SecondOrder Circuits 7 3 Dr Holbert April 19

Second-Order Circuits (7. 3) Dr. Holbert April 19, 2006 ECE 201 Lect-21 1

2 nd 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 2 nd order differential equation. ECE 201 Lect-21 2

Important Concepts • The differential equation • Forced and homogeneous solutions • The natural frequency and the damping ratio ECE 201 Lect-21 3

A 2 nd 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. ECE 201 Lect-21 4

Applications Modeled by a 2 nd Order RLC Circuit • Filters – A lowpass filter with a sharper cutoff than can be obtained with an RC circuit. ECE 201 Lect-21 5

The Differential Equation i (t) + vr(t) – R vs(t) + – + vc(t) C – vl(t) + – L KVL around the loop: vr(t) + vc(t) + vl(t) = vs(t) ECE 201 Lect-21 6

Differential Equation ECE 201 Lect-21 7

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: ECE 201 Lect-21 8

Important Concepts • The differential equation • Forced and homogeneous solutions • The natural frequency and the damping ratio ECE 201 Lect-21 9

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. ECE 201 Lect-21 10

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. ECE 201 Lect-21 11

Complementary Solution ECE 201 Lect-21 12

Characteristic Equation • To find the complementary solution, we need to solve the characteristic equation: • The characteristic equation has two rootscall them s 1 and s 2. ECE 201 Lect-21 13

Complementary Solution • Each root (s 1 and s 2) contributes a term to the complementary solution. • The complementary solution is (usually) ECE 201 Lect-21 14

Important Concepts • The differential equation • Forced and homogeneous solutions • The natural frequency and the damping ratio ECE 201 Lect-21 15

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 0 – It determines how fast sinusoids wiggle. ECE 201 Lect-21 16

Roots of the Characteristic Equation The roots of the characteristic equation determine whether the complementary solution wiggles. ECE 201 Lect-21 17

Real Unequal Roots • If > 1, s 1 and s 2 are real and not equal. • This solution is overdamped. ECE 201 Lect-21 18

Overdamped ECE 201 Lect-21 19

Complex Roots • If < 1, s 1 and s 2 are complex. • Define the following constants: • This solution is underdamped. ECE 201 Lect-21 20

Underdamped ECE 201 Lect-21 21

Real Equal Roots • If = 1, s 1 and s 2 are real and equal. • This solution is critically damped. ECE 201 Lect-21 22

Example i (t) 10 W vs(t) + – 769 p. F 159 m. H • This is one possible implementation of the filter portion of the IF amplifier. ECE 201 Lect-21 23

More of the Example For the example, what are and 0? ECE 201 Lect-21 24

Even More Example • = 0. 011 • 0 = 2 p 455000 • Is this system over damped, under damped, or critically damped? • What will the current look like? ECE 201 Lect-21 25

Example (cont. ) • The shape of the current depends on the initial capacitor voltage and inductor current. ECE 201 Lect-21 26

Slightly Different Example i (t) 1 k. W vs(t) + – 769 p. F 159 m. H • Increase the resistor to 1 k. W • What are and 0? ECE 201 Lect-21 27

More Different Example • = 2. 2 • 0 = 2 p 455000 • Is this system over damped, under damped, or critically damped? • What will the current look like? ECE 201 Lect-21 28