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In sinusoidal PWM, a low-frequency sine reference is compared against a high-frequency triangular carrier. Whenever the sine is above the carrier the top switch is on, so the duty cycle of each switching pulse follows the sine wave. Around zero-crossing the duty is 50%; at the sine peak it reaches its maximum; at the trough its minimum. The instantaneous duty is d(θ) = 0.5×(1 + ma·sinθ).
| Quantity | Formula |
|---|---|
| Instantaneous duty | d(θ) = 0.5(1 + ma·sinθ) |
| Maximum duty (θ=90°) | dmax = 0.5(1 + ma) |
| Minimum duty (θ=270°) | dmin = 0.5(1 − ma) |
| Pulse on-time | ton = d(θ) / fsw |
The duty swings symmetrically around 0.5. With ma = 1 it spans the full 0–100% range; with smaller ma it stays closer to 50%, producing a smaller output voltage. This averaged duty, filtered by the load inductance, reconstructs the sine wave at the inverter output.
An inverter leg can only connect its output to one DC rail or the other — there is no intermediate voltage available. Sinusoidal PWM produces a sine anyway by varying how long it spends on each rail, cycle by cycle:
d(t) = ½(1 + m sin ωt)
With a modulation index m = 0.8, the duty cycle traces 0.5 at the zero crossing, 0.7 at 30°, 0.9 at the positive peak and 0.1 at the negative peak. Averaged over each switching period, the output follows the reference sine; the switching itself appears as ripple at the carrier frequency, which the load inductance and any output filter remove.
Raising the carrier pushes the switching harmonics further above the fundamental, making them easier to filter and leaving a cleaner output with a smaller filter. But every switching transition costs energy in the devices, so switching loss rises roughly in proportion to carrier frequency. The choice is a direct trade between filter size and efficiency, and it is usually made by finding the lowest carrier that gives acceptable output quality.
Keeping the carrier an odd multiple of three times the fundamental in a three-phase inverter has a useful side effect: the carrier-related harmonics become triplen, so they are identical in all three phases and cancel in the line-to-line output.
Both devices in a leg must never conduct simultaneously, so the gate signals include a short dead time where both are off. During that interval the output voltage is set not by the modulator but by the direction of the load current, which forces it to whichever rail the freewheeling diode connects.
The result is a small voltage error on every switching edge, always in the direction that opposes current flow. It accumulates into low-order harmonic distortion — most visible as a flat spot near the current zero crossing — and it is worst at low output voltage, where the dead time is a large fraction of the pulse width. Dead-time compensation in the control software is standard practice for exactly this reason.
The instantaneous duty is d(θ)=0.5×(1+ma·sinθ), where ma is the amplitude modulation ratio and θ is the electrical angle of the sine reference.
At the sine zero-crossing the reference equals the carrier centre, so the pulse is symmetric — on for half the period and off for half. This produces zero average output at that instant.
They occur at the sine peak and trough: dmax=0.5(1+ma) at 90° and dmin=0.5(1−ma) at 270°. With ma=1 they reach 100% and 0%.
ma sets how far the duty swings away from 50%. A larger ma gives a wider swing and larger output voltage; a smaller ma keeps the duty near 50% with a smaller output.
Multiply the duty by the switching period: ton=d(θ)/fsw=d(θ)×Tsw. For example, 90% duty at 10 kHz (100 µs period) gives a 90 µs on-time.
Bipolar SPWM switches the full bridge between +Vdc and −Vdc; unipolar switching adds a third level (0 V), halving the effective harmonic content and doubling the apparent switching frequency at the output. The per-leg duty formula is the same.
Not physically. If ma>1 the ideal formula would exceed 0–100%, but the hardware clamps the duty at the rails — this clamping is what causes overmodulation distortion.
Each of the three legs uses the same formula but with references 120° apart: da=0.5(1+masinθ), db at θ−120°, dc at θ−240°. The line-to-line voltages then form a balanced three-phase set.
mf=fcarrier/foutput is the number of carrier cycles per output cycle. A high mf gives more PWM pulses per sine cycle, pushing harmonics to high frequencies that the load inductance easily filters.
Yes. A small dead-time is inserted between the top and bottom switch turn-on to prevent shoot-through, which slightly reduces the effective on-time and distorts the output near zero-crossings, especially at low ma.
One duty value per switching pulse, i.e. mf points per output cycle. A higher carrier frequency needs a larger table but produces a smoother sine and lower audible noise.
No. SPWM compares each phase to a carrier independently. Space-vector PWM (SVPWM) treats the three phases together and adds a common-mode component, using the DC bus about 15% more effectively while producing a similar per-leg duty pattern.
d(t) = ½(1 + m sin ωt) for a half-bridge leg, where m is the modulation index. The duty sits at 0.5 at the zero crossing and swings symmetrically either side of it.
By varying the fraction of each switching period spent connected to each DC rail. Averaged over a switching period the output follows the sinusoidal reference; the switching ripple itself is filtered by load inductance and any output filter.
Higher carriers push switching harmonics further from the fundamental, allowing a smaller output filter, but increase switching losses proportionally. Choose the lowest carrier that gives acceptable output quality.
A short interval in which both devices in a leg are held off, to guarantee they never conduct simultaneously and short the DC link. It is essential for safety but introduces output voltage error.
During the dead interval the output is determined by load current direction rather than by the modulator, producing a small voltage error on every edge. These errors accumulate into low-order harmonics, worst at low output voltage where dead time is a large fraction of the pulse width.
In a three-phase inverter this makes the carrier-related harmonics triplen, so they are identical in all three phases and cancel in the line-to-line output, improving the spectrum at no extra cost.
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