**Star/Delta Connections**

**1. **Three equal impedances each having a resistance of 20 Ω and reactance of 15 Ω are connected in star to 400 V, 3- phase, 50 Hz system. Calculate:

*(i) *The line current,

*(ii) *The power factor, and

*(iii) *The power consumed.

[**Ans**. 9.24 A; 0.8 (lag) ; 5120 W]

**2. **Three equal impedances are star-connected to a 3-phase, 50 Hz supply. If the resistance and reactance of each branch are 25 Ω and 38 Ω respectively, calculate:

*(i) *The line current, and

*(ii)*The power consumed.

[**Ans**. 5.28 A; 2086 W]

**3. **Three resistances of 20 Ω each are connected in star across the 400 V, 3-phases A.C. supply. Calculate :

*(i) *The line and phase currents,

*(ii) *Phase voltages, and

*(iii) *Total power taken

[**Ans.** (i) 11.55 A; (ii) 231 V ; (iii) 8.0 kW]

**4. **A star-connected, 3-phase load consists of three similar impedances. When the load is connected to a 3-phase, 500 V, 50 Hz supply, the line current is 28.85 A and the power factor is 0.8 lagging, calculate:

*(i) *The total power taken by the load, and

*(ii) *The resistance of each phase of the load .

[**Ans**. 20 kW ; 8 Ω]

**5. **Three non-inductive resistances each of 50 Ω are connected in star across 400 V, 3-phase A.C. supply. Calculate the current through each. Calculate the current if they were connected in delta across the same supply.

[**Ans**. 4.62 A ; 8 A]

**6. **Three identical coils are connected in star to a 200 V (line voltage), 3-phase, A.C. supply and each coil takes 200 W. The power factor is 0.8 (lagging), calculate:

*(i) *The line current,

*(ii) *Impedance, and

*(iii) *Resistance and inductance of each coil.

[**Ans**. 2.165 A ; 53.334 Ω, 42.662 Ω, 0.102 H]

**7. **In a three-phase, 3-wire system with star-connected load the impedance of each phase is (6 + *j8) *Ω. If the line voltage is 230 V, calculate:

*(i) *The line current, and

*(ii) *The power absorbed by each phase.

[**Ans.** 13.3 A; 1067 W]

**8. **A balanced star-connected load of (8 + *j6) *Ω per phase is connected to 3-phase, 230 V supply. Find:

*(i) *Line current,

*(ii) *Power factor,

*(iii) *Power,

*(iv) *Reactive volt-amperes, and

*(v) *Total volt-amperes.

[**Ans**. 13.28 A; 0.8 (lag); 4.232 W; 3174; 5290]

**9. **A star-connected, 5000 V, 3-phase alternator is supplying 3,000 kW at power factor of 0.8. Calculate the active and reactive components of the current in each phase.

[**Ans**. 346.2 A; 260 A]

**10. **In a 3-phase, 4-wire system, two phases have currents of 10 A and 6 A in lagging power factors of 0.8 and 0.6 respectively, while the third phase is open-circuited. Calculate the current in the neutral and sketch the vector diagram.

[**Ans.** 7 *-*73° 24']

**11. **In a star-connected load each phase consists of a resistance of 100 Ω in parallel with a capacitor of capacitance 31.8µF*. *When it is connected to a 416 V, 3-phase, 50 Hz supply, calculate:

*(i) *The line current,

*(ii) *The power factor,

*(iii) *The power absorbed, and

*(iv) *The total kVA.

[**Ans**. 3.39 A; 0.707 (leading); 1.728 kW; 20443 kVA]

**12. **Three identical coils connected in delta across 400 V, 50 Hz, 3- phase supply takes a line current of 17.32 A at power factor 0.5 lagging. Determine:

*(i) *The current in each phase, and

*(ii) *Resistance, inductance and impedance of each phase winding.

[**Ans**. 10 A; 20 Ω ; 0.11 H, 40 Ω]

**13. **A 220 V, 3-phase voltage is applied to a balanced delta-connected 3-phase load of phase impedance (15 *+j20) *Ω Find:

*(i) *The phasor current in each line.

