Question

The power factor of wattless current is
A. Infinity
B. 1
C. Zero
D. $\dfrac{1}{2}$

Answer 1

- Hint: First we have to know what is the power factor. Then to learn what is wattless current. Then, we can find the power factor for the wattless current. We should already have some idea of the phase difference of voltage and current.

Formula used: $P_{avg}=V_{rms}.I_{rms}. \cos\phi$

Complete step-by-step solution -
We are going to see what is wattless current and what is the power factor. For this, let's take any ac circuit. Let’s take its source voltage to be $V=V_0.\sin\omega t$ . Here, $\omega$ is the angular frequency of the ac source. And also assume that the current in the circuit is given by $I=I_0.\sin(\omega t+\phi)$
$\phi$ is the phase difference between voltage and current.
The power dissipation,
$\begin{align}
  & P=VI={{V}_{0}}.{{I}_{0}}.\sin \omega t.\sin (\omega t+\phi ) \\
 & P=\dfrac{1}{2}{{V}_{0}}.{{I}_{0}}(\cos \phi -\cos (2\omega t+\phi )) \\
\end{align}$
Now, taking time average over a complete cycle, we know that the second cosine function will lead to zero. So, average power is given by,
\[\begin{align}
  & {{P}_{avg}}=\dfrac{1}{2}{{V}_{0}}.{{I}_{0}}.\cos \phi \\
 & {{P}_{avg}}=\dfrac{{{V}_{0}}}{\sqrt{2}}.\dfrac{{{I}_{0}}}{\sqrt{2}}\cos \phi \\
 & {{P}_{avg}}={{V}_{rms}}.{{I}_{rms}}.\cos \phi \\
\end{align}\]
In case of purely inductive circuits, current lags behind voltage by $\dfrac{\phi}{2}$ . Hence, $\phi=-\dfrac{\phi}{2}$. So, average power ${{P}_{avg}}={{V}_{rms}}>{{I}_{rms}}.\cos (-\dfrac{\phi }{2})=0\text{ watt}$
Similarly, for purely capacitive circuit or capacitive inductive circuit, $P_{avg}=0$ watt.
The currents in these circuits are called wattless current. Since average power is zero and watt is the unit of measuring power.
Power factor is $\cos\phi$. For, wattless current, its value is $\cos\dfrac{\phi}{2}=0$.
So, option C is the correct answer.

Additional information:
In the case of LCR circuits, the value of $\phi$ depends on values of R, C, L and $\omega$. Power factor can have a value ranging from 0 to 1. In case of resonance, $\phi=0$. It makes the power factor equal to one. Hence, maximum power is obtained.

Note: Remember this few thing:
1. Power factor can never have a value greater than one, also it can’t be negative.
2. Know that, $\cos(-\phi)=\cos\phi$
3. If there’s no resistance in a circuit, its current is called wattless. But in practical, all circuits have resistance and wattless current is not possible.