{Outline}

Plant Model for Speed Control

Stator Voltage Equations for the VSI Drive

{Stator Voltage Equations for the VSI Drive}

Expressing Rotor Currents in Terms of Stator Quantities

From the field-orientation conditions, the rotor \(dq\) currents are:

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

Substituting into the Stator Equations

\begin{align*} v\qs &= (R_s + \sigma L_s\, p)\,\iqs + \sigma L_s\,\omegas\,\ids + \frac{\Lm}{\Lr}\,\omegas\,\lambdar \\[4pt] v\ds &= (R_s + \sigma L_s\, p)\,\ids - \sigma L_s\,\omegas\,\iqs + \frac{\Lm}{\Lr}\,p\lambdar \end{align*}

where the leakage coefficient \(\sigma = 1 - \Lm^2/(\Ls\Lr)\).

In steady state: \(\ids = \ifld = \) const, \(p\ids = 0\)
The torque-producing current \(\iT = \iqs\) is the control variable
The \(q\)-axis equation defines the plant for torque-current control

Simplified Plant and DC Motor Analogy

{Simplified \(q\)-Axis Plant Model}

\(q\)-Axis Voltage Equation at Constant Flux

With \(\ifld\) and \(\lambdar\) constant in steady state:

\[ v\qs = (R_s + L_a\, p)\,\iT + \omegas\,L_s\,\ifld \]

where the effective transient inductance:

\[ L_a = \sigma L_s = L_s - \frac{\Lm^2}{\Lr} \]

The last term \(\omegas L_s \ifld\) is a speed-dependent back-EMF disturbance — cancelled by feed-forward.

Transfer Function of Torque Channel

After back-EMF feed-forward cancellation:

\[ \frac{\iT(s)}{v\qs'(s)} = \frac{1/R_s}{1 + sT_a}, \qquad T_a = \frac{L_a}{R_s} \]

A first-order plant — mathematically identical to a DC motor armature circuit.

Analogy to DC Motor

ll@{}} IM vector control	DC motor
\(\iT\)	\(i_a\) (armature)
\(\ifld\)	\(i_f\) (field)
\(L_a = \sigma\Ls\)	\(L_a\)
\(R_s\)	\(R_a\)

Current Loop Modelling

Current Controller Design

{Current Controller Design — Pole-Zero Cancellation}

Inner Current Loop Structure

Lecture Figure — Block diagram of the inner PI current control loop

PI Current Controller

\[ G_c(s) = K_s\,\frac{1 + sT_s}{sT_s} \]

PI zero placed at \(s = -1/T_s\)
Set \(T_s = T_a = L_a/R_s\) to cancel the plant pole
Pole-zero cancellation leaves a pure integrator in the open loop, ensuring zero steady-state error

Closed Current-Loop Transfer Function

After pole cancellation (\(T_s = T_a\)):

\[ G_i(s) = \frac{K_i}{1 + sT_i} \]

where:

\[ K_i = \frac{K_s\,K_{in}\,T_a}{T_{ar}}, \quad T_i = T_1 \]

and \(T_{ar} = T_a + T_{in}\) combines the plant and measurement time constants.

Model Reduction

The higher-order current loop is reduced to a first-order approximation — essential for systematic speed-controller synthesis.

Complete Current Loop Transfer Function

{Complete Current Loop — Exact and Simplified Forms}

Open-Loop Transfer Function (Exact)

\[ G_i(s) = \frac{K_s\,K_{in}\,T_s\,(1+sT_m)}{T_{ar}\,T_m\,(1+sT_1)(1+sT_2)} \]

where \(T_1\) and \(T_2\) are the closed-loop time constants. Simplified near crossover (\(T_1 < T_2 < T_m\)):

\[ G_i(s) \approx \frac{K_i}{1 + sT_i}, \quad K_i = \frac{K_s\,K_{in}\,T_1}{T_{ar}} \]

Validation of Approximation

Bode plot comparison between exact and simplified current loop:

Gain discrepancy at high frequency (acceptable)
Phase responses almost identical in the frequency range of interest
Both models cross 0\,dB at the same \(\omega_c\)
Simplified model sufficient for speed-loop design

Speed Loop Design - Symmetric Optimum

Speed Loop Architecture

{Speed Loop Architecture and Mechanical Subsystem}

Speed Loop Structure

Mechanical Subsystem Transfer Function

\[ G_m(s) = \frac{K_m}{s(1 + sT_m)} = \frac{P/(2J)}{s\!\left(1+s\tfrac{B}{J}\right)} \]

\[ T_m = J/B \quad\text{(mechanical time constant)} \]

Speed-Loop Open-Loop Function

For the simplified model (first-order current loop):

\[ G_f(s) = \frac{K_m\,K_s\,(1+sT_s)}{s\,T_s\cdot s\,T_m\cdot \tfrac{1}{K_i}(1+sT_i)(1+sT_\omega)} \]

Design goal: choose \(K_s\) and \(T_s\) for desired bandwidth and phase margin while ensuring adequate disturbance rejection.

Symmetric Optimum Design

{Symmetric Optimum — Concept and Design Procedure}

What is the Symmetric Optimum?

The symmetric optimum is a frequency-domain design criterion that maximises the phase margin at the gain crossover frequency for a given closed-loop bandwidth, while ensuring zero steady-state error to step disturbances.

