# AC Circuits | Power Analysis

Electric utilities, electronic, and communication systems involve transmission of power from one point to another and all devices and gadgets have a maximum amount of electrical power they can safely operate with. Thus, power analysis is of paramount importance. In DC circuits, power is described as the product of voltage and current. In AC circuits, we have three different types of power: real, reactive, and apparent power. We shall begin with instantaneous and average power and proceed to other AC power concepts.

## Instantaneous and Average Power

*Instantaneous Power*

The instantaneous power (in watts) is the power consumed at any particular point in time. It is the power absorbed by an element at a specific instant of time and is the product of the instantaneous voltage across the element and the instantaneous current through it.

Following the passive sign convention,

Consider the general case of instantaneous power absorbed by a combination of circuit elements fed by a sinusoidal source, as in Figure 1.

Figure 1: Passive linear network connected to a sinusoidal source

We write the voltage and current at the terminals as

where *V _{m}* and

*I*are the amplitudes, and

_{m}*v*and

*i*are the phase angles of the voltage and current, respectively. Multiplying

*v(t)*with

*i(t)*and applying a product identity, the instantaneous power absorbed by the circuit is

This result shows that the instantaneous power has two components:

- The first component is constant and its value is dependent on the phase difference between the voltage and the current. Later, we will see that this represents the average power. It is responsible for an amplitude shift in the power as shown in Figure 2.
- The second part is time-dependent. It is a sinusoidal function with a frequency of , which is twice the angular frequency of the voltage or current.

Figure 2: Instantaneous power

*p(t)*entering a circuit

A sketch of is shown in Figure 2, where is the period of voltage or current. We readily observe that is periodic, , and has a period of , since its frequency is twice that of voltage or current.

is either positive or negative in different parts of a cycle. - When , power is absorbed by the circuit. Power is transferred from the source to the circuit.
- When , power is absorbed by the source. Storage elements such as capacitors and inductors allow power transfer from the circuit to the source.

The instantaneous power is time-varying and is difficult to measure, hence it is more convenient to deal with the *average power*.

## Average, Active, or Real Power

The average power *P* (in watts) is the instantaneous power *p(t)* averaged over one period. Average power is sometimes referred to as active power, real power or simply power.

The average power can be obtained when both voltage and current are in the time domain or when both are in the frequency domain.

*Time domain*

Substituting *p(t)* in eq.(3) into eq.(4) gives

- The first integrand is a constant, hence the average is the same constant.
- The second integrand is a sinusoid. The area under the sinusoid during the positive half-cycle is compensated by the area under during the following negative half-cycle; the second term is zero and

- Since , what is relevant is the
*phase difference*between the voltage and current. - While
*p(t)*is time-varying,*P*, on the other hand, does not depend on time. To find the instantaneous power, it is essential to have both*v(t)*and*i(t)*in the time domain.

- Since , what is relevant is the

*Frequency domain*

The phasor forms of the voltage *v(t)* and current *i(t)* in eq.(2) are and , respectively. *P* can be calculated using eq.(6) or using phasors **V** and **I**. To use phasors, we notice that

From eq 6, the real part of this expression is the average power *P*. Thus,

*Resistive and Reactive Loads*

- When , he voltage and current are in phase. This implies that the load or circuit is purely resistive (R) and

where . This shows that*a purely resistive circuit absorbs power at all times*. - When , we have a purely reactive circuit (L or C ), and

This shows that

*a purely reactive circuit absorbs no average power.*

A resistive load (R) absorbs all circuit power; while a reactive load (L or C ) alternately absorbs power from and returns power to the source, resulting in zero average power.

## Maximum Average Power Transfer

Consider the circuit in Fig. 3, where an AC circuit is connected to a load which is represented by its Thevenin equivalent. The load is represented by *Z _{L}*.

Figure 3: Finding the maximum average power transfer: (a) circuit with load, (b) the Thevenin equivalent

The Thevenin impedance *Z _{Th}* and the load impedance

*Z*in rectangular form are

_{L}The current through the load is

From eq.(9), the average power delivered to the load is

We adjust the load parameters *R*_{L} and *X*_{L} so that eq.(14) yields the maximum value of *P*. To do this we set and equal to zero. From eq.(14),

- Setting to zero gives

- Setting to zero gives

Combining eq.(15) and eq.(16) , we can infer that for maximum average power transfer, we must select a *Z _{L}* such that

*X*and

_{L}= -X_{Th}*R*, i.e.,

_{L}= R_{Th}*The load impedance Z_{L} is equivalent to the complex conjugate of the source impedance for maximum average power transfer. In other words, Z_{L} = Z_{Th}*; where Z_{Th}* is the complex conjugate of the Thevenin impedance.
*

