Induction Motor Drive Performance: Load Interaction, Power Factor, Closed-Loop Control & Efficiency

SECTION 01

Lecture 6B — Scope and Objectives

Lecture 6A Recap

Air-gap power partition: rotor copper loss \(= s\,P_{ag}\); ceiling \(\eta\leq(1-s)\).
SCR waveforms: non-sinusoidal for \(\alpha>\phi\); harmonics worsen with \(\alpha\).
Firing angle \(\alpha\) reduces rms stator voltage; control only for \(\alpha > \phi\).
Stability criterion: operating point must lie to the right of peak torque (low-slip side).
Conduction angle \(\beta\): transcendental equation (Newton–Raphson).
Open-loop speed range \(\approx 6\,\%\) for fan load.

This Lecture

Load torque models and rotor current profile vs. speed
Stator current vs. speed: thermal design
Input power factor and reactive power of the phase-controlled drive
Closed-loop cascade PI controller design
Efficiency: phase control vs. external resistance
Practical drive applications
Transition: SER principle and power circuit

SECTION 02

Load Torque Models and Rotor Current Profile

General Load Torque Model: \(T_L = B_L\,\omega_r^{\,k}\)

Load torque exponent \(k\) and corresponding load types
\(k\)	Load Type
\(0\)	Constant torque (press, crane hoist)
\(1\)	Viscous (linear) friction
\(2\)	Centrifugal load (pump, fan)

Rotor current at each operating point — using the approximation \(I_r \propto T_e / \omega_r\) at low slip:

\[I_r \propto \frac{B_L\,\omega_r^k}{\omega_r} = B_L\,\omega_r^{k-1}\]

Location and relative magnitude of peak rotor current for different load types
Load Type	\(I_{r,\max}\) Slip	Relative Magnitude
\(k=1\) (viscous)	\(s = 1/3\)	intermediate
\(k=2\) (pump/fan)	\(s = 1/5\)	lower

Stator current (phasor sum):

\[\dot{I}_s = \dot{I}_r + \dot{I}_m, \qquad \dot{I}_m = \frac{E_{ag}}{jX_m}\]

Key design implication: Peak stator current occurs at an intermediate speed, not at standstill. The current limiter must be set at or below this peak to protect motor windings and SCR devices.

Thermal constraint: Phase control introduces harmonic currents that produce additional winding losses with no useful average torque. The rms stator current — not its fundamental alone — governs the thermal rating.

SECTION 03

Thermal Limits and NEMA D Motor Selection

Feasible Closed-Loop Speed Range

The region where \(I_s \leq I_{s,\max}\) defines the safe operating band.
Below a minimum speed, \(I_s\) rises back above the limit even under closed-loop control; the drive must not be operated below \(\omega_{\min}\).
Typical usable range: 70–100% of synchronous speed (dependent on load type).

NEMA D motor advantage:

High \(R_r\) \(\Rightarrow\) peak torque near \(s \approx 1\) (near standstill).
Larger stable region on the torque–speed curve.
Peak rotor current reduced and occurs at lower speed.
Trade-off: higher rotor \(I^2R\) losses at full load; lower full-load efficiency compared with NEMA B.

Stator current versus speed curves comparing NEMA B and NEMA D motors showing wider safe operating band for NEMA D in phase-controlled drives — Stator current vs. speed — NEMA B vs. NEMA D; NEMA D keeps \(I_s\) below the rated limit over a wider speed range

SECTION 04

Input Power Factor of the Phase-Controlled Drive

Why Power Factor is Degraded

Stator current \(i_s(t)\) is non-sinusoidal.
Displacement power factor \(\cos\phi_1\): phase lag of the fundamental current component relative to supply voltage.
Distortion factor \(D_f\): accounts for harmonic content.

\[D_f = \frac{I_{s1}}{I_{s,\text{rms}}} \leq 1\]

\[\mathrm{PF} = D_f \cdot \cos\phi_1 \leq \cos\phi_1\]

Consequence: As \(\alpha\) increases (speed reduced), both \(\cos\phi_1\) and \(D_f\) decrease simultaneously. A drive at 50% speed can draw a total power factor as low as 0.3–0.4.