*(ii) *What is the power consumed per phase?

*(iii) *What is the phasor sum of the three line currents? Why does it have this value?

[**Ans**. 15.24 A; 1161.6 W; zero]

**14. **Calculate *(i) *line current and *(ii) *the total power absorbed when three coils each having a resistance of 10 Ω and reactance of 7 Ω are connected *(a) *in star and *(b) *in delta across a 400 V, 3-phase supply.

[**Ans**. 18.93 A, 10750 W; 56.7 A, 32250 W]

**15. **A delta-connected balanced 3-phase load is supplied from a 3-phase 400 V supply. The line current is 20 A and power taken by the load is 10000 W. Find:

*(i) *Impedance in each branch ; and

*(ii) *The line current, power factor and power consumed if the same load is connected in star.

[**Ans.** 34.6Ω, 20/3 A, 0.7217, 3330 W]

**16. **A balanced 3-phase load consists of resistances of 4 0 each. Determine the total power when the resistances connected to 400 V supply are:

(i) Star connected

(ii) Delta connected.

[**Ans**. 40 kW; 120 kW]

**17. **Three non-inductive resistances, each of 1000 are connected in star across 400 V supply. Calculate the current through each. What would be the current.through each if they are connected in delta across the same supply.

[**Ans**. 2.31 A; 4 A]

**18. **Three 100 Ω non-inductive resistances are connected in *(i) *star *(ii) *delta across a 400 V, 50 Hz, 3-phase mains. Calculate the power taken from the supply system in each case. In the event of one of the three resistances opened, what would be the value of the total power taken from the mains in each of the two

[**Ans**. 1600 W; 4800 W]

**19. **A 3-phase delta connected load; each phase of which has an inductive reactance of 400 and a resistance of 25 Ω, is fed from the secondary of a 3-phase star-connected transformer, which has phase voltage of 240 V. Draw the circuit diagram of the system and calculate:

*(i) *The current in each phase of the load,

*(ii) *The voltage across each phase of the load,

*(iii) *The current in the transformer secondary windings, and

*(iv) *The total power taken from the supply and its power factor.

[**Ans**. 8.8 A; 415.7 V; 15.3 A; 5820 W]

**20. **Three similar resistors are connected in star across. 400 V, 3-phase supply. The line current is 5 A. Calculate the value of each resistor. To what value line voltage be changed to obtain the same current with the resistors connected in delta?

[**Ans**. 46.2 Ω; 133,3 V]

**15. **A 3-phase, star-connected system with 230 V between each phase and neutral has a resistance of 4, 5 and 6 Ω respectively in three phases. Calculate:

*(i) *The current flowing in each phase,

*(ii) *The neutral current, and

*(iii) *The total power absorbed.

[**Ans**. 57.5 A, 46 A, 38.3 A; 16.71 A; 32610 W]

**22. **In a 3-phase, 4 wire system there is a balanced 3-phase motor load taking 20 kW at a power factor of 0.8 lagging while lamps connected between phase conductors and the neutral are taking 5, 4 and 10 kW respectively. Voltage between line conductors is 430 V. Calculate the current in each conductor and in the neutral wire of the feeder supplying these loads.

[**Ans**. 51.2 A; 47.5 A; 80 A; 22.4 A]

**Power Measurement in 3-phase Circuits**

**22. **The power input to a 3-phase induction motor is read by two wattmeters. The readings are 860 W and 240 W. What is the power factor of the motor?

[**Ans**. 0.7155 (lag)]

**23. **While performing a load test on a 3-phase wound rotor induction motor by two wattmeters method, the readings obtained on two wattmeters were + 12.5 kW and – 4.8 kW and the line voltage was 440 V, Calculate :

*(i) *Power drawn by the motor,

*(ii) *Power factor, and

*(iii) *Line current.

