Lecture 18•Review:

•First order circuit natural response•Forced response of first order circuits•Step response of first order circuits

•Examples•Related educational modules:

–Section 2.4.4, 2.4.5

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:

y(0) = y0

u(t) y(t)System

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:

• Governing equation:

• Example circuit:y(0) = y0

Au0(t) y(t)System

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:

1. Satisfy the original differential equation as t2. 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