Mitigation measures:

Shunt power-factor correction capacitors at motor terminals.
AC line reactors to reduce harmonic current injection to the supply.
12-pulse converter topology for large drives (Lecture 3).

Input power factor versus firing angle alpha showing displacement power factor and total power factor both decreasing as alpha increases in phase-controlled IM drive — Input power factor vs. firing angle \(\alpha\): displacement PF and distortion factor both degrade with increasing \(\alpha\)

SECTION 05

Closed-Loop Cascade Controller

Cascade PI Speed–Current Controller

Inner Current Loop (Fast)

Regulates stator rms current to reference \(I_s^*\).
Protects SCR devices and motor windings from over-current.
Bandwidth set several times larger than the outer speed loop.

Outer Speed Loop (Slow)

PI controller converts speed error to current demand \(I_s^*\).
Soft-start ramp limits \(\omega_r^*\) to control acceleration current.
Limiter prevents \(I_s^*\) from exceeding \(I_{s,\max}\).
Enables stable operation in open-loop-unstable high-slip regions.

SECTION 06

Controller Design Principles — Plant Model

Inner Loop Plant and PI Tuning

The motor stator winding is modelled as a first-order electrical lag:

\[G_I(s) = \frac{K_I}{1 + s\,T_I}, \qquad T_I = \frac{L_{eq}}{R_{eq}}\]

The PI current controller:

\[C_I(s) = K_{pc}\!\left(1+\frac{1}{s\,T_{ic}}\right)\]

tuned so that the inner closed-loop bandwidth \(\omega_{c,I} \gg \omega_{c,\omega}\) (speed bandwidth).

Outer loop plant (speed): Once the inner loop is closed and fast, the mechanical sub-system is approximated as an integrator:

\[G_\omega(s) \approx \frac{K_t}{J\,s} \qquad (T_e \approx K_t\,I_s^*,\;J = \text{inertia})\]

Design Rules

Inner loop bandwidth \(\geq 5\times\) outer loop bandwidth to justify the cascade approximation.
Inner integral time \(T_{ic} \approx T_I\) (pole-zero cancellation).
Outer integral time set by required speed-error settling time.
Anti-windup on both PI stages to handle saturation during run-up.

Practical Note

Exact pole-zero cancellation is not achievable in practice; however the cascade structure is robust to parameter variations because the inner loop largely masks motor impedance changes from the outer loop.

SECTION 07

Efficiency of the Phase-Controlled Drive

Efficiency Expressions and Comparison

Approximate upper bound (stator resistance and core losses neglected):

\[\eta \;\approx\; 1-s \;=\; \frac{\omega_r}{\omega_s}\]

Complete efficiency expression:

\[\eta = \frac{T_e\,\omega_r}{P_{ag} + 3I_s^2 R_s + P_{\text{core}}}\]

Phase control vs. external resistance:

Resistance control: slip power dissipated in external resistors; motor rotor losses are lower.
Phase control: all slip power dissipated inside the rotor — motor runs hotter.
Both share the same fundamental efficiency ceiling.

Efficiency versus speed comparison curves for phase control, external resistance control and slip-energy recovery showing SER achieving significantly higher efficiency at reduced speeds — Efficiency vs. speed: phase control vs. external resistance vs. SER

Fundamental limitation: Both methods share the ceiling \(\eta \leq (1-s)\). To surpass this, slip power must be returned to the supply — topic of Lecture 3.

SECTION 08

Applications of Phase-Controlled IM Drives

Variable-Speed Drives (Closed-Loop)

Centrifugal pumps and fans (30–100% speed range)
Compressors and blowers
Extruders and injection-moulding machines
Mine hoists, ski-lifts, conveyors
Wire-drawing and textile machinery

Soft-Starters

Limit starting inrush to \({\leq}\,3{\times}I_{\text{rated}}\)
Replaces star-delta starters: smoother torque, less mechanical shock
Solid-state — no moving contacts

Other Industrial Uses

Resistance heating furnaces
Light dimmers and UV-curing lamps
AC voltage regulators for sensitive loads

💡

Why Phase Control Remains Relevant

Cost-Effective for Limited Speed Ranges

Despite being largely superseded by PWM-inverter drives for new installations, phase-controlled drives remain cost-effective for large-motor applications where the required speed range is limited (\({\leq}\,30\,\%\) of rated speed).

