Homework2_ solutions

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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).
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online