Lecture 13 Secondorder Circuits 1 Hungyi Lee Secondorder

Lecture 13 Second-order Circuits (1) Hung-yi Lee

Second-order Circuits • A second order-circuit contains two independent energy-storage elements (capacitors and inductors). Capacitor + inductor 2 Capacitors 2 inductors

Second-order Circuits • Steps for solving by differential equation (Chapter 9. 3, 9. 4) • 1. List the differential equation (Chapter 9. 3) • 2. Find natural response (Chapter 9. 3) • There is some unknown variables in the natural response. • 3. Find forced response (Chapter 9. 4) • 4. Find initial conditions (Chapter 9. 4) • 5. Complete response = natural response + forced response (Chapter 9. 4) • Find the unknown variables in the natural response by the initial conditions

Solving by differential equation Step 1: List Differential Equation

Systematic Analysis Mesh Analysis

Systematic Analysis Mesh Analysis Find i. L: Find v. C:

Systematic Analysis Node Analysis

Systematic Analysis Node Analysis Find v. C: v. C=v Find i. L:

v 1 Example 9. 6 Find i 2 v 1: v 2

v 1 v 2 Example 9. 6 Find i 2 Target: Find v 2 from the left equations Equations for v 1 and v 2 Then we can find i 2

v 1 Example 9. 6 Find i 2 Find v 2

v 1 v 2 Example 9. 6 Find i 2 Replace with

Example 9. 7 • Please refer to the appendix

Summary – List Differential Equations

Solving by differential equation Step 2: Find Natural Response

Natural Response • The differential equation of the second-order circuits: y(t): current or voltage of an element α = damping coefficient ω0 = resonant frequency

Natural Response • The differential equation of the second-order circuits: Focus on y. N(t) in this lecture

Natural Response y. N(t) looks like: Characteristic equation

Natural Response Real Overdamped Critical damped λ 1, λ 2 is Complex Underdamped Undamped

Solving by differential equation Step 2: Find Natural Response Overdamped Response

Overdamped Response λ 1, λ 2 are both real numbers y. N(t) looks like

Overdamped Response

Solving by differential equation Step 2: Find Natural Response Underdamped Response

Underdamped

Underdamped y. N(t) should be real. Euler's formula:

Underdamped y. N(t) should be real. Euler's formula: (no real part)

Underdamped Memorize this! a and b will be determined by initial conditions

Underdamped L and θ will be determined by initial conditions

Underdamped

Solving by differential equation Step 2: Find Natural Response Undamped Response

Undamped is a special case of underdamped.

Solving by differential equation Step 2: Find Natural Response Critical Damped Response

Critical Damped Overdamped Underdamped Critical damped Not complete

Critical Damped (Problem 9. 44)

Solving by differential equation Step 2: Find Natural Response Summary

Summary Fix ω0, decrease α (α is positive): Overdamped Critical damped Decrease α, smaller R Underdamped Undamped Decrease α, increase R

Fix ω0, decrease α (α is positive) The position of the two roots λ 1 and λ 2. α=0 Undamped

Homework • 9. 30 • 9. 33 • 9. 36 • 9. 38

Thank You!

Answer • 9. 30: v 1’’ + 3 v 1’ + 10 v 1 = 0 • 9. 33: y. N=a e^(-0. 5 t) + b te^(-0. 5 t) • 9. 36: y. N=a e^(4 t) + b e(-6 t) • 9. 38: y. N=2 Ae^(3 t) cos (6 t+θ) or y. N=2 e^(3 t) (acos 6 t + bsin 6 t) • In 33, 36 and 38, we are not able to know the values of the unknown variables.

Appendix: Example 9. 7

Example 9. 7 Mesh current: i 1 and ic

Example 9. 7 (1): (2) – (1):

Example 9. 7

Appendix: Figures from Other Textbooks

Undamped