[**Ans**. 7.7 kW; 0.2334; 43.3 A]

**24. **The input power to a 3-phase delta-connected motor was measured by two wattmeters method. The readings were 20.8 kW and – 6.8 kW and the line voltage was 400 V. Calculate:

*(i) *Input power,

*(ii) *Power factor, and

*(iii) *Line current.

[**Ans**. 14 kW; 0.281; 72 A]

**25. **A 3-phase, 500 V motor load has a power factor of 0.4*. *Two wattmeters connected to measure the power show the input to be 30 kW. Find the reading on each instrument.

[**Ans**. 35 kW; -5 kW]

**26. **The power in a 3-phase circuit is measured by two wattmeters. If the total power is 100 kW and power factor is 0.66 leading, what will be the reading of each wattmeter? For what power factor will one of the wattmeters read zero?

[**Ans**. 17.15 kW; 82.85 kW; p.f. = 0.5]

**27. **Two wattmeters used to measure the power input in a 3-phase circuit indicate 1000 W and 500 W respectively. Find the power factor of the circuit,

*(i) *when both wattmeter readings are positive; and

*(ii) *when the latter is obtained by reversing the current coil connection.

[**Ans.** 0.866 (lag); 0.1889 (lag)]

**28. **Two wattmeters connected to read the total power in a 3-phase system supplying a balanced load read 10.5 kW and – 2.5 kW respectively, Calculate the total power and the power factor. Draw suitable phasor diagram, explain the significance of

*(i) *equal wattmeter readings; and

*(ii) *a zero reading on one wattmeter.

[**Ans**. 8 kW; 0.3348 (lag) *(i) *unity p.f., *(ii) *0.5 p.f.]

**30. **The power input to a 1000 V, 3-phase induction motor running on full-load is measured by two wattmeters which indicate 300 kW and 100 kW respectively. Determine:

*(i) *Input,

*(ii) *Line current,

*(iii) *Power factor, and

*(iv) *Output if the efficiency of the motor is 92%.

[**Ans**. 400 kW; 303.5 A; 0.756 (lag); 368 kW]

**31. **Two wattmeters are used for measuring the power input and the power factor of an over-excited synchronous motor. If the readings of the meters are – 2.0 kW and + 7.0 kW respectively, calculate the input and power factor of the motor.

[**Ans**. 5 kW; 0.3057 (leading)]

**32. **The power input to a 2-kV, 50 Hz, 3-phase motor running on full-load at an efficiency of 90 per cent is measured by two wattmeters which indicate 300 kW and 100 kW respectively. Calculate :

*(i) *Input,

*(ii) ** *Power factor,

*(iii) *Line current, and

*(iv) *Output.

[**Ans**. 400 kW; 0.756 (lag); 153 A; 360kW]

**33. **In a balanced three-phase system power is measured by two-wattmeters method and the ratio of the two wattmeter readings is 2 : 1 Find the power factor of the system.

[**Ans**. 0.866 (log)]

**34. **The power input of a synchronous motor is measured by two wattmeters both of which indicate 50 kW. If the power factor of the motor be changed to 0.866 leading, determine the readings of the two wattmeters, the total input power remaining the same. Draw the vector diagram for the second condition of load.

[**Ans**. 33.33 kW; 66.67 kW]

**35. **Each phase of a 3-phase delta-connected load consists of an impedance Z = 20 *60° *Ω. The line voltage is 440 V at 50 Hz. Calculate the power consumed by each phase impedance and the total power. What will be the readings of the two wattmeters connected?

[**Ans**. 4840 W; 14520 W; 14520 W. zero]

**36. **A 3-phase 3-wire, 415 V supplies a balanced load of 20 A at power factor 0.8 lagging. Two wattmeter are used to measure power. Calculate:

*(i) *Power,

*(ii) *Reading of wattmeter No 1, and

*(iii) *Reading of wattmeter No.2.

[**Ans**. 11.5 kW; 8240 W; 3260 W]

**37. **A balanced load is supplied from a 3-phase, 400 V, 3-wire system whose power is measured by two wattmeters. If the total power supplied is 26 kW at 0.75 power factor lagging, find the readings of each) of the two wattmeters.

