Three-phase motors · Hardware Academy

How three currents make a motor turn

A field that turns by itself

Put three identical windings in a stator, 120° apart, and drive them with three currents that are also 120° apart in time: iA = Im·cos ωt, iB = Im·cos(ωt − 120°), iC = Im·cos(ωt + 120°). Each winding's field only grows and shrinks along its own axis. Add the three vectors, though, and the pulsating parts cancel: the sum always has length 1.5·Im and points at angle ωt. A rotating magnetic field appears with no moving parts. Nikola Tesla patented the polyphase induction motor built on this idea in 1888, and Galileo Ferraris found it independently. Swap any two phases and the field turns the other way, which is all reverse gear is in an EV.

Poles and synchronous speed

Wind the stator so the pattern repeats and you get more poles. A machine with p poles has p/2 pole pairs, and its field advances one pole pair per electrical cycle, so it turns at the synchronous speed n_s = 120·f/p rpm. A 2-pole motor on 60 Hz mains turns 3,600 rpm, a 4-pole motor on 50 Hz 1,500 rpm. A traction inverter makes whatever frequency it needs: a 4-pole motor at 400 Hz turns 12,000 rpm, and an 8-pole motor at 18,000 rpm needs 1,200 Hz. More poles mean shorter end turns and thinner back iron, but a higher electrical frequency for the same speed, which raises iron loss and asks more of the inverter.

Induction motors: torque from slip

An induction rotor is a squirrel cage: aluminum or copper bars in a laminated steel core, shorted by end rings, with no magnets and no electrical connection. If it turns slower than the field, the field sweeps through the bars, induces currents, and those currents push the rotor along. The lag is the slip, s = (n_s − n)/n_s, typically 1-5% at rated load; at zero slip nothing is induced and there is no torque. Rotor currents run at the slip frequency s·f, a few hertz, and the rotor copper loss equals slip times the air-gap power, so the rotor runs hot and is hard to cool. The stator must also supply magnetizing current, which costs efficiency at light load. In exchange you get a rugged motor with no rare earths and no back-EMF when unpowered. The 2008 Tesla Roadster used a 4-pole induction motor of about 185 kW that spun to 14,000 rpm, and the Model S (2012) and Model X (2015) used copper-rotor induction motors.

Permanent-magnet synchronous motors

Replace the cage with neodymium-iron-boron (NdFeB) magnets and the rotor needs no induced current: it locks to the field and turns at exactly synchronous speed. With no magnetizing current and almost no rotor copper loss, PM motors reach 95-97% peak efficiency and stay efficient at light load. In a surface PM motor torque is simply T = Kt·Iq, where Iq is the stator current component 90° ahead of the magnet axis and Kt = 1.5 × pole pairs × magnet flux linkage. Four pole pairs and 0.05 Wb give Kt = 0.3 N·m/A, so 400 A makes 120 N·m.

The magnets bring three costs. They need rare earths, with dysprosium or terbium added for heat resistance. They can be demagnetized: NdFeB coercivity falls about 0.5-0.6% per °C, so a hot magnet hit by a strong reverse field loses flux for good. And they always generate back-EMF in proportion to speed. Above base speed the back-EMF approaches what the inverter can produce from the DC bus, so the controller injects negative d-axis current to oppose the magnets. This field weakening holds power roughly constant up to top speed, at the price of extra current and reverse field on the magnets.

Synchronous reluctance and Tesla's hybrid

A synchronous reluctance motor has no magnets and no cage, just a steel rotor cut with flux barriers so it is far easier to magnetize along one axis than the other. The rotor turns to align its easy axis with the field, giving reluctance torque proportional to (Ld − Lq)·Id·Iq. It is cheap and runs cool but has a poor power factor. Tesla's Model 3 (2017) combined the two in an interior-PM synchronous reluctance motor: NdFeB magnets buried in the flux barriers, so magnet torque and reluctance torque add and fewer magnets are needed. In dual-motor Model 3s the rear motor is this PM machine and the front is induction, because an induction motor that is switched off spins with no magnet drag.

