Induction Motor Slip-Energy Recovery Drives: Analysis, Harmonics & Drive Design

SECTION 01

Lecture 6C — Scope and Objectives

Lectures 6A–6B Recap

Phase (stator voltage) control: \(\eta \leq (1-s)\).
Power factor degrades severely with increasing \(\alpha\).
SER principle: diode bridge + DC choke + SCR inverter return \(sP_{ag}\) to supply.
Speed equation: \(s = (n_t V_s/V_r)\,|\cos\alpha|\).
SER requires a wound-rotor IM with accessible slip rings.

This Lecture

Per-phase equivalent circuit (stator-referred)
Torque expression and DC-machine analogy: \(T_e = K_t I_{dc}\)
SER performance and efficiency
Harmonic currents: sixth-harmonic pulsating torque; 5th and 7th orders
Supply-side harmonic distortion and mitigation strategies
Transformer turns-ratio design
Starting procedure and converter ratings
Closed-loop \(I_{dc}\) and speed control

SECTION 02

Per-Phase Equivalent Circuit — Stator-Referred

SER Per-Phase Equivalent Circuit

Rotor-loop KVL (referred to stator, per phase):

\[I_r'\!\left(\frac{R_r'+R_f'}{s}+jX_r'\right) = E_s - \frac{n_t V_s\cos\alpha}{a\,s}\]

where \(E_s\) is the stator-referred air-gap EMF and the second term is the referred back-EMF of the SCR inverter.

Parameter referral (stator-to-rotor turns ratio \(a\)):

\[R_f' = a^2 R_f,\quad R_r' = a^2 R_r,\quad X_r' = a^2 X_r\]

\(R_f'\): referred DC-link resistance (choke + diode forward drop).
The back-EMF term opposes the rotor current, controlling the DC-link current \(I_{dc}\) and hence slip.

SECTION 03

Electromagnetic Torque — DC-Machine Analogy

General Torque Expression

Stator impedance neglected:

\[T_e = \frac{3(P/2)}{\omega_s} \Bigl[I_r'^{\,2}R_r' -\frac{n_t V_s\cos\alpha}{a}\cdot I_r'\Bigr]\cdot\frac{1}{s}\]

Neglecting rotor leakage \(X_r'\) (valid for small operating slips) and using the 6-pulse diode bridge relation \(I_r' = (\sqrt{6}/\pi)\,a\,I_{dc}\):

⚡

DC-Machine Analogy — Core Result

Torque Proportional to DC-Link Current

\[\boxed{T_e \approx K_t\,I_{dc}}\]

\[K_t = \frac{3\sqrt{6}}{\pi}\cdot a\cdot\frac{P/2}{\omega_s}\cdot\frac{L_m}{L_s} \quad[\mathrm{N\,m\,A}^{-1}]\]

Air-Gap Flux Interpretation and Practical Significance

\[\Psi_m \approx \frac{E_s}{\omega_s} \approx \text{const.}\]

The stator supply voltage holds the air-gap flux approximately constant — analogous to the field flux in a DC machine.

Practical significance:

Simple single-variable torque control.
No coordinate transformation required.
No three-phase current regulation needed.
Controller design mirrors a standard phase-controlled DC drive.
Only one current transducer (\(I_{dc}\)) needed in the inner loop.

SECTION 04

SER Drive — Performance and Efficiency

Efficiency Comparison at \(\omega_r = 0.6\;\text{p.u.}\)

SER drive torque-speed curves at different inverter firing angles showing high efficiency operation maintained across the speed range — SER drive torque–speed curves at different inverter firing angles

Key performance results:

High motor power factor maintained across the speed range (motor operates near rated flux at all speeds).
Recovered slip power increases as speed decreases \(\Rightarrow\) efficiency greatly improved compared with phase control.

Efficiency comparison of drive schemes at 60% speed
Drive Scheme	Efficiency \(\eta\)
Slip-energy recovery	\(\approx 86\,\%\)
Phase control	\(\approx 60\,\%\)
External resistance	\(\approx 45\,\%\)

Take-away: SER achieves \(\approx 86\,\%\) efficiency at \(60\,\%\) speed — more than 25 percentage points better than phase control alone.

SECTION 05

Harmonic Currents and Sixth-Harmonic Pulsating Torque

Fourier Expansion of Rotor Current (6-Pulse Diode Bridge)

\[i_r(t) = \frac{2\sqrt{3}}{\pi}I_{dc} \Bigl[\cos\omega t -\tfrac{1}{5}\cos 5\omega t +\tfrac{1}{7}\cos 7\omega t -\cdots\Bigr]\]

Characteristic orders: \(n = 6k \pm 1\), \(k = 1,2,3,\ldots\) — Magnitudes: \(I_{r,n} = I_{r,1}/n\).

