 Solution

ID: past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2018/9/d/solution

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

τ = RC

and

M

be the positive output extremum in the periodic steady state. The square-wave response of an RC low-pass filter has

M = V_{0} tanh \frac{T}{4 τ}, V_{out} (t) = {V_{0} - (V_{0} + M) e^{- t / τ}, - V_{0} + (V_{0} + M) e^{- (t - T /2) / τ}, 0 \leq t < T /2, T /2 \leq t < T,

extended periodically. The extremum formula follows by imposing

V_{out} (0) = - M

and

V_{out} (T /2) = M

. It makes the three regimes below quantitative, including their finite amplitudes.

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