{Outline}

Sources of Parameter Mismatch

{Why Motor Parameters Deviate from Controller Values}

Physical Causes of Variation

Rotor resistance \(\Rr\): varies up to \(\pm 50\%\) with temperature (\(\approx 0.4\%/\text{°C}\) for copper); the most significant source of mismatch in practice
Magnetising inductance \(\Lm\): decreases under magnetic saturation (depends on flux level); typical variation \(\pm 20\)–\(30\%\)
Rotor inductance \(\Lr\): saturation and leakage changes; similar order of magnitude to \(\Lm\)
Stator resistance \(\Rs\): temperature-dependent (primarily affects direct vector control)
Incorrect initial values: poor machine identification or inaccurate nameplate data

Consequences of Mismatch

Parameter mismatch between motor and controller causes:

Coupling between flux and torque channels — decoupling is lost
Rotor flux deviates from commanded value
Electromagnetic torque deviates from command
Nonlinear, operating-point-dependent torque characteristic
Oscillations in flux and torque with settling time \(\approx \Tr\)

Parameterisation of Mismatch

{Mismatch Ratios and Error Expressions}

Normalised Mismatch Parameters

Define dimensionless mismatch ratios:

\[ \alpha = \frac{\Rr^*}{\Rr} \quad\text{(rotor resistance ratio)} \]

\[ \beta = \frac{\Lm^*}{\Lm} \quad\text{(magnetising inductance ratio)} \]

Perfect match: \(\alpha = \beta = 1\)
\(\alpha < 1\): controller uses a lower \(\Rr^*\) than actual (motor hotter than assumed)
\(\alpha > 1\): controller uses a higher \(\Rr^*\) than actual (motor colder than assumed)

Steady-State Error Expressions

Actual rotor flux relative to commanded value:

\[ \frac{\lambdar}{\lambdar^*} = \beta\, \sqrt{\frac{1 + (\omegasl^*\Tr)^2}{1 + (\alpha\omegasl^*\Tr)^2}} \]

Actual torque relative to commanded value:

\[ \frac{\Te}{T_e^*} = \alpha\beta\, \frac{1 + (\omegasl^*\Tr)^2}{1 + (\alpha\omegasl^*\Tr)^2} \]

Both expressions depend on the operating slip speed \(\omegasl^*\).

Sensitivity - Open Outer Speed Loop

Flux Error

{Flux Error — Open Speed Loop}

Effect of \(\alpha\) Variation on Rotor Flux

\[ \frac{\lambdar}{\lambdar^*} = \beta\, \sqrt{\frac{1 + (\omegasl^*\Tr)^2}{1 + (\alpha\omegasl^*\Tr)^2}} \]

Physical interpretation:

At zero slip: \(\lambdar/\lambdar^* = \beta\) (only \(\Lm\) mismatch matters; \(\alpha\) is immaterial)
At high slip: \(\lambdar/\lambdar^* \approx \beta/\alpha\)
\(\alpha < 1\) (hot motor): flux increases beyond command
\(\alpha > 1\) (cold motor): flux decreases from command
Flux error is relatively mild compared to torque error

Graphical Interpretation

Plotting \(\lambdar/\lambdar^*\) versus \(\alpha\) for various slip speeds:

Curves are nearly flat for small slip
Deviation amplifies with increasing slip speed
At rated slip, a 50\% \(\Rr\) error produces approximately 10–20\% flux error

Flux error is less critical than torque error in most operating conditions.

Torque Error

{Torque Error — Open Speed Loop}

Effect of \(\alpha\) on Electromagnetic Torque

\[ \frac{\Te}{T_e^*} = \alpha\beta\, \frac{1 + (\omegasl^*\Tr)^2}{1 + (\alpha\omegasl^*\Tr)^2} \]

Key observations:

The denominator grows as \(\alpha^2\) for large \(\omegasl^*\Tr\)
Numerator grows only as \((\omegasl^*)^2\) — slower
Torque error is more severe than flux error
At rated conditions, a 50\% \(\Rr\) error can produce 40–60\% torque error
Error magnitude increases with operating slip speed

Nonlinearity Introduced by Mismatch

With parameter mismatch, the actual torque is no longer proportional to the torque command:

\[ \Te = f\!\left(T_e^*,\,\alpha,\,\beta,\,\omegasl^*\right) \]

The torque–command characteristic becomes nonlinear and speed-dependent
The closed-loop speed controller partially compensates but cannot eliminate the torque oscillations

{Transient Behaviour with Parameter Mismatch}

Torque Transient with \(\alpha \ne 1\)

During a step torque command:

Ideal (\(\alpha = 1\)): flux constant, torque steps instantaneously
With mismatch (\(\alpha \ne 1\)):
1. The torque-producing current \(\iT\) excites the rotor flux dynamics
2. Rotor flux oscillates with time constant \(\sim \Tr\)
3. Torque oscillates correspondingly
4. Settling time \(\approx 3\Tr\) (0.3–3\,s typically)

Oscillation amplitude increases with \(|\alpha - 1|\).

