Alternating Current

# Power Consumption

Power is the rate at which electrical energy is consumed in an electric circuit. In DC circuits, power $(P)$ is the product of voltage $(V)$ and current $(I)$ i.e. $P=VI$ Here, $V$ and $I$ are in same phase. But this is not so in an ac circuit because voltage and current vary continuosly with time and they are generally not in the same phase. Hence, power in the circuit also varies with time.

Thus, power of an ac circuit is defined by the instantaneous power of the circuit. Hence, instantaneous power $P$ of an ac circuit is defined as the product of instantaneous voltage $E$ and instantaneous current $I$. $\therefore P=EI$

The instantaneous alternating emf at instant $t$ is given by $E=E_0\sin\omega t$

## Power Consumption in LCR Circuit

In the LCR circuit, let the current $I$ lag behind the voltage $E$ by an angle of $\theta$. The instantaneous alternating current at instant $t$ is given by $I=I_0\sin(\omega t-\theta)$

Hence, the instantaneous power at instant $t$ is given by $P=E_0\sin\omega t\,.\,I_0\sin(\omega t-\theta)$ $=E_0I_0\sin\omega t\:\sin(\omega t-\theta)$ $=\frac{E_0I_0}{2}.2\sin\omega t\:\sin(\omega t-\theta)$ $\therefore P=\frac{E_0I_0}{2}[\cos\theta-\cos(2\omega t-\theta)]$

If this instantaneous power is assumed to remain constant for a small time $dt$, then small amount of work done $dW$ in this time is given by $dW=P\:dt$

Hence, the total work done in one complete cycle is given by $W=\int_0^TP\:dt$

$=\frac{E_0I_0}{2}\int_0^T[\cos\theta-\cos(2\omega t-\theta)]\:dt$ $=\frac{E_0I_0}{2}\left[\int_0^T\cos\theta\:dt-\int_0^T\cos(2\omega t-\theta)\:dt\right]$ $=\frac{E_0I_0}{2}\left[\cos\theta[t]_0^T-\left[\frac{\sin(2\omega t-\theta)}{2\omega}\right]_0^T\right]$ $=\frac{E_0I_0}{2}\left[T\cos\theta-\left[\frac{\sin(2\omega T-\theta)}{2\omega}-\frac{\sin(-\theta)}{2\omega}\right]\right]$

$\text{Here,}\;\sin(2\omega T-\theta)=\sin\left(2\omega×\frac{2π}{\omega}-\theta\right)$ $=\sin(4π-\theta)=\sin(-\theta)$ $\therefore W=\frac{E_0I_0}{2}\left[T\cos\theta-\left[\frac{\sin(-\theta)}{2\omega}-\frac{\sin(-\theta)}{2\omega}\right]\right]$ $W=\frac{E_0I_0}{2}[T\cos\theta-0]$ $\therefore W=\frac{E_0I_0}{2}\cos\theta T$

Since power varies from instant to instant so average power over a complete cycle is considered. The average power in the inductive circuit over a complete cycle is given by $P_{\text{av}}=\frac{W}{T}=\frac{E_0I_0}{2}\cos\theta=\frac{E_0}{\sqrt{2}}\frac{I_0}{\sqrt{2}}\cos\theta$ $\therefore P_{\text{av}}=E_{\text{rms}}I_{\text{rms}}\cos\theta$

where, $E_{\text{rms}}=\frac{E_0}{\sqrt{2}}$ is the rms value of emf and $I_{\text{rms}}=\frac{I_0}{\sqrt{2}}$ is the rms value of current. [Root Mean Square (RMS) Value of AC]

Hence, average power over a complete cycle in an inductive circuit is the product of rms values of emf and current and cosine of the phase angle between the voltage and current. The term $\cos\theta$ is known as power factor. $\text{Average power}=\text{Apparent power}×\text{Power factor}$ where, $\text{Apparent power}=E_{\text{rms}}I_{\text{rms}}$

## Special Cases

### Power Consumption across Resistor only

For resistor, $V$ and $I$ are in same phase i.e. $\theta=0°$. [From: AC Through a Resistor (AC Through Circuit Elements)] $\therefore P_{\text{av}}=E_{\text{rms}}I_{\text{rms}}$ Hence, $\text{True power} = \text{Apparent power}$.

### Power Consumption across Inductor only

In an inductor, voltage $V$ leads the current $I$ by a phase angle of $\frac{π}{2}$ i.e. $\theta=\frac{π}{2}$. [From: AC Through an Inductor (AC Through Circuit Elements)] $\therefore P_{\text{av}}=E_{\text{rms}}I_{\text{rms}}\cos\frac{π}{2}=0$ Hence, no power is consumed in an inductor.

### Power Consumption across Capacitor only

In a capacitor, voltage $V$ lags behind the current $I$ by a phase angle of $\frac{π}{2}$ i.e. $\theta=\frac{π}{2}$. [From: AC Through a Capacitor (AC Through Circuit Elements)] $\therefore P_{\text{av}}=E_{\text{rms}}I_{\text{rms}}\cos\frac{π}{2}=0$ Hence, no power is consumed in a capacitor.

## Power Factor

The factor by which the apparent power has to be multiplied to obtain the average power or true power of the circuit is called power factor. $\therefore\text{Power factor}=\cos\theta$ i.e. the cosine of phase angle between voltage and current is called power factor. $\text{True power}=\text{Apparent power}×\text{Power factor}$ $\therefore\text{Power factor}=\frac{\text{True power}}{\text{Apparent power}}$

The power factor of an ac circuit may also be defined as the ratio of resistance and impedance of the circuit. $\therefore\cos\theta=\frac{R}{Z}$ Its value ranges from $0$ to $1$.

For an ac circuit containing only resistor, $\cos\theta=1$ For an ac circuit containing only inductor or capacitor, $\cos\theta=0$ [AC Through Circuit Elements]

For LR circuit, $cos\theta=\frac{R}{\sqrt{R^2+X_L^2}}=\frac{R}{\sqrt{R^2+\omega^2L^2}}$ For CR circuit, $\cos\theta=\frac{R}{\sqrt{R^2+X_C^2}}=\frac{R}{\sqrt{R^2+\frac{1}{\omega^2C^2}}}$ For LCR circuit, $\cos\theta=\frac{R}{\sqrt{R^2+(X_L-X_C)^2}}$$=\frac{R}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}}$