dq-axis model for SPM and IPM motors. Adjust pole pairs, inductances, PM flux linkage, and operating point to compute torque, losses, and efficiency instantly.
Motor Topology
Type
Pole pairs p
Electrical Parameters
Stator resistance Rs
mΩ
Ld (d-axis inductance)
mH
Lq (q-axis inductance)
mH
PM flux linkage ψPM
Wb
Operating Point
Speed n
rpm
d-axis current Id
A
q-axis current Iq
A
—
Torque Te (N·m)
—
Efficiency η (%)
—
Copper loss Pcu (W)
—
Back EMF E (V)
—
MTPA angle (°)
—
Power factor cosφ
Torque–speed characteristic (current operating point ●)
Torque components vs Iq (fixed Id, current point ●)
What exactly is this "dq-axis" model? Why do we need a special coordinate system just to analyze a motor?
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Great question! Basically, it's a mathematical trick to make our lives easier. In reality, the three-phase currents in the stator windings are constantly changing with rotor position, which is messy to analyze. The dq-transform locks the viewpoint onto the spinning rotor itself. The 'd' (direct) axis aligns with the permanent magnet's north pole, and the 'q' (quadrature) axis is 90 degrees ahead. In this rotating frame, the AC currents look like steady DC values, which are much simpler to control. Try changing the Id and Iq sliders in the simulator—you're directly commanding the motor in this simplified, transformed space.
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Wait, really? So the torque equation up there has two parts. What's the difference between "magnet torque" and "reluctance torque"?
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Exactly! They are two different physical mechanisms for producing force. The magnet torque comes from the interaction between the permanent magnet's flux ($\psi_{PM}$) and the q-axis current ($i_q$). It's like a standard permanent magnet motor. The reluctance torque is extra! It comes from the rotor's magnetic saliency—the fact that $L_d$ and $L_q$ are different. The rotor is designed to be magnetically "lumpy," so it has a preferred alignment with the stator field, creating additional torque. In the simulator, select an SPM (Surface PM) type—you'll see $L_d = L_q$, so reluctance torque is zero. Switch to IPM (Interior PM) to see a difference and unlock that second term.
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So how do engineers use this in practice? If I want maximum efficiency, should I just maximize both torque terms?
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Not quite—it's a balancing act! Maximizing torque often means pushing a lot of current, which increases $I^2R$ losses in the copper windings (that's the Rs parameter). The real art of motor design is finding the optimal combination of Id and Iq for a given required torque. This is called Maximum Torque Per Ampere (MTPA) control. For an IPM motor, you often use negativeId to exploit the reluctance torque fully. Play with the sliders: you'll see the total torque change, and you can find the current pair that gives the most torque for the least total current magnitude, which directly impacts the efficiency calculated by the tool.
Physical Model & Key Equations
The core of PMSM performance prediction is the electromagnetic torque equation derived from the dq-axis model. It separates the two fundamental torque-producing mechanisms.
$T_e$: Electromagnetic Torque (Nm) $p$: Number of Pole Pairs (sets the motor's base speed) $\psi_{PM}$: Permanent Magnet Flux Linkage (Wb) - strength of the rotor magnets $i_d, i_q$: d-axis and q-axis currents (A) - the control inputs in the rotating frame $L_d, L_q$: d-axis and q-axis inductances (H) - define the magnetic saliency of the rotor
To understand losses and efficiency, we calculate the copper losses from the stator resistance and the applied currents. The total input power is the sum of output power and these losses.
$P_{cu}$: Copper Losses (W) - heat generated in the windings. $R_s$: Stator Resistance per phase (Ω) - a key loss parameter. $\omega$: Mechanical angular speed (rad/s), related to speed n in RPM. $\eta$: Efficiency (%) - the ultimate design metric.
Frequently Asked Questions
For SPM (Surface Permanent Magnet), since Ld ≈ Lq, the reluctance torque term (Ld - Lq)id iq is nearly zero, and torque is determined solely by magnet torque. For IPM (Interior Permanent Magnet), since Ld < Lq, reluctance torque is generated, and by applying a negative id current through MTPA control, efficient torque can be achieved. This tool supports both types.
When the back EMF exceeds the power supply voltage, the voltage required to drive the motor becomes insufficient, making current control impossible (the limit of flux-weakening control). This tool calculates E in real time from the rotational speed and flux linkage, allowing you to verify control limits in advance. During design, it is recommended to keep E at 90% or less of the power supply voltage.
