{Outline}

Derivation of the Indirect Vector-Control Scheme

Rotor Equations and Orientation Condition

{Starting Point — Rotor Voltage Equations}

Rotor Equations in the Synchronous Reference Frame

Assuming a current-source inverter (stator dynamics neglected):

\begin{align*} R_r i_{qr}^e + \frac{d\lambda_{qr}^e}{dt} + \omegasl\lambda_{dr}^e &= 0 \\[4pt] R_r i_{dr}^e + \frac{d\lambda_{dr}^e}{dt} - \omegasl\lambda_{qr}^e &= 0 \end{align*}

where \(\omegasl = \omegas - \omegar\) is the slip speed, and the rotor flux linkages are:

\begin{align*} \lambda_{qr}^e &= \Lm\,\iqs + \Lr\,i_{qr}^e \\[4pt] \lambda_{dr}^e &= \Lm\,\ids + \Lr\,i_{dr}^e \end{align*}

Field-Orientation Condition

Align the \(d\)-axis of the synchronous frame with the rotor flux phasor:

\[ \lambdar = \lambda_{dr}^e, \qquad \lambda_{qr}^e = 0, \qquad \frac{d\lambda_{qr}^e}{dt} = 0 \]

This reduces the coupled 2-D rotor equations to two independent first-order systems.

Decoupled Equations after Orientation

{Decoupled Rotor Equations after Field Orientation} Substituting the orientation conditions into the rotor equations:

Slip-Speed Equation — from \(q\)-axis

\[ \omegasl = \frac{\Rr\,\iqs}{\lambdar} = \frac{1}{\Tr}\cdot\frac{\iqs}{\ids} \]

where the rotor time constant \(\Tr = \Lr/\Rr\). This is the key equation: the commanded slip speed is proportional to the ratio of torque-producing to field-producing stator current.

Rotor Flux Dynamics — from \(d\)-axis

Using \(i_{dr}^e = (\lambdar - \Lm\ids)/\Lr\):

\[ \Tr\,\frac{d\lambdar}{dt} + \lambdar = \Lm\,\ids \]

Rotor flux \(\lambdar\) is a first-order lag response to the field-producing current \(\ids = \ifld\)
Time constant \(= \Tr\) (typically 0.1–1\,s)
This is the slow control channel

Significance

The two equations are completely decoupled when \(\lambda_{qr}^e = 0\) — torque can be changed instantaneously without disturbing the rotor flux.

Torque Equation and Transfer Functions

{Electromagnetic Torque and Control Transfer Functions}

Electromagnetic Torque Expression

\[ \Te = \frac{3P}{4}\cdot\frac{\Lm}{\Lr}\,\lambdar\,\iqs = K_{te}\,\lambdar\,\iT \]

where \(K_{te} = \tfrac{3P\Lm}{4\Lr}\). Key result: torque is the product of two independently controllable quantities:

\(\lambdar\) — controlled via \(\ids\) (slow)
\(\iT = \iqs\) — torque-producing current (fast)

Transfer Functions of the Two Channels

Flux channel (slow):

\[ \frac{\lambdar(s)}{\ifld(s)} = \frac{\Lm}{1 + s\Tr} \]

Torque channel (fast, at constant \(\lambdar\)):

\[ \frac{\Te(s)}{\iT(s)} = K_{te}\,\lambdar \approx \text{const} \]

Torque responds instantaneously to \(\iT\)
Flux responds with time constant \(\Tr\)
No coupling between channels when \(\lambda_{qr}^e = 0\)

Coordinate Transformation and Field Angle

{Coordinate Transformation for Indirect Control}

Three-Phase to \(dq\) Transformation at Field Angle \(\thetaf\)

\[ \begin{bmatrix} \iqs \\ \ids \end{bmatrix} = \frac{2}{3} \begin{bmatrix} \cos\thetaf & \cos\!\left(\thetaf-\tfrac{2\pi}{3}\right) & \cos\!\left(\thetaf+\tfrac{2\pi}{3}\right)\\[4pt] \sin\thetaf & \sin\!\left(\thetaf-\tfrac{2\pi}{3}\right) & \sin\!\left(\thetaf+\tfrac{2\pi}{3}\right) \end{bmatrix} \begin{bmatrix}i_{as}\\i_{bs}\\i_{cs}\end{bmatrix} \]

Field Angle Computation (Indirect)

\[ \thetaf = \theta_r + \theta_{sl}, \qquad \theta_{sl} = \int\!\omegasl\,\mathrm{d}t = \int\!\frac{\iqs}{\Tr\,\ids}\,\mathrm{d}t \]

\(\theta_r\) measured by shaft encoder or resolver
\(\theta_{sl}\) computed from motor parameters and commanded currents

No Flux Sensor Required

Field angle is estimated, not measured
Only rotor position \(\theta_r\) is needed
Accuracy depends on \(\Tr = \Lr/\Rr\)
Parameter mismatch causes imperfect decoupling (analysed in detail in Lecture~5)

The Indirect Vector-Control Scheme

Current Commands

{Current Commands from the Vector Controller}

Commanded Quantities (Derived from Rotor Equations)

Field-producing current command:

\[ \ifld^* = \frac{(1 + s\Tr^*)}{\Lm^*}\,\lambdar^* \]

The lead term \((1+s\Tr^*)\) compensates the rotor-flux first-order lag — enabling fast flux step-changes. Torque-producing current command:

\[ \iT^* = \frac{T_e^*}{K_{te}^*\,\lambdar^*} \]

