VSI-Fed IM: Steady-State Performance & V/f Control | Frequency-Controlled IM Drives

SECTION 01

Learning Outcomes

After this lecture you will be able to:

Use the per-phase equivalent circuit of the IM to derive air-gap power, torque, and maximum torque expressions.
Explain why maintaining a constant \(V_s/f_s\) ratio preserves rated torque capability across the entire speed range.
Sketch the V/f voltage command profile, identifying the constant-torque region, the low-speed IR-boost, and the field-weakening region.
Describe the closed-loop slip-speed control scheme and state how it improves upon open-loop V/f.
Explain constant air-gap flux control and compare it with simple V/f control in terms of accuracy and complexity.
Apply the V/f design equations to a given motor specification.

SECTION 02

Per-Phase Equivalent Circuit of the Induction Motor

Steady-State T-Equivalent Circuit (per phase, fundamental)

Per-phase T-equivalent circuit of the induction motor showing stator resistance Rs, stator leakage Xls, magnetising reactance Xm, and referred rotor branch Rr/s with Xlr — Per-phase T-equivalent circuit of the induction motor (fundamental frequency)

\(R_r/s\) is the total referred rotor resistance; mechanical power is extracted from \(R_r(1-s)/s\)
\(R_c\) represents core (iron) losses; often omitted in drive analysis for simplicity
This circuit is valid for the fundamental frequency only — harmonics require separate circuits (see Lecture 7D)
The stator voltage \(V_s\) drives current through the stator impedance (\(R_s + jX_{ls}\)) and the parallel combination of the magnetising branch and rotor branch

SECTION 03

Power & Torque Relations

Air-Gap Power, Rotor Losses, and Electromagnetic Torque

Power Flow Relations

\[P_{ag} = 3\,I_r^2\,\frac{R_r}{s} \quad \text{(air-gap power)}\]

\[P_{cu,r} = s \cdot P_{ag} \quad \text{(rotor copper loss)}\]

\[P_{mech} = (1-s) \cdot P_{ag} \quad \text{(mechanical output power)}\]

\[T_e = \frac{P_{ag}}{\omega_{s,mech}}\]

where \(\omega_{s,mech} = 4\pi f_s / P\) is the synchronous mechanical speed.

Electromagnetic Torque (exact expression)

\[T_e = \frac{3P}{2\,\omega_s} \cdot \frac{|E_1|^2\,(R_r/s)}{(R_r/s)^2 + X_{lr}^2}\]

Valid when stator resistance and leakage are neglected (\(E_1 \approx V_s\)). The air-gap EMF \(E_1\) is the voltage across the magnetising branch.

Maximum (Breakdown) Torque

\[T_{max} = \frac{3P}{4\,\omega_s} \cdot \frac{|E_1|^2}{X_{lr}}, \qquad s_{T_{max}} = \frac{R_r}{X_{lr}}\]

The slip at maximum torque \(s_{T_{max}}\) is independent of stator voltage — only rotor parameters matter.

Physical Insight: Flux Linkage and Torque

Key Relationship: Air-Gap Flux and Maximum Torque

\[T_{max} \propto \frac{|E_1|^2}{\omega_s^2} \propto \lambda_m^2\]

Keeping the air-gap flux linkage \(\lambda_m = \text{const}\) is equivalent to keeping \(|E_1|/\omega_s = \text{const}\). This ensures rated torque capability at every speed. Neglecting stator impedance: \(E_1 \approx V_s\), so \(V_s/f_s = \text{const}\) achieves this approximately.

SECTION 04

Constant Volts-per-Hertz (V/f) Control

Why Maintain Constant V/f?

Air-gap flux linkage: \(\lambda_m \propto E_1/\omega_s\)
Neglecting stator impedance drop: \(E_1 \approx V_s\)
Keeping \(V_s/f_s = \text{const} \;\Rightarrow\; \lambda_m = \text{const} \;\Rightarrow\; \text{rated torque at all speeds}\)
Without this: flux collapses at low speed → weak motor, very poor torque production

V/f Voltage Command (peak phase voltage)

\[V_s^* = V_o + K_{vf}\,f_s^*\]

where \(K_{vf} = \hat{V}_{ph,rated}/f_{rated}\) is the V/Hz slope and \(V_o \approx I_{rated}\,R_s\) is the low-speed IR-drop boost voltage.

Without \(V_o\): the stator resistance drop at low frequencies consumes a large fraction of \(V_s\), leaving insufficient voltage across the magnetising branch → flux collapses → torque drops to near zero at low speeds.

DC Bus Voltage Requirement for Six-Step VSI

Relationship Between \(V_{dc}\) and Rated Phase Voltage

For the six-step VSI: \(\hat{V}_{as,1} = \dfrac{2}{\pi}V_{dc} \approx 0.637\,V_{dc}\). Therefore:

\[V_{dc} = \frac{\hat{V}_{ph,rated}}{0.637} = \frac{\sqrt{2}\,V_{ph,rms}}{0.637}\]

For a PWM-VSI, the DC bus voltage must be at least this value. In practice, a diode rectifier from the AC supply provides a fixed \(V_{dc}\), and PWM adjusts the effective output voltage.

