11.1Linearity and Time Invariance:If a first-order linear system has no initial stored energy (i.e. initial conditions are allzero iL(0+)=vc(0+)=0) and several sources are applied, the resultant differential equation will be of the form:dxdt Tx f t f t f t f tn 11 2 3( ) ( ) ( ) ( )The solution is then the sum of the responses to each individual source:x t x t x t x tn( ) ( ) ( ) ( ) 1 2 If a time invariant system has no initial stored energy and the application of a sourcesignal is delayed by t0 seconds, then the response will be identical to the non-delayed response except for a delay of t0 seconds.Time invariance implies that:If dxdt Tx f t 1( ) results in solution x t( ) then dxdt Tx f t t 10( ) results in solution x t t( )0.11.2More implications of linearity:In general for any linear system, if a linear operation is performed on the source (or sources) to create a new source, then the new response is the response of the original source (or sources) with the same linear operation applied. Since differentiation and integration are linear operations, then the derivative of the unit ramp response will be the unit step response, and the derivative to the unit step response will be the unit impulse response.Examples: For a given linear and time-invariant system with no initial store energy, the unit ramp response is given by: v t ttu tramp( ) exp ( ) 6 2 62Find the response if source is a) a unit step, b) a unit impulse, c) function below:Show a) v ttu tstep( ) exp ( ) 2 32b) v ttu t timp( ) exp ( ) ( ) 32 25 1250tvs(t)11.3c) v t ttu tres( ) exp ( ) 6 2 62 6 2 5 6525 5 2 312212( ) exp( )( ) exp( )( )ttu ttu tExample: Find v0 for the circuit below:5 10 10 0.3Fvs(t)+vo(t)-a) when v t r t r t r ts( ) ( ) ( ) ( ) 2 1 2b) when v t u t u ts( ) ( ) ( ) 2 2 1c) when v t ts( ) ( ) 3 2Show a) v ttu ttu ttu t02 144 1141 2 1242( ) exp ( ) exp( )( ) exp( )( ) b) v ttu ttu t04141( ) exp ( ) exp( )( ) 11.4c) v ttu t t038242322( ) exp( )( ) ( )
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