Transient Response

Wednesday, 3 April 2019

4:48 PM

First order circuits - contains resistive elements and a capacitor, or resistive elements and an inductor… and an op-amp or two… "First order" - Order of the differential equation due to RC or RL circuit, analysed in terms of either a voltage or a current Capacitor - 𝑖﷐𝑡﷯=﷐𝑑𝑞﷐𝑡﷯﷮𝑑𝑡﷯=𝐶﷐𝑑𝑣﷐𝑡﷯﷮𝑑𝑡﷯ Inductor - 𝑣﷐𝑡﷯=𝐿﷐𝑑𝑖﷐𝑡﷯﷮𝑑𝑡﷯ Depends on the equations for energy storage elements: Example first-order circuits: ﷐𝑣﷮𝑠﷯﷐𝑡﷯=𝑅𝑖+𝐿﷐𝑑𝑖﷮𝑑𝑡﷯ ﷐𝑑𝑖﷮𝑑𝑡﷯+﷐𝑅﷮𝐿﷯𝑖=﷐1﷮𝐿﷯﷐𝑣﷮𝑠﷯ Express in descending orders of derivative, with highest order having a coefficient of 1 (convention) Ink Drawings ﷐𝑣−﷐𝑣﷮𝑠﷯﷮𝑅﷯+𝐶﷐𝑑𝑣﷐𝑡﷯﷮𝑑𝑡﷯=0 ﷐𝑑𝑣﷐𝑡﷯﷮𝑑𝑡﷯+﷐1﷮𝑅𝐶﷯𝑣﷐𝑡﷯=﷐1﷮𝑅𝐶﷯﷐𝑣﷮𝑠﷯(𝑡) First order linear circuits produce linear differential equations with constant coefficient, with the form: … Where 𝑥(𝑡) is a circuit quantity (either voltage or current) … Where ﷐𝑎﷮0﷯ is a constant … Where 𝑓(𝑡) is a forcing function ﷐𝑑𝑥﷐𝑡﷯﷮𝑑𝑡﷯+﷐𝑎﷮0﷯𝑥﷐𝑡﷯=𝑓﷐𝑡﷯ Find the solution of the homogeneous equation ﷐𝑑𝑥﷐𝑡﷯﷮𝑑𝑡﷯+﷐𝑎﷮0﷯𝑥﷐𝑡﷯=0 Solution: ﷐𝑥﷮𝑛﷯(𝑡) is called the natural response, and is independent of the source applied. Solving 1st Order: ﷐𝑑𝑥﷮𝑑𝑡﷯+﷐𝑎﷮0﷯𝑥=0, the general solution has the form ﷐𝑥﷮𝑛﷯﷐𝑡﷯=﷐𝐴﷮1﷯﷐𝑒﷮−﷐𝑎﷮0﷯𝑡﷯=﷐𝐴﷮1﷯﷐𝑒﷮−﷐𝑡﷮τ﷯﷯ where τ=﷐1﷮﷐𝑎﷮0﷯﷯ Look for a solution to the forced response (by trial and error) If 𝑓﷐𝑡﷯ = constant, then we assume a forced response solution of the form ﷐𝑥﷮𝑓﷯﷐𝑡﷯ = constant The complete solution 𝑥﷐𝑡﷯=﷐𝑥﷮𝑛﷯﷐𝑡﷯+﷐𝑥﷮𝑓﷯﷐𝑡﷯ Use the initial conditions (i.e. 𝑥(0)) to determine any unknowns/constants Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings           Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings                      Ink Drawings Ink Drawings Ink Drawings Ink Drawings                Ink Drawings              Ink Drawings Ink Drawings Ink Drawings Ink Drawings 
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Second-order: Combination of resistive elements and two-energy storage elements inductors and or capacitors) refers to the order of the differential equation in terms of voltage or current Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings   Ink Drawings Ink Drawings Ink Drawings       Ink Drawings Ink Drawings Ink Drawings Ink Drawings     Ink Drawings Ink Drawings Ink Drawings Ink Drawings    

 

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