Question

# Show that the average power in an ac circuit is P= IVcos Q , where Q...

Show that the average power in an ac circuit is P= IVcos Q , where Q is the phase difference between voltage and current. (CosQ is called lne power factor.)

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

Applying the trigonometric product-to-sum identity of:

and ? = ?v – ?i (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 ? is the phase difference between the two waveforms. Therefore we can express the instantaneous power as being:

Instantaneous AC Power Equation

This equation shows us that the instantaneous AC power has two different parts and is therefore the sum of these two terms. The second term is a time varying sinusoid whose frequency is equal to twice the angular frequency of the supply due to the 2? part of the term. The first term however is a constant whose value depends only on the phase difference, ? between the voltage, (V) and the current, (I).

As the instantaneous power is constantly changing with the profile of the sinusoid over time, this makes it difficult to measure. It is therefore more convenient, and easier on the maths to use the average or mean value of the power. So over a fixed number of cycles, the average value of the instantaneous power of the sinusoid is given simply as:

where V and I are the sinusoids rms values, and ? (Theta) is the phase angle between the voltage and the current. The units of power are in watts (W).

The AC Power dissipated in a circuit can also be found from the impedance, Z of the circuit using the voltage, Vrms or the current, Irms flowing through the circuit as shown.

AC Power Example No1

The voltage and current values of a 50Hz sinusoidal supply are given as: vt = 240 sin(?t +60o)Volts and it = 5 sin(?t -10o)Amps respectively. Find the values of the instantaneous power and the average power absorbed by the circuit.

From above, the instantaneous power absorbed by the circuit is given as:

Applying the trigonometric identity rule from above gives:

The average power is then calculated as:

You may have noticed that the average power value of 205.2 watts is also the first term value of the instantaneous power p(t) as this first term constant value is the average or mean rate of energy change between the source and load.

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