Chapter 10 — AC Power Analysis
Chapter Overview (click to expand)
AC power analysis distinguishes between three types of power — real, reactive, and apparent — that together characterize how efficiently energy is delivered to a load. This chapter develops the mathematics of complex power and the power triangle, and explains power factor correction, an essential technique for reducing wasted energy in electrical systems. **Key Takeaways** 1. Real (average) power P, measured in watts, represents energy actually consumed; reactive power Q, measured in VARs, represents energy oscillating back and forth between source and reactive elements. 2. Apparent power S is the product of RMS voltage and current, and equals the magnitude of complex power S = P + jQ. 3. Power factor (cos θ) measures how effectively a load uses apparent power; capacitor banks are added to correct a lagging power factor and reduce transmission losses.10.1 Instantaneous Power
Instantaneous power is the power at any specific instant:
For sinusoidal signals: (p(t) = \dfrac{V_m I_m}{2}\cos(\theta_v - \theta_i) + \dfrac{V_m I_m}{2}\cos(2\omega t + \theta_v + \theta_i)]
This reveals two components: a constant term (the average power) and an oscillating term at twice the signal frequency. The oscillating component means power flows back and forth between the source and reactive elements every quarter cycle — it can actually go negative, meaning energy flows back to the source.
10.2 Real Power (Average Power)
Real power (also called average power or active power) is the time-averaged power over one complete cycle. It represents the power that actually does useful work — heating elements, turning motors, producing light and sound.
Measured in watts (W). \(\theta = \theta_v - \theta_i\) is the phase angle between voltage and current.
The factor \(\cos\theta\) is the power factor — it determines how much of the apparent power does useful work. Your electric bill is based on real power (kilowatt-hours).
Definition: Real (Average) Power Real power \(P\) is the time-averaged power consumed by a load, measured in watts (W): \(P = V_{rms} I_{rms} \cos\theta = I_{rms}^2 R\). It represents energy that is actually converted into heat, light, or mechanical work. The power factor \(\cos\theta\) tells you what fraction of the apparent power is real — your electricity bill charges for real power, not apparent power.
| Phase Angle θ | cos θ | Interpretation |
|---|---|---|
| 0° | 1.00 | Pure resistive — all power is real |
| 30° | 0.866 | Moderate reactive component |
| 60° | 0.50 | Mostly reactive |
| 90° | 0.00 | Pure reactive — zero real power |
Diagram: RMS Value Calculator
10.3 Reactive Power
Reactive power (Q) represents energy oscillating between the source and reactive elements without being consumed. It's measured in volt-amperes reactive (VAR).
Inductive loads: \(Q > 0\) (lagging). Capacitive loads: \(Q < 0\) (leading).
Reactive power doesn't do useful work, but it still requires current to flow — meaning larger wires, transformers, and generators must be sized to carry it. This is why utilities penalize poor power factor.
Common Mistake: Thinking Reactive Power Is "Wasted" Reactive power \(Q\) is not wasted — it is energy that oscillates back and forth between the source and reactive elements (inductors and capacitors) but is never consumed. The problem is that it requires real current to flow through real wire resistance, causing genuine power loss (\(I^2 R\)) in the distribution system. So reactive power itself isn't wasted, but the extra current it demands does cause real losses in transmission lines.
10.4 Apparent Power
Apparent power (S) is the product of RMS voltage and RMS current — the total "power handling" demanded from the source. Measured in volt-amperes (VA).
Apparent power determines wire sizing, transformer ratings, generator capacity, and circuit breaker ratings — equipment must handle the full current regardless of power factor.
Example: A motor draws 10 A from a 120 V source at PF = 0.8: - \(S = 120 \times 10 = 1{,}200\) VA - \(P = 1{,}200 \times 0.8 = 960\) W - \(Q = 1{,}200 \times 0.6 = 720\) VAR
The motor does only 960 W of work, but the supply must have capacity for 1,200 VA.
10.5 Complex Power
Complex power combines real and reactive power into a single complex number:
where \(\mathbf{I}^*\) is the complex conjugate of the current phasor.
Key Formula: Complex Power
Use the complex conjugate of the current phasor (\(\mathbf{I}^*\)), not the current itself. This ensures the real part gives \(P = I^2 R\) and the imaginary part gives \(Q = I^2 X\). Forgetting the conjugate gives a result with opposite sign for \(Q\).
| Quantity | Symbol | Units | Formula |
|---|---|---|---|
| Real Power | \(P\) | W | \(V I \cos\theta\) |
| Reactive Power | \(Q\) | VAR | \(V I \sin\theta\) |
| Apparent Power | \(S\) | VA | \(VI = \|\mathbf{S}\|\) |
| Complex Power | \(\mathbf{S}\) | VA | \(P + jQ\) |
10.6 The Power Triangle
The power triangle shows the geometric relationship between P, Q, and S — they form a right triangle:
- Hypotenuse: Apparent power S (VA)
- Horizontal leg: Real power P (W)
- Vertical leg: Reactive power Q (VAR)
- Angle θ: Phase angle between voltage and current
The power triangle is identical in shape to the impedance triangle (\(|Z|\), R, X).
10.7 Power Factor: Leading and Lagging
Power factor (PF) = \(\cos\theta = P/S\). It ranges from 0 to 1 and measures how efficiently apparent power converts to real work.
