Alternating Currents

Summary
Alternating Current

• AC and DC Current:
A current that changes its direction periodically is called alternating current (AC). If a current
maintains its direction constant it is called direct current (DC).

• Average value:
t2

Average value of a function, from t1 to t2 is defined as < f >=

∫ t1
f .dt
t 2 − t1
• Root Mean Square Value:
t2

Root Mean Square Value of a function, from t1 to t2 is defined as f rms =

∫
t1
f .dt
t 2 − t1
• Power Consumed of Supplied in an ac Circuit:
instantaneous power P consumed by the device= Vi
= ( Vm sin ωt ) ( Im sin ( ωt + φ ) )

2π

Average powder consumed in a=

cycle
∫=
o
Pdt
ω
1
Vm I m cos φ
2π 2
ω
Vm I m
= .= .cos φ Vrms I rms cos φ.
2 2
Here cos φ is called power factor.
• Some Definitions:
The factor cos φ is called power factor.
I m sin φ is called wattless current

447
@aakashallen
 Alternating Currents

Vm Vrms
Impedance Z is defined as=
Z =
Im I rms
ωL is called inductive reactance and is denoted by XL
1
is called capacitive reactance and is denoted by XC
ωL
• Purely Resistive Circuit:

Vs Vm sin ωt
I
= = = I m sin ωt
R R
V V
I m = m ⇒ I rms = rms
R R
2
Vrms
= < P > Vrms=
I rms cos φ
R
• Purely Capacitive Circuit:

dq d ( CV ) d ( CVm sin ωt ) Vm Vm
I= = = = CVm ω cos ωt= cos ωt= cos ωt= I m cos ωt
dt dt dt 1/ ωC XC
1
XC = and is called capacitive reactance.
ωC

IC leads VC by π / 2 Diagrammatically (phasor diagram) it is represented as

.
Since φ= 90°, < P >= Vrms I rms cos φ= 0 .
• RLC Series Circuit with An ac Source:

@aakashallen
448
 Alternating Currents

From the phasor diagram

( IR ) + ( IX L − IX C ) = ( R ) + ( X L − X C ) = ( R ) + ( XL − XC )
2 2 2 2 2 2
V= IZ Z=
I ( XL − XC ) ( XL − XC )
=tan φ =
IR R
• Resonance:
Amplitude of current (and therefore Irms also) in an RLC series circuit is maximum for a given
value of Vm and R , if the impedance of the circuit is minimum, which will be when XL-XC =
0. This condition is called resonance. So at resonance: XL-XC =0.
1 1
Or ωL
= = or ω Let us denote this ω as ωr
ωC LC

XL XC
Quality factor: =
Q =
R R
Re sonance freq. ωR fR
Q
= = =
Band width R f 2 − f1
where f1 & f2 are half power frequencies.
• Transformer;
A transformer changes an alternating potential difference from one value to another of greater
or smaller value using the principle of mutual induction. For an ideal transformer
Es Ns Ip
= = where denotations have their usual meanings. ESN and I are the emf, number
E p N p Is
of turns and current in the coils.

NS > N P ⇒ ES > E P → step up transformer.

NS < N P ⇒ ES < E P → step dpwn transformer.

449
@aakashallen
Alternating Currents

Energy Losses ln Transformer are due to

1. Resistance of the windings
2. Eddy Current.
3. Hysteresis.
4. Flux Leakage.

@aakashallen
450
 Alternating Currents

Practical Questions
ALTERNATING CURRENT

1. In an a.c. circuit, the instantaneous e.m.f. and current are given by

 π
=e 100 sin 30 t = and i 20sin  30t −  .
 4
In once cycle of a.c., the average power consumed by the circuit and the wattles current
are, respectively : (2018)
(a) 50, 0
(b) 50, 10
1000
(c) ,10
2
50
(d) ,0
2

1
2. For an RLC circuit driven with voltage of amplitude Vm and frequency ω0 = the
LC
current exhibits resonance. The quality factor, Q is given by :
(2018)
CR
(a)
ω0
ω0 L
(b)
R
ω0 R
(c)
L
R
(d)
(ω0C )

3. An ideal capacitor of capacitance 0.2 μF is charged to a potential difference of

10 V. The charging
battery is then disconnected. The capacitor is then connected to an ideal inductor
of self inductance
0.5 mH. The current at a time when the potential difference across the capacitor
is 5V is : (2018)
(a) 0.15 A
(b) 0.17 A
(c) 0.34 A
(d) 0.25 A

