1. Lecture 17. System Response II
Dr. Holbert
F06-Lect1
Lecture 17. System Response II
Poles & Zeros
Second-Order Circuits
LCR Oscillator circuit: An example
Transient and Steady States
EEE 202

2. Pole-Zero Plot
For a pole-zero plot place "X" for poles and "0" for zeros using real-imaginary axes
Poles directly indicate the system transient response features
Poles in the right half plane signify an unstable system
Consider the following transfer function Re Im

3. Second-Order Circuits+ –
vc(t)
vr(t) L
vl(t)
i(t)
KVL around the loop:
vr(t) + vc(t) + vl(t) = vs(t)

4. Second-Order Circuits
In general, a second-order circuit is described by
For zero-initial conditions, the transfer function would be

5. Characteristic Equation & Poles
The denominator of the transfer function is known as the characteristic equation
To find the poles, we solve :
which has two roots: s1 and s2

6. Real and Unequal Roots: Overdamped
If  > 1, s1 and s2 are real and not equal
The amplitude decreases exponentially over time. This solution is overdamped

7. Complex Roots: Underdamped
If  < 1, s1 and s2 are complex
Define the following constants:
This solution is underdamped

8. Real and Equal Roots If  = 1, then s1 and s2 are real and equal
This solution is critically damped

9. An Example i(t) 10W + 769pF vs(t) – 159mH
Dr. Holbert
F06-Lect16
An Example
10W
769pF
vs(t)
i(t)
159mH + –
This is one possible implementation of the filter portion of an intermediate frequency (IF) amplifier
IF stands for intermediate frequency
EEE 202

10. An Example (cont’d.) Note that w0 = 2pf = 2p (455,000 Hz)
Dr. Holbert
F06-Lect16
An Example (cont’d.)
Note that w0 = 2pf = 2p (455,000 Hz)
Is this system overdamped, underdamped, or critically damped?
What will the current look like?
ζ < 1, so the system is underdamped; and the response is a decaying exponential with sinusoid
EEE 202

11. An Example (cont’d) 1kW 769pF vs(t) i(t) 159mH + –
Increase the resistor to 1kW
Exercise: what are z and w0?
The natural (resonance) frequency does not change: w0 = 2p(455,000 Hz)
But the damping ratio becomes z = 2.2
Is this system overdamped, underdamped, or critically damped?
What will the current look like?

12. A Summary Damping Ratio Poles (s1, s2) Damping ζ > 1
Real and unequal
Overdamped
ζ = 1
Real and equal
Critically damped
0 < ζ < 1
Complex conjugate pair set
Underdamped
ζ = 0
Purely imaginary pair
Undamped

13. Transient and Steady-State Responses
The steady-state response of a circuit is the waveform after a long time has passed, and depends on the source(s) in the circuit
Steady
State
Response
Transient
Response
Transient
Response
Steady-State
Response 13

14. Class Examples
P7-6, P7-7, P7-8