Bird J. Electrical Circuit Theory and Technology

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206 Electrical Circuit Theory and Technology

D 2500 C 10000  8660

21 160 D 145.5V

Using the sine rule,

100

sin

145.5

sin150

from which sin D

100sin150

145.5

D 0.3436

and  D arcsin0.3436 D 20

D 0.35 radians, and lags v

Hence v

D v

C v

D 145.5sin.!t − 0.35/ V

Problem 15. Find a sinusoidal expression for i

C i

 of

Problem 13, (a) by drawing phasors, (b) by calculation.

(a) The relative positions of i

and i

at time t D 0 are shown as phasors

in Figure 14.11(a). The phasor diagram in Figure 14.11(b) shows the

resultant i

,andi

is measured as 26 A and angle  as 19

or 0.33

rads leading i

Hence, by drawing, i

= 26sin.!t Y 0.33/ A

(b) From Figure 14.11(b), by the cosine rule:

Figure 14.11

D 20

C 10

 22010cos 120



from which i

D 26.46 A

By the sine rule:

sin

26.46

sin120

from which  D 19.10

(i.e. 0.333 rads)

Hence, by calculation i

= 26.46sin.!t Y 0.333/ A

Problem 16. Two alternating voltages are given by v

120sinωt volts and

D 200sinωt  /4 volts. Obtain sinu-

soidal expressions for

 v

(a) by plotting waveforms, and (b) by

resolution of phasors.

(a) v

D 120sinωt and v

D 200sinωt  /4 are shown plotted in

Figure 14.12. Care must be taken when subtracting values of

ordinates especially when at least one of the ordinates is negative.

For example

at 30

, v

 v

D 60 52 D 112 V

at 60

, v

 v

D 104  52 D 52 V

at 150

, v

 v

D 60 193 D133 V and so on

Alternating voltages and currents 207

Figure 14.12

The resultant waveform,

D v

 v

, is shown by the broken line in

Figure 14.12. The maximum value of

is 143 V and the waveform

is seen to lead

by 99

(i.e. 1.73 radians)

Hence, by drawing,

= v

− v

= 143sin.!t Y 1.73/ volts

(b) The relative positions of

and v

are shown at time t D 0 as phasors

in Figure 14.13(a). Since the resultant of

 v

is required, v

drawn in the opposite direction to C

and is shown by the broken

line in Figure 14.13(a). The phasor diagram with the resultant is

shown in Figure 14.13(b) where 

is added phasorially to v

By resolution:

Sum of horizontal components of

and v

D 120cos0

C 200 cos135

D21.42

Sum of vertical components of

and v

D 120sin0

C 200sin135

D 141.4

From Figure 14.13(c), resultant



[21.42

C 141.4

] D 143.0,

and tan

141.4

21.42

D tan6.6013, from which



D arctan6.6013 D 81

and

 D 98

or 1.721 radians

Figure 14.13

208 Electrical Circuit Theory and Technology

Hence, by resolution of phasors,

= v

− v

= 143.0sin.!t Y 1.721/ volts

Further problems on the combination of periodic functions may be found

in Section 14.8, problems 16 to 19, page 211.

14.7 Rectiﬁcation

The process of obtaining unidirectional currents and voltages from

alternating currents and voltages is called rectiﬁcation. Automatic

switching in circuits is carried out by devices called diodes.

Using a single diode, as shown in Figure 14.14, half-wave rectiﬁcation

is obtained. When P is sufﬁciently positive with respect to Q, diode D

is switched on and current i ﬂows. When P is negative with respect to

Q, diode D is switched off. Transformer T isolates the equipment from

direct connection with the mains supply and enables the mains voltage to

be changed.

Two diodes may be used as shown in Figure 14.15 to obtain full wave

rectiﬁcation. A centre-tapped transformer T is used. When P is sufﬁ-

ciently positive with respect to Q, diode D

conducts and current ﬂows

(shown by the broken line in Figure 14.15). When S is positive with

respect to Q, diode D

conducts and current ﬂows (shown by the contin-

uous line in Figure 14.15). The current ﬂowing in R is in the same direc-

tion for both half cycles of the input. The output waveform is thus as

shown in Figure 14.15.