Lagging power factor — current lags voltage (inductive loads: motors, transformers, ballasts): - \(\theta > 0\), \(Q > 0\), load absorbs VARs from source
Leading power factor — current leads voltage (capacitive loads: correction capacitors, some power supplies): - \(\theta < 0\), \(Q < 0\), load supplies VARs to circuit
| Power Factor | Description | Typical Load |
|---|---|---|
| 1.0 | Unity | Resistive heaters, incandescent bulbs |
| 0.95 lagging | Excellent | Well-corrected industrial motors |
| 0.85 lagging | Good | Typical industrial mix |
| 0.70 lagging | Poor | Uncompensated motors |
| 0.50 lagging | Very poor | Lightly loaded motors |
10.8 Power Factor Correction
Power factor correction improves PF by adding capacitors in parallel with inductive loads. The capacitors supply reactive power locally, reducing the reactive current drawn from the source.
Benefits: reduced line current, less I²R loss, increased infrastructure capacity, avoided utility penalties.
Pro Tip: Power Factor Correction Always Uses Capacitors in Parallel Industrial loads (motors, transformers) are inductive and draw lagging current — they need VARs supplied from the grid. Adding a capacitor in parallel with the load provides those VARs locally, reducing the reactive current the utility must supply. Always connect the correction capacitor in parallel with the load, never in series — series connection changes the impedance seen by the load and alters circuit behavior unpredictably.
Correction formula — to improve from \(\theta_1\) to \(\theta_2\):
Example: A 10 kW load at PF = 0.7 lagging from 240 V, 60 Hz. Correct to PF = 0.95:
\(\theta_1 = 45.57°\), \(\tan\theta_1 = 1.020\) ; \(\theta_2 = 18.19°\), \(\tan\theta_2 = 0.329\)
\(Q_C = 10{,}000(1.020 - 0.329) = 6{,}910\) VAR
\(C = 6{,}910 / (2\pi \times 60 \times 240^2) = 319\) μF
A 320 μF capacitor reduces reactive demand from the utility by 6,910 VAR.
10.9 Power in Resistors, Capacitors, and Inductors
Each passive component has distinct power behavior:
Resistors — voltage and current are in phase (\(\theta = 0\)):
Capacitors — current leads voltage by 90°, no real power consumed:
Inductors — current lags voltage by 90°, no real power consumed:
| Element | Real Power P | Reactive Power Q | PF |
|---|---|---|---|
| Resistor | \(I^2 R\) | 0 | 1 |
| Capacitor | 0 | \(-V^2/X_C\) | 0 leading |
| Inductor | 0 | \(+V^2/X_L\) | 0 lagging |
10.10 Maximum Power Transfer in AC
For maximum power transfer from a source with internal impedance \(Z_s = R_s + jX_s\) to load \(Z_L\):
The load impedance must be the complex conjugate of the source impedance.
At conjugate matching: \(R_L = R_s\) (equal resistances) and \(X_L = -X_s\) (reactances cancel). The result is resonance — maximum current and maximum power transfer — but efficiency is only 50%. This trade-off is acceptable in RF and audio signal systems; power delivery systems optimize for efficiency instead.
10.11 Efficiency and Power Gain
Efficiency:
| Device | Typical Efficiency |
|---|---|
| Power transformer | 95–99% |
| AC induction motor | 80–95% |
| LED bulb | 35–50% |
| Incandescent bulb | 2–5% |
| Switching power supply | 85–95% |
Power gain measures amplification:
10.12 Worked Example: Complete Power Analysis
Problem: A series RL load with R = 30 Ω and \(X_L = 40\) Ω is connected to a 120 V RMS, 60 Hz source. Find all power quantities and the capacitor needed for unity PF correction.
Step 1 — Impedance and current:
\(Z = 30 + j40 = 50\angle 53.13°\) Ω \(\quad I_{rms} = 120/50 = 2.4\) A
Step 2 — Powers:
[P = I^2 R = (2.4)^2(30) = 172.8 \text{ W}] [Q = I^2 X_L = (2.4)^2(40) = 230.4 \text{ VAR}] [S = I^2|Z| = (2.4)^2(50) = 288 \text{ VA} \qquad PF = P/S = 0.6 \text{ lagging}]
Step 3 — Power factor correction to unity:
\(Q_C = -230.4\) VAR needed. \(C = 230.4 / (2\pi \times 60 \times 120^2) = 42.5\) μF
After correction: current drops from 2.4 A to \(172.8/120 = 1.44\) A — a 40% reduction!
10.13 Chapter Summary
The three power quantities:
| Quantity | Symbol | Units | Formula |
|---|---|---|---|
| Real (active) | P | W | \(V_{rms} I_{rms}\cos\theta\) |
| Reactive | Q | VAR | \(V_{rms} I_{rms}\sin\theta\) |
| Apparent | S | VA | \(V_{rms} I_{rms}\) |
Power triangle: \(S^2 = P^2 + Q^2\), \(\quad PF = P/S = \cos\theta\)
Power factor correction: add \(C = Q_C/(\omega V^2)\) in parallel with inductive load
Per-element power: - Resistors: only P (real power) - Capacitors: only Q < 0 (supply VARs) - Inductors: only Q > 0 (absorb VARs)
AC maximum power transfer: \(Z_L = Z_s^*\) (conjugate matching), efficiency = 50%