# RL circuit with switches

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The circuit shown below can be used to regulate the current through an inductor. (The switches model transistors.) Typical applications include the regulation of currents in motors, solenoids and loud speakers, all of which have inductive windings. We will analyze the circuit assuming that it operates in a cyclic manner with switching period T. During the first part of each period, which lasts for a duration DT, switches S1 and S4 are closed while switches S2 and S3 are open. During the second part of each switching period, which lasts for a duration (1 − D)T, switches S1 and S4 are open while switches S2 and S3 are closed. Note that 0 \le D \le 1.

###A. Assume that a new switching period characterized by a given D and T begins at t = 0. In terms of the unknown i(0), determine i(t) for 0 \le t \le DT.

**For this and following questions, enter i(0) as i0 and exp() for the exponential, i.e., exp(5) for e^5.**

###B. Next, determine i(t) for DT \le t \le T.

###C. Assume further that the circuit has operated long enough to reach a cyclic steady state by t = 0 such that i(t + T) = i(t). Use your result from Part (B) to determine i(0). Note that with this result, and that from Part (A), i(t) is completely determined.

###D.
Find the average value of i(t) over the period 0 \le t \le T. *Hint: is it necessary to average the result from Part A, or is there a faster method to find the average?*