2.4 Impedance, Resonance, Power Factor, and Three-Phase Systems

2.4 Impedance, Resonance, Power Factor, and Three-Phase Systems

Practical aircraft AC electrical systems contain combinations of resistance ($R$), inductance ($L$), and capacitance ($C$). When these three elements interact under alternating current, analyzing voltage, current, and power requires vector addition and complex number analysis. This section covers total circuit impedance ($Z$), electrical resonance ($f_0$), power factor efficiency, and the 3-phase 115/200 VAC polyphase distribution systems standard on aircraft.

Impedance ($Z$) and Vector RLC Analysis

Impedance ($Z$) is the total combined opposition to the flow of alternating current in an AC circuit containing both resistance and reactance. Like resistance and reactance, impedance is measured in ohms ($\Omega$).

Because resistance ($R$) dissipates energy in phase ($0^\circ$), inductive reactance ($X_L$) leads by $+90^\circ$, and capacitive reactance ($X_C$) lags by $-90^\circ$, reactances directly oppose each other along the vertical imaginary axis. The net reactance ($X$) is:

$X = X_L - X_C$

Resistance and net reactance must be combined vectorially using the Pythagorean Theorem to find total impedance ($Z$):

$Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + X^2}$

The circuit phase angle ($\theta$) between total applied voltage and total line current is calculated via trigonometry:

$\theta = \arctan\left(\frac{X_L - X_C}{R}\right) = \arctan\left(\frac{X}{R}\right)$

• If $X_L > X_C$: The circuit is inductive (voltage leads current, $\theta > 0^\circ$).
• If $X_C > X_L$: The circuit is capacitive (current leads voltage, $\theta < 0^\circ$).
• If $X_L = X_C$: The circuit is purely resistive (voltage and current in phase, $\theta = 0^\circ$).

Worked Formula Example: Series RLC Circuit

An aircraft radio power filter has $R = 40\ \Omega$, $X_L = 90\ \Omega$, and $X_C = 60\ \Omega$ connected in series across 115 VAC RMS, 400 Hz power.

Step 1: Net Reactance $X = X_L - X_C = 90\ \Omega - 60\ \Omega = +30\ \Omega\text{ (Inductive)}$

Step 2: Calculate Impedance ($Z$) $Z = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50.0\ \Omega$

Step 3: Calculate Line Current ($I$) and Phase Angle ($\theta$) $I = \frac{V}{Z} = \frac{115\text{ VAC}}{50\ \Omega} = 2.30\text{ Amperes}$ $\theta = \arctan\left(\frac{30}{40}\right) = \arctan(0.75) = +36.87^\circ$

Series and Parallel Electrical Resonance ($f_0$)

Resonance occurs in an RLC circuit at a specific frequency where inductive reactance equals capacitive reactance exactly ($X_L = X_C$). The frequency at which this balance occurs is the resonant frequency ($f_0$).

Setting $2\pi f_0 L = \frac{1}{2\pi f_0 C}$ and solving for $f_0$ yields the universal resonant frequency equation:

$f_0 = \frac{1}{2\pi \sqrt{L \cdot C}}$

Where $f_0$ is in Hz, $L$ in Henrys, and $C$ in Farads.

Series Resonance

In a series RLC circuit at resonance ($f_0$):

• Net reactance $X_L - X_C = 0\ \Omega$.
• Total impedance drops to its absolute minimum, equal solely to resistance ($Z_{min} = R$).
• Line current reaches its absolute maximum ($I_{max} = V/R$).
• High reactive voltages build up across $L$ and $C$ equal to $Q \cdot V_{in}$, where $Q = X_L / R$ is the circuit Quality Factor. Series resonant circuits act as acceptor filters.

Parallel Resonance (Tank Circuits)

In a parallel RLC circuit (LC tank circuit) at resonance ($f_0$):

• Reactive currents circulating back and forth between inductor and capacitor cancel line current.
• Total circuit impedance reaches its absolute maximum ($Z_{max} = \frac{L}{C \cdot R}$).
• External line current drawn from the source drops to its absolute minimum.
• Parallel resonant circuits act as rejector filters, widely used in aircraft VOR, ILS, and VHF radio tuning stages.

Power Factor, True Power, and Apparent Power

In AC circuits containing reactive components, power calculations must separate energy converted into physical work from energy stored temporarily in reactive fields.

1. True Power ($P_{true}$)

Power dissipated as thermal heat or converted into mechanical work by resistive components. Measured in Watts (W):

$P_{true} = I^2 \cdot R = V \cdot I \cdot \cos(\theta)$

2. Apparent Power ($S_{apparent}$)

The total raw volt-ampere capacity delivered by the aircraft generator. Measured in Volt-Amperes (VA) or kVA:

$S_{apparent} = V_{rms} \cdot I_{rms}$

3. Reactive Power ($Q_{reactive}$)

Power bounced back and forth between source and reactive fields ($L$ and $C$). Measured in Volt-Amperes Reactive (VAR):

$Q_{reactive} = V \cdot I \cdot \sin(\theta)$

4. Power Factor ($PF$)

Power Factor is the ratio of True Power to Apparent Power, representing electrical system efficiency:

$PF = \frac{P_{true}}{S_{apparent}} = \cos(\theta)$

Power Factor ranges from $0$ (purely reactive) to $1.0$ (100% purely resistive). A low power factor (e.g., $0.70$ lagging due to heavy motor loads) forces generators and wiring to carry excess reactive current, causing parasitic $I^2 R$ heat losses in aircraft wiring.

Aircraft Three-Phase AC Systems (115/200 VAC, 400 Hz)

Modern multi-engine aircraft utilize Three-Phase (3-Phase) AC Power Systems to supply heavy electrical loads (hydraulic pumps, galley ovens, windshield anti-ice, cargo doors). A 3-phase alternator contains three separate armature windings spaced $120^\circ$ apart around the stator, producing three distinct AC voltage sine waves separated by $120^\circ$ phase displacement.

Advantages of 3-Phase Aircraft Power

1. Constant Power Transfer: Total 3-phase power delivery never drops to zero during any part of the cycle.
2. Reduced Conductor Weight: 3-phase systems deliver 1.732 times more power than single-phase systems using smaller copper wire sizes.
3. Self-Starting AC Motors: 3-phase AC induction motors create a rotating magnetic field naturally, eliminating heavy starting capacitors or centrifugal switches.

Polyphase Configurations: Wye (Y) and Delta ($\Delta$)

Standard aircraft 4-wire Wye distribution provides two selectable voltage levels:

• 115 VAC Line-to-Neutral: Supplies single-phase avionics instruments, lighting, and flight deck displays.
• 200 VAC Line-to-Line: Supplies heavy 3-phase motors and Transformer-Rectifier Units (TRUs).

Avionics Trap: Phase Sequence and Rotation Errors

Avionics Trap: Three-phase AC motors rely on the correct phase sequence (A-B-C) to determine mechanical rotation direction. If an avionics technician accidentally swaps any two phase wires (e.g., changing to A-C-B) during generator replacement or TRU maintenance, 3-phase fuel pumps and cooling fans will rotate in reverse! Reverse pump rotation starves engine fuel manifolds and causes rapid avionics bay overheating without tripping circuit breakers.