MOTOR-04 — Back EMF, Inductance and Resistance

ASSUMPTION — This lesson examines the three fundamental electrical parameters governing an electric machine as an electrical load driven by a high-power inverter: Back Electromotive Force (Back EMF), stator phase resistance ($R_s$), and phase inductances ($L_d, L_q$). These parameters form the core inputs for FOC current regulator tuning and inverter voltage limit analysis.

In this lesson, you will learn:

• The physical origin of Back EMF generated in stator coils by rotating permanent magnet flux
• Stator resistance ($R_s$) thermal dynamics and how winding heating alters current regulation
• The distinction between direct ($L_d$) and quadrature ($L_q$) axis inductances, and the saliency ratio ($\xi = L_q / L_d$)
• The machine dynamic voltage equations and the maximum modulation ceiling imposed by DC bus voltage
• Functional safety hazards surrounding Uncontrolled Generator Operation (UGO) during high-speed inverter faults

1. Back Electromotive Force (Back EMF)

By Faraday’s law of electromagnetic induction, a time-varying magnetic flux linking a coil induces an opposing voltage across its terminals:

$e = -N \cdot \frac{d\Phi}{dt}$

In a permanent magnet machine (PMSM/BLDC), rotating rotor magnets produce an alternating magnetic flux linkage through the stationary stator teeth. This generates a counter-voltage in the windings proportional to rotational speed, termed Back EMF:

$E_{rms} = K_e \cdot \omega_e = K_e \cdot (p \cdot \omega_m)$

Where:

• $K_e$: Back EMF voltage constant ($V_{rms} / (\text{rad/s})$)
• $\omega_e$: Electrical angular frequency ($\text{rad/s}$)
• $\omega_m$: Rotor mechanical angular velocity ($\text{rad/s}$)
• $p$: Pole-pair count of the machine

FACT — Back EMF directly opposes the terminal voltage impressed by the inverter. As the rotor accelerates, Back EMF scales linearly. To drive current into the motor, the inverter must apply a terminal voltage exceeding the instantaneous Back EMF ($V_{inv} > E$). When Back EMF approaches the maximum voltage deliverable by the inverter under the prevailing DC bus, the machine reaches its Base Speed.

2. Stator Resistance (Rs) and Copper Temperature Dynamics

Stator winding resistance ($R_s$) is a function of total copper conductor length, active cross-sectional area, and operating temperature:

$R_s(T) = R_0 \cdot [1 + \alpha_{Cu} \cdot (T - T_0)]$

• $R_0$: Phase resistance at $20^\circ\text{C}$ reference (typically $10 - 50\text{ m}\Omega$ in modern EV traction stators)
• $\alpha_{Cu} \approx +0.00393 / ^\circ\text{C}$: Temperature coefficient of resistance for copper
• $T$: Actual winding temperature

EXAMPLE — A hairpin stator with $R_0 = 20\text{ m}\Omega$ at $20^\circ\text{C}$ operating under severe sustained hill-climbing reaches $140^\circ\text{C}$: $R_s(140^\circ\text{C}) = 20 \cdot [1 + 0.00393 \cdot (140 - 20)] = 29.43\text{ m}\Omega$ Phase resistance increases by 47.2%. At $300\text{ A}$ RMS phase current, total 3-phase resistive dissipation ($3 \cdot I^2 R$) climbs from $5.4\text{ kW}$ to $7.9\text{ kW}$. Motor control algorithms must continuously adapt current controller PI gains to compensate for this thermal shift.

3. Machine Inductances: Ld and Lq Saliency

Inductance quantifies the magnetic flux linkage established per unit of stator current ($L = \lambda / i$). In salient machines, geometric asymmetry in the rotor causes magnetic permeance to vary with rotor angle:

• d-Axis (Direct Axis): Aligned with the magnetic pole of the permanent magnet. Sintered NdFeB magnets have a relative permeability near unity ($\mu_r \approx 1.05$), behaving magnetically like an extended air gap. Magnetic reluctance along the d-axis is high.
• q-Axis (Quadrature Axis): Positioned between adjacent magnet poles, passing entirely through high-permeability laminated rotor steel. Reluctance along the q-axis is significantly lower.

Because inductance is inversely proportional to reluctance ($L \propto 1/\mathcal{R}$):

$L_q > L_d$

• Saliency Ratio ($\xi$): $\xi = \frac{L_q}{L_d}$
• Surface PM Machine (SPMSM): $L_d \approx L_q \implies \xi \approx 1.0$ (Non-salient)
• Interior PM Machine (IPMSM): $L_q \approx 1.5 - 2.5 \cdot L_d \implies \xi \approx 1.5 - 2.5$ (Strong saliency)

FACT — The condition $L_q > L_d$ enables IPMSMs to produce substantial Reluctance Torque alongside permanent magnet torque (MOTOR-11). FOC control firmware exploits this through Maximum Torque Per Ampere (MTPA) vector control, delivering 20%–30% higher torque for equivalent current.

