Lecture 5C: Torque-Speed Characteristics & Parameter Measurement | Polyphase Induction Machines

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Key Results from Lectures 5A and 5B

✓Established Results

Slip: \(s = \dfrac{n_s - n_r}{n_s}\)
Per-phase equivalent circuit with \(R_r'/s\) rotor branch
Power ratio: \(P_{ag} : P_{r,Cu} : P_{conv} = 1 : s : (1-s)\)
Torque: \(T_{em} = \dfrac{P_{ag}}{\omega_s} = \dfrac{3\,I_r'^{\,2}\,R_r'}{s\,\omega_s}\)
Power factor always lagging; poor at light loads

Unanswered Questions

How does \(I_r'\) vary with slip? What shape does \(T_{em}(s)\) take? Where is the maximum torque? How do we measure the circuit parameters? — All answered in this lecture.

SECTION 01

Thévenin Equivalent — Why and How

To find \(I_r'\) as a function of slip, we replace everything to the left of the air-gap nodes with its Thévenin equivalent — a simple series circuit remains, making torque vs. slip analysis tractable.

Thévenin Parameters

\[\left|V_{th}\right| \approx V_s \cdot \frac{X_m}{X_s + X_m} \approx 0.95\text{–}0.98\,V_s\] \[Z_{th} = R_{th} + jX_{th} = \frac{jX_m(R_s + jX_s)}{R_s + j(X_s + X_m)}\]

Typical: \(R_{th} \approx R_s\), \(\quad X_{th} \approx X_s\)

Thévenin equivalent circuit of the induction motor showing the simplified series loop used for torque-slip analysis — Fig. 1 — Thévenin equivalent: \(V_{th}\), \(Z_{th}\) in series with rotor branch \(R_r'/s + jX_r'\).

💡 The Thévenin circuit is a single series loop — rotor current \(I_r'\) can be written in one clean expression, making torque vs. slip analysis straightforward.

SECTION 02

Rotor Current & Fundamental Torque-Slip Equation

Rotor Current Magnitude

\[Z_{total} = \left(R_{th} + \frac{R_r'}{s}\right) + j(X_{th} + X_r')\] \[I_r' = \frac{V_{th}}{\sqrt{\left(R_{th} + \dfrac{R_r'}{s}\right)^2 + (X_{th} + X_r')^2}}\]

Fundamental Torque-Slip Equation

\[T_{em} = \frac{3\,V_{th}^2\,\dfrac{R_r'}{s}}{\omega_s\!\left[\left(R_{th} + \dfrac{R_r'}{s}\right)^2 + (X_{th} + X_r')^2\right]}\]

Low-Slip Linear Approximation

At small \(s\): \(R_r'/s \gg (R_{th},\; X_{th}+X_r')\), so the denominator \(\approx (R_r'/s)^2\):

\[T_{em} \approx \frac{3V_{th}^2}{\omega_s\,R_r'} \cdot s = K \cdot s\]

Torque is directly proportional to slip in the normal operating region — the motor is self-regulating.

Table 1 — Torque Behaviour at Key Slip Values
Slip	\(R_r'/s\)	Effect on \(T_{em}\)
\(s = 1\)	\(= R_r'\) (small)	Standstill torque
\(0 < s < s_{max}\)	Moderate	Torque rising
\(s = s_{max}\)	Optimal	Peak (breakdown) torque
\(s \to 0\)	\(\to \infty\)	\(T \to 0\)

SECTION 03

The Complete Torque-Slip Curve

Complete torque-slip characteristic curve showing motoring, generating, and plugging regions with key operating points — Fig. 2 — Complete torque-slip curve: five key points from generating (\(s<0\)) through motoring (\(0<s<1\)) to plugging (\(s>1\)).

