Dual-Mode Torque Control for Low Torque Ripple Across the Speed Range in a 60 kW Switched Reluctance Motor Drive for Electric Vehicles: A Simulation Study

2.1 Machine and drive

The studied machine is a three-phase 24/16 SRM rated at 60 kW, sized for the traction of a passenger electric vehicle. Its rated and circuit parameters are listed in Table 1. The 24-stator/16-rotor pole combination gives a rotor pole pitch of 22.5° mechanical, which the model uses as the periodic base for the rotor angle; the magnetic tables are tabulated over a 45° span of rotor position. Each phase is fed by an asymmetric half-bridge converter from a 500 V DC bus — the standard converter for SRM drives, because it allows each phase to be magnetized, freewheeled or demagnetized independently [1].

Table 1. Switched reluctance motor (SRM) and drive parameters

2.2 Electromagnetic model

The SRM is represented by the standard flux-linkage model. For each phase, the terminal-voltage equation is:

$v=R_s i+\frac{d \psi}{d t}$    (1)

where, v is the phase voltage imposed by the converter, R_s the phase resistance, i the phase current and ψ the flux linkage. The flux linkage is therefore obtained by integration,

$\psi(t)=\int\left(v-R_s i\right) d t$    (2)

The phase current is recovered from the flux linkage and the rotor position θ through the inverse magnetization characteristic,

$i=f_i(\psi, \theta)$    (3)

and the instantaneous electromagnetic torque of the phase from

$T=f_T(i, \theta)$    (4)

The two functions f_i and f_T of Eqs. (3) and (4) are two-dimensional lookup tables. f_T is the static-torque table T(i,θ); f_i is obtained by inverting the flux-linkage table ψ(i,θ): for every rotor angle the (current → flux) curve is inverted by cubic-spline interpolation and resampled on a regular flux grid.

The flux-linkage and static-torque tables were obtained by magnetostatic finite-element characterization of the 24/16 machine in Simcenter Motorsolve (switched-reluctance module): the phase current and the rotor position were swept over the grid below and the flux linkage and static torque computed at each grid point. The material data of the finite-element model are used as supplied, and no thermal dependence of the magnetization data is represented; experimental verification of the tables on the prototype machine is part of the planned hardware phase. The tables span 0–400 A in steps of 10 A (41 current levels) and 0–45° mechanical (two pole pitches) at a nominal resolution of 1.5° (31 rotor positions, with additional support points around the unaligned position at 23.5°), i.e. 41 × 31 points per table. Both tables are evaluated at run time with cubic-spline interpolation inside the tabulated grid. The interpolation error was quantified by leave-one-out cross-validation over the interior current lines of the grid: the reconstruction error does not exceed 0.45% of full scale for the flux-linkage table (0.035% root-mean-square, RMS) and 0.06% of full scale for the static-torque table (0.005% RMS), so the interpolated maps contribute negligibly to the uncertainty of the torque estimator and of the current-reference mapping. Outside the tabulated grid a bounded linear extension of the last tabulated characteristics is used; as shown in Section 2.3, converter overcurrent protection confines the drive to the tabulated region in normal operation, and the reported results are insensitive to the choice of extension (≤0.5 percentage points at the single operating corner where a brief excursion occurs). The magnetic characteristics are shown in Figure 2: the flux linkage saturates strongly near the aligned position, and the static torque is a markedly nonlinear function of both current and position — the very nonlinearity responsible for the torque ripple.

The total electromagnetic torque is the sum of the three phase torques,

$T_e=\sum_{k=1}^3 T_k$     (5)

and the mechanical motion follows

$J \frac{d \omega}{d t}=T_e-T_L-K_f \omega$    (6)

where, J is the rotor inertia, ω the mechanical speed, T_L the load torque and K_f the viscous-friction coefficient. The rotor position of each phase is obtained by integrating the speed and reduced modulo the 22.5° pole pitch.

