1• Linearity and Response ClassificationECE 201: Lecture 19Borja PeleatoBackground• “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 separately2Response 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.3Linearity• The zero-state response can be further sub-divided 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 factor45Example6789101112Example
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