Mechanism of sixth-harmonic pulsating torque:

The 5th harmonic rotates backward at \(5\omega_s\); its speed relative to the fundamental stator flux is \(6\omega_s\).
The 7th harmonic rotates forward at \(7\omega_s\); its relative speed is also \(6\omega_s\).
Both together produce a torque pulsation at \(6\omega_s\) (sixth harmonic of supply frequency).

Sixth-harmonic pulsating torque ratio:

\[\left|\frac{T_{6h}}{T_e}\right| \approx \frac{2}{3}\,X_{\ell,\,\mathrm{pu}}\]

Large machines: \(X_\ell \leq 2\,\%\) p.u. \(\Rightarrow\) \(|T_{6h}| \leq 1.3\,\%\,T_e\)
Small machines: \(X_\ell \leq 5\,\%\) p.u. \(\Rightarrow\) \(|T_{6h}| \leq 3.3\,\%\,T_e\)

Sixth-harmonic torque causes minor vibration and acoustic noise. For fan and pump loads this is negligible in practice.

DC-link choke \(L_{dc}\): Sized to suppress sixth-harmonic ripple in the DC-link current, keeping \(I_{dc}\) smooth and torque ripple small.

SECTION 06

Individual Harmonic Torques — 5th and 7th Orders

Average Torque Components from 5th and 7th Harmonic Currents

Fifth-harmonic slip and torque ratio:

The 5th harmonic current rotates backward; its effective slip referred to synchronous speed is:

\[s_5 = \frac{6-s}{5} \approx \frac{6}{5} \quad(\text{for small }s)\]

The 5th harmonic produces a braking average torque; its ratio is:

\[\left|\frac{T_{e5}}{T_e}\right| \approx \frac{1}{25}\cdot\frac{s}{|s-6|} \;\ll 1\,\%\]

Seventh-harmonic: The 7th harmonic rotates forward; its effective slip is \(s_7 = (s-6)/7 \approx -6/7\). It produces a small forward average torque, even smaller in magnitude than the 5th.

Key conclusion: Both 5th and 7th individual harmonic average torques remain below 1% of rated torque at all operating slips. Entirely negligible for fan, pump, and compressor loads.

Summary of Harmonic Effects

6th-harmonic pulsating torque: \(\leq 3.3\,\%\) for typical machines.
Individual 5th/7th average torques: \(< 1\,\%\) of \(T_e\).
Dominant practical concerns: additional motor copper and iron losses from harmonic currents; DC-link current ripple (mitigated by \(L_{dc}\)); supply-side line current harmonics.

SECTION 07

Supply-Side Harmonic Distortion and Mitigation

Origin of Supply-Side Harmonics

Both the diode bridge and the SCR inverter are 6-pulse converters drawing non-sinusoidal currents from the AC supply. The characteristic harmonic orders are:

\[n = 6k \pm 1, \quad k = 1, 2, 3, \ldots \;\Rightarrow\; 5,\,7,\,11,\,13,\,17,\,19,\ldots\]

with magnitudes \(I_n \approx I_1 / n\) for an ideal quasi-square waveform.

Effects on the supply system:

Voltage distortion at the point of common coupling (PCC).
Interference with other sensitive loads sharing the same bus.
Additional losses in supply transformers and cables.
THD may violate grid codes (e.g. IEEE 519, IEC 61000-3-2).

Passive Mitigation Strategies

AC-line reactor (3–5% impedance): reduces harmonic current amplitude and protects SCRs from supply voltage spikes.

12-pulse configuration: Two 6-pulse bridges fed from a dual-secondary transformer with 30° phase shift. Cancels 5th and 7th harmonics; only \(11,\,13,\,\ldots\) remain.

Tuned passive filter: LC branch tuned to the dominant harmonic (usually 5th); also provides reactive power compensation simultaneously.

📐

12-Pulse Advantage

THD Reduction Without Active Electronics

Current THD at PCC reduces from \(\approx 29\,\%\) (6-pulse) to \(\approx 10\,\%\) (12-pulse) with no active electronics.

