{Outline}

Need for Flux Weakening

Voltage Constraint Analysis

{Voltage Constraint and the Need for Flux Weakening}

Inverter Voltage Limit

The stator voltage phasor magnitude is bounded by the inverter DC-link voltage:

\[ V_s \le V_{s,\max} = \frac{V_{dc}}{\sqrt{3}} \quad\text{(six-step limit)} \]

In the synchronous frame (neglecting \(\Rs\)):

\[ V_s \approx \omegas\sqrt{(\sigma\Ls\,\iqs)^2 + \!\left(\sigma\Ls\,\ids + \tfrac{\Lm}{\Lr}\lambdar\right)^2} \]

As \(\omegas\) rises above base speed \(\omega_b\), the back-EMF term \(\omegas(\Lm/\Lr)\lambdar\) pushes \(V_s\) beyond \(V_{s,\max}\) unless \(\lambdar\) is reduced.

Flux-Weakening Law

Reduce the rotor flux inversely with synchronous speed:

\[ \lambdar^* \propto \frac{1}{\omegas} \quad (\omegas > \omega_b) \]

This keeps \(V_s \approx V_{s,\max}\) while extending the speed range.

Constant-Power Characteristic

With \(\lambdar \propto 1/\omegas\) and \(\iT \approx\) const:

\[ \Te \propto \lambdar\cdot\iT \propto \frac{1}{\omegas} \]

\[ P = \Te\cdot\omegar \approx \text{const} \]

Power remains approximately constant — ideal for traction.

Operating Regions

{Operating Regions of the Vector-Controlled Drive}

Stator-Flux-Linkage-Controlled Schemes

Normalised Voltage Equations

{Normalised Stator Voltage Equations in Flux Weakening}

Stator Equations in Synchronous Frame (Normalised, Steady State)

Neglecting stator resistive drops and setting \(p = 0\):

\begin{align*} v_{qsn}^e &\cong \omega_{sn}\,\lambda_{dsn}^e \\[4pt] v_{dsn}^e &\cong -\omega_{sn}\,\lambda_{qsn}^e \end{align*}

Stator voltage phasor magnitude:

\[ v_{sn} = \omega_{sn}\sqrt{(\lambda_{qsn}^e)^2 + (\lambda_{dsn}^e)^2} = \omega_{sn}\,\lambda_{sn} \]

Stator Flux in Flux Weakening

\[ \lambda_{sn} = \frac{v_{sn}}{\omega_{sn}} \]

To maintain \(v_{sn}\) at its rated value as \(\omega_{sn}\) increases, the stator flux \(\lambda_{sn}\) must be reduced proportionally to \(1/\omega_{sn}\).

Key Result for Stator-Flux Control

With the stator flux aligned to the \(d\)-axis (\(\lambda_{qsn}^e = 0\)):

Torque: \(T_{en} = i_{qsn}^e\,\lambda_{sn}\)
Air-gap power: \(P_{an} = v_{sn}\,i_{qsn}^e\)
If \(i_{qsn}^e = \) const and \(v_{sn} = \) const, then \(P_{an} = \) const

Constant-power operation throughout flux weakening.

Air-Gap Power in the Flux-Weakening Region

{Constant Air-Gap Power in the Flux-Weakening Region}

Air-Gap Power Derivation

The normalised air-gap power:

\[ P_{an} = T_{en}\cdot\omega_{sn} = i_{qsn}^e\,\lambda_{sn}\,\omega_{sn} = i_{qsn}^e\,v_{sn}^e \]

With \(v_{sn}^e = v_{sn} = \) const (rated voltage maintained):

\[ P_{an} = v_{sn}\,i_{qsn}^e = \text{const} \quad \text{if } i_{qsn}^e = \text{const} \]

Maintaining \(i_{qsn}^e\) constant maintains constant power
Achieved by holding \(I_{s,\max}\) and adjusting flux such that \(i_{qsn}^e = \iT\) remains at its rated value

Stator-Flux Weakening Algorithm

Monitor \(|v_s|\) (measured or estimated from model)
When \(|v_s| \to V_{s,\max}\): initiate flux weakening
Set \(\lambdas^* = V_{s,\max}/\omega_s\)
Hold \(q\)-axis current at its limiter value
\(d\)-axis current adjusts automatically via flux controller
Speed range limited only by the \(\iT\) capability

