Question

# The circuit shown below contains two switches which open and close perfectly out of phase with one another (only one is open at any given time, and once one closes the other immediately opens, and vice-versa). The switches alternate at a frequency of10kHz, and each switch spends 50% of the time open and 50% of the time closed (i.e.each switch is open for 100 microseconds and then closed for 100 microseconds). The initial inductor current (i (0)) is 400mA. Assume that at t= 0, half of the first switching cycle is already complete (i.e. the first switch period is 50 microseconds instead of 100microseconds), and that S, is in the closed position and S, is in the open position.

In terms of forced and natural responses (i (t) = , + in(t)), write an equation for the instantaneous inductor current during each part of the full (100 microsecond) switching cycle (two equations total). You must include the steady-state current (forced response att = 0), length of each time period, and time constant for each switching period. Use"I," for the current at the beginning of each switching period.

In the space below, plot the inductor current and inductor voltage (using the polarities specified in the diagram) between t= 0 and t = 650µs (i.e. from the beginning to the seventh switch transition). Label the instantaneous current and voltage values and elapsed time at each switch transition. Over a number of switching cycles, the inductor current waveform tends to stabilize and reach a fixed maximum and minimum value over each full switching cycle. Answer the following two questions under the assumption that the circuit has already reached this equilibrium state. What is the approximate average inductor current (over both switching periods)? \mathrm{I}_{\text {Lavg }} \cong What is the approximate maximum voltage across the 502 resistor (ve(t)) during any part of the switching period? \mathrm{V}_{\mathrm{Rmax}} \cong Since the maximum voltage seen by the 502 resistor is greater than the source voltage,and since KVL must be satisfied, what does this imply about the inductor voltagewave form?

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