E.m.f. Equation-Turn Ratio
1. The no-load ratio required in a single-phase 50 Hz transformer is 6600/300 V. If the maximum value of flux in the core is to be about 0.09 weber, find the number of turns in each winding.
[Ans. N_{1} = 300 ; N_{2} = 15]
2. A 20 kVA single-phase transformer has 200 turns on the primary and 40 turns on the secondary. The primary is connected to 1000 V, 50-Hz supply. Determine :the secondary voltage on open circuit;
the currents flowing through the two windings on full-load ; and
the maximum value of flux.
[Ans. (i) = 200 V ; (ii) = 20 A, 100 A ; (iii) = 0.0225 Wb]
3. A single-phase transformer is connected to a 230 V, 50-Hz supply. The net cross-sectional area of the core is 50 cm^{2}. The number of turns in the primary is 46 and in the secondary 80. Determine :
- Transformation ratio ;
- Peak value of flux density in the core ; and
- E.m.f. in the secondary winding
[Ans. 0.1739 ; 0.4504 T ; 40 V]
4. A 6600/440 V single-phase 600 kVA transformer has 1200 primary turns. Find :
- Transformation ratio ;
- Secondary turns ;
- Voltage/turn; and
- Secondary current when it supplies a load of 400 kW at 0.8 power factor lagging.
5. Find the primary and secondary turns of a 3300/300 V, single-phase, 50-H2, 30 kVA transformer if the flux in the core is to be about 0.06 Wb. Also determine the primary and secondary currents if the losses are to be neglected.
[Ans. 264 ; 24 ; 100 A ; 9.09 A]
6. The voltage/turn of a single-phase transformer is 1.1 V, when the primary winding is connected to a 220 V, 50-Hz A.C. supply, the secondary voltage is found to be 600 V. Find : Primary and secondary turns ; and
Core area if the maximum flux density is 1.2 T
[Ans. 200 ; 600 ; 42.29 cm^{2}]
Transformer on No-Load
7. A 2000/200 V single-phase transformer gives 0.5 A and 40 W as ammeter and wattmeter readings when supply is given to the low voltage winding and high voltage winding is kept open. Find :
a. The magnetizing component,
b. The iron loss component, and
c. The power factor of no-load current,
[Ans. 0.4582 A ; 0.2 A ; 0.4 lagging]
8. Find
(i) active component and reactive components of no-load current and ;
(ii) no-load current of a 230 V/115 V single-phase transformer if the power input on no-load to the high-voltage winding is 70 W and power factor of no-load current is 0.25 lagging.
[Ans. 0.3044 A ; 1.179 A ; 1.2176 A]
9. A single-phase transformer has 500 turns on the primary and 40 turns on the secondary winding. The mean length of the magnetic path in the core is 150 cm and the joints are equivalent to an air-gap of 0.1 mm. When a potential difference of 3000 V is applied to the primary, maximum flux density is 1.2 T. Calculate :
- The cross-sectional are of the core,
- No-load secondary voltage,
- The no-load current drawn by the primary, and
- Power factor on no-load.
Given that AT/cm for a flux density of 1.2 T in the iron to be 5, the corresponding iron loss to be 2 W/kg at 50 Hz and density of iron as 7.8 g/cm^{3}.
[Ans. 225 cm^{2} ; 240 V ; 1.208 A ; 0.1475 ]
Transformer on Load
10. A 230 V/115 V single phase transformer takes a no-load current of 1.7 A at a power factor of 0.18 lagging with low voltage winding kept open. If the low voltage winding is now loaded to take a current of 13 A at 0.8 power factor lagging find the current taken by high voltage winding.
[Ans. 7.834 A]
11. A transformer has a primary winding of 800 turns and a secondary winding of 200 turns. When the load current on the secondary is 80 A at 0.8 power factor lagging, the primary current is 25 A at 0.707 power factor lagging. Determine graphically or otherwise the no-load current of the transformer and its phase with respect to the voltage.
[Ans. I_{0} = 5.93 A ; Φ_{0}=73.3^{0}]
12. A 8:1 step down, single-phase transformer takes a no-load current of 0.6 A at a power factor of 0.8 lagging with 1.v. winding kept open. If the secondary is connected to a load taking a current of 80 A at0.8 power factor lagging ; and
0.8 power factor leading find the primary current and power factor.
