# Relaxation oscillator

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This problem studies the square-wave relaxation oscillator shown below. The oscillator employs a comparator with hysteretic positive feedback so that its output, v_{\rm OUT}, switches between the comparator power supplies of \pm V_{\rm S} depending upon the sign of its differential input; for simplicity, the comparator power supplies are not shown. Except for an output limited by its power supplies, the comparator is otherwise ideal. For a discussion of hysteresis, see the previous exercise.

###Part 1
Assume that v_{OUT} = V_{S}. Determine the value of v_{+}. **Use VS for V_S.**

###Part 2 Suppose at t=t_\circ that v_{\rm OUT} = V_{\rm S}, and that v_{-} is given and satisfies v_{\rm -}(t_\circ) < v_{\rm +}. Derive an expression for v_{-}(t) that is valid from t=t_\circ until the time at which v_{\rm OUT} switches to v_{\rm OUT} = -V_{\rm S} as v_{-} reaches v_{+}. You need not determine the actual switching time.

**Use vto for v_-(t_o), to for t_o, and VS for V_S.**

###Part 3 Of the four plots below, pick the one which best matches the steady-state output voltage v_{OUT}.

###Part 4

Determine the steady-state period T of oscillation.