1• Second order circuits: RLC with sourcesECE 201: Lecture 22Borja PeleatoReview• In the previous two lectures we saw the method to solve a HOMOGENEOUS (no independent terms) second order differential equation:– Build and solve the characteristic equation (second order polynomial)– Find general form for the response based on the sign of the type of roots– Determine the constants in the response using the initial conditions• RLC circuits WITHOUT external DC sources are defined by a HOMOGENEOUS equation and can be solved with the above method 2RLC with sources• If external DC sources are applied, the differential equation becomes• The solution is very simple: x(t) = xn(t) + X where– xn(t) is the solution to the homogeneous equation– X is an additional term independent of time (if you plug x(t) in the above equation, you see X=F/c)• Since X is independent of time, we can simply “measure” it after a very long time, so in practice– xn(t) = solution to the circuit when all the independent sources are deactivated (studied in previous lectures)– X = solution to the circuit in DC (inductors become short circuits and capacitors open circuits), once all the transient effects have disappeared• x(t) can be any voltage/current in the circuit3Example 1456789101112Example
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