The MTPA (Maximum Torque per Ampere) angle is the current phase angle that produces the maximum torque for a given current. This tool calculates it analytically by partially differentiating the torque equation with respect to id and iq. By setting the current command according to this angle, copper loss is minimized, enabling high-efficiency operation. This is especially important in IPM motor design.
Ld and Lq are essential parameters for calculating torque and reluctance torque. If unknown, please input estimated values based on the motor geometry (for SPM, Ld ≈ Lq; for IPM, Ld < Lq). Since the results of this tool vary depending on the input values, using measured values or FEM analysis results enables more accurate design.
Real-World Applications
Electric Vehicles (EVs): IPM-SynRM motors (Interior Permanent Magnet Synchronous Reluctance Motors) are the dominant choice in modern EVs, like those from Tesla and Toyota. Engineers use the dq-model to optimize the mix of magnet and reluctance torque, maximizing power density and efficiency across the entire speed range, which directly translates to longer driving range.
Industrial Drives & Robotics: High-performance servo motors for CNC machines and robotic arms often use PMSMs for their precise control and high torque density. The dq-current control allows for extremely fast and accurate torque response, enabling smooth motion and high positional accuracy.
Home Appliances: Advanced washing machines, air conditioner compressors, and refrigerator fans increasingly use PMSM motors for their superior efficiency and quiet operation. The design calculator helps optimize for lower Rs and optimal current angles to meet strict energy efficiency regulations.
Aerospace Actuation: Fly-by-wire systems in aircraft use fault-tolerant PMSM designs for actuating control surfaces. The dq-model is crucial for predicting performance under extreme conditions and designing controls that ensure reliability and safety.
Common Misconceptions and Points to Note
First, the misconception that "inductance is a fixed value." In actual motors, values like Ld and Lq change significantly with increasing current due to magnetic saturation. Get into the habit of running simulations in the tool for both "rated current" and "overload current" to check how much the torque constant varies. For example, during field-weakening control in an interior permanent magnet motor where a large negative Id is applied, Ld in particular saturates and fluctuates, becoming a cause for discrepancy between calculated values and actual machine performance.
Next, designs that chase only the "peak efficiency point" on the efficiency map. While a 95% point feels great, what's important in practical operation is the average efficiency across the entire typical operating region. For an electric vehicle, for instance, broadly evaluate the characteristics generated by the tool to see if the mid-speed, mid-torque region, corresponding to city driving, forms a high-efficiency "hill."
Finally, the simplistic judgment that "it's okay as long as the back-EMF doesn't exceed the voltage." A safety margin is essential. Battery voltage fluctuates with load changes, and the maximum usable voltage also differs depending on the inverter's modulation method (e.g., sinusoidal PWM vs. overmodulation). For the back-EMF $E$ calculated by the tool, aim to keep it sufficiently lower than at least $1/\sqrt{3}$ times the DC link voltage (the maximum output for space vector modulation). For instance, with a 300V supply voltage, a practical design would have the back-EMF significantly below 170V.
Enter pole pairs (typically 2–8 for industrial PMSMs) in sPNum field
Input rated power in kW (sP; e.g., 15 kW for servo drives, 55 kW for pump motors) in sP field
Set stator resistance in ohms (sRs; measure via DC test or datasheet; 0.8–2.5 Ω for 11 kW machines) in sRsNum
Define d-axis inductance (sLd) and q-axis inductance (sLq) in mH—IPM motors show Lq > Ld (reluctance torque) while SPM types have Ld ≈ Lq
Run simulation to compute electromagnetic torque, copper losses (I²R), iron losses, and efficiency (%)
Worked Example
Design a 7.5 kW SPM motor: 4 pole pairs, Rs=1.2 Ω, Ld=18 mH, Lq=19 mH, rated speed 1500 rpm, peak current 25 A. Calculator returns electromagnetic torque ≈48 Nm, copper loss P_cu=750 W (25²×1.2), iron loss estimated ≈400 W at rated flux, overall efficiency ≈92%. For IPM variant with same power but Ld=15 mH, Lq=28 mH, reluctance torque contribution adds 8–12 Nm, reducing required permanent magnet flux by 15–20%.
Practical Notes
Measure stator resistance (sRs) with calibrated multimeter at 20°C; temperature coefficient adds 0.4% per °C for copper windings in marine/automotive duty
Copper losses dominate below 50% load; iron losses become significant at full speed; always cross-check thermal rise against 155°C Class F insulation rating