= Periodic RC pulse-train baseline
{title2=$V_b/V_0=\beta(1-\alpha)/(1-\alpha\beta)$}
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 are
$$
b_* =V_0\frac{\beta(1-\alpha)}{1-\alpha\beta},\qquad
v_{p,*}=V_0\frac{1-\alpha}{1-\alpha\beta},\qquad
b_n-b_*=q^n(b_0-b_*).
$$
This 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 $\langle v\rangle=\langle u\rangle=(T/P)V_0$. 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 gives
$$
\frac{b_*}{V_0}=r-\frac{r(1-r)}2\frac P\tau+O((P/\tau)^2),\qquad
\frac{v_{p,*}}{V_0}=r+\frac{r(1-r)}2\frac P\tau+O((P/\tau)^2).
$$
Thus 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.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed pp. 1–2]{ref} Threshold placement relative to the repeatable minimum and maximum determines acceptance and <missing comparator reset in an RC pulse train>.
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