= Threshold-crossing time of a finite rectangular pulse through an RC load
{title2=$t_\uparrow=-\tau\log(1-a)$}
= RC threshold-crossing time
{c}
{synonym}
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 equation>
$$
\tau\frac{dv}{dt}+v=v_{\rm in}(t),\qquad
v(t)=\begin{cases}
V_0(1-e^{-t/\tau}),&0\le t\le T,\\
V_0(1-e^{-T/\tau})e^{-(t-T)/\tau},&t>T.
\end{cases}
$$
The output is a <continuous function> and reaches its <maximum> at the pulse's falling edge:
$$
v_{\rm peak}=V_0(1-e^{-T/\tau}).
$$
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 $V_{\rm th}=aV_0$ with $0<a<1$. Solving the rising <exponential function> using a <natural logarithm> gives
$$
\boxed{t_\uparrow=-\tau\log(1-a),\qquad T_{\min}=t_\uparrow.}
$$
If $T<T_{\min}$, the output never reaches the threshold. At $T=T_{\min}$, it touches the threshold only at its peak; the ideal time spent \i[strictly above] threshold is zero. A nonzero high interval in an ideal non-inverting <electronic comparator> requires \b[$T>T_{\min}$]. 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> gives
$$
t_\downarrow=T+\tau\log\frac{1-e^{-T/\tau}}{a},\qquad
W=t_\downarrow-t_\uparrow
=T+\tau\log\frac{(1-e^{-T/\tau})(1-a)}{a}.
$$
Here $W$ 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 $V_{\rm th}=0.50\,\mathrm V$, so $\tau=1\,\mathrm{\mu s}$. A $0.40\,\mathrm{\mu s}$ input peaks at only $0.3297\,\mathrm V$ and is rejected. A $2.00\,\mathrm{\mu s}$ input peaks at $0.8647\,\mathrm V$ and gives
$$
\boxed{t_\uparrow=0.6931\,\mathrm{\mu s},\quad
t_\downarrow=2.5477\,\mathrm{\mu s},\quad W=1.8546\,\mathrm{\mu s}.}
$$
These are model calculations, not recorded instrument traces.
\Image[/rc-finite-pulse-threshold.png]
{title=Pulse duration determines whether an <RC circuit> reaches a threshold and how long it stays above it}
{description=Original calculation for $\tau=1\,\mathrm{\mu s}$ and a half-height threshold. The short pulse is rejected, the boundary-duration pulse just touches the threshold, and the longer pulse yields two crossings. The right panel shows the corresponding ideal above-threshold duration.}
A threshold near the asymptotic input height is also especially sensitive to <noise> and threshold <measurement uncertainty>. Holding the \i[absolute] input height $V_0$ and $\tau$ fixed, the <threshold timing sensitivity of an RC step response> is
$$
\frac{\partial t_\uparrow}{\partial V_{\rm th}}=\frac{\tau}{V_0-V_{\rm th}}.
$$
For the same $1\,\mathrm V$, $1\,\mathrm{\mu s}$ 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 $70.5\,\mathrm{\mu s}$ component-value <time constant>.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed pp. 1–2, component values and decay/phase measurements]{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.
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