Current angle γ vs rotor position: a toy synchronous reluctance motor

This page is a teaching toy for one question that trips up almost everyone who starts simulating synchronous motors: if torque depends on both the rotor position and the phase of the currents, do I need to sweep both to know how much torque my motor makes? The short answer is no. In a running motor the controller ties the currents to the rotor, which leaves one meaningful knob, the current angle γ. Everything else here exists to make that visible.

1. The short version
2. The machine being modeled
3. Mechanical vs electrical angles
4. From three phases to d and q
5. Where the torque comes from
6. The two modes
7. Reading each panel
8. About the toy ripple
9. What a real machine adds
10. Mapping this onto a field simulation
11. A note on axial-flux machines
12. Glossary
13. References and further reading

The short version

Rotor position is not something you choose. The rotor visits every position as it spins, so anything that depends on position gets averaged. What you do choose, through the motor controller, is where the stator's current vector points relative to the rotor. That angle is γ. Pick a current magnitude, pick γ, and in an ideal machine the torque is a single constant number no matter where the rotor is. So the useful curve is torque vs γ, and its peak is the most torque you can get per amp.

Sweeping rotor position still has a job: at a fixed γ, it shows how much the torque wiggles around its average. That wiggle is torque ripple. If you only care about average torque, you can skip the position sweep; you cannot skip the γ sweep.

The machine being modeled

The drawing shows a 4-pole (p = 2 pole pairs), 12-slot, 3-phase synchronous reluctance motor (SynRM). The stator has one slot per pole per phase, in the classic pattern A, −C, B, −A, C, −B, repeated twice around. Slot colours show the current in each coil side: warm for current flowing out of the page, cool for current flowing in, stronger colour for more current.

The rotor has no magnets and no windings. Its curved slots are flux barriers: air (or other non-magnetic material) that makes it easy for magnetic flux to travel along one direction, the d-axis, and hard along the direction 90 electrical degrees away, the q-axis. The rotor therefore has a high inductance L_d along d and a low inductance L_q along q. The black tick labelled d marks one d-axis; the fainter ticks mark the other three.

A reluctance motor makes torque the way a piece of iron is pulled into a magnetic field: the rotor tries to turn its easy axis toward the stator's field. If the stator field keeps rotating just ahead of the d-axis, the rotor keeps getting pulled around.

Mechanical vs electrical angles

With p pole pairs, the magnetic pattern repeats p times per revolution. One full electrical cycle happens every 360°/p of mechanical rotation:

θ_e = p · θ_m

Here p = 2, so turning the rotor 180° mechanically is 360° electrically. γ, φ, the phase currents and the dq frame are all in electrical degrees. The rotor position slider is in mechanical degrees. Keeping these straight removes most of the confusion in motor math.

From three phases to d and q

Three balanced, sinusoidal phase currents 120° apart add up to a single current vector of constant length that rotates in the stator. In this toy the phase currents are:

i_a = I cos(θ_e + γ),   i_b = I cos(θ_e + γ − 120°),   i_c = I cos(θ_e + γ + 120°)

The Clarke transformation turns the three phase currents into two stationary components (α, β). The Park (dq0) transformation then rotates those into a frame that turns with the rotor, with d along the rotor's easy axis and q along its hard axis. In the amplitude-invariant form used here:

i_α = (2/3)(i_a − ½ i_b − ½ i_c),   i_β = (1/√3)(i_b − i_c) i_d = i_α cos θ_e + i_β sin θ_e,   i_q = −i_α sin θ_e + i_β cos θ_e

Plugging the phase currents in gives the payoff of the whole transformation:

i_d = I cos γ,   i_q = I sin γ

The rotor position has dropped out. Seen from the rotor, the three AC phase currents are just two constant (DC) numbers set by I and γ. This is why motor controllers work in the dq frame, and it's the heart of field-oriented control (FOC, also called vector control).

Where the torque comes from

Each axis has its own inductance, so the flux linkages are:

ψ_d = L_d i_d,   ψ_q = L_q i_q

Electromagnetic torque is the cross product of flux linkage and current, scaled by the number of pole pairs:

T = (3/2) · p · (ψ_d i_q − ψ_q i_d) = (3/2) · p · (L_d − L_q) · i_d i_q T = (3/4) · p · (L_d − L_q) · I² · sin 2γ

A few things fall straight out of this:

• Torque needs saliency. If L_d = L_q, ψ is parallel to i and torque is zero at every γ. The design goal for a SynRM rotor is to make L_d − L_q (and the ratio L_d/L_q) as large as possible.
• Torque scales with I², so doubling the current quadruples the torque in this unsaturated model.
• The best angle for torque per amp is γ = 45°, where sin 2γ = 1. This operating strategy is called MTPA, maximum torque per ampere.
• γ between 0° and 90° gives motoring torque (pulling the rotor forward). γ between −90° and 0° gives negative torque (braking or generating).

