AC power and filters
In AC circuits, voltage and current may be out of phase, so power has real and reactive parts. The same frequency dependence lets engineers build filters.
Instantaneous and average power
With voltage and current sinusoids out of phase by angle \(\theta\), the average (real) power is \(P = V_{rms}I_{rms}\cos\theta\), in watts. Resistors absorb real power; ideal inductors and capacitors absorb none on average, storing energy during part of each cycle and returning it during another.
Real, reactive, and apparent power
Apparent power \(S = V_{rms}I_{rms}\) is measured in volt-amperes (VA). Reactive power \(Q = V_{rms}I_{rms}\sin\theta\), in volt-amperes reactive (var), represents energy sloshing back and forth. They form a power triangle: \(S^2 = P^2 + Q^2\). Complex power \(\mathbf{S} = P + jQ\) combines them; inductive loads absorb positive Q, capacitive loads negative Q.
Power factor
Power factor is \(\cos\theta = \frac{P}{S}\), between 0 and 1, described as lagging for inductive loads such as motors and leading for capacitive ones. A low power factor means larger currents for the same real power, increasing losses in cables and requiring bigger equipment, so utilities may charge large industrial users for poor power factor.
Power factor correction
Adding capacitors in parallel with inductive loads supplies reactive power locally, cancelling much of the load's reactive demand and raising the power factor towards 1. Factories use capacitor banks; modern electronic power supplies use active power factor correction circuits to draw current in phase with the voltage.
Filters
Filters pass some frequencies and attenuate others. Low-pass filters pass low frequencies and block high ones, removing noise; high-pass filters do the reverse, blocking DC and hum; band-pass filters pass a range, as in radio tuning; and band-stop or notch filters reject a range, such as 50 or 60 Hz mains hum.
The RC low-pass filter
A resistor in series and a capacitor to ground form a first-order low-pass filter. Its transfer function is \(H(j\omega) = \frac{1}{1 + j\omega RC}\). At the cutoff frequency \(f_c = \frac{1}{2\pi RC}\), the output amplitude falls to \(\frac{1}{\sqrt{2}}\), about 70.7% or -3 dB, and the phase shift is -45 degrees. Above cutoff, the output falls by 20 dB per decade of frequency. Swapping R and C makes a high-pass filter.
Decibels and Bode plots
Engineers express gain in decibels: \(20\log_{10}\left|\frac{V_{out}}{V_{in}}\right|\) dB for voltage ratios. A factor of 10 in voltage is 20 dB; a factor of 2 is about 6 dB; half power is -3 dB. Bode plots graph gain in dB and phase against frequency on a logarithmic axis, showing filter behaviour at a glance, with straight-line approximations for quick sketches.
Worked example
A motor draws 10 A RMS from 230 V RMS at a power factor of 0.8 lagging. Apparent power \(S = 230 \times 10 = 2{,}300\) VA; real power \(P = 2{,}300 \times 0.8 = 1{,}840\) W; reactive power \(Q = 2{,}300 \times 0.6 = 1{,}380\) var. Adding a capacitor supplying 1,380 var would raise the power factor to 1, cutting the current for the same real power to \(\frac{1{,}840}{230} = 8\) A.
Key takeaways
Average power is P = V_rms I_rms cos(theta); ideal L and C absorb none on average. Apparent power S (VA), reactive power Q (var), and real power P (W) satisfy S^2 = P^2 + Q^2. Power factor cos(theta) = P/S is lagging for inductive loads; low values raise currents and losses, so capacitors or active circuits correct it. Filters are low-pass, high-pass, band-pass, or band-stop. An RC low-pass filter has H = 1/(1 + j omega RC) and cutoff f_c = 1/(2 pi RC), where gain is -3 dB and phase -45 degrees, falling 20 dB per decade above. Decibels and Bode plots describe frequency response.
Flashcard
P = V_rms I_rms cos(theta)
Check your understanding
A load draws 5 kVA of apparent power at a power factor of 0.6. What real power does it use?