Lecture 18 Review: Forced response of first order circuits First order circuit natural response Forced response of first order circuits Step response of first order circuits Examples Related educational materials: Chapter 7.4, 7.5

Natural response of first order circuits – review Circuit being analyzed has a single equivalent energy storage element Circuit being analyzed is “source free” Any sources are isolated from the circuit during the time when circuit response is determined Circuit response is due to initial energy storage Circuit response decays to zero as t

First order circuit forced response – overview Now consider the response of circuits with sources Notes: We will typically write our equations in terms of currents through inductors and voltages across capacitors The above circuits are very general; consider them to be the Thévenin equivalent of a more complex circuit

RC circuit forced response

RL circuit forced response

First order circuit forced response – summary Forced RC circuit response: Forced RL circuit response:

General first order systems Block diagram: Governing differential equation:

Active first order system – example Determine the differential equation relating Vin(t) and Vout(t) for the circuit below

Active first order system – example Determine the differential equation relating Vin(t) and Vout(t) for the circuit below

Step Response – introduction Our previous results are valid for any forcing function, u(t) In this course, we will be mostly concerned with a couple of specific forcing functions: Step inputs Sinusoidal inputs We will defer our discussion of sinusoidal inputs until later

Applying step input Block diagram: Example circuit: Governing equation: Example circuit:

First order system step response Solution is of the form: yh(t) is homogeneous solution Due to the system’s response to initial conditions yh(t)0 as t yp(t) is the particular solution Due to the particular forcing function, u(t), applied to the system y (t) yp(t) as t

First order system – homogeneous solution Assume form of solution: Substitute into homogeneous D.E. and solve for s : Homogeneous solution:

First order system – particular solution Recall that the particular solution must: Satisfy the original differential equation as t Have the same form as the forcing function As t:

First order system particular solution -- continued As t, the original differential equation becomes: The particular solution is then

First order system step response Superimpose the homogeneous and particular solutions: Substituting our previous results: K1 and K2 are determined from initial conditions and steady-state response;  is a property of the circuit

Example 1 The switch in the circuit below has been open for a long time. Find vc(t), t>0

Example 1 – continued Circuit for t>0:

Example 1 – continued again Apply initial and final conditions to determine K1 and K2 Governing equation: Form of solution:

Example 1 – checking results