Efficiency maps, cooling and windings

Motor makers plot efficiency as contours over the torque-speed plane. A PM motor's map has a 96-97% island at mid speed and moderate torque, falling off at high torque, where copper loss grows with current squared, and at high speed, where iron loss and field-weakening current grow. Drive cycles spend most of their time at low torque, which is exactly where PM beats induction. Heat sets continuous power, so modern drive units spray oil, which does not conduct, straight onto the end windings and pump it through a hollow rotor shaft. Many stators now use hairpin windings: rectangular copper bars bent like hairpins, pushed into the slots and laser-welded. They fill about 70% of the slot against roughly 45% for round wire, cutting resistance and improving heat transfer, but they suffer more AC loss at high frequency, so designers split them into four to eight thin layers per slot. BorgWarner's HVH motors, originally Remy's High Voltage Hairpin line, are an early example.

A 2-pole stator at the instant phase A carries its peak current. Each winding pushes flux along its own axis; tip to tail, the three contributions add to 1.5 × Im along phase A's axis.

Interactive: spin up the field

Three currents, one rotating field

Each winding's field only pulses along its own axis. Their sum, the violet arrow, keeps a constant length and turns at the electrical frequency. Pick a rotor and watch how it follows.

Synchronous speed (120·f/p): 1,500 rpm
Rotor speed: 1,425 rpm
Rotor current frequency (s·f): 2.50 Hz
Slip: 5.0%

The rotor turns slower than the field, so the field sweeps through the bars and induces current. The green mark drifts backwards at the slip rate.

Rotor

Electrical frequency50 Hz

Poles (for the speed readout)

Slip5 %

The drawing is a 2-pole machine, so one electrical cycle is one mechanical turn; a p-pole machine turns 2/p of a revolution per cycle, which is where n_s = 120·f/p comes from. Animation slowed 200×. Bar colors show the current induced in each rotor bar (orange out of the page, cyan into it).

Inside a traction motor

An interior-PM motor taken apart along its shaft. Drag to orbit the 3D model; slide Explode to separate the housing, stator, rotor and end caps.

Explode70%

Stator: 24 slots of rectangular hairpin bars in a stack of thin steel laminations (thin sheets cut eddy-current loss). Rotor: 8 NdFeB magnets buried in the steel, which is what makes it an interior-PM machine. Oil sprayed on the end turns and pumped through the hollow shaft carries the heat away.

In practice

Practice

Synchronous speed at 400 Hz

A 4-pole traction motor is fed at 400 Hz. The car has a 9:1 reducer and 0.67 m diameter tires.

What is the synchronous speed?
If it is an induction motor running at 3% slip, how fast does the rotor turn, and at what frequency do its rotor currents flow?
For a PM motor at synchronous speed, how fast is the car going?
What frequency would an 8-pole motor need for the same shaft speed?

text

1) n_s = 120 × f / p = 120 × 400 / 4 = 12,000 rpm
   (400 electrical rev/s ÷ 2 pole pairs = 200 rev/s × 60 = 12,000 rpm)

2) n = n_s × (1 − s) = 12,000 × 0.97 = 11,640 rpm
   f_rotor = s × f = 0.03 × 400 = 12 Hz

3) wheel = 12,000 / 9 = 1,333 rpm = 22.2 rev/s
   v = 22.2 × π × 0.67 m = 46.8 m/s ≈ 168 km/h

4) f = n × p / 120 = 12,000 × 8 / 120 = 800 Hz

Answer: 12,000 rpm synchronous. The induction rotor turns at 11,640 rpm with 12 Hz currents in its bars; the PM rotor turns at exactly 12,000 rpm, which through 9:1 and a 0.67 m tire is about 168 km/h. An 8-pole motor would need 800 Hz for the same speed, so its inverter must also switch faster to keep enough PWM pulses per electrical cycle.

Real-world scenario

Track day leaves a PM motor 8% weaker

A customer takes a rear-drive performance EV with an interior-PM motor to a track day: six back-to-back hot laps on a 32 °C afternoon. The drive unit never logged a fault. Back on the road the car feels flat. At the service visit, a timed run shows 0-100 km/h has slipped from 4.4 s to 4.8 s, and the inverter's torque estimate no longer matches the measured axle torque.

