EECS314 Hmwk6 Soln

# EECS314 Hmwk6 Soln - EECS 314 Fall 2008 Homework set 6...

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Unformatted text preview: EECS 314 Fall 2008 Homework set 6 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above © 2008 Alexander Ganago Page 1 of 4 Problem 1 The RC circuit response to a pulse The Big Picture Assuming that before the arrival of the pulse, at time t =0- the circuit was under DC steady-state conditions, one can easily find that the capacitor voltage at time t >0 is expressed as V C = V S ⋅ 1 − e − t τ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ where the time constant is τ = R ⋅ C When the pulse begins, the capacitor is gradually charging. Note that the circuit does not “know” that the pulse will end: its response, according to the equation above, is exactly as it would be for a switch closed forever or an infinitely long pulse. When the pulse ends, the capacitor is gradually discharging. Remember the continuity demand! In the circuit above, the discharge has the same time constant: recall that a voltage source of zero voltage (at time t > Δ t ) acts as a short circuit. The Assignment Part 1 (10 points) Given R = 240 Ω , determine the range of capacitances such that the capacitor voltage V C reaches at least 3.5 V within the pulse duration of 200 ps (1 picosecond = 10-12 sec). This duration of a clock pulse corresponds to the frequency of 2.5 GHz, typical of today’s computers. Your result: _______ C ≤ 6.92 x 10-13 F = 692 fF _____________ We know that the voltage across the capacitor is expressed as ܸ ஼ ൌ ܸ ௌ ሺ1 െ ݁ ି ௧ ఛ ሻ Part 1 2 3 Total EECS 314 Fall 2008 Homework set 6 Student’s name ___________________________ Discussion section # _______ (Last, First, write legibly, use ink) (use ink) Instructor is not responsible for grading and entering scores for HW papers lacking clear information in the required fields above © 2008 Alexander Ganago Page 2 of 4 Given that V S = 5 V, τ = RC, R = 240 Ω , and we want the V C ≥ 3.5 V at the end of the pulse (i.e. t = 200 ps), we obtain: ܸ ஼ ሺݐ ൌ 200 ݌ݏሻ ൒ 3.5 ܸ 5 ൬1 െ ݁ ି ଶ଴଴௣௦ ஼ൈଶସ଴Ω ൰ ൒ 3.5 ܸ Solving for C, we obtain: C ≤ 6.92 x 10-13 F = 692 fF Part 2 (10 points) Given C = 3.5 pF, determine the range of resistances such that the capacitor voltage V C reaches at least 3.5 V within the pulse duration of 200 ps. Your result: ________ R ≤ 47.5 Ω ____________ Similar to Part 1. Given that V S = 5 V, τ = RC, C = 3.5 pF, and we want the V C ≥ 3.5 V at the end of the pulse (i.e. t = 200 ps), we obtain: ܸ ஼ ሺݐ ൌ 200 ݌ݏሻ ൒ 3.5 ܸ 5 ൬1 െ ݁ ି ଶ଴଴௣௦ ோൈଷ.ହ௣ி ൰ ൒ 3.5 ܸ Solving for R, we obtain: R ≤ 47.5 Ω Problem 1, continued Part 3 (20 points) Assume Δ t = 2 ⋅ τ and sketch on the grids below: • the capacitor voltage, and • the resistor voltage. EECS 314 Fall 2008 Homework set 6...
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## This note was uploaded on 09/23/2009 for the course EECS EECS314 taught by Professor Ganago during the Winter '09 term at University of Michigan.

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EECS314 Hmwk6 Soln - EECS 314 Fall 2008 Homework set 6...

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