@aakashallen
451
Alternating Currents

4. A power transmission line feeds input power at 2300V to a step-down

transformer with its primary
windings having 4000 turns giving the output power at 230V. If the current in the
primary of the
transformer is 5A and its efficiency is 90%, then the output current would be :
(2018)
(a) 45A
(b) 50A
(c) 20A
(d) 25A

5. A pure inductance of 1 H is connected across a 110 V, 70 Hz source. Find the

reactance, and the peak value of current.
(a) 439.6 Ω, 0.354 A
(b) 489.7 Ω, 0.423 A
(c) 391.2 Ω, 0372 A
(d) 350.0 Ω, 0.472 A

6. The average current of sinusoidally varying alternating current of peak value 5 A and
T T
initial phase zero, between the instants t = to t = is
8 4
14.14
(a) A
π
17.32
(b) A
π
12.33
(c) A
π
10.00
(d) A
π

7. When a 100 V dc is applied across a coil, a current of 1 A flows through it. When a 100
V ac of frequency 50 Hz is applied to the same coil, the current is 0.5 A. The inductance
of the coil is
(a) 0.75 H
(b) 10.65 H
(c) 0.55 H
(d) 10.45 H

8. A 50 W, 100 V electric lamp is to be operated on a 200 V, 50 Hz electric mains. The

capacitance required in series with the lamp is

452
@aakashallen
Alternating Currents

(a) 7.2 μF
(b) 8.2 μF
(c) 9.2 μF
(d) 10.2 μF

9. An alternating emf of frequency 50 Hz is applied to a series circuit of resistance 20 Ω,

an inductance of 100 mH and a capacitance of 30 μF. Then
(a) voltage leads the current by tan-1 (3.73)
(b) current leads the applied voltage by tan-1 (3.73)
(c) voltage leads the current by tan-1 (1.73)
(d) current leads the applied voltage by tan-1 (1.73)

10. A capacitor of 10 μF and an inductor of 1H are joined in series. An ac of 50 Hz is

applied to this combination. The impedance of the combination is
(a) 5.67 Ω
(b) 4.16 Ω
(c) 7.02 Ω
(d) 3.21 Ω

11. A 60 Hz, 230 V rms voltage is applied on an inductance of 0.265 H. At t = 0, voltage is

also zero. The equation for voltage is:
(a) V = 230sin12π t
(b) V = 230 2 sin 60π t
230
(c) V = sin 60π t
2
(d) V = 230 2 sin120π t

12. An LCR series circuit with 100 Ω resistance is connected to an ac source of 200 V and
angular frequency 300 rad s-1. When only the capacitance is removed, the current lags
behind the voltage by 600. When only the inductance is removed, the current leads the
voltage by 600. Calculate the current in LCR circuit.
(a) 2 A
(b) 2.5 A
(c) 3 A
(d) 1.75 A

13. A series LCR circuit containing a resistance of 120 Ω has resonance frequency 4 × 105
rad s-1. The voltages, at resonance, across the resistance and inductance are 60 V and 40
V respectively. The values of L and C respectively are

453
@aakashallen
Alternating Currents

(a) 0.3 mH, 0.0195 μF

(b) 0.1 mH, 0.4525 μF
(c) 0.2 mH, 0.03125 μF
(d) 0.4 mH, 0.5125 μF

14. An LCR circuit has inductance 10 mH, resistance 3 Ω and capacitance 1 μF connected
in series to a source of V = 15 cos ωt volts. The current amplitude, at a frequency that is
10 % lower than the resonance frequency, is
(a) 0.611 A
(b) 0.523 A
(c) 0.704 A
(d) 0.821 A

15. A box B and a coil K are connected in series, with an ac source of variable frequency.
The peak emf of the source is a constant and is 10 V. Box B contains a capacitance of 1
μF in series with a resistance of 32 Ω. Coil K has self inductance 4.9 mH and a resistance
of 68 Ω. The frequency is adjusted so that the maximum current flows in B and K. Find
the impedance of B at this frequency
(a) 88 Ω
(b) 99 Ω
(c) 66 Ω
(d) 77 Ω

16. A current of 4 A flows in a coil when connected to a 12 V dc source. If the same coil
is connected to a 12 V ac source (ω = 50 rad s-1), a current of 2.4 A flows in the circuit.
The inductance of the coil is
(a) 80 mΩ
(b) 90 mΩ
(c) 60 mΩ
(d) 72 mΩ

17. A choke coil is needed to operate an arc lamp at 160 V (rms) and 50 Hz. The arc lamp
has an effective resistance of 5 Ω when running at 10 A (rms). The inductance of the
choke coil is
(a) 67.8 mH
(b) 48.4 mH
(c) 71.2 mH
(d) 53.5 mH

