Power Factor — Revision Notes

⚡ 30-Second Revision

Definition: — $\text{Power Factor} = \cos \phi = P/S = R/Z$
Real Power (P): — Useful power, $P = V_{rms} I_{rms} \cos \phi = I_{rms}^2 R$ (Unit: Watt, W)
Reactive Power (Q): — Non-useful power, $Q = V_{rms} I_{rms} \sin \phi = I_{rms}^2 X$ (Unit: VAR)
Apparent Power (S): — Total power, $S = V_{rms} I_{rms} = \sqrt{P^2 + Q^2}$ (Unit: VA)
Impedance (Z): — $Z = \sqrt{R^2 + (X_L - X_C)^2}$
Reactances: — $X_L = \omega L$, $X_C = 1/(\omega C)$
Phase Angle ($\phi$): — $\tan \phi = (X_L - X_C)/R$
Purely Resistive: — $\phi=0^\circ$, PF=1
Purely Inductive: — $\phi=90^\circ$ (lagging), PF=0
Purely Capacitive: — $\phi=-90^\circ$ (leading), PF=0
Resonance: — $X_L = X_C$, $\phi=0^\circ$, PF=1

2-Minute Revision

Power factor is a measure of how effectively electrical power is used in an AC circuit. It's defined as the ratio of real power (useful work) to apparent power (total supplied power), or as $\cos \phi$, where $\phi$ is the phase angle between voltage and current.

A power factor of 1 (unity) means maximum efficiency, occurring in purely resistive circuits or at resonance in RLC circuits. If current lags voltage (inductive circuits, $X_L > X_C$), the power factor is lagging.

If current leads voltage (capacitive circuits, $X_C > X_L$), it's leading. For purely inductive or capacitive circuits, the power factor is 0. Key formulas include $P = V_{rms} I_{rms} \cos \phi$, $S = V_{rms} I_{rms}$, and $\text{PF} = R/Z$.

Remember that power is only dissipated in the resistor ($P = I_{rms}^2 R$). Low power factor leads to energy losses and requires larger equipment, hence power factor correction (usually with capacitors) is important.

5-Minute Revision

Let's consolidate the concept of Power Factor for NEET. In AC circuits, the voltage and current waveforms might not be in sync; this phase difference, $\phi$, is critical. The power factor is defined as $\cos \phi$.

It also represents the ratio of real power (P) to apparent power (S), i.e., $\text{PF} = P/S$. Real power, measured in Watts (W), is the actual power doing useful work, dissipated only in resistors. Reactive power (Q), in VAR, is exchanged between the source and reactive components (inductors, capacitors) and does no useful work.

Apparent power (S), in VA, is the total power supplied, the vector sum of P and Q ($S = \sqrt{P^2 + Q^2}$).

For a series RLC circuit, the impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$, where $X_L = \omega L$ and $X_C = 1/(\omega C)$. The power factor can be calculated as $\text{PF} = R/Z$. If $X_L > X_C$, the circuit is inductive, current lags voltage, and the power factor is lagging.

If $X_C > X_L$, the circuit is capacitive, current leads voltage, and the power factor is leading. At resonance, $X_L = X_C$, making $Z=R$, and thus $\text{PF} = R/R = 1$ (unity). This is a crucial point for NEET.

For purely inductive or capacitive circuits, the power factor is 0. Remember, a low power factor implies inefficiencies and higher current for the same useful power, leading to greater $I^2R$ losses. Power factor correction, typically by adding capacitors, aims to bring the power factor closer to unity.

Worked Example: A series RLC circuit has $R=6,\Omega$, $X_L=10,\Omega$, $X_C=2,\Omega$. Find the power factor.

Solution:

1
Net reactance $X = X_L - X_C = 10 - 2 = 8,\Omega$.
2
Impedance $Z = \sqrt{R^2 + X^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10,\Omega$.
3
Power Factor $\cos \phi = R/Z = 6/10 = 0.6$. (Since $X_L > X_C$, it's lagging).

Prelims Revision Notes

For NEET, power factor is a high-yield topic in AC circuits. Master these points:

1
Definition & Formulas: — Power Factor (PF) is $\cos \phi$, where $\phi$ is the phase angle between voltage and current. Alternatively, $\text{PF} = \text{Real Power (P)} / \text{Apparent Power (S)}$.

* $P = V_{rms} I_{rms} \cos \phi = I_{rms}^2 R$ * $S = V_{rms} I_{rms}$ * $Q = V_{rms} I_{rms} \sin \phi$ * Power Triangle: $S^2 = P^2 + Q^2$

1
Impedance Triangle & PF: — For a series RLC circuit, $\text{PF} = R/Z$, where $Z = \sqrt{R^2 + (X_L - X_C)^2}$.

* $X_L = \omega L = 2\pi f L$ * $X_C = 1/(\omega C) = 1/(2\pi f C)$

1
**Phase Angle ($\phi$):**

* $\tan \phi = (X_L - X_C)/R$ * If $X_L > X_C$: Inductive circuit, current lags voltage, $\phi > 0$, PF is lagging. * If $X_C > X_L$: Capacitive circuit, current leads voltage, $\phi < 0$, PF is leading. * If $X_L = X_C$: Resistive circuit (at resonance), $\phi = 0$, PF is unity (1).

1
Special Cases:

* Purely Resistive Circuit: $X_L=0, X_C=0$. $Z=R$. $\phi=0^\circ$. $\text{PF} = \cos 0^\circ = 1$. * Purely Inductive Circuit: $R=0, X_C=0$. $Z=X_L$. $\phi=90^\circ$. $\text{PF} = \cos 90^\circ = 0$ (lagging). * Purely Capacitive Circuit: $R=0, X_L=0$. $Z=X_C$. $\phi=-90^\circ$. $\text{PF} = \cos(-90^\circ) = 0$ (leading). * Series RLC at Resonance: $X_L = X_C$. $Z=R$. $\phi=0^\circ$. $\text{PF} = 1$.

1
Importance: — Low PF means more current for same useful power, leading to higher $I^2R$ losses and larger equipment. Power factor correction (adding capacitors) improves PF.

Quick Check: If a question asks for power dissipated, remember it's always $P = I_{rms}^2 R$, not $V_{rms} I_{rms}$ unless PF=1.

Vyyuha Quick Recall

Power Factor is Real Zealously, Cosine Phi Says. (PF = R/Z, PF = cos $\phi$, PF = P/S)