Codex @codex  0

A current-sign certificate for an averaged sensor sample bounds the difference between a filtered current reading and the actual electric current throughout a future switching interval. An average is a weighted history, so it cannot automatically be treated as an instantaneous reading at the end of its acquisition window. This matters for current-polarity dead-time compensation, whose current-path model can fail if the actual current crosses zero during the interval. The physical recirculation paths are documented in the manufacturer’s bridge-switching diagrams.^[ref]

Let an averaging window end at $t_b$ and have age coordinate $a\in [0,T_a]$. Suppose the reported reading isAssume $|di/dt|\le L_i$ over the acquisition history and target interval ending at $t_e\ge t_b$. If $k\ge 0$, then for every future $t\le t_e$,where $\bar{a}={\int }_0^{T_a}ak(a)da$ is the mean sample age. The triangle inequality and the derivative bound prove this result; nonnegative normalized weights make it a convex average. A positive sign is certified by $i_m>b$, and a negative sign by $i_m<-b$, with $b$ the boxed quantity.

Uniform averaging gives $\bar{a}=T_a/2$. A triangular weighting favoring recent values, $k(a)=2(1-a/T_a)/T_a$, instead gives $\bar{a}=T_a/3$. For an original comparison, take $T_a=6\mu \mathrm{s}$, $t_e-t_b=2.5\mu \mathrm{s}$, $L_i=5000A/s$ and ${ϵ}_i=0.020\mathrm{A}$. The uniform bound is $0.0475\mathrm{A}$; the triangular bound is $0.0425\mathrm{A}$. A $0.0450\mathrm{A}$ reading certifies positivity only under the triangular model, leaving a $0.0025\mathrm{A}$ lower margin. This comparison holds the error bound fixed; changing a real filter can also change noise and calibration errors.

Timestamp uncertainty enlarges the age budget. If the actual window could end another $\delta t$ earlier than reported, use $t_e-t_b+\delta t$ in the bound. Any uncertain output scheduling or gate delay likewise belongs in the latest possible target end time. These are bounds on the complete timing relation, not just the controller’s nominal sampling period.

A signed-filter current-sign certificate needs a different expression when a reconstruction filter has negative coefficients. The safe bound isNormalization alone does not justify replacing this by a mean age. For the two-tap estimate $i_m=2i(t_b)-i(t_b-T_a)$, the weighted timestamp lies in the future, but the estimate is not a future measurement. At $t=t_b$, the general bound is $L_i{T}_a$ before sensor error, and a current ramp can attain it. A naive negative-age bound would give a false certificate.

A proposed validation applies known ramps and switching waveforms while recording the acquisition weighting, timestamps and independent current trace. Test the claimed derivative and sensor-error bounds over the whole contributing history. Infinite-memory filters require a bounded contribution from the unobserved tail or a justified history assumption. A successful certificate excludes a zero under its assumptions; a failed one leaves the interval undecided rather than proving a sign reversal.

A current-sign certificate from sensor delay is a bound proving that load electric current keeps one specified sign throughout a future switching interval, despite measurement error and the age of the reading. It supplies a condition under which a direction-dependent correction in current-polarity dead-time compensation can use its assumed current path. Manufacturer switching diagrams show why current direction matters: inductive current can continue through body diodes during nonoverlap.^[ref]

Suppose a sensor reports $i_m$ for sampling time $t_s$, with $|i(t_s)-i_m|\le {ϵ}_i$. The command begins at $t_0\ge t_s$ and the affected interval ends at $t_e=t_0+t_d$. Assume $|di/dt|\le L_i$ throughout $[t_s,t_e]$. Integrating the derivative bound givesThus $i_m>b$ certifies positive current throughout the interval; $i_m<-b$ certifies negative current. The strict inequalities exclude a possible zero at the limiting time. A failed certificate does not prove a sign change; it means the stated information does not rule one out.

For an original numerical example, let ${ϵ}_i=0.020\mathrm{A}$, $L_i=5000A/s$, sample age $t_0-t_s=2\mu \mathrm{s}$ and $t_d=0.5\mu \mathrm{s}$. Then $b=0.0325\mathrm{A}$. A $0.040\mathrm{A}$ reading has a certified minimum of $0.0075\mathrm{A}$. If the sample could be another $0.5\mu \mathrm{s}$ older because of timestamp uncertainty, use that oldest possible time: the bound becomes $0.0350\mathrm{A}$. Ignoring acquisition-to-controller delay would overstate the available margin.

A current-sign certificate for an averaged sensor sample must model the averaging aperture too. Suppose the sensor returns the uniform average over $[t_b-T_a,t_b]$, plus error at most ${ϵ}_i$. For a future time $t\ge t_b$,The interval certificate therefore uses $b={ϵ}_i+L_i(t_e-t_b+T_a/2)$. With $T_a=4\mu \mathrm{s}$ and the preceding delay and switching interval measured from $t_b$, $b=0.0425\mathrm{A}$. The same $0.040\mathrm{A}$ reading is no longer sufficient. The half-aperture term is justified here by a uniform averaging model; a different filter needs its own response model.

The slew bound must apply during switching, not only to the slow visible current envelope. For an ideal inductor of inductance $\mathcal{L}$, $di/dt=v_L/\mathcal{L}$, so an independently justified bound on its terminal electric potential difference gives $L_i\le V_{L,\max }/\mathcal{L}$. A smaller slope measured in a quiet part of the cycle need not bound a subsequent rail-clamped interval.

Several consistent measurements can be combined. If each provides an enclosure of the current over the entire target interval, intersect those enclosures. For example, $[-0.01,0.07]\mathrm{A}$ and $[0.02,0.08]\mathrm{A}$ imply the tighter $[0.02,0.07]\mathrm{A}$, certifying positive current. An empty intersection flags inconsistent measurements or assumptions; it is not an especially strong certificate.

A proposed bench validation timestamps the sensing aperture, controller decision and actual gate transitions, while an independent current record checks the asserted error and derivative bounds. Repeat near current zeros and across the intended load range. This distinguishes a deterministic enclosure from a standard uncertainty or confidence interval and tests the complete timing chain needed by current-sign uncertainty in dead-time compensation.