Table 2 — Five Key Points on the Torque-Slip Curve
Point	Slip	Significance
Synchronous	\(s = 0,\; T = 0\)	No electromagnetic induction; theoretical only
Rated operating	\(s \approx 0.04\)	Normal stable operating point; self-regulating
Breakdown	\(s = s_{max}\)	Maximum / pull-out torque; stability boundary
Starting	\(s = 1\)	Locked-rotor; high current, moderate torque
Generating	\(s < 0\)	External prime mover drives rotor above \(n_s\)

SECTION 04

Stable vs. Unstable Operating Regions

✅ Stable Region (\(0 < s < s_{max}\))

Self-regulating mechanism:

Load torque \(T_L\) increases
Motor decelerates ⇒ slip \(s\) increases
In linear region: \(T_{em} = Ks\) ⇒ \(T_{em}\) increases
New equilibrium: \(T_{em} = T_L\) at slightly higher slip
Speed barely changes — stiff speed characteristic

SR Speed regulation \(\text{SR} \approx s_{rated}/(1-s_{rated}) \approx 2\text{–}5\%\)

⚠️ Unstable Region (\(s_{max} < s < 1\))

Positive feedback — runaway to stall:

Load torque \(T_L > T_{em}\) momentarily
Motor decelerates ⇒ slip increases
Now past \(s_{max}\): \(T_{em}\) decreases with increasing \(s\)
Gap between \(T_L\) and \(T_{em}\) grows
Motor stalls (comes to rest)

Motor must never be operated continuously in this region.

🛡️

Design Safety Margin

\(T_{max} = (2.0\text{ to }3.0) \times T_{rated}\)

Ensures the motor survives transient overloads without stalling. This generous margin is deliberately designed in by the manufacturer.

SECTION 05

Maximum Torque — Derivation & Significance

Setting \(dT_{em}/ds = 0\) and solving (treating \(V_{th}\), \(R_{th}\), \(X_{th}\), \(R_r'\), \(X_r'\) as constants):

Slip at Maximum Torque

\[s_{max} = \frac{R_r'}{\sqrt{R_{th}^2 + (X_{th}+X_r')^2}}\]

Simplified (\(R_{th} \ll X_{th}+X_r'\)):

\[s_{max} \approx \frac{R_r'}{X_{th}+X_r'}\]

Maximum (Pull-Out) Torque

\[T_{max} \approx \frac{3\,V_{th}^2}{2\,\omega_s\,(X_{th}+X_r')}\]

⭐

The Critical Independence Result

\(T_{max}\) is independent of \(R_r'\) — \(s_{max}\) is proportional to \(R_r'\)

These two facts explain the entire family of torque-slip curves and the wound-rotor motor starting strategy. Changing rotor resistance shifts the peak left or right on the slip axis, but never changes the height of the peak.

SECTION 06

Wound-Rotor Starting Strategy & Family of Curves

🚀Wound-Rotor Starting Sequence

Start with high external \(R_{ext}\) so that \(s_{max} = 1\) ⇒ \(T_{start} = T_{max}\)
As motor accelerates, cut resistance in steps — each step restores \(T_{em} \approx T_{max}\)
Rated speed: all resistance removed; low-\(R_r'\) run curve gives high efficiency

📊Reading the Curve Family

All curves share the same \(T_{max}\)
Higher \(R_r'\) shifts peak to larger slip
Low \(R_r'\): peak near synchronous speed; high efficiency, low \(T_{start}\)
Optimal \(R_r'\): \(s_{max} = 1\) ⇒ \(T_{start} = T_{max}\)

Family of torque-slip curves showing how increasing rotor resistance shifts the slip at maximum torque without changing its magnitude — Fig. 3 — Effect of rotor resistance: \(s_{max} \propto R_r'\) while \(T_{max}\) remains constant.

Fig. 3 — Increasing \(R_r'\) shifts \(s_{max}\) toward standstill while \(T_{max}\) remains constant.

Wound-rotor motor torque-speed curves during starting showing step-by-step external resistance reduction — Fig. 4 — Wound-rotor starting: resistance cut in steps keeps \(T_{em} \approx T_{max}\) throughout acceleration.

SECTION 07

Starting Torque & Current

Starting Torque (\(s = 1\))

\[T_{start} = \frac{3\,V_{th}^2\,R_r'}{\omega_s\!\left[(R_{th}+R_r')^2 + (X_{th}+X_r')^2\right]}\] \[\frac{T_{start}}{T_{max}} = \frac{2\,s_{max}}{1+s_{max}^2}\]

⚠️ Starting Current Problem

At \(s = 1\), the rotor branch impedance is at its minimum. Typically:

\[I_{start} = 5\text{–}7 \times I_{rated}\]

High inrush current with only moderate torque — the fundamental starting problem addressed in Lecture 5D.

