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ECE201 Lect-21 1

Second-Order Circuits (7.3)

Dr. Holbert

April 19, 2006

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ECE201 Lect-21 2

2nd Order Circuits

Any circuit with a single capacitor, a single

inductor, an arbitrary number of sources,

and an arbitrary number of resistorsis acircuit of order 2.

Any voltage or current in such a circuit is

the solution to a 2nd order differentialequation.

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ECE201 Lect-21 3

Important Concepts

The differential equation

Forced and homogeneous solutions The natural frequency and the damping

ratio

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ECE201 Lect-21 4

A 2nd Order RLC Circuit

The source and resistor may be equivalent

to a circuit with many resistors and sources.

R

Cvs(t)

i(t)

L

+

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ECE201 Lect-21 5

Applications Modeled by a 2nd

Order RLC Circuit

Filters

A lowpass filter with a sharper cutoffthan can be obtained with an RC circuit.

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ECE201 Lect-21 6

The Differential Equation

KVL around the loop:

vr(t) + vc(t) + vl(t) = vs(t)

R

Cvs(t)

+

vc(t)

+

vr(t)

L

+vl(t)

i(t)

+

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ECE201 Lect-21 7

Differential Equation

)()(1)(

)( tvdxxiCdt

tdiLtRi s

t

dt

tdv

Lti

LCdt

tdi

L

R

dt

tid s )(1)(1)()(

2

2

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ECE201 Lect-21 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 currentin such a circuit is the solution to a differential

equation of the following form:

)()()(

2)( 2

002

2

tftidt

tdi

dt

tid

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ECE201 Lect-21 9

Important Concepts

The differential equation

Forced and homogeneous solutions The natural frequency and the damping

ratio

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ECE201 Lect-21 10

The Particular Solution

The particular (or forced) solution ip(t)is

usually a weighted sum off(t)and its first

and second derivatives.

Iff(t)is constant, then ip(t)is constant.

Iff(t)is sinusoidal, then ip(t)is sinusoidal.

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ECE201 Lect-21 11

The Complementary Solution

The complementary (homogeneous) solution

has the following form:

Kis a constant determined by initial conditions.

sis a constant determined by the coefficients of

the differential equation.

st

c Keti )(

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ECE201 Lect-21 12

Complementary Solution

02 2002

2

ststst

Kedt

dKe

dt

Ked

02 2002 ststst KesKeKes

02 2002 ss

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ECE201 Lect-21 13

Characteristic Equation

To find the complementary solution, we

need to solve the characteristic equation:

The characteristic equation has two roots-

call thems1ands2.

02 2002 ss

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ECE201 Lect-21 14

Complementary Solution

Each root (s1ands2) contributes a term to

the complementary solution.

The complementary solution is (usually)

tsts

c eKeKti 21

21)(

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ECE201 Lect-21 15

Important Concepts

The differential equation

Forced and homogeneous solutions

The natural frequency and the damping

ratio

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ECE201 Lect-21 16

Damping Ratio () and

Natural Frequency (0) The damping ratiois .

The damping ratio determines what type of

solution we will get:

Exponentially decreasing (>1)

Exponentially decreasing sinusoid (< 1)

The natural frequencyis 0

It determines how fast sinusoids wiggle.

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ECE201 Lect-21 17

Roots of the Characteristic

EquationThe roots of the characteristic equation

determine whether the complementary

solution wiggles.

12001 s

12002 s

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ECE201 Lect-21 18

Real Unequal Roots

If > 1,s1ands2are realand not equal.

This solution is overdamped.

tt

c eKeKti

1

2

1

1

200

200

)(

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ECE201 Lect-21 19

Overdamped

0

0.2

0.4

0.6

0.8

1

-1.00E-06

t

i(t)

-0.2

0

0.2

0.4

0.6

0.8

-1.00E-06

t

i(t)

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ECE201 Lect-21 20

Complex Roots

If < 1,s1ands2are complex.

Define the following constants:

This solution is underdamped.

tAtAeti ddtc sincos)( 21

02

0 1 d

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ECE201 Lect-21 21

Underdamped

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.00E-05 1.00E-05 3.00E-05

t

i(t)

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ECE201 Lect-21 22

Real Equal Roots

If = 1,s1ands2are realand equal.

This solution is critically damped.

tt

c teKeKti 00

21)(

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ECE201 Lect-21 23

Example

This is one possible implementation of the

filter portion of the IF amplifier.

10W

769pFvs(t)

i(t)

159mH

+

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ECE201 Lect-21 24

More of the Example

dt

tdv

Lti

LCdt

tdi

L

R

dt

tid s )(1)(1)()(

2

2

)()()(

2)( 2

002

2

tftidt

tdi

dt

tid

For the example, what are and 0?

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ECE201 Lect-21 25

Even More Example

= 0.011

0= 2p455000

Is this system over damped, under damped,

or critically damped?

What will the current look like?

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ECE201 Lect-21 26

Example (cont.)

The shape of the current depends on the

initial capacitor voltage and inductor

current.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.00E-05 1.00E-05 3.00E-05

t

i(t)

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ECE201 Lect-21 27

Slightly Different Example

Increase the resistor to 1kW

What are and 0?

1kW

769pFvs(t)

i(t)

159mH

+

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ECE201 Lect-21 28

More Different Example

= 2.2

0= 2p455000

Is this system over damped, under damped,

or critically damped?

What will the current look like?

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ECE201 Lect-21 29

Example (cont.)

The shape of the current depends on the

initial capacitor voltage and inductor

current.

0

0.2

0.4

0.6

0.8

1

-1.00E-06

t

i(t)

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ECE201 Lect-21 30

Damping Summary

Roots (s1, s2) Damping

>1 Real and unequal Overdamped

=1 Real and equal Critically damped

0

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Class Example

Learning Extension E7.9