Alternating Voltage Applied to a Resistor :
The applied alternating voltage is given by,
$V = V_0 \sin \omega t$
Let ( I ) be the current in the circuit at any instant ( t ).
As per Ohm's Law, potential difference across the resistor,
$V = IR$
or
$I = \frac{V}{R}$
Using eqn. (i), we get,
$I = \frac{V_0 \sin \omega t}{R}$
or
$I = I_0 \sin \omega t$
where,
$I_0 = \frac{V_0}{R}$
$I_0$ is the peak value of alternating current.
Phasor diagram for pure resistive circuit shows that the phase difference between current ( I ) and voltage ( V ) is zero.
Instantaneous Power :
The instantaneous power dissipated in resistor as per Joule’s heating effect is given by
$P = I^2 R = (I_0^2 \sin^2 \omega t) R$
Average Power :
$P_{av} = \text{average of } I^2 R$
$P_{av}= \text{average of } I_0^2 R \sin^2 \omega t$
Since,
$\sin^2 \omega t = \tfrac{1}{2}(1 - \cos 2\omega t)$
Average of ( $\cos 2\omega t$) over a full cycle = 0.
Therefore,
$\langle \sin^2 \omega t \rangle = \tfrac{1}{2}$
Substituting,
$P_{av} = \tfrac{1}{2} I_0^2 R$
Now,
$I_{rms} = \frac{I_0}{\sqrt{2}} \quad \Rightarrow \quad I_0 = \sqrt{2} I_{rms}$
So,
$P_{av} = I_{rms}^2 R$
RMS (Root Mean Square) Values :
To express AC power in the same form as DC power ( $P = I^2 R$), a special value of current is defined and used. It is called root mean square (rms) or effective current and is denoted by ( $I_{rms}$ ) or ( I).
It is defined by
$I_{rms} = \sqrt{\bar{i^2}} = \sqrt{\tfrac{1}{2} i_0^2} = \frac{i_0}{\sqrt{2}} = 0.707 \, i_0 \tag{7.6}$
In terms of ( I ), the average power is
$P = \bar{P} = \tfrac{1}{2} i_0^2 R = I^2 R \tag{7}$
Similarly, rms voltage or effective voltage is defined by
$V_{rms} = \frac{v_0}{\sqrt{2}} = 0.707 \, v_0 \tag{7.8}$
It is customary to measure and specify rms values for AC quantities. For example, the household line voltage of 220 V is an rms value with a peak voltage of
$v_0 = \sqrt{2} V = (1.414)(220 \, V) = 311 \, V$
In fact, the rms current ( I ) is the equivalent DC current that would produce the same average power loss as the alternating current.
$P = \frac{V^2}{R} = VI \quad \text{(since } V = IR \text{)}$