Power in AC Circuit: The Power Factor

Instantaneous Power in an AC Circuit

For a series RLC circuit driven by a sinusoidal voltage $v = v_m \sin(\omega t)$, the resulting current is $i = i_m \sin(\omega t + \phi)$, where $\phi$ is the phase angle between voltage and current.

The instantaneous power supplied by the source is the product of instantaneous voltage and current:

Using the trigonometric identity $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$, this becomes:

The first term $\frac{v_m i_m}{2} \cos \phi$ is constant (time-independent). The second term $\frac{v_m i_m}{2} \cos(2\omega t + \phi)$ oscillates at twice the source frequency.

Average Power Over a Cycle

The average power over a complete cycle is found by averaging the instantaneous power. The time-dependent term $\cos(2\omega t + \phi)$ averages to zero over a cycle (its positive and negative halves cancel). Therefore, the average power $P$ is:

Since $v_m = \sqrt{2} V$ and $i_m = \sqrt{2} I$ (where $V$ and $I$ are RMS values), we can write:

This is the key result for average power in an AC circuit.

The Power Factor

The quantity $\cos \phi$ is called the power factor. It determines what fraction of the apparent power ($VI$) is actually dissipated as real power.

• Physical meaning: The power factor tells us how much the current and voltage are in phase. A power factor of 1 means they are perfectly in phase (maximum power transfer). A power factor of 0 means they are $90^\circ$ out of phase (no net power dissipation).

Alternative Expression for Power

Using $V = IZ$ (Ohm's law for AC circuits), the average power can also be written as:

Since $\cos \phi = \frac{R}{Z}$ for a series RLC circuit, this simplifies to:

This confirms that power is dissipated only in the resistor, regardless of the presence of inductors or capacitors.

Special Cases

1. Pure Resistive Circuit ($\phi = 0$, $\cos \phi = 1$): Maximum power dissipation occurs. $P = VI = I^2 R$.

2. Purely Inductive or Capacitive Circuit ($\phi = \pm 90^\circ$, $\cos \phi = 0$): No power is dissipated, even though current flows. This current is called wattless current.

3. LCR Series Circuit (non-resonant): Power is dissipated only in the resistor. The power factor $\cos \phi = \frac{R}{Z}$ is less than 1.

4. Resonance in LCR Circuit ($X_L = X_C$, $\phi = 0$, $\cos \phi = 1$): Maximum power is dissipated, equal to $I^2 R$. The impedance is minimum ($Z = R$).

Power Factor Improvement …