[Ans. 10.49 A, 0.7996 lagging ; 9.817 A, 0.8333 leading]
13. A 20 kVA, 2000/200 V single phase, 50 Hz transformer has a primary resistance of 2.5 Ω and reactance of 4.8 Ω. The secondary resistance and reactance are 0.01 Ω and 0.018 Ω respectively. Find
- Equivalent resistance referred to primary
- Equivalent impedance referred to primary
- Equivalent resistance, reactance and impedance referred to secondary, and
- Total copper loss of the transformer.
[Ans. (i) 3.5 Ω, (ii) 7.47 Ω, (iii) 0.035, 0.066 Ω, 0.747 Ω ; (iv) 350 W]
14. A 50 kVA, 4400/220 V transformer has R_{1} = 3.45 Ω, R_{2} = 0.009 Ω. The values of reactances are X_{1} = 5.2 Ω and X_{2}= 0.015 Ω. Calculate for the transformer :
- Equivalent resistance as referred to primary
- Equivalent resistance as referred to secondary
- Equivalent reactance as referred to both primary and secondary,
- Equivalent impedance as referred to both primary and secondary, and
- Total copper loss, first using individual resistances of the two windings and secondly, using equivalent resistance as referred to each side.
[Ans. (i) 7.05 Ω, (ii) 0.0176 Ω, (iii) 11.2 Ω, (iv) 13.23 Ω ; 0.331 Ω, (iv) 909 W]
15. A single phase transformer has the following data :
Turn ratio = 19.5 : 1
R_{1} = 25 Ω
X_{1 }= 100 Ω
R_{2}= 0.06 Ω
X_{2}= 0.25 Ω
No load current = 1.25 A leading the flux by 30^{0}.
16. The high voltage and low voltage windings of a 1100/110 V single-phase 50 Hz transformer has resistances of 2.4 Ω and 0.02 Ω and reactances 1 Ω and 0.009 Ω respectively. The low voltage winding is connected to a load having an impedance of (3+j2)Ω. Determine :
- Current in l.v. winding
- Current in h.v. winding
- The load voltage
- Power consumed by the load.
[Ans. (i) 30.11 A; (ii) 3.011 A ; 108.5 V (iv) 2720.7 W]
Equivalent Circuit and O.C. and S.C.Tests
17. The parameters of a 2300/230 V, 50 Hz transformer are given below :
R_{1} = 0.286 Ω | R_{2}’ = 0.319 Ω |
X_{1} = 0.73 Ω | X_{2}’ = 0.73 Ω |
R_{0} = 250 Ω | R_{0} = 1250 Ω |
The secondary load impedance Z_{L} = 0.387 + j0.29. Solve the exact equivalent circuit with normal voltage across the primary.
[Ans. η=78.8%, % Regulation=2.7%]
18. A 230 V, 3 kVA single-phase transformer has an iron loss of 100 W at 40-Hz and 70 W at 30-Hz. Find the hysteresis and eddy current losses at 50-Hz.
[Ans. 91.67 W, 41.67 W]
19. When a transformer is supplied at 400 V, 50-Hz the hysteresis loss is found to be 300 W and eddy current loss is found to be 250 W. Determine the hysteresis loss and eddy current loss when the transformer is supplied at 800 V, 100Hz.
[Ans. 600 W ; 1000 W]
20. When the primary of a transformer rated at 2200 V, 50-Hz is supplied at 2200 V, 50-Hz the wattmeter gives a reading of 1200 W on no-load. When it is supplied at 1100 V, 25-Hz, the wattmeter gives a reading of 400 W on no-load. If the wattmeter is connected in the input circuit find the hysteresis loss and eddy current loss at normal voltage and frequency.
[Ans. 400 V ; 800 W]
21.A 4400 V, 50-Hz transformer has a hysteresis loss of 1000 W, eddy current loss of 1500 W and full-load copper loss of 3500 W. If the transformer is supplied at 6600 V, 75-Hz, what will be the losses? Assume that the full-load current remains the same
[Ans. P_{h} = 1500 W, P_{e} = 3375 W]
22. A 50 Hz, single-phase transformer has a turn ratio of 6. The resistances are 0.9 Ω, 0.03 Ω and reactances are 5 Ω and 0.13 Ω for high-voltage and low-voltage windings respectively. Find :
The voltage to be applied to the h.v. side to obtain full-load current of 200 A in the l.v. winding on short-circuit.
The power factor on short-circuit.
Draw the equivalent circuit and show therein all the values.
[Ans. 329.3 V, 0.2]
23. A 200/2000 V transformer is fed from a 200 V supply. The total winding resistance and leakage reactance as referred to low voltage side are 0.16 Ω and 7.0 Ω respectively. The representing core loss is 400 Ω and the magnetizing reactance is 231 Ω. A load of impedance 596 + j444 Ω is connected across the secondary terminals. Calculate :
- Input current
- The secondary terminal voltage
- Primary power factor.