The factor 3/2 comes from the amplitude-invariant scaling used above. With the power-invariant version of the transformation it disappears; both conventions are common, so check which one a reference uses before comparing numbers.

The two modes

Controller-locked (γ held)

This is what a real drive does. A position sensor (or an estimator) reads θ_m, and the controller sets the phase currents so the current vector always sits at angle γ ahead of the d-axis. As the rotor turns, the currents rotate with it. Drag the rotor or press Spin rotor: the MMF arrows follow the rotor, i_d and i_q stay fixed, and the torque trace is a flat line.

In a full drive, an outer speed loop decides how much torque it needs and sets I accordingly; the choice of γ for that I comes from a rule like MTPA. The γ slider stands in for that rule.

Frozen DC currents

Here the phase currents are frozen at a fixed stator-frame angle φ, like holding a stepper motor in place. Now moving the rotor changes γ:

γ = φ − p · θ_m

Drag the rotor slowly and watch the dot in the torque-vs-γ panel: it slides along the same sin 2γ curve. That is why "hold the excitation fixed, sweep the rotor, find the peak torque" gives the right answer for peak torque. It is a γ sweep in disguise. The peak is the pull-out torque: push past it and the rotor slips a pole.

Press Spin rotor in this mode and the torque swings positive and negative, completing 2p = 4 cycles per revolution and averaging to exactly zero. Fixed currents cannot drive a spinning rotor. The controller has to track the rotor, which is the whole point of controller-locked mode.

Switching modes keeps the physical state continuous: the currents and rotor stay where they were, and only the meaning of the top slider changes.

Reading each panel

Machine

Stator slots coloured by current, rotor with flux barriers, orange arrows for the stator MMF (the direction the stator's field points, two arrows because there are two north poles), and a curved black arrow for the torque on the rotor. A counter-clockwise arrow means the rotor is pushed toward increasing θ_m, so spinning forward with positive torque is motoring.

Torque vs γ

The ideal (3/4)p(L_d−L_q)I² sin 2γ curve for the current settings. The orange dot is the present operating point; the dashed line marks the MTPA peak. This is the one curve a design comparison really needs at a given current.

Torque vs rotor position

Points traced as you move or spin the rotor. Each time a setting other than position changes, a new trace starts and older ones fade, so you can stack several γ values and see them as separate horizontal bands. The dashed line is the true one-revolution average for the current settings, not the average of the points you happened to trace.

Phase currents

i_a, i_b, i_c as functions of the current-vector angle (θ_e + γ in locked mode, φ in frozen mode). The vertical orange line marks where the drive is now, and the dots are the instantaneous phase currents. In locked mode the line sweeps as the rotor turns; in frozen mode it stays put.

Rotor (dq) frame

The current vector (orange) at angle γ from the d-axis, and the flux-linkage vector (purple), which bends toward d because L_d > L_q. The shaded parallelogram's area is ψ × i, proportional to torque. Try setting L_q equal to L_d and watch it collapse.

State

The numbers behind the pictures: electrical angle, γ, i_d, i_q, ψ_d, ψ_q, instantaneous torque, and the one-revolution average (highlighted), which is what "the motor's torque" usually means.

About the toy ripple

The ripple slider adds a made-up term to show what ripple looks like, not to predict it:

ΔT = r · T_pk · cos(6 θ_e)

A 6th electrical harmonic is a natural choice for a balanced 3-phase machine, and for this 12-slot, 4-pole layout the slot pattern also repeats every 60 electrical degrees. Real torque ripple comes from the discrete slots, the flux barrier layout, saturation, and non-sinusoidal currents from the inverter, and its size and harmonic content depend strongly on the geometry. It is commonly quoted as (T_max − T_min) / T_avg at a given operating point. Notice that adding ripple leaves the average unchanged: that's why average torque and ripple can be studied separately.

What a real machine adds

• Saturation. Iron saturates, so L_d falls as i_d grows. That shifts the MTPA angle above 45°; for steel SynRMs it often lands somewhere around 55–65° at rated current. Low-permeability rotor materials saturate less and stay closer to 45°.
• Cross-saturation. Current on one axis changes the inductance of the other, so ψ_d depends on i_q too. Linear L_d, L_q can't capture this; flux maps can.
• Voltage limits. At speed, the back-EMF terms ω·ψ grow until the inverter runs out of voltage. Steady-state dq voltages are v_d = R i_d − ω ψ_q and v_q = R i_q + ω ψ_d, where ω is the electrical speed. Past that point the controller moves γ further from d to reduce flux (field weakening), trading torque for speed.
• Losses and power factor. Copper loss, iron loss, and a SynRM's modest power factor all matter for a real design and are absent here.
• Spatial harmonics and ripple, as described above.

