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Analog switch on-resistance is the conducting path's effective electrical resistance at a specified signal level, supply and temperature. In a locally resistive model, $R_{\mathrm{on}}=\Delta v/i$. It is neither an ideal short circuit nor necessarily constant as the signal moves. Complementary switch devices reduce signal-dependent resistance variation, as the manufacturer's circuit and curves illustrate.^[ref]

An original example drives a $1.0\mathrm{k\Omega }$ load from an ideal source through $R_{\mathrm{on}}=30\Omega$. The voltage divider givesIf the analog switch on-resistance rises to $50\Omega$ elsewhere in the signal range, the ratio becomes $0.95238$, about $1.90\%$ below its first value. One gain correction cannot make both ratios exact. Switch on-resistance flatness therefore matters separately from the minimum resistance; variation with the signal can create nonlinearity.

The same resistance affects dynamics. A stipulated constant $50\Omega$ discharging $2.0\mathrm{nF}$ has RC circuit time constant $0.10\mu \mathrm{s}$ and requires $\tau \log 100\approx 0.461\mu \mathrm{s}$ to reduce an initial residual to one percent. This is a chosen single-pole model; amplifier dynamics can extend the actual integrator reset settling time.

On a bench, measure the terminal drop at known electric current while sweeping the signal level within the permitted range. Separate a nearly constant series loss from its signal-dependent variation. The source resistance and load belong in that measurement; the switch's control propagation delay is not its conducting-path resistance.

An electrical circuit is a network of components and connections used to transport electric charge and exchange energy. Resistors dissipate energy, capacitors store energy in an electric field, and inductors store energy in a magnetic field.

A lumped-element model represents the circuit by branch electric currents and terminal electric potential differences. Kirchhoff's current law expresses charge balance at connections, while Kirchhoff's voltage law relates the branch voltages around loops. ^[ref] For constant ideal components, with current entering the positive-voltage terminal,Combining these relations produces algebraic equations or time-dependent differential equations.

For example, an RC circuit charging an initially uncharged capacitor from a fixed voltage source $V_0$ obeysThe product $RC$ is its time constant. This transient connects energy storage to the rate of the circuit's response. ^[ref] When propagation along connections matters, a transmission line description can replace a single lumped model.

The exponential settling bound for RC sinusoidal turn-on measures the decaying error relative to the persistent sinusoid's amplitude, rather than dividing by a sinusoid that repeatedly crosses zero. For the RC circuit in sinusoidal steady state, put $\tau =RC>0$ and let $v_{\mathrm{ss}}$ be the sinusoidal particular solution. Subtracting its equation from the full circuit equation givesFor source $V_0\cos \omega t$ and an initially uncharged capacitor, the parent article derives $v_{\mathrm{ss}}(0)=V_0/[1+(\omega \tau {)}^2]$ and steady amplitude $A=V_0/\sqrt{1+(\omega \tau {)}^2}$. Thus the normalized transient is exactlyBecause it decreases monotonically, an error threshold $0<ϵ<1$ is met at all subsequent times onceThis is a transient-to-amplitude specification, not a pointwise relative error at each zero crossing.

For an original comparison, choose $\tau =2\mathrm{ms}$, $\omega \tau =3$, and $ϵ=0.01$. The stated cosine turn-on needs $t_{ϵ}\approx 6.908\mathrm{ms}$. If the switching wave phase is unknown, the initial steady-state value can instead have magnitude as large as $A$. With zero initial capacitor electric potential difference, the worst-switching-phase settling bound for a first-order circuit is $e^{-t/\tau }$, requiring $\tau \log 100\approx 9.210\mathrm{ms}$. The shorter first result depends on the specified switching phase.

An unknown stored initial energy changes the leading mismatch again; the same exponential identity still applies. A bench test must therefore record the switching phase, initial capacitor electric potential difference, and the chosen normalization. MIT's RC laboratory measures the natural decay with square-wave excitation and then compares sinusoidal amplitudes and phases on an oscilloscope.^[ref]

An integrator reset establishes the charge on an integrating amplifier's feedback capacitor before the next measurement. Without it, the next output includes the previous initial condition, and repeated input pulses or input offset voltage can eventually cause amplifier clipping. Resetting makes consecutive measurement intervals comparable; it does not reconstruct a pulse already lost to amplifier clipping.