24. Determine the approximate equivalent circuit of a given 200/2000 V single-phase 40 kVA transformer having the following test results :
O.C. test : 200 v, 6.4 A, 384 W …. on l.v. side
S.C.test : 78 V, 20 A, 620 W …. on h.v. side.
[Ans. R_{01} = 0.0155 Ω, X_{01} = 0.358 Ω, R_{0} = 104.2 Ω, X_{0} = 32.75 Ω ]
25. Determine the approximate circuit of a 200/400 V, 50-Hz, single-phase transformer from the following test data:
O.C. test (l.v. side) : 200 V, 0.7 A, 70 W
S.C. test (h.v. side) : 15 V, 10 A, 85 W
Calculate the secondary voltage when delivering 5 kW at 0.8 power factor, lagging, the primary voltage being 200 V.
[Ans. R_{0} =571.4 Ω, X_{0} = 330 Ω, R_{01}=0.21 Ω, X_{01}=0.31 Ω, V_{2}=377.8 V]
25. A single-phase, 10 kVA, 500/250 V, 50-Hz transformer has the following constants :
Resistance : Primary 0.2 Ω, Secondary 0.5 Ω
Reactance : Primary 0.4 Ω, Secondary 0.1 Ω
Resistance or equivalent exciting circuit referred to primary, R_{0} = 1500 Ω, reactance of the equivalent exciting circuit referred to primary, X_{0} =750 Ω.
What would be the readings of the instruments when the transformer is connected for the open-circuit and short-circuit tests?
[Ans. O.C. test: 500 V, 0.745 A, 167 W;S.C. test: 46.8 V, 20 A, 880 W]
26. Find the approximate equivalent circuit of a single-phase 400/4400 V transformer having the following test readings:
O.C. test (l.v. side) : 400 V, 5.2 A, 600 W
S.C. test (h.v. side) : 155 V, 50 A, 1850 W
[Ans. R_{0}=266.7Ω, X_{0} = 80.34 Ω, Z_{01} = 0.02563 Ω,R_{01}=0.006115 Ω, X_{01}=0.02468 Ω]
27. The following readings were taken in open-circuit and short-circuit tests on a single-phase 20 kVA, 2000/200 V transformer :
O.C. test (l.v. side) : 220V, 2 A, 120 W
S.C. test (h.v. side) : 30 V, 10 A, 140 W
Determine:
(a) Equivalent circuit referred to l.v. side
(b) Secondary terminal voltage as full-load at 0.8 power factor leading.
[Ans. (a) R_{0}=333.3 Ω, X_{0}=104.8 Ω,R_{02}=0.014 Ω, X_{02}=0.2654 Ω; (ii) 200.47 V]
Regulation and Efficiency of a Transformer
28. The corrected instrument reading obtained from open and short-circuit tests on 10 kVA, 450/120 V, 50-Hz transformer are:
O.C. Test:
V_{1} = 120 V, I_{1} = 4.2 A, W_{1} = 80 W; V_{1}, W_{1} and I_{1} were read on the low-voltage side.
S.C. Test:
V_{1} = 9.65 V ; I_{1} = 22.2 A, W_{1} = 120 W; with the low-voltage side.
Calculate:
(a) The equivalent circuit (approximate) constants.
(b) Efficiency and voltage regulation for 0.8 lagging power.
(c) The efficiency at half full-load and 0.8 lagging power factor load.
[Ans. (a) R_{0} = 25.30 Ω, X_{0} = 409 ; (b) 97.57%, 2.04% ; (c) 97.34%]
29. The primary and secondary winding resistance of a 40 kVA, 6600/250 V single-phase transformer are 10 Ω and 0.05 Ω respectively. The equivalent leakage reactance as referred to the primary winding is 35 Ω. Find the full-load regulation for load power factors of
- unity, ;
- 0.8 lagging ; and
- 0.8 leading
[Ans. (a) 2.202% ; (b) 3.69% ; (c) 0.8 leading.
30. The percentage resistance and reactance of a transformer are 2.5% and 4% respectively. Find the approximate regulation on full-load at
- Unity power factor,
- 0.8 ;power factor lagging, and
- 0.8 power factor leading.
[Ans. (a) 2.5% ; (b) 4.4% (c) -0.4%]
31. A single-phase 100 kVA, 2000/200 V, 50 Hz transformer has impedance drop of 10% and resistance drop of 5%.
- Find the regulation at full-load, 0.8 power factor lagging.