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Figure 2. Finite-element-computed flux-linkage and static-torque characteristics of the 24/16 switched reluctance motor (SRM)

2.3 Discrete-time implementation and verification

The complete drive — flux integrators, lookup tables, converter, mechanical equation and controllers — is implemented in discrete time with a forward-Euler scheme at a sample period T_s = 10 μs for the hysteresis controllers and T_s = 1 μs for the fixed-frequency PWM controllers, so that the 20 kHz carrier is resolved. The flux integrator of Eq. (2) is clamped to non-negative values to represent the blocking action of the converter diodes, and a converter overcurrent protection at 400 A — the boundary of the tabulated magnetic map, and a realistic peak-current setting for a 60 kW converter — blocks any positive phase voltage while the phase current is at or above the limit. With this protection active, all steady-state operating points reported in this paper remain inside the tabulated magnetic region; the single exception is the 300 rpm, 200 N·m corner, where the sampled (10 μs) protection allows brief excursions to at most 450 A, about 12% beyond the tabulated current grid. Replacing the linear table extension by a hard saturation-aware clamp at this corner changes the reported ripple by no more than 0.5 percentage points, so the conclusions do not depend on the extrapolation method. The model was verified in three steps. First, a timestep-convergence check: refining the plant integration step from 10 μs to 1 μs while holding the control update at its 100 kHz digital rate changes the steady-state ripple of the hysteresis DITC drive at the rated point by less than 0.3 percentage points (8.8% versus 9.1%), confirming that the reported figures are not integration artefacts. Second, internal consistency: the mean electromagnetic torque matches the applied load torque at every settled operating point, and the phase currents and flux linkages remain within the tabulated maps as stated above. Third, a cross-check against an earlier Simulink reference of the same drive at the matched rated point (200 N·m, 2500 rpm), where the steady-state speed agrees within about 13% (roughly 300 rpm); the residual difference is attributable to the detailed semiconductor and dead-time models of the reference converter versus the idealized switching used here. A systematic multi-point cross-validation of the model over the full operating envelope is ongoing and will accompany the experimental phase of the project. The torque ripple percentages reported in the following sections should therefore be read as model-based predictions whose absolute values may shift on hardware — particularly through dead-time, device on-state voltage drops and finite switching transitions, quantified in Section 5.4 — pending the experimental confirmation that Section 5 identifies as the essential next step.

All results are obtained with the discrete-time model of Section 2 at a rated load of 200 N·m unless stated otherwise. Torque ripple is quantified by Eq. (12) as the steady-state peak-to-peak torque excursion expressed as a percentage of the mean torque,

$\tau_{\text {ripple }}=\frac{T_{\max }-T_{\min }}{T_{\text {avg }}} \times 100 \%$   (12)

4.1 Machine characteristics

Figure 2 shows the finite-element-computed magnetic characteristics used by the model. The flux-linkage curves exhibit the expected behaviour: near the aligned position the flux linkage saturates strongly above about 100 A, whereas near the unaligned position it remains almost linear. The static torque is strongly nonlinear in current and varies strongly with rotor position: it rises steeply with current, is largest near mid-stroke, and vanishes at the aligned and unaligned positions where the rotor sits in magnetic equilibrium. It is precisely this nonlinearity, combined with the discrete energization of the phases, that produces the torque ripple addressed in the remainder of the paper.

4.2 Controller comparison and torque ripple

The two controllers were run on the identical drive at a speed reference of 2500 rpm with a 200 N·m load. For SMC the firing angles were set to their optimum (turn-on 2.5°, turn-off 14.5°); for DITC the coordinate-descent loop returned the angles given in Section 3.3. The steady-state torque waveforms are compared in Figure 3 and the ripple figures are summarized in Table 2.

Figure 3. Steady-state torque of the sliding-mode control (SMC) and direct instantaneous torque control (DITC) drives (2500 rpm, 200 N·m)

Table 2. Steady-state torque ripple of the sliding-mode control (SMC) and direct instantaneous torque control (DITC) drives (2500 rpm, 200 N·m)

The first result concerns the sliding-mode speed controller. Despite its nonlinear law, the SMC leaves the steady-state torque ripple at ≈23.6%. This is not a tuning artefact. With the current-controlled inner loop, the drive operates close to its torque–speed ceiling: during each stroke the hysteresis current loop saturates against the available DC-bus voltage, so the torque waveform is dictated by the machine and the firing angles, not by the speed controller. A more elaborate speed-loop law therefore cannot, by itself, alter the steady-state ripple. The practical implication is important: for this drive the choice of speed-loop controller is secondary, and torque ripple reduction must come from the inner loop.

