Inverters & PWM formulas

Inverters and pulse width modulation. 15 formulas for GATE EE, IN Power Electronics, each with when to use it. Each is explained in the topic notes.

All 15 Inverters & PWM formulas

SQUARE-WAVE RMS OUTPUT (FULL-BRIDGE)

V_{o, rms} = V_{s}

Half-bridge: the output is $\pm V_{s} /2$ , so the rms is $V_{s} /2$ .

FOURIER SERIES OF THE FULL-BRIDGE SQUARE WAVE

v_{o} (t) = n = 1, 3, 5, \dots \sum \frac{4 V _{s}}{nπ} sin (nω t)

Time origin at a rising edge. Only odd harmonics; the nth has peak $4 V_{s} / (nπ)$ . For the half-bridge use $V_{s} /2$ in place of $V_{s}$ .

FUNDAMENTAL RMS (FULL-BRIDGE)

V_{1, rms} = \frac{4 V _{s}}{2 π} = 0.9 V_{s}

Half-bridge: $0.45 V_{s}$ .

TOTAL HARMONIC DISTORTION OF THE SQUARE WAVE

THD = \frac{V _{o, rms}^{2} - V _{1, rms}^{2}}{V _{1, rms}^{2}} = \frac{π ^{2}}{8} - 1 \approx 0.483

About 48.3 percent, the same for half- and full-bridge. The harmonic factor of the nth harmonic is $HF_{n} = V_{n, rms} / V_{1, rms}$ .

LOAD CURRENT WITH A SERIES R-L LOAD

I_{n, rms} = \frac{V _{n, rms}}{R ^{2} + ( nω L ) ^{2}}, P = n \sum I_{n, rms}^{2} R

Steady state. Each harmonic is treated separately, with the impedance at its own frequency $nω$ . Higher harmonics see a larger reactance, so the current is closer to a sine wave than the voltage. For a lossless inverter the average dc source current is $I_{s} = P / V_{s}$ .

QUASI-SQUARE RMS AND HARMONICS

V_{o, rms} = V_{s} \frac{σ}{π}, V_{n, peak} = \frac{4 V _{s}}{nπ} sin \frac{nσ}{2} (n odd)

$σ$ is the width of each output pulse in radians per half-cycle. Even harmonics are absent.

MULTIPLE-PULSE (UNIFORM) PWM RMS

V_{o, rms} = V_{s} \frac{p δ}{π}

Full-bridge output of $p$ equal pulses of height $V_{s}$ per half-cycle, each $δ$ radians wide ( $p δ \leq π$ ). With $p = 1$ and $δ = σ$ this is the quasi-square result.

SIX-STEP 180°: FUNDAMENTAL AND HARMONICS

V_{LL 1, rms} = \frac{6}{π} V_{s} = 0.78 V_{s}, V_{p h 1, rms} = \frac{2}{π} V_{s} = 0.45 V_{s}

For 180° conduction. The nth harmonic ( $n = 6 k \pm 1$ ) has rms $1/ n$ of the fundamental, so $V_{LL n, rms} = 0.78 V_{s} / n$ . Triplen harmonics are absent from both phase and line voltages.

LINE-TO-PHASE RELATION

V_{LL, rms} = 3 V_{p h, rms}

Balanced load. Holds for the total rms and for the fundamental when the phase voltage has no triplen harmonics, as in both conduction modes here.

MODULATION RATIOS

m_{a} = \frac{V _{control}}{V _{tri}}, m_{f} = \frac{f _{c}}{f _{m}}

$m_{a}$ is the amplitude modulation ratio; $m_{f}$ is the frequency modulation ratio.

FUNDAMENTAL IN THE LINEAR REGION ($M_A\LE1$)

V_{1, peak} = m_{a} V_{s} (full-bridge), V_{a o 1, peak} = m_{a} \frac{V _{s}}{2} (half-bridge or one leg)

Large $m_{f}$ ; the full-bridge result is the same for bipolar and unipolar switching. The fundamental grows linearly with $m_{a}$ and its frequency is $f_{m}$ .

THREE-PHASE SPWM LINE VOLTAGE (LINEAR REGION)

V_{LL 1, rms} = \frac{3}{2} \cdot \frac{m _{a} V _{s}}{2} = 0.612 m_{a} V_{s}

Each leg gives a pole voltage of peak $m_{a} V_{s} /2$ , and the line voltage is $3$ times its rms. $V_{s}$ is the dc bus voltage, $m_{a} \leq 1$ , sinusoidal references with no third-harmonic injection.

CSI OUTPUT CURRENT

I_{1, rms} = \frac{4 I _{d}}{2 π} = 0.9 I_{d} (single-phase), I_{L 1, rms} = \frac{6}{π} I_{d} = 0.78 I_{d} (three-phase)

Single-phase: a square wave of current $\pm I_{d}$ . Three-phase: each line current is a 120°-wide block of height $I_{d}$ , the same shape as the 180°-mode VSI line voltage, with harmonics of order $6 k \pm 1$ .

V/F FLUX AND TORQUE RELATIONS

Φ \propto \frac{V}{f}, T_{m a x} \propto (\frac{V}{f})^{2}, N_{s} = \frac{120 f}{P}

Stator impedance drop neglected. Constant $V / f$ gives constant flux and roughly constant breakdown torque below base speed. $P$ is the number of poles and $f$ is in Hz.

SPEED FROM ENCODER PULSE COUNT

n_{rpm} = \frac{60 Δ c}{K Δ t}

$Δ c$ is the count in time $Δ t$ seconds and $K$ is the counts per revolution ( $N$ , or $4 N$ with 4x decoding). Counting pulses in a fixed time suits high speed, while timing between pulses suits low speed.

Where this was asked: 2020 (1), 2022 (2), 2024 (1), 2025 (1), 2026 (1). See the 6 previous-year questions on Inverters & PWM. This shows what was asked, not what will be asked.

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Inverters & PWM: the notes and 6 solved previous-year questions