Setting *R _{L} = R_{Th} *and

*X*in eq.(14) gives us the maximum average power as

_{L}= -X_{Th}In the case of a load that is purely real, the condition that must be satisfied for maximum power transfer is derived from eq.(16) by setting XL=0; that is,

*For a purely resistive load, the load resistance is equal to the Thevenin impedance for maximum average power transfer.*

## Effective or RMS Value

*Effective value* is a concept that arises from the necessity to measure the effectiveness of a voltage or current source in delivering power. For current, the effective value of ac current *i* is equal to dc current *I* that delivers the same average power to the load as *i.*

Figure 4: Finding the effective current: (a) ac circuit, (b) dc circuit

Our goal is to find the current *I*_{eff} that will transfer the same power to the resistor *R* as the sinusoid *i*.

- In the ac circuit, the average power absorbed by the resistor is

- In the dc circuit, the power absorbed by the resistor is

Equating the expressions in eqs.(20) and (21) and solving for *I*_{eff},

We can find the effective value of voltage the same way as current; that is,

From eq.(23), the effective value is the root of the mean of the square of the periodic signal. For this reason, the effective value is also known as the rms value (root-mean-square value); and we can write

The rms value of a periodic function x(t) is generally given by

*The root mean square (rms) of a periodic signal is its effective value.* We first find the square of *x(t)* and then find the mean of that, and then the square root of the mean. The rms value of a constant is equal to the constant itself.

For the sinusoid , the effective or rms value is

Similarly, for ,

Eqs. (26) and (27) are only valid for sinusoidal signals.

The average power in eq.(6) can be written in terms of the rms values:

The average power absorbed by *R* in eq.(9) can also be written as

Sinusoidal voltage or current is usually specified in terms of its peak value or its rms value. In power analysis, it is often convenient to express voltage and current in terms of their rms values.

## Apparent Power and Power Factor

We can express the average power in eq.(28) as a product of two terms:

The product *V _{rms}I_{rms}* is referred to as the apparent power

*(S)*, while the factor is called the power factor (pf).

*Apparent Power*

The apparent power (in VA) is the product of *V _{rms}* and

*I*.

_{rms}- It seems apparent that the apparent power S should be the voltage-current product, as in dc resistive circuits.
- Apparent power is measured in volt-amperes or VA

*Power Factor*

The power factor is the ratio of the average power to the apparent power *(P/S)*. It is the cosine of the phase difference between voltage and current and also the cosine of the angle of the load impedance.

- The power factor is dimensionless, since it is the ratio of two powers.
- The angle is called the power factor angle, as it is the angle whose cosine is the power factor. Given
**V**as the voltage across a load and**I**as the current through it, the angle of the load impedance is the power factor angle:

Alternatively, since

the impedance is

- From eq.(32), the power factor can be thought of as the factor by which the apparent power is multiplied to obtain the real or average power.
- Possible pf values range from 0 to 1.
- Extreme cases:
- For a purely resistive load, v and i are in phase and . Reactive power is zero and the apparent power is equal to the average power.
- For a purely reactive load and . The average power is zero.

For pf values between 0 and 1, pf is said to be

*leading*or*lagging*:- Leading pf means that current leads voltage, implying a capacitive load. The power factor angle is negative and falls within the interval .
- Lagging pf means that current lags voltage, implying an inductive load. The power factor angle is positive and falls within the interval .

## Complex Power

The complex power **S** is the vector sum of average (or real) power and reactive power and is used to find the total effect of parallel loads. It contains all the information related to the power absorbed by a given load.

Consider the ac load in fig.5. Given the phasor form and of voltage *v(t)* and current *i(t)*, the complex power absorbed is .

Figure 5: The voltage and current phasors associated with a load

In terms of the rms values, recall eq.(34) to write **S** as

assuming the passive sign convention. We can further rewrite this into

- We observe that the apparent power is the magnitude of
**S***(V*; hence,_{rms}I_{rms})**S**is measured in volt-amperes (VA). - The angle of the complex power is the pf angle.

