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An inverter turns DC into AC by switching. In sinusoidal PWM (SPWM) a sine reference is compared with a triangular carrier; the amplitude modulation ratio ma is the ratio of the sine peak to the carrier peak. In the linear region (ma ≤ 1) the fundamental output voltage is directly proportional to ma, which is how the inverter (or VFD) controls the AC voltage.
| Inverter | Fundamental output (ma ≤ 1) |
|---|---|
| 1φ Half-bridge | V1(peak) = ma·Vdc/2 → V1(rms) = ma·Vdc/(2√2) |
| 1φ Full-bridge | V1(peak) = ma·Vdc → V1(rms) = ma·Vdc/√2 |
| 3φ (line-to-line RMS) | VLL(rms) = ma·(√3/(2√2))·Vdc = 0.612·ma·Vdc |
Above ma = 1 the inverter enters overmodulation; the output keeps rising but no longer linearly, and low-order harmonics appear. The limit is the six-step (square-wave) mode, where the single-phase full bridge reaches (4/π)·Vdc/√2 ≈ 0.90·Vdc RMS and the three-phase line-to-line RMS reaches 0.78·Vdc.
In sinusoidal PWM, the modulation index is the ratio of the reference (control) waveform amplitude to the triangular carrier amplitude:
m = Vcontrol / Vcarrier
The duty cycle of a half-bridge then follows d(t) = ½(1 + m sin ωt). At m = 0.8, duty swings between 0.1 and 0.9 across the fundamental cycle, sitting at 0.5 at each zero crossing.
While m stays at or below 1 the output fundamental is linearly proportional to m, and the switching harmonics sit in a tidy cluster around the carrier frequency where a modest output filter removes them easily. This predictability is the entire appeal of PWM.
Push m above 1 and the reference exceeds the carrier peak, so for part of the cycle there is no crossing at all and the leg simply saturates to one rail. Fundamental output keeps rising, but no longer in proportion, and low-order harmonics — the fifth, seventh and so on — appear in the output. Those are much harder to filter than the carrier-frequency cluster, because they sit close to the fundamental. At the extreme the modulator degenerates into six-step square-wave operation, which gives the maximum possible fundamental at the cost of the worst harmonic spectrum.
A three-phase inverter can do better than m = 1. Adding a third-harmonic component to the reference flattens its peaks without changing the line-to-line output, because third harmonics are identical in all three phases and cancel in the line-to-line difference. This allows the modulation index to reach 2/√3 ≈ 1.155 while still operating linearly — roughly 15% more output voltage from the same DC link, at no cost in harmonic quality. Space-vector modulation achieves the same ceiling by a different route and is the usual choice in modern drives.
The amplitude modulation ratio ma is the ratio of the peak of the sinusoidal reference to the peak of the triangular carrier in SPWM. It directly scales the fundamental output voltage in the linear region (ma ≤ 1).
In the linear region the fundamental output is proportional to ma. For a single-phase full bridge V1(rms)=ma·Vdc/√2, so doubling ma doubles the output (up to ma=1).
It is 0 ≤ ma ≤ 1. Within this range the output rises linearly with ma and the harmonics stay near the switching frequency, making them easy to filter.
When ma > 1 the reference exceeds the carrier for part of the cycle, so the output stops rising linearly and low-order harmonics appear. It boosts the fundamental beyond the linear limit at the cost of distortion.
The absolute maximum is the six-step (square-wave) mode: for a single-phase full bridge (4/π)·Vdc/√2 ≈ 0.90·Vdc RMS, and for a three-phase inverter 0.78·Vdc line-to-line RMS.
For SPWM the phase peak is ma·Vdc/2. The line-to-line value is √3 times the phase, and RMS divides the peak by √2, giving √3/(2√2)=0.612 times ma·Vdc.
A full bridge applies the whole DC bus across the load and gives twice the output of a half bridge, which only swings between +Vdc/2 and −Vdc/2. Hence the full-bridge fundamental is ma·Vdc/√2 versus ma·Vdc/(2√2).
mf = fcarrier/foutput is the ratio of the switching (carrier) frequency to the output frequency. A high, odd, integer mf pushes harmonics to high frequencies and keeps the waveform symmetric.
Rearrange the formula. For a three-phase inverter, Vdc = VLL(rms)/(0.612·ma). Designers usually pick ma around 0.85–0.95 to leave margin, which sets the required bus voltage.
Yes. Space-vector modulation (SVPWM) uses the bus about 15% more effectively than basic SPWM, reaching a line-to-line RMS of roughly 0.707·Vdc at the edge of the linear region versus 0.612·Vdc for SPWM.
A VFD keeps a constant volts-per-hertz ratio, so as it raises the output frequency it raises ma to raise the voltage, until ma reaches its limit at base speed. Above that, the motor runs in the field-weakening (constant-power) region.
Zero fundamental output. With no reference signal the inverter produces equal positive and negative pulses that average to zero, so no net AC voltage appears at the output.
The ratio of the control (reference) waveform amplitude to the carrier waveform amplitude in a PWM modulator. It sets how much of the available DC-link voltage appears in the output fundamental.
m ≤ 1 for plain sinusoidal PWM. Within it, output fundamental voltage is directly proportional to m and the switching harmonics stay clustered around the carrier frequency, where they are easy to filter.
The reference exceeds the carrier peak, so parts of the cycle have no crossings and the leg clamps to a rail. Output keeps rising but non-linearly, and low-order harmonics such as the fifth and seventh appear, which are far harder to filter.
Adding a third-harmonic component to the reference waveform to flatten its peaks. Because third harmonics are identical in all three phases, they cancel in the line-to-line output, so the modulation index can reach about 1.155 while remaining linear.
About 15%, since the usable modulation index rises from 1 to 2/√3 ≈ 1.155. Space-vector modulation reaches the same limit by a different method.
A higher carrier pushes switching harmonics further from the fundamental, so the output filter can be smaller. The cost is higher switching loss in the devices, which is the usual trade-off between filter size and efficiency.
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