Power in AC Circuits

In a direct current circuit, the power consumed is simply the product of the dc voltage times the DC current, VxI and is measured in watts. However, we can not calculate it in a similar manner for reactive AC circuits.

Electrical power is the “rate” at which energy is being consumed in a circuit and as such all electrical and electronic components and devices have a limit to the amount of electrical power that they can safely handle. For example, a 1/4 watt resistor or a 20 watt amplifier.

Electrical power can be time-varying either as a DC quantity or as an AC quantity. The amount of power in a circuit at any instant of time is called the instantaneous powerand is given by the well-known relationship of P = VI. So one watt (which is the rate of expending energy at one joule per second) will be equal to the volt-ampere product of one volt times one ampere.

Then the power absorbed or supplied by a circuit element is the product of the voltage, V across the element, and the current, I flowing through it. So if we had a DC circuit with a resistance of “R” ohms, the power dissipated by the resistor in watts is given by any of the following generalised formulas:

Electrical Power

 

Where: V is the dc voltage, I is the dc current and R is the value of the resistance.

So power within an electrical circuit is only present when both the voltage and current are present, that is no open-circuit or closed-circuit conditions. Consider the following simple example of a standard resistive dc circuit:

DC Resistive Circuit

Electrical Power in an AC Circuit

In a DC circuit, the voltages and currents are generally constant, that is not varying with time as there is no sinusoidal waveform associated with the supply. However in an AC circuit, the instantaneous values of the voltage, current and therefore power are constantly changing being influenced by the supply. So we can not calculate the power in AC circuits in the same manner as we can in DC circuits, but we can still say that power (p) is equal to the voltage (v) times the amperes (i).

Another important point is that AC circuits contain reactance, so there is a power component as a result of the magnetic and/or electric fields created by the components. The result is that unlike a purely resistive component, this power is not only consumed but instead is stored and then returned back to the supply as the sinusoidal waveform goes through one complete periodic cycle.

Thus, the average power absorbed by a circuit is the sum of the power stored and the power returned over one complete cycle. So a circuits average power consumption will be the average of the instantaneous power over one full cycle with the instantaneous power, p defined as the multiplication of the instantaneous voltage, v by the instantaneous current, i. Note that as the sine function is periodic and continuous, the average power given over all time will be exactly the same as the average power given over a single cycle.

Let us assume that the waveforms of the voltage and current are both sinusoidal, so we recall that:

Sinusoidal Voltage Waveform

 

As the instantaneous power is the power at any instant of time, then:

 

Applying the trigonometric product-to-sum identity of:

 

and θ = θv – θi (the phase difference between the voltage and the current waveforms) into the above equation gives:

 

Where V and I are the root-mean-squared (rms) values of the sinusoidal waveforms, v  and i respectively, and θ is the phase difference between the two waveforms.