ LI2
ωC r
Case 1 P= I
2
l
Zcos O 9. Magnetic energy= 1
ω = 1 = ωo Variation of peak current with applied > r, XL > XC rms _
Elastic U =1 kx2 2 2
2 q
_
LC
frequency
current lags l =0
Purely Resistive circuit - O l =1
,COS O Elastic U = 2C
o ωo
In resonance Maximum power dissipation
L-C-R Series Circuit Case 2 TRANSFORMERS
L C R V = VR (applied voltage = voltage across
resistance) Purely inductive or capacitive circuit- “Device which raises or lowers
VL VC VR Z = R (impedance is minimum and equal to l =90
O 0
l
cos O=0 voltage in ac circuits through mutual NP NS
output
Input
resistance) No power is dissipated even though a induction”.Transformer can increase
i V=Vosinωt or decrease voltage or secondary coil

Voltmeter connected across VL & VC will show current is flowing in the circuit current but not both simultaneously.
~ Primary coil
the same reading Case 3
V = VO sin t
Voltmeter connected commonly across inductor LCR Series circuit
VR = iOR, VL = iO XL , VC = iOX C & capacitor shows no reading l non zero in R-L,C-R,or CLR circuit.
O soft iron core
Assuming VL > VC for drawing phasor Here V = 0
P=Vrms Irms Cos O
l EQUATIONS
Voltage phasor diagram V
VL = VC Vs -Voltage in secondary
VL VC VR
V N IP Vp -Voltage in primary
VL
VL - VC VN = VR Case 4 1) _S = _S = _
VP NP IS Ns -No of turns in secondary
VR Power dissipation at resonance Pout V I
io Vnet = VR 2) Efficiency η= _ = _
S S Np -No of turns in primary
VR imax = l
XL-XC=0 or O=0 l
=> cos O=1 ⇒Z=R P in
IVP P Ip -Current in primary
~ Z R
P=I Z = I R
2 2 3) For ideal transformer, η=1 Is -Current in secondary
VC
VR, io Vnet Maximum power is dissipiated in a circuit
at resonance.
V = VR2 + (VL - VC) 2 TRANSFORMER TYPES LOSSES IN TRANSFORMER
Here i = io sin ( t - ) LC OSCILLATIONS step up step down 1) Cu loss (I2R loss)
→ To minimise, windings are made of thick
(since VL is leading ) transformer transformer Cu wires (high resistance)
I
APPLICATION OF RESONANT
Impedance Triangle
CIRCUIT
Capacitor stores 2) Eddy current loss
c → To minimise Cores are laminated
XL - XC XL - XC electrical energy O/P
O/P

Z Z Tuning mechanism of a radio or TV set q L I/P

I/P 3) Hysteresis loss
1.Antenna of radio accepts signals Inductor stores → select material of narrow hysteresis loop
2.Signal acts as an AC source in tuning the radio magnetic energy → Cores of transformer is made of soft iron
R,io R
XL - XC
3.In tuning, capacitance of capacitor is NP < NS NP > NS 4) Magnetic flux linkage
tan = varied such that the resonant frequency < > → To minimise, secondary winding is kept
Z = R + ( XL - XC )
2 2
R of the circuit becomes nearly equal to the When connected, charge on the VP VS VP VS
inside the primary winding
frequency of the radio signal received. capacitor and current in the inductor IP > IS IP < IS
5) Humming loss
So, the simple is largely amplified and perform electrical oscillations RP < RS RP > RS

distinctly heard between each other.