Flying Capacitor Inverter Sizing and Natural Balancing — 800 V

For each cell, $s_1,s_2$ are 0/1 switching functions. With $e=v_C-V_{dc}/2$ and $d=s_1-s_2$, Kirchhoff's equations give $i_C=d i$ and $v_o=V_{dc}(s_1+s_2-1)/2-d e$ for a midpoint-referenced load. Equal duty and a 180° carrier shift generate alternating capacitor charge and discharge intervals.

For a locally constant current, the longest charging interval is $aT_s$, where $a=\min(D,1-D)$. Therefore $\Delta v_{C,pp}=|i|a/(f_s C)$ and the terminal-ripple bound adds $2|i|R_{ESR}$. A 1,441-point fundamental sweep finds the required effective capacitance for the specified sinusoidal current and ripple budget. Divide by the minimum capacitance fraction to obtain nominal capacitance. The separate all-duty frozen-current estimate uses $a=1/2$, equivalent to $C\ge I_{pk}/(2f_s\Delta v)$ before ESR and derating. This comparison assumes prescribed current; it does not size the coupled fixed-duty circuit. RMS capacitor current follows $\langle i_C^2\rangle=\langle2a i^2\rangle$.

Natural balancing follows the switching-function harmonic framework of Wilkinson et al. (2006) and McGrath and Holmes (2009), adapted here to one leg with the entered load impedance. For the 32 odd carrier groups through order 63, $m=1,3,\ldots,63$, the carrier coefficient is $d_m(\theta)=2\sin[m\pi D(\theta)]/(m\pi)$. A 256-point FFT gives its fundamental sidebands $d_{mn}$. The model evaluates the load at every retained frequency $mf_s+nf_0$, rather than only at $f_s$. With both frequency signs included, $G_b=\sum_{m,n}|d_{mn}|^2\Re\{Y(m\omega_s+n\omega_0)\}$, $C\dot e=-G_b e$, and $\tau=C/G_b$. The optional impedance is $Z(s)=R+sL+(R_p sL_p)/(R_p+sL_p)$.

Fixed-duty mode also solves the full switched equations $L\dot i=v_{dc}(s_1+s_2)/2-d e-(R+d^2R_{ESR})i$ and $C\dot e=d i$. Exact matrix exponentials are composed at the PWM edges. Half-cycle symmetry, $i(t+T_s/2)=i(t)$ and $e(t+T_s/2)=-e(t)$, gives the periodic initial state from $(J-P_{1/2})x_0=b_{1/2}$, avoiding the ill-conditioned full-cycle fixed-point solve. The dominant Floquet multiplier gives $\tau=-T_s/\ln\rho$. For real multipliers the small eigenvalue deficit is found from $I-M$ and evaluated with log1p; for a complex-conjugate pair the log modulus follows the exact Liouville determinant. This decay time replaces the harmonic approximation in fixed-duty mode. The periodic current averages are checked against the independent Maple simulations in Kadyrov et al. (2018), attached pp. 12–13.

Fixed-duty capacitance sizing evaluates each candidate with the coupled switched circuit, including its changed current. It brackets and bisects the terminal-ripple limit on the high-capacitance branch, with lower effective capacitance $\max(0.1\,\mu\mathrm{F}\,k_C,\,[a/(f_s)]^2/L)$ and an upper nominal limit of 1 F. The branch floor keeps the LC phase during a capacitor conduction interval at or below one radian; a resonant current-collapse solution is not selected as a design. Terminal extrema include switching edges and the stationary points of the damped RLC solutions, rather than only the plotted samples. The required nominal value is rounded upward to 0.01 µF and is re-evaluated by substitution; its predicted ripple is checked against the limit. Fixed-duty comparison curves rebuild the transition matrix for half and twice the capacitance; linear time-constant scaling is used only for the harmonic model.

Two cells, one flying capacitor, complementary ideal switches, equal cell duty ratios and 180° phase-shifted triangular carriers. Sinusoidal mode uses $D=(1+M\sin\theta)/2$ and the entered sinusoidal phase current; it allows externally imposed current or back EMF. The harmonic load is linear and passive. The added parallel RL branch is a frequency-response fit, not an identified induction-motor model.

The sinusoidal sizing and switching-period animation freeze phase current within each cell period; include expected switching-current excursions in the peak-current input. The sinusoidal balancing envelope is the slow, small-imbalance harmonic average. Fixed-duty mode instead computes actual coupled current and capacitor-voltage ripple, and sizes a candidate bank with that coupled solution. Its animated orbit is steady state. Initial imbalance sets the recovery envelope and voltage-stress allowance, not that periodic orbit.

Capacitance fraction applies to all calculations. ESR is constant: it contributes terminal ripple and loss, and is included in the exact fixed-duty state equations. It is omitted from the sinusoidal harmonic balancing feedback. The overshoot allowance is supplied independently.

Intended for capacitor selection and natural-balancing assessment of a single three-level leg. The sinusoidal harmonic model retains the 32 odd carrier groups through order 63, with 256 Fourier bins per group (the center and 255 sideband bins); the switching/fundamental ratio must be at least 130 to keep these groups separated. The last eight retained groups are monitored as a truncation indicator, not a rigorous error bound. Very resistive loads may require more orders. The fundamental sizing sweep uses 0.25° spacing. The animation uses 801 samples; fixed-duty orbit waveforms use 80 exact state advances per nonzero PWM interval. The sizing solver includes continuous terminal-voltage extrema even when a waveform sample misses a peak. RMS current and displayed peak-current estimates still use the plotted samples.

Sinusoidal balancing assumes the imbalance changes slowly relative to a fundamental cycle; a warning appears below five fundamental periods per time constant. The exponential curve is a mean-envelope estimate. Fixed-duty decay uses the dominant Floquet multiplier, so an oscillatory transient is represented by its envelope. A log-modulus decrement no greater than 10⁻¹⁰ per cycle is treated as numerically unresolved: the time readouts and affected curves are withheld, a corrective condition is shown, and no negative recovery time is reported. A 70-digit independent evaluation of the review's weak-damping case gives a decrement about 6.23 × 10⁻¹⁶ per cycle, below dependable double-precision resolution. These are not precharge or startup simulations. A purely reactive harmonic load has no finite dissipative balancing time; the plot then retains the initial imbalance.

Check the load impedance over the relevant switching-frequency range, including winding resistance, skin/proximity effects and any external filter. A single-leg impedance is not automatically the transfer impedance of a three-phase floating-star load. The 2009 motor identification procedure and its experimental motor parameters are not reproduced here.

Switch stress is bounded by $V_{dc}/2+|e_0|+\Delta v_{terminal,pp}/2+V_{spike}$. This assumes a centered ripple and ideal complementary operation; it excludes bus transients, dead time, unequal gate delays, parasitic LC ringing and faults. Capacitor RMS rating, dielectric derating and thermal behavior must be checked with component data. Natural balancing does not establish a safe initial capacitor voltage before switching.

Fixed-duty sizing returns a verified point on the stated high-capacitance branch within 0.1 µF to 1 F nominal, rather than a global minimum across resonant branches. The all-duty frozen-current estimate is only a comparison for the current of the installed bank. The capacitor chart shows ripple versus fundamental angle in sinusoidal mode and the exact dielectric/terminal voltage versus switching time in fixed-duty mode. The switching-period lanes have separate physical A/V scales, with the capacitor DC reference printed on its voltage axis.