It is possible to express the complex power in terms of the load impedance **Z**, as in eq.(35). Thus, **V**_{rms} = **ZI**_{rms}. Substituting this into eq.(36) gives

From eq.(37), we notice that

- The real power
*P*is the average power in watts (W) delivered to a load; it is the only useful power and the actual power dissipated by the load. - The reactive power
*Q*in volt-ampere reactive (VAR) is a measure of the energy reflected back and forth between the source and the load.- Energy storage elements neither absorb nor supply power. They exchange power back and forth with the rest of the network, the same way the reactive power bounces between the load and the source.
- for resistive loads (unity pf)
- for capacitive loads (leading pf)
- for inductive loads (lagging pf)

- Energy storage elements neither absorb nor supply power. They exchange power back and forth with the rest of the network, the same way the reactive power bounces between the load and the source.

*As a complex number the complex power has real and imaginary parts. The real part of complex power S is real power P and the imaginary part is reactive power Q. Its magnitude is the apparent power and the cosine of its phase angle is the pf.*

The complex power holds all the pertinent power information with a given load. It allows us to obtain the real and reactive powers from voltage and current phasors.

*Power Triangle*m/h4>

**S**, *P*, and *Q* can be represented in the form of a triangle, known as the power triangle, shown in fig.6(a). This is similar to the impedance triangle illustrated in fig.6(b), which shows the relationship between the impedance (**Z**), resistance (*RM*), and reactance (*XM*).

Figure 6: (a) Power triangle, (b) impedance triangle

The power triangle has four elements:

- the power factor angle
- the apparent power
*S*which can also be the complex power**S**; the hypotenuse - the real power
*P*; adjacent to - the reactive power
*Q*; opposite

As shown in fig.7,

- When
**S**lies in the first quadrant,- load is inductive
- power factor angle is positive
- pf is lagging

- When S lies in the fourth quadrant,
- load is capacitive
- power factor angle is negative
- pf is leading

Figure 7: Power triangles in the first and fourth quadrants

Complex power can also lie in the second or third quadrant. This implies a negative power factor, when the load generates true power which then flows to the source.

## Conservation of AC Power

Power is the rate at which energy is converted or transferred. In a closed electrical circuit, t*he complex power, real power, and reactive power of the sources equal the respective power sums of the individual loads. *

Consider the circuit in fig.8(a) with two load impedances **Z**_{1} and **Z**_{2} connected in parallel to an ac source **V**. KCL gives

Figure 8: An ac voltage source supplying loads connected in (a) parallel and (b) series

From this point, all values of voltages and currents are assumed to be rms values unless otherwise specified. The complex power supplied by the source is

where **S**_{1} and **S**_{2} are the complex powers delivered to loads **Z**_{1} and **Z**_{2}, respectively. If the loads are connected in series as shown in fig. 8(b), KVL yields

The complex power from the source is

where **S**_{1} and **S**_{2} are the complex powers delivered to loads **Z**_{1} and **Z**_{2}, respectively.

From eq.(42) and eq.(44), the total power from the source is the total power delivered to the load. For N loads,

## Power Factor Correction

Domestic and industrial loads are typically inductive and operate at a low lagging power factor. As power factor is an indication of energy efficiency, it is advantageous to bring its value closer to unity.

*Power factor correction is the process of compensating for lagging or leading current by connecting reactive elements in parallel to the load, reducing the reactive power. *

Common: When there is a lagging power factor caused by high inductive loads, capacitors are utilized as the reactive element.

Rare: When there is a leading power factor caused by high capacitive loads, inductors can be utilized as the reactive element.

As shown in fig.9(a), an inductive load is represented as an inductor-resistor combination in series. Since most loads are inductive, the power factor is improved by connecting a capacitor in parallel with the load, as shown in fig.9(b).

Figure 9: (a) Original inductive load and (b) improved power factor with a capacitor

Figure 10: Effect of adding a capacitor in parallel with the inductive load

- Adding the capacitor reduced the phase angle between the voltage and current from to , thereby increasing the power factor (recall eq.32).
- We also notice from the magnitudes of the vectors that with the same supplied voltage, the circuit in fig.9(a) draws a larger current
*I*compared to the current I drawn by the circuit in fig.9(b)._{L}- Larger currents result in increased power losses . Therefore, it is beneficial to minimize current.
- By choosing the appropriate capacitor size, the current can be made to be in phase with the voltage; and .

Another way of looking at power factor correction is through the power triangle in fig.(11).

Figure 11: Power triangle illustrating power factor correction

Let *S*_{1} be the apparent power of the original inductive load. Then,

But from eq.(38), . The value of the required capacitance *C* is

The real power *P* is not affected by the power factor correction because the capacitor consumes zero real power.

As mentioned before, capacitive loads are also possible. This means that there is a leading power factor. For power factor correction, an inductor should be connected across the load. The required inductance *L* can be calculated from

where is the difference between the new and old reactive powers.

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