[**Ans**. 19.62 kW; 6.38 kW]

**38. **A 3-phase 4-wire, star-connected system supplies only non-inductive load. The current in line *R *is 8 A, the current in line Y is 10 A and the current in line *B *is 6 A. The voltage from each line to neutral is 120 V. Find:

*(i) *Voltage shown by each of the three wattmeters, and

*(ii) *Power taken by the lighting load.

[**Ans**. 960 W, 1200 W, 720 W ; 2880 W]

**39. **Three identical coils, each having a resistance of 20 Ω and reactance of 20 Ω are connected in *(i) *star (ii) delta across 440 V, 3-phase supply. Calculate for each method of connection the line current and reading on each of the two wattmeters connected to measure power.

[**Ans**. 8.98 A; 3817.5 W, 1022.5 W; 26.95 A; 12452.5 W; 3067.5 W]

**40. **A balanced star-connected load, each having a resistance of 10 Ω and inductive reactance of 30 Ω is connected to a 400 V, 50 Hz supply. The phase sequence is RYB. Two wattmeters connected to read total power have their current coils connected in the red and blue lines respectively. Calculate the reading of each wattmeter. Draw the circuit and vector diagram.

[**Ans.** W_{1} = -585 W ; W_{2} = 2185 W]

**41. **Three equal star-connected inductors take 8 kW at power factor 0.8 when connected to a 460 V, 3ɸ, 3-wire supply. Give the connection diagram of 2 single-phase wattmeters to measure the power in the circuit. Allocate the reading of each wattmeter. Find the line currents if one of the inductor is short-circuited. Draw the vector diagrams of the currents and voltages under the condition.

[**Ans**. 57.32 W, 2268 W; 21.74 A, 21.74 A]

**42. **A 3-phase, 3-wire balanced load with a lagging power factor is supplied at 400 V (between lines). A single phase wattmeter (scaled in kW) when connected with its current coil in the R-line and the voltage coil between R and Y lines gives a reading of 6 kW. When the same terminals of the same voltage coil are switched over to Y-line and B-line, the current-coil connections remaining the same, the reading of the wattmeter remains unchanged. Calculate the line current and power factor of the load. Phase sequence is R → *Y *→ *B. *

[**Ans**. 30 A; 0.866 (lag)]

**43. **A wattmeter reads 5.54 kW when its current coil is connected in red phase and its potential coil is connected between the neutral and red phase of a symmetrical3-phase system supplying a balanced load of 30 A at 400 V. What will be the reading of the instrument if the current coil connections remains unchanged but the potential coil be connected between blue and yellow phase? What does this figure represent?

[**Ans**. 7 kVAR]

**44. **A 3-phase, 434 V, 50 Hz supply is connected to a 3-phase star-connected induction motor and synchronous motor. Impedance of each phase of induction motor is 1.25 + *j2.17 *Ω. The 3-phase synchronous motor is over-excited and it draws a current of 120 A at 0.87 leading power factor. Two wattmeters are connected in usual manner to measure power drawn by the two motors. Calculate:

*(i) *Reading on each wattmeter, and

*(ii) *Combined power factor.

[**Ans**. 63970 W, 52050 W; 0.9845 (lag)]

**45. **A star-connected balanced load is supplied from a 3-phase balanced supply with a line voltage of 416 V

at a frequency of 50 Hz. Each phase ofload consists of a resistance and a capacitor joined in series and

the readings on two wattmeters connected to measure the total power supplied are 782 W and 1980 W,

both positive. Calculate:

*(i) *The power factor of the circuit,

*(ii) *The line current, and

*(iii) *The capacitance of each capacitor

[**Ans**. 0.8 (leading); 4.8 A; 106 µF]

**Unbalanced Load Circuits**

**46. **A 400 V*, *3-phase supply has delta-connected load having 100 Ω between R and Y, 318 mH between Y and B and 31.8 µF between B and *R. *Find the line currents for the phase sequence *RYB. *Calculate the value of star-connected balanced resistor for the same power.

