## Iii5 Power In Ac Circuits

This section considers three aspects of power in ac circuits. First, the case of a circuit containing resistance and inductance is discussed, followed by the introduction of the power triangle for circuits containing resistance and inductance. Finally, power-factor improvement by the use of capacitors is presented.

III.5.1 Power in a Circuit Containing Both Resistance and Inductance

Figure III.13 reviews this situation through a circuit drawing and the voltages and currents shown in the complex plane. Meters are in place that read the effective or rms voltage V across the complex load and the effective or rms line current I.

Power is usually thought of as the product of voltage and the current in a circuit. The question is: The current I times which voltage will yield the correct or true power? This is an important question, since Figure III.13b shows three voltages in the complex plane.

Each of the three products may be taken, and each

Fig. III.13 (a) Circuit having resistance and inductance; meters are in place to measure the line current I and the voltage V. (b) Relationship between the various voltages and the line current for this circuit.

has a name and a meaning. Taking the ammeter reading I times the voltmeter reading yields the apparent power. The apparent power is the load current-load voltage product without regard to the phase relationship of the current and voltage. This figure by itself is meaningless:

If the voltmeter could be connected across the resistor only, to measure vR, then the line current-voltage product would yield the true power, since the current and voltage are in phase.

Usually, this connection cannot be made, so the true power of a load is measured with a special meter called a wattmeter that automatically performs the following calculation.

Note that in Figure III.13b, the circuit voltage V and the resistance voltage VR are related through the cosine of 6. The third product that could be taken is called imaginary power or VAR, the voltampere reactive product.

P(imaginary) = IvL

This is the power that is alternately stored and given up by the inductor to maintain its magnetic field. None of this reactive power is actually used.

If the voltages in the foregoing examples were measured in kilovolts, the three values computed would be the more familiar:

P(apparent) = kVA

P(imaginary) = kVAR

This discussion, together with Figure III.13b, leads to the power triangle.

### III.5.2 The Power Triangle

The power triangle consists of three values, kVA, kW, and kVAR, arranged in a right triangle. The angle between the line current and voltage, 0, becomes an important factor in this triangle. Figure III.14 shows the power triangle.

To emphasize the relationship between these three quantities, an example may be helpful. Suppose that we have a circuit with inductive characteristics and using a voltmeter, ammeter, and wattmeter the following values are measured:

line current = 10 A

line voltage = 240 V

From this information we should be able to determine the kVA, 0, and the kVAR.

The kVA can be computed directly from the voltmeter and ammeter readings:

Fig. III.14 Power triangle for an inductive load. The angle 0 is the angle of displacement between the line voltage and the line current.

Looking at the triangle in Figure III.14 and recalling some basic trigonometry, we have and 0 is the angle whose cosine equals 0.625. This can be looked up in a table or calculated using a hand calculator that computes trig functions:

Again referring to the power triangle and a little trig, we see that kVAR = kVA sin 0

Figure III.15 puts all these measured and calculated data together in a power triangle.

Of particular interest is the ratio kW/kVA. This ratio is called the power factor (PF) of the circuit. So the power factor is the ratio of true power to apparent power in a circuit. This is also the cosine of the angle 0, the angle of displacement between the line voltage and the line current. To improve the power factor, the angle 0 must be reduced. This could be accomplished by reducing the kVAR side of the triangle.

Fig. III.15 Organization of the measured and computed data of the example into a power triangle.

Fig. III.15 Organization of the measured and computed data of the example into a power triangle.

### III.5.3 Power-Factor Improvement

Recall that inductive reactance and capacity reactance are plotted in opposite directions on the imaginary axis, j. Thus it should be no surprise to consider that kVAR produced by a capacitive load behave in an opposite way to kVAR produced by inductive loads. This is the case and is the reason capacitors are commonly added to circuits having inductive loads to improve power factor (reduce the angle 0).

Suppose in the example being considered that enough capacity is added across the load to offset the effects of 90% of the inductive load. That is, we will try

Fig. III.16 (a) Inductive circuit with capacity added to correct power factor. (b) Power vectors showing the relationship among kW, kVAR inductive, and kVAR capacitive.

to improve the power factor by better than 90%. Figure III.16 shows the circuit arrangement with the kW and kVAR vectors drawn to show their relationship.

Following the example through, consider Figure III.17, where 90% of the kVAR inductive load has been neutralized by adding the capacitor.

Working with the modified triangle in Figure III.17, we can compute the new 0, call it 02.

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