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CHAPTER 17 SINGLE-PHASE SERIES A.C. CIRCUITS
Exercise 108, Page 299
1. Calculate the reactance of a coil of inductance 0.2 H when it is connected to (a) a 50 Hz,
(b) a 600 Hz and (c) a 40 kHz supply.
(a) Inductive reactance, = 62.83
(b) Inductive reactance, = 754
(c) Inductive reactance, = 50.27 k
2. A coil has a reactance of 120 in a circuit with a supply frequency of 4 kHz. Calculate the
inductance of the coil.
hence, inductance, L = = 4.77 mH
3. A supply of 240 V, 50 Hz is connected across a pure inductance and the resulting current is 1.2 A.
Calculate the inductance of the coil.
Inductive reactance, = 200 Ω
hence, inductance, L = = 0.637 H
4. An e.m.f. of 200 V at a frequency of 2 kHz is applied to a coil of pure inductance 50 mH.
Determine (a) the reactance of the coil, and (b) the current flowing in the coil.
(a) Inductive reactance, = 628
(b) Current, I = = 0.318 A
© John Bird Published by Taylor and Francis 268
5. A 120 mH inductor has a 50 mA, 1 kHz alternating current flowing through it. Find the p.d.
across the inductor
Inductive reactance, = 753.982
P.d. across inductor, = 37.7 V
6. Calculate the capacitive reactance of a capacitor of 20 μF when connected to an a.c. circuit of
frequency (a) 20 Hz, (b) 500 Hz, (c) 4 kHz
(a) Capacitive reactance, = 397.9
(b) Capacitive reactance, = 15.92
(c) Capacitive reactance, = 1.989
7. A capacitor has a reactance of 80 when connected to a 50 Hz supply. Calculate the value of
the capacitor.
Capacitive reactance, from which, capacitance, C =
= 39.79 F
8. Calculate the current taken by a 10 F capacitor when connected to a 200 V, 100 Hz supply.
Capacitive reactance, = 159.155
Current, I = = 1.257 A
© John Bird Published by Taylor and Francis 269
9. A capacitor has a capacitive reactance of 400 when connected to a 100 V, 25 Hz supply.
Determine its capacitance and the current taken from the supply.
Capacitive reactance, from which, capacitance, C =
= 15.92 F
Current, I = = 0.25 A
10. Two similar capacitors are connected in parallel to a 200 V, 1 kHz supply. Find the value of
each capacitor if the current is 0.628 A.
= 318.47
i.e. , hence, total capacitance, = 0.50 F
Since for parallel connection of capacitors, , then = 0.25 F
i.e. each capacitor has a capacitance of 0.25 F
© John Bird Published by Taylor and Francis 270
Exercise 109, page 302
1. Determine the impedance of a coil which has a resistance of 12 and a reactance of 16
Impedance, Z = = 20
2. A coil of inductance 80 mH and resistance 60 is connected to a 200 V, 100 Hz supply.
Calculate the circuit impedance and the current taken from the supply. Find also the phase angle
between the current and the supply voltage.
Inductive reactance, = 50.265
Impedance, Z = = 78.27 (see impedance triangle in the diagram below)
Current, I = = 2.555 A
From the impedance triangle,
hence the circuit phase angle, = = 39.95 lagging
3. An alternating voltage given by v = 100 sin 240t volts is applied across a coil of resistance 32
and inductance 100 mH. Determine (a) the circuit impedance, (b) the current flowing, (c) the p.d.
across the resistance, and (d) the p.d. across the inductance.
(a) Inductive reactance, = 24
© John Bird Published by Taylor and Francis 271
Circuit impedance, Z = = 40
(b) Current flowing, I = = 1.77 A (Note r.m.s. current = 0.707 maximum value)
(c) P.d. across the resistance, = 56.64 V
(d) P.d. across the inductance, = 42.48 V
4. A coil takes a current of 5 A from a 20 V d.c. supply. When connected to a 200 V, 50 Hz a.c.
supply the current is 25 A. Calculate the (a) resistance, (b) impedance and (c) inductance of the
coil.
(a) From a d.c. circuit, resistance, R = = 4
(b) From an a.c. circuit, impedance, Z = = 8
(c) From the impedance triangle,
from which, = 6.9282
Also, from which, inductance, L = = 22.05 mH
5. A resistor and an inductor of negligible resistance are connected in series to an a.c. supply. The
p.d. across the resistor is 18 V and the p.d. across the inductor is 24 V. Calculate the supply voltage
and the phase angle between voltage and current.
