12.3 Non-Sinusoidal Waves & Single/Three-Phase Principles

12.3 Non-Sinusoidal Waves & Single/Three-Phase Principles

Quick Answer: Square and triangular waves are periodic AC-like shapes but not sines—do not use 0.707 blindly. Single-phase AC is one sinusoidal (or other) voltage source. Three-phase AC uses three equal voltages 120° apart, giving smoother power and forming the basis of aircraft AC generation.

Syllabus 3.13 closes by widening two ideas: (1) not every exam waveform is a sine, and (2) practical power systems often use more than one phase. Both ideas reappear in topics 3.17 AC generators and 3.15 transformers.

Non-Sinusoidal Periodic Waves

A waveform is periodic if it repeats with period T. It is sinusoidal only if that shape is a sine (or cosine). Square, triangular, sawtooth, and many distorted aircraft voltage waveforms are periodic but non-sinusoidal.

Why Module 3 cares: RMS and average definitions (root-mean-square of the instantaneous function; mean of absolute value) still apply. The numerical factors 0.707 and 0.637 are sine-specific. Blindly converting “peak ÷ √2” on a square wave is wrong.

Square wave (symmetrical about zero)

A symmetrical square wave of amplitude A stays at +A for half a period and −A for half a period (ideal vertical edges).

Worked example 1 — square RMS. Square wave ±50 V.

V_peak = 50 V; V_RMS = 50 V (not 50/√2).

Heating in R matches a 50 V DC source, not 35 V.

Worked example 2 — contrast with sine. Sine with V_peak = 50 V has V_RMS ≈ 35.4 V. Same peak, different RMS and heating.

Triangular wave (symmetrical about zero)

An ideal symmetrical triangle rises and falls linearly between −A and +A.

Worked example 3 — triangle RMS. Triangular voltage, peak 30 V.

V_RMS = 30 / √3 ≈ 17.3 V.

Sine with same peak: 30/√2 ≈ 21.2 V. Triangle delivers less RMS (and less heating in R) than a sine of equal peak.

Practical distorted waves

Real aircraft buses can show flat-topping, notches from converters, or harmonics. True-RMS meters measure the actual root-mean-square. Average-responding meters calibrated assuming a sine mis-read non-sine waves. For Module 3 calculations, use the wave type the stem states; default training assumption remains sine unless told otherwise.

Comparison table

Form factor (RMS / average of |v|) differs by shape—another reminder that sine tables are not universal.

Single-Phase AC Principles

Single-phase AC means one alternating voltage source for the load (classically two conductors: line and neutral, plus protective earth in installations).

Properties Module 3 emphasises:

1. One voltage waveform (or one pair of lines) feeds the load.
2. Instantaneous power to a resistive load is pulsating—p = v²/R goes to zero twice per cycle when v crosses zero.
3. Generation can be a single winding on an alternator; distribution is simple.
4. For a given power, currents are higher than in a comparable multi-phase feed—conductor sizing and torque ripple in motors are reasons industry prefers three-phase for large loads.

Worked example 4 — single-phase power. 115 V RMS single-phase into 23 Ω pure resistance.

I_RMS = 115/23 = 5 A; P = 115 × 5 = 575 W average. Instantaneous power still hits zero twice each cycle.

Aircraft utilization equipment may be single-phase loads hung between one phase and neutral (or between two phases) even when the source is three-phase. Distinguish source system from load connection.

Three-Phase AC Principles

Three-phase AC uses three voltages of equal RMS magnitude and the same frequency, phase-displaced by 120° (one-third of a cycle):

v_R = V_m sin(ωt)

v_Y = V_m sin(ωt − 120°)

v_B = V_m sin(ωt − 240°)

(Colour letters vary by standard—R/Y/B, A/B/C, L1/L2/L3—the 120° geometry is what matters.)

Worked example 5 — time between phase peaks. At 400 Hz, T = 2.5 ms. Separation of 120°:

Δt = (120/360) × 2.5 ms = 0.833 ms between successive phase peaks.

Worked example 6 — sum of balanced instantaneous voltages. For ideal balanced three-phase sine voltages, v_R + v_Y + v_B = 0 at every instant. That identity underpins balanced star connection behaviour (neutral current zero for balanced loads).

Line vs phase (preview only)

Detailed star (wye) and delta voltage/current ratios belong with 3.17 / 3.15. For AC theory:

• Phase voltage — voltage of one phase winding (or phase-to-neutral in star).
• Line voltage — voltage between two lines.

In a balanced star: V_line = √3 × V_phase (RMS). In delta: line voltage equals phase voltage. Remember the existence of the √3 relationship; full network drills come later.

Worked example 7 — star preview. Phase voltage 115 V RMS, balanced star.

V_line = 115 × √3 ≈ 199 V RMS (often quoted near 200 V line on some training diagrams).

Single-phase vs three-phase summary

Linking Back to Sine Skills

Whether the system is single- or three-phase, each phase voltage is still usually treated as a sine for Module 3:

• Frequency still sets period (400 Hz → 2.5 ms).
• Each phase still has peak = RMS × √2.
• Phase difference between phases is 120°, not to be confused with the phase difference between voltage and current on one loaded phase (that depends on R, L, C — topic 3.14).

Exam scenario — mixed ideas. A stem shows three 400 Hz traces shifted by one-third period and asks frequency and displacement: answer 400 Hz and 120°. Another stem gives a square wave peak and asks RMS: answer equals the peak, not peak/√2.

Study Close for Topic 3.13

You should now:

1. Convert f ↔ T and read phase lead/lag.
2. Convert peak ↔ RMS ↔ average for sines, and refuse those factors on square/triangle without adjustment.
3. State why aircraft use 400 Hz, and that meters quote RMS.
4. Contrast single-phase with three-phase 120° systems at principle level.

Topic 3.14 applies these waves to R, L, and C together—impedance, phase angle between V and I, and power factor—using the same RMS and phase vocabulary built here.