Synchronous reluctance motor: torque from saliency, with no magnets

A rotor that only has a preferred direction

In a synchronous machine the rotor locks to the rotating stator field. A permanent-magnet or wound-field rotor does so with its own flux. A reluctance rotor has no flux of its own; it is simply a piece of steel shaped so that magnetic flux finds a low-reluctance path along one axis and a high-reluctance path along the perpendicular axis.

A piece of iron in a magnetic field turns to align with the field along its easiest path, the way a compass needle aligns, but without needing to be a magnet. If the field rotates, the rotor follows it at synchronous speed, with a load angle that grows with the load. There is no slip, so the rotor has no copper loss and stays cool.

Because the rotor has no cage, the motor has no starting torque from the line. It must be driven by an inverter that raises the field speed from zero.

d and q axes, Ld and Lq

The rotor is described in a frame that turns with it. The d axis is the direction of easy flux, with a high inductance Ld, and the q axis is the direction of hard flux, with a low inductance Lq. The ratio ξ = Ld / Lq is the saliency ratio. The stator current is split into id, which makes the flux along d, and iq, which is the torque-producing part.

The electromagnetic torque is

T = 1.5 × p × (Ld − Lq) × id × iq

where p is the number of pole pairs and the currents are peak phase values. Torque is made only by the difference Ld − Lq. If the rotor were round (Ld = Lq), the torque would be zero, however much current you push in.

Example: p = 2, Ld = 60 mH, Lq = 10 mH, with a peak current of 10 A at 45° from the d axis (id = iq = 7.07 A, so id × iq = 50 A²):

T = 1.5 × 2 × 0.050 × 50 = 7.5 N·m

At the same peak current with Ld = Lq the torque is zero, and doubling the difference doubles the torque.

Flux barriers: how the saliency is built

A rotor with a simple shape would only be weakly salient. Modern SynRM rotors are laminations cut with several curved slots, called flux barriers, around each q axis. Air in the slots blocks the flux in the q direction and so reduces Lq, while thin steel paths between the barriers let the flux in the d direction pass, keeping Ld high. Saliency ratios of the order of 5 to 10 are often quoted for well-designed rotors with several barrier layers. The thin iron bridges that hold the rotor together are a mechanical compromise: they leak a little q flux but give strength at speed.

Saturation matters, too. When id is high the iron near the d axis saturates, Ld drops and the saliency shrinks, which is why the real ratio of an operating motor is lower than the no-load one.

Control: MTPA and the power factor limit

For a given torque, many combinations of id and iq are possible. The drive picks the one that suits the aim:

• Maximum torque per ampere (MTPA): the least current for a torque. In a linear, lossless machine the current vector is at 45° from the d axis, and saturation moves it to a larger angle.
• Maximum power factor: a steeper current angle, about atan(√ξ), for the best cos φ, at the price of more current for the same torque.
• Constant id: a fixed flux current, like a simple V/f drive. It is simple but the angle drifts away from the optimum with load.

The SynRM must be magnetised by the stator current, since the rotor has no flux. All of the flux comes through reactive current, so the power factor is limited. In a lossless machine

cos φ ≤ (ξ − 1) / (ξ + 1)

and at MTPA it is (ξ − 1) / √(2 × (ξ² + 1)). For ξ = 6 the ceiling is 5/7 ≈ 0.71, and at MTPA it is about 0.58. For ξ = 3 the ceiling is 0.50 and MTPA gives about 0.45. This is why high saliency matters, and why SynRMs are sized for a slightly larger inverter than comparable PM machines.

Above base speed the inverter voltage is the limit. The drive reduces id (field weakening) and raises iq, and if the load cannot be held the speed has to fall.

Efficiency, cost and where it is used

• No rotor copper loss and no magnet loss, so the rotor runs cool, bearings last longer and the machine can reach a higher efficiency class than an induction motor of the same frame, or be smaller for the same output. Some manufacturers offer SynRM and inverter packages in IE4 class with standard induction-motor frame sizes.
• No rare-earth magnets: no cost or supply risk, no demagnetisation at high temperature, and no back-EMF if the inverter fails (a safety advantage).
• Lower power factor and lower torque density than interior permanent-magnet machines, with a slightly larger inverter.
• Typical uses: pumps, fans and compressors with a variable-speed drive, conveyors and machine tools.

A SynRM is a good replacement for induction motors when a drive is already there, and it does not need any rare-earth magnets.

How the simulator models the SynRM

The simulator solves the steady-state d–q equations: v_d = R × id − ω_e × Lq × iq and v_q = R × iq + ω_e × Ld × id, with T = 1.5 × p × (Ld − Lq) × id × iq. Ld saturates with id, but never falls below Lq, so ξ = 1 gives exactly zero torque. The model is calibrated so that, at the reference saliency ξ = 6, rated current at the best angle gives rated torque. Changing ξ changes only Lq = Ld0 / ξ, so torque per ampere scales with (1 − 1/ξ).

The stator resistance is set for a copper loss of 6% of rated power, plus iron and friction losses (about 3.5% of rated power for iron). The strategies are MTPA, Max power factor and Constant id. The current limit is 1.5 times rated, the ξ slider runs from 1 to 12, and the voltage limit follows from the drive supply voltage. If the voltage limit stops the commanded speed, the rotor settles at the highest speed the load allows, and with no solution even at zero speed the motor is stalled.

Not modelled: cross-saturation, the effect of iq on Lq, the current regulator, PWM harmonics and dead time.

Try it in the synchronous reluctance motor simulator

• Set the Saliency ratio ξ = Ld/Lq (range 1 to 12) to 1. The torque falls to zero and a low-saliency warning appears. Raise it towards 6 and 12, and watch the Torque against current angle chart gain a taller peak.
• Set the Control strategy to MTPA and read Current angle from d-axis, id (flux) and iq (torque). Switch to Max power factor and Constant id and compare the current angle and the current needed.
• Open the Power factor against saliency ratio chart and compare the limit curve (ξ−1)/(ξ+1), the MTPA curve and the present machine. The Power factor ceiling readout shows the limit for the current ξ. The low-saliency warning appears below ξ = 3.
• Raise the Speed command (up to 200% of base speed) above 100%. The Field weakening readout becomes Active, and if the load is large the Speed, limited by voltage readout shows the actual and the commanded speed.
• In 3D, the rotor has no magnets or winding, only the flux barrier slots, and it follows the field lagging by an angle that grows with load. Blue and orange bars mark the d axis and the q axis.

The lesson is that torque is the saliency, and the power factor is its price.

Try it in the simulator