Power inverter

A power inverter converts electrical power from a direct current input to an alternating current output. Its switches reverse or modulate the connection to the DC supply, while its control and filtering determine the output waveform. It transfers energy from its input; the switching operation itself is not an energy source. The DOE overview explains the switching-and-filtering principle and distinguishes equipment designed for independent operation from equipment relying on a grid reference.^[ref]

An ideal H-bridge uses four controlled switches around a two-terminal load. One diagonal pair applies $+V_d$ across the load, and the opposite pair applies $-V_d$. In a stipulated square-wave mode with equal half-periods,Its root mean square magnitude is $V_d$, but it is not sinusoidal. Its Fourier series isThe fundamental RMS magnitude is $2\sqrt{2}{V}_d/\pi$. Thus a $48\mathrm{V}$ input gives a $48\mathrm{V}$ RMS ideal square-wave output, but only about $43.22\mathrm{V}$ RMS in the fundamental. Equal total RMS does not imply equal waveform or equal response of every electrical load.

For a $12\Omega$ purely resistive load directly across this ideal square-wave output, mean active power is $V_d^2/R=192\mathrm{W}$. The fundamental contributes $8V_d^2/({\pi }^{2}R)\approx 155.63\mathrm{W}$ and the other harmonic components supply the remainder. The harmonic active power sum accounts for that difference. Adding a filter changes the circuit; the directly connected resistor calculation should not then be reused unchanged.

Rapid switching also permits averaged control. If a short switching interval applies $+V_d$ for fraction $d$ and $-V_d$ for the rest, its average output is $(2d-1)V_d$. Choosing $d(t)=(1+m\sin \omega t)/2$, with $0\le m\le 1$, makes that local average sinusoidal. The actual waveform still contains switching components that the load and filter must handle.

A proposed low-voltage demonstration records DC input electric potential difference and electric current, output waveform and load electric current, comparing average input and output power. Output frequency, total RMS, fundamental RMS and waveform distortion answer different questions. For islanded microgrid operation, control of a usable local supply is an additional capability, not a consequence of DC-to-AC conversion alone.

An H-bridge is a four-switch arrangement that can apply either polarity of a DC supply across a two-terminal electrical load. Two switches form each side, with the load joining their midpoint nodes. Power inverters and bidirectional electric motor drives use this switching geometry. The primary TI lesson illustrates its components, forward/reverse states and inductive-current paths.^[ref]

In an idealized switching model, let $q_A,q_B\in \{0,1\}$ select the lower or upper rail at midpoint A and B, with supply magnitude $V_d$. ThenOpposite choices give $+V_d$ or $-V_d$ across the load. Equal choices give zero terminal difference even though both midpoints can be at the upper rail. This zero-output state does not mean every switch is open.

Each leg's upper and lower devices must not simultaneously provide a low-resistance path between supply rails: that is H-bridge shoot-through. Bridge switching dead time introduces a nonoverlap interval during a transition. Inductive load electric current still needs a path during that interval, often involving diodes or controlled conducting devices. The actual current path depends on the bridge state and current direction.

An original inductor example shows why reversing the commanded terminal polarity need not instantly reverse electric current. Neglect winding electrical resistance and opposing motor electromotive force, and take $L=10\mathrm{mH}$ with initial current $i=2\mathrm{A}$. Applying $v_{AB}=-12\mathrm{V}$ givesCurrent reaches zero only after about $1.667\mathrm{ms}$ if this voltage is maintained. Initially $v_{AB}i=-24\mathrm{W}$: the inductive load is returning stored energy through the bridge rather than absorbing it. Its initial magnetic energy is $L{i}^2/2=0.020\mathrm{J}$.

During rapid ideal bipolar switching, applying $+V_d$ for fraction $d$ and $-V_d$ for the rest gives interval-average output $(2d-1)V_d$. With $V_d=12\mathrm{V}$ and $d=0.75$, the average is $6\mathrm{V}$. This average does not specify instantaneous current or heating without a load model.

