102_1_lecture5

102_1_lecture5 - UCLA Fall 2010 Systems and Signals Lecture...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: UCLA Fall 2010 Systems and Signals Lecture 5: Time Domain System Analysis October 11, 2010 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 1 Time Domain Analysis of Continuous Time Systems Todays topics Zero-input and zero-state responses of a system Impulse response LTI System response to arbitrary inputs Convolution Properties of convolution EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 2 System Equation The System Equation relates the outputs of a system to its inputs. Example from last time: the system described by the block diagram + +- Z a x y has a system equation y + ay = x. In addition, the initial conditions must be given to uniquely specifiy a solution. EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 3 Solutions for the System Equation Solving the system equation tells us the output for a given input. The output consists of two components: The zero-input response, which is what the system does with no input at all. This is due to initial conditions, such as energy stored in capacitors and inductors. t H t y ( t ) x ( t ) = EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 4 The zero-state response, which is the output of the system with all initial conditions zero. t H y ( t ) x ( t ) t If H is a linear system, its zero-input response is zero. Homogeneity states if y = F ( ax ) , then y = aF ( x ) . If a = 0 then a zero input requires a zero output. t H x ( t ) = y ( t ) = t EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 5 Example: Solve for the voltage across the capacitor y ( t ) for an arbitrary input voltage x ( t ) , given an initial value y (0) = Y . +- R C y ( t ) +- x ( t ) i ( t ) From Kirchhoffs voltage law x ( t ) = Ri ( t ) + y ( t ) Using i ( t ) = Cy ( t ) RCy ( t ) + y ( t ) = x ( t ) . This is a first order LCCODE, which is linear with zero initial conditions. First we solve for the homogeneous solution by setting the right side (the input) to zero RCy ( t ) + y ( t ) = 0 . EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 6 The solution to this is y ( t ) = Ae- t/RC which can be verified by direct substitution. To solve for the total response, we let the undetermined coefficient be a function of time y ( t ) = A ( t ) e- t/RC . Substituting this into the differential equation RC A ( t ) e- t/RC- 1 RC A ( t ) e- t/RC + A ( t ) e- t/RC = x ( t ) Simplying A ( t ) = x ( t ) 1 RC e t/RC which can be integrated from t = 0 to get A ( t ) = Z t x ( ) 1 RC e /RC d + A (0) EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 7 Then y ( t ) = A ( t ) e- t/RC = e- t/RC Z t x ( ) 1 RC e /RC d + A (0) e- t/RC = Z t x ( ) 1 RC e- ( t- ) /RC d + A (0) e- t/RC At t = 0 , y (0) = Y , so this gives A (0) = Y y ( t ) = Z t x ( ) 1 RC e- ( t- ) /RC d | {z } zero- state response + Y e- t/RC | {z } zero- input response ....
View Full Document

Page1 / 41

102_1_lecture5 - UCLA Fall 2010 Systems and Signals Lecture...

This preview shows document pages 1 - 9. Sign up to view the full document.

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