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Homework2_ solutions

# Homework2_ solutions - EE 341 Fall 2012 EE 341 Homework...

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EE 341 Fall 2012 EE 341 – Homework Chapter 2 2.1 The electrical circuit shown in Fig. P2.1 consists of two resistors R1 and R2 and a capacitor C. (i) Determine the differential equation relating the input voltage v(t) to the output voltage y(t). (ii) Determine whether the system is (a) Linear, (b) time-invariant; (c) memoryless; (d) causal, (e) invertible, and (f) stable. (i) Applying Kirchoff’s current law to node 1 gG±² − ³G±² ´1 + gG±² ´2 + µ ¶g ¶± = 0 ¶g ¶± + ´1 + ´2 µ´1´2 gG±² = ³G±² µ´1 (ii) Determine whether the system is (b) Linear For v1(t) applied as the input, the output is y1(t), also for v2(t) applied the output is y2(t): µ´1 ·¸¹ ·º + »¹¼»½ »½ g1G±² = ³1G±² and µ´1 ·¸½ ·º + »¹¼»½ »½ g2G±² = ³2G±² Also let, ³3G±² = ¾³1G±² + ¿³2G±² or µ´1 ·¸À ·º + »¹¼»½ »½ g3G±² = ¾³1G±² + ¿³2G±² Substituting v1(t) and v2(t) we’ll have µ´1 ¶g3 ¶± + ´1 + ´2 ´2 g3G±² = ¾[µ´1 ¶g1 ¶± + ´1 + ´2 ´2 g1G±²] + ¿[µ´1 ¶g2 ¶± + ´1 + ´2 ´2 g2G±²] µ´1 ¶g3 ¶± + ´1 + ´2 ´2 g3G±² = µ´1[ ¶G¾g1 + ¿g2² ¶± + ´1 + ´2 ´2 G¾g1 + ¿g2²] So g3G±² = ¾g1G±² + ¿g2G±² System is linear. (c) time-invariant For v(t-t0) applied as the input, the output y1(t) is: ¶g1 ¶± + ´1 + ´2 µ´1´2 g1G±² = 1 µ´1 ³G± − ±0² Substitute τ =t-t0 (so dt=d τ ) ¶g1G τ + ±0² + ´1 + ´2 µ´1´2 g1G + ±0² = 1 µ´1 ³G ²

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EE 341 Fall 2012 So y1( τ )=y( τ -t0) The system is time-invariant (d) memoryless gG±² = 1 ³´1 µ ¶G·²¸· − ´1 + ´2 ³´1´2 ¹ º» µ gG·²¸· ¹ º» The output y(t) as t =t0 is gG±² = 1 ³´1 µ ¶G·²¸· − ´1 + ´2 ³´1´2 ¹¼ º» µ gG·²¸· ¹¼ º» It is clear that the output depends on previous values of the input v(t).
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