Exponential settling bound for RC sinusoidal turn-on ($e^{-t/\tau }/\sqrt{1+(\omega \tau {)}^2}$)

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 $v_{\mathrm{ss}}$ be the sinusoidal particular solution. Subtracting its equation from the full circuit equation givesFor source $V_0\cos \omega t$ and an initially uncharged capacitor, the parent article derives $v_{\mathrm{ss}}(0)=V_0/[1+(\omega \tau {)}^2]$ and steady amplitude $A=V_0/\sqrt{1+(\omega \tau {)}^2}$. Thus the normalized transient is exactlyBecause it decreases monotonically, an error threshold $0<ϵ<1$ is met at all subsequent times onceThis 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 $ϵ=0.01$. The stated cosine turn-on needs $t_{ϵ}\approx 6.908\mathrm{ms}$. 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 $\tau \log 100\approx 9.210\mathrm{ms}$. 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.^[ref]

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 $v_{\mathrm{ss}}(t)=A\cos (\omega t+\psi )$, 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 givesSince $\cos \psi$ ranges from minus one to one,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 $0<ϵ<1$ is guaranteed thereafter ifThere 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 $ϵ=0.02$. The worst-case settling time is $4\log 75\approx 17.270\mathrm{ms}$. Assuming an uncharged capacitor instead would give $4\log 50\approx 15.648\mathrm{ms}$. 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.^[ref] The result assumes the same linear first-order model throughout switching; additional modes or a time-varying component require another transient bound.