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UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences EE 42 / 100 Problem Set 8 Spring 2008 1. Calculate the inductance between nodes (a,b). 2. The graph below shows the current applied to an 8 mH inductor as a function of time. (a) Derive expressions for the voltage across the inductor for t = 0 . . . 3 ms, t = 3 . . . 8 ms, and t = 8 . . . 9 ms. (b) Derive expressions for the power delivered to the inductor as a function of time. (c) Derive expressions for the energy stored on the inductor as a function of time. (d) Plot the results from parts (a) to (c). 5 Current i L [mA] 4 3 2 1 0 0 1 2 3 4 5 6 Time [ms] 1 7 8 9 10 3. The current through the inductor is initially zero. Accurately plot the value of vo (t) for the source voltage shown below. Calculate the voltage at each breakpoint. Assume that the opamp is ideal. Vs [V] 4 3 2 1 0 t [ns] -1 -2 -3 -4 0 1 2 3 4 5 6 2 7 8 9 10 11 4. Assume a laptop requires i1 = 1 A at v1 = 16 V. The circuit below shows a simple (and bad) solution to power this device directly from 110 V. (a) Find the correct value of R1 . (b) Calculate the power P1 dissipated by the laptop, and the total power Ptot delivered by the source Vs . (c) Calculate the efficiency of the circuit, PPtot1 in percent. (d) Assume that 50,000 people at UC Berkeley use a laptop each day for 3 hours. Calculate the energy wasted each day in kWh (W × hour). (e) Assuming a price of $ 0.12 per kWh, what is the cost to power the laptops per day? How much money could be saved each day if better circuits with 100% efficiency were used instead? 3 5. In this problem you will design a “step down converter” that efficiently converts the input voltage Vs = 110 V to a lower value vo = 16 V. These circuits are employed in the familiar “power bricks” used to power laptops and gadgets from the grid. In the diagram below the switch is controlled by a periodic waveform. It is closed for a period T1 , followed by a period T2 = T − T1 during which it is open. The diode D1 acts like a “valve”: current passes for vd ≥ 0 V; for vd < 0 V the diode behaves like an open circuit. (a) Assuming that the circuit behaves as intended, i.e. vo = 16 V draw a simplified circuit when the switch is closed. (b) Repeat (a) when the switch is open. (c) Derive expressions for the change of the current Δi L through the inductor when the switch is opened and closed. These expressions are functions of Vs , vo , L1 , T1 , and T2 . (d) In steady state, the change of the inductor current Δi L when the switch is closed and the change when the switch is opened must be equal. Use this condition to derive an expresvo as a function of T1 and T2 . sion for V s (e) Find T1 and T2 for T = T1 + T2 = 5 μs. (f) During period T2 when the switch is open the capacitor C1 supplies power to the load X L . Assuming that the load draws 1 A dimension C1 such that vo drops by no more than 0.5 V during T2 . (g) What is the efficiency of this circuit, defined as the ratio of power dissipated by the load X L divided by the power delivered by the source Vs ? Hint: which elements in the circuit dissipate power? (h) The efficiency of actual realizations of this circuit is 10 . . . 20 %. Explain what nonidealities in the circuit are responsible for dissipation. (i) Verify your design with circuit simulation. Practical realizations contain additioal circuitry to dynamically adjusts the duty cycle 4