Consider an inverting integrating amplifier with the input held at zero during reset. Place an analog switch across its feedback capacitor $C$. In the approximation of a stable ideal operational amplifier maintaining a virtual ground, a closed switch with constant on-state electrical resistance $R_{\mathrm{on}}$ gives the RC circuit relationThis exponential decay gives an integrator reset settling time for residual magnitude at most $\epsilon$:For stipulated values $C=10\mathrm{nF}$, $R_{\mathrm{on}}=10\mathrm{k\Omega }$ and $|V_{\mathrm{old}}|=0.50\mathrm{V}$, the reset time constant is $100\mu \mathrm{s}$. Reaching a residual of $1\mathrm{mV}$ requires about $621\mu \mathrm{s}$ in this model. Actual amplifier drive, stability and recovery from overload can change the response. A switch's control delay also differs from the full circuit's settling time.^[ref]

Opening the reset switch can itself disturb the initial condition. Analog switch charge injection transfers a small charge to the connected capacitor, producing a reset charge-injection pedestal of magnitudeFor a stipulated $10\mathrm{pC}$ deposited on $10\mathrm{nF}$, this step is $1\mathrm{mV}$. Its sign depends on the circuit and charge direction. A longer reset suppresses the preceding exponential residual but need not suppress the new step created when the switch opens. Analog Devices illustrates this capacitive coupling mechanism and the resulting output step separately from retained-charge settling.^[ref]

A practical experiment repeats zero-input measurements with several reset durations and reads the output at the same delay after switch opening. A decreasing residual indicates incomplete discharge; a persistent offset can instead include reset charge-injection pedestal, input offset voltage and subsequent drift. Repeated readings help separate the reproducible offset from noise. The ACF2101 provides a concrete historical implementation: separate hold and reset controls disconnect the input and discharge the integration capacitor, and its datasheet shows the sequence of hold, integration and reset states.^[ref]

The reset interval also creates integrator reset dead time: charge arriving while the circuit is being reset may not contribute to the next measured integral. Pulse timing relative to the reset therefore matters. A perfectly repeatable baseline alone does not establish that every input pulse was captured.

Jeffrey Lang is an electrical engineer whose work concerns electromechanical conversion power balance, control and small-scale devices. MIT's Research Laboratory of Electronics identifies the same professor as Jeffrey H. Lang and describes research on electric motors, electrical generators and microelectromechanical systems, together with teaching in circuits and electromagnetism.^[ref]

The identified MIT profile provides an externally hosted portrait of Jeffrey H. Lang.

For an original motor-scale illustration, a shaft delivering torque $\tau =2\mathrm{mN}\mathrm{m}$ at angular velocity $\omega =1000\mathrm{rad}{\mathrm{s}}^{-1}$ delivers $P_m=\tau \omega =2\mathrm{W}$. If electrical input is $3\mathrm{W}$ and the operating point is steady, the remaining $1\mathrm{W}$ must be accounted for by losses or another specified output. During acceleration some input can instead increase rotational kinetic energy. Reporting only shaft speed would not establish either output power or conversion efficiency.

With Anant Agarwal, Lang is credited on the MIT course material State and Memory by Anant Agarwal and Jeffrey Lang. It explains how a capacitor's state summarizes the part of past input needed to predict its future response.^[ref] For a series RC circuit with a constant later input $V_f$,Two different earlier input histories that leave the same initial capacitor electric potential difference give the same subsequent response under the same future input. For stipulated $RC=1\mathrm{ms}$, $v_C(0)=1\mathrm{V}$ and $V_f=3\mathrm{V}$, the value after one time constant is $3-2/e\approx 2.2642\mathrm{V}$. This original numerical example shows what it means for the state to retain enough of the past without retaining the entire waveform.

Karl K. Berggren is an electrical engineer whose research includes superconductivity-based electrical circuits, electronic devices and single-photon detectors for quantum applications. MIT's Research Laboratory of Electronics identifies him as a professor of electrical engineering and physics and as head of its quantum nanostructures and fabrication group.^[ref] These research areas connect the behavior of very small devices to the circuits needed to read and control them.