Supply voltage, V = = 30 V
© John Bird Published by Taylor and Francis 272
Tan ϕ = from which,
phase angle between voltage and current, ϕ = = 53.13º lagging (i.e. current lags voltage in an inductive circuit)
6. A coil of inductance 636.6 mH and negligible resistance is connected in series with a 100
resistor to a 250 V, 50 Hz supply. Calculate (a) the inductive reactance of the coil, (b) the
impedance of the circuit, (c) the current in the circuit, (d) the p.d. across each component, and
(e) the circuit phase angle.
The circuit is shown in the diagram below.
(a) Inductive reactance of coil, = 200
(b) Impedance, Z = = 223.6 (from impedance triangle)
(c) Current, I = = 1.118 A
(d) Voltage across resistance, = 111.8 V
Voltage across inductance, = 223.6 V
(e) From impedance triangle,
© John Bird Published by Taylor and Francis 273
from which, circuit phase angle, = = 63.43 lagging
Exercise 110, page 304
1. A voltage of 35 V is applied across a C-R series circuit. If the voltage across the resistor is 21 V, find the
voltage across the capacitor.
Supply voltage, V = i.e.
i.e.
from which, voltage across the capacitor, = 28 V
2. A resistance of 50 is connected in series with a capacitance of 20 F. If a supply of 200 V,
100 Hz is connected across the arrangement find (a) the circuit impedance, (b) the current
flowing, and (c) the phase angle between voltage and current.
The circuit diagram is shown below.
(a) Capacitive reactance, = 79.577
Impedance, Z = = 93.98
(b) Current, I = = 2.128 A
© John Bird Published by Taylor and Francis 274
1 1LX 200tan tanR 100
(c) from which, phase angle, = = 57.86 leading
3. A 24.87 μF capacitor and a 30 resistor are connected in series across a 150 V supply. If the current
flowing is 3 A find (a) the frequency of the supply, (b) the p.d. across the resistor and (c) the p.d. across the
capacitor.
(a) Impedance, Z = = 50 Ω
Also, impedance, Z = i.e. 50 =
from which, and = 40 Ω
Capacitive reactance, from which,
frequency, f = = 160 Hz
(b) P.d across the resistor, = 90 V
(c) P.d across the capacitor, = 120 V
4. An alternating voltage v = 250 sin 800t volts is applied across a series circuit containing a 30
and 50 F capacitor. Calculate (a) the circuit impedance, (b) the current flowing, (c) the p.d.
across the resistor, (d) the p.d. across the capacitor, and (e) the phase angle between voltage and
current.
The circuit is shown below.
© John Bird Published by Taylor and Francis 275
(a) Capacitive reactance, = 25
Impedance, Z = = 39.05
(b) Current, I = = 4.526 A
(c) P.d across the resistor, = 135.8 V
(d) P.d across the capacitor, = 113.2 V
(e) from which, phase angle, = = 39.81 leading
5. A 400 resistor is connected in series with a 2358 pF capacitor across a 12 V a.c. supply.
Determine the supply frequency if the current flowing in the circuit is 24 mA.
The circuit is shown below.
Impedance, Z = = 500
From the impedance triangle (as in the previous problem),
© John Bird Published by Taylor and Francis 276
from which, capacitive reactance,
Hence, 300 =
from which, supply frequency, f = = 225 kHz
Exercise 111, Page 307
1. A 40 μF capacitor in series with a coil of resistance 8 and inductance 80 mH is connected to a 200 V,
100 Hz supply. Calculate (a) the circuit impedance, (b) the current flowing, (c) the phase angle between
voltage and current, (d) the voltage across the coil, and (e) the voltage across the capacitor.
The circuit diagram is shown below.
(a) Inductive reactance,
Capacitive reactance,
Impedance, Z = = 13.18
(b) Current flowing, I = = 15.17 A
© John Bird Published by Taylor and Francis 277
(c) from which,
phase angle, = = 52.63 lagging
(d)
Voltage across coil, = 772.1 V
(e) Voltage across capacitor, = 603.6 V
2. Find the values of resistance R and inductance L in the circuit shown.
Circuit impedance, Z =
Hence, resistance, R = 131
and Now, = 79.577
Hence, from which,
i.e. and inductance, L = = 0.545 H
3. Three impedances are connected in series across a 100 V, 2 kHz supply. The impedances
comprise: (i) an inductance of 0.45 mH and resistance 2
(ii) an inductance of 570 H and 5 resistance, and
© John Bird Published by Taylor and Francis 278
(iii) a capacitor of capacitance 10 F and resistance 3
Assuming no mutual inductive effects between the two inductances calculate (a) the circuit
impedance, (b) the circuit current, (c) the circuit phase angle and (d) the voltage across each
impedance. Draw the phasor diagram.
The circuit is shown below.
Total resistance, = 2 + 5 + 3 = 10
Total inductance,
The simplified circuit is shown below.
