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

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

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

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

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

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

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

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

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

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

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?

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

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

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