# Basic Three-Phase and AC Power:

**Phase and AC Power:**

- Using the instantaneous power relationship, prove that two watt-meter can be used to measure power in a 3-phase 3-wire system.

- In the two watt-meter method, prove that one watt-meter will measure zero power if the power factor angle was 60.

- A single phase load is drawing 8 kW from a 240V power supply at 0.8 PF lagging. Calculate the load current and impedance.

- A Y-connected load with P=8 kW and Q=6 kVAr from a 480-V 3-phase supply. Calculate the load PF and the load current. Calculate the capacitance required to increase the PF to 0.95. Plot the value of the capacitance against the PF within the range 0.8-1.0.

- A load consisting of three identical impedances is connected to a three-phase 220-V source.

- Determine the phase and line currents
- Determine the total power P, reactive power Q, voltampere (apparent power) S, and the power factor PF

- Three equal impedances of are connected in Y across a 220-V 3-phase supply.

- Determine the phase voltage
- Determine the phase and line current
- Determine the total power P, reactive power Q, voltampere (apparent power) S, and the power factor PF

- In problem 5, size the capacitor needed to improve the power factor at the service entrance to 0.98

- In problem 6, size the capacitor needed to improve the power factor at the service entrance to 0.98

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**3-Phase Induction Motors**

- A 460-V, 3-phase, 50Hz, 8-pole, Wye-connected induction motor has the following equivalent circuit parameters per phase;

Mechanical losses = 1-kW

Full load slip s = 0.03

Using the exact equivalent circuit, calculate the followings:

- The speed of the rotating field
- The speed of the motor
- The rotor frequency
- The rotor current referred to the stator
- The no-load current
- The ratio of the no-load to the full-load current
- The stator current
- The stator emf
- The power and magnetizing components of the no-load current
- The rotor cupper losses
- The input power factor PF
- The full-load torque
- The maximum torque
- The speed at which maximum torque will occur
- The starting torque
- The net output horse power
- The motor efficiency

- Repeat problem (1) using the approximate equivalent circuit

- Repeat problem (1) while neglecting the stator impedance

- (a) Determine the conditions for delivering maximum torque at starting of a three-phase induction motor.

- A 500-V, 3-phase, 8-pole, 50Hz, Y-connected induction motor has the following equivalent circuit parameters;
*R*_{1 }= 13ohms,*R*_{2}^{` =}0.13ohms,*x*_{1 }= 0.6 ,*x*_{2}^{`}=0.6 .The magnetizing branch_{admittance }*Y*= 0.004-_{m}*j*0.05 referred to the stator (primary) side. The full-load slip is 5% and the rotational losses (mechanical losses) is 890 W. Using the approximate equivalent circuit; determine the full-load electromagnetic torque, the shaft torque, the stator input current, the total input power and input power factor, the gross mechanical output, the rotor copper losses and efficiency. - Describe the no-load (light running) and locked rotor tests of a 3-phase induction motor.

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**T.L Models:**

- Define voltage regulation of transmission line. Provide the formula used to calculate the percentage voltage regulation of a power transmission line.

- Determine the ABCD constants of a T-section model of a power transmission line.

- A short 3-phase, 33-kV power transmission line delivers a load of 7-MW at a power factor of 0.85 lagging and 33-kV. If the series impedance of the line is 20+j30 Ohms/phase, calculate

- The ABCD constants (parameters)
- The sending end voltage
- The load angle
- The voltage regulation
- The transmission efficiency

- A 275-kV, 3-phase power transmission line of length 300 miles is rated 850-A. The values of the resistance, inductance, and susceptance per phase per mile are 0.125-, 1.7-mH, and 5.92, respectively. The receiving-end voltage is 275-kV when full-load is transmitted at 0.85 PF lagging. Using the long line model, calculate

- The ABCD constants
- The sending end voltage
- The load angle
- The voltage regulation
- The transmission efficiency

- Repeat problem (5) using the -section model. Compare the voltage regulation and transmission efficiency to those obtained in (5)

**Transformers:**

- A 50kVA, 1-phase, 60 Hz, 4400/200 V distribution transformer has a maximum efficiency at full load. The transformer gave the following test results

Open-circuit test: 4400 V applied to HT side, power taken 320 W.

Short-circuit test: Secondary short-circuited, power taken 320 W at full load.

Calculate the transformer energy efficiency (all-day efficiency) for the load cycle over 24 hour period given that the transformer works at full-load for 4 hours, at half load for 8 hours, at quarter load for 6 hours and at no-load for 6 assuming a unity power factor.

- In reference to the transformer in problem (1), determine the approximate equivalent circuit model of the transformer.

- In reference to the transformer in problem (1), determine the percentage voltage regulation at full load and 0.8 PF lagging, 0.8 PF leading, and unity PF.

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**Category**: Homework Help