Why Can a Motor’s Maximum Torque Drop as Its Speed Increases?
CODER MŨ RƠM · BLDC / PMSM 101 · PART 1
Back-EMF, current limits, and voltage limits: three things to understand before diving into motor control algorithms.
When I first started learning about brushless motors, I jumped straight into FOC. Suddenly, I was looking at coordinate transformations, PI loops, and block diagrams. But there was a simpler question I still could not answer clearly: what does the controller actually do to make the motor produce torque?
This series starts with questions like that. I will use models, diagrams, and worked examples to build an understanding of motor control. Let’s begin with a familiar observation: a motor can pull hard at low speed, yet its maximum torque drops as it spins faster.
To understand why, imagine holding a permanent-magnet DC motor and turning its shaft by hand.
▶ Watch the introductory video · 3 minutes 35 seconds — This article adds models and their operating assumptions to the explanation in the video.
1. A spinning motor also generates voltage
Turn the shaft, and a voltage appears across the motor’s terminals. Turn it faster, and that voltage increases. The same electromagnetic mechanism can take electrical energy and produce motion, or take mechanical motion and generate electricity.
This induced voltage is still present when the motor is powered and running. In motoring operation, it opposes the effect of the applied voltage on current. That is why it is called back electromotive force, or back-EMF.
If we treat the magnet flux as constant, the relationship is:
e = Kₑ ω
Here, e is the back-EMF, ω is angular speed, and Kₑ is the motor constant under the convention being used. In this model, doubling the speed doubles the back-EMF.
In this first article, I use a permanent-magnet DC motor to make the voltage–current relationship clear. BLDC and PMSM motors follow the same induction principle, but their phase voltages vary with rotor position. Extending the discussion to three phases also requires looking at the waveforms and how the controller coordinates the phases. [1]
2. A 24 V supply does not leave all 24 V available to drive current through resistance
Connect a resistor to a supply, and you can calculate current using Ohm’s law. A spinning motor also has winding inductance and back-EMF. Its simplified electrical model becomes:
v = Ri + L di/dt + e
The applied motor voltage must account for three terms: the resistive drop Ri, the voltage L di/dt needed to change the current, and the back-EMF e. Rearranging the equation shows directly how current changes:
di/dt = (v − Ri − e) / L
For example, at the same applied voltage and instantaneous current, a higher back-EMF reduces the rate at which current rises. If the expression in parentheses becomes negative, the current falls.
For a DC motor with steady current, di/dt is zero. Then:
i = (v − e) / R
This is what “back-EMF opposes the supply” means in concrete terms: in this steady state, the difference v − e determines the current through the resistance. Energy is conserved as well. Multiplying the steady-state equation by current gives vi = Ri² + ei: some input power becomes winding heat, while ei is the power converted between the electrical and mechanical domains. [1]
3. As speed rises, the controller increases voltage to maintain current
At this point, it is tempting to conclude that current must fall whenever speed rises. That is true in the model with a fixed applied voltage. A current controller, however, can actively increase the voltage applied to the motor.
Consider a hypothetical example: a maximum output voltage of 24 V, a motor resistance of 1 Ω, and a current limit of 5 A. For now, ignore power-stage voltage drops, current ripple, and temperature changes. To maintain 5 A, the required voltage is:
v = e + 1 Ω × 5 A = e + 5 V
Back-EMF | Voltage required to maintain 5 A | What the controller can deliver |
4 V | 9 V | Can maintain 5 A |
14 V | 19 V | Can still maintain 5 A |
19 V | 24 V | Just reaches the voltage limit |
22 V | 27 V | Insufficient voltage; at 24 V, only 2 A can be maintained |
Figure 1. Voltage required to maintain 5 A in the DC example. The final bar represents a 27 V requirement, above the available 24 V, so the actual current cannot remain at 5 A.
As back-EMF rises from 4 V to 14 V, the controller maintains current by raising the applied voltage from 9 V to 19 V. Current tracking only becomes limited when the required voltage exceeds the available output.
