 Threshold-crossing time of a finite rectangular pulse through an RC load

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.