- At what power factor is the regulation zero?
[Ans. 9.196% ; 0.866 leading]
32. The high voltage of a single-phase 220 kVA 4400/220 V transformer takes a current of 30 A and power of 1200 W at 75 V when the low voltage winding is short-circuited. Determine:
- The voltage to be applied to the high voltage winding on full-load 0.8 power factor lagging if the secondary terminal voltage is to be kept at 220 V, and
- % regulation.
[Ans. (a) 4506 V; (b) 2.2355%]
33. A 20 kVA, 2200/220 V 50-Hz distribution transformer is tested for efficiency and regulation as follows;
O.C. test (l.v. side) : 220 V, 4.2 A, 148 W.
S.C. test (h.v. side) : 86 V, 10.5 A, 360 W.
Determine:
(i) Core loss
(ii) Equivalent resistance referred to primary.
(iii)Equivalent resistance referred to secondary.
(iv) Equivalent reactance referred to primary.
(v) Equivalent reactance referred to secondary.
(vi) Regulation of transformer at 0.8 power factor lagging current.
[Ans. 148 W, 3.26 Ω, 0.0326 Ω, 7.51 Ω, 0.0751 Ω, 2.95%, 97.4%, 97.3%]
34. A 100 kVA single-phase transformer has a full-load primary current of 400 A and total resistance referred to primary is 0.006 Ω. If the iron loss amount to 500 W, find the efficiency on full-load and half-load at
- unity power factor, and
- 0.8 power factor.
[Ans. 98.58%, 98.56%, 98.22%, 98.19%]
Maximum Efficiency
35. A 50 kVA, single phase transformer has an iron loss of 400 W and full-load copper loss of 800 W.
- Find the load at which maximum efficiency occurs and the value of maximum efficiency at unity power factor.
- If the maximum efficiency occurs at 80% full load, find the new core loss and full-load copper loss assuming that total full-load loss is constant.
[Ans. (a) 70.7% of full-load, 97.79% ; (b) P_{i} = 468.36 W, P_{c} = 731.64 W]
36. A 1100/220 V transformer has a primary resistance of 0.25 Ω and secondary resistance of 0.03 Ω. If iron loss amounts to 250 W, determine the secondary current at which maximum efficiency occurs and find the maximum efficiency at 0.8 power factor.
[Ans. 111.8 A ; 97.58%]
37. A single-phase 250 kVA transformer has an efficiency of 96% on full-load at 0.8 power factor and on half full-load at 0.8 power factor. Find:
- Iron loss, and
- full-load copper loss.
[Ans. P_{i} = 2.767 kW, P_{c} = 5.533 kW]
38. The efficiency of a 300 kVA, single-phase transformer is 97.8% when delivering full-load at 0.8 power factor lagging and 98.4% when delivering half full-load at unity power factor. Determine the efficiency at 80% of full-load at 0.8 power factor lagging.
[Ans. 97.95%]
39. A 12 kVA, 400/200 V single phase 50 Hz transformer has maximum efficiency of 95% at 85% of full-load at unity power factor. Determine the efficiency at full-load at 0.8 power factor lagging.
[Ans. 93.76%]
40. A 50 kVA, 2000/250 V single-phase transformer has resistance of 1.1 Ω and 0.015 Ω and reactance of 4.4 Ω and 0.06 Ω for the high voltage and low-voltage windings respectively. It has a maximum efficiency of 96.05% at 80% of full-load at unity p.f. The magnetizing current for the h.v. side at 2000 V is 1.25 A. Find the readings of suitable instruments for open circuit test and short circuit tests, supply being given to h.v. side in both cases.
[Ans. O.C. test : 2000 V, 1.316 A, 824 W
S.C. test: 212.3 V, 25 A, 1287 W]
41. The maximum efficiency of a single-phase 240 kVA, 2000/250 V transformer occurs at 70% of full-load and is equal to 98% of 0.8 p.f. lagging. Determine the efficiency and regulation on full-load at 0.8 p.f. lagging if the impedance of the transformer is 8 per cent.
[Ans. R_{01} = 105.3 Ω, X_{0} = 33.11 Ω,
R_{02} = 0.01538 Ω, X_{02} = 0.02451 Ω]
42. The maximum efficiency of a single-phase 240 kVA, 2000/250 V transformer occurs at 70% of full-load and is equal to 98% at 0.8 p.f. lagging. Determine the efficiency and regulation on full-load at 0.8 p.f. lagging if the impedance of the transformer is 8 per cent.
