Network Functions Definition, examples , and general property Poles, zeros, and frequency response Poles, zeros, and impulse response Physical interpretation of poles and zeros Application to oscillator design Symmetry properties

)(sH

Definition, Examples , and General Property

zero state responseNetwork Function ( )

inputH s

L

L

( )( )

( )

B sH s

A s

or

( ) ( )B s b tL

where

( ) ( )A s a tL and s j

0RRZRResistor

90LLjZL Inductor

Capacitor 9011

CCjZc

Sinusoidal steady state driving point impedance = special case of a network function

R

sL

1

sC

input

response

i

v

Example 1 Parallel RC circuit

Ci G v

1( ) ( )V s I s

G sC

( ) 1( )

( )

V sH s

I s G sC

Example 2 Low pass filter

2

0

( )( )

( )

I sH s

E s

C0e R

1L 2L1i

Mesh analysis gives

1 1 2 0

1 1 L s I I E

Cs Cs

1 2 2

1 10 I L s R I

Cs Cs

Solve for I2

02 3 2

1 2 1 1 2

( )

EI

L L Cs RL Cs L L s R

23 2

0 1 2 1 1 2

1( )

( )

IH s

E L L Cs RL Cs L L s R

General Property

For any lumped linear time-invariant circuit

10 1 m-1

10 1 n-1

b b b( )( ) =

( ) a a a

m mm

n nn

s s s bP sH s

Q s s s s a

1

1

( )( )

( )

m

iin

jj

s zH s K

s p

Poles, Zeros, and Frequency Response

( )( ) ( ) = ( ) j H jH s H j H j e magnitude

phase

( ) ( ) ( ) + ( )j ln H j ln H j j H j

Gain (nepers)

20log ( )H jGain (dB)

Example 3 RC Circuit Frequency Response

Ci R v

11( )

1CH s

G sC s RC

No finite zero

Pole at s = 1/RC

1 1( )

1( )H j

C j RC

2 2

1 1 1 1( )

1 1( ) ( )H j

C Cj RC RC

1( ) tanH j RC

( )H j

R

2R

1

RC

0 2

RC

3

RC

( )H j

45

1

RC0

2

RC

3

RC

90

2 2

1 1

1( )CRC

1tan RC

0

1j

RC

1j

RC

1

RC

X

S-plane

Im[s]

Re[s]

pole j

1

45

1d

1j

RC

1 1

1 1 1 1 1( )

1( )H j

C C d Cdj RC

11( ) tanH j RC

(0)H RAt 0

(0) 0H

( ) 0H At

( ) 90H

Example 4 RLC Circuit Frequency Response

2

1( )

( / ) 1/

1 0

[ ( )][ ( )]d d

sH s

C s G C s LC

s

C s j s j

Ci R v

L

1( ) ( )

1V s I s

G sC Ls

Zero at s = 0

Complex conjugate poles at ds j

01( )

( ) ( )d d

jH j

C j j j j

( ) 0 ( ) ( )d dH j j j j j j

1

1 1

1( )

lH j

C d d

1 1 1( )H j

0

dj

dj

X S-plane

Im[s]

Re[s]poles

j

1

1d

dj

X

O

1

dj

zero

11l

1d

(0) 0H At 0

(0) 0H

( ) increases with H j For and 0

( ) 90H j d

( ) is maximum at dH j 1 mind

1 1 1( )

2 2d

dd

H j RC C G

( ) 0dH j 1 90

( ) 0H j At

( ) 90H j

1( ) is proportional to H j

For d

General Case

10 1 m-1

10 1 n-1

b b b( )=

a a a

m mm

n nn

s s s bH s

s s s a

0 1 2 2

0 1 2 3 3

b ( )( )( )( )=

a ( )( )( )( )

s z s z s zH s

s p s p s p s p

1 2 20

0 1 2 3 3

b( ) =

a

j z j z j zH j

j p j p j p j p

0 1 2 2

0 1 2 3 3

b( ) =

a

l l lH j

d d d d

Example 3 zeros 4 poles

01 2 2

0

1 2 3 3

b( )= ( ) ( ) ( )

a

( ) ( ) ( ) ( )

H j j z j z j z

j p j p j p j p

01 2 2 1 2 3 3

0

b( )= ( ) ( )

aH j

0

X S-plane

Im[s]

Re[s]

j

3

2d3p

X

1

3p

11l

3dX X

1p2p

O

O

O

2z

1z

2z

1d

2

3

2l

2l

2

3d

2

Poles, Zeros, and Impulse Response

Example 5 RC Circuit11

( )1CH s

G sC s RC

1[ ( )] ( )H s h t L

1( ) ( )

tRCh t u t e

C

See section 6 Chapter 4 for derivation of h(t)

0

1

1X