A proposed low-voltage observation records both midpoint potentials, differential load electric potential difference and load electric current. Comparing them through reversal distinguishes a changed switch command from changed current direction. Driver delays, device drops, recirculation and the supply's ability to accept returned energy determine how closely a real bridge follows the ideal states.

H-bridge shoot-through is unintended conduction directly between the supply rails through devices in the same bridge leg. It differs from the intended load-current path through opposite legs of an H-bridge. Overlapping device conduction during switching can create that rail-to-rail path even when the intended logical states never command it. The manufacturer application note explicitly identifies cross-conduction as shorting the supply and introduces nonoverlap timing to prevent it.^[ref]

An original lumped fault-loop model shows why a short overlap can matter. Let a previously unenergized parasitic loop have electrical resistance $R$, inductance $L$ and constant supply electric potential difference $V_d$. During overlap,soFor $t\ll \tau$, the initial ramp is $i\simeq V_{d}t/L$; the steady resistive value $V_d/R$ is not reached instantaneously.

Take stipulated $V_d=24\mathrm{V}$, $R=0.10\Omega$ and $L=100\mathrm{nH}$, giving $\tau =1\mu \mathrm{s}$. An overlap lasting $100\mathrm{ns}$ reaches about $22.84\mathrm{A}$. The supply deliversSome of this energy remains in the loop's magnetic field at turn-off; it should not all be labeled heat dissipated in one switch. A $500\mathrm{ns}$ overlap in the same ideal model reaches about $94.43\mathrm{A}$, illustrating the nonlinear dependence on duration. Device and supply behavior can invalidate these constant-parameter assumptions.

Bridge switching dead time addresses timing overlap by allowing one conduction path to cease before the complementary one starts. An inductive load still needs a recirculation path while the controlled devices are off. TI's primary bridge lesson illustrates current paths during that interval.^[ref]

A proposed diagnostic compares actual device control waveforms and supply-current transients with load current during transitions, using a limited-energy test arrangement. A large load current by itself does not identify shoot-through; the timing and path establish the mechanism. The calculated fault-loop pulse is an original model, not an asserted device rating.

Bridge switching dead time is an intentional nonoverlap interval between complementary conduction commands in a power-switching leg. It allows the outgoing device to cease conducting before the incoming device conducts, reducing H-bridge shoot-through. The applicable interval depends on propagation and device switching delays, not only on the timestamps of ideal logic commands. Nexperia's primary bridge discussion introduces the delay between turning off one pair and turning on the other, while retaining a path for inductive current.^[ref]

For an original timing bound, command the outgoing device off at $t=0$ and the incoming device on at $t=t_d$. Let ${\tau }_{\mathrm{off},\max }$ be the latest possible end of outgoing conduction, and ${\tau }_{\mathrm{on},\min }$ the earliest incoming conduction delay after its own command. A sufficient nonoverlap condition in this delay model isWith an additional positive timing margin $m$, chooseThe two switching directions can require different bounds. The inputs must cover the operating conditions for which the guarantee is claimed.

Suppose the latest turn-off completion is $80\mathrm{ns}$ and the earliest turn-on delay is $20\mathrm{ns}$. Zero modeled overlap requires at least $60\mathrm{ns}$ of commanded separation; a stipulated $20\mathrm{ns}$ extra margin gives $t_d=80\mathrm{ns}$. Incoming conduction then starts no earlier than $100\mathrm{ns}$, leaving at least $20\mathrm{ns}$ after the outgoing device has ceased. These numbers illustrate the bound rather than a particular gate-driver specification.

Longer separation also changes the output waveform. In an original half-bridge example with rails 0 and $V_d$, suppose one nominal high interval per period is shortened by exactly $t_d$, with the removed interval clamped to the low rail. The mean electric potential difference changes byFor $V_d=24\mathrm{V}$, $T=50\mu \mathrm{s}$ and $t_d=100\mathrm{ns}$, this is $-48\mathrm{mV}$. The dead-time output-voltage error in a complete bridge depends on which intervals are replaced and on load-current direction; this one-pulse calculation does not supply a universal coefficient.