Inductive reactance,
Capacitive reactance,
(a) Impedance, Z = = 11.12
(b) Current, I = = 8.99 A
© John Bird Published by Taylor and Francis 279
(c) from which,
phase angle, = = 25.92 lagging
(d)
Voltage across first impedance, = 53.92 V
Voltage across second impedance, = 78.53 V
from earlier
Voltage across third impedance, = 76.46 V
4. For the circuit shown below determine the voltages and if the supply frequency is 1 kHz.
Draw the phasor diagram and hence determine the supply voltage V and the circuit phase angle.
and lagging
Voltage, = = 26.0 V at 67.38 lagging
© John Bird Published by Taylor and Francis 280
and leading
Voltage, = = 67.05 V at 72.65 leading
The voltages are shown in the phasor diagram (i) below.
(i) (ii)
The supply voltage V is the phasor sum of voltages and . V is shown by the length ac in
diagram (ii).
In triangle abc, b = 180 - 72.65 – 67.38 = 39.97
Using the cosine rule,
from which, ac = 50 V
Using the sine rule, from which,
from which, and from diagram (ii), leading.
Hence, supply voltage, V = 50 V at 53.14 leading
© John Bird Published by Taylor and Francis 281
Exercise 112, Page 310
1. Find the resonant frequency of a series a.c. circuit consisting of a coil of resistance 10 and inductance
50 mH and capacitance 0.05 μF. Find also the current flowing at resonance if the supply voltage is 100 V.
Resonant frequency, = 3.183 kHz
At resonance, current, I = = 10 A
2. The current at resonance in a series L-C-R circuit is 0.2 mA. If the applied voltage is 250 mV at
a frequency of 100 kHz and the circuit capacitance is 0.04 F, find the circuit resistance and
inductance.
At resonance, current, I = i.e. resistance, R = = 1.25 k
At resonance, resonant frequency, i.e. and © John Bird Published by Taylor and Francis 282
Hence, inductance, L = = 63.3 H
3. A coil of resistance 25 and inductance 100 mH is connected in series with a capacitance of
0.12 F across a 200 V, variable frequency supply. Calculate (a) the resonant frequency, (b) the
current at resonance and (c) the factor by which the voltage across the reactance is greater than
the supply voltage.
(a) Resonant frequency, = 1.453 kHz
(b) At resonance, current, I = = 8 A
(c) Q-factor = = 36.51
4. A coil of 0.5 H inductance and 8 resistance is connected in series with a capacitor across a 200 V, 50 Hz
supply. If the current is in phase with the supply voltage, determine the capacitance of the capacitor and the
p.d. across its terminals.
If the current is in phase with the supply voltage, then the circuit is resonant.
At resonance, i.e. from which,
capacitance, C = = 20.26 μF
P.d. across the capacitor terminals,
= 3928 V = 3.928 kV
5. Calculate the inductance which must be connected in series with a 1000 pF capacitor to give a resonant
© John Bird Published by Taylor and Francis 283
frequency of 400 kHz.
Resonant frequency,
from which, and
and LC = and inductance, L =
= 158 H or 0.158 mH
6. A series circuit comprises a coil of resistance 20 and inductance 2 mH and a 500 pF capacitor.
Determine the Q-factor of the circuit at resonance. If the supply voltage is 1.5 V, what is the
voltage across the capacitor?
Q-factor = = 100
Q = hence, voltage across the capacitor, = 150 V
© John Bird Published by Taylor and Francis 284
Exercise 113, Page 314
1. A voltage v = 200 sin t volts is applied across a pure resistance of 1.5 k. Find the power dissipated in the
resistor.
Power dissipated in the resistor, P =
Current, I = = 0.09428 A (note that in the formula for power I has to be the r.m.s. value)
Hence, power dissipated = = 13.33 W
2. A 50 F capacitor is connected to a 100 V, 200 Hz supply. Determine the true power and the
apparent power.
Capacitive reactance, = 15.915
© John Bird Published by Taylor and Francis 285
Current, I = = 6.283 A
True power, P = V I cos = (100)(6.283) cos 90 = 0
Apparent power, S = V I = (100)(6.283) = 628.3 VA
3. A motor takes a current of 10 A when supplied from a 250 V a.c. supply. Assuming a power
factor of 0.75 lagging find the power consumed. Find also the cost of running the motor for 1
week continuously if 1 kWh of electricity costs 12.20 p.
P = V I cos = (250)(10)(0.75) since power factor = cos
= 1875 W
Energy = power time = (1.875 kW)(7 24) = 315 kWh
Hence, cost of running motor for 1 week = 315 12.20 = 3843 p = £38.43
4. A motor takes a current of 12 A when supplied from a 240 V a.c. supply. Assuming a power
factor of 0.70 lagging find the power consumed.