4. Equivalent Circuit and DC Bus Voltage Ceiling

In the synchronously rotating d-q reference frame, the machine terminal voltage equations are:

$V_d = R_s \cdot I_d + L_d \cdot \frac{dI_d}{dt} - \omega_e \cdot L_q \cdot I_q$ $V_q = R_s \cdot I_q + L_q \cdot \frac{dI_q}{dt} + \omega_e \cdot (L_d \cdot I_d + \Psi_m)$

Where:

• $\Psi_m$: Permanent magnet flux linkage ($Wb$)
• $-\omega_e L_q I_q$ and $+\omega_e L_d I_d$: Cross-coupling speed voltages
• $\omega_e \Psi_m$: Fundamental Back EMF voltage term

Inverter Voltage Ceiling

Using Space Vector PWM (SVPWM), the maximum fundamental line-to-neutral voltage amplitude deliverable without overmodulation is:

$V_{max} = \frac{V_{dc}}{\sqrt{3}}$

Under steady-state operation, the synthesized terminal voltage vector cannot exceed this boundary:

$V_d^2 + V_q^2 \le V_{max}^2 = \left(\frac{V_{dc}}{\sqrt{3}}\right)^2$

FACT — As speed ($\omega_e$) rises, the $\omega_e \Psi_m$ term expands, pushing the voltage demand against $V_{max}$. To prevent current collapse and extend operating speed, the inverter injects negative d-axis current ($I_d < 0$). This demagnetizing current counteracts magnet flux $\Psi_m$, collapsing net back-EMF and allowing the rotor to double or triple its base speed within the same DC bus limit (MOTOR-19).

5. Uncontrolled Generator Operation (UGO) and Safety

In automotive functional safety (ISO 26262 ASIL-D), a primary hazard is high-speed inverter shutdown:

• If an inverter shuts down all active switching at 18,000 RPM (e.g., cruising at 200 km/h), the machine’s open-circuit Back EMF exceeds the nominal DC bus voltage: $V_{EMF,LL} = \sqrt{3} \cdot \omega_e \cdot \Psi_m > V_{dc}$
• The freewheeling body diodes of the inverter power modules act as an uncontrolled full-wave diode bridge, rectifying Back EMF directly into the vehicle’s DC bus.
• This creates severe overvoltage risks for DC link capacitors and battery cells, while generating massive uncontrolled regenerative braking torque that can destabilize the vehicle chassis.

FACT — To neutralize UGO hazards, automotive inverters incorporate hardware-level Active Short Circuit (ASC) protection. Upon fault detection, the gate driver stage commands either all three low-side or all three high-side switches to turn ON simultaneously. This shorts the motor terminals together, circulating Back EMF currents harmlessly within the stator windings and isolating the high-voltage DC bus.

6. Frequently Asked Questions (FAQ)

Why do $L_d$ and $L_q$ vary with operating current?

FACT — As stator current rises, the iron core enters magnetic saturation, reducing permeability. At peak torque, high $I_q$ current heavily saturates the q-axis flux paths, reducing $L_q$ by 30% to 50%. Inverter firmware utilizes 2D lookup tables ($L_d(I_d, I_q)$ and $L_q(I_d, I_q)$) to dynamically adjust decoupling and current controller gains in real time.

Why is stator resistance so low in traction machines (e.g., 20 mΩ)?

FACT — Automotive traction motors carry phase currents up to $500 - 800\text{ A}$ RMS. If phase resistance were $0.5\text{ }\Omega$, $500\text{ A}$ would generate $375\text{ kW}$ of heat, destroying the motor within seconds. Milliohm-level resistance is indispensable to achieve 96%+ efficiency and prevent thermal runaway.

7. Summary

• Back EMF is induced in stator coils by rotating magnet flux, scaling linearly with speed and opposing inverter applied voltage.
• Base speed occurs when Back EMF reaches the inverter’s maximum voltage output ceiling ($V_{dc} / \sqrt{3}$).
• Stator resistance ($R_s$) increases with temperature (+0.4%/°C), increasing copper losses under sustained high loads.
• In IPMSMs, $L_q > L_d$ creates magnetic saliency, enabling reluctance torque generation.
• Uncontrolled Generator Operation (UGO) at high speed is safely mitigated via hardware Active Short Circuit (ASC).

8. References and Verification Note

• Lipo, T.A. — Introduction to AC Machine Design, IEEE Press / Wiley.
• Bose, B.K. — Modern Power Electronics and AC Drives, Prentice Hall.
• SAE J1715 — Hybrid Electric Vehicle (HEV) and Electric Vehicle (EV) Terminology.

Next Lesson

• MOTOR-05 — Stator, Rotor and Windings: Hairpin winding manufacturing, slot fill factor, lamination stacking, and insulation systems.