Electromagnetic Field Simulation of Field Weakening Control
Theory and Physics
What is Field Weakening Control
Field weakening... does it mean intentionally weakening the magnet's force? Isn't that wasteful?
It might seem wasteful at first glance, but to spin above the base speed, you run into the inverter's voltage limit. By flowing negative d-axis current to cancel the magnet's flux, you suppress the back EMF and achieve high-speed rotation. It's an essential technology for EV highway driving.
Huh? Don't EVs change speed with gears?
Many EVs don't have a transmission; they are so-called single-gear. While an engine car would shift into 5th or 6th gear for high-speed driving, an EV motor uses field weakening control to spin up to 3-4 times its rated speed to cope. In other words, field weakening control is a technology that gives the motor a "virtual gear change."
I see, so you change how you use the current instead of gears! But doesn't torque drop when you weaken the magnet's force?
Exactly. In the field weakening region, torque decreases inversely with speed. That is, the output (= torque × speed) becomes an almost constant "constant power region." For cruising at 100 km/h on a highway, the required torque is low, so this is sufficient.
Field Weakening Control is a current control technique for achieving high-speed operation beyond the base speed in permanent magnet synchronous motors (IPMSM/SPMSM). By flowing negative d-axis current ($i_d < 0$) to electrically cancel the permanent magnet flux $\psi_m$ and reduce the back EMF, high-speed rotation within the inverter's output voltage constraint is made possible.
dq-Axis Voltage Equations and Constraints
Specifically, what equations constrain the control?
Let's start from the dq-axis voltage equations for an IPMSM. Assuming steady state ($di/dt = 0$), we get:
Here, $R_s$ is the winding resistance, $L_d, L_q$ are the dq-axis inductances, $\omega_e$ is the electrical angular velocity, and $\psi_m$ is the permanent magnet flux linkage. Neglecting the voltage drop due to $R_s$ as it is small in the high-speed region, the voltage limit condition can be summarized as follows:
Simultaneously, the current limit condition imposed by the inverter's current rating applies:
So, there are upper limits for both voltage and current, and the motor must operate within those ranges.
Yes. And this is important: when you look at the voltage limit equation on the $i_d$-$i_q$ plane, it becomes an ellipse. As the speed increases, $\omega_e$ increases and the ellipse shrinks. So, in the high-speed region, the feasible current operating area becomes narrower and narrower.
Voltage Limit Ellipse and Current Limit Circle
Rewriting the voltage limit condition on the $i_d$-$i_q$ plane yields:
This is an ellipse with center $(-\psi_m/L_d, \, 0)$ and major/minor axes of $V_{\max}/(\omega_e L_d)$, $V_{\max}/(\omega_e L_q)$. For IPMSM, since $L_d < L_q$, the d-axis direction becomes the minor axis. As the speed rises, the ellipse shrinks, and its intersection area with the current limit circle $i_d^2 + i_q^2 = I_{\max}^2$ changes.
Why is the ellipse's center shifted to $(-\psi_m/L_d, 0)$ instead of the origin?
Because the permanent magnet flux $\psi_m$ always exists in the d-axis direction, it "consumes" that much voltage. Even with zero d-axis current, back EMF is generated by the magnet flux. Therefore, the ellipse's center shifts to the position $i_d = -\psi_m/L_d$. Field weakening control is essentially about moving the operating point towards this ellipse center.
MTPA Control and MTPV Control
I often hear about MTPA and MTPV, what's the difference?
Simply put, they are two optimal current strategies used in different speed ranges.
MTPA (Maximum Torque Per Ampere): A strategy that selects the current vector angle $\beta$ that produces maximum torque for a given current amplitude, within the current limit circle. For the IPMSM torque equation:
it utilizes the reluctance torque term $(L_d - L_q) i_d i_q$, thus operating in the $i_d < 0$ region. The MTPA trajectory forms a hyperbolic-like curve on the $i_d$-$i_q$ plane.
MTPV (Maximum Torque Per Voltage): A strategy that selects the operating point on the voltage limit ellipse that produces maximum torque. After reaching the voltage limit in the high-speed region, the operating point moves along the ellipse's edge. The condition for the MTPV trajectory is:
So, does the operating point shift depending on speed: MTPA trajectory → on the voltage limit ellipse → MTPV trajectory?
Exactly. In the low-speed region, operation is on the MTPA trajectory for highest efficiency; near the base speed, it reaches the voltage limit and moves on the ellipse; and in even higher speed regions, it transitions to the MTPV trajectory. Smoothly switching between these three regions is where control skill comes in, and in CAE, we calculate the optimal current point for every speed-torque combination to create an "efficiency map."
