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

RECAP

Key Results from Lectures 5A and 5B

✓Established Results

Slip: s = (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 = P_ag/ω_s = 3I_r′²R_r′/(s·ω_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

|V_th| ≈ V_s · X_m / (X_s + X_m) ≈ 0.95–0.98 V_s

Z_th = R_th + jX_th = jX_m(R_s+jX_s) / (R_s+j(X_s+X_m))

Typical: R_th ≈ R_s, X_th ≈ 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 = (R_th + R_r′/s) + j(X_th + X_r′)

I_r′ = V_th / √[(R_th + R_r′/s)² + (X_th + X_r′)²]

Fundamental Torque-Slip Equation

T_em = [3 V_th² (R_r′/s)] / {ω_s [(R_th + R_r′/s)² + (X_th + X_r′)²]}

Low-Slip Linear Approximation

At small s: R_r′/s ≫ (R_th, X_th+X_r′), so the denominator ≈ (R_r′/s)²:

T_em ≈ [3V_th² / (ω_s·R_r′)] · s = K · 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 → 0	→ ∞	T → 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 ≈ 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 SR ≈ s_rated/(1−s_rated) ≈ 2–5%

⚠️ 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 to 3.0) × 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 = R_r′ / √[R_th² + (X_th+X_r′)²]

Simplified (R_th ≪ X_th+X_r′): s_max ≈ R_r′ / (X_th+X_r′)

Maximum (Pull-Out) Torque

T_max ≈ 3V_th² / [2ω_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 ≈ 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 ∝ R_r′ while T_max remains constant.

Fig. 1 — 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 ≈ T_max throughout acceleration.

SECTION 07

Starting Torque & Current

Starting Torque (s = 1)

T_start = 3 V_th² R_r′ / {ω_s [(R_th+R_r′)² + (X_th+X_r′)²]}

T_start/T_max = 2·s_max / (1+s_max²)

⚠️ Starting Current Problem

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

I_start = 5–7 × 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 ≈ 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 = V_DC / (2 · I_DC) (Y-connection)
R_s = (3/2) · V_DC/I_DC (Δ-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 ≈ 1–10 Ω
Large motors (>100 kW): R_s ≈ 0.01–0.1 Ω
AC skin effect correction factor: k_ac ≈ 1.1–1.3

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 ≈ 0), so R_r′/s → ∞ and the rotor branch carries negligible current. The magnetising branch dominates.

Measurements & Equations

Measure: V_s (phase), I_NL, P_NL (per phase)

cosφ₀ = P_NL / (V_s · I_NL)
I_c = I_NL·cosφ₀ ⇒ R_c = V_s/I_c
I_m = I_NL·sinφ₀ ⇒ X_m = V_s/I_m

Key Insight

No-load current I_NL is mostly reactive (I_m ≫ I_c) — the no-load power factor is very low (≈ 0.1–0.3)
The wattmeter reading P_NL captures both core losses (P_core = V_s²/R_c) and friction & windage losses
At s ≈ 0: the rotor branch is effectively open circuit; all current flows through R_c∥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 = V_sc/I_sc
cosφ_sc = P_sc/(V_sc·I_sc)
R_sc = Z_sc·cosφ_sc = R_s + R_r′
X_sc = Z_sc·sinφ_sc = X_s + X_r′
R_r′ = R_sc − R_s
X_s ≈ X_r′ = X_sc/2 (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 ≈ Ks — linear, self-regulating. Motor is a "constant speed" machine in this region.

3. The Key Independence

s_max ∝ 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).