Power consumed, P = V I cos = (240)(12)(0.70) since power factor = cos
= 2016 W or 2.016 kW
5. A transformer has a rated output of 100 kVA at a power factor of 0.6. Determine the rated power output and
the corresponding reactive power.
VI = 100 kVA = 100 × 103 and p.f. = 0.6 = cos
Power output, P = VI cos = (100 × 103)(0.6) = 60 kW
Reactive power, Q = VI sin
If cos = 0.6, then = cos 0.6 = 53.13
Hence sin = sin 53.13o = 0.8
© John Bird Published by Taylor and Francis 286
Hence reactive power, Q = (100 × 103)(0.8) = 80 kvar
6. A substation is supplying 200 kVA and 150 kvar. Calculate the corresponding power and power
factor.
Apparent power, S = V I = VA and reactive power, Q = V I sin = var
Hence, sin = from which, sin =
and = = 48.59
Thus, power, P = V I cos = cos 48.59 = 132 kW
and power factor = cos = cos 48.59 = 0.66
7. A load takes 50 kW at a power factor of 0.8 lagging. Calculate the apparent power and the reactive power.
True power P = 50 kW = VI cos and power factor = 0.8 = cos
Apparent power, S = VI = = = 62.5 kVA
Angle = cos 0.8 = 36.87o hence sin = sin 36.87o = 0.6
Hence, reactive power, Q = VI sin = 62.5 × 103 × 0.6 = 37.5 kvar
8. A coil of resistance 400 and inductance 0.20 H is connected to a 75 V, 400 Hz supply.
Calculate the power dissipated in the coil.
Inductive reactance, = 502.65
Impedance, Z = = 642.39
Current, I = = 0.11675 A
From the impedance triangle, and = 51.49
© John Bird Published by Taylor and Francis 287
Hence, power dissipated in coil, P = V I cos = (75)(0.11675) cos 51.49 = 5.452 W
Alternatively, P = = 5.452 W
9. An 80 resistor and a 6 μF capacitor are connected in series across a 150 V, 200 Hz supply. Calculate
(a) the circuit impedance, (b) the current flowing and (c) the power dissipated in the circuit.
(a) Capacitive reactance, = 132.63
Impedance, Z = = 154.9
(b) Current, I = = 0.968 A
(c) From the impedance triangle, and = 58.90
Hence, power dissipated in coil, P = V I cos = (150)(0.968) cos 58.90 = 75 W
Alternatively, P = = 75 W
10. The power taken by a series circuit containing resistance and inductance is 240 W when
connected to a 200 V, 50 Hz supply. If the current flowing is 2 A find the values of the
resistance and inductance.
Power, P = i.e. 240 = from which, resistance, R = = 60
Impedance, Z = = 100
From the impedance triangle,
from which, = 80
i.e. 2 f L = 80 from which, inductance, L = = 0.255 H or 255 mH
© John Bird Published by Taylor and Francis 288
11. The power taken by a C-R series circuit, when connected to a 105 V, 2.5 kHz supply, is 0.9 kW and the
current is 15 A. Calculate (a) the resistance, (b) the impedance, (c) the reactance, (d) the capacitance,
(e) the power factor, and (f) the phase angle between voltage and current.
(a) Power, P = i.e. from which, resistance, R = = 4
(b) Impedance, Z = = 7
(c) From the impedance triangle,
from which, capacitive reactance, = 5.745
(d) Capacitive reactance, i.e. 5.745 =
from which, capacitance, C = = 11.08 μF
(e) Power factor, p.f. = = 0.571
(f) Tan ϕ =
and the phase angle between voltage and current, ϕ = = 55.15º leading
12. A circuit consisting of a resistor in series with an inductance takes 210 W at a power factor of
0.6 from a 50 V, 100 Hz supply. Find (a) the current flowing, (b) the circuit phase angle, (c) the
resistance, (d) the impedance and (e) the inductance.
(a) Power, P = V I cos i.e. 210 = (50) I (0.6) since p.f. = cos
Hence, current, I = = 7 A
(b) If cos = 0.6 then circuit phase angle, = = 53.13 lagging © John Bird Published by Taylor and Francis 289
(c) Power, P = i.e. 210 = from which, resistance, R = = 4.286
(d) Impedance, Z = = 7.143
(e) From the impedance triangle,
from which, = 5.71425
i.e. 2 f L = 5.71425 from which, inductance, L = = 9.095 mH
13. A 200 V, 60 Hz supply is applied to a capacitive circuit. The current flowing is 2 A and the
power dissipated is 150 W. Calculate the values of the resistance and capacitance.
Power, P = i.e. 150 = from which, resistance, R = = 37.5
Impedance, Z = = 100
From the impedance triangle,
from which, = 92.702
i.e. 92.702 = from which, capacitance, C = = 28.61 F
© John Bird Published by Taylor and Francis 290
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