🛡️Breakdown Torque Safety Factors

Motor Type	\(T_{max}/T_{rated}\)
General purpose	2.0–2.5
High-torque	2.5–3.0
Wound rotor	2.5–3.5

SECTION 08

NEMA Design Classifications

A single squirrel-cage motor cannot simultaneously achieve low starting current (needs low \(R_r'\)), high starting torque (needs high \(R_r'\)), and high rated efficiency (needs low \(R_r'\)). NEMA classifications codify standard compromises for different industrial applications.

🔬 Physical Mechanism — Deep-Bar / Double-Cage: At standstill (high rotor frequency), current crowds to top of bar via skin effect ⇒ high effective \(R_r'\) ⇒ high \(T_{start}\). Near rated speed (low rotor frequency), current fills full bar ⇒ low \(R_r'\) ⇒ high efficiency.

Table 3 — NEMA Design Class Summary
Class	\(T_{st}/T_r\)	Rated Slip	Best For
A	1.0–1.5	<5%	Fans, pumps — low starting torque loads
B (Industry Standard)	≥1.5	<5%	Compressors, conveyors — general purpose
C	2.0–2.5	<5%	Crushers, reciprocating compressors
D	>2.5	7–11%	Presses, cranes — high starting torque

Torque-speed curves for NEMA design classes A, B, C, and D showing differences in starting torque and rated slip — Fig. 5 — NEMA design classes A–D: Class B is the industry standard (>90% of installations).

Class B — Industry Standard

Deep-bar or double-cage rotor achieves both limited starting current and adequate starting torque. Over 90% of new industrial motor installations use Class B.

📌Class D — Trade-off

Inherently high \(R_r'\) gives maximum starting torque, but rated slip of 7–11% means significant rotor copper loss at full load — efficiency is poor.

SECTION 09

Measurement of Motor Parameters — Overview

Before using the equivalent circuit for calculation, we need numerical values for all six parameters. Manufacturers seldom provide all of them — they must be extracted experimentally using three standard tests.

Table 4 — Three Standard Tests
Test	Operating Condition	Parameters Extracted
DC Resistance	Standstill, DC supply	\(R_s\)
No-Load	Rated \(V\) & \(f\); free shaft (\(s \approx 0\))	\(R_c\), \(X_m\)
Locked-Rotor	Low \(V\), rated \(I\); \(s = 1\)	\(R_r'\), \(X_s\), \(X_r'\)

📋 Test Philosophy: Each test drives the motor to an extreme operating condition where one part of the circuit dominates and the rest can be neglected — making the mathematics tractable. Standardised under IEEE 112 / IEC 60034.

Circuit diagrams for the three standard parameter measurement tests: DC resistance, no-load, and locked-rotor — Fig. 6 — Three standard tests: DC resistance (→ \(R_s\)), no-load (→ \(R_c\), \(X_m\)), locked-rotor (→ \(R_r'\), \(X_s\), \(X_r'\)).

SECTION 10

Test 1 — DC Resistance Test (Extracts \(R_s\))

🔌Procedure

Apply DC voltage \(V_{DC}\) between two stator line terminals with rotor at standstill
Measure DC current \(I_{DC}\)
Two stator phase windings in series (for Y-connection)

Stator Resistance Formula

\[R_s = \frac{V_{DC}}{2\,I_{DC}} \qquad \text{(Y-connection)}\] \[R_s = \frac{3}{2}\cdot\frac{V_{DC}}{I_{DC}} \qquad \text{(\(\Delta\)-connection)}\]

Why DC Excitation?

DC eliminates inductance — only resistance appears in the measurement. Any AC voltage would cause an additional voltage drop across \(X_s\), giving an incorrect (higher) resistance reading.

⚠️ Always perform DC test first. \(R_s\) is subtracted in the locked-rotor test to isolate \(R_r'\). An error here propagates into every subsequent calculation.

Typical Values & Correction

Small motors (<5 kW): \(R_s \approx 1\text{–}10\ \Omega\)
Large motors (>100 kW): \(R_s \approx 0.01\text{–}0.1\ \Omega\)
AC skin effect correction factor: \(k_{ac} \approx 1.1\text{–}1.3\)

SECTION 11

Test 2 — No-Load Test (Extracts \(R_c\) and \(X_m\))

Apply rated three-phase voltage at rated frequency with shaft uncoupled (no mechanical load). Motor runs near synchronous speed (\(s \approx 0\)), so \(R_r'/s \to \infty\) and the rotor branch carries negligible current. The magnetising branch dominates.

