= Exponential settling bound for RC sinusoidal turn-on
{title2=$e^{-t/\tau}/\sqrt{1+(\omega\tau)^2}$}
The <exponential settling bound for RC sinusoidal turn-on> measures the decaying error relative to the persistent sinusoid's <amplitude>, rather than dividing by a sinusoid that repeatedly crosses zero. For the <RC circuit> in <sinusoidal steady state>, put $\tau=RC>0$ and let $v_{\rm ss}$ be the sinusoidal <particular solution>. Subtracting its equation from the full circuit equation gives
$$
v_C(t)-v_{\rm ss}(t)=\{v_C(0)-v_{\rm ss}(0)\}e^{-t/\tau}.
$$
For source $V_0\cos\omega t$ and an initially uncharged <capacitor>, the parent article derives $v_{\rm ss}(0)=V_0/[1+(\omega\tau)^2]$ and steady <amplitude> $A=V_0/\sqrt{1+(\omega\tau)^2}$. Thus the normalized transient is exactly
$$
\frac{|v_C(t)-v_{\rm ss}(t)|}{A}
=\frac{e^{-t/\tau}}{\sqrt{1+(\omega\tau)^2}}.
$$
Because it decreases monotonically, an error threshold $0<\epsilon<1$ is met at all subsequent times once
$$
t\ge t_\epsilon=\tau\max\!\left\{0,\log\frac{1}{\epsilon\sqrt{1+(\omega\tau)^2}}\right\}.
$$
This is a transient-to-amplitude specification, not a pointwise relative error at each zero crossing.
For an original comparison, choose $\tau=2\,\mathrm{ms}$, $\omega\tau=3$, and $\epsilon=0.01$. The stated cosine turn-on needs $t_\epsilon\approx6.908\,\mathrm{ms}$. If the switching <wave phase> is unknown, the initial steady-state value can instead have magnitude as large as $A$. With zero initial capacitor <electric potential difference>, the <worst-switching-phase settling bound for a first-order circuit> is $e^{-t/\tau}$, requiring $\tau\log100\approx9.210\,\mathrm{ms}$. The shorter first result depends on the specified switching phase.
An unknown stored initial <energy> changes the leading mismatch again; the same exponential identity still applies. A bench test must therefore record the switching phase, initial capacitor <electric potential difference>, and the chosen normalization. MIT's RC laboratory measures the natural decay with square-wave excitation and then compares sinusoidal amplitudes and phases on an <oscilloscope>.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed pp. 1–2, natural response and sinusoidal observations]{ref}
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