8.3 Power in AC circuits

Power Types in AC Circuits

In DC circuits, power is straightforward: voltage times current. AC circuits are more complicated because voltage and current can be out of phase with each other. This phase difference creates three distinct types of power you need to track: real, reactive, and apparent. Understanding how these relate to each other is central to analyzing any AC system.

Power factor tells you how efficiently a circuit converts supplied power into useful work. A low power factor means the source has to deliver more current than necessary, wasting energy along the way. Engineers use power factor correction to fix this problem in real electrical systems.

Real, Reactive, and Apparent Power

Real power (also called active power, symbol $P$) is the power actually consumed by resistive elements and converted into useful work or heat. It's measured in watts (W).

$P = V_{rms} I_{rms} \cos(\theta)$

where $\theta$ is the phase angle between voltage and current. In a purely resistive circuit, $\theta = 0$, so $\cos(\theta) = 1$ and all the supplied power is real power.

Reactive power (symbol $Q$) is the power that sloshes back and forth between the source and the reactive components (inductors and capacitors). It's measured in volt-ampere reactive (VAR).

$Q = V_{rms} I_{rms} \sin(\theta)$

Reactive power doesn't do useful work. Instead, it sustains the magnetic fields in inductors and electric fields in capacitors. Even though it delivers zero net energy over a full cycle, it still forces the source to supply extra current, which is why it matters.

• Inductive loads (motors, transformers) draw positive reactive power (current lags voltage)
• Capacitive loads draw negative reactive power (current leads voltage)

Apparent power (symbol $S$) is the total power the source delivers, combining both real and reactive components. It's measured in volt-amperes (VA).

$S = V_{rms} I_{rms}$

Apparent power represents the full burden on the source. If the power factor were a perfect 1.0, all of the apparent power would be real power.

Complex Power

Complex power gives you a single expression that captures both real and reactive power at once:

$S = P + jQ$

Here $j$ is the imaginary unit. The real part is the real power, and the imaginary part is the reactive power.

A few key relationships follow from this:

• The magnitude of complex power equals the apparent power: $|S| = \sqrt{P^2 + Q^2} = V_{rms} I_{rms}$
• The angle of complex power is the phase angle: $\theta = \tan^{-1}(Q/P)$
• You can also compute complex power directly from phasors: $S = \tilde{V} \cdot \tilde{I}^*$, where $\tilde{I}^*$ is the complex conjugate of the current phasor

Power Factor and Correction

Power Factor

Power factor is the ratio of real power to apparent power:

$pf = \frac{P}{S} = \cos(\theta)$

It ranges from 0 to 1. A power factor of 1 means the load is purely resistive and all supplied power does useful work. A power factor near 0 means almost all the power is reactive, and very little useful work gets done.

You'll often see power factor described as leading or lagging:

• Lagging power factor: current lags voltage (inductive load, the most common case)
• Leading power factor: current leads voltage (capacitive load)

This distinction matters because it tells you what kind of correction is needed.

Power Triangle and Correction

The power triangle is a right triangle that visually connects all three power types:

• The horizontal leg is real power $P$
• The vertical leg is reactive power $Q$
• The hypotenuse is apparent power $S$
• The angle between $P$ and $S$ is the phase angle $\theta$

This triangle is just a geometric version of $S = P + jQ$, and it makes the relationships easy to see at a glance.

Power factor correction reduces the reactive power in a circuit, bringing the power factor closer to 1. Here's how it works in practice:

1. Identify whether the load is inductive or capacitive (most industrial loads are inductive)

2. Add a compensating component in parallel with the load to offset the reactive power

   • For inductive loads: add capacitors (they supply leading reactive power that cancels the lagging reactive power)
   • For capacitive loads: add inductors (less common in practice)

3. Calculate the required reactive power compensation: $Q_c = P(\tan\theta_{old} - \tan\theta_{new})$

4. Size the capacitor accordingly: $C = \frac{Q_c}{\omega V_{rms}^2}$

The benefits are real: lower current draw from the source, reduced $I^2R$ losses in transmission lines, and better voltage regulation. Utilities actually charge industrial customers penalties for low power factor, so this has direct financial impact.

AC Circuit Calculations

RMS Values and Calculations

RMS (Root Mean Square) values represent the effective DC-equivalent value of an AC signal. An AC voltage with an RMS value of 120 V delivers the same heating power to a resistor as a 120 V DC source would.

For sinusoidal waveforms, the conversion is simple:

$V_{rms} = \frac{V_{peak}}{\sqrt{2}} \approx 0.707 \cdot V_{peak}$

$I_{rms} = \frac{I_{peak}}{\sqrt{2}} \approx 0.707 \cdot I_{peak}$

For example, a wall outlet in the US has $V_{rms} = 120$ V, which means the peak voltage is about 170 V.

All the power formulas in AC analysis use RMS values. Here's a summary of the key calculation relationships:

• Purely resistive circuit ($\theta = 0$): $P = V_{rms} I_{rms}$ and $Q = 0$
• General AC circuit: $P = V_{rms} I_{rms} \cos(\theta)$, $Q = V_{rms} I_{rms} \sin(\theta)$, $S = V_{rms} I_{rms}$
• Ohm's law with impedance: $V_{rms} = I_{rms} Z$, where $Z$ is the magnitude of the circuit impedance
• Power in terms of impedance: $P = I_{rms}^2 R$ and $Q = I_{rms}^2 X$, where $R$ is resistance and $X$ is reactance

That last pair of formulas is especially useful when you know the current and the impedance components but not the phase angle directly.