PWM Techniques and Harmonic Control

No. That figure is correct, expected, and not the number that matters.

Why it is so high. The output is still a two-level waveform switching between \( \pm V_s/2 \), so its total RMS is \( V_s/2 \) whatever \( m_a \) is. The fundamental is only \( m_aV_s/(2\sqrt2) \). At \( m_a = 0.8 \):

\[ \text{THD} = \frac{\sqrt{(V_s/2)^2 - (0.8V_s/2\sqrt2)^2}}{0.8V_s/2\sqrt2} = 108\% \]

Six-step, by contrast, spends the whole cycle at full amplitude, so more of its RMS is fundamental.

Why the comparison is nevertheless misleading. THD counts amplitude and ignores frequency entirely, and frequency is the whole point:

• Six-step: 20% at 250 Hz, 14% at 350 Hz — right next to the fundamental, and inescapable without a large filter.
• PWM at \( m_f = 40 \): nothing below about 1.9 kHz.

The number that matters is current THD. An inductive load attenuates the \( n \)th harmonic by roughly \( n \), so:

• Six-step: 20% voltage at the 5th → 4% current. Total current THD 6–10%.
• PWM: 30% voltage at the 41st → 0.7% current. Total current THD 2–5%.

What to quote instead of terminal THD:

• Current THD, which is what heats the machine and pulsates the torque.
• Weighted THD (WTHD), which divides each harmonic by its order before summing — precisely to account for the load's filtering. On WTHD, PWM beats six-step decisively, as it should.

Quoting unweighted voltage THD for a PWM inverter is a category error, and a common one.

Because the added harmonic is common to all three legs, and a floating neutral makes anything common invisible to the load.

The geometric part. A pure sine peaks at 1.000 and touches the modulator's ceiling only instantaneously. Adding \( -\tfrac16\sin3\omega t \) flattens the top:

\[ \max\left(\sin\theta - \tfrac16\sin3\theta\right) = 0.866 \]

The fundamental component is unchanged at 1.000 but the peak is now 0.866 — so the whole reference can be scaled by \( 1/0.866 = 1.155 \) before it touches the ceiling again. The fundamental is now 1.155.

The electrical part, which is why it is free. The same third harmonic is added to all three legs, in phase. From Chapter 17:

\[ v_{ab} = v_{aN} - v_{bN} \]

and any component identical in \( v_{aN} \) and \( v_{bN} \) subtracts to zero. The third harmonic moves the load's star point up and down relative to the DC link, but produces no voltage across any phase.

So the load sees a pure sine of 1.155 times the amplitude that plain SPWM could deliver, and no third harmonic at all.

Three conditions that must hold:

1. The neutral must be isolated. Connect it and the third harmonic becomes real, driving large triplen currents.
2. All three legs must get the same injection. Any asymmetry leaves a residue the load does see.
3. The load must be three-phase. A single-phase inverter has no floating star point and gains nothing.

The generalisation worth carrying. Any zero-sequence component can be added freely — third harmonic, a triangular wave, or the min–max function. All reach the same 1.155 limit, because the limit is set by how flat you can make the reference, not by which function does the flattening. Space vector modulation (Chapter 19) turns out to be another member of this family rather than a separate technique.

Two independent symmetry conditions happen to point at the same set of values.

Odd, so the even harmonics vanish. An odd \( m_f \) gives the waveform half-wave symmetry: \( v(\theta + \pi) = -v(\theta) \). Any such waveform has only odd harmonics and no DC term.

With an even \( m_f \), the second half cycle is not the negative of the first. Even harmonics appear, and so does a DC offset — which in a motor is the serious one, since a fraction of an ohm of winding resistance turns a few volts of DC into a large unidirectional current that heats the stator and produces no torque.

A multiple of three, so the carrier cancels. If \( m_f \) is a multiple of three, the carrier-frequency component appears identically in all three legs — it is a triplen — so it disappears between lines.

This matters more than it sounds, because the carrier component is the largest harmonic in the whole spectrum, around 0.82 of the fundamental. Choosing \( m_f \) correctly removes it entirely, for free.

So: 15, 21, 27, 33, 39, 45. That is why a drive's carrier options are 3.15 kHz rather than 3.0 kHz.

What happens with a non-integer \( m_f \). This is the usual case in practice — asynchronous modulation with a fixed carrier and a variable output frequency. The pattern never repeats, so:

• Sub-harmonics appear at the beat frequencies. These are worse than high-order harmonics because the motor's impedance is lowest at low frequency, so a small sub-harmonic voltage makes a large current.
• A slowly drifting DC component appears as the pattern precesses.

Why it is usually tolerable anyway. Above about \( m_f = 21 \), these effects are small enough to ignore, and a fixed carrier is far easier to implement. Good drives switch to synchronous modulation only at high output frequencies, where \( m_f \) would otherwise become small — which is why a large drive's tone is sometimes heard to step as it accelerates.

From two independent halvings that multiply together.

Halving one: the step size.