Symptoms

At the same q-axis current, measured torque is about 8% below the delivery value, and it does not recover after a night of cooling.
A coast-down test with the inverter off shows lower back-EMF: open-circuit line voltage at 6,000 rpm has dropped from 250 V to 230 V.
The stator winding thermistors peaked at 140 °C, well inside the 180 °C insulation rating, so no derate was ever triggered.
The loss is the same at every speed, not only near top speed.

Interview questions

screenHow do three phase currents create a rotating magnetic field?

Three windings sit 120° apart around the stator, and the inverter drives them with currents 120° apart in time. Each winding makes a field that only pulses along its own axis, but when you add the three vectors the pulsing parts cancel and the sum has a constant length, 1.5 times one phase's peak, pointing in a direction that turns at the electrical frequency. With more poles the pattern repeats around the stator, so it turns mechanically at 120·f/p rpm. Swapping any two phases reverses the rotation, which is all reverse gear is in an EV.

screenWhat is slip, and why does an induction motor need it?

Slip is the fraction by which the rotor turns slower than the field, s = (n_s − n)/n_s. An induction rotor has no magnets; its bars only carry current while the field sweeps past them, so at exactly synchronous speed there would be no induced current and no torque. At rated load slip is typically 1-5%, and in that range torque rises roughly in proportion to slip. The rotor currents run at the slip frequency s·f, and the rotor copper loss equals slip times the air-gap power, which is why induction rotors run hot and are hard to cool.

deepWhy did Tesla move to permanent-magnet reluctance motors, and why do its dual-motor cars still keep an induction motor?

The 2008 Roadster and the original Model S and X used induction motors: no rare earths, rugged, and no back-EMF or drag when unpowered. But an induction motor needs magnetizing current and has rotor copper loss, so it is less efficient at the light loads where cars spend most of their time. The Model 3 (2017) introduced an interior-PM synchronous reluctance motor, where magnets buried in the rotor's flux barriers add magnet torque to reluctance torque, which raises efficiency across the drive cycle. In a dual-motor Model 3 the front motor stays induction, because when it is not needed it can be switched off completely and spins with no magnet losses. The PM motor does the efficient everyday work, and the induction motor adds peak power at almost no cost when idle.

deepWhat is field weakening, and what risks does it create in a PM motor?

Back-EMF in a PM motor rises in proportion to speed, and above base speed it would exceed the voltage the inverter can make from the DC bus, so current and torque would collapse. Field weakening injects negative d-axis current whose flux opposes the magnets, lowering the net flux linkage so the motor can keep accelerating at roughly constant power. The first risk is to the magnets: that reverse field adds to the demagnetizing stress, and hot NdFeB has less coercivity, so it can be pushed past the knee and lose flux permanently. The second is a fault at high speed: if the inverter stops switching, the full back-EMF can exceed the bus voltage and drive uncontrolled current through the diodes into the pack, so drives use an active short circuit, all three low-side switches on, as the safe state.

screenWhy are hairpin windings popular in EV motors?

Hairpins are rectangular copper bars bent into a U, pushed axially into the slots and laser-welded at the other end. Rectangular conductors pack far better than round wire, about 70% slot fill against roughly 45%, so DC resistance and copper loss fall and heat flows more easily into the lamination stack. They also suit automated insertion and welding, which is faster and more repeatable than winding round wire. The catch is AC loss: at high frequency, current crowds into part of a thick bar through skin and proximity effects, so designers use more, thinner layers, typically four to eight per slot.

Quiz

References and further reading

Where the ideas in this lesson come from, and what to read next. Books are the standard texts; papers are the original sources; standards are what employers work to.

Book
Austin Hughes and Bill Drury (2019). Electric Motors and Drives: Fundamentals, Types and Applications. Newnes, 5th edition.
Read: The induction-motor and permanent-magnet chapters.
Book
R. Krishnan (2010). Permanent Magnet Synchronous and Brushless DC Motor Drives. CRC Press.
Read: The PM synchronous machine model.
Book
J. R. Hendershot and T. J. E. Miller (2010). Design of Brushless Permanent-Magnet Machines. Motor Design Books.
Read: How PM motors are designed and cooled.
Paper
Felix Blaschke (1972). The Principle of Field Orientation as Applied to the New Transvector Closed-Loop Control System for Rotating-Field Machines. Siemens Review 34, 217-220.
Read: The origin of the dq frame idea.