Question

Explain the meaning of power factor and wattless current in alternating current circuits.

Hint:

Start from average AC power \( P = V_{rms} I_{rms}\cos\phi \); the factor \( \cos\phi = R/Z \) is the power factor, and the current component \( I\sin\phi \) that transfers no net power is the wattless current.

Answer 1

Step 1: Average power in an AC circuit.
In a general AC circuit the voltage and current differ in phase by an angle \( \phi \). If \( V_{rms} \) and \( I_{rms} \) are the root mean square values, the average (true) power consumed is
\[ P_{avg} = V_{rms}\, I_{rms}\, \cos\phi \]

Step 2: Power factor.
The term \( \cos\phi \) in the above expression is called the power factor of the circuit. It is the ratio of the true power actually consumed to the apparent power \( V_{rms} I_{rms} \):
\[ \text{Power factor} = \cos\phi = \frac{\text{True power}}{\text{Apparent power}} = \frac{R}{Z} \]
where \( R \) is the resistance and \( Z \) is the impedance of the circuit. It is a pure number lying between 0 and 1. For a pure resistor \( \phi = 0 \) so \( \cos\phi = 1 \); for a pure inductor or capacitor \( \phi = 90^\circ \) so \( \cos\phi = 0 \).

Step 3: Wattless current.
The current in an AC circuit can be resolved into two components: one along the voltage, \( I\cos\phi \) (which consumes power), and one perpendicular to the voltage, \( I\sin\phi \). The component \( I\sin\phi \) does not consume any average power because the average power associated with it is \( V_{rms}(I\sin\phi)\cos 90^\circ = 0 \). This component is called the wattless current (idle current).

Step 4: Condition.
When the circuit is purely inductive or purely capacitive, \( \phi = 90^\circ \), so \( \cos\phi = 0 \) and the whole current becomes wattless; no net power is dissipated even though current flows.
\[\boxed{\text{Power factor } = \cos\phi = R/Z,\quad \text{Wattless current } = I\sin\phi}\]