Figure 14.14 Figure 14.15

Four diodes may be used in a bridge rectiﬁer circuit, as shown in

Figure 14.16 to obtain full wave rectiﬁcation. As for the rectiﬁer shown

in Figure 14.15, the current ﬂowing in R is in the same direction for both

half cycles of the input giving the output waveform shown.

To smooth the output of the rectiﬁers described above, capacitors

having a large capacitance may be connected across the load resistor

R. The effect of this is shown on the output in Figure 14.17.

Alternating voltages and currents 209

Figure 14.16 Figure 14.17

14.8 Further problems

on alternating voltages

and currents

Frequency and periodic time

1 Determine the periodic time for the following frequencies:

(a) 2.5 Hz (b) 100 Hz (c) 40 kHz

[(a) 0.4 s (b) 10 ms (c) 25

µs]

2 Calculate the frequency for the following periodic times:

(a) 5 ms (b) 50

µs (c) 0.2 s

[(a) 0.2 kHz (b) 20 kHz (c) 5 Hz]

3 An alternating current completes 4 cycles in 5 ms. What is its

frequency? [800 Hz]

A.c. values of non-sinusoidal waveforms

4 An alternating current varies with time over half a cycle as follows:

Current (A) 0 0.7 2.0 4.2 8.4 8.2 2.5 1.0 0.4 0.2 0

time (ms) 0 1 2 3 4 5 6 7 8 9 10

The negative half cycle is similar. Plot the curve and determine:

(a) the frequency (b) the instantaneous values at 3.4 ms and 5.8 ms

[(a) 50 Hz (b) 5.5 A, 3.4 A (c) 2.8 A (d) 4.0 A]

5 For the waveforms shown in Figure 14.18 determine for each (i) the

frequency (ii) the average value over half a cycle (iii) the rms value

(iv) the form factor (v) the peak factor.

[(a) (i) 100 Hz (ii) 2.50 A (iii) 2.88 A (iv) 1.15 (v) 1.74

(b) (i) 250 Hz (ii) 20 V (iii) 20 V (iv) 1.0 (v) 1.0

(d) (i) 250 Hz (ii) 25 V (iii) 50 V (iv) 2.0 (v) 2.0]

210 Electrical Circuit Theory and Technology

Figure 14.18

6 An alternating voltage is triangular in shape, rising at a constant

rate to a maximum of 300 V in 8 ms and then falling to zero at a

constant rate in 4 ms. The negative half cycle is identical in shape

to the positive half cycle. Calculate (a) the mean voltage over half a

cycle, and (b) the rms voltage [(a) 150 V (b) 170 V]

A.c. values of sinusoidal waveforms

7 Calculate the rms value of a sinusoidal curve of maximum value

300 V [212.1 V]

8 Find the peak and mean values for a 200 V mains supply

[282.9 V, 180.2 V]

9 A sinusoidal voltage has a maximum value of 120 V. Calculate its

rms and average values. [84.8 V, 76.4 V]

10 A sinusoidal current has a mean value of 15.0 A. Determine its

maximum and rms values. [23.55 A, 16.65 A]

v = V

sin.!t ± f/

11 An alternating voltage is represented by

v D 20sin 157.1 t volts.

Find (a) the maximum value (b) the frequency (c) the periodic time.