[**Ans**. 7.73 15° A; 7.73 165° A; 4 270° A; 100Ω]

**47. **Three load (31 + j59), (30 – j40) and (80 + j60) 0 are connected in delta across a 400 V, 3-phase supply. Find the phase currents, line currents and power.

[**Ans**. 3 -62.3° A; 466.9° A; 83.1°; A;

4.78 -76 A, 1.038 - 80.35 A,

5.819 - 256.8° ; 1078.9 W ]

**48. **A 3-phase, 400 V, 50 Hz system has the following load connected in delta: between Rand Y lines a non-reactive resistor of 100 Ω, between Y and *B *lines a coil having a reactance of 66 Ω and a resistance of 20 Ω between B and *R *lines a loss free capacitor of 30 µF. Calculate:

*(i) *The phase currents, and

*(ii) *The line currents.

Assume the phase sequence to RYB.

[**Ans.** 4 0°A ; 6.324 - 191.56°: A: 3.774 210° A;

7.5 14.6° A, 10.275 173° A, 4.3 - 47.13° A]

**49. **A balanced 3-phase system supplies an unbalanced 3-phase *delta-connected *load made up of two resistors 100 Ω and 200 Ω and a reactor having an inductance of 0.3 H with negligible resistance. *E _{L}* = 100 V at 50 Hz. Calculae:

*(i) *The total power in the system, and

*(ii) *The total volt-amperes reactive.

[**Ans. **150 W ; 106 (lag)]

**50. **A 400 volt 3-phase, 4-wire system has the loads 12 – *j16, *12 + *j0 *and 8 + *j6 *Ω connected in star. Find the line currents, neutral current arid total power.

[**Ans**. 11.55 *53.1° *A; 19.25 *-120° *A; 23.09 *83.to *A;

*I _{N}= *15.47

*-*90.3° A; 10310.11 W]

**51. **A 3-phase 3-wire system having a 254 V line-to-neutral has the following loads connected between the respective lines and neutral: *Z _{R}= *10

*0°*Ω,

*Z*= 10

_{Y}*37°*Ω,

*Z*= 10

_{B}*-*53° Ω. Calculate the current in the neutral wire and power taken by each load when phase sequence is

*(i) **RYB *

*(ii) **RBY.*

[**Ans**.(i) 24.46 16° 15', 6452 W ; 5162 W ; 3871

(ii) 38.5 -177.3' ; Power same as in (i)]

**52. **A 3-phase star connected system with 230 V between each phase and neutral has resistances of 4 Ω, 5 Ω and 6 Ω respectively in the three phases. Calculate:

*(i) *Current flowing in each phase,

*(ii) *Neutral current, and

*(iii) *Total power absorbed.

[**Ans**. 57.5 A, 46 A, 38.33 A; 16.71 A; 32620 W]

**53. **A 440/254 V, 3-phase, 4-core supplies an unbalanced load represented by the following impedances in ohms between *R, Y *and *B *phases respectively and the neutral 16 + *j12, *14 – *j21 *and 25 O. The phase sequence is *RYE. *Calculate:

*(i) *The current in each conductor of the cable, and

*(ii) *Readings of three wattmeters connected in each line and neutral.

[**Ans**. 12.7 A, 10.06 A, 10.16 A; 2580 W, 1413 W; 2580 W]

**54. **Calculate the voltage across each of the following star-connected impedances when they are connected to a 400 V, 3-phase system, *Z _{R} =j20 *Ω,

*Z*Ω,

_{Y}=j1*ZB*= –

*j1*Ω.

Also find the potential of the star point above ground.

The phase sequence is *RYB.*

[**Ans**. (4000 + *j6928) *V ; (3600 + *j6928) *V ;

* *(3800 + *j7274.4) *V; - 8000 V]

**55. **An unbalanced star-connected load comprising of the impedances *Z _{R} *= (10 +

*j0), Z*= (3 +

_{Y}*j4), ZB*= (0 –

*j10)*is connected across a 3-phase 200 V circuit with balanced voltages. Find:

*(i) *Three line currents, and

*(ii) *Voltage across each impedance.