Measurements & Equations

Measure: \(V_s\) (phase), \(I_{NL}\), \(P_{NL}\) (per phase)

\[\cos\phi_0 = \frac{P_{NL}}{V_s\,I_{NL}}\] \[I_c = I_{NL}\cos\phi_0 \;\Rightarrow\; R_c = \frac{V_s}{I_c}\] \[I_m = I_{NL}\sin\phi_0 \;\Rightarrow\; X_m = \frac{V_s}{I_m}\]

Key Insight

No-load current \(I_{NL}\) is mostly reactive (\(I_m \gg I_c\)) — the no-load power factor is very low (\(\approx 0.1\text{–}0.3\))
The wattmeter reading \(P_{NL}\) captures both core losses (\(P_{core} = V_s^2/R_c\)) and friction & windage losses
At \(s \approx 0\): the rotor branch is effectively open circuit; all current flows through \(R_c \parallel jX_m\)

SECTION 12

Test 3 — Locked-Rotor Test (Extracts \(R_r'\), \(X_s\), \(X_r'\))

Rotor is mechanically locked (\(s = 1\)). Apply reduced three-phase voltage until rated current flows. At \(s = 1\), \(R_r'/s = R_r'\) (minimum rotor impedance), and \(X_m\) carries negligible current. The circuit reduces to a series combination: \((R_s+R_r') + j(X_s+X_r')\).

Measurements & Equations

Measure: \(V_{sc}\), \(I_{sc}\), \(P_{sc}\) (per phase)

\[Z_{sc} = \frac{V_{sc}}{I_{sc}}, \qquad \cos\phi_{sc} = \frac{P_{sc}}{V_{sc}\,I_{sc}}\] \[R_{sc} = Z_{sc}\cos\phi_{sc} = R_s + R_r'\] \[X_{sc} = Z_{sc}\sin\phi_{sc} = X_s + X_r'\] \[R_r' = R_{sc} - R_s\] \[X_s \approx X_r' = \frac{X_{sc}}{2} \qquad \text{(NEMA A, B, D)}\]

Table 5 — NEMA Split Ratios for \(X_{sc}\)
NEMA Class	\(X_s/X_{sc}\)	\(X_r'/X_{sc}\)
A, B, D	0.50	0.50
C	0.30	0.70

Practical Notes

For large motors, use a 25 Hz supply — at 50 Hz, skin effect inflates \(R_r'\) above its true value at rated frequency
Apply reduced voltage to avoid thermal damage from the locked-rotor current
The equal-split \(X_s = X_r'\) is an approximation valid for NEMA A, B, and D motors

SECTION 13

Lecture Summary

Table 6 — Complete Parameter Extraction Summary
Test	Condition	Measurements	Extracted	Formula
DC Resistance	Standstill, DC	\(V_{DC},\; I_{DC}\)	\(R_s\)	\(R_s = V_{DC}/(2I_{DC})\)
No-Load	Rated \(V\), \(f\); free shaft	\(V_s,\; I_{NL},\; P_{NL}\)	\(R_c\), \(X_m\)	\(R_c=V_s/I_c\), \(X_m=V_s/I_m\)
Locked-Rotor	Low \(V\), rated \(I\); \(s=1\)	\(V_{sc},\; I_{sc},\; P_{sc}\)	\(R_r'\), \(X_s\), \(X_r'\)	\(R_r'=R_{sc}-R_s\)

1. Thévenin Equivalent

Simplifies full circuit to a series loop; gives tractable \(I_r'(s)\) expression for all operating conditions.

2. Torque-Slip — Low Slip Region

\(T_{em} \approx Ks\) — linear, self-regulating. Motor is a "constant speed" machine in this region.

3. The Key Independence

\(s_{max} \propto R_r'\) (shifts the peak)
\(T_{max}\) independent of \(R_r'\) (height unchanged)

4. NEMA Classes A–D

Different torque-speed shapes via rotor bar geometry (skin effect); Class B is the industry standard (>90% of installations).

5. Test Sequence Is Critical

DC resistance test first → \(R_s\)
No-load test → \(R_c\), \(X_m\)
Locked-rotor test last → \(R_r'\), \(X_s\), \(X_r'\)

Next: Lecture 5D

Starting methods (DOL, Y-Δ, autotransformer, soft starters, VFD) and speed control methods (pole changing, voltage control, rotor resistance, V/f control with affinity laws).