ECE 201 Lecture 19 Borja Peleato Linearity and Response Classification 1 Background Response and output are the same thing the voltage at a node or the current in a branch whatever we are trying to find response is more common for the time varying case We saw that the output of a resistive circuit has a linear relationship with its inputs independent sources Output can be decomposed as a sum of components one due to each input superposition If one of the inputs is scaled by a factor k the corresponding component is scaled by the same factor k linearity The same applies to first order circuits because derivatives are also linear but the initial conditions need to be dealt with separately 2 Response classification The complete response of a first order circuit has two components Zero input response response if all the INDEPENDENT sources are de activated voltage sources become shorts current sources become open circuits the initial conditions remain unchanged This gives the response to the initial conditions without sources In most cases the capacitors inductors will discharge and the response will be a decreasing exponential However dependent sources could change this Zero state response response if all the initial conditions are de activated set to 0 Capacitors are assumed to start with no voltage inductors with no current This gives the response to the sources without initial conditions Complete resp Zero input resp Zero state resp 3 Linearity The zero state response can be further subdivided into multiple components one for each input independent source If the value of an input changes the corresponding component is also scaled by the same factor See lecture on Linearity and superposition The zero input response is linear respect to the initial conditions If the initial voltage current of the capacitor inductor is scaled the corresponding component is also scaled by the same factor 4 5 Example 6 7 8 9 10 11 12 Example 2 13
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