 Residual baseline in RC pulse-width discrimination

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.