AC Power Analysis

Keywords

average power, instantaneous power, maximum average power, effective value, root-mean-square value, apparent power, power factor, power factor angle, complex power, real power, reactive power, quadrature power, volt-ampere reactive, power factor correction, shunt capacitance, shunt inductance

Related Concepts

alternating current, load, Thevenin’s theorem, reactance, resistance

Instantaneous and Average Power

The instantaneous power (in watts) is the power at any instant of time, and is the product of the instantaneous voltage across the element and the instantaneous current through it. It is mathematically expressed as

Helpful trigonometric identity

Power is the rate at which an element absorbs energy. When the power is positive, it implies that the circuit absorbed power. Conversely, if it is negative, the source absorbed the power—power moves from circuit to source.

Because the instantaneous power is not constant, the average power is used for convenience. This quantity pertains to the average of the instantaneous power over one period. A key difference between the instantaneous power and the average power is that the former depends on time, while the latter doesn’t. The average power is mathematically expressed as

INFO

A resistive load () absorbs power at all times, while a reactive load ( or ) absorbs no average power.

Maximum Average Power Transfer

The value of the maximum power delivered to the load is equal to the power’s value when the load impedance is equivalent to the Thevenin impedance ().

Summary of Relevant Formulas

Description	Formula
Thevenin impedance
Load impedance
Current through the load
Average power delivered to the load
Load impedance needed for the maximum average power
Maximum average power

Effective or RMS Value

The effective value of a periodic current (or ac current) is the dc current that delivers equal average power to the resistor as the periodic current. It is crucial for measuring a source’s effectiveness in delivering power to a resistor.

Because finding the effective value results in the root of the mean of the square of the periodic signal, the effective value became known as the root-mean-square value or rms value, thus

For any periodic function , the rms value is given by

Summary of the Other Relevant Formulas

Description	Formula
Effective value of
Effective value of
Average power in terms of rms values
Average power absorbed by a resistor

Sinusoidal voltage or current ( or ) is often specified in terms of their rms value or maximum value because their average value is 0; however, power industries express the phasor magnitudes in terms of rms values instead of peak values.

Apparent Power and Power Factor

The product of the rms values of voltage and current is known as the apparent power (expressed in ). Because it feels like it is apparent that the power should be the product of voltage and current (like in dc circuits), the quantity became known as the apparent power.

while the factor is known as the power factor (pf). This quantity is also the ratio between of the average power to the apparent power. The angle, or argument of the cosine of the power factor, is referred to as the power factor angle

TIP

The power factor is equivalent to the load impedance when is the voltage across the load and is the current through it.

Different pf values and cases depending on the load:

1. purely resistive load: , , and the apparent power is equal to the average power.
2. purely reactive load: , , and the average power is 0.

In both cases, pf is either leading or lagging; the former happens when current leads voltage (capacitive load), and the latter happens when the current lags voltage (inductive load).

Complex Power

Power engineers developed the concept of complex power () to find the total effect of parallel loads. Complex power contains all important information related to the power absorbed by a particular load (measured in ).

The magnitude of the complex power is equivalent to the apparent power, while its angle is equivalent to the power factor angle. We can express the complex power in terms of the load impedance , hence

When expressed in rectangular form, we can obtain the real () and imaginary () parts of the complex power

The real part is known as the average or real power (in Watts); it depends on the load’s resistance . On the other hand, the imaginary part is known as the reactive or quadrature power (in volt-ampere reactive ); it depends on the load’s reactance .

and

and are used instead of Watts for the purpose of distinguishing the real power from the other types.

The real power is the only useful power, and is dissipated by the load. In contrary, the reactive power measures the lossless energy exchange between the load and the source. Notice that:

1. for resistive loads (unity pf).
2. for capacitive loads (leading pf).
3. for inductive loads (lagging pf).

Summary of Relevant Formulas

Description	Formula
Complex power
Apparent power
Real power
Reactive power
Power factor

Conservation of AC Power

The total power supplied by the source is equal to the total power delivered to the load, thus

In other words, the different powers (i.e., complex, real, and reactive) of the sources is equal to the respective sums of the different powers of the individual loads—the total is the sum of the individual components.

Power Factor Correction

Most loads are inductive and operate at a low lagging power factor; nonetheless, we can increase its power factor, despite its immutable nature, through power factor correction. In simple terms, it can allow us to make the power factor closer to unity by adding a reactive element in parallel with the load. For example, we can improve the power factor of an inductive load by installing a capacitor parallel to it.

Power factor correction on an inductive load results in a lesser current drawn; hence, it decreases power losses. This means that power factor correction, or making the power factor closer to unity, can make it cheaper and more efficient to use electrical appliances.

Example Application of Power Factor Correction

1. If we have an inductive load with the apparent power , then our real power is and our reactive power is .
2. If we desire converting the power factor from to without altering the real power, we alter the reactive power into by using a shunt capacitor, which is equivalent to .
3. The formula for the shunt capacitance is

• If a shunt inductance is needed instead, we use the formula

Sources

1. Fundamentals of Electric Circuits by Charles K. Alexander and Matthew N.O. Sadiku (Chapter 11)