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# Power in AC Circuit – Active Power, Reactive Power, Apparent Power

In electrical and electronic circuits, the **power **is one of the most significant quantities used to analyze the circuits for practical applications. The **electrical power** defined as the time rate of expanding or absorbing energy in a circuit, i.e.,

$$\mathrm{Power,P=\frac{Energy \;expanded\; or\; absorbed (𝑊)}{Time(𝑡)}}.....(1)$$

This article is meant for explaining **power relations in AC circuits**. Where, an AC circuit is one which is excited from a source alternating voltage.

## Instantaneous Power in an AC Circuit

The value of electric power in an AC circuit measured at a certain instant of time is known as
**instantaneous power**. It is generally expressed by a small case letter $p$. In general, the
instantaneous power in an AC circuit is obtained by multiplying the instantaneous voltage and
instantaneous current, i.e.,

$$\mathrm{Instantaneous \;power, 𝑝 = \upsilon. i\;\;}....(2)$$

Consider any AC circuit, if the instantaneous voltage and current in the circuit are given by,

$$\mathrm{\upsilon=V_{m}\sin\left({wt}\right)}....(3)$$

$$\mathrm{i=I_{m}\sin\left({wt-\phi}\right)}....(4)$$

Where, $\mathrm{\phi}$ is the phase angle between the voltage and current at any instant. Where, $\mathrm{\phi}$ is negative when current lags the voltage, positive when current leads the voltage and zero when current and voltage are in same phase.

Hence, by the definition, the instantaneous power is given by,

$$\mathrm{p=vi=V_{m}\sin(wt).I_{m}\sin(wt-\phi)}$$

$$\mathrm{\Rightarrow\; p=\frac{1}{2}\times2\times\; V_{m}I_{m}\sin wt \sin (wt-\phi)}$$

$$\mathrm{\Rightarrow\; p=\frac{V_{m}I_{m}}{2}[\cos\phi-\cos(2wt-\phi)]}$$

$$\mathrm{\therefore\; p=\frac{V_{m}I_{m}}{2}\cos\phi-\frac{V_{m}I_{m}}{2}\cos(2wt-\phi)}......(5)$$

Here, the second term on RHS of equation (5) contains a double frequency term and the magnitude of the average value of this term is zero it is because the average of a sinusoidal quantity over a complete cycle is zero. Thus, the instantaneous power consists only the first term of the equation (5), that is

$$\mathrm{P=\frac{1}{2}\;V_{m}I_{m}\cos\phi......(6)}$$

This term is the **average power** in the AC circuit. Also, the average power in AC circuit may be expressed in terms of RMS values of voltage and current as,

$$\mathrm{P=\frac{V_{m}}{\sqrt{2}}\frac{I_{m}}{\sqrt{2}}\cos\phi}$$

$$\mathrm{\therefore \;P=VI\cos\phi.....(7)}$$

Where, $\mathrm{\cos\phi}$ is known as **power factor** of the circuit.

There are following three types of electric powers in an AC electric circuit −

- Active power
- Reactive power
- Apparent Power

## Active Power

The **active power** is that amount of the total electric power in an AC electric circuit which actually consumed or utilized. It is also called as **true power** or **real power**. The active power is measured in Watts (W). The larger units of active power are kilowatt (kW), mega-watt (MW), gigawatt (GW) and so on.

Technically, when in an AC electric circuit the phase angle becomes zero, i.e. the power factor becomes unity, then the power consumed in the circuit is called **active power**. This happens in case of the resistive load. Thus, the active power is given by,

$$\mathrm{P=VI\cos\phi=VI\cos0^{\circ}}$$

$$\mathrm{\therefore Active\;Power,P=VI}$$

In practice, the active power is used to specify ratings of electrical loads such motors, bulbs, irons, etc.

## Reactive Power

The **reactive power** is that amount of total electrical power which remains unused in the AC electric circuit and flows back and forth in the electrical system from load to source and *viceversa*. It is denoted by letter *Q* and is measured in **Volt Ampere Reactive (VAR)**.

The reactive power in AC circuit may also be defined as the product of RMS values of voltage and current with the sine of the phase angle, i.e.,

$$\mathrm{Q=VI\sin\phi}$$

The reactive power is also known as **wattles power** or **quadrature power**. For an inductive load, the reactive power consumed is the **lagging reactive power** and that consumed by the capacitor is the **leading reactive power**. Therefore, there are two AC circuit elements namely inductor and capacitor that responsible for the flow of reactive power in the circuit.

The reactive power is responsible for operating of all electromagnetic machines such as motors, generators, etc., because it produces required magnetic excitation in these machines.

## Apparent Power

The total power produced by a source of alternating current is the **apparent power**. It is measured as the product of RMS values of voltage and current. The apparent power is denoted by letter S and is measured in **Volt-Ampere (VA)**.

$$\mathrm{Apparent\;power,\;S=VI}$$

Also, the apparent power is given by phasor sum of active power and reactive power, i.e.

$$\mathrm{S=P+jQ}$$

In practice, the apparent power is used to specify the ratings of those electrical devices that act as sources and transmitters of power like generators, alternators, transformers, etc.

## Numerical Example

If the RMS values of current and voltage in an AC circuit are 220 V and 5 A. If there is phase difference of 60° between voltage and current. Determine active power, reactive power and apparent power in the circuit.

### Solution

Given data,

Voltage,

*V*= 220VCurrent,

*I*= 5 APhase angle, \phi = 60°

Therefore, the active power in the given circuit is,

$$\mathrm{P=VI\cos\phi}$$

$$\mathrm\Rightarrow{P=220\times5\times\cos60}$$

$$\mathrm{\therefore\;P=550W}$$

The reactive power flowing in the circuit is,

$$\mathrm{Q=VI\sin\phi}$$

$$\mathrm{Q=220\times5\times\sin60}$$

$$\mathrm{\therefore Q = 952.63 \;VAR}$$

The apparent power supplied to the circuit is,

$$\mathrm{S = VI = 220\times5}$$

$$\mathrm{\therefore S = 1100 \;VA}$$

## Conclusion

In this article, we discussed the three main types of electric powers namely, active power, reactive power and apparent power in an AC circuit. The main reason behind this classification of powers is that in an AC circuit, the electric power depends on the power factor.

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