One could object that this comparison confounds the outer control law with the inner-loop realization: the SMC drive uses a two-level current hysteresis, whereas DITC uses a three-level torque hysteresis with two-mode commutation. To isolate the outer law, a second baseline was therefore simulated in which the SMC speed controller drives a fixed-frequency PWM current-tracking inner loop under exactly the same converter constraint as the PWM-DITC drive of Section 4.7 (20 kHz carrier, 320 A current limit, identical demagnetization logic, the SMC’s own firing angles). Under this matched switching constraint the SMC drive produces 33.5% ripple at the rated point, and between 54% and 58% over the 300–1500 rpm range, versus 7.0–12.3% for PWM-DITC under the identical constraint (Section 4.7). The gap therefore does not originate in the switching realization: a current-reference controller regulates the wrong variable, because at constant current the torque of an SRM still varies strongly with rotor position, whereas DITC regulates the torque itself.

DITC acts exactly there. By regulating the instantaneous total torque directly, DITC reduces the steady-state ripple to 8.8% at the model’s native 100 kHz control rate — an almost threefold improvement over the SMC value — at the same rated operating point. Figure 3 makes the mechanism visible: the SMC torque waveform swings widely between strokes, whereas the DITC waveform is held within a much narrower band around the demand. The residual 8.8% is set by the rate at which the torque-hysteresis loop can act; on a 60 kW drive that rate is bounded by the achievable switching frequency, which motivates the fixed-frequency PWM-DITC realization of Section 4.7.

4.3 Low-speed operation

Torque ripple in an SRM is generally most severe at low speed, an operating region that an EV uses continuously in urban driving and during launch. Figure 4 reports the steady-state ripple of the DITC and SMC drives over the 300–2500 rpm range at a 200 N·m load. Two trends are visible. First, DITC outperforms SMC by a wide margin at every speed, by a factor of about three at rated speed and four to five over the low and mid-range: the SMC ripple stays near 48–50% over most of the range, whereas the DITC ripple lies between 9% and 16%. Second, the DITC ripple itself increases as the speed decreases — from 8.8% at 2500 rpm to 15.9% at 300 rpm. The cause is physical: at low speed the back electromotive force (EMF) is small, so the phase current and hence the torque change very rapidly when a converter state is applied, and the discrete hysteresis controller therefore overshoots its band by more per step. Low-speed operation is consequently the weakest point of hysteresis DITC. It remains markedly better than the SMC baseline, but the result indicates that, for an EV that must deliver smooth torque from standstill, a faster torque loop or a PWM-DITC realization would be needed to keep the low-speed ripple within single digits.

Figure 4. Torque ripple versus speed for direct instantaneous torque control (DITC) and sliding-mode control (SMC) at 200 N·m

4.4 Robustness to inertia mismatch

The effective inertia seen by an EV traction motor varies with vehicle load and driving conditions, so the controller must tolerate a mismatch between the design inertia and the actual one. Figure 5 shows the enhanced DITC subjected to a four-to-one inertia variation, from one-half to twice the nominal rotor inertia, during a start-up followed by a load step from 100 to 250 N·m. The speed response remains stable and free of overshoot for every inertia: a lighter rotor simply accelerates faster and a heavier one slower, as the physics dictates, and the load disturbance is rejected cleanly in all cases. The steady-state torque ripple, shown in the lower panel, stays between 7.4% and 8.6% across the whole inertia range — a spread of barely one percentage point. This weak sensitivity is a direct consequence of the control structure: the torque ripple is set by the inner torque-hysteresis loop, which acts on electrical quantities and does not see the rotor inertia, while the inertia affects only the outer speed loop. It should be noted that this result concerns the steady-state electromagnetic torque ripple of the simulated drive over the tested 0.5–2.0 times inertia range only; it is not claimed to extend to all EV drivetrain conditions, where compliant couplings, gear backlash and load-side dynamics may interact with the drive in ways the present rigid single-inertia model does not represent.