454
@aakashallen
Alternating Currents

18. An ac circuit draws 550 W power from a 220 V, 50 Hz source. The power factor is
0.8, with the current lagging behind the applied voltage. The capacitance C required to
be connected in series with this circuit, to increase the power factor to 1.0, is
(a) 85.4 μF
(b) 75.4 μF
(c) 65.4 μF
(d) 55.4 μF

19. A 750 Hz, 20 V source is connected to a resistance of 100 Ω, an inductance of 0.1803

H and capacitance of 10 μF all in series. If thermal capacity of resistor is 2 J0 C-1, the time
in which the resistor will get heated by 100 C is
(a) 5.8 min
(b) 6.8 min
(c) 4.8 min
(d) 7.8 min

20. An L-C circuit (L = 0.01 H, C = 1 μF) is connected to a variable frequency source as

shown in the figure. The best sketch that represents the current variation as the
frequency is changed from 1 kHz to 2 kHz is

(a)

(b)

(c)

455
@aakashallen
Alternating Currents

(d)

21. For the circuit shown in the figure, current in the inductor is 0.8 A and that in the
capacitor is 0.6 A (both peak values). The current (peak) drawn from the source is

(a) 0.1 A
(b) 0.2 A
(c) 0.3 A
(d) 0.4 A

2 800
22. A series LCR circuit contains a 50 Ω resistance, H inductance and μF
π 19π
capacitance connected across a 220 V, 50 Hz ac source. The net reactance of the circuit
is
(a) 200 Ω
(b) 237.5 Ω
(c) 37.5 Ω
(d) 50 Ω

 NS 
23. The transformer ratio   of a transformer is 50. The input power across the
 NP 
primary at 220 V is 17.6 kW. The resistance of the secondary coil is 3.7 Ω. If the
efficiency is 80%, the rate of heat loss in the secondary is
(a) 4.06 W
(b) 7.06 W

456
@aakashallen
Alternating Currents

(c) 5.06 W
(d) 6.06 W

24. Which of the following graphs shown, correctly represents the variation of
impedance Z of an L - R circuit with frequency (v) of applied voltage?
(a)

(b)

(c)

(d)

25. The variation of voltage across a pure capacitor with time is represented in the
figure. Which of the graphs shown represents the variation of current with time?

(a)

(b)

457
@aakashallen
Alternating Currents

(c)

(d)

26. The equations of voltage and current corresponding to a series L-C-R circuit are
π
and i 5  3000t −  ampere respectively. If the inductance is
V = 141.4 sin (3000 t) volt=
 4
10 mH, then, the resistance is
(a) 20 Ω
(b) 10 Ω
(c) 30 Ω
(d) 40 Ω

27. A current depends on time (t) as i = i1 sin ωt + i2 cosωt . What is its rms value for one
complete cycle ?
(a) i1i2
i12 + i22
(b)
2
i12 + i22
(c)
3
i12 − i22
(d)
2

28. The figure shows three circuits with identical batteries, inductors (L), and resistors
(R). Rank the circuit in the decreasing order, of the current through the battery

458
@aakashallen
 Alternating Currents

(i) Just after the switch is closed and

(ii) A long time after the switch is closed
(a) ( i ) ib > ic =
> ia ( ia 0 )       ( ii )=
ib > ic ia
(b) ( i ) ib > ic < ia (ia ≠ 0)   ( ii ) ib > ic > ia
(c) ( i ) ib = ic = ia (ia = 0)    ( ii ) ib < ic < ia
(d) ( i ) ib = ic > ia (ia ≠ 0)    ( ii ) ib > ic > ia

29. In an oscillating LC circuit [L = 50 mH and C = 4 μF], the current is initially maximum.

How long will it take before the capacitor is fully discharged for the first time?
(a) 0.5 ms
(b) 0.6 ms
(c) 0.7 ms
(d) 0.8 ms

1
30. The power factor of a series RL circuit is . If the frequency of the applied ac is
2
doubled, the power factor becomes
1
(a)
3
1
(b)
5
1
(c)
7
1
(d)
11

@aakashallen
459
 Alternating Currents

ANSWER KEYS
1. (c) 2. (b) 3. (b) 4. (a) 5. (a) 6. (a) 7. (c) 8. (c) 9. (b) 10. (b) 11. (d) 12. (a) 13. (c) 14. (c) 15.
(d) 16. (a) 17. (b) 18. (b) 19. (a) 20. (d) 21. (b) 22. (c) 23. (d) 24. (d) 25. (a) 26. (a) 27. (b)
28. (a) 29. (c) 30. (b)

“Detail solutions are mentioned in the content library”

460
@aakashallen