12.1 Sinusoidal Waveforms — Phase, Period, Frequency

12.1 Sinusoidal Waveforms — Phase, Period, Frequency

Quick Answer: A sinusoidal AC quantity repeats in cycles. Period T is the time for one cycle; frequency f = 1/T. Instantaneous value follows v = V_peak sin(ωt + φ) with ω = 2πf. Phase locates the sine relative to a time origin; phase difference compares two same-frequency waves (lead/lag). Aircraft AC is typically 115 V RMS at 400 Hz.

CAAS SAR-66 Module 3 topic 3.13 AC theory follows DC machines (3.12) and precedes R, L, and C circuits (3.14). Everything later—reactance, power factor, transformers, and AC generators—assumes you can read a sine wave: how often it repeats, where it sits in time, and what “in phase” means. This section builds that language.

Why AC Is Drawn as a Sine Wave

In a simple AC generator, a coil rotates at constant speed in a uniform magnetic field. The induced EMF is proportional to the rate of flux cutting. For constant rotation in a uniform field, that rate varies as the sine (or cosine) of the rotor angle. Plotting induced voltage against time therefore yields a sinusoidal waveform—the ideal shape assumed in Module 3 unless a stem specifies otherwise.

A sine wave is completely described by:

1. Amplitude (peak value) — how tall the wave is.
2. Frequency (or period) — how fast it repeats.
3. Phase — where the wave sits relative to t = 0 or to another wave.

Amplitude arithmetic (peak, RMS, average) is Section 12.2. Here the focus is time and phase.

Cycle, Period, and Frequency

Cycle

One cycle is one complete pattern of the waveform that returns to the same value and the same direction of change. For a sine starting at zero going positive:

• 0° → rising through zero
• 90° → positive peak
• 180° → falling through zero
• 270° → negative peak
• 360° → back to zero rising again — one cycle complete

Angles may be stated in degrees (0° to 360°) or radians (0 to 2π). One cycle = 360° = 2π rad.

Period (T)

Period T is the time duration of one cycle. SI unit: second (s). On an oscilloscope, T is the horizontal distance for one full repetition multiplied by the time/division setting.

Frequency (f)

Frequency f is the number of cycles per second:

f = 1 / T

Worked example 1 — period from frequency. Aircraft AC at f = 400 Hz:

T = 1 / 400 = 0.0025 s = 2.5 ms.

Worked example 2 — frequency from period. A sine wave has T = 20 ms = 0.020 s.

f = 1 / 0.020 = 50 Hz (utility frequency in many regions).

Worked example 3 — compare times. At 60 Hz, T = 1/60 ≈ 16.7 ms. At 400 Hz, T = 2.5 ms. The aircraft wave completes a cycle about 6.7 times faster than a 60 Hz utility wave (400/60 ≈ 6.67).

Angular frequency

ω = 2πf converts frequency into radians per second—the natural rate used inside the sine argument.

At 400 Hz: ω = 2π × 400 = 800π rad/s ≈ 2513 rad/s.

At 50 Hz: ω = 2π × 50 = 100π rad/s ≈ 314 rad/s.

Module 3 rarely asks you to compute ω numerically, but you must recognise that ωt advances one full cycle when ωt increases by 2π.

Instantaneous Sinusoidal Expression

For a sinusoidal voltage:

v = V_m sin(ωt + φ)

Current uses the same form: i = I_m sin(ωt + φ_i).

Special cases:

• If φ = 0 and the sine starts at the origin: v = V_m sin(ωt).
• If the wave is a cosine at the origin: v = V_m cos(ωt) = V_m sin(ωt + 90°), because cos θ = sin(θ + 90°).

Worked example 4 — instantaneous value. V_m = 170 V, f = 400 Hz, φ = 0. Find v at t = 0.625 ms.

ωt = 2π × 400 × 0.000625 = 2π × 0.25 = π/2 rad = 90°.

v = 170 sin(90°) = 170 V (positive peak).

Worked example 5 — zero crossings. Same wave, zeros when sin(ωt) = 0 → ωt = 0°, 180°, 360°, … Positive-going zero at the start of each cycle; negative-going zero at mid-cycle.

Phase and Phase Difference

Phase of one waveform

Phase answers: where is the sine relative to the time origin? The constant φ shifts the whole wave left or right without changing frequency or amplitude.

• φ > 0 (with the +φ convention above) advances the argument — the wave reaches a given point earlier than the φ = 0 reference (often called a leading phase for that single-wave description).
• φ < 0 delays the wave — it lags the φ = 0 reference.

Always read the stem’s sign convention. Some texts write v = V_m sin(ωt − φ) with φ defined as a lag angle. The physics is the same; the algebra must match the formula given.

Phase difference between two waveforms

Maintenance and Module 3 care most about phase difference—the constant angular gap between two same-frequency sinusoids (for example voltage and current in an AC circuit, or two phase voltages of a generator).

If

v = V_m sin(ωt)

i = I_m sin(ωt − θ)

then current lags voltage by angle θ (common inductive case). If i = I_m sin(ωt + θ), current leads voltage by θ (common capacitive case).

Time shift linked to phase: For frequency f, a phase difference of θ degrees corresponds to a time shift:

Δt = (θ / 360°) × T = θ / (360° × f)

Worked example 6 — time from phase. At 400 Hz, T = 2.5 ms. A current lags voltage by 90°.

Δt = (90/360) × 2.5 ms = 0.625 ms.

Worked example 7 — phase from time. Two 50 Hz voltages; wave B peaks 5 ms after wave A.

T = 20 ms. Fraction of cycle = 5/20 = 0.25 → phase lag of B behind A = 90°.

Aircraft Context: 115 V, 400 Hz

Large civil aircraft commonly distribute 115 V AC (RMS line-to-neutral on many systems) at 400 Hz. Why 400 Hz rather than 50 or 60 Hz?

Module 3 will not ask you to design a 400 Hz generator here, but using 400 Hz in period and phase-shift calculations is a standard exam habit. Ground utility power (50/60 Hz) is a different frequency world—do not mix periods.

Many training stems also quote 28 V DC for DC buses. Keep the two systems mentally separate: DC has no frequency; aircraft AC almost always implies f = 400 Hz unless the question says otherwise.

Reading Waveforms on Paper or Scope

Exam diagrams often show two sine waves on one time axis. Method:

1. Confirm both have the same period (same frequency). Phase difference is undefined in the simple constant sense if frequencies differ.
2. Pick a clear reference event (for example positive-going zero or positive peak) on wave A.
3. Measure how far wave B’s corresponding event is to the right (later → lag) or left (earlier → lead).
4. Convert the horizontal fraction of one period into degrees: fraction × 360°.

Amplitude on the vertical axis is independent of phase. A small wave can lead a large wave; lead/lag is about timing, not height.

Summary Table for Syllabus 3.13 Time Vocabulary

Master cycle ↔ T ↔ f, the sine expression, and lead/lag from diagrams. Section 12.2 then converts the vertical scale of the same sine into peak, average, and RMS values used on every aircraft AC nameplate.