Figure 5. Robustness of the direct instantaneous torque control (DITC) drive to rotor-inertia mismatch over the tested 0.5–2.0 × range

4.5 Speed-loop enhancement

The separability of the inner torque loop and the outer speed loop, already invoked in Section 4.4, can be exploited deliberately to improve the speed-loop dynamics without any penalty on torque ripple. The property is demonstrated here with a load-step test at speed, a disturbance the drive experiences continuously in traction (gradients, road irregularities): with the drive settled at 2000 rpm and 200 N·m, the load torque is stepped to 300 N·m. In the initial DITC implementation the outer speed loop carried untuned gains (K_p = 2, K_i = 60); re-tuning them to K_p = 6, K_i = 40 — values giving a well-damped closed speed loop — produces the comparison of Figure 6 and Table 3. With the original gains the speed dips by 43.8 rpm and re-enters the ±1% band only after 34 ms; with the re-tuned gains the dip is below 1 rpm and the speed never leaves the band — a reduction of the disturbance excursion by more than a factor of forty. The lower panel of Figure 6 confirms that the steady-state torque behaviour is untouched: before the step the ripple is 8.4% and 9.4% for the two gain sets, and after the step it is 7.6% and 7.5% respectively — identical within the stroke-to-stroke variation — because the inner torque loop has not been modified. All currents in this test remain within the 400 A converter limit and hence within the tabulated magnetic map.

Figure 6. Direct instantaneous torque control (DITC) outer-loop re-tuning under a 200→300 N·m load step at 2000 rpm

Table 3. Direct instantaneous torque control (DITC) outer-loop re-tuning under a 200→300 N·m load step at 2000 rpm

This is worth emphasizing as a design guideline. In the original current-controlled drive (Section 4.2), outer-loop tuning was ineffective because the saturated inner loop dominated the response. Under DITC the inner loop tracks the torque demand quickly and accurately, so the outer speed loop sees a clean “torque-in, speed-out” plant and can be tuned aggressively for fast, well-damped disturbance rejection. The speed dynamics and the torque ripple can thus be designed independently — a practical advantage of DITC for an EV drive, where both a responsive pedal feel and low torque ripple are required.

4.6 Representative composite speed–load test profile

To assess the drive under conditions representative of EV operation, the enhanced DITC (Section 4.5) was run through the composite speed–load test profile of Figure 7. The profile is defined manually at the drive level: it is not derived from a vehicle-dynamics model, and no vehicle mass, wheel radius, gear ratio, road slope or aerodynamic-drag parameter is assumed. It should therefore be read as a representative composite test profile that exercises the controller, not as a standardized EV drive cycle. The profile chains the four regimes that a traction drive must handle: a low-speed crawl at 300 rpm, an acceleration to 1500 rpm followed by an urban cruise, a second acceleration to 2200 rpm for highway operation during which the load is stepped from 200 to 300 N·m to represent a gradient or a head-wind, and a deceleration back to 800 rpm. The run is started from an already settled crawl, so that the response shown is free of the start-up transient.

Figure 7. Enhanced direct instantaneous torque control (DITC) over the representative composite speed–load test profile

The three panels report the speed, the torque and the speed-tracking error. The measured speed tracks the reference to within 1.7% in every steady segment (0.2–0.4% at the cruise and highway points, 1.6% at the 300 rpm crawl, where one rpm is already 0.3%) and within about 3.7% momentarily during the reference ramps. The torque panel shows the electromagnetic torque together with the load torque: the electromagnetic torque rises above the load during each acceleration — the net torque needed to accelerate the rotor inertia, peaking at 311 N·m and always within the 400 A converter limit — sits on the load during cruise, follows the 200-to-300 N·m load step without sustained error, and falls below the load during the deceleration. The torque ripple stays at the level established in Section 4.3: 8.2% at the 300 N·m highway segment, 10.8% at the urban cruise, 13.9% at the 800 rpm cruise and 15.6% at the 300 rpm crawl. The drive thus retains accurate speed tracking and tight torque regulation across the complete profile.

4.7 Fixed-frequency pulse-width-modulated direct instantaneous torque control

Hysteresis DITC reduces the rated-speed ripple to 8.8% (Section 4.2), but it does so through a fast torque-hysteresis loop, and on a 60 kW power stage the achievable rate is limited: the insulated-gate devices switch realistically at only 10–20 kHz, and the hysteresis law moreover produces a variable, load-dependent switching frequency that complicates the design of the electromagnetic-interference (EMI) filter and the thermal management. A practical realization must therefore decouple the torque-control action from the switching frequency of the power devices and fix the latter at a value chosen by the designer.