During nonoverlap, the electric current of an inductor cannot disappear instantly. Its path can pass through diodes or other permitted conducting devices, and the midpoint potential depends on that path. A proposed test records both actual transition timing and the resulting terminal waveform under the intended load conditions. It tests the timing margin and output distortion separately, instead of treating an increased programmed delay as a complete characterization.

Dead-time output-voltage error is the difference between the intended switching-cycle average electric potential difference and the actual average produced when nonoverlap intervals alter the terminal waveform. Bridge switching dead time reduces H-bridge shoot-through, but during it an inductive load still carries electric current. Its recirculation path helps determine which rail clamps the output. The primary manufacturer discussion shows the bipolar/unipolar switching states and diode paths during that interval.^[ref]

An original event-accounting model avoids assuming a universal sign. Let an ideal half-bridge output take values 0 and $V_d$ over period $T$. During disjoint affected intervals of lengths $\Delta t_k$, suppose the intended normalized level is $q_k$ and the actual clamped level is $c_k$, each either 0 or 1. ThenAn interval replaced by the same rail contributes no error. An intended high interval replaced by low contributes negatively; the reverse contributes positively. Device drops require additional voltage terms.

For a stipulated single-pulse switching model, assume the load current keeps one sign throughout a cycle and that ideal recirculation clamps the midpoint to the low rail for current leaving the midpoint, or to the high rail for current entering it. With one effective high interval shortened or extended by $t_d$,For $V_d=24\mathrm{V}$, $T=50\mu \mathrm{s}$ and $t_d=100\mathrm{ns}$, the magnitude is $48\mathrm{mV}$. The corresponding ideal duty error magnitude is $t_d/T=0.002$. In a full H-bridge, subtract the two measured midpoint averages; their errors can reinforce or partly cancel according to their states.

Current-polarity dead-time compensation would add an opposing correction in that specified model. Near a current zero crossing, however, the sign may change within the interval, and the assumed clamping model can fail. Blindly applying the same signed offset throughout a crossing need not correct the actual waveform.

A proposed low-voltage test records both switch commands, actual midpoint potential and load electric current, integrates the difference from the intended waveform over a cycle, and compares it with the event sum. Repeat across current directions and pulse widths. Separate switching delay, diode drops and uncertain timing rather than fitting every discrepancy to one nominal dead-time number. The formulas above are illustrative switching models, not specifications for a particular driver.

Current-polarity dead-time compensation adjusts a switching command to oppose the dead-time output-voltage error predicted from the direction of load electric current. During bridge switching dead time, an inductive current can flow through a MOSFET body diode; the resulting rail clamp makes the cycle-average error depend on current direction. The manufacturer’s bridge-state diagrams establish that current-path mechanism.^[ref]

Consider an original ideal half-bridge model with rail separation $V_d$, switching period $T$, duty fraction $D$, and effective dead-time fraction $\delta =t_d/T$. Assume current has one nonzero sign throughout the affected interval and that device drops are negligible. Let positive current leave the midpoint, so that the adopted clamping model givesFor a desired normalized average $D_{\ast }$, chooseSubstitution cancels the average error in this model. The correction changes the commanded pulse duration; it does not remove the nonoverlap needed to avoid H-bridge shoot-through. Different full-bridge switching patterns require applying the appropriate event model to each leg before subtracting the terminal averages.

For $V_d=48\mathrm{V}$, $T=50\mu \mathrm{s}$ and $t_d=0.5\mu \mathrm{s}$, $\delta =0.01$ and the uncompensated error magnitude is $0.48\mathrm{V}$. At $D_{\ast }=0.40$, use $D_{\mathrm{cmd}}=0.41$ for positive current and $0.39$ for negative current. Each gives the intended $19.2\mathrm{V}$ average under the stated assumptions.

Command limits matter. Suppose the allowed duty range ends at $0.98$ because a minimum off-time is required. With positive current and $D_{\ast }=0.975$, the formula asks for $0.985$. Clipping to $0.98$ leaves an actual normalized average of $0.97$, an error of $-0.24\mathrm{V}$ at this bus potential difference. Thus a compensation law can be correct locally while the requested average lies outside its achievable range. Minimum pulse durations impose further limits.