MIT's identified faculty profile also links an externally hosted portrait of Karl K. Berggren.

His 6.200 Notes: Sinusoidal Steady State by Karl K. Berggren explain why complex numbers simplify linear differential equations driven by sinusoidal inputs. For a series RC circuit observed across its capacitor, the equation $RCd{v}_C/dt+v_C=v_{\mathrm{in}}$ gives the late-time frequency responseThe notes obtain this result by solving for a complex exponential and taking its real part.^[ref]

For an original illustration, $R=1\mathrm{k\Omega }$ and $C=1\mu \mathrm{F}$ give a time constant of $1\mathrm{ms}$. At $\omega =1000\mathrm{rad}{\mathrm{s}}^{-1}$, $H=(1-i)/2$: a one-volt peak sinusoidal input produces a $1/\sqrt{2}$-volt peak output lagging by $\pi /4$. A separate term proportional to $e^{-t/(RC)}$ is the transient and depends on the starting state. Distinguishing that term from the forced sinusoid is what makes the steady-state calculation experimentally useful. This numerical illustration applies his teaching derivation; it is not a reported device measurement.

Parametric model uncertainty allows specified parameters to vary while retaining a chosen mathematical model structure. It records a nominal value and an allowed set, such as an interval, for each uncertain quantity. Repeated occurrences of the same physical parameter must remain linked. MathWorks's uncertain-real-parameter model explicitly represents nominal values, ranges and named repeated parameters.^[ref]

An original RC circuit example uses the first-order transfer functionAssume every resistance/capacitance pair in those positive intervals is allowed. Since their product increases with either factor, the time constant has the exact rangeThe nominal product is1millisecond, so separate symmetric10percent component ranges give an asymmetric time-constant range of minus19percent to plus21percent. The angular frequency at which this model's magnitude falls to $1/\sqrt{2}$ is ${\omega }_c=1/\tau$, giving approximately $826.4$ to $1234.6\mathrm{rad}{\mathrm{s}}^{-1}$. Reciprocal conversion reverses the endpoints.

This family changes response speed while preserving the modeled DC gain $H(0)=1$. A guarantee of correct DC gain therefore does not establish a particular transient or bandwidth. Conversely, an extra pole from unmodeled dynamics is absent from every member of this first-order family, regardless of how widely $R$ and $C$ are allowed to vary.

Parameter dependencies can restrict the family further. If the same uncertain resistor appears in several coefficients, independently varying its appearances changes the physical model. The mass example in control model uncertainty derives one such coefficient relation. Parameter bounds are also not a probability distribution: an interval describes what must be covered, while probability assignments require additional information.

Robust stability and robust performance can then be assessed over this specified family. The result applies to the model structure and bounds actually supplied, not automatically to aging, temperature changes or other effects outside them.

An RC circuit contains electrical resistance and capacitance. For a series resistor $R$ charging a capacitor $C$ from a constant voltage $V$, Kirchhoff's voltage law and $i=Cd{v}_C/dt$ give $RCd{v}_C/dt+v_C=V$.

With an initially uncharged capacitor, $v_C(t)=V(1-e^{-t/(RC)})$. The time constant $\tau =RC$ is the time to reach $1-e^{-1}$ of the final voltage. With the source removed, the same model predicts exponential discharge. Measuring this response provides a simple way to determine an unknown capacitance or to construct a smoothing low-pass filter.

RC pulse-width discrimination converts an input-duration test into a threshold observation on an RC circuit. For a relaxed first-order load with time constant $\tau >0$, a positive rectangular input of height $V_0$ and duration $T$ reaches peak output $V_0(1-e^{-T/\tau })$. An ideal electronic comparator has a nonzero interval above fixed threshold $0<V_{\mathrm{th}}<V_0$ precisely whenEquality only touches the threshold at the peak. The parent threshold-crossing time of a finite rectangular pulse through an RC load derives the full response and the rejected-pulse boundary.