Commanded slip speed:

\[ \omegasl^* = \frac{1}{\Tr^*}\cdot\frac{\iT^*}{\ifld^*} = \frac{\Lm^*\,\iT^*}{\Tr^*\,\lambdar^*} \]

Three-Phase Current Synthesis

Stator current phasor magnitude:

\[ i_s^* = \sqrt{(\ifld^*)^2+(\iT^*)^2} \]

Inverse \(dq \to abc\) transform:

\[ \begin{bmatrix}i_{as}^*\\i_{bs}^*\\i_{cs}^*\end{bmatrix} = i_s^*\begin{bmatrix} \sin\theta_s^*\\\sin(\theta_s^*-\tfrac{2\pi}{3})\\\sin(\theta_s^*+\tfrac{2\pi}{3}) \end{bmatrix} \]

where \(\theta_s^* = \thetaf^* + \theta_T^*\) and \(\theta_T^* = \tan^{-1}\!\left(\iT^*/\ifld^*\right)\).

{Block Diagram — Indirect Vector Controller}

Implementation Details

Flux Command and Field Weakening

{Complete Implementation — Flux Command and Speed Controller}

Flux Reference Schedule

\[ \lambdar^* = \begin{cases} \lambda_b & |\omegar| \le \omega_b \\[4pt] \dfrac{\omega_b}{|\omegar|}\lambda_b & |\omegar| > \omega_b \end{cases} \]

Below base speed: constant-flux (constant-torque) region
Above base speed: flux reduced \(\propto 1/\omegar\) (constant-power region)
Extends the speed range without exceeding inverter voltage or current ratings

Torque Command from Speed Controller

Speed error \(\to\) PI controller \(\to\) \(T_e^*\)
Torque limiter clamps \(T_e^*\) to safe levels
Anti-windup in the PI prevents integrator saturation

Signal Flow Summary

\(\lambdar^*\) and \(T_e^*\) from outer loop
Vector controller computes \(\ifld^*\), \(\iT^*\), \(\omegasl^*\)
Phasor resolver: \(\theta_s^*\), \(i_s^*\)
Inverse transform: \(i_{abc}^*\)
CSI or VSI enforces \(i_{abc}^*\)

Unity Transfer Functions

When motor and controller parameters match exactly:

\[ \frac{\lambdar(s)}{\lambdar^*(s)} = 1, \quad \frac{\Te(s)}{T_e^*(s)} = 1 \]

Perfect decoupling is confirmed.

CSI versus VSI

{Current-Source vs.\ Voltage-Source Indirect Vector Controller}

Current-Source Inverter (CSI)

Natural current tracking — inherently high bandwidth
Stator dynamics negligible (source impedance \(\gg\) motor)
Three-phase current commands enforced directly
Simple hysteresis or predictive current control
Well-suited to low-power servo drives

Voltage-Source Inverter (VSI)

Stator voltage equations must be explicitly included
\(dq\) current controllers produce voltage commands
Cross-coupling terms require feed-forward cancellation:
\[ v_{qs}^{e*} = v_{PI,qs} + \underbrace{\omegas\sigma\Ls\ids + \omegas\frac{\Lm}{\Lr}\lambdar}_{\text{feed-forward terms}} \]
PWM synthesises the voltage commands
Dominant topology in industrial drives (IGBT-based)

Time Constants of the Two Channels

{Time Constants of the Two Control Channels}

Flux Channel — Slow Response

\[ \frac{\lambdar(s)}{\ifld(s)} = \frac{\Lm}{1 + s\Tr} \]

Time constant: \(\Tr = \Lr/\Rr \approx 0.1\)–\(1\)\,s
Analogous to the field time constant of a DC machine
Limits the rate of flux step changes
The lead compensator in \(\ifld^*\) partially overcomes this limitation:
\[ \ifld^*(s) = \frac{(1+s\Tr^*)}{\Lm^*}\,\lambdar^*(s) \]

Torque Channel — Fast Response

\[ \frac{\Te(s)}{\iT(s)}\bigg|_{\lambdar=\mathrm{const}} = K_{te}\,\lambdar \approx \text{const} \]

Torque time constant \(\approx \sigma\Ls/R_s^{\mathrm{eff}}\) (a few milliseconds)
Practically instantaneous response
Full rated torque achievable at standstill
This is the primary advantage over scalar control

Bandwidth Hierarchy

Torque bandwidth \(\gg\) Speed bandwidth \(\gg\) Flux bandwidth. This separation of time scales is by design and is essential for stable cascade control.

Summary

{Summary — Lecture 3}

Key Equations

ll@{}} Variable	Expression
Flux command	\(\lambdar^* = \Lm^\ifld^ / (1+s\Tr^*)\)
Torque command	\(T_e^* = K_{te}^\lambdar^\iT^*\)
Slip speed	\(\omegasl^* = \iT^* / (\Tr^\ifld^)\)
Field angle	\(\thetaf^* = \theta_r + \int\omegasl^*\,dt\)
Stator angle	\(\theta_s^* = \thetaf^* + \theta_T^*\)

Critical Parameter Dependency

Accuracy of indirect vector control depends entirely on:

\[ \Tr^* = \frac{\Lr^*}{\Rr^*} \approx \Tr = \frac{\Lr}{\Rr} \]

Mismatch in \(\Rr\) (which varies with temperature) degrades decoupling — the subject of Lectures 5 and 6.

Industrial Significance

Indirect vector control dominates industrial AC servo drives owing to its robustness, simplicity, and the elimination of flux sensor hardware.