Periodic RC pulse-train baseline ($V_b/V_0=\beta (1-\alpha )/(1-\alpha \beta )$)

A periodic RC pulse-train baseline is the repeatable pre-pulse capacitor voltage reached after startup. It need not be zero even when the input low level is zero. With high duration $T$, low duration $D$, period $P=T+D$, and $\alpha =e^{-T/\tau }$, $\beta =e^{-D/\tau }$, the recurrence in residual baseline in RC pulse-width discrimination has contraction factor $q=\alpha \beta =e^{-P/\tau }$. Its fixed baseline and peak areThis is a deterministic periodic state, not a random noise floor. The startup error is reduced by a specified fraction $\eta$ after $n\ge (\tau /P)\log (1/\eta )$ cycles, rounded upward.

The mean supplies an independent check. Integrating $\tau \dot{v}+v=u$ over a full period makes the derivative term vanish, so $⟨v⟩=⟨u⟩=(T/P)V_0$. The mean depends on duty cycle while the baseline and ripple depend on period relative to the time constant. For $P\ll \tau$ at fixed $r=T/P$, expansion givesThus a rapid train approaches a level near its duty-cycle average, rather than isolated relaxed pulses.

For an original fast-train example, choose $V_0=1\mathrm{V}$, $\tau =1\mathrm{ms}$, $P=0.20\mathrm{ms}$ and $r=0.30$. The exact baseline is about $0.27929\mathrm{V}$, the peak $0.32127\mathrm{V}$, and the mean exactly $0.30\mathrm{V}$. The startup deviation contracts by $e^{-0.2}$ each cycle; reducing it below one percent of its initial value takes 24 cycles, or $4.8\mathrm{ms}$.

A pulse-generator/oscilloscope test should therefore distinguish the first pulses from the repeatable train, and record minima, maxima and mean rather than only input width. The natural-response measurement in the primary RC laboratory checks the time constant used here.^[ref] Threshold placement relative to the repeatable minimum and maximum determines acceptance and missing comparator reset in an RC pulse train.