This is a duration discriminator only while the input height is controlled. At a fixed duration, the same condition isA larger input can pass the threshold even when a smaller pulse of exactly the same duration fails. The minimum accepted duration decreases with height, with derivative $-\tau V_{\mathrm{th}}/[V_0(V_0-V_{\mathrm{th}})]$.

For an original example, choose $\tau =2\mu \mathrm{s}$, $V_{\mathrm{th}}=0.40\mathrm{V}$, and $T=1\mu \mathrm{s}$. The minimum height for a nonzero comparator interval is about $1.0166\mathrm{V}$. A $0.8\mathrm{V}$ pulse peaks at about $0.3148\mathrm{V}$ and is rejected; a $1.2\mathrm{V}$ pulse peaks at about $0.4722\mathrm{V}$ and is accepted. Treating this observation as a width-only decision would conflate two different inputs.

A pulse-generator experiment therefore sweeps width at a recorded height and threshold, measures the actual capacitor waveform on an oscilloscope, and allows the initial state to relax. A residual baseline in RC pulse-width discrimination changes the next pulse's peak and invalidates the relaxed-state formula. Probe and comparator input capacitance also belong in the effective time constant. The natural-decay apparatus in MIT's RC laboratory supplies a direct component-level check of that time constant.^[ref] Comparator propagation delay and hysteresis are additional dynamics, not part of the ideal RC acceptance inequality.

A capacitor need not finish discharging before the next pulse arrives. This residual baseline in RC pulse-width discrimination changes the next pulse's RC threshold-crossing time even if its applied height and duration are unchanged. The relaxed-state acceptance formula assumes an initial output of zero; a repeated pulse train can violate that assumption systematically.

Take a first-order RC circuit with time constant $\tau =RC$, input high level $V_0>0$, zero input low level and initial capacitor output $V_b$. During a pulse of duration $T$,For $0\le V_b<V_{\mathrm{th}}<V_0$, solving for the rising threshold givesA finite interval above threshold requires $T>t_{\uparrow }$. At equality the output only touches the threshold at its peak. A positive residual $V_b$ makes the crossing earlier than the zero-baseline crossing. If the output already exceeds the threshold at the start, this rising-crossing formula no longer describes a new edge.

The input of a repeated pulse train is a periodic function with high duration $T$, low duration $D>0$ and period $P=T+D$. Put $\alpha =e^{-T/\tau }$ and $\beta =e^{-D/\tau }$. The next baseline is the previous peak multiplied by $\beta$. Writing $b_n$ for the baseline before pulse $n+1$ gives the linear recurrence relationBecause $0<\alpha \beta =e^{-P/\tau }<1$, the deviation from its fixed point is a decaying geometric sequence:This periodic RC pulse-train baseline is a repeatable state of the circuit, rather than random readout noise. In the periodic regime, separate rising and falling comparator transitions require $b_{\ast }<V_{\mathrm{th}}<V_{p,\ast }$. If $V_{p,\ast }\le V_{\mathrm{th}}$, there is no finite high interval. If $b_{\ast }\ge V_{\mathrm{th}}$, there is no finite low interval: missing comparator reset in an RC pulse train can turn repeated input pulses into one sustained output state.

For an original calculation, choose $\tau =1\mu \mathrm{s}$, $V_0=1\mathrm{V}$, $V_{\mathrm{th}}=0.50\mathrm{V}$, $T=0.40\mu \mathrm{s}$ and $D=0.60\mu \mathrm{s}$. An isolated relaxed pulse peaks at only $0.32968\mathrm{V}$ and is rejected. The repeatable train instead hasIts rising threshold occurs $0.35595\mu \mathrm{s}$ after each pulse starts. The falling crossing is $T+\tau \log (V_{p,\ast }/V_{\mathrm{th}})=0.44219\mu \mathrm{s}$, giving an above-threshold duration of $0.08624\mu \mathrm{s}$. Starting from zero baseline, the first three pulses are rejected, but the fourth peaks at $0.51199\mathrm{V}$ and crosses. A change in pulse acceptance need not mean that the source pulse itself changed.