Current-sign uncertainty in dead-time compensation is particularly important near a zero crossing. If an estimated sign $\hat{\sigma }$ is used while the actual sign is $\sigma$, the residual error, before command clipping, isA wrong opposite sign doubles the uncompensated error magnitude: $0.96\mathrm{V}$ in the example. Current-sensor delay, noise and a sign reversal within the switching interval can all invalidate a sign held constant for the cycle. Using a zero correction in an uncertain interval avoids that particular wrong-sign doubling, but does not cancel the original error.

A proposed bench comparison records gate commands, midpoint electric potential difference and load electric current together, then integrates each switching cycle with and without compensation. Compare positive-current, negative-current and crossing intervals separately, and repeat near pulse limits. This separates the average-error cancellation being tested from preservation of nonoverlap, device voltage drops and imperfect knowledge of switching delay.

Current-sign uncertainty in dead-time compensation occurs when the direction of measured load electric current does not reliably identify its direction during the switching interval being corrected. The recirculation path during bridge switching dead time depends on that direction. Manufacturer switching diagrams show how the body diodes provide paths when the controlled devices are off.^[ref]

Use the ideal half-bridge model in current-polarity dead-time compensation: bus separation $V_d$, period $T$, effective delay $t_d$, and an actual nonzero current sign $\sigma \in \{-1,1\}$ held constant throughout that interval. Let $a=V_d{t}_d/T$. A commanded additive correction $au$, where $-1\le u\le 1$, leaves average output errorA correct sign gives zero error with $u=\sigma$. An opposite sign gives magnitude $2a$, twice the uncompensated magnitude. For $V_d=48\mathrm{V}$, $t_d=0.5\mu \mathrm{s}$ and $T=50\mu \mathrm{s}$, $a=0.48\mathrm{V}$ and a wrong full-sign correction gives $0.96\mathrm{V}$ error. These conclusions assume unclipped commands and the stated rail-clamping model.

A current-sign certificate from sensor delay combines measurement and motion bounds. Suppose a reading $i_m$ refers to time $t_0-\tau$, has absolute error at most ${ϵ}_i$, and the true current satisfies $|di/dt|\le L_i$ from then through the end of $[t_0,t_0+t_d]$. For every time in that switching interval,This follows by adding the measurement error to the integrated derivative bound. A positive sign is certified if $i_m>b$; a negative sign if $i_m<-b$. Within the remaining band, neither sign is certified by those bounds. At equality, a zero can still occur, so strict sign certification requires a strict inequality.

For an original example, take ${ϵ}_i=0.020\mathrm{A}$, $L_i=5000A/s$, $\tau =2\mu \mathrm{s}$ and $t_d=0.5\mu \mathrm{s}$. Then $b=0.0325\mathrm{A}$. A $0.060\mathrm{A}$ reading certifies positive current, whereas $0.025\mathrm{A}$ permits currents from $-0.0075$ to $0.0575\mathrm{A}$. A sign selected solely from the latter reading can therefore be wrong. A bound covering only the sensing instant would miss the delay contribution.

Robust and probabilistic choices answer different questions. If both signs are possible and nothing else is known, the worst-case error is $a(1+|u|)$; its minimum occurs at $u=0$. If instead a validated conditional probability $p=\mathrm{Pr}(\sigma =1)$ is available and mean squared error is the objective, differentiation givesFor $p=0.75$, the optimal fractional correction is $u=0.5$ and the predicted RMS error is about $0.416\mathrm{V}$ for the bus example. That is a probability-model prediction, not a worst-case sign guarantee.

A proposed validation records sensor timestamps, actual load electric current, gate timing and midpoint electric potential difference on a common time base. Compare integrated cycle errors in certified and uncertified sign intervals, including crossings within the dead time. A dead band or Schmitt trigger can reduce noisy toggling, but holding a previous sign through a genuine crossing can prolong a wrong correction; the relevant test is whether the current-path assumptions and timing bounds hold.

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.

