Download presentation

1
**Second-Order Circuits (6.3)**

Dr. Holbert November 20, 2001 ECE201 Lect-23

2
2nd Order Circuits Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2. Any voltage or current in such a circuit is the solution to a 2nd order differential equation. ECE201 Lect-23

3
**Important Concepts The differential equation**

Forced and homogeneous solutions The natural frequency and the damping ratio ECE201 Lect-23

4
A 2nd Order RLC Circuit i (t) R + – vs(t) C L The source and resistor may be equivalent to a circuit with many resistors and sources. ECE201 Lect-23

5
**Applications Modeled by a 2nd Order RLC Circuit**

Filters A lowpass filter with a sharper cutoff than can be obtained with an RC circuit. ECE201 Lect-23

6
**The Differential Equation**

vr(t) i (t) + – R + + – vc(t) vs(t) C – vl(t) – + L KVL around the loop: vr(t) + vc(t) + vl(t) = vs(t) ECE201 Lect-23

7
**Differential Equation**

ECE201 Lect-23

8
**The Differential Equation**

Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form: ECE201 Lect-23

9
**Important Concepts The differential equation**

Forced and homogeneous solutions The natural frequency and the damping ratio ECE201 Lect-23

10
**The Particular Solution**

The particular (or forced) solution ip(t) is usually a weighted sum of f(t) and its first and second derivatives. If f(t) is constant, then ip(t) is constant. If f(t) is sinusoidal, then ip(t) is sinusoidal. ECE201 Lect-23

11
**The Complementary Solution**

The complementary (homogeneous) solution has the following form: K is a constant determined by initial conditions. s is a constant determined by the coefficients of the differential equation. ECE201 Lect-23

12
**Complementary Solution**

ECE201 Lect-23

13
**Characteristic Equation**

To find the complementary solution, we need to solve the characteristic equation: The characteristic equation has two roots-call them s1 and s2. ECE201 Lect-23

14
**Complementary Solution**

Each root (s1 and s2) contributes a term to the complementary solution. The complementary solution is (usually) ECE201 Lect-23

15
**Important Concepts The differential equation**

Forced and homogeneous solutions The natural frequency and the damping ratio ECE201 Lect-23

16
**Damping Ratio () and Natural Frequency (0)**

The damping ratio is . The damping ratio determines what type of solution we will get: Exponentially decreasing ( >1) Exponentially decreasing sinusoid ( < 1) The natural frequency is w0 It determines how fast sinusoids wiggle. ECE201 Lect-23

17
**Roots of the Characteristic Equation**

The roots of the characteristic equation determine whether the complementary solution wiggles. ECE201 Lect-23

18
**Real Unequal Roots If > 1, s1 and s2 are real and not equal.**

This solution is overdamped. ECE201 Lect-23

19
Overdamped ECE201 Lect-23

20
**Complex Roots If < 1, s1 and s2 are complex.**

Define the following constants: This solution is underdamped. ECE201 Lect-23

21
Underdamped ECE201 Lect-23

22
**Real Equal Roots If = 1, s1 and s2 are real and equal.**

This solution is critically damped. ECE201 Lect-23

23
Example i (t) 10W + – vs(t) 769pF 159mH This is one possible implementation of the filter portion of the IF amplifier. ECE201 Lect-23

24
More of the Example For the example, what are z and w0? ECE201 Lect-23

25
**Even More Example z = 0.011 w0 = 2p455000**

Is this system over damped, under damped, or critically damped? What will the current look like? ECE201 Lect-23

26
Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current. ECE201 Lect-23

27
**Slightly Different Example**

i (t) 1kW + – vs(t) 769pF 159mH Increase the resistor to 1kW What are z and w0? ECE201 Lect-23

28
**More Different Example**

z = 2.2 w0 = 2p455000 Is this system over damped, under damped, or critically damped? What will the current look like? ECE201 Lect-23

29
Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current. ECE201 Lect-23

30
Damping Summary ECE201 Lect-23

31
**Class Examples Learning Extension E6.9 Learning Extension E6.10**

ECE201 Lect-23

Similar presentations

OK

Sinusoidal Steady-state Analysis Complex number reviews Phasors and ordinary differential equations Complete response and sinusoidal steady-state response.

Sinusoidal Steady-state Analysis Complex number reviews Phasors and ordinary differential equations Complete response and sinusoidal steady-state response.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on buildings paintings and books Ppt on conservation of environment for kids Download ppt on recruitment and selection process of infosys Download ppt on graphical user interface Ppt on state of indian economy Ppt on going places by a.r.barton Quantum dot display ppt online Ppt on e waste management in india Scrolling message display ppt on ipad Business plan ppt on event management