This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: University of Minnesota Dept. of Electrical and Computer Engineering EE3015 – Signals and Systems Discussion Session #5: CT convolution Consider the following RC circuit, with input voltage x ( t ) and output voltage y ( t ). ) ( ) ( ) ( t x t y dt t dy RC = + 1. Find and sketch the output voltage when the input is x ( t )= Vu ( t ). The circuit is assumed to be at rest for t <0. The homogeneous equation is ) ( ) ( = + t y dt t dy RC h h . The solution is of the form st h Ae t y = ) ( f o r t >0 Plugging it in the homogeneous equation, = + st st Ae RCAse . Dividing by st Ae we obtain RC s RCs 1 1 − = ⇒ = + So the homogeneous solution is RC t h Ae t y / ) ( − = for t >0. Now, consider a particular solution of the form B t y p = ) ( f o r > t . Substituting in the differential equation, V B RCB = + ⋅ . Hence, V B = . The complete solution is the sum of the homogeneous and particular solutions. V Ae t y t y t y RC t p h − = + = − / ) ( ) ( ) ( f o r t> 0 Using now the initial rest condition, V A V Ae y = ⇒ − = ) ( and the output is given by ( ) RC t p h e V t y t y t y / 1 ) ( ) ( ) ( − − = + = for t >0 Or, equivalently, ( ) ) ( 1 ) ( / t u e V t y RC t − − = . x ( t ) C R y ( t ) t V y ( t ) 2. Find and sketch the impulse response h ( t ) of the circuit. The circuit can be seen as an LTI system. As we know, we can find the impulse response of the circuit by differentiating the step response....
View Full Document
- Fall '08
- Volt, LTI system theory, RC, RC RC RC