= Worst-switching-phase settling bound for a first-order circuit
The <worst-switching-phase settling bound for a first-order circuit> maximizes the initial transient over the source's unknown switching <wave phase>. For the stable <RC circuit>, let its sinusoidal <particular solution> be $v_{\rm ss}(t)=A\cos(\omega t+\psi)$, where $A>0$, and let $u_0$ be the initial capacitor <electric potential difference>. The error identity in <exponential settling bound for RC sinusoidal turn-on> gives
$$
\frac{|v_C(t)-v_{\rm ss}(t)|}{A}
=\frac{|u_0-A\cos\psi|}{A}e^{-t/\tau},\qquad \tau=RC.
$$
Since $\cos\psi$ ranges from minus one to one,
$$
\sup_\psi|u_0-A\cos\psi|=|u_0|+A.
$$
For a validated initial bound $|u_0|\le U$, the sharp uniform bound is therefore $(1+U/A)e^{-t/\tau}$. It is attained by choosing the extreme initial value and the opposite steady-state <wave phase>. A normalized error tolerance $0<\epsilon<1$ is guaranteed thereafter if
$$
t\ge\tau\log\frac{1+U/A}{\epsilon}.
$$
There is no division by the instantaneous <sinusoid>, so its zero crossings cause no singular relative-error specification.
For an original example, choose $\tau=4\,\mathrm{ms}$, $A=0.5\,\mathrm V$, $U=0.25\,\mathrm V$, and $\epsilon=0.02$. The worst-case settling time is $4\log75\approx17.270\,\mathrm{ms}$. Assuming an uncharged capacitor instead would give $4\log50\approx15.648\,\mathrm{ms}$. That shorter guarantee requires the additional initial-state information.
A phase-locked test can measure a smaller mismatch, even zero for a specially matched initial condition, without contradicting the worst-case result. A valid comparison records both the switching <wave phase> and the capacitor's initial state. The MIT laboratory's square-wave natural-decay observation and sinusoidal <amplitude>/<wave phase> measurement provide the two underlying bench measurements.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed pp. 1–2]{ref} The result assumes the same linear first-order model throughout switching; additional modes or a time-varying component require another transient bound.
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