PWM Techniques and Harmonic Control

The linear region. While the reference peak stays below the carrier peak, every carrier period produces a pulse whose width is proportional to the reference:

Exactly proportional, with no trigonometry — which is the whole reason PWM displaced quasi-square control. A regulator that wants 5% more output simply asks for 5% more \(m_a\).

What lies beyond. Raising the reference above the carrier means some carrier intersections are missed, and pulses begin to merge:

More than a quarter again — a substantial amount to leave on the table when Set 26's drive was 13% short. But it is not free.

The whole range:

The fourth column is what is being bought and sold. Every volt gained past \(m_a = 1\) comes with low-order harmonics that PWM existed to remove — and by the square-wave limit the spectrum is exactly the one Set 25, Problem 1 started with.

Why the relationship goes nonlinear. In the linear range each carrier period contributes a pulse; past the limit some do not:

Near the peaks of the reference the output simply stays clamped at the rail for several carrier periods together. Those missing transitions are what produce the low-order harmonics — and they also mean the switching loss falls in overmodulation, which is occasionally useful.

The three-phase version uses the same idea against a different baseline:

Injection has already collected two-thirds of the available gain within the linear range, leaving only 10.3% for overmodulation to fight over. That is why a modern drive uses injection first and overmodulates only in field weakening.

Why odd. An odd carrier ratio makes the modulated waveform half-wave symmetric:

And with them the DC component. An even carrier ratio leaves a DC offset in the output — small, but enough to saturate an output transformer or produce a standing torque in a machine, and entirely avoidable at zero cost.

Why a multiple of three. If \(m_f\) is divisible by 3, the carrier harmonic appears identically in all three phases:

The largest single harmonic in the spectrum — 81.8 V of the 200 V link in Set 25's example — disappears from every line voltage. Only the sidebands remain, and they are smaller. So the standard choices are 15, 21, 27, 33, 39: odd and divisible by three.

Synchronous versus asynchronous. A synchronous modulator locks the carrier to a fixed multiple of the output frequency; an asynchronous one runs the carrier at a fixed frequency regardless:

The critical row is subharmonics. If the carrier is not an integer multiple of the output, the waveform does not repeat every output cycle — and the difference appears as components below the fundamental. In a motor those produce low-frequency torque pulsations and a slow mechanical oscillation, which is far worse than any high-frequency harmonic.

The rule. Subharmonic amplitude scales roughly as \(1/m_f\), so it becomes negligible when the carrier ratio is large:

Applied to this drive at a fixed 4 kHz carrier:

A general-purpose drive is asynchronous throughout its normal range and switches to synchronous modulation only at high output frequency — which is precisely where it is also overmodulating. Traction drives, which spend their lives there, are synchronous throughout.

The interaction with loss and filtering:

Set 26's drive found switching loss already at 63% of the total at 4 kHz, so 16 kHz would take the inverter from 98.1% to about 94%. Silence is expensive, and large drives choose noise.

The waveform. Instead of comparing against a carrier, the switching instants are computed offline. With two notches per quarter cycle at \(\alpha_1\) and \(\alpha_2\), the Fourier coefficients are:

Two unknowns, so two conditions can be imposed. Choosing to null the fifth and seventh uses both — leaving the fundamental to be whatever it turns out to be, with no voltage control. Set 25, Problem 2 met the same arithmetic with one angle.

The equations:

Transcendental and coupled — no closed form exists. They are solved numerically once, offline, and the angles stored in a lookup table. That offline solution is the defining feature of the method: nothing is computed in real time.

The solution:

The resulting spectrum, normalised to \(V_{dc}\):

Two observations. The fundamental is 1.188 times the link — well above the 1.0 that SPWM manages, because the waveform is closer to a square wave. And the eliminated energy has not disappeared: the 11th and 13th are now larger than the 5th and 7th were.

Why that redistribution is still a win. In a motor the harmonic current is limited by the leakage reactance, which rises with frequency:

Comparing against the six-step spectrum's 5th harmonic: a 27% voltage harmonic at the 13th produces only 62% of the current that a 20% harmonic at the 5th did. Pushing distortion up in frequency reduces the current it causes even when the voltage grows — and current is what heats the machine and pulsates the torque.

Where SHE belongs:

The last two rows confine it. SHE angles are valid for one modulation index, so a variable-speed drive needs a table across the whole range and interpolation between entries — and the waveform cannot be changed mid-cycle without breaking the elimination. It is therefore the method of choice for high-power, low-switching-frequency, slowly-varying applications: traction inverters, large industrial drives, grid-connected converters where each switching transition costs kilowatts.

Locate each requirement on the utilisation scale:

So rated operation sits exactly at the linear ceiling of third-harmonic injection, with no margin at all, and anything above it requires overmodulation. That is the normal condition of every mains-fed drive.