This is achieved by a fixed-frequency PWM-DITC. The torque-hysteresis comparator is replaced by a proportional–integral torque regulator whose output is a duty ratio; the duty is sampled once per carrier period and applied to the regulation phase through a pulse-width modulator running at a fixed carrier frequency f_pwm, which becomes the switching frequency of the devices. The incoming phase is excited through the same converter, and a converter current limit of 320 A — a standard hardware protection — bounds the phase current and regularizes the commutation. The scheme is the torque-control counterpart of classical PWM current control [5] and requires no hardware beyond the existing asymmetric half-bridge: it is a firmware-level change to the same drive.

The PWM-DITC controller was implemented in the discrete-time model of Section 3 and first evaluated at the rated point of 2500 rpm and 200 N·m. Table 4 and Figure 8 report the steady-state torque ripple as a function of the fixed carrier frequency. The ripple decreases substantially as the carrier is raised, from 26.7% at 10 kHz to 11.0% at 40 kHz — the trend is clear though not strictly monotonic, with a minor upturn near 25 kHz — and is 12.3% at a practical 20 kHz — about half the 23.6% of the SMC baseline. The essential point is that the carrier frequency is now a free design parameter: the engineer trades switching loss against torque ripple on a known, broadly monotonic curve, instead of accepting the uncontrolled and load-dependent rate of the hysteresis law. With the silicon-carbide devices now common in traction inverters a 30–40 kHz carrier is realistic, placing the ripple at about 11%.

Figure 8. Pulse-width-modulated direct instantaneous torque control (PWM-DITC) torque ripple versus the fixed carrier frequency (2500 rpm, 200 N·m)

Table 4. Pulse-width-modulated direct instantaneous torque control (PWM-DITC) torque ripple versus the fixed carrier frequency (2500 rpm, 200 N·m)

Note: PWM = pulse-width-modulated, DITC = direct instantaneous torque control, SMC = sliding-mode speed controller.

In summary, the fixed-frequency PWM-DITC offers a practical route to a designer-controlled switching frequency. It is a firmware-level upgrade of the same magnet-free machine and converter, requiring no extra hardware; it fixes the switching frequency at a chosen 10–40 kHz, removing the uncontrolled, load-dependent rate of the hysteresis law; and at a practical 20 kHz it brings the torque ripple to 12.3%, about half the SMC baseline, at the rated operating point. Two points must be stated plainly. First, this 12.3% is obtained with a simple proportional–integral torque regulator; a predictive or deadbeat torque regulator, or a higher silicon-carbide carrier, would lower it further. Second, the present single-duty PWM-DITC has been demonstrated at the rated operating point: holding a low torque ripple across the whole speed range — and in particular at the low speeds where an EV spends much of its time — calls for a dedicated low-speed controller. Section 4.8 develops one, based on a per-phase TSF, and combines it with the present PWM-DITC into a speed-scheduled dual-mode drive. The multilevel-converter route reported to reduce the ripple further [7, 11] remains a subject of future work. All results reported in this section are obtained in simulation and call for experimental confirmation on a high-power SRM test bench.

4.8 Dual-mode operation across the speed range

The low-speed sweep of Figure 4 showed that the torque ripple of hysteresis DITC, although well below the SMC baseline, grows steadily as the speed falls — from 8.8% at 2500 rpm to 15.9% at 300 rpm — and Section 4.7 showed that the fixed-frequency PWM-DITC, while it bounds the switching frequency, leaves a comparable rated-speed ripple. Neither realization is strongest at low speed, which is precisely where an EV operates during launch and urban driving. This motivates a dedicated low-speed controller, combined with the PWM-DITC into a speed-scheduled dual-mode drive.

