Past exam of the physics 2 paper of the University of Oxford / 2018 / 9 / d

For explicit sketches take a symmetric, 50% duty cycle square wave with levels $\pm V_0$. The source does not specify levels or duty cycle; a unipolar 50% wave follows by adding its mean and rescaling. Let $\tau =RC$ and $M$ be the positive output extremum in the periodic steady state. The square-wave response of an RC low-pass filter hasextended periodically. The extremum formula follows by imposing $V_{\mathrm{out}}(0)=-M$ and $V_{\mathrm{out}}(T/2)=M$. It makes the three regimes below quantitative, including their finite amplitudes.

When $\omega RC\ll 1$, each half-cycle lasts many time constants, so $M\simeq V_0$. The output nearly follows the two input levels, with exponentially rounded transitions lasting order $RC$. It remains continuous rather than making ideal vertical jumps.

When $\omega RC\gg 1$, the capacitor voltage changes little during each half-cycle. Thus ${\dot{V}}_{\mathrm{out}}\simeq V_{\mathrm{in}}/(RC)$, giving an approximately triangular output. Its extrema areThe finite but small triangle is centered on zero for the chosen zero-mean input. A nonzero input mean shifts the output mean by the same amount.

When $\omega RC$ is of order one, the output consists of visibly curved exponential charging and discharging arcs, with neither long flat plateaus nor nearly straight sides. At $\omega RC=1$,The output is continuous and lags the abrupt input transitions; its precise shape is the periodic solution given in the parent part.