Maximum-Torque-Per-Ampere Control of an Outer-Rotor In-Wheel IPMSM: Discrete Design, Implementation and Benchmarking against id=0 Vector Control

In a distributed-drive electric vehicle, each wheel carries its own motor. The mechanical differential, the drive shafts and the gearbox disappear, and every wheel torque can be commanded independently, which benefits vehicle stability, regenerative braking and chassis packaging[1-3]. Among the machines considered for this duty, the outer-rotor IPMSM stands out for its high torque density at low speed, short axial length and mechanical robustness [4,5]. With a unity transmission ratio, the torque produced per ampere of the in-wheel machine translates directly into vehicle acceleration, gradeability and thermal margin. Unequal direct- and quadrature-axis inductances make the electromagnetic torque of an IPMSM consist of two parts, the PM alignment torque and the reluctance torque: Setting id = 0 keeps the current vector on the q-axis and discards the reluctance term altogether. MTPA control, in contrast, biases the vector into the second quadrant (id < 0) so that the largest possible torque is obtained per ampere, which reduces copper loss, relaxes the inverter current rating and eases the thermal budget [6-9]. Reported MTPA realizations fall into four categories: on-line searching[10], signal-injection self-tuning[11], off-line fitted look-up tables [12], and analytical feed-forward allocation [13-15]. The first two add computation or acoustic-noise overhead; look-up tables demand extensive off-line characterization. Analytical allocation expresses id as an explicit function of iq, which suits low-cost automotive microcontrollers best. Research that addresses the specific constraints of in-wheel operation, low base speed, many pole pairs, restricted dc-bus utilization of a wheel-integrated inverter, and the discrete-implementation effects of a low switching frequency, is still scarce. Most comparison studies, moreover, verify the MTPA benefit only at isolated operating points rather than over a continuous load sweep. This paper contributes (i) complete, reproducible implementation parameters for an analytical-MTPA cascaded controller on a 1.5 kW, 4-pole-pair OR-IPMSM fed from a 560 V bus; (ii) a pole-placement PI design that explicitly accounts for the one-sample delay of a 6 kHz digital realization, together with back-calculation anti-windup and feed-forward decoupling; and (iii) a quantitative comparison with id=0 control over a 2–14 N·m load sweep, reporting both current and copper-loss reductions and verifying the theoretical MTPA trajectory against steady-state simulation data.

The machine under study is a three-phase star-connected OR-IPMSM; its parameters are collected in Table 1. With the usual assumptions of sinusoidal back-EMF, negligible cross-saturation and no iron loss, the rotor-reference-frame voltage and motion equations describe the motor dynamics. At the control level, the outer-rotor topology does not alter the d-q model; its influence enters only through the parameter set of Table 1. Let delta_L = Lq - Ld > 0. Taking the negative root gives the explicit allocation law, which stays monotone over the whole motoring quadrant. In operation, the speed-loop PI first produces iq*, and Eq. (5) then sets id* directly. At any given torque the MTPA operating point requires a visibly smaller current magnitude than its id=0 counterpart. Obtaining (id*, iq*) from a torque command by inverse solving would instead require a fourth-order polynomial[14]. The route adopted here bypasses that burden: along the MTPA curve, allocating id* from iq* is algebraically equivalent and remains numerically well-behaved in either floating- or fixed-point arithmetic.

The drive keeps the classical cascaded arrangement. The outer speed PI converts the speed error into iq*. The MTPA law maps iq* to id*. Two inner PI loops regulate id and iq; their outputs, augmented by the decoupling terms, are returned to the stationary frame by the inverse Park transformation and passed to a seven-segment SVPWM that commands the two-level three-phase inverter. Once the decoupling terms are active, each current loop reduces to the first-order plant Gx(s)=1/(Rs+sLx), x in {d,q}. Choosing Kpx/Kix = Lx/Rs cancels the plant pole, so that the open loop becomes wc/s with wc = 800 rad/s (Kpd = 1.86, Kpq = 3.10, Kid = Kiq = 88). The total digital delay of the 6 kHz loop introduces a phase lag of about 11.5 deg at crossover, leaving a phase margin of about 78 deg; wc is about 1/47 of the sampling angular frequency. The speed loop is closed at wcs = 20 rad/s using the PM torque constant Kt = 0.12 N·m/A, i.e., Kp_w = 1.67 and Ki_w = 6.67. Under MTPA, the effective torque per unit q-axis current exceeds Kt (about 0.34 N·m/A at 10 N·m), so the same gains yield a higher effective speed-loop bandwidth. Sensitivity runs further show that varying Kp_w between 1.67 and 3.5 leaves the startup transient overshoot-free. Both current PI outputs are clamped to +/-323.3 V, and their integrators are protected by back-calculation anti-windup with tracking gain Kb = Ki/Kp; the speed-loop output is limited to +/-120 A. Feed-forward decoupling compensation is applied. The seven-segment SVPWM derives the sector number from the signs of three projection variables, computes the adjacent-vector dwell times from the X/Y/Z expressions, rescales them proportionally during over-modulation, and assembles center-symmetric duty cycles. The 560 V dc bus sets the maximum linear modulation voltage at Udc/sqrt(3). Gate pulses for the two switches of a leg are obtained by comparing the duty reference with a symmetric 6 kHz triangular carrier; the low-side pulse is the strict logic inversion of the high-side one, so leg shoot-through is ruled out by construction.