At low speed the drive uses a TSF current controller, shown together with the dual-mode structure in Figure 9. The outer speed loop produces the same total torque demand T*. A TSF distributes this demand among the three phases: phase k receives a torque reference f_k(θ) T^*, where the position-dependent share functions f_k are cosine (raised-sine) profiles with a rise, a flat and a fall region. With the three phases offset by one third of the pole pitch (7.5°) the shares sum to unity at every rotor position, so the phase torques add up to the demand. Each conduction window spans θ_d = 12° (turn-on 4.5°, turn-off 16.5° measured from the aligned position within the 22.5° stroke); the rise and fall regions span θ_ov = 4.5° each and overlap the adjacent phases, with a 3° flat region in between. The PWM carrier is fixed at f_pwm = 20 kHz, and the flux loop uses a feed-forward-plus-proportional gain k_ψ = 3 × 10⁴ V/Wb. Each per-phase torque reference is converted to a current reference through the inverse of the static-torque table, $i_k^*=T^{-1}\left(f_k T^*, \theta\right)$. Each phase current is then regulated by a fixed-frequency loop: because the phase flux linkage is the integral of (v-R_si), the controller drives the flux to the reference $\psi\left(i_k^*, \theta\right)$ with a feed-forward-plus-proportional law, and a pulse-width modulator running at a fixed carrier applies the converter voltage. Each phase therefore follows a smooth current reference at a designer-chosen switching frequency.

TSF current control depends on the phase current following its shaped reference. At low speed the DC bus has ample voltage headroom to do so; as the speed rises, the rotor sweeps the conduction window faster and the finite 500 V bus can no longer force the current along the reference. The TSF controller is therefore effective only at low speed — a known property of current-controlled SRM drives — and is not a full-envelope solution on its own.

The drive consequently operates in two modes, selected by speed (Figure 9). Below a base speed of 600 rpm the TSF-PWM current controller is active; above it the fixed-frequency PWM-DITC of Section 4.7 takes over. Both modes run at a 20 kHz switching frequency. Figure 10 reports the resulting steady-state torque ripple across the 300–2500 rpm range, against hysteresis DITC alone. At the rated 200 N·m load the TSF controller holds the ripple between 4.2% at 300 rpm and 9.9% at 600 rpm — at 300 rpm nearly a four-fold reduction relative to the 15.9% of hysteresis DITC — and above 600 rpm the PWM-DITC keeps the ripple within a 7–12.3% band, so the dual-mode drive holds the torque ripple within roughly 4–12% over the whole speed range at a fixed 20 kHz switching frequency.

Figure 10. Torque ripple of the dual-mode drive across the speed range at three load levels (50, 200 and 300 N·m), 20 kHz switching

To establish the torque–speed envelope rather than a single load line, the sweep was repeated at a high-demand load of 300 N·m and a light load of 50 N·m (Figure 10). At 300 N·m the dual-mode drive performs at least as well as at rated load: the TSF mode delivers 10.0–14.8% over 300–600 rpm and the PWM-DITC 2.7–13.0% over 700–2500 rpm, with the full 300 N·m mean torque delivered at every point within the 320 A limit. At 50 N·m the picture is more nuanced and is reported here deliberately. First, the nominal PWM-DITC configuration cannot regulate this light load at all: its build-up zone applies the full bus voltage unconditionally for 3° every stroke, which alone exceeds a 50 N·m demand, so the build-up gating (already present in the control law as an option) must be enabled below roughly 100 N·m. Second, even with gating enabled, the percentage ripple of the PWM mode grows large at light load — of the order of 90–160% of the 50 N·m mean, corresponding to an absolute peak-to-peak excursion of about 45–80 N·m set by the per-stroke torque granularity of a 60 kW machine — whereas the TSF mode holds 9–16% (an absolute excursion below 8 N·m) and, because the currents are small, remains able to track its shaped references up to about 1000 rpm instead of 600 rpm. The practical conclusion is that the optimal mode boundary is load-dependent: it lies near 600 rpm at rated and high torque, but moves up to roughly 1000 rpm at light load, so a deployed drive should schedule the switching locus in the torque–speed plane rather than on speed alone. The dual-mode structure itself accommodates this directly, since the boundary is a firmware threshold. It does not reach uniformly single-digit ripple: the residual high-speed ripple is set by the 20 kHz switching frequency, and lowering it further without raising that frequency would require the multilevel converter noted above. As with the other results reported here, the dual-mode drive is evaluated in simulation and calls for experimental confirmation.

