= Current-sign certificate from sensor delay
{title2=$|i_m|>\epsilon_i+L_i(t_e-t_s)$}
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.<Using power MOSFETs in DC motor control applications by Nexperia>[IAN50004, §4.1, Figures 5–6]{ref}
Suppose a sensor reports $i_m$ for sampling time $t_s$, with $|i(t_s)-i_m|\le\epsilon_i$. 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 gives
$$
|i(t)-i_m|\le\epsilon_i+L_i(t-t_s)\le b,
\qquad b=\epsilon_i+L_i(t_e-t_s).
$$
Thus $i_m>b$ 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 $\epsilon_i=0.020\,\mathrm A$, $L_i=5000\,\mathrm{A/s}$, sample age $t_0-t_s=2\,\mathrm{\mu s}$ and $t_d=0.5\,\mathrm{\mu s}$. 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 $0.5\,\mathrm{\mu s}$ 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 $\epsilon_i$. For a future time $t\ge t_b$,
$$
\left|i(t)-\frac1{T_a}\int_{t_b-T_a}^{t_b}i(s)\,ds\right|
\le\frac{L_i}{T_a}\int_{t_b-T_a}^{t_b}(t-s)\,ds
=L_i(t-t_b+T_a/2).
$$
The interval certificate therefore uses $b=\epsilon_i+L_i(t_e-t_b+T_a/2)$. With $T_a=4\,\mathrm{\mu s}$ 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>.
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