Module 3.13, 3.14 and 3.16 AC theory, R-L-C circuits and filters

Knowledge levels

Sub-module	A	B1 / B2 / B2L	B3
3.13 AC theory	1	2	1
3.14 Resistive, capacitive and inductive circuits	—	2	1
3.16 Filters	—	1	—

Category A is examined only on 3.13 here.

3.13 The sine wave

• Cycle: one complete wave. Period T: time for one cycle (s). Frequency f = 1/T (hertz, Hz).
• Aircraft AC systems are typically 115 V / 200 V, 400 Hz, three-phase (some newer aircraft use variable frequency, about 360-800 Hz). The higher frequency allows smaller, lighter transformers and motors.
• Phase: the position of a wave in its cycle, measured in degrees (360° per cycle). Two waves can be "in phase" or one can lead/lag the other.

Values of a sine wave

Value	Relation to peak (Vₚ)
Peak (maximum)	Vₚ
Peak-to-peak	2 Vₚ
RMS (effective)	0.707 Vₚ (Vₚ / √2)
Average (over half cycle)	0.637 Vₚ (2Vₚ/π)
Instantaneous	v = Vₚ sin θ

• Form factor = RMS / average = 1.11; peak (crest) factor = peak / RMS = 1.414.
• AC meters read RMS unless stated otherwise. RMS is the DC value that would produce the same heating in a resistor.

Worked example

Aircraft AC bus 115 V RMS.

• Peak = 115 × 1.414 = 162.6 V
• Peak-to-peak = 325.2 V
• Average = 162.6 × 0.637 = 103.6 V

At 400 Hz, period T = 1/400 = 2.5 ms.

Other waveforms

• Square wave: switches between two levels; contains the fundamental plus odd harmonics. RMS = peak.
• Triangular / sawtooth waves: used in timebases and sweep circuits; RMS of a triangle = Vₚ/√3.

Single-phase and three-phase

• Three-phase: three voltages 120° apart. More efficient power transmission, constant total power, simpler self-starting motors.
• Star (wye): V_line = √3 × V_phase (e.g. 115 V phase → 200 V line); I_line = I_phase.
• Delta: V_line = V_phase; I_line = √3 × I_phase.

3.14 Phase relationships in R, L and C

Component	Opposition	Phase
Resistor	R (Ω), independent of f	V and I in phase
Inductor	X_L = 2πfL (rises with f)	current lags voltage by 90°
Capacitor	X_C = 1 / (2πfC) (falls with f)	current leads voltage by 90°

Key fact: CIVIL - in a Capacitor, I leads V; in an inductor (L), V leads I.

Impedance

• Impedance Z = total opposition to AC (Ω).
• Series RL or RC: Z = √(R² + X²). Series RLC: Z = √(R² + (X_L − X_C)²).
• Phase angle: tan φ = X / R; cos φ = R / Z.
• I = V / Z.

Worked example - series RL

R = 30 Ω, X_L = 40 Ω, supply 100 V.

• Z = √(30² + 40²) = 50 Ω
• I = 100 / 50 = 2 A
• cos φ = 30 / 50 = 0.6 → φ = 53.1° (current lagging)

Resonance

• In series RLC, when X_L = X_C the reactances cancel: Z = R (minimum), current is maximum and in phase with voltage.
• Resonant frequency f₀ = 1 / (2π√(LC)).
• Parallel resonance: impedance is maximum, line current minimum (used in tuned circuits/rejector circuits).

Power in AC circuits

Power	Symbol	Unit	Formula
True (real) power	P	watt (W)	P = V I cos φ = I²R
Reactive power	Q	volt-ampere reactive (VAR)	Q = V I sin φ
Apparent power	S	volt-ampere (VA)	S = V I

• Power factor = P / S = cos φ. Pure resistance: PF = 1. Pure L or C: PF = 0 (no real power consumed; energy oscillates).
• Generators and transformers are rated in kVA because their heating depends on current regardless of power factor.

Worked example - power

Using the RL circuit above (100 V, 2 A, cos φ = 0.6):

• S = 100 × 2 = 200 VA
• P = 200 × 0.6 = 120 W (check: I²R = 4 × 30 = 120 W ✓)
• Q = 200 × 0.8 = 160 VAR

Worked example - reactance

A 10 µF capacitor at 400 Hz.

• X_C = 1 / (2π × 400 × 10 × 10⁻⁶) ≈ 39.8 Ω
• At 800 Hz it halves to about 19.9 Ω.