The full drive is built in MATLAB/Simulink (R2022b). Power stage and control algorithms execute at a common discrete step of 1/6000 s inside a discrete powergui environment, so the effective control delay equals exactly one carrier period. The machine is represented by the built-in Simscape Electrical PMSM model with the data of Table 1; the controller settings are listed in Table 2.

All results in this section are exported from the discrete Simulink model; steady-state values are time-averages over the last 0.2 s of each run. Accelerating from rest toward 200 rad/s without load, both strategies reach the reference with small overshoot. At no load the reluctance torque is negligible and law (5) yields id* ≈ 0, so both strategies draw practically the same current; the residual steady-state currents in Table 3 are dominated by a ripple-induced bias in the current measurement chain and are not used for the efficiency comparison, which is carried out under load in Section 5.3. At 1500 r/min, the load is stepped from 5 N·m to 10 N·m at t = 0.5 s. Under id=0 control the speed dips by 20.2 rad/s and recovers within 0.123 s; under MTPA control the dip is only 7.5 rad/s and never leaves the ±5% error band. This difference is mainly attributable to the larger effective torque per q-axis current under MTPA, which raises the effective speed-loop gain for the same PI settings (Section 3.2). At 1500 r/min, six load points are selected within 2–14 N·m (2/6/8/10/12/14 N·m, all within the ±120 A limit). For each point the model runs for 0.9 s, the load is applied at 0.35 s after the speed settles, and the mean over 0.7–0.9 s is taken. Fig. 4 plots the stator-current magnitude against load torque; the simulated points agree well with the theoretical curves. The stator-current magnitude Is is computed from time-averaged d-q current values over the observation window. The simulated d-axis currents follow the MTPA law within 3.5% for 6–14 N·m (0.3% at 10 N·m). For 6–14 N·m, the simulated reductions (41.1%–60.3%) agree with theory (45.2%–60.5%) within about 4 percentage points. At 14 N·m the copper loss falls from 2026.6 W to 318.8 W, an 84.3% reduction that nearly equals the theoretical ceiling of 84.4%. The saving grows with load because the reluctance term scales with id*iq and Eq. (5) biases the vector further into the negative-id region as iq rises. It should be noted that the machine of Table 1 has a small PM flux linkage relative to its saliency (delta_L * |id| exceeds psi_f for |id| > 12.8 A), so the reluctance torque dominates at medium and high load; the reductions reported here are therefore larger than those expected for machines with a stronger magnet. With the stator-current magnitude limited to 120 A, id=0 control cannot exceed Kt * 120 = 14.4 N·m. Along the MTPA trajectory, the same current magnitude yields about 78 N·m in the linear magnetic model (id ≈ -81.7 A, iq ≈ 87.9 A). This value is only an upper bound: at such current levels magnetic saturation strongly reduces Lq, and the practical gain must be re-evaluated with a saturated machine model. Moreover, the present implementation limits iq* rather than |is|, so a current-magnitude limiter is required before this overload capability can be exploited.

An explicit analytical MTPA scheme for an outer-rotor in-wheel IPMSM has been implemented in discrete form and benchmarked against conventional id=0 control. The allocation law of Eq. (5) removes any need for on-line polynomial solving and costs almost nothing computationally. Pole-placement PI tuning that respects the one-sample delay of the 6 kHz loop, combined with back-calculation anti-windup, provides well-damped current loops and satisfactory robustness. Across the 2–14 N·m sweep at equal torque, MTPA lowers the stator current by 23.0%–60.5% in theory and 14.6%–60.3% in simulation, and the copper loss by as much as 84.3%, with a smaller speed dip under load disturbance. Future work will validate the scheme on a hardware prototype and extend the allocation law with magnetic-saturation and cross-coupling corrections for the full speed range.

This research was supported by the ANCHOR program through the Gyeongbuk ANCHOR CENTER, funded by the Ministry of Education (MOE) and Gyeongsangbuk-do, Republic of Korea. (Project No. 2026-ANCHOR-15-107)