The mode switch itself is not free of dynamics, and it must be benign in both directions. Figure 11 shows the dual-mode drive crossing the boundary during a slow reference ramp at 200 N·m load, in acceleration and in deceleration. In acceleration the controller switches from TSF to PWM-DITC the first time the speed reaches the upper hysteresis threshold of 620 rpm; the total torque exhibits a single peak of about 284 N·m — an excursion of +84 N·m above the demand — and re-enters the pre-switch ripple band within about 1.3 ms. In deceleration the controller switches back from PWM-DITC to TSF at the lower threshold of 580 rpm; the torque briefly dips by about 102 N·m below the demand and recovers within about 1.8 ms, with no overshoot above the demand. In both directions the transient is contained to a single electrical period, the speed deviation is negligible, and the ±20 rpm hysteresis band prevents mode chattering, since each transient pushes the speed away from the opposite threshold. The transients reflect the change in current shape between the two controllers — the TSF mode tracks shaped current references whereas the PWM-DITC re-seeds its torque regulator at entry — and their magnitude is comparable to one hysteresis-DITC torque excursion at low speed. This handover behavior is taken up in Section 5, where the switched dual-mode is compared with a single-controller architecture that avoids the discrete switch.

Figure 11. Mode-transition transients at 200 N·m: (a) torque-sharing function (TSF) to pulse-width-modulated direct instantaneous torque control (PWM-DITC) at 620 rpm, +84 N·m peak settling in ≈1.3 ms; (b) back to TSF at 580 rpm, −102 N·m dip recovering in ≈1.8 ms

This paper has presented a simulation-based study of torque ripple minimization for a 60 kW three-phase 24/16 SRM intended for EV traction; every result reported here is obtained from a discrete-time simulation model, and no experimental measurement is included. The model of the complete drive was built from the machine’s finite-element-computed magnetic tables, and two control strategies — sliding-mode speed control and DITC with two-mode commutation — were compared on the identical drive, with the firing angles tuned by a coordinate-descent optimization loop.

The main findings are as follows. For this actuator-limited drive the speed-loop controller is secondary: even a well-tuned sliding-mode speed controller leaves the steady-state torque ripple at ≈23.6% — and at 33.5% even under the same fixed-frequency PWM constraint as the proposed drive — because the saturated inner current loop, not the speed law, dictates the torque waveform. Torque ripple reduction must therefore come from the inner loop, and DITC delivers it: regulating the instantaneous total torque directly lowers the ripple to 8.8% at the native control rate at the rated operating point. The DITC drive tracks a representative composite speed–load test profile — crawl, accelerations, highway cruise and a load step — accurately; its torque ripple, while three to five times lower than the SMC baseline over the speed range, deteriorates at low speed, which is identified as the weakest operating point of hysteresis DITC. The steady-state torque ripple is weakly affected by a four-to-one rotor-inertia mismatch in the tested model, indicating that the inner loop is insensitive to this mechanical parameter under the simulated conditions. Finally, because the inner torque loop and the outer speed loop are separable, re-tuning only the speed loop reduced the speed excursion under a rated 200-to-300 N·m load step from 43.8 to below 1 rpm — holding the speed within its ±1% band throughout — with no change in torque ripple.

As a practical realization, Section 4.7 implemented a fixed-frequency PWM-DITC, in which a torque regulator drives a pulse-width modulator at a designer-chosen carrier frequency. At a practical 20 kHz the rated-speed torque ripple is 12.3%, about half the SMC baseline, decreasing to 11.0% at 40 kHz; the switching frequency is thereby fixed by the designer instead of being left to the load-dependent rate of the hysteresis law. To hold a low ripple across the full speed range, and in particular at the low speeds that dominate urban driving, Section 4.8 added a torque-sharing-function current controller for the low-speed regime and combined it with the PWM-DITC into a speed-scheduled dual-mode drive: the TSF controller operates below 600 rpm and the PWM-DITC above it. The dual-mode drive holds the torque ripple within roughly 4–12% across the 300–2500 rpm range at rated load and remains effective at a 300 N·m high-demand load, all at a fixed 20 kHz switching frequency, eliminating the low-speed ripple growth of DITC alone; at light load the optimal mode boundary moves up in speed, so a deployed drive should schedule the mode switch in the torque–speed plane. Section 5 set out the requirements for a hardware realization — a SiC power stage, a position transducer accurate to a fraction of a degree, an FPGA-class digital controller, and standard parameter and dead-time compensation — and identified the single-controller architecture, with a natural transition into single-pulse operation, as the simpler choice for deployment. A multilevel-converter realization, which would lower the residual high-speed ripple without raising the switching frequency, and experimental validation on a high-power SRM test bench are the subjects of future work.