Physical Mechanism

The \(q\)-axis rotor flux equation, when orientation is lost:

\[ \Tr\,\frac{d\lambda_{qr}^e}{dt} + \lambda_{qr}^e = \Lm\,\iqs - (\omegasl^*\Tr)\,\lambda_{dr}^e \]

The right-hand side acts as a disturbance input driving \(\lambda_{qr}^e\) away from zero. This couples the torque channel with the rotor flux dynamics — the defining characteristic of loss of field orientation.

Sensitivity - Closed Outer Speed Loop

Speed Loop Effects

{Effect of Parameter Mismatch in Speed-Controlled Drive}

How the Speed Loop Modifies Sensitivity

Speed error \(\to\) PI controller \(\to\) \(T_e^*\) adjusted
The speed loop partially corrects for torque deficiency by demanding a higher \(T_e^*\)
Steady-state speed error: zero (PI integrator)
Transient torque oscillations still occur and can:
- Cause current saturation in the inverter
- Demand excessive peak current from the supply
- Produce speed ripple when rotor inertia \(J\) is small

Temperature and Saturation Effects

Temperature rise (increasing \(\Rr\), \(\alpha\) increases):

Reduces torque per unit slip command
Speed controller demands more \(T_e^*\)
Higher stator current \(\to\) more heating — a positive thermal feedback!

Saturation (reducing \(\Lm\), \(\beta\) decreases):

Flux drops below command
Torque reduced proportionally
Higher \(\ifld^*\) demanded to restore flux

Air-Gap Power Sensitivity

{Air-Gap Power Sensitivity to Rotor Resistance}

Air-Gap Power Expressions

Actual air-gap power:

\[ P_a = \frac{3I_r^2\,\Rr}{s_{\text{act}}} \]

Reference air-gap power (from controller commands):

\[ P_a^* = \frac{\omegas\,T_e^*}{P/2} \]

At \(\alpha = 1\): \(P_a = P_a^*\) (exact match)
\(\alpha \ne 1\): up to 30\% power deviation possible at twice the rated \(\Rr\)
Uncompensated: power increases monotonically with \(\Rr\) — overheating risk in thermally stressed environments

Compensation Requirement

Uncompensated: output power drifts significantly across the operating temperature range
Compensated: power remains at \(P_a^*\) across the full \(\Rr\) variation range

This justifies the need for the parameter compensation strategies developed in Lecture~6.

Summary Table

{Summary of Parameter-Sensitivity Effects}

lccc@{}} Parameter change	Primary effect	Open loop	Closed speed loop
\(\Rr\) increases (\(\alpha > 1\))	Reduced torque/flux	\(T_e < T_e^*\)	Speed error \(\to 0\); more current
\(\Rr\) decreases (\(\alpha < 1\))	Increased torque/flux	\(T_e > T_e^*\)	Over-torque; possible saturation
\(\Lm\) decreases (\(\beta < 1\))	Flux drops	\(\lambdar < \lambdar^*\)	Higher \(\ifld^*\) demanded
\(\Lr\) decreases	Slip angle mismatch	\(\omegasl^*\) error	Torque oscillations in transients

Design Rule for Robust Operation

Parameter sensitivity is minimised at low slip speeds (i.e., high flux, low torque fraction of rating). High-torque, high-slip operation amplifies all mismatch effects. The term \(\omegasl^*\Tr\) is the key non-dimensional sensitivity parameter.

Summary

{Summary — Lecture 5}

Key Findings

Motor–controller parameter mismatch destroys the field-orientation decoupling
Torque error is more severe than flux error for the same level of parameter mismatch
Both errors worsen with increasing slip speed and increasing \(|\alpha - 1|\)
The closed speed loop eliminates steady-state speed error but cannot eliminate torque oscillations
Temperature rise (increasing \(\Rr\)) is the most problematic variation in industrial practice

Preview — Lecture 6

Methods to compensate for parameter sensitivity:

Direct axis-alignment monitoring
Reactive power measurement method
Rotor flux deviation detection
Model reference adaptive system (MRAS)
Online rotor-resistance identification