A useful design bound controls timing error rather than merely calling the circuit discharged. Put $a=V_{\mathrm{th}}/V_0$ and suppose $0\le V_b/V_0\le \epsilon <a$. Relative to the relaxed crossing,Since a nonnegative pulse train in this model has $V_p\le V_0$, a low interval $D\ge \tau \log (1/\epsilon )$ guarantees that baseline bound. Combining the inequalities yields a sufficient quiet interval required for an RC baseline timing budget $\delta >0$:For a 1% baseline allowance, $D\ge 4.6052\tau$ limits the crossing advance to $0.01005\tau$, or $10.05\mathrm{ns}$ when $\tau =1\mu \mathrm{s}$. The bound is conservative because it uses the largest possible peak.

A bench test holds pulse height, duration and threshold fixed while sweeping repetition period with a pulse generator. Record the actual input and capacitor output on an oscilloscope, including the baseline immediately before each pulse. Probe capacitance and effective source electrical resistance belong in $\tau$. MIT's laboratory uses repetitive square-wave excitation to measure RC decay, providing a direct way to check the underlying time constant.^[ref] The calculations above assume ideal input edges and zero input low level; comparator hysteresis and switching dynamics require their own model.

The sinusoidal steady state of a stable linear time-invariant system driven by a sinusoid is its persistent response at the driving frequency, after the relevant transient response has decayed. A phasor calculation finds that response; it does not generally satisfy the circuit's switch-on initial conditions. The late-time assumption is explicit in Berggren's treatment of sinusoidal circuit analysis.^[ref]

Using peak phasors, write the source as $v_s(t)=\mathrm{Re}({\hat{V}}_s{e}^{i\omega t})$. If $H(i\omega )$ is the relevant transfer function, its particular solution isThe complete response adds a homogeneous solution determined by the initial conditions. Approaching the displayed expression requires the modes affecting the measured output to decay; an unchanged amplitude and a fixed phase shift then characterize its frequency response.

For an original turn-on example, drive a series RC circuit with $v_s(t)=V_0\cos \omega t$ for $t\ge 0$, measure the capacitor's electric potential difference, and take $v_C(0)=0$. With time constant $\tau =RC>0$, Kirchhoff's voltage law gives the first-order linear differential equationPut $x=\omega \tau$. Substitution verifies the sinusoidal particular solution $V_0(\cos \omega t+x\sin \omega t)/(1+x^2)$. Adding $A{e}^{-t/\tau }$ and imposing the initial condition fixes $A=-V_0/(1+x^2)$, givingIts sinusoidal part has amplitude $V_0/\sqrt{1+x^2}$ and lags by $\arctan x$. The exponential settling bound for RC sinusoidal turn-on, relative to that amplitude, isWith chosen $R=1\mathrm{k\Omega }$, $C=1\mu \mathrm{F}$, $V_0=1\mathrm{V}$ and $\omega =1000rad/s$, we have $\tau =1\mathrm{ms}$ and $x=1$. At turn-on the phasor approximation predicts 0.5 volt while the capacitor is initially at zero. After 5 milliseconds, their difference is about 3.37 millivolts, or 0.476 percent of the sinusoidal amplitude. A 1 percent settling time threshold for this specific switch-on is about $4.259\tau$.

A useful measurement compares a long, low-frequency square-wave response with sinusoidal responses at several frequencies. An oscilloscope can identify the exponential decay and then measure amplitude ratios and phase shifts. MIT's laboratory uses that sequence to connect the time constant to the sinusoidal response; probe loading belongs in the measured circuit model.^[ref]

Undamped resonant forcing has no bounded sinusoidal steady state. For an ideal series LC circuit, let $q$ be electric charge and ${\omega }_0=1/\sqrt{LC}$. At resonance, the equationhas the particular solution $q_p(t)=V_{0}t\sin ({\omega }_{0}t)/(2L{\omega }_0)$. Its growing envelope prevents a bounded attracting sinusoidal response. This ideal counterexample explains why decay of the relevant homogeneous solutions is part of the physical steady-state assumption.

An electronic comparator registers a pulse only if its input reaches the chosen threshold. For an initially uncharged capacitor driven through an electrical resistance $R>0$, a short rectangular pulse may end before that happens. The resulting threshold-crossing time of a finite rectangular pulse through an RC load depends on threshold and pulse duration, as well as the time constant $\tau =RC$; it is not determined by group delay alone.

