Bird J. Electrical Circuit Theory and Technology

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Series and parallel networks 45

Figure 5.7

Problem 5. Two resistors are connected in series across a 24 V

supply and a current of 3 A ﬂows in the circuit. If one of the

resistors has a resistance of 2  determine (a) the value of the

other resistor, and (b) the p.d. across the 2  resistor. If the circuit

is connected for 50 hours, how much energy is used?

The circuit diagram is shown in Figure 5.8

Figure 5.8

(a) Total circuit resistance R D

D 8 

Value of unknown resistance, R

D 8  2 D 6 Z

(b) P.d. across 2  resistor, V

D IR

D 3 ð2 D 6V

Alternatively, from above,



C R



V D



2 C 6



24 D 6V

Energy used D power ð time

D V ð I ð t

D 24 ð 3W50 h

D 3600 Wh D 3.6kWh

5.3 Parallel networks

Figure 5.9 shows three resistors, R

, R

and R

connected across each

other, i.e., in parallel, across a battery source of V volts.

In a parallel circuit:

(a) the sum of the currents I

, I

and I

is equal to the total circuit

current, I,i.e.I

= I

Y I

,and

(b) the source p.d., V volts, is the same across each of the resistors.

From Ohm’s law:

and I D

where R is the total circuit resistance.

Since I D I

C I

then,

Figure 5.9

46 Electrical Circuit Theory and Technology

Dividing throughout by V gives:

This equation must be used when ﬁnding the total resistance R of a parallel

circuit. For the special case of two resistors in parallel

C R

Hence R =

Y R



i.e.

product

sum



Problem 6. For the circuit shown in Figure 5.10, determine (a) the

reading on the ammeter, and (b) the value of resistor R

P.d. across R

is the same as the supply voltage V.

Hence supply voltage, V D 8 ð 5 D 40 V

(a) Reading on ammeter, I D

D 2A

(b) Current ﬂowing through R

D 11  8  2 D 1A

Hence, R

D 40 Z

Figure 5.10

Problem 7. Two resistors, of resistance 3  and 6 , are

connected in parallel across a battery having a voltage of 12 V.

Determine (a) the total circuit resistance and (b) the current ﬂowing

in the 3  resistor.

The circuit diagram is shown in Figure 5.11.

(a) The total circuit resistance R is given by

2 C 1

Hence, R D

D 2 Z

Figure 5.11

Series and parallel networks 47



Alternatively, R D

C R

3 ð 6

3 C6

D 2 Z



(b) Current in the 3  resistance, I

D 4A

Problem 8. For the circuit shown in Figure 5.12, ﬁnd (a) the value

of the supply voltage V and (b) the value of current I.

Figure 5.12

(a) P.d. across 20  resistor D I

D 3 ð 20 D 60 V, hence supply

voltage V

= 60 V since the circuit is connected in parallel.

(b) Current I

D 6AI I

D 3A

D 1A

Current I D I

C I

and hence I D 6 C 3 C 1 D 10 A

Alternatively,

1 C 3 C 6

Hence total resistance R D

D 6 

Current I D

D 10 A

Problem 9. Given four 1  resistors, state how they must be

connected to give an overall resistance of (a)

 (b) 1  (c) 1



(d) 2

, all four resistors being connected in each case.

Figure 5.13

(a) All four in parallel (see Figure 5.13),

since

, i.e., R D



Figure 5.14

(b) Two in series, in parallel with another two in series (see

Figure 5.14), since 1  and 1  in series gives 2 , and 2  in

parallel with 2  gives:

2 ð 2

2 C2

D 1 

the three in parallel,

, i.e., R D

 and

 in series

with 1  gives 1



Figure 5.15

48 Electrical Circuit Theory and Technology

Figure 5.16

(d) Two in parallel, in series with two in series (see Figure 5.16),

since for the two in parallel

R D

1 ð 1

1 C 1

, and

, 1  and 1  in series gives 2



Problem 10. Find the equivalent resistance for the circuit shown

in Figure 5.17.

Figure 5.17

, R

and R

are connected in parallel and their equivalent resistance R

is given by:

6 C 3 C 1

Hence R D

D 1.8 

The circuit is now equivalent to four resistors in series and the equivalent

circuit resistance D 1 C 2.2 C 1.8 C 4 D 9 Z

5.4 Current division

For the circuit shown in Figure 5.18, the total circuit resistance, R

given by:

C R

and V D IR

D I



C R



Figure 5.18

Current I



C R





Y R



.I/

Similarly,

current I



C R





Y R



.I/

Summarizing, with reference to Figure 5.18



Y R



.I/ and I



Y R



.I/

Problem 11. For the series-parallel arrangement shown in

Figure 5.19, ﬁnd (a) the supply current, (b) the current ﬂowing

through each resistor and (c) the p.d. across each resistor.

