{Outline}

Steady-State Analysis

Steady-State Operating Equations

{Steady-State Conditions in Indirect Vector Control}

Steady-State Simplification (\(p = d/dt = 0\))

Setting all time derivatives to zero in the rotor equations:

\begin{align*} \lambdar^* &= \Lm^*\,\ifld^* \\[4pt] \omegasl^* &= \frac{1}{\Tr^*}\cdot\frac{\iT^*}{\ifld^*} = \frac{\Lm^*\,\iT^*}{\Tr^*\,\lambdar^*} \end{align*}

Stator current magnitude in steady state:

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

Stator Currents and Voltages (Synchronous Frame)

\begin{align*} V\qs &= R_s I\qs + \omegas\sigma\Ls I\ds + \omegas\tfrac{\Lm}{\Lr}\lambdar \\[3pt] V\ds &= R_s I\ds - \omegas\sigma\Ls I\qs \end{align*}

Stator voltage phasor:

\[ V_s = \sqrt{V\qs^2 + V\ds^2} \]

Rotor Currents in Steady State

From the orientation condition \(\lambda_{qr}^e = 0\):

\[ I_{dr}^e = 0, \qquad I_{qr}^e = -\frac{\Lm}{\Lr}\,I\qs \]

The rotor carries only the torque-component of stator current, reflected through the turns ratio.

Operating Regions

{Steady-State Operating Regions}

Constant-Torque Region (below base speed)

\(\lambdar^* = \lambda_b\) = constant
\(\ifld^* = \lambda_b / \Lm^*\) = constant
Torque command limited by rated stator current
Back-EMF proportional to speed; voltage headroom available

Constant-Power (Flux-Weakening) Region (above base speed):

\(\lambdar^* = (\omega_b/|\omegar|)\lambda_b\) decreasing
\(V_s \approx V_{s,\max}\) (voltage limit reached)
Maximum available torque \(\propto 1/\omegar\)

{Stator Voltage Constraint — Why Flux Weakening is Necessary}

Voltage Limit Constraint

At rated speed and rated flux, the stator voltage can exceed the maximum inverter output:

\[ V_s = \sqrt{V\qs^2 + V\ds^2} > V_{s,\max} \]

The dominant contribution is the back-EMF: \(\omegas(\Lm/\Lr)\lambdar\)
To maintain \(V_s \le V_{s,\max}\), the flux must be reduced as \(\omegas\) increases
Transition occurs at base speed \(\omega_b\) where \(V_s = V_{s,\max}\)

Practical Design Note

The relationship between phase and line voltage:

\[ V_{LL} = \sqrt{3}\cdot\sqrt{2}\cdot V_s \]

Even a 5–10\% voltage excess forces flux weakening to begin below the rated electrical speed. Proper drive design sets \(\omega_b\) slightly below the electrical rated speed to provide sufficient headroom.

Dynamic Simulation Framework

Motivation and Model Components

{Purpose and Structure of Dynamic Simulation}

Motivation

Determine the transient response of the indirect vector-controlled drive to evaluate its suitability for a target application
Avoid costly prototype build-and-test cycles
Explore performance under different load profiles, reference trajectories, and parameter variations
Validate controller design prior to hardware implementation

System Components Modelled

Speed controller (PI with anti-windup)
Vector controller (flux and torque computation)
Inverter with switching logic (PWM / hysteresis)
Induction machine (full nonlinear equations)
Load (constant, fan, or pump torque profile)

Full Nonlinear Machine Model

{Full Nonlinear Induction Machine Model}

Complete Machine Equations in the Synchronous Reference Frame

Stator voltage equations:

\begin{align*} v\qs &= \Rs\,\iqs + \frac{d}{dt}\!\left(\Ls\,\iqs + \Lm\,i_{qr}^e\right) + \omegas\!\left(\Ls\,\ids + \Lm\,i_{dr}^e\right)\\[3pt] v\ds &= \Rs\,\ids + \frac{d}{dt}\!\left(\Ls\,\ids + \Lm\,i_{dr}^e\right) - \omegas\!\left(\Ls\,\iqs + \Lm\,i_{qr}^e\right) \end{align*}