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 signed-filter current-sign certificate guarantees the sign of an actual load current from a filtered estimate whose coefficients can be negative. Such a filter may extrapolate or sharpen a waveform, but its output is not necessarily a convex average of past currents. The sign must hold over the intended switching interval before it can justify current-polarity dead-time compensation; the relevant recirculation paths depend on current direction.^[ref]

Suppose $\hat{i}={\sum }_j{a}_{j}i(t_j)+\eta$, with ${\sum }_j{a}_j=1$, $|\eta |\le ϵ$ and the Lipschitz bound $|i(t)-i(s)|\le L_i|t-s|$ over all relevant times. ThenThe absolute coefficients are essential. To certify an interval, maximize the right-hand side over its times. A positive sign follows if the estimate exceeds that bound; a negative sign follows if it is less than the negative bound. If individual sample errors are bounded by ${ϵ}_j$, their contribution is at most ${\sum }_j|a_j|{ϵ}_j$ rather than a signed cancellation.

The generic bound can be conservative because all samples come from the same current history. For the two-tap predictor $\hat{i}=2i(0)-i(-T_a)+\eta$ and future time $\delta \ge 0$, rewrite the error asThis gives the sharper resultinstead of the direct absolute-weight bound $ϵ+L_i(3\delta +T_a)$. It is sharp under only a Lipschitz current assumption: a history proportional to $|t|$ changes slope at the central sample and attains the positive error limit. The Lipschitz assumption permits that corner even though a derivative need not exist at the corner itself.

For an original example, take $L_i=5000A/s$, $T_a=6\mu \mathrm{s}$, a target interval $0\le t\le 2\mu \mathrm{s}$, and aggregate estimate error $ϵ=0.020\mathrm{A}$. The sharper interval bound is $0.060\mathrm{A}$; the direct absolute-weight bound is $0.080\mathrm{A}$. A reading $\hat{i}=0.065\mathrm{A}$ therefore certifies at least $0.005\mathrm{A}$ throughout the interval using the sharper proof. A signed timestamp average would misleadingly place the predictor at time $+T_a$; it is an extrapolation, not a measured future sample.

A current-sign certificate with a curvature bound can exploit stronger information. For the target time $T_a$ and a twice differentiable current with $|i^{\prime \prime }|\le M_i$ on $[-T_a,T_a]$,Taylor’s theorem, or integrating the second derivative against its triangular weight, proves the inequality. At $T_a=6\mu \mathrm{s}$ and stipulated $M_i=10^{8}A/{\mathrm{s}}^2$, the model error bound is $0.0036\mathrm{A}$ before sample errors. This smaller number is justified only if the curvature bound also covers switching transitions; an affine-ramp fit alone does not establish it.

A proposed validation injects known ramps and waveforms with changing slope, records the actual filter coefficients and sampling times, and checks the resulting enclosures against an independent current trace. Account for sensor saturation, timing uncertainty and the history used by the filter. A sharper mathematical bound and a stronger physical assumption are different ways of obtaining a smaller certificate margin.

A current-sign certificate with a curvature bound uses a bound on the second derivative of an electric current to certify the sign of an extrapolated current. An affine ramp is predicted exactly by two correctly timed samples, so its uncertainty should depend on curvature rather than counting the entire ramp as error. Current direction matters physically because it selects recirculation paths during switching dead time.^[ref]

Let samples at $-T$ and $0$ produce measurements $y_{-}=i(-T)+e_{-}$ and $y_0=i(0)+e_0$, with $|e_{-}|\le {ϵ}_{-}$ and $|e_0|\le {ϵ}_0$. For a future time $t\ge 0$, use the linear extrapolationIf $i$ is twice continuously differentiable and $|i^{\prime \prime }(s)|\le M$ throughout $[-T,t]$, the polynomial interpolation error givesThe model term follows by applying the remainder to the two-node interpolant: $i(t)-[(1+t/T)i(0)-(t/T)i(-T)]=i^{\prime \prime }(\xi )t(t+T)/2$ for some $\xi \in (-T,t)$. The sample-error terms use absolute coefficients, as required by the signed-filter current-sign certificate. At $t=T$, the model contribution is $M{T}^2$ and equal sample bounds contribute $3ϵ$.