SECTION 09

Why Recover Slip Energy? — Principle

Slip-Energy Recovery Principle

In every slip-controlled induction-motor drive:

\[P_{cu,r} = s\,P_{ag}\]

This fraction of air-gap power is dissipated as heat. Can it instead be returned to the supply?

SER principle: Insert a controllable back-EMF source \(E_r\) in the rotor circuit. It extracts \(s\,P_{ag}\) from the rotor; a power converter feeds this energy back to the AC supply. Speed is set by adjusting \(|E_r|\).

\[\eta_{\text{phase ctrl}} \approx (1-s) \;\xrightarrow{\text{with recovery}}\; \eta_{\text{SER}} \gg (1-s)\]

Machine requirement: The rotor circuit must be externally accessible \(\Rightarrow\) Wound-rotor IM with slip rings. Squirrel-cage machines cannot be used.

SER power circuit diagram showing wound-rotor induction motor with diode bridge rectifier, DC-link choke inductor, SCR line inverter and isolating transformer returning slip power to the AC supply — SER power circuit: diode bridge rectifier + DC-link choke + SCR line inverter + isolating transformer

The converter is rated at only \(s_{\max}\times P_{\text{motor}}\) — a significant cost advantage for limited speed ranges.

SECTION 10

Slip-Energy Recovery — DC-Link Voltage Balance

DC-Link Voltages (6-Pulse Bridges)

Diode bridge output: \(V_{dc} = 1.35\,s V_r\) (referred to rotor side)
SCR line-inverter voltage (\(\alpha > 90°\), inverting): \(V_{inv} = -1.35\,n_t V_s\,|\cos\alpha|\)

Voltage balance in steady state (neglecting DC-link resistance drop):

\[V_{dc} + V_{inv} = 0 \;\Longrightarrow\; 1.35\,s V_r = 1.35\,n_t V_s\,|\cos\alpha|\]

\[\boxed{s = \frac{n_t\,V_s}{V_r}\,|\cos\alpha|}\]

Speed is set by the firing angle \(\alpha\) of the SCR inverter (\(\alpha > 90°\)).

Practical Control Range

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

The upper limit of \(\alpha\) is set by the commutation margin angle required to reliably turn off the SCR inverter thyristors.

Key Cost Advantage

For a 30% speed range, the converter is rated at only 30% of the motor nameplate power — far cheaper than a full-rated variable-frequency drive.

Sign convention note: For \(\alpha > 90°\) the SCR bridge operates in inverter mode (negative average voltage), which opposes the diode-bridge output and controls the DC-link current, hence the slip.

SECTION 11

Summary and Preview

What We Covered

\(T_L = B_L\omega_r^k\): rotor current peaks at intermediate slip, not at standstill.
Thermal rating governed by rms stator current over the full speed range.
Power factor degrades with \(\alpha\): both displacement PF and distortion factor fall; shunt capacitors or line reactors are needed.
Cascade PI controller: inner current loop + outer speed loop + soft-start.
\(\eta \approx (1-s)\): phase control more efficient than resistor control, but both are fundamentally bounded.
SER principle: wound-rotor IM + diode bridge + DC choke + SCR inverter + transformer.
Speed equation: \(s = (n_t V_s / V_r)|\cos\alpha|\).

Preview — Lecture 6C

Per-phase equivalent circuit of the SER drive (stator-referred)
Key result: \(T_e = K_t\,I_{dc}\) (DC-machine analogy)
Torque–speed performance and efficiency
Harmonic currents: sixth-harmonic pulsating torque
Supply-side harmonic distortion and mitigation
Starting procedure and converter ratings
Closed-loop control of the SER drive