1Current vectorEnter the stator current however you have it: as its d-/q-axis split (Id, Iq), or as a magnitude and angle. Amplitude-invariant convention — Id, Iq and the magnitude are all peak values, and |Is|_peak = √(Id²+Iq²) equals the peak phase current (RMS = peak/√2). The angle γ is measured from the d-axis; Id<0 is the flux-weakening / reluctance-assist direction.

Enter from

Id (A, peak)Negative = flux-weakening / reluctance-assist.

Iq (A, peak)Torque-producing component.

2Motor parametersThese drive the torque, MTPA and back-EMF outputs. Presets are illustrative representative values for each machine class — replace with your motor's characterised parameters for real work. A surface-PM machine has no saliency (Lq = Ld), so its MTPA is simply Id = 0.

Preset

Motor type

Salient: Lq > Ld gives reluctance torque and a non-trivial MTPA angle.

Pole pairs

PM flux linkage λpm (Wb)Peak per-phase. Ke(V·s/rad) = λpm; line-line back-EMF const scales with √3.

Ld (mH)

Lq (mH)

3Operating speedOptional — adds the electrical frequency, phase back-EMF, and ideal mechanical shaft power at this speed. Back-EMF is the open-circuit magnet voltage ωe·λpm; mechanical power is T·ωm with no losses subtracted.

Include speed-dependent outputs

Mechanical speed (rpm)

Current conversions

Id (peak)

-20A

Iq (peak)

80A

|Is| peak

82.5A

|Is| rms

58.3A

Angle γ (from d)

104°

Advance β (from q)

14°

Space-vector diagram

TorqueThe magnet term (λpm·Iq) is the surface-PM torque; the reluctance term ((Ld−Lq)·Id·Iq) is the extra torque a salient rotor produces from a negative Id. Together they give the total shaft torque before losses.

Magnet torque	29.76 N·m
Reluctance torque	2.88 N·m
Total torque	32.64 N·m

MTPA comparison (same |Is|)Maximum-torque-per-ampere is the Id/Iq split that extracts the most torque from a given current magnitude. This compares your entered operating point against the MTPA-optimal point at the same |Is| — the gap is torque (and copper loss) left on the table.

MTPA Id* / Iq*	-26.2 / 78.2 A
MTPA current angle γ*	108.6°
MTPA torque	32.77 N·m
vs entered point	+0.4% available

At 6,000 rpmElectrical frequency f_e = p·n/60; open-circuit phase back-EMF Ê = ωe·λpm; ideal mechanical power P = T·ωm (lossless).

Electrical frequency	400 Hz
Back-EMF (peak / rms, per phase)	155.8 / 110.2 V
Mechanical power	20.51 kW

Reference & assumptions

New to Field-Oriented Control? Read the guide: Id/Iq & Field-Oriented Control: The Park Transform and MTPA — a plain-language explainer of the standard behind this calculator.

Uses the amplitude-invariant Clarke/Park transform (the standard motor-control convention), so Id, Iq and the current magnitude are all peak-amplitude quantities and the d-q vector length equals the peak phase current: |Is|_peak = √(Id² + Iq²), with |Is|_rms = |Is|_peak / √2. The current angle γ is measured counter-clockwise from the d-axis (rotor flux axis); the advance angle β = γ − 90° is the same vector measured from the q-axis, the convention many drives use. Torque is the standard salient-PMSM expression T = (3/2)·p·[λpm·Iq + (Ld − Lq)·Id·Iq] — a magnet term plus a reluctance term that only exists when the rotor is salient (Lq ≠ Ld). Maximum-torque-per-ampere uses the closed-form d-axis solution Id* = [λpm − √(λpm² + 8·(Lq−Ld)²·|Is|²)] / [4·(Lq−Ld)] at a fixed current magnitude, which reduces to Id* = 0 for a non-salient (surface-PM) machine. Back-EMF is the open-circuit magnet voltage ωe·λpm and mechanical power is T·ωm. Idealisations: linear magnetics (constant Ld, Lq and λpm — no magnetic saturation or d-q cross-coupling), purely sinusoidal quantities, and no stator-resistance, iron, or mechanical losses in the power and back-EMF figures. Real machines saturate, so Ld, Lq and λpm all vary with operating point; treat these as first-order estimates and use characterised look-up-table parameters for accurate torque and MTPA work. Applies to permanent-magnet synchronous machines — an induction machine's field orientation is a different, slip-based model.

Validated: a 3-4-5 current triangle (Id=3A, Iq=4A) round-trips through the magnitude/angle conversion exactly (5.000 A, 53.13°, back to 3/4A). Torque matches hand calculation exactly for both a non-salient case (magnet-only, no reluctance term) and a salient interior-PM case (both terms non-zero). The MTPA closed form was checked two ways: it returns exactly Id=0 for a non-salient machine, and for a salient one, its predicted operating point was independently confirmed to actually maximize torque by a brute-force 0.5°-resolution sweep of the same current-magnitude circle — the closed form and the numerical search landed on the same torque to 4 significant figures.

Calculation steps

1. Current-vector magnitude and angle (amplitude-invariant dq)

|Is|_peak = √(Id² + Iq²), |Is|_rms = |Is|_peak / √2, γ = atan2(Iq, Id)

Id = -20 A, Iq = 80 A

|Is| = 82.46 A peak = 58.31 A rms, γ = 104° from d-axis (β = 14° from q-axis)

2. Electromagnetic torque (magnet + reluctance)

T = (3/2)·p·[ λpm·Iq + (Ld − Lq)·Id·Iq ]

p = 4, λpm = 0.062 Wb, Ld = 0.22 mH, Lq = 0.52 mH

T = 29.76 (magnet) + 2.88 (reluctance) = 32.64 N·m

3. MTPA operating point at the same |Is|

Id* = [ λpm − √(λpm² + 8·(Lq−Ld)²·|Is|²) ] / [ 4·(Lq−Ld) ], Iq* = √(|Is|² − Id*²)

|Is| = 82.46 A peak

Id* = -26.24 A, Iq* = 78.18 A (γ* = 108.6°), T* = 32.77 N·m — 0.4% more torque than the entered point

4. Speed-dependent quantities

f_e = p·n/60, Ê = ωe·λpm, P_mech = T·ωm

n = 6,000 rpm, p = 4

f_e = 400 Hz, back-EMF = 155.8 V peak (110.2 V rms/phase), P_mech = 20.51 kW

Id / Iq Current Vector Calculator