Vector Control and Field Orientation

The rotor time constant:

A sixth of a second. The rotor cage is a short-circuited winding of substantial inductance and low resistance, so any change in its flux linkage decays slowly — and there is no terminal at which to force it faster.

What that means for a torque command. In a V/f drive, torque is commanded by changing the frequency, which changes the slip:

Half a second to establish a new torque. For a pump that is irrelevant — the hydraulic system is slower still. For a servo, a spindle, a traction drive or anything that must reject a load disturbance, it is useless.

Compare the timescales in play:

Nearly two orders of magnitude separate the current from the flux. That separation is the opportunity: if the flux could be held constant while only the current changes, torque would respond at the current's timescale rather than the flux's.

Which is precisely what vector control does. Resolve the stator current into two components relative to the rotor flux:

Hold \(i_d\) constant and the flux never changes, so its slow dynamics are simply never excited. Torque is then commanded through \(i_q\) alone, which responds in milliseconds.

The resulting improvement:

A hundredfold improvement, and no change to the machine or the inverter — the hardware of Set 26, Problem 6 serves either. The entire difference is in what the controller computes.

The one thing that stays slow. Establishing the flux from zero still takes \(3\tau_r\):

So a vector drive cannot produce torque instantly from a cold start — it must first build the rotor flux, which takes half a second. Drives handle this by pre-magnetising: applying \(i_d\) with \(i_q = 0\) before the brake releases. It is also why field weakening is a slow manoeuvre, and why a drive commanded rapidly through base speed can briefly lose orientation.

The torque equation in rotor-flux coordinates:

Compare Set 33's DC machine: \(T = K_a\Phi I_a\) with \(K_a\Phi = 1.321\). The form is identical — a constant times a current — and that identity is the entire point of the transformation.

The torque-producing current:

The flux-producing current, from the steady-state relation:

Note that this is the magnetising current, and in a V/f drive it is drawn automatically as reactive current. Here it is commanded explicitly, which means it can also be reduced — the basis of field weakening and of efficiency optimisation at light load.

The stator current is their vector sum, because they are perpendicular by construction:

Within 2% of the nameplate current computed in Set 34 by an entirely different route — power, voltage, power factor and efficiency. The dq model and the equivalent circuit are describing the same machine.

What the two components mean physically:

The second row is why the technique works at all. An induction machine has no field winding, so \(i_d\) is a synthetic field current — carried by the same three wires as \(i_q\) and separable only because the controller knows the flux angle.

The transformations that make it possible:

Both are pure algebra; the difficulty is entirely in \(\theta_e\), the rotor flux angle. It cannot be measured — there is no terminal at the rotor flux — so it must be computed, which is Problem 3 and the whole practical content of vector control.

The problem. The Park transformation needs \(\theta_e\), the angle of the rotor flux — which is not the rotor's mechanical angle, and cannot be measured:

The rotor position \(\theta_r\) is available from an encoder. The slip angle is not — but it can be computed from the currents the controller is itself commanding. That is indirect field orientation.

The slip relation. Requiring the rotor flux to lie entirely on the d-axis forces a specific slip:

Against Set 34's rated slip frequency of 2.0 Hz, computed from the equivalent circuit — agreement to within the difference between the two models' assumptions. The machine's rated slip frequency and its field-orientation slip are the same physical quantity.

The complete IFOC scheme:

Note that step 6 hands a voltage vector straight to the space vector modulator — no conversion to three-phase references and back. That fit is why SVM and vector control are always paired.

Now the weakness. The slip computation depends on \(R_r\), which is a rotor temperature measurement in disguise:

A 50% error in the commanded slip frequency — and the error is in the integrated angle, so it does not merely scale the torque. It rotates the entire dq frame away from the actual flux.

What detuning does:

The third row is the subtle one. Field orientation's whole benefit was that \(i_q\) does not affect the flux; once the frame is misaligned that guarantee fails, and a torque step drags the flux with it — reintroducing the 169 ms dynamics the scheme was built to avoid.

The remedies:

Direct field orientation observes the flux from the stator voltage and current rather than computing the slip — which removes the \(R_r\) dependence but introduces an integration that drifts at low speed, and a dependence on \(R_s\) instead. Most drives use indirect orientation with adaptation, and Problem 5 explains why.

The premise. The stator flux moves in the direction of the applied voltage vector:

So applying one of the six active vectors of Set 28 pushes the stator flux directly, and the torque — which depends on the angle between stator and rotor flux — changes immediately. No current loop, no transformation to a rotating frame, no modulator.