Rotor-Flux-Linkage-Controlled Schemes

Rotor-Flux Weakening Strategy

{Rotor-Flux Weakening in Indirect Vector Control}

Voltage Constraint in the Rotor-Flux Frame

In the indirect vector-control scheme with rotor flux aligned to the \(d\)-axis:

\begin{align*} v\qs &\approx \omegas\!\left(\sigma\Ls\,\ids + \tfrac{\Lm}{\Lr}\lambdar\right)\\[4pt] v\ds &\approx -\omegas\,\sigma\Ls\,\iqs \end{align*}

As \(\omegas\) increases, \(v\qs\) grows primarily through the back-EMF term \(\omegas(\Lm/\Lr)\lambdar\).

Flux-Weakening Law

\[ \lambdar^* = \frac{V_{s,\max}}{\omegas}\cdot\frac{\Lr}{\Lm} \quad (\omegas > \omega_b) \]

Equivalently, the field-producing current command becomes:

\[ \ifld^* = \frac{\lambdar^*}{\Lm^*} = \frac{V_{s,\max}}{\omegas\,\Lm^*} \]

Implementation in Indirect VC

A voltage limiter detects when the VSI approaches saturation
Alternatively: an open-loop schedule \(\lambdar^* = f(1/\omegas)\) is pre-programmed
The slip-speed command during flux weakening becomes:
\[ \omegasl^* = \frac{\iT^*}{\Tr^*\,\ifld^*(t)} \]
where \(\ifld^*(t)\) is now time-varying
The lead compensator in \(\ifld^*\) ensures fast flux response during flux-change transients

Torque Capability and Optimisation

{Torque Capability and Optimal Flux in the Flux-Weakening Region}

Maximum Torque vs.\ Speed

With stator current limit \(I_{s,\max}\) and voltage limit \(V_{s,\max}\):

\[ T_{e,\max}(\omegas) = K_{te}\,\lambdar(\omegas)\,I_{T,\max} \]

where:

\[ \lambdar(\omegas) = \frac{V_{s,\max}}{\omegas}\cdot\frac{\Lr}{\Lm}, \quad I_{T,\max} = \sqrt{I_{s,\max}^2 - \ifld^{*2}} \]

As \(\omegas \to \infty\): \(\ifld^* \to 0\), \(I_{T,\max} \to I_{s,\max}\), but torque \(\to 0\) because \(\lambdar \to 0\) faster.

Optimal Flux for Maximum Torque

The flux value maximising torque at a given speed is found from:

\[ \frac{\partial \Te}{\partial \lambdar}\bigg|_{I_s = I_{s,\max}} = 0 \]

Solution:

\[ \lambdar^{\mathrm{opt}} = \frac{\Lm\,I_{s,\max}}{\sqrt{2}} \]

This corresponds to:

\[ \ifld = \iT = \frac{I_{s,\max}}{\sqrt{2}} \]

i.e., a \(45°\) torque angle — the maximum torque per ampere condition.

{Practical Flux-Weakening Controller Block Diagram}

Flux-weakening is triggered when measured \(|v_s| \to V_{s,\max}\)
The lead compensator \((1 + s\Tr^*)\) in \(\ifld^*\) ensures fast flux response during the constant-to-weakening transition

Summary

{Summary — Lecture 7}

Key Concepts

Flux weakening extends the speed range above \(\omega_b\) by reducing \(\lambdar\) in proportion to \(1/\omegas\), keeping \(V_s \le V_{s,\max}\)
Approximately constant output power is maintained if the torque-producing current is held constant
Stator-flux scheme (direct VC): flux reduced to \(\lambdas^* = V_{s,\max}/\omegas\); elegant constant-power law
Rotor-flux scheme (indirect VC): flux reduced via updated \(\ifld^*\) schedule; more sensitive to \(\Rr\) mismatch
Optimal flux for maximum torque: \(\ifld = \iT = I_{s,\max}/\sqrt{2}\)

Applications Demanding Flux Weakening

Electric vehicle traction drives (2–4\(\times\) base speed)
Machine tool spindle drives (up to 10\(\times\) base speed)
Centrifuges and high-speed blowers
Wind turbine generators (variable wind speed)

Important Reminder

Flux weakening in indirect VC increases the commanded slip speed, amplifying sensitivity to \(\Rr\) mismatch. Robust parameter compensation (Lecture~6) is therefore even more critical in the flux-weakening region.