Space Vector Modulation

The eight states. Each leg is either high or low, giving \(2^3 = 8\) combinations. Six produce a non-zero output; two — all-high and all-low — short the load and produce nothing:

The six active vectors are 60° apart and equal in magnitude, so their tips form a regular hexagon. Two distinct states at the origin is the redundancy that makes SVM flexible — Problem 4.

The magnitude, from the space-vector transform of the three phase voltages:

Each vertex represents applying the full link voltage across one phase and half of it across the other two — the six-step waveform of Set 26, seen instant by instant rather than as a waveform.

The inscribed circle. A rotating reference must be reproducible at every angle, so it must stay inside the hexagon — and the binding constraint is the midpoint of each side, at 30° from a vertex:

The linear limit of space vector modulation, and it is a purely geometric statement: the largest circle inscribed in a hexagon of circumradius \(R\) has radius \(R\cos30^\circ\). No Fourier analysis was involved.

Translate to voltages. The space vector's magnitude is the peak phase voltage:

Which is exactly the injection result:

Two completely different arguments — one trigonometric, one geometric — giving the identical number. They are the same statement: a sinusoidal reference constrained by its own peak is a circle constrained by the hexagon's vertex distance, and relaxing that to the inscribed circle is what both methods do.

Where SPWM sits on the diagram. Its limit corresponds to a smaller circle:

Sinusoidal PWM restricts the reference to a circle of radius \(V_{dc}/2\), which is \(\cos30^\circ\) of the inscribed circle — and \(1/\cos30^\circ = 1.1547\) is precisely the injection boost of Set 26. The whole 15.5% is the difference between two circles.

The principle: volt-second equivalence. The inverter cannot produce the reference vector, but it can produce the two adjacent vertices and the origin. Spending the right fractions of each switching period on each gives the right average:

Exactly the volt-second balance of Set 1, applied to a vector instead of a scalar. The load's inductance does the averaging, as it always has.

The modulation index, normalised so that \(m = 1\) at the inscribed circle:

Comfortably inside the linear range. Note that this \(m\) is the SVM convention, not the carrier-based \(m_a\) — the two differ by \(2/\sqrt3\), which is a common source of confusion when comparing sources.

Resolve the reference onto the two vertices. Applying the sine rule to the triangle formed by \(\vec V_{ref}\) and the two adjacent vectors:

Equal, as symmetry demands — the reference sits exactly halfway between \(\vec V_1\) at 0° and \(\vec V_2\) at 60°. That symmetry is a useful check on any dwell-time calculation.

The zero time is whatever is left:

Thirteen per cent of every period spent producing nothing. As the reference grows the zero time shrinks, reaching zero exactly when the reference touches the hexagon boundary — which at \(\theta = 30^\circ\) is the inscribed circle at \(m = 1\).

Verify by reconstructing the average. Resolving both vectors onto the reference direction:

And the perpendicular components cancel, since the two dwell times are equal and the vectors are symmetric about the reference. The synthesis is exact in the average.

How the dwell times behave across a sector:

The active time is largest in the middle of a sector and smallest at its edges — the opposite of what intuition suggests, and it follows directly from the hexagon: the boundary is nearest at the side midpoint, so a given reference magnitude uses proportionally more of the available time there.

The dwell times say nothing about order. \(T_1\), \(T_2\) and \(T_0\) could be applied in any sequence and the average would be the same — but the number of switching transitions would not:

A good sequence changes only one leg at each transition. That halves the switching count compared with a careless ordering, and it is entirely free — a matter of choosing an order.

The seven-segment sequence for sector 1, symmetric about the centre of the period:

Note that the zero time is split: a quarter at each end in state 000, and a half in the middle in state 111. Using both zero states is what makes single-leg transitions possible throughout.

Count the transitions. Each boundary changes exactly one leg:

Each device switches on once and off once per period — the minimum possible for a scheme that uses both zero states. A sequence that jumped from 100 to 111 directly would need two legs to change at once, doubling the count for no benefit.

Why symmetry matters. Placing the segments symmetrically about the period centre makes each leg's waveform a centred pulse:

The last row is a practical gift. With centred pulses, sampling the current at the middle of the period catches it exactly halfway through its ripple — giving the average value without any filtering. Every digital drive uses this.

The five-segment alternative. Dropping one zero state entirely:

Which is exactly the discontinuous modulation of Set 27, Problem 5 — and here the mechanism is transparent. Choosing to spend all the zero time in 000 rather than splitting it means leg C stays low throughout the period. The zero-sequence freedom of the carrier view is, in the vector view, simply the choice of how to divide \(T_0\).

The two schemes side by side:

Start from what SPWM does in vector terms. Three independent sinusoidal references, each compared against a carrier, produce a rotating reference vector — but the constraint is applied to each phase separately:

A circle of radius 200 V — smaller than the inscribed circle's 230.9. SPWM is leaving the region between the two circles unused, and that region is entirely reachable.