Take a positive input electric potential difference $V_0$ for $0\le t<T$, zero input outside that interval, and output $v(t)$ across a capacitance $C>0$. Include the source's effective series electrical resistance in $R$. The initial condition is $v(0)=0$. Kirchhoff's voltage law and the capacitor current relation give the first-order linear differential equationThe output is a continuous function and reaches its maximum at the pulse's falling edge:The input jumps there, but an ideal capacitor's stored electric charge cannot change discontinuously under the finite currents of this model.

Choose a fixed threshold $V_{\mathrm{th}}=a{V}_0$ with $0<a<1$. Solving the rising exponential function using a natural logarithm givesIf $T<T_{\min }$, the output never reaches the threshold. At $T=T_{\min }$, it touches the threshold only at its peak; the ideal time spent strictly above threshold is zero. A nonzero high interval in an ideal non-inverting electronic comparator requires $T>T_{\min }$. This is RC pulse-width discrimination: different input durations can give a missing pulse or a finite output pulse even when their input heights are equal.

For $T>T_{\min }$, solving the falling exponential decay givesHere $W$ is the time above threshold for an RC-filtered rectangular pulse, before any electronic comparator switching dynamics. It is not generally the input duration $T$. Just above $T_{\min }$, differentiation gives $dW/dT=1/(1-e^{-T/\tau })$ and hence $W\approx (T-T_{\min })/a$ to first order; a half-height threshold gives slope two at the detection boundary.

For an original numerical example, take $R=1\mathrm{k\Omega }$, $C=1\mathrm{nF}$, $V_0=1\mathrm{V}$, and $V_{\mathrm{th}}=0.50\mathrm{V}$, so $\tau =1\mu \mathrm{s}$. A $0.40\mu \mathrm{s}$ input peaks at only $0.3297\mathrm{V}$ and is rejected. A $2.00\mu \mathrm{s}$ input peaks at $0.8647\mathrm{V}$ and givesThese are model calculations, not recorded instrument traces.

A threshold near the asymptotic input height is also especially sensitive to noise and threshold measurement uncertainty. Holding the absolute input height $V_0$ and $\tau$ fixed, the threshold timing sensitivity of an RC step response isFor the same $1\mathrm{V}$, $1\mu \mathrm{s}$ model, a $1\mathrm{mV}$ effective threshold perturbation gives approximately $2\mathrm{ns}$ of timing change at $0.50\mathrm{V}$, but $10\mathrm{ns}$ at $0.90\mathrm{V}$, by first-order uncertainty propagation. The rectangular pulse must still last long enough to reach the selected threshold; changing $a$ together with $V_0$ is a different sensitivity calculation.

A bench test uses a pulse generator and an oscilloscope to record both the applied electric potential difference and the capacitor output. Sweep $T$ while keeping $V_0$ and the actual threshold fixed, and allow enough time between pulses for the measured initial output to be negligible at the desired precision. Include probe and comparator-input capacitance in $C$. MIT's laboratory measures RC decay with a $47\mathrm{nF}$ capacitor, a $1.5\mathrm{k\Omega }$ resistor and square-wave excitation; that gives a directly measurable $70.5\mu \mathrm{s}$ component-value time constant.^[ref] The pulse-duration sweep proposed here extends that decay measurement to a threshold observation. A real electronic comparator's input offset voltage, hysteresis and finite switching time must be separated from the ideal RC crossing times; a Schmitt trigger uses different rising and falling thresholds.

The worst-switching-phase settling bound for a first-order circuit maximizes the initial transient over the source's unknown switching wave phase. For the stable RC circuit, let its sinusoidal particular solution be $v_{\mathrm{ss}}(t)=A\cos (\omega t+\psi )$, where $A>0$, and let $u_0$ be the initial capacitor electric potential difference. The error identity in exponential settling bound for RC sinusoidal turn-on givesSince $\cos \psi$ ranges from minus one to one,For a validated initial bound $|u_0|\le U$, the sharp uniform bound is therefore $(1+U/A)e^{-t/\tau }$. It is attained by choosing the extreme initial value and the opposite steady-state wave phase. A normalized error tolerance $0<ϵ<1$ is guaranteed thereafter ifThere is no division by the instantaneous sinusoid, so its zero crossings cause no singular relative-error specification.