The two overmodulation regions. They differ in what the modulator is doing:

In region I the modulator is still switching every carrier period but cannot reproduce the reference near its peaks. In region II it abandons whole intervals, dwelling on a single state — which is where the low-order harmonics grow rapidly.

The gain-versus-distortion trade:

Ten per cent more voltage for thirty-one per cent distortion. Whether that is a good trade depends entirely on what the extra voltage is for — and in field weakening, it is for speed.

Why field weakening makes it acceptable. Above base speed the machine's flux is reduced and the frequency raised:

A 31% voltage distortion at 100 Hz produces half the harmonic current that the same distortion would at 50. So the operating region where overmodulation is needed is also the region where its consequences are mildest — a genuinely fortunate alignment, and the reason six-step operation at high speed is standard rather than exceptional.

What it costs elsewhere:

The third row is what makes overmodulation awkward for a controller. Inside the linear range the loop sees a constant gain; in region II the gain falls towards zero as six-step is approached, so a current regulator tuned for the linear range becomes sluggish and can lose control entirely. Modern drives handle it with an explicit inverse mapping from desired to commanded modulation index.

The insight from Set 26, Problem 4 was that any signal added to all three references cancels in the line voltages. Third-harmonic injection used that freedom to flatten the peak. It can be used for something else entirely:

If a reference is pinned at the carrier peak, that leg never crosses the carrier — so it never switches. It is clamped to the rail, dissipating only conduction loss, while the other two legs carry the modulation.

How much can be clamped. By symmetry, each of the three legs takes a turn:

A third off the switching loss, from a scheme that produces exactly the same line voltages. Applied to Set 26's drive, that is 31 W of the 92 W switching budget — taking the inverter from 98.1% to 98.5%, or allowing the carrier to rise by half for the same loss.

But the saving is not exactly a third, and this is the interesting part. Switching loss depends on the current at the switching instant:

So the placement of the clamped interval should follow the current, not the voltage — and since the phase angle between them depends on the load, the optimum shifts with power factor. Generalised DPWM schemes take the power factor angle as an input and position the clamping accordingly.

What it costs:

Rows two and three cross over at around \(m = 0.6\). Above it discontinuous modulation is better on both counts — less loss and less ripple — because fewer, larger voltage pulses produce less current ripple than many small ones once the modulation depth is high. Below it the continuous scheme wins on ripple.

The natural strategy is therefore to switch schemes with operating point:

Which most modern drives do, changing over on the fly. The transition is seamless because both schemes produce identical line voltages — only the common-mode component differs, and nothing downstream can detect it.

The fourth row of the table is a bonus. Clamping a leg to a rail means it is at a known potential rather than switching:

The 4 A spikes of Set 26, Problem 5 occur at every transition, so removing a third of the transitions removes a third of them. Discontinuous modulation therefore improves efficiency, ripple at high modulation, and bearing life simultaneously.

Map the operating range against the constraints established so far:

Five regions with five different binding constraints — which is why no single modulation scheme serves the whole range.

Region 1, below 5 Hz. Set 25, Problem 5 found the dead-time error to be a fixed voltage:

Twenty-eight per cent distortion at creep speed. The response is to lower the carrier here — halving \(f_s\) to 2 kHz halves the error — and to apply dead-time compensation. Continuous SVM, because ripple quality matters more than efficiency at a twentieth of rated power.

Regions 2 and 3. The crossover of Problem 5 governs:

Below 0.6 the continuous scheme gives lower ripple; above it the discontinuous scheme gives both lower ripple and 33% less switching loss. The changeover is seamless, since both produce identical line voltages.

Region 4, base speed. This is where Set 26 found the design constraint:

Exactly at the linear limit of SVM with no margin — so any supply sag pushes the drive into overmodulation at rated speed. That is normal, and it is why drives are specified with a slight voltage derating rather than a hard guarantee.

Region 5, field weakening. Two things happen together:

So above about 150 Hz the modulator locks the carrier to a synchronous multiple — \(m_f = 21\), then 15, then 9 as the frequency rises — and progressively overmodulates. At the top of the range it is running six-step: one state per 60°, no modulation at all, and the highest possible output voltage.

The complete strategy:

Five regions, four transitions, all handled in software with no hardware change. Every transition is triggered by a constraint identified in Sets 25 to 27, and all of them are invisible to the motor except through the waveform quality.

What this reveals about modulation. The scheme is not a property of the inverter:

The hardware of Set 26, Problem 6 is fixed: six IGBTs, a link capacitor, a diode bridge. Everything in this problem is a choice made in the controller, and the difference between a good drive and a poor one at the same power rating is almost entirely here.