Relay Controller
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In the circuit shown below, a transistor (modeled as an ideal switch), together with an extra resistor having resistance R_X, is used to control the current i_R through the winding of a relay. Here, the relay is modeled as a series inductor and resistor having inductance L_R and resistance R_R, respectively.
- Assume that the circuit is at rest for t < 0. Then, at t = 0, the transistor turns on (the switch closes). Determine i_R(t) for t ≥ 0 given that i_R(t < 0) = 0.
- Next, at t = T, the transistor turns off (the switch opens). Assume that T is very long so that the transient from Part (A) has reached a steady state. Determine both the relay current i_R(t) and the transistor voltage v_T(t) for t ≥ T.
- Plot your answers for parts (A) and (B): i_R(t ≥ 0), and v_T(t ≥ 0). Make sure to clearly label the graph axes and the important points on the graph.
- The relay control circuit would be less expensive without the external resistor, which may be “removed” from the circuit by considering the limit as R_X approaches infinity. Why might such a cost reduction be unwise?
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