Power in AC Circuit — Revision Notes

⚡ 30-Second Revision

Instantaneous Power — $P(t) = V(t)I(t)$
Average Power — $P_{avg} = V_{rms}I_{rms}cosphi$
RMS Values — $V_{rms} = V_0/sqrt{2}$, $I_{rms} = I_0/sqrt{2}$
Power Factor — $cosphi = R/Z$
Impedance — $Z = sqrt{R^2 + (X_L - X_C)^2}$
Reactances — $X_L = omega L$, $X_C = 1/(omega C)$
Phase Angle — $anphi = (X_L - X_C)/R$
Pure R Circuit — $phi = 0^circ$, $cosphi = 1$, $P_{avg} = V_{rms}I_{rms}$
Pure L/C Circuit — $phi = pm 90^circ$, $cosphi = 0$, $P_{avg} = 0$ (Wattless current)
Resonance ($X_L = X_C$) — $phi = 0^circ$, $cosphi = 1$, $Z=R$, $P_{avg,max} = V_{rms}^2/R$

2-Minute Revision

Power in AC circuits differs significantly from DC due to the time-varying nature of voltage and current, and the phase difference ($phi$) between them. Instantaneous power, $P(t) = V(t)I(t)$, fluctuates, but the useful power is the average power, $P_{avg} = V_{rms}I_{rms}cosphi$.

Here, $V_{rms}$ and $I_{rms}$ are root mean square values, and $cosphi$ is the power factor. The power factor, which is $R/Z$, indicates how efficiently power is used; a value of 1 means maximum efficiency (purely resistive circuit or resonance), while 0 means no useful power is consumed (purely inductive or capacitive circuit, leading to wattless current).

In LCR series circuits, calculate $X_L = omega L$ and $X_C = 1/(omega C)$, then impedance $Z = sqrt{R^2 + (X_L - X_C)^2}$. At resonance ($X_L = X_C$), $Z=R$, $phi=0^circ$, and power is maximum ($P_{avg} = V_{rms}^2/R$).

Remember, only resistors dissipate average power; inductors and capacitors store and release energy.

5-Minute Revision

To master power in AC circuits, start with the core concept: average power. While instantaneous power $P(t) = V(t)I(t)$ is always true, it's the average power over a cycle, $P_{avg} = V_{rms}I_{rms}cosphi$, that's practically important.

Here, $V_{rms}$ and $I_{rms}$ are the effective values ($V_0/sqrt{2}, I_0/sqrt{2}$), and $cosphi$ is the power factor. The phase angle $phi$ is crucial: it's the angle by which current leads or lags voltage.

For a purely resistive circuit, $phi=0^circ$, $cosphi=1$, and $P_{avg} = V_{rms}I_{rms}$. For purely inductive or capacitive circuits, $phi=pm 90^circ$, $cosphi=0$, and $P_{avg}=0$. This is the concept of wattless current – current flows, but no net power is consumed.

In a series LCR circuit, first calculate the inductive reactance ($X_L = omega L$) and capacitive reactance ($X_C = 1/(omega C)$). Then find the impedance $Z = sqrt{R^2 + (X_L - X_C)^2}$. The power factor is $cosphi = R/Z$.

The average power can also be calculated as $P_{avg} = I_{rms}^2 R$, emphasizing that power is dissipated only in the resistive component. A key scenario is resonance, where $X_L = X_C$. At resonance, $Z=R$, $phi=0^circ$, $cosphi=1$, and the circuit dissipates maximum power, $P_{avg,max} = V_{rms}^2/R$.

Understanding power factor correction (adding capacitors to inductive loads) is also important for conceptual questions. Always use RMS values for average power calculations unless specifically asked for peak power.

Prelims Revision Notes

1
Instantaneous Power — $P(t) = V(t)I(t)$. Varies with time, can be negative.
2
Average Power — $P_{avg} = V_{rms}I_{rms}cosphi$. This is the useful power.
3
RMS Values — $V_{rms} = V_0/sqrt{2}$, $I_{rms} = I_0/sqrt{2}$. Use these for average power.
4
Power Factor ($cosphi$) — Ratio of real power to apparent power. $cosphi = R/Z$. Varies from 0 to 1.
5
Phase Angle ($phi$) — Angle between $V_{rms}$ and $I_{rms}$. $anphi = (X_L - X_C)/R$.

* Pure Resistor: $phi = 0^circ$, $cosphi = 1$. $P_{avg} = V_{rms}I_{rms} = I_{rms}^2 R = V_{rms}^2/R$. * Pure Inductor: $phi = -90^circ$ (current lags), $cosphi = 0$. $P_{avg} = 0$ (wattless current). * Pure Capacitor: $phi = +90^circ$ (current leads), $cosphi = 0$. $P_{avg} = 0$ (wattless current).

1
LCR Series Circuit

* Inductive Reactance: $X_L = omega L = 2pi f L$ * Capacitive Reactance: $X_C = 1/(omega C) = 1/(2pi f C)$ * Impedance: $Z = sqrt{R^2 + (X_L - X_C)^2}$ * RMS Current: $I_{rms} = V_{rms}/Z$ * Average Power: $P_{avg} = I_{rms}^2 R = V_{rms}I_{rms}(R/Z)$

1
Resonance — Occurs when $X_L = X_C$. At resonance:

* $Z = R$ (minimum impedance) * $phi = 0^circ$, $cosphi = 1$ (unity power factor) * $I_{rms}$ is maximum ($I_{rms,max} = V_{rms}/R$) * $P_{avg}$ is maximum ($P_{avg,max} = V_{rms}^2/R$)

1
Power Factor Correction — Adding capacitors in parallel to inductive loads to increase the power factor closer to 1, reducing current and losses.

Vyyuha Quick Recall

P-A-W: Power Always Watts (only in Resistors). Remember the formula: Power = Voltage * In-phase Current. (P = V_rms * I_rms * cos phi). The 'C' in 'Current' reminds you of 'cos phi'.