What SVM does differently. It does not constrain individual phases at all; it constrains only the vector:

The full 15.5%, and the ratio is \(1/\cos30^\circ\) because that is the ratio between the inscribed circle and the circle of radius \(V_{dc}/2\). Nothing about harmonics enters.

Where the freedom comes from. The zero vector has two realisations, and the split between them is arbitrary:

Because both states produce zero differential output. But they produce very different common-mode voltages — \(-V_{dc}/2\) and \(+V_{dc}/2\) from Set 26, Problem 5. So choosing the split is choosing a common-mode voltage, which is choosing a zero-sequence component.

The equal split of the seven-segment sequence corresponds to a specific zero-sequence waveform. Working out the resulting leg reference:

A triangular wave at three times the fundamental frequency, of amplitude \(V_{dc}/4\) — whose Fourier series is dominated by a third harmonic of amplitude very close to one-sixth of the fundamental, plus small ninth and fifteenth terms.

The comparison:

All three of the lower rows reach the same 0.707, because all three place the reference on the inscribed circle. They differ only in which zero-sequence waveform they use to get there — and therefore in switching loss, ripple and common-mode behaviour, not in output capability.

Why SVM is nonetheless preferred over explicit injection:

The fourth row is decisive in a modern drive. Field-oriented control computes a voltage demand as a vector in the same reference frame; handing that directly to a space vector modulator avoids converting to three phase references and back.

What goes wrong beyond the circle. Between the inscribed circle and the hexagon, a reference is reachable at some angles and not others:

The dwell-time equations demand more time than the period contains, which is geometrically the statement that the reference has left the hexagon. Something has to give, and how it gives defines the overmodulation regions.

Region I — the reference is projected onto the boundary. Whenever \(T_1+T_2 > T_s\), both are scaled down:

The angle is preserved, the magnitude is clipped to the hexagon edge. The reference trajectory becomes a circle with the corners flattened — and near the sector centres, where the hexagon is nearest, it follows the straight edge instead of the circle.

Region II — the angle is distorted too. As the demand rises further, clipping the magnitude alone is insufficient:

The trajectory stops being continuous: it dwells at \(\vec V_1\), then jumps along the edge, then dwells at \(\vec V_2\). The dwell fraction grows with the demand until it fills the whole sector.

The six-step limit. When the dwell fills each sector completely, the inverter applies exactly one state per 60°:

Which matches Set 27, Problem 4 exactly — and note that the fundamental at six-step (254.6 V peak) exceeds the hexagon's vertex-to-centre distance would suggest, because averaging over a sector recovers more than the geometry of a single instant.

The complete range:

Read the fourth column. Overmodulation is exactly the region where the zero time has been exhausted — the modulator has no idle time left to give up, so it must start distorting instead. That is a cleaner statement of the limit than any harmonic argument.

What the controller must do about it. Two problems appear:

The gain from command to output collapses, so a current regulator tuned for the linear range becomes ineffective. And (ii) the produced vector no longer equals the commanded one in either magnitude or angle, so a field-oriented controller loses the orthogonality it depends on. Practical drives apply an explicit inverse map from desired to commanded \(m\), and freeze or detune the current loop in deep overmodulation.

The per-period computation, executed in an interrupt at the period centre:

Steps 6 and 7 are the ones omitted from textbook descriptions and required in every real implementation. Neither is optional.

The compare values. With an up-down counter of period \(T_s/2\), each leg's compare register sets when its output changes. For sector 1 with the durations of Problem 3:

Three numbers, loaded once per period, and the timer hardware generates the complete seven-segment sequence including its symmetry. That mapping — from a vector to three compare values — is the entire runtime cost of SVM.

Dead time is inserted by the hardware, not the algorithm:

Modern PWM peripherals insert it automatically on every complementary pair, delaying each turn-on by \(t_d\). The algorithm computes ideal times and the hardware perturbs them — which is exactly the distortion of Set 25, Problem 5, and must be compensated in the reference, not in the timer.

The minimum pulse width constraint. A pulse shorter than the dead time cannot be produced at all:

Near the sector boundaries one dwell time approaches zero — at \(\theta = 0\), \(T_2 = 0\) exactly. And near \(m = 1\) the zero time approaches zero. Both situations demand pulses the hardware cannot make.

The three ways of handling it:

Dropping is standard and its error is second-order: near a sector boundary the dropped vector was contributing almost nothing anyway. The third method is used in high-performance drives, where the accumulated error over many periods would otherwise produce a small but persistent current distortion.

The full implementation budget:

The sixth row is the reward for the symmetric sequence of Problem 3: at the centre of the period all three legs are in the zero state and the current is exactly at its average, so a single sample gives the fundamental with no filtering and no phase lag. That is why the interrupt is placed there.