SECTION 08

Transformer Turns-Ratio Design Principle

Design Constraint and Turns-Ratio Selection

The speed equation \(s = (n_t V_s/V_r)\,|\cos\alpha|\) must be satisfied across the entire speed range with \(\alpha\) constrained to:

\[90° < \alpha \leq \alpha_{\max} \approx 155°\]

Selecting \(n_t\):

At minimum speed (\(s = s_{\max}\), \(\alpha = \alpha_{\max}\)):

\[n_t = \frac{s_{\max}\,V_r}{V_s\,|\cos\alpha_{\max}|}\]

Larger \(n_t\) widens speed range but increases converter VA rating.
Smaller \(n_t\) reduces converter cost but narrows the controllable speed band.

Design principle: \(n_t\) is chosen as the smallest value that achieves the required \(s_{\max}\) while keeping \(\alpha \leq \alpha_{\max}\). This simultaneously minimises transformer apparent power rating, SCR inverter voltage rating, and no-load reactive current.

Practical speed-range rule:

\[s_{\max} = 1 - \frac{\omega_{\min}}{\omega_s}\]

A 70–100% speed range gives \(s_{\max} = 0.30\), so the converter is rated at 30% of motor power.

SECTION 09

Starting Procedure and Converter Ratings

The Starting Problem and Auxiliary Rotor-Resistance Solution

At standstill (\(s = 1\)): all air-gap power flows through the rotor as slip power. If the SER converter must handle this, its rating equals the full motor rating — defeating the economic advantage of the scheme.

Solution: Auxiliary rotor-resistance starting

Insert external resistors in the rotor circuit at standstill; gate pulses to the SER converter are blocked.
Motor accelerates on its wound-rotor characteristic with resistance control.
When speed reaches \(\omega_{\min}\) (lower edge of the SER speed range), rotor resistors are switched out.
SER converter takes over and regulates speed within its designed range.

Thermal note: Starting resistors absorb the full rotor energy during acceleration. Liquid cooling is mandatory for large machines (MW class).

Converter Ratings at Maximum Slip \(s_{\max}\)

\[P_{\text{conv}} = s_{\max}\,P_{ag}\]

\[V_{D,\mathrm{pk}} = \sqrt{2}\,V_r^{(L)},\quad I_{D,\mathrm{rms}} \approx \sqrt{\tfrac{2}{3}}\,I_{dc}\]

\[V_{\mathrm{SCR,pk}} = \sqrt{6}\,n_t V_s \approx 2.45\,n_t V_s\]

\[I_{\mathrm{SCR,avg}} = \tfrac{1}{3}\,I_{dc}\]

Economic benefit: For \(s_{\max} = 0.30\), converters rated at only 30% of motor power. Significant cost reduction vs. a full-rated variable-frequency drive. Primary economic justification for SER drives in large pump systems.

SECTION 10

Closed-Loop Control — SER Drive

Cascade \(I_{dc}\)–Speed Controller

Inner Loop — \(I_{dc}\) (Torque) Control

Since \(T_e \propto I_{dc}\), current control is equivalent to torque control.
PI controller output is the SCR firing angle \(\alpha\).
Firing angle clamped at \(\alpha \leq \alpha_{\max} \approx 155°\) to preserve the commutation margin.
Fast response; time constant \(\tau \approx L_{dc}/R_{dc}\).

Outer Loop — Speed Regulation

PI controller generates \(I_{dc}^*\) (current/torque demand).
Structure mirrors a phase-controlled DC drive — same design methodology applies directly.
One-quadrant operation only (subsynchronous motoring); PI output floored at zero.
Soft-start limiter on \(\omega_r^*\) prevents over-current during run-up.

SECTION 11

Summary and Preview

What We Covered

Stator-referred equivalent circuit with referred back-EMF of the SCR inverter.
\(T_e = K_t I_{dc}\): SER torque is structurally identical to a DC machine.
SER efficiency \(\approx 86\,\%\) at \(60\,\%\) speed vs. \(\approx 60\,\%\) for phase control.
6th-harmonic pulsating torque: \(\leq 3.3\,\%\,T_e\) for typical machines.
5th/7th individual average harmonic torques: \(< 1\,\%\) of \(T_e\).
Supply-side harmonics (\(n = 6k\pm1\)): mitigated by line reactors, 12-pulse topology, or tuned passive filters.
Turns ratio \(n_t\) chosen to cover the required \(s_{\max}\) at minimum converter rating.
Auxiliary resistor starting required; liquid cooling for MW-class machines.
Cascade \(I_{dc}\)–\(\omega_r\) controller mirrors DC drive design.

Preview — Lecture 6D

Historical context: Kramer and Scherbius drives
Static Scherbius drive: bidirectional converter for four-quadrant operation
Supersynchronous operation: power flow and speed range
Industrial application: large pump drives and affinity laws
Wind energy: DFIG concept, MPPT and reactive power control
Unified comparison: all three drive types side by side