SiC Inverter Losses and Drive-Cycle Junction Temperature

The vehicle balance is $F=m k_{rot}a+\tfrac12\rho C_dA v^2+mg(C_{rr}\cos\alpha+\sin\alpha)$. Gear efficiency converts wheel demand to motor torque. A surface-PM model with $i_d=0$ gives $i_q=T/(1.5p\psi)$, $v_q=R_s i_q+\omega_e\psi$ and the inductive voltage $\omega_eL_s i_q$. Each speed segment is linear. Frequency changes continuously inside it and electrical angle is integrated exactly: $\theta=\theta_0+2\pi f_0t+\pi\dot f t^2$.

For signed phase current, ideal sinusoidal-PWM duty is corrected by $f_{sw}t_d\,\mathrm{sign}(i)$ before gate blanking. Forward channel, synchronous reverse channel, two dead-time diode intervals and switching energies are counted separately for all six positions. Compensated-duty headroom is checked at both ends of each speed interval with that interval's torque. Resistance is quadratic through the three temperature inputs. Switching energies scale with current, voltage and temperature. Reverse voltage is limited to the lesser of channel and diode drop.

All 24 Foster states satisfy $\dot x_k=(R_kP(T_j,i,\theta)-x_k)/\tau_k$, with $T_j=T_c+\sum x_k$. Direct exponential steps hold power linearly between endpoints and solve the nonlinear endpoint temperature implicitly. Instantaneous frequency and integrated angle are used through acceleration, launch, load changes and stall. Mean electrical frequency does not propagate a rotating thermal interval.

An FFT solution supplies a periodic reference. Its Fourier derivatives and nonlinear ODE residual are bounded across each frequency cell using cubic Bernstein coefficients and angular-curvature bounds. Positive RC comparison propagates all six devices' state-error enclosures. A rotating shortcut is accepted only when its peak upper bound cannot exceed the resolved peak plus the selected numerical target. Otherwise direct integration continues. Low-load and regenerative intervals may instead use nonnegative-power comparison bounds; current polarity restricts each channel's possible gate duty. Zero-current intervals cool exactly.

Balanced sinusoidal currents; a nonsalient surface-PM motor with zero direct-axis current, no field weakening, no saturation and a hot winding resistance held fixed. The motor follows the prescribed speed exactly; it has no torque or current limiter. Regenerative braking supplies the selected fraction of negative wheel torque. Ordinary stops command zero motor torque; only the explicitly added initial stall applies locked-load torque.

Equal current sharing among parallel dies and identical devices in all six positions. A regulated case temperature is imposed independently of inverter loss, and all junctions initially equal the case temperature. Foster branches belong to a single die. Default electrical data: Infineon IMZ120R030M1H, datasheet v2.2, pp. 5 and 7, 18 V gate drive, 2 Ω external gate resistance, 800 V and 25 A switching reference. The maximum Foster fit is from p. 13; its resistance sum is 0.6604 K/W versus the rounded 0.66 K/W rating.

This is a carrier-averaged semiconductor model: electrical-frequency heating and locked-rotor heat allocation are retained, while individual PWM temperature spikes, switching-node ringing, gate-driver loss, bus/capacitor loss, magnetic loss and PWM current ripple are excluded. Energy scaling is an engineering approximation, especially near zero current. Default turn-on data include the opposing diode's recovery; it is not added twice. Ideal current-polarity compensation is assumed, including at stall; real controllers may disable or limit it.

Every EPA 1 Hz speed sample is retained. Acceleration is its segment slope; torque and road-force evaluation are held at the segment's starting sample. Speed, frequency and angle continue to change within the segment. Standard uses 5 torques × 8 frequencies × 64 reference angles; fine uses 7 × 10 × 128. Direct steps use at most 8 ms and four samples per electrical period in standard, versus 4 ms and eight in fine; below 3 Hz the limits are 4 ms and 2 ms. Coarser standard time and angular grids reduce runtime and can shift a narrow peak. The independent Radau launch regression differs by less than 0.08 K at the baseline; this does not establish a universal integration-error bound.

Peak temperature and same-device swing are upper estimates. The numerical bracket combines propagated shortcut enclosures with a 0.08 K standard or 0.02 K fine time-step allowance; shortcut acceptance targets are 0.5 K and 0.1 K. These are numerical settings, not device tolerances. Compare fine resolution for small design margins. Analytical comparison bounds cover shortcut intervals; direct-step accuracy is assessed by numerical refinement and the independent launch/stall regressions.

The charts condense the full history into 128 bins, retaining enclosure extrema and energy-weighted mean losses. Thermal bounds can be wider than the actual fundamental oscillation. Device colors interpolate sampled enclosure centers, so use the peak and bracket for brief launch events. The angle chart is a frozen operating-point reference; at stall it compares alternative locked angles. Its smooth picture uses pchip between solved samples. Loss energy in skipped intervals uses periodic loss quadrature.

Fixed case temperature excludes spreading, device-to-device thermal coupling, coolant dynamics and heatsink sizing. Foster states are fitted mathematical states, not physical layers. Device fits are calibrated over 25–175 °C. Field weakening, overmodulation, current limiting, SOA and avalanche are excluded. Thermal swing supports package-specific lifetime assessment; no service-life prediction or rainflow count is made.