For an original example, take $T=6\mu \mathrm{s}$, $M=10^{8}A/{\mathrm{s}}^2$ and each sample error bounded by $0.002\mathrm{A}$. At $t=T$,If $y_0=0.030\mathrm{A}$ and $y_{-}=0.040\mathrm{A}$, the predicted current is $0.020\mathrm{A}$, certifying actual current at least $0.0104\mathrm{A}$ at that future instant. To certify the entire interval $0\le t\le T$, check $\hat{i}(t)-B(t)>0$ everywhere. In this example the prediction decreases linearly and $B$ increases, so the minimum margin occurs at $T$ and the same lower bound applies throughout.

The curvature limit is a physical assumption. A switching event, sensor clipping or a slope discontinuity can invalidate it even when a fitted ramp looks smooth at the sample times. A bounded first derivative alone does not imply any finite $M$. Timing uncertainty in a curvature-based current-sign certificate also matters: two accurate current values at incorrectly modeled times need not extrapolate an affine ramp correctly.

A proposed test supplies known smooth ramps and curved waveforms, records sample timestamps and checks the independently measured future waveform against the enclosure. Test transitions separately. The result is a deterministic sign guarantee only when the sample and curvature bounds are deterministic; substituting one standard uncertainty for an absolute error bound changes the claim.

Timing uncertainty in a curvature-based current-sign certificate adds the effect of uncertain sample and decision times to a bound on extrapolated electric current. Two accurate readings do not identify a future current precisely unless their timing relative to that future instant is also bounded. The physical purpose is to establish the current direction during the intended switching interval, whose recirculation paths depend on that direction.^[ref]

Let nominal sample times be $-T$ and $0$, and a nominal target time be $t\ge 0$. Actual sample times are $-T+{\delta }_{-}$ and ${\delta }_0$, and actual target time is $t+{\delta }_t$, with absolute timing bounds ${\Delta }_{-},{\Delta }_0,{\Delta }_t$. Measurements areAssume $|e_j|\le {ϵ}_j$, $|i^{\prime }|\le S$ and $|i^{\prime \prime }|\le M$ over all nominal and possible actual times and the intervals connecting them. Comparing each actual sample with its nominal-time value using the first-derivative bound, then applying the current-sign certificate with a curvature bound, givesThe triangle inequality proves this conservative enclosure without assuming independent or random timing errors. Its units are current: a timing error multiplied by a current-slew bound gives an amplitude error. A small curvature bound alone is insufficient, because a steep affine ramp has zero curvature but remains sensitive to relative timing.

For an original example, use $T=t=6\mu \mathrm{s}$, $M=10^{8}A/{\mathrm{s}}^2$, $S=5000A/s$ and ${ϵ}_0={ϵ}_{-}=0.002\mathrm{A}$. The amplitude-plus-curvature allowance is $0.0096\mathrm{A}$. With ${\Delta }_0={\Delta }_{-}=50\mathrm{ns}$ and ${\Delta }_t=100\mathrm{ns}$, timing addsThe full bound is $0.01085\mathrm{A}$. A prediction $0.020\mathrm{A}$ therefore certifies a positive target current of at least $0.00915\mathrm{A}$. To certify a whole switching interval, minimize $\hat{i}(t)-B_{\mathrm{time}}(t)$ over that interval rather than checking only one instant.

Shared-clock cancellation in current extrapolation can make the separate timing allowances conservative. If both samples and the target undergo exactly the same time translation, an affine ramp is still extrapolated exactly. The relevant error is relative timing; a common shift of the acquisition clock and a separately timed switching command need not cancel. Report which clocks, triggers and transport delays define the target.

A proposed check uses a known current ramp and an independently timed reference trace, deliberately perturbs sample and command timing, and tests whether the resulting errors stay inside the enclosure. Repeat near switching transitions. Slew and curvature assumptions must cover those transitions, not only the quiet portion used to fit a smooth waveform.