The control law is a lookup table:

Two hysteresis comparators and a 36-entry table — that is the entire controller. There is no PI regulator anywhere in the torque path, and no PWM: the selected vector is applied directly until a comparator changes state.

Why it is faster. Count what stands between a torque command and a change in the inverter's output:

The two rows in bold are the difference. A current loop necessarily takes several switching periods to change a current; DTC changes the applied voltage on the very next sample, which is the fastest anything can happen in a switching converter.

What it costs:

The first row propagates into the next two and is DTC's main practical objection: an EMI filter designed against a fixed carrier, as Set 32 did, has nothing fixed to work against. The fifth row is shared with sensorless FOC and is the subject of Problem 5.

The three schemes side by side:

Modern drives blur these lines: space-vector-modulated DTC adds a modulator back to fix the variable frequency, and predictive control generalises the vector selection over a longer horizon. The three columns are landmarks rather than a complete taxonomy.

The voltage model. The stator flux can be found by integrating the terminal quantities:

Elegant, and it needs only one parameter. From the stator flux and the currents, the rotor flux angle follows — and hence the rotor position, without any sensor. It works well at speed and fails completely at low speed, for a reason that can be quantified exactly.

Why it fails. The integrand is a difference of two terms, and their relative size depends entirely on frequency:

The resistive drop is larger than the signal being estimated. Subtracting a 13.6 V correction from a 13.6 V measurement to find an 8.3 V quantity leaves nothing but the error in \(R_s\).

The crossover frequency:

Below about 1.6 Hz the resistive drop dominates and estimation degrades rapidly; practical drives quote a minimum of 1–3 Hz for sensorless operation, which agrees. Note that this frequency is a property of the machine — specifically of its per-unit stator resistance — not of the algorithm.

Three compounding problems at low speed:

The third row connects directly to Set 25, Problem 5: the drive does not actually apply the voltage it commanded, and the discrepancy is worst at exactly the low modulation index where the estimator is already struggling. Sensorless drives therefore need dead-time compensation before they need a better observer.

The technique that works to zero speed:

If the machine has any magnetic saliency — a difference between \(L_d\) and \(L_q\) — the injected carrier produces a current response whose magnitude varies with rotor position. Demodulating that response gives position without depending on back EMF at all, so it works perfectly at standstill.

Its own limitations:

The first row explains where the technique is actually used. An interior permanent magnet machine, with its buried magnets, has strong saliency and responds beautifully; a cage induction motor has only the small saturation-induced saliency, so injection works but marginally. That is one reason Set 36's IPMSM has displaced the induction machine in traction.

When to fit an encoder:

The third row is the elevator of Set 33 again, now with an induction machine. A hoist must produce full torque at exactly zero speed to hold a load while the brake releases — the one condition under which every sensorless method fails — so an encoder is not a refinement but a safety requirement.

The per-period sequence, at a 5 kHz sampling rate synchronised to the PWM centre — the sampling instant Set 28 established:

The bandwidths, each a decade below the one inside it — the cascade principle of Set 33:

The flux loop is deliberately slow because the plant is slow — there is no point demanding a faster response than \(1/\tau_r\) can deliver, and trying only saturates the voltage.

The decoupling terms. The d and q axes are not actually independent in the voltage equations — rotation couples them:

The \(\omega_e\) terms are speed-dependent disturbances that each axis injects into the other, and they grow with speed. Adding their computed values to the PI outputs — feedforward decoupling — leaves each current loop seeing a clean first-order plant, which is what the pole-cancellation design of Set 33 assumed.

Without decoupling, the picture at rated speed:

A 136 V disturbance in the q-axis voltage equation — more than half the phase voltage. The PI would eventually reject it, but only within its bandwidth, so a speed transient produces a large transient current error. Feedforward removes it before the loop ever sees it.

The limits that must be enforced:

The first row's priority matters. When the current limit binds, reducing \(i_q\) costs torque but reducing \(i_d\) costs flux — and losing flux takes 169 ms to recover while losing torque is instantly reversible. So \(i_d\) is protected and \(i_q\) absorbs the limit.

What the whole structure achieves:

Every entry in the last column matches or beats the DC drive of Set 33 — in a machine with no brushes, no commutator and no maintenance. Twenty microseconds of arithmetic per period is what replaced a mechanical commutator, and it is why the DC drive has essentially disappeared.