A Nice, Relaxing Problem
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This problem studies the circuit below, a "relaxation oscillator" that produces a square wave on its output. This oscillator employs positive feedback so that its output, v_{\rm out}, switches between the op-amp power supply rails of +V_{\rm S} and -V_{\rm S} depending on the sign of v_+ - v_-.
Remember that when the op-amp is in positive feedback, we cannot make the ideal op-amp assumption; rather, in this configuration, the op-amp behaves much like a comparator, effectively setting its output either to +V_{\rm S} or -V_{\rm S}.
Answer the following questions about this circuit.
Part A
Suppose that, at time t=t_0, v_{\rm out}(t_0) = V_{\rm S}, and that v_-(t_0) is given and satisfies v_-(t_0) < 0. Derive an expression for v_-(t) that is valid from t = t_0 until the time at which v_{\rm out} switches to -V_{\rm S}. You do not need to determine the actual switching time. You may answer in terms of V_{\rm S}, v_-(t_0), and any other values from the circuit or mathematical constants that you might need.
Part B
Below is shown a graph of v_{\rm out}(t) over two cycles of operation. On these same axes, sketch and clearly label v_+(t) and v_-(t) on the same scales.
Part C
Determine the steady-state period T of the square-wave oscillator in terms of the given constants.
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