With constant flux, a DC motor’s electromagnetic torque is approximately proportional to current:
Tₑ = Kₜ i
In this model, maintaining current means maintaining torque. When the maximum current falls from 5 A to 2 A, the maximum torque falls to 40% of its previous value.
These numbers are for explanation only. In a three-phase BLDC/PMSM system, battery voltage, line-to-line voltage, and phase voltage must be distinguished. The available output voltage also depends on the modulation method. A 24 V battery voltage cannot simply be treated as the phase voltage in every calculation.
4. Why does the torque envelope often have a flat region?
At low speed, the controller usually has enough voltage available to maintain a high current. Torque capability is then limited by the permissible current of the motor and power stage, together with thermal conditions. If the current limit stays constant, maximum torque can remain nearly constant over a range of speeds.
As speed rises to the point where the available output voltage can no longer maintain the desired current, the system becomes voltage-limited. The transition depends on the motor, supply voltage, and control strategy. In a PMSM system, the corresponding operating threshold is commonly called the base speed. [2]
Figure 2. Maximum-current envelope calculated from the example’s DC model, without field weakening: iₘₐₓ = min[5, max(0, 24 − e)] A for R = 1 Ω. Because e is proportional to speed, the horizontal axis also represents speed on a fixed scale. This is not a measured characteristic of a particular PMSM.
This figure also shows why the assumptions behind a plot matter. The torque curve of a DC motor at fixed applied voltage differs from the available torque envelope of a current-controlled system. Field weakening, power limits, and thermal limits change the characteristic further.
Another common source of confusion: high torque does not necessarily mean high speed. The mechanical relationship is:
J dω/dt = Tₑ − T_load − T_loss
The torque remaining after load and loss torques are subtracted determines angular acceleration. If motor torque just balances them, speed stays constant. A motor holding a stationary load can still produce substantial torque. Likewise, an electric vehicle’s initial surge depends on the control command, load, and drivetrain; throttle feel alone does not reveal the shape of its torque envelope.
5. How does field weakening extend the speed range?
In e = Kₑω, I assumed constant flux. If the controller reduces the effective flux, the voltage required at a given speed can fall. That is the basis of field weakening.
In a PMSM, the controller produces a stator-current component that opposes part of the rotor magnet flux. Under the usual control convention, this is negative d-axis current. The aim is to adjust the combined flux during operation, not to permanently demagnetize the magnets. [3]
The trade-off is that field-weakening current uses part of the system’s current capacity. In a simple surface-mounted PMSM model, this leaves less current available for torque production. In an interior permanent-magnet motor, the torque relationship is more complex because reluctance torque also contributes. Both cases must operate within current and voltage limits. [4]
Field weakening therefore extends the speed range, but does not guarantee that maximum torque stays unchanged. In a region of approximately constant mechanical power, P = Tω shows that torque must fall as speed rises. The width of that region depends on the design and control of the particular system.
The Toyota Prius is one application example: an ORNL report on the 2004 Prius motor notes that field weakening or current-angle adjustment is needed over a substantial high-speed range. [5]
6. Would a higher supply voltage solve the problem?
It can help. More available output voltage lets the controller maintain current at higher speeds before reducing torque or moving further into field weakening. But the entire system must be designed for that voltage.
A higher voltage changes the requirements for transistor and capacitor ratings, insulation, and battery configuration. Switching losses may also increase when other conditions remain comparable. The wider speed range must be weighed against cost, heat, and durability. Raising voltage does not automatically increase torque in a region already limited by current.
References
University of Michigan — DC Motor Speed: System Modeling. Electrical and mechanical equations, torque, and back-EMF for a DC motor.
MathWorks — Field-Weakening Control. Torque regions, base speed, and voltage limits.
Microchip — Equation-based Flux Weakening. Negative d-axis current and effective flux.
MathWorks — Field-Weakening Control (with MTPA) of PMSM. Current constraints and torque capability.
The figures were calculated and drawn from the illustrative model described in this article; they are not experimental data. Original video: Cốt Đơ Mũ Rơm.
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