Rotor voltage equations:

\begin{align*} 0 &= \Rr\,i_{qr}^e + \frac{d}{dt}\!\left(\Lr\,i_{qr}^e + \Lm\,\iqs\right) + \omegasl\!\left(\Lr\,i_{dr}^e + \Lm\,\ids\right)\\[3pt] 0 &= \Rr\,i_{dr}^e + \frac{d}{dt}\!\left(\Lr\,i_{dr}^e + \Lm\,\ids\right) - \omegasl\!\left(\Lr\,i_{qr}^e + \Lm\,\iqs\right) \end{align*}

Mechanical equation:

\[ J\,\frac{d\omegar}{dt} = \Te - T_L - B\omegar, \qquad \Te = \frac{3P}{4}\,\Lm\!\left(\iqs\,i_{dr}^e - \ids\,i_{qr}^e\right) \]

All nonlinearities preserved — no linearisation applied
Numerically integrated with 4th-order Runge–Kutta method

Simulation Results

CSI Drive Response

{Current-Source Indirect Vector Control — Transient Response}

Step Speed Reversal — CSI Drive

Observable from simulation traces:

Speed \(\omega_m\): smooth, fast tracking with no oscillatory response
Rotor flux \(\lambda_m\): remains essentially constant during speed transients — decoupling maintained
Torque \(T_e\): fast, high-magnitude response with well-defined limiter clipping during reversal
Stator current \(i_{as\)}: magnitude varies cleanly; frequency changes smoothly with speed

Sinusoidal Speed Reference — CSI

The drive tracks a sinusoidal speed reference closely:

No phase lag in flux response
Torque in phase with the speed derivative
Stator current envelope follows the speed amplitude

Demonstrates servo-grade bandwidth consistent with high-performance motion control applications.

VSI Drive Response

{Voltage-Source Indirect Vector Control — Transient Response}

Step Speed Reversal — VSI Drive

Key observations:

Faster speed command response compared to CSI (higher current bandwidth from inner \(dq\) current loops)
Torque command \(T_e^*\) shows ideal limiter clipping
Actual torque \(T_e\) tracks the command closely
Flux \(\lambda_m\) remains constant throughout the reversal — flux and torque channels remain decoupled
Stator current waveform shows natural PWM switching ripple

VSI vs.\ CSI — Performance Comparison

lcc@{}} Feature	VSI	CSI
Current control	Inner PI	Direct
Torque bandwidth	Higher	Moderate
Flux regulation	Excellent	Excellent
Switching losses	Moderate	Lower
Industrial use	Dominant	Less common

Performance Assessment

{Assessment of Vector-Controlled Drive Performance}

Performance Metrics Achieved

Decoupling: flux and torque independently controllable
Torque bandwidth: \(>100\)\,Hz (set by current loop)
Speed bandwidth: 10–50\,Hz (limited by inertia)
Flux bandwidth: set by \(\Tr\) (deliberately slow)
Zero-speed torque: full rated torque at standstill
Four-quadrant operation: regenerative braking included

Conditions for Ideal Performance

All stated metrics assume:

Motor and controller parameters exactly matched
Accurate rotor position measurement
Sufficient inverter voltage headroom
Operation within the linear magnetic range

Real drives must contend with parameter variation — the subject of Lectures 5 and 6.

Summary

{Summary — Lecture 4}

Steady-state analysis confirms that the indirect vector controller produces correct flux and torque when parameters are matched; the stator voltage at rated speed may necessitate flux weakening
Two operating regions: constant-torque below \(\omega_b\) and constant-power (flux-weakening) above \(\omega_b\)
Dynamic simulation uses the complete nonlinear induction machine model — stator, rotor, and mechanical equations — together with the full drive chain
Simulation confirms: rotor flux remains constant during torque transients (decoupling verified), torque response is achieved in a few milliseconds, and speed tracking is smooth
VSI drives show slightly higher bandwidth than CSI drives due to the inner \(dq\) current regulation loops
Ideal performance degrades with parameter mismatch — quantified rigorously in Lecture~5