Question

The power factor may be equal to 1 for

• A. pure inductor
• B. pure capacitor
• C. pure resistor
• D. an LCR circuit

Answer 1

Answer: (C), (D)

an LCR circuit

Power factor of an ac circuit

1. The angle between the voltage and current phasors is what is known as the power factor of an ac circuit.
2. Therefore, we must take the cosine of the phase angle in order to determine the power factor of an ac circuit.
3. Consequently, we can get the ac circuit's power factor using the formula,

$powerfactor=pf=\cos \theta [\theta =phaseangleoftheaccircuit]$

The explanation of the correct options:

In case of option D

1. In an LCR combination circuit, $Z$ is the impedance of the circuit or the total resistance offered by the circuit, it is given as $Z=\sqrt{R^2+{(X_L-X_C)}^2}$ where, $X_L,X_C$ are the inductive reactance and the capacitive reactance.
2. Then the power due to the circuit is given as $P={I_{rms}}^{2}Z=\frac{{\left(Z\right)}^2}{V_{rms}}Z$ which is called the apparent power.
   Then the power factor is defined as the ratio of real power to the apparent power in the circuit. This value varies from $0-1.$
   $P.F=\frac{R.P}{A.P}$
   If the power factor is $0$, then there is some impedance in the circuit.
   If the power factor is $1$, then the circuit is said to be resistive in nature $Z=R$ , and here maximum power is used to do work.
3. Therefore, the power factor is one for an LCR circuit.

In case of option C

1. In a purely resistive circuit, the current passing through the resistance is always in phase with the implemented voltage.
2. This means that the angle between voltage and current is $0^o$∘and hence
   $$\begin{gathered} powerfactor=pf=\cos \theta \\ pf=\cos 0^o \\ pf=1 \end{gathered}$$
3. This indicates that resistance will only utilize active power.
4. So, the power factor of a pure resistive ac circuit is 1.

The explanation of the incorrect options:

In case of option A

1. In a purely inductive circuit, the current by the inductor lags the used voltage by ${90}^o$.
2. Therefore, the angle between voltage and current is 90 degrees, and hence
   $$\begin{gathered} powerfactor=pf=\cos \theta \\ pf=\cos {90}^o \\ pf=0 \end{gathered}$$
3. Therefore, the power factor of an ac circuit having pure inductive load is 0. An ideal pure inductive load is not possible in the real case.

In case of option B

1. Similarly, the current through the capacitor leads to the voltage by ${90}^o$. Therefore
   $$\begin{gathered} powerfactor=pf=\cos \theta \\ pf=\cos {90}^o \\ pf=0 \end{gathered}$$
2. Therefore, the power factor of an ac circuit having a pure capacitive load is 0. The pure capacitive load is possible only in ideal cases.

Hence, option C is the correct option.