[Ans. 97.81%, 5.699%]
43. A 5 kVA, 230/115 V transformer takes 1.2 A and 75 W when 230 V is applied to h.v. winding and l.v. winding is kept open. It takes 21.75 A and 150 W when the l.v. winding is short-circuited and 17.4 V is applied to the h.v. winding. Find:
- The no-load current as a fraction of full-load input current at 0.8 p.f. lagging ;
- Percentage regulation on full-load input current at 0.8 p.f. lagging ; and
- The load at which maximum efficiency occurs and the maximum efficiency at unity p.f.
[Ans. 5.162% of full-load input current ; 6.566%; 70.7% of full load, 95.94%]
44. The efficiency of a 20 kVA, 2000/200 V transformer is 96.8% at full-load at unity p.f. and 96% at full-load at unity p.f. and 96% at 60% of0.8 p.f. lagging ; and 0.8 p.f. leading if the impedance is 7 per cent.
[Ans. (a) 5.469%, (b) -2.385%]
45. A 100 kVA, 2000/200 V single-phase transformer takes a current of 50 A and 2400 W at 100 V when the low voltage winding is short circuited. Determine the low-voltage and percentage regulation when delivering full-load current at 0.8 p.f. leading,, the supply voltage being 2000 V.
[Ans. 200.142 V; -0.071%]
46. A single-phase, 25 kVA transformer has an iron loss of 240 W and full-load copper loss of 600 WFind the load at which maximum efficiency occurs and maximum efficiency at 0.8 p.f.
If the maximum efficiency occurs at 70% of full-load, find the core loss and copper loss assuming the total loss to be the same as in the previous case.
[Ans. (a) 63.25% of full-load, 96.36%,
276.3 W, 563.7 W]
All-Day Efficiency
47. A 20 kVA, 2000/200 V transformer has an iron loss of 300 W and full-load copper loss of 400 W. During the day it is loaded as follows:
No. of hours |
Load | Power factor |
8 |
¼ load | 0.5 |
6 |
Full-load |
0.8 |
6 | ¾ load |
unity |
4 | No Load |
- |
Find all-day efficiency,
[Ans. 94.84%]
48. A 30 kVA transformer has got a maximum efficiency of 97% at 80% of load at unity p.f. During the day it is loaded as follows:
No. of hours |
Load | Power factor |
10 |
4.4 kW | 0.6 lag |
9 | 20 kW |
0.8 lag |
5 | 24 kW |
0.85 lag |
Find the all-day efficiency.
[Ans. 95.68%]
49. The all-day efficiency of 200 kVA transformer is 96% of full-load when it is loaded as follows:
No. of hours |
Load | Power factor |
12 |
120 kW | 0.8 |
8 | 150 kW |
Unity |
4 | No-Load |
- |
If maximum efficiency of this transformer occurs at 80% of full-load, find the iron loss and full-load copper loss.
[Ans. P_{i} = 2.645 kW, P_{c} = 4.133 kW]
50. A auto transformer supplies a load of 5 kW at 125 V and at unity power factor. If the primary voltage is 250 V, determine
- Transformer ratio,
- Secondary current,
- Primary current,
- Number of turns across secondary if the total number of turns is 250.
- Power transformed, and
- Power conducted directly from the supply mains to load
[Ans. ½, 40 A, 20 A, 125, 9.5 kW, 2.5 kW]
51. The primary and secondary voltages of an auto-transformer are 500 V and 400 V respectively. Show with the aid of diagram the current distribution in the windings when the secondary current is 100 A and calculate the economy in copper
[Ans. 80 per cent]
52. A 200/250 V auto-transformer draws power from a 200 V line and supplies a 5 kW load with a power factor of 0.8 lagging. A second load of 2 kW is supplied at unity power factor from 100 V winding. Neglecting losses, calculate the current drawn by the transformer from the 200 V line and its power factor.
[Ans. 42.64 A, 0.898]
53. A 5 kVA, single-phase 50-Hz transformer has full-load efficiency of 95 per cent and an iron loss of 50 W. The transformer is now connected as an auto-transformer to a 220 V supply. If it delivers a 5 kW load at unity power factor to a 110 V circuit, calculate the efficiency of the operation and the current drawn by the high-voltage side.
[Ans. 79.27%, 23.2 A]
54. A 480/120 V, 5 kVA two winding transformer is to be used as an auto-transformer to supply power at 480 V from 600 V source. Draw the connection diagram and determine the kVA capacity as an auto-transofrmer.
[Ans. 25 kVA]
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