• Bipolar drives both legs from one comparator, so leg B is always the inverse of leg A. The output jumps straight from \( +V_s \) to \( -V_s \) — a step of \( 2V_s \).
• Unipolar gives each leg its own comparator with references 180° apart, so the legs switch at different instants. The output passes through zero: steps of \( V_s \).

Since \( di/dt = v/L \), half the step means half the current slope.

Halving two: the interval. Two independent legs each switching at \( f_c \) means the output changes at \( 2f_c \). Half the time to build ripple.

Half the slope for half the time gives a quarter of the ripple.

\[ \Delta I \propto \frac{\Delta V}{L}\cdot\Delta t \;\Longrightarrow\; \frac{1}{2}\times\frac{1}{2} = \frac{1}{4} \]

And the devices are unaffected. Each still switches at \( f_c \), so switching loss is identical. This is a genuinely free improvement, which is why every solar inverter and UPS uses it.

The spectrum confirms it. The lowest harmonic group moves from around \( m_f \) to around \( 2m_f \) — the output behaves as though the carrier had been doubled, without doubling anyone's loss.

The cost, and it is real in one application. Because the legs switch independently, the common-mode voltage — the average of the two terminals — now moves at \( f_c \) instead of staying constant. In a transformerless grid-tied solar inverter, that moving common-mode voltage drives leakage current through the PV array's capacitance to earth, which safety standards limit strictly. The H5 and HERIC topologies exist precisely to keep unipolar's ripple benefit while holding the common-mode voltage still.

Efficiency, and probably a great deal of it. The carrier frequency is not a free quality knob.

What scales with \( f_c \):

• Switching loss: \( P_{sw} \approx \tfrac12 V_sI(t_r + t_f)f_c \) per device — linear in \( f_c \).
• Gate drive loss: \( Q_gV_gf_c \) — also linear.
• Diode recovery loss: \( Q_{rr}V_sf_c \) — often larger than the transistor's own switching loss in an IGBT bridge.

Put numbers on a 10 kW drive. At 4 kHz the switching loss might be 200 W. At 16 kHz it is 800 W — an extra 600 W to remove from the same heatsink, and efficiency falling from about 96% to 92%.

What was gained in exchange:

• Ripple current quartered (\( \Delta I \propto 1/f_c \)) — but it was already only a few percent at 4 kHz.
• Inaudible operation, since 16 kHz is at the edge of hearing.
• Slightly lower motor iron loss from the smaller ripple.

So 600 W was spent to remove a few percent of ripple and some acoustic noise. In a 10 kW drive that is a poor trade.

How real drives resolve it:

1. Automatic derating. Most industrial drives reduce their continuous current rating as the carrier is raised — a 15 kW drive at 4 kHz may only be rated 11 kW at 16 kHz. The setting is available, but it is not free, and the manual says so.
2. Load-dependent carrier. High when lightly loaded (quiet, and switching loss is small in absolute terms), low at full load (efficient, and mechanical noise masks the whine).
3. Better devices. SiC MOSFETs switch several times faster with no reverse recovery, so 16 kHz costs what 4 kHz costs in silicon. This is the main commercial argument for wide-bandgap devices in drives.

The sensible default for a general-purpose industrial drive is 2–8 kHz, choosing the low end for efficiency and the high end for acoustics — and reading the derating curve before changing it.

Because at high speed the harmonics cost little and the extra voltage is worth a great deal.

What is gained. Pushing from \( m_a = 1.155 \) to full six-step raises the line fundamental from \( 0.707V_s \) to \( 0.780V_s \) — 10% more. For a motor whose back-EMF rises with speed, that is 10% more speed before field weakening becomes necessary.

Why the cost is small at speed:

1. Inertia filters the torque ripple. The 5th and 7th produce a pulsation at \( 6f_1 \). At 50 Hz that is 300 Hz, far above any drivetrain resonance and easily absorbed by the rotor's inertia. At 5 Hz it would be 30 Hz — squarely in the problem band.
2. The load filters the current. Harmonic currents fall as \( 1/n^2 \), so a 20% 5th-harmonic voltage produces about 4% current — a 0.16% extra loss.
3. Acoustics are masked. At motorway speed, road and wind noise cover the 300 Hz tone completely.

Why the alternative is worse. Without over-modulation, reaching the same speed requires more field weakening — injecting negative d-axis current to reduce the back-EMF. That current produces copper loss, no torque, and risks demagnetising the magnets if a fault interrupts it. Over-modulation is the cheaper way to buy the same speed.

Where it must not be used:

• Low speed, where the ripple frequency falls into the mechanical resonance band and the machine runs visibly roughly.
• Precision servos, where torque ripple becomes position error.
• Grid-tied inverters, where the 5th and 7th would be injected into the network and would breach IEEE 519 immediately.

The control cost, which is the part usually underestimated. In the linear region \( V_1 \propto m_a \), so the current loop's gain is constant. In over-modulation the relationship saturates, and the gain falls towards zero as six-step is approached. A loop tuned for the linear region becomes sluggish and can wind up. Production drives pre-distort \( m_a \) through a lookup table to linearise it, and add anti-windup — which is real engineering effort, spent to buy 10% of speed.