Design Formulas

For a plant \(G_p(s) = K_p/[s(1+s\tau)]\), the symmetric optimum yields PI parameters:

\[ T_s = a^2\tau, \quad K_s = \frac{1}{2a\,K_p\,\tau} \quad (a \ge 2) \]

Crossover frequency: \(\omega_c = 1/(a\tau)\)
Phase margin with \(a=2\): \(\varphi_m \approx 36°\)
Phase margin with \(a=3\): \(\varphi_m \approx 53°\) (recommended choice)
Parameter \(a\) trades off bandwidth against stability margin

Why ``Symmetric''?

The Bode phase plot is symmetric about the crossover frequency \(\omega_c\):

The PI zero contributes equal phase lead below \(\omega_c\)
The plant pole contributes equal phase lag above \(\omega_c\)
This symmetry maximises the phase margin for a given crossover frequency
Optimises disturbance rejection bandwidth

Speed-Loop Transfer Functions

{Speed-Loop Transfer Functions — Exact and Simplified}

Closed-Loop Speed Transfer Functions

Exact speed-loop transfer function:

\[ G_{\omega e}(s) = \frac{\omega_r(s)}{\omega_r^*(s)} = \frac{G_{\omega f}(s)}{1 + G_{\omega f}(s)\,G_\omega(s)} \]

Simplified (using first-order current-loop model):

\[ G_{\omega s}(s) = \frac{G_f(s)}{1 + G_f(s)\,G_\omega(s)} \]

With reference pre-filter \(1/(1+sT_s)\):

\[ G_{\omega ss}(s) = \frac{G_{\omega s}(s)}{1+sT_s} \]

Verification

Bode plot comparison confirms:

Gain difference at high frequencies (higher-order terms)
Phase responses nearly identical in the bandwidth of interest
Crossover at the same \(\omega_c\)
Simplified model is adequate for controller synthesis

Reference Pre-filter

{Reference Pre-filter and Step Response Characteristics}

Reference Pre-filter

A first-order low-pass filter is added to the speed reference to eliminate the PI-zero overshoot in the tracking response:

\[ F(s) = \frac{1}{1 + sT_f}, \quad T_f = T_s \]

Cancels the PI zero in the closed-loop numerator
Converts the tracking response to a well-damped second-order Butterworth-like shape
The disturbance rejection response is unchanged (pre-filter is not in the disturbance path)

Step Response Characteristics (\(a=3\))

Rise time: \(\approx 3/\omega_c\)
Overshoot: \(< 5\%\) (with pre-filter)
Settling time: \(\approx 8/\omega_c\)
Load step recovery: \(\approx T_m\)

Without pre-filter: overshoot \(\approx 43\%\) due to PI zero contribution to the closed-loop numerator.

Design Recommendation

Always include the reference pre-filter when the symmetric optimum is used for speed-loop design — it transforms a fast but oscillatory response into a well-damped one at no cost to disturbance rejection.

Complete Drive Architecture

{Complete Indirect Vector-Controlled Drive with Speed Controller}

{Step-by-Step Design Procedure and Parameter Expressions}

Design Procedure

Current loop: design PI gains using pole-zero cancellation (\(T_s = T_a\), \(K_s\) from bandwidth specification)
Model reduction: reduce current loop to first-order \(K_i/(1+sT_i)\)
Speed loop plant: combine reduced current loop, mechanical subsystem, and speed sensor model
Symmetric optimum: apply design formulas with \(a = 3\) (recommended)
Reference pre-filter: set \(T_f = T_s\)
Verification: simulate step response and load disturbance rejection; compare exact vs.\ simplified

Summary of Parameter Expressions

\begin{align*} T_a &= L_a / R_s &&\text{(stator transient TC)} \\ T_m &= J / B &&\text{(mechanical TC)} \\ T_{ar} &= T_a + T_{in}&&\text{(combined TC)} \\ T_i &= T_1 &&\text{(current loop TC)} \\ T_s &= a^2 T_i &&\text{(speed PI zero TC)} \\ K_{\omega s} &= \frac{1}{2a\,K_m\,K_i\,H_\omega\,T_i} && \end{align*}

Typical Values with \(a = 3\)

Phase margin \(\approx 53°\)\\ Bandwidth \(\omega_c = 1/(3T_i)\)\\ Good disturbance rejection and reference tracking

Summary

{Summary — Lecture 8 and Course Overview}

Key Results of Lecture 8

The torque channel of the vector-controlled IM is modelled as a first-order \(R_s\)-\(L_a\) plant — identical in form to a DC motor armature circuit
Inner PI current controller with pole-zero cancellation simplifies to \(K_i/(1+sT_i)\)
Speed loop designed via symmetric optimum maximises phase margin for a given bandwidth
Reference pre-filter \(1/(1+sT_s)\) eliminates PI-zero overshoot without affecting disturbance rejection
Complete cascade: reference filter \(\to\) speed PI \(\to\) vector controller \(\to\) current PI \(\to\) VSI \(\to\) IM

Course — Complete Overview

cl@{}} Lecture	Topic
1	Introduction \	Principle
2	Direct Vector Control
3	Indirect Vector Control
4	Steady State \	Dynamics
5	Parameter Sensitivity
6	Parameter Compensation
7	Flux-Weakening Operation
8	Speed-Controller Design

Course Take-Away

Vector control transforms the induction motor into a high-performance, independently-controllable flux-and-torque machine — enabling AC drives to surpass DC machines in precision, speed range, and robustness.