SECTION 05

V/f Operating Profile: Constant Torque and Field-Weakening Regions

Two Operating Regions

V/f voltage profile: constant-torque and field-weakening regions

Region 1: Below Base Frequency \(f_b\) — Constant Torque

\(V_s/f_s = K_{vf}\) (constant slope)
\(\lambda_m = \text{const}\)
\(T_{max} = \text{const}\) (constant torque capability)
Low-speed boost \(V_o\) applied to compensate \(R_s\) drop

Region 2: Above \(f_b\) — Field Weakening (Constant Power)

\(V_s = V_{rated}\) (voltage clamped at rated value)
\(\lambda_m \propto 1/f_s\) (flux decreasing)
\(T_{max} \propto 1/f_s^2\) (torque capability falls)
Constant-power region: \(P = T_e \cdot \omega_r \approx \text{const}\) if \(T_e \propto 1/\omega_r\)

SECTION 06

Closed-Loop V/f: Constant Slip-Speed Control

Limitation of Open-Loop V/f

Pure V/f has no speed feedback
Speed regulation error of ±2–5% under load variation
At sudden load increase: motor slows but inverter frequency stays fixed → slip grows uncontrolled
Risk of exceeding breakdown torque → stalling

Slip-Speed Control Law

Closed-loop slip-speed control block diagram

Frequency Synthesis

\[\omega_s^* = \omega_r + \omega_{sl}^* \quad \text{(stator frequency command)}\]

\[\omega_{sl}^* = K_{sl}\,T_e^* \quad \text{(slip-speed command proportional to torque)}\]

\(\omega_{sl}^*\) is limited to \(\omega_{sl,max} = R_r/L_{lr}\) to prevent stalling. The speed measurement \(\omega_r\) comes from a shaft encoder or speed observer.

Improvements over Open-Loop V/f

Speed regulation improved to <1%
Faster dynamic response to load transients
Slip limiter prevents breakdown and stalling
Better low-speed torque capability
Still scalar — flux and torque not independently regulated

SECTION 07

Constant Air-Gap Flux Control

Why Go Beyond Simple V/f?

Fixed V/f ignores the stator impedance drop \(\mathbf{I}_s(R_s + jX_{ls})\)
At low frequencies this drop is significant relative to \(V_s\); air-gap flux is not truly constant
Air-gap EMF from KVL: \(E_1 = V_s - I_s(R_s + jX_{ls})\)
Solution: measure \(I_s\) in real time and compensate the voltage command accordingly

Constant-\(\lambda_m\) Voltage Command

Desired air-gap EMF: \(E_1^* = \lambda_m^*\,\omega_s\)

\[\mathbf{V}_s^* = \lambda_m^*\,\omega_s + \mathbf{I}_s\,(R_s + jX_{ls})\]

More accurate than fixed \(V_o\) boost, especially at very low speeds (\(f_s < 5\) Hz). Requires real-time current measurement each PWM cycle.

Key Result and Transition to FOC

With Constant Air-Gap Flux Control

Peak torque is nearly equal at all frequencies, providing excellent speed range and low-speed torque. This constant-flux scalar strategy leads naturally to Field-Oriented Control (FOC), which achieves full decoupling of the flux and torque channels and is covered in Chapter 8.

SECTION 08

Comparison of Scalar V/f Control Strategies

Three Strategies Side-by-Side

Comparison of scalar V/f control strategies for VSI-fed induction motor drives
Feature	Const. V/Hz	Const. Slip Speed	Const. \(\lambda_m\)
Speed feedback	None	Speed sensor	\(V_s\) and \(I_s\) sensors
Speed accuracy	Low (±2–5%)	Medium (<1%)	High
Flux control	Approximate	Approximate	Exact (real-time compensation)
Low-speed torque	Poor	Better	Best
Complexity	Low	Medium	High
Cost	Lowest	Medium	Higher
Best application	Fans, pumps	General purpose	High performance

Limitation of All Scalar Methods

All scalar (V/f) methods regulate magnitude only — flux and torque cannot be independently controlled. Dynamic torque-step response is slow (typically >50 ms). For fast dynamics, use Field-Oriented Control (FOC) or Direct Torque Control (DTC) — covered in Chapter 8.

SECTION 09

Design Example: V/f Drive for a 5 hp Motor

Motor Data and Design Steps

Given Motor Parameters

5 hp, 200 V (L-L), 60 Hz, 3-phase, 4-pole
\(R_s = 0.277\,\Omega\), \(X_{ls} = 0.554\,\Omega\), \(X_m = 20.3\,\Omega\), \(R_r = 0.183\,\Omega\), \(X_{lr} = 0.841\,\Omega\)

Step 1: V/f Constant (peak phase basis)

\[K_{vf} = \frac{\hat{V}_{ph,rated}}{f_{rated}} = \frac{\sqrt{2} \times 115.5}{60} = 2.72 \text{ V/Hz}\]

Step 2: Low-speed boost voltage

\[V_o \approx I_{rated}\,R_s \approx 27.6 \times 0.277 \approx 7.7 \text{ V} \;\;(\approx 6.7\%)\]

Step 3: Maximum allowable slip speed

\[\omega_{sl,max} = \frac{R_r}{L_{lr}} = \frac{0.183}{0.841/377} \approx 82 \text{ rad/s}\]

Step 4: DC bus voltage at rated output

\[V_{dc} = \frac{\sqrt{2} \times V_{ph,rms}}{0.637} = \frac{\sqrt{2} \times 115.5}{0.637} \approx 256 \text{ V}\]

Step 5: Slip at maximum torque

\[s_{T_{max}} = \frac{R_r}{X_{lr}} = \frac{0.183}{0.841} \approx 0.22\]

Design Summary

\(K_{vf} = 2.72\) V/Hz | \(V_o = 7.7\) V | \(V_{dc} = 256\) V | \(\omega_{sl,max} = 82\) rad/s | \(s_{T_{max}} = 0.22\)