(d) What is the angular velocity of the phasor representing this wave-

form? [(a) 20 V (b) 25 Hz (c) 0.04 s (d) 157.1 rads/s]

12 Find the peak value, the rms value, the periodic time, the frequency

and the phase angle (in degrees and minutes) of the following alter-

nating quantities:

(a)

v D 90sin400t volts

[90 V, 63.63 V, 5 ms, 200 Hz, 0

]

(b) i D 50sin100t C0.30 amperes

[50 A, 35.35 A, 0.02 s, 50 Hz, 17

lead]

[200 V, 141.4 V, 0.01 s, 100 Hz, 23

lag]

13 A sinusoidal current has a peak value of 30 A and a frequency of

60 Hz. At time t D 0, the current is zero. Express the instantaneous

current i in the form i D I

sinωt [i D 30sin 120t]

14 An alternating voltage

v has a periodic time of 20 ms and a maximum

value of 200 V. When time t D 0,

v D75 volts. Deduce a sinu-

soidal expression for

v and sketch one cycle of the voltage showing

important points. [

v D 200sin100t  0.384]

15 The instantaneous value of voltage in an a.c. circuit at any time t

seconds is given by

v D 100sin50t  0.523 V. Find:

(a) the peak-to-peak voltage, the periodic time, the frequency and

the phase angle

Alternating voltages and currents 211

(b) the voltage when t D 0

(d) the times in the ﬁrst cycle when the voltage is 60 V

(e) the times in the ﬁrst cycle when the voltage is 40 V, and

(f) the ﬁrst time when the voltage is a maximum.

Sketch the curve for one cycle showing relevant points.

[(a) 200 V, 0.04 s, 25 Hz, 29

lagging

(b) 49.95 V (c) 66.96 V (d) 7.426 ms, 19.23 ms

(e) 25.95 ms, 40.71 ms (f) 13.33 ms]

Combination of periodic functions

16 The instantaneous values of two alternating voltages are given by

D 5sinωt and v

D 8sinωt  /6. By plotting v

and v

on the

same axes, using the same scale, over one cycle, obtain expressions

for (a)

C v

and (b) v

 v

[(a) v

C v

D 12.58sinωt  0.325 V

(b)

 v

D 4.44sinωt C 2.02 V]

17 Repeat Problem 16 by resolution of phasors.

18 Construct a phasor diagram to represent i

C i

where i

D 12sinωt

and i

D 15sinωt C /3. By measurement, or by calculation, ﬁnd

a sinusoidal expression to represent i

C i

[23.43 sinωt C 0.588]

19 Determine, either by plotting graphs and adding ordinates at inter-

vals, or by calculation, the following periodic functions in the form

v D V

sinωt š 

(a) 10sinωt C 4sinωt C /4 [13.14sinωt C 0.217]

(b) 80sinωt C /3 C 50sinωt  /6

[94.34sinωt C 0.489]

Assignment 4

This assignment covers the material contained in chapters 13

and 14.

The marks for each question are shown in brackets at the end of

each question.

1 Find the current ﬂowing in the 5  resistor of the circuit shown

in Figure A4.1 using (a) Kirchhoff’s laws, (b) the Superposition

theorem, (c) Th

evenin’s theorem, (d) Norton’s theorem. Demonstrate

that the same answer results from each method. Find also the current

ﬂowing in each of the other two branches of the circuit. (27)

10 V

3 V

1 Ω

5 Ω

2 Ω

Figure A4.1

2 A d.c. voltage source has an internal resistance of 2  and an open

circuit voltage of 24 V. State the value of load resistance that gives

maximum power dissipation and determine the value of this power.

(5)

3 A sinusoidal voltage has a mean value of 3.0 A. Determine it’s

maximum and r.m.s. values. (4)

4 The instantaneous value of current in an a.c. circuit at any time t

seconds is given by: i D 50 sin100t  0.45 mA. Determine

(a) the peak to peak current, the periodic time, the frequency and

the phase angle (in degrees and minutes)

(b) the current when t D 0

(d) the ﬁrst time when the current is a maximum.