The phase sequence is *RYB *

[**Ans**. (- 20.2 + *j4.54) *; (7.24 + *j5.48) *; (12.78 – *j10.02);*

(-200.2 + *j45.4) *; (- 0.2 + *j45.4) *; (- 100.2 -)127.8)]

**56. **An unbalanced star-connected load is supplied from a symmetrical 3-phase, 440 V, 3-wire system. Till branch impedances of the-load are *Z _{1}*= 5

*30°*Ω,

*Z*= 10

_{2}*45°*Ω and

*Z*= 10

_{3}*60°*Ω. Assuming the positive phase sequence of

*RYB*determine the line currents and voltage across each impedance. Draw the phasor diagram.

[**Ans**. 35.75 *-71.3° *A; 32.77 *156.1° *A; 27.67 *48° *A,

178.7 -41.3°V, 327.7 201.1°V, 276.7 108°V]

** 57. **An unbalanced star-connected load has branch impedances of

*Z*= 1030° Ω,

_{1}*Z*= 30° Ω

_{2}*-*45° Ω,

*Z*= 20

_{3}*60*° Ω and is connected across a balanced 3-phase, 3-wqire supply of 200 V. Find the line current and the voltage across each impedance using star-delta conversion method.

[**Ans**. 5.39 8° 32' A, 15.79 -148° 6' A,

10.73 44° 16' A, 53.9 38° 32' V,

157.9 -193.6°, 214.6 104° 16']

**58. **A 300 V (line) 3-phase supply feeds a star-connected load consisting of non-inductive resistors of 15, 6 and 10 Ω connected to the *R, Y *and *B *lines respectively. The phase sequence is *RYB. *Calculate the voltage across each resistor,

[**Ans**. (195 - *j78) *V, (- 105 - *j78) *V, (4.5 + *j182)* V]

**59. **A 400 V, 3-phase, 3-wire symmetrical system *RYB *supplies a star-connected 1000 whoso branch circuit impedances are: *Z _{R} *=

*10*

*+j10; Z*= 20

_{Y}*+ j20*-;

*Z*

_{B}= 0 –*j10.*

Determine the current in each branch.

Take the phase sequence as R → Y → B.

[**Ans**. 22.55 -57°45’ ; 6.48-133.7° ; 29109° 45’]

**Power Factor Improvement**

**60. **At 50-MVA, 11 kV 3-phase alternator supplies full-load at a lagging power factor of 0.7. What would be percentage increase in earning capacity if the power factor is increased to 0.95.

[**Ans**. 35.7 per cent]

**61. **A 3-phase, 37.3 kW, 440 V, 50 Hz induction motor operates on full-load with an efficiency of 89% and a, power factor of 0.85 lagging. Calculate the total kVA rating of capacitors required to raise the full-load power factor to 0.95 lagging. What will be the capacitance per phase if the capacitors are:

*(i) *Delta-connected, and

*(ii) *Star-connected.

[**Ans**. 12.19 kVA; 66.8 µF ; 200.4 µF]

**62. **Three impedance coils, each having a resistance of 20 Ω and a reactance of 150 are connected in star to a 400 V, 3-phase, 50 Hz supply. Calculate:

*(i) *line current,

*(ii) *Power supplied, and

*(iii) *The power factor.

If three capacitors, each of the same capacitance are connected in delta to the same supply as to form parallel circuit with the above impedance coils, calculate the capacitance of each capacitor to obtain a resultant power factor of 0.95 lagging.

[**Ans**. 9.24 A, 5120 W, 0.8 (lag) ; 14.32 µF]

**63. **A balanced 3-phase load, each element of which consist of a 5 Ω resistance in series with an inductance of reactance 3 Ω, is connected in star across mains whose line potential difference is 400 V and frequency 50 Hz. What is the power consumed by the system?

Three equal capacitors are connected across the mains in the mesh. What should their capacitance be to cause the total line current to be in phase with the line voltage?

[**Ans**. 23.55 kW; 93.8µF/phase]

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