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:
u(t)
y(0) = y0
System
y(t)
• 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:
y(0) = y0
Au0(t)
System
y(t)
• Governing equation:
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 t
2. 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