Figure 5.19

Series and parallel networks 49

(a) The equivalent resistance R

of R

and R

in parallel is:

6 ð 2

6 C2

D 1.5 

The equivalent resistance R

of R

, R

and R

in series is:

D 2.5 C 1.5 C 4 D 8 

Supply current I D

200

D 25 A

(b) The current ﬂowing through R

and R

is 25 A

The current ﬂowing through R



C R



I D



6 C 2



D 6.25 A

The current ﬂowing through R



C R



I D



6 C 2



D 18.75 A

(Note that the currents ﬂowing through R

and R

must add up to

the total current ﬂowing into the parallel arrangement, i.e. 25 A)

p.d. across R

, i.e., V

D IR

D 252.5 D 62.5V

p.d. across R

, i.e., V

D IR

D 251.5 D 37.5V

p.d. across R

, i.e., V

D IR

D 254 D 100 V

Hence the p.d. across R

D p.d. across R

D 37.5V

Figure 5.20

Problem 12. For the circuit shown in Figure 5.21 calculate (a) the

value of resistor R

such that the total power dissipated in the circuit

is 2.5 kW, and (b) the current ﬂowing in each of the four resistors.

Figure 5.21

50 Electrical Circuit Theory and Technology

(a) Power dissipated P D VI watts, hence 2500 D 250I

I D

2500

250

D 10 A

From Ohm’s law, R

250

D 25 , where R

is the

equivalent circuit resistance.

The equivalent resistance of R

and R

in parallel is

15 ð 10

15 C10

150

D 6 

The equivalent resistance of resistors R

and R

in parallel is equal

to 25   6 , i.e., 19 

There are three methods whereby R

can be determined.

Method 1

The voltage V

D IR, where R is 6 , from above,

i.e. V

D 106 D 60 V

Hence V

D 250 V  60 V D 190 V D p.d. across R

D p.d. across R

190

D 5 A. Thus I

D 5 A also, since I D 10 A

Thus R

190

D 38 

Method 2

Since the equivalent resistance of R

and R

in parallel is 19 ,

19 D

38R

38 C R



i.e.

product

sum



Hence

1938 C R

 D 38R

722 C 19R

D 38R

722 D 38R

 19R

D 19R

Thus R

722

D 38 Z

Method 3

When two resistors having the same value are connected in parallel

the equivalent resistance is always half the value of one of the

resistors. Thus, in this case, since R

D 19  and R

D 38 , then

D 38  could have been deduced on sight.

Series and parallel networks 51

(b) Current I



C R



I D



15 C 10



10





10 D 4A

Current I



C R



I D



15 C 10



10





10 D 6A

From part (a), method 1, I

= I

= 5A

Problem 13. For the arrangement shown in Figure 5.22, ﬁnd the

current I

Figure 5.22

Commencing at the right-hand side of the arrangement shown in

Figure 5.22, the circuit is gradually reduced in stages as shown in

Figure 5.23(a)–(d).

From Figure 5.23(d) I D

4.25

D 4A

From Figure 5.23(b) I



9 C 3



I D





4 D 3A

Figure 5.23

From Figure 5.22 I



2 C 8



I

 D





3 D 0.6A

52 Electrical Circuit Theory and Technology

5.5 Wiring lamps in

series and in parallel

Series connection

Figure 5.24 shows three lamps, each rated at 240 V, connected in series

across a 240 V supply.

(i) Each lamp has only

240

V, i.e., 80 V across it and thus each lamp

glows dimly.

(ii) If another lamp of similar rating is added in series with the other

three lamps then each lamp now has

240

V, i.e., 60 V across it

and each now glows even more dimly.

Figure 5.24

(iii) If a lamp is removed from the circuit or if a lamp develops a fault

(i.e. an open circuit) or if the switch is opened then the circuit is

broken, no current ﬂows, and the remaining lamps will not light up.

(iv) Less cable is required for a series connection than for a parallel one.

The series connection of lamps is usually limited to decorative lighting

such as for Christmas tree lights.

Parallel connection

Figure 5.25 shows three similar lamps, each rated at 240 V, connected in

parallel across a 240 V supply.

Figure 5.25

(i) Each lamp has 240 V across it and thus each will glow brilliantly

at their rated voltage.

(ii) If any lamp is removed from the circuit or develops a fault (open

circuit) or a switch is opened, the remaining lamps are unaffected.

(iii) The addition of further similar lamps in parallel does not affect the

brightness of the other lamps.

(iv) More cable is required for parallel connection than for a series one.

The parallel connection of lamps is the most widely used in electrical

installations.

Problem 14. If three identical lamps are connected in parallel and

the combined resistance is 150 , ﬁnd the resistance of one lamp.

Let the resistance of one lamp be R, then,

150

, from which, R D 3 ð 150 D 450 Z

Problem 15. Three identical lamps A, B and C are connected in

series across a 150 V supply. State (a) the voltage across each lamp,

and (b) the effect of lamp C failing.

Series and parallel networks 53

(a) Since each lamp is identical and they are connected in series there

150

V, i.e. 50 V across each.

(b) If lamp C fails, i.e., open circuits, no current will ﬂow and lamps

A and B will not operate.

5.6 Further problems on

series and parallel

networks

1 The p.d’s measured across three resistors connected in series are

5 V, 7 V and 10 V, and the supply current is 2 A. Determine (a) the

supply voltage, (b) the total circuit resistance and (c) the values of

the three resistors. [(a) 22 V (b) 11  (c) 2.5 ,3.5,5]

2 For the circuit shown in Figure 5.26, determine the value of V

.If

the total circuit resistance is 36  determine the supply current and

the value of resistors R

, R

and R

[10 V, 0.5 A, 20 ,10,6]

Figure 5.26

3 When the switch in the circuit in Figure 5.27 is closed the reading

on voltmeter 1 is 30 V and that on voltmeter 2 is 10 V. Determine

the reading on the ammeter and the value of resistor R

[4 A, 2.5 ]

Figure 5.27

4 Two resistors are connected in series across an 18 V supply and a

current of 5 A ﬂows. If one of the resistors has a value of 2.4 

determine (a) the value of the other resistor and (b) the p.d. across

the 2.4  resistor. [(a) 1.2  (b) 12 V]

5 Resistances of 4  and 12  are connected in parallel across a

9 V battery. Determine (a) the equivalent circuit resistance, (b) the

supply current, and (c) the current in each resistor.

[(a) 3  (b)3A(c)

2.25 A, 0.75 A]

6 For the circuit shown in Figure 5.28 determine (a) the reading on

the ammeter, and (b) the value of resistor R.[2.5A,2.5]

7 Find the equivalent resistance when the following resistances are

connected (a) in series, (b) in parallel

(i) 3  and 2  (ii) 20 k and 40 k

(iii) 4 , 8  and 16  (iv) 800 , 4k and 1500 

[(a) (i) 5  (ii) 60 k (iii) 28  (iv) 6.3 k

(b) (i) 1.2  (ii) 13

k (iii) 2

 (iv) 461.5 k]

Figure 5.28

8 Find the total resistance between terminals A and B of the circuit

shown in Figure 5.29(a) [8 ]

9 Find the equivalent resistance between terminals C and D of the

circuit shown in Figure 5.29(b) [27.5 ]

10 Resistors of 20 ,20 and 30  are connected in parallel. What

resistance must be added in series with the combination to obtain a

total resistance of 10 . If the complete circuit expends a power of

0.36 kW, ﬁnd the total current ﬂowing. [2.5 ,6A]

54 Electrical Circuit Theory and Technology

Figure 5.29

11 (a) Calculate the current ﬂowing in the 30  resistor shown in

Figure 5.30

(b) What additional value of resistance would have to be placed in

parallel with the 20  and 30  resistors to change the supply

current to 8 A, the supply voltage remaining constant.

[(a) 1.6 A (b) 6 ]

12 Determine the currents and voltages indicated in the circuit shown

in Figure 5.31.

D 5A,I

D 2.5A,I

D 1

A, I

D 3A,I

D 2A,V

D 20 V, V

D 5V,V

D 6V]

Figure 5.30

13 Find the current I in Figure 5.32. [1.8 A]

Figure 5.31 Figure 5.32

14 If four identical lamps are connected in parallel and the combined

resistance is 100 , ﬁnd the resistance of one lamp. [400 ]

15 Three identical ﬁlament lamps are connected (a) in series, (b) in

parallel across a 210 V supply. State for each connection the p.d.

across each lamp. [(a) 70 V (b) 210 V]