For an original example, choose $\tau =4\mathrm{ms}$, $A=0.5\mathrm{V}$, $U=0.25\mathrm{V}$, and $ϵ=0.02$. The worst-case settling time is $4\log 75\approx 17.270\mathrm{ms}$. Assuming an uncharged capacitor instead would give $4\log 50\approx 15.648\mathrm{ms}$. That shorter guarantee requires the additional initial-state information.

A phase-locked test can measure a smaller mismatch, even zero for a specially matched initial condition, without contradicting the worst-case result. A valid comparison records both the switching wave phase and the capacitor's initial state. The MIT laboratory's square-wave natural-decay observation and sinusoidal amplitude/wave phase measurement provide the two underlying bench measurements.^[ref] The result assumes the same linear first-order model throughout switching; additional modes or a time-varying component require another transient bound.

The zero-input response is the output due to a system's initial internal state when its externally applied input is set to zero. It describes memory left by earlier excitation. For a linear time-invariant system, linearity separates this from the zero-state response due to the subsequent input.

For constant matrices in $\dot{x}=Ax+Bu$, $y=Cx+Du$, setting $u=0$ and $x(0)=x_0$ givesThe matrix exponential solves the homogeneous state equation. With an arbitrary input, the complete solution adds the zero-state response:A zero initial output $C{x}_0=0$ alone does not establish that the whole state is zero.

For an RC circuit with capacitor difference $v(0)=V_0$ and no applied source, $RC\dot{v}+v=0$, so $v(t)=V_0{e}^{-t/(RC)}$. The capacitor energy falls as $C{V}_0^2{e}^{-2t/(RC)}/2$; the resistor receives the dissipated energy. The primary MIT lecture explicitly distinguishes the capacitor's initial state, zero external input and zero initial state.^[ref]

At a time constant $RC=2\mathrm{s}$ and $V_0=3\mathrm{V}$, the modeled difference after two seconds is $3/e\approx 1.104\mathrm{V}$. Zero input has not meant zero output or zero stored energy. A direct experiment charges the same capacitor to different known initial differences, removes the charging source and records the decay across a fixed resistor. The amplitudes should scale with $V_0$ while the normalized decay remains the same in the linear model.

A zero-input response need not decay in every system: an ideal undamped oscillator retains its initial motion, and an unstable homogeneous system can amplify a nonzero initial state. The transfer function describes the zero-state input-output map, so its product with a zero input cannot represent a separate nonzero initial-state contribution.

The zero-state response is the output produced by an applied input when the system starts with zero internal state. For a linear time-invariant system it separates the effect of the input from the zero-input response caused by an initially stored state. A zero initial output alone need not mean zero internal state.

For a causal linear time-invariant system with impulse response $h$, the zero-state response is the convolutionAn instantaneous feedthrough term, if present, is represented by a Dirac delta term in $h$. Equivalently, for constant matrices in $\dot{x}=Ax+Bu$, $y=Cx+Du$ with $x(0)=0$,For nonzero $x(0)=x_0$, linearity adds the zero-input response $C{e}^{At}{x}_0$. Here the matrix exponential propagates the state left by the earlier input.

For example, $\tau \dot{y}+y=u$ with $\tau >0$, $y(0)=0$ and a constant input $u(t)=A$ for $t\ge 0$ has the step response $y_{\mathrm{zs}}=A(1-e^{-t/\tau })$. At one time constant it has reached about $63.2\%$ of its final value. If the initial value instead equals $y_0$, the total output is $A(1-e^{-t/\tau })+y_0{e}^{-t/\tau }$. A charged capacitor in an RC circuit provides a direct physical example: discharge it before repeating the same input step to isolate the zero-state response, and compare with a run from a known initial electric potential difference.^[ref]

With compatible transforms, the zero-state input-output relation is $Y=HU$, where $H$ is the transfer function. For a nonzero initial state, a unilateral Laplace transform includes additional initial-condition terms. Those terms cannot be absorbed into an input-independent transfer function multiplying $U$.