Sketch the current for one cycle showing relevant points. (14)

15 Single-phase series a.c.

circuits

At the end of this chapter you should be able to:

ž draw phasor diagrams and current and voltage waveforms for

(a) purely resistive (b) purely inductive and (c) purely

capacitive a.c. circuits

ž perform calculations involving X

D 2fL and X

2fC

ž draw circuit diagrams, phasor diagrams and voltage and

impedance triangles for R–L, R–C and R–L –C series a.c.

circuits and perform calculations using Pythagoras’ theorem,

trigonometric ratios and Z D

ž understand resonance

ž derive the formula for resonant frequency and use it in

calculations

ž understand Q-factor and perform calculations using

or V









ž understand bandwidth and half-power points

ž perform calculations involving f

 f

 D

ž understand selectivity and typical values of Q-factor

ž appreciate that power P in an a.c. circuit is given by

P D VI cos or I

R and perform calculations using these

formulae

ž understand true, apparent and reactive power and power factor

and perform calculations involving these quantities

Figure 15.1

15.1 Purely resistive a.c.

circuit

In a purely resistive a.c. circuit, the current I

and applied voltage V

are

in phase. See Figure 15.1.

214 Electrical Circuit Theory and Technology

15.2 Purely inductive a.c.

circuit

In a purely inductive a.c. circuit, the current I

lags the applied voltage

by 90

(i.e. /2 rads). See Figure 15.2.

In a purely inductive circuit the opposition to the ﬂow of alternating

current is called the inductive reactance, X

= 2pfL Z

where f is the supply frequency, in hertz, and L is the inductance, in

henry’s. X

is proportional to f as shown in Figure 15.3.

Figure 15.2

Figure 15.3

Problem 1. (a) Calculate the reactance of a coil of inductance

0.32 H when it is connected to a 50 Hz supply. (b) A coil has a

reactance of 124  in a circuit with a supply of frequency 5 kHz.

Determine the inductance of the coil.

(a) Inductive reactance, X

D 2fL D 2500.32 D 100.5 Z

(b) Since X

D 2fL, inductance L D

2f

124

25000

H D 3.95 mH

Problem 2. A coil has an inductance of 40 mH and negligible

resistance. Calculate its inductive reactance and the resulting current

if connected to (a) a 240 V, 50 Hz supply, and (b) a 100 V, 1 kHz

supply.

(a) Inductive reactance, X

D 2fL D 25040 ð 10

3

 D 12.57 Z

Current, I D

240

12.57

D 19.09 A

(b) Inductive reactance, X

D 2100040 ð 10

3

 D 251.3 Z

Current, I D

100

251.3

D 0.398 A

15.3 Purely capacitive

a.c. circuit

In a purely capacitive a.c. circuit, the current I

leads the applied voltage

by 90

(i.e. /2 rads). See Figure 15.4.

In a purely capacitive circuit the opposition to the ﬂow of alternating

current is called the capacitive reactance, X

2pfC

where C is the capacitance in farads.

varies with frequency f as shown in Figure 15.5.

Single-phase series a.c. circuits 215

Figure 15.4

Problem 3. Determine the capacitive reactance of a capacitor

of 10

µF when connected to a circuit of frequency (a) 50 Hz

(b) 20 kHz

(a) Capacitive reactance X

2fC

25010 ð 10

6



25010

D 318.3 Z

(b) X

2fC

220 ð 10

10 ð 10

6



220 ð 10

10

D 0.796 Z

Figure 15.5

Hence as the frequency is increased from 50 Hz to 20 kHz, X

decreases

from 318.3  to 0.796  (see Figure 15.5).

Problem 4. A capacitor has a reactance of 40  when operated

on a 50 Hz supply. Determine the value of its capacitance.

Since X

2fC

, capacitance C D

2fX

25040

µF D 79.58 mF

Problem 5. Calculate the current taken by a 23 µF capacitor when

connected to a 240 V, 50 Hz supply.

Current I D



2fC



D 2fCV D 25023 ð10

6

240

D 1.73 A

Further problems on purely inductive and capacitive a.c. circuits may be

found in Section 15.12, problems 1 to 8, page 234.

15.4 R–L series a.c.

circuit

In an a.c. circuit containing inductance L and resistance R, the applied

voltage V is the phasor sum of V

and V

(see Figure 15.6), and thus

the current I lags the applied voltage V by an angle lying between 0

and

(depending on the values of V

and V

), shown as angle .Inany