Physics of Demagnetization Risk
I heard that if you make the d-axis current too large, the magnet can break. Is that true?
It's not so much breaking as "demagnetizing." Neodymium magnets have a demagnetization curve (the second quadrant of the B-H curve), and if the operating point exceeds the knee point (the inflection point of the curve), the magnetic force irreversibly decreases. In field weakening control, large negative d-axis current is applied, so there is a risk that the magnetic flux density inside the magnet drops close to the knee point.
Temperature is also a factor, right? Summer seems dangerous...
That's precisely the most dangerous part. The coercivity of neodymium magnets drops sharply with rising temperature. Coercivity that was 2000 kA/m at 20°C can become less than half at 150°C. Therefore, in CAE, we always check the magnet's operating point under worst-case conditions (maximum temperature × maximum d-axis current × transient current overshoot).
Demagnetization judgment is performed by comparing the magnetic flux density distribution inside the magnet obtained from FEM with the demagnetization curve corresponding to the temperature. Special attention is required at magnet corners (edges), where magnetic flux density tends to drop locally due to concentration of the demagnetizing field.
Why the Tesla Model 3 Motor Design is the "Textbook of Field Weakening"
The Tesla Model 3's rear motor (IPMSM type) has a maximum speed of about 18,000 rpm against a rated speed of about 5,000 rpm. That means it spins up to 3.6 times the base speed using field weakening. What makes this possible is the utilization of reluctance torque through its magnet-embedded structure (V-shape arrangement) and the high voltage utilization ratio enabled by its SiC inverter. It can be inferred that during the design phase, an enormous number of FEM analyses for demagnetization margin under field weakening conditions were performed. Similar design processes are becoming standardized among Japanese automotive OEMs as well, with FEM-circuit coupled analysis in JMAG or Ansys Maxwell playing a central role.
Derivation Process of dq-Axis Voltage Equations
- d-axis voltage $v_d = R_s i_d + L_d \frac{di_d}{dt} - \omega_e L_q i_q$: The time derivative of the d-axis flux linkage $\lambda_d = L_d i_d + \psi_m$ plus the resistive drop. The term $-\omega_e L_q i_q$ is the speed EMF due to rotation of the q-axis flux (cross-coupling term).
- q-axis voltage $v_q = R_s i_q + L_q \frac{di_q}{dt} + \omega_e(L_d i_d + \psi_m)$: The time derivative of the q-axis flux linkage $\lambda_q = L_q i_q$ plus the back EMF $\omega_e(L_d i_d + \psi_m)$ due to rotation of the d-axis flux. This back EMF term is the fundamental constraint of field weakening control.
- Torque equation $T = \frac{3}{2}p(\psi_m i_q + (L_d - L_q)i_d i_q)$: The first term is the magnet torque (interaction of permanent magnet flux and q-axis current), the second term is the reluctance torque (utilizing magnetic anisotropy due to saliency $L_d \neq L_q$). For IPMSM, $L_q > L_d$, so $i_d < 0$ contributes positively to reluctance torque.
Application Limits and Assumptions of Field Weakening Control
- Steady-state assumption: The voltage limit ellipse is drawn for steady state $di/dt = 0$. During transients, the ellipse fluctuates, potentially causing instantaneous voltage overrun.
- Effect of magnetic saturation: Under high load, $L_d$, $L_q$ change depending on current. Large errors occur if the nonlinear inductance map obtained from FEM is not used.
- Temperature dependence: $\psi_m$ decreases with rising magnet temperature, and $R_s$ increases with winding temperature. Efficiency maps must be created for a wide temperature range.
- Spatial harmonics: In concentrated winding motors, spatial harmonic flux exists due to winding arrangement, which can degrade the accuracy of the dq-axis model.
- Effect of iron loss: Increasing d-axis current changes the amplitude of flux density variation in the core, altering iron loss distribution. Equivalent circuit models may be insufficiently accurate.
Typical Values of Key Parameters (for EV IPMSM)
| Parameter | Typical Value | Remarks |
|---|---|---|
| Permanent magnet flux linkage $\psi_m$ | 0.05–0.15 Wb | Decreases by approx. -0.1%/°C with temperature rise |
| d-axis inductance $L_d$ | 0.1–0.5 mH | Current dependent (decreases with saturation) |
| q-axis inductance $L_q$ | 0.3–1.5 mH | For IPMSM, $L_q > L_d$ |
| Saliency ratio $L_q/L_d$ | 2–8 | Larger ratio means larger reluctance torque |
| Base speed | 3,000–6,000 rpm | Point where voltage utilization reaches 100% |
| Maximum speed / Base speed ratio | 3–5 | Width of the field weakening constant power region |