Mapping this onto a field simulation

If you are running a magnetostatic finite element (or similar) solver, the toy suggests a workflow that grows with your needs:

1. Average torque at one current. Fix |I|. Sweep γ over roughly 0° to 90° electrical, doing one static solve per γ at any rotor position, with phase currents set from I, γ, and that position. The peak of T(γ) is your MTPA torque. Torque vs γ is smooth with one peak, so a handful of points and a curve fit, or a golden-section search, is enough.
2. Ripple. At the best γ, keep the currents locked to the rotor (controller-locked mode) and step the rotor through one ripple period. A good first guess for that period is 360°/LCM(slots, poles) mechanical, which is 30° for this 12-slot, 4-pole layout; check that the result actually repeats. The average of this sweep refines step 1, and max minus min is the ripple.
3. Everything else: flux maps. Compute ψ_d(i_d, i_q) and ψ_q(i_d, i_q) on a grid of currents (each averaged over a ripple period if you like). Torque, MTPA curves, inductance saturation curves, voltage limits and efficiency maps then come from the equations above, without new solves. This is the approach used by open-source tools like SyR-e.

For extracting torque from a field solution, the usual method is the Maxwell stress tensor integrated over a surface in the middle of the air gap. It is sensitive to mesh quality, so a good check is to compare it with (3/2)p(ψ_di_q − ψ_qi_d) computed from the solver's flux linkages, or with a virtual-work (co-energy) estimate from two nearby rotor positions. Agreement within a few percent means the gap mesh is probably fine.

A note on axial-flux machines

In an axial-flux motor the air gap is a flat annulus rather than a cylinder, and flux crosses it parallel to the shaft. None of the dq math changes: d and q axes, γ, the torque equation and the modes all apply as-is. The practical differences are in the field solution. The stress-tensor surface becomes a plane at mid-gap, torque is the integral of r × (tangential force) over that annulus, and because the geometry changes with radius, full 3D solves or several 2D "slices" at different radii are common.

Glossary

d-axis, q-axis

The rotor's easy and hard magnetic directions, 90 electrical degrees apart. The dq frame rotates with the rotor.

γ (current angle)

Angle of the stator current vector measured from the rotor's d-axis, in electrical degrees. Some texts measure it from the q-axis instead, which shifts every formula by 90°.

φ (excitation angle)

Angle of the current vector in the stationary stator frame. Used here only for frozen mode.

Pole pairs, p

Number of north-south pairs on the rotor. Sets the ratio between electrical and mechanical angles.

MMF

Magnetomotive force: the "push" that stator currents give to magnetic flux. Its direction is the orange arrows.

Flux linkage, ψ

Total magnetic flux passing through a winding, weighted by turns. Inductance is flux linkage per amp.

Saliency

The difference between L_d and L_q. No saliency, no reluctance torque.

MTPA

Maximum torque per ampere: choosing γ to get the most torque from a given current.

FOC

Field-oriented control, the standard way to run synchronous motors by regulating i_d and i_q in the rotor frame.

Torque ripple

Periodic variation of torque around its average as the rotor turns.

Cogging torque

Position-dependent torque with no current applied. A permanent-magnet effect; an ideal pure SynRM has none.

References and further reading

• Reluctance motor (Wikipedia) — overview of reluctance machines, including synchronous reluctance.
• Field-oriented control (Wikipedia) — how drives regulate d and q currents.
• Direct-quadrature-zero (Park) transformation (Wikipedia) — derivation and conventions.
• Clarke transformation (Wikipedia) — three phases to α, β.
• Reactances of synchronous machines (Wikipedia) — d- and q-axis quantities in synchronous machines.
• Torque ripple and cogging torque (Wikipedia).
• Torque ripple glossary entry (Emetor) — how ripple is usually defined and quoted.
• Maxwell stress tensor (Wikipedia) — the standard way to pull torque out of a field solution.
• Axial flux motor (Wikipedia).
• SyR-e (GitHub) — open-source MATLAB/Octave toolkit for designing and evaluating synchronous reluctance and PM machines with FEMM, including flux maps and MTPA.
• SyR-e tutorial (GitHub) — beginner-friendly walkthrough covering torque, ripple and flux maps.

This is a toy: linear inductances, ideal sinusoidal currents, and a made-up ripple term. Use it to build intuition, not to size a motor.