D . C . G E N E R A T O R S
U = E − IaRa
where U is the terminal voltage, E is the generated e.m.f., Ia is the armature current, and Ra is the armature resistance.
EXAMPLE Calculate the e.m.f. generated by a shunt generator which is delivering 15 A at a terminal voltage of 440 V. The armature circuit resistance is 0.15 , the resistance of the shunt field is 300 , and a voltage drop of 2 V occurs at the brushes.
The circuit is shown in Figure 92.
U = 440 V Ro= 0.15 Ω Rs= 300 Ω 15 A Io Is Fig. 92
To find the shunt field current, U = Is× Rs
where Is is shunt field current, and Rs is shunt field resistance.
∴ 440= Is× 300
∴ Is= 440
300 = 1.47A
Total armature current= 15 + 1.47 = 16.47 A Neglecting the voltage drop at the brushes,
U = E − IaRa
∴ 440 = E − 16.47 × 0.15 = E − 2.47
∴ 440 + 2.47 = E E= 442.47 V
Allowing for the voltage drop at the brushes, generated e.m.f .= 442.47 + 2
= 444.47 = 444 V
D . C . M O T O R S
U = E + IaRa
where U is the terminal voltage, E is the back e.m.f., Ia is the armature current, and Ra is the circuit resistance.
EXAMPLE Calculate the back e.m.f. of a d.c. motor which is taking an armature current of 25 A from a 220 V supply.
The resistance of its armature is 0.2 . U = E + IaRa ∴ 220= E + 25 × 0.2 = E + 5.0 ∴ 220− 5 = E ∴ E= 215 V A L T E R N A T O R S A N D S Y N C H R O N O U S M O T O R S f = n × p
where f is the frequency in hertz,
n is the speed in revolutions per second, and p is the number of pairs of poles.
EXAMPLE 1 Calculate the number of poles in an alternator which generates 60 Hz at a speed of 5 rev/s.
f = n × p
∴ 60= 5 × p
∴ p=60
5 = 12
∴ the machine has 2× p = 24 poles
EXAMPLE 2 Calculate the speed at which a four-pole synchronous motor will run from a 50 Hz supply.
f = n × p
∴ 50= n × 2 (4 poles gives 2 pairs)
∴ n=50
2 = 25 rev/s
I N D U C T I O N M O T O R S
Percentage slip= ns− nr
ns × 100%
where ns is the synchronous speed, and nr is the actual speed of the rotor.
The synchronous speed nscan be determined from the relationship
f = ns× p
as in the case of the synchronous motor.
EXAMPLE Calculate the actual speed of a six-pole
cage-induction motor operating from a 50 Hz supply with 7% slip. f = ns× p ∴ 50= ns× 3 ∴ ns=50 3 = 16.7 rev/s Percentage slip= ns− nr ns × 100 ∴ 7= 16.7− nr 16.7 × 100 0.07= 16.7− nr 16.7 ∴ 0.07× 16.7 = 16.7 − nr ∴ nr= 16.7 − 0.07 × 16.7 = 15.5 rev/s I N S U L A T I O N R E S I S T A N C E
The insulation resistance of a cable is inversely proportional to its length.
EXAMPLE 1 The insulation resistance measured between the cores of a certain twin cable 100 m long is 1000 M. Calculate the insulation resistance of 35 m of the same cable.
The shorter length will have a higher value of insulation resistance because the path for the leakage current has less cross-sectional area (Figure 93). Core Core Insulation 100 m 35 m Tester Fig. 93 Insulation resistance of 100 m = 1000 M
∴ insulation resistance of 35 m= 1000 ×100 (larger) 35 (smaller) = 2857
EXAMPLE 2 The insulation resistance measured between the cores of a certain twin cable is 850 M. Calculate the insulation resistance obtained when two such cables are connected (a) in series, (b) in parallel.
It is seen from Figure 94 that the effect in both cases is the same, i.e. to increase the c.s.a. of the leakage-current path through Tester
850 MΩ 850 MΩ
Tester (a) Cables in series
(b) Cables in parallel Core Core Core Core Core Core Insulation Fig. 94
the insulation. The insulation resistance in either case is thus 850 M
2 = 425 M
EXERCISE 21
1. What is meant by the expression ‘back e.m.f.’ of a direct-current motor? In what way does the back e.m.f. affect the starting of a direct-current motor?
A direct-current motor connected to a 460 V supply takes an armature current of 120 A on full load. If the armature circuit has a resistance of 0.25 , calculate the value of the back e.m.f. at this load. (CGLI)
2. A d.c. machine has an armature resistance of 8 . Calculate
(a)the back e.m.f. when it is operating from a 110 V supply and taking an armature current of 2.5 A;
(b)the e.m.f. generated when the machine is running as a generator and delivering 2 A at a terminal voltage of 110 V. (Neglect the field current.)
3. A d.c. motor connected to a 460 V supply has armature resistance of 0.15 . Calculate
(a)the value of the back e.m.f. when the armature current is 120 A,
(b)the value of the armature current when the back e.m.f. is 447.4 V. (CGLI)
4. Explain briefly, with the aid of diagrams, the differences between series, shunt, and compound d.c. generators. A d.c. shunt generator delivers a current of 96 A at 240 V. The armature resistance is 0.15 , and the field winding has a resistance of 60 . Assuming a brush contact drop of 2 V, calculate (a) the current in the armature, (b) the generated e.m.f. (CGLI)
5. Calculate the speed at which an eight-pole alternator must be driven in order to generate 50 Hz.
6. Calculate the frequency of the voltage generated by a four-pole alternator when it is running at (a) 16 rev/s, (b) 12 rev/s.
7. Determine the speed at which a six-pole synchronous motor will run from the 50 Hz mains.
8. The synchronous speed of an induction motor is 750 rev/min. The motor actually runs at 715 rev/min. Calculate the percentage slip.
9. A four-pole induction motor is operating at 24 rev/s from a 50 Hz supply. Calculate the percentage slip.
10. A cage-induction motor having six poles operates with a 4.5% slip from a 50 Hz supply. Calculate the actual rotor speed.
11. Calculate the full-load torque of a 30 kW six-pole 50 Hz induction motor, assuming that the slip at full load amounts to 5%.
12. Explain the term ‘insulation resistance’. Describe, with wiring diagram, a suitable instrument for measuring insulation resistance.
Calculate the insulation resistance of a 100 m coil of insulated cable. The insulation resistance of 1 km of the same cable is given as 2500 M.
13. The insulation resistance of 1000 m of two-core cable is 1500 M. Calculate the insulation resistance of
(a)100 m (b) 200 m (c) 400 m (d) 600 m (e) 800 m and plot a graph showing the relationship between cable length and insulation resistance.
14. Explain the term ‘insulation resistance of an installation’. Describe, with connection diagram, the working of an instrument suitable for measuring insulation resistance.
Three separate circuits are disconnected from a distribution board and tested for insulation resistance to earth. The respective values are 40 M, 60 M, and 300 M. What is the combined insulation resistance to earth? (CGLI)
15. The insulation resistance measured between the cores of a certain twin cable is 950 . Calculate the insulation resistance of three identical cables connected in parallel.
16. The resistance of an armature circuit of a motor is 1.2 . The current through it is 15 A and the terminal voltage is 200 V. The generated e.m.f. is
(a)218 V (b) 182 V (c) 13.3 V (d) 125 V
17. An alternator generates 400 Hz at a speed of 500 rev/min. The number of pairs of poles is
(a)12 (b) 48 (c) 6 (d) 3
18. The insulation resistance measured between the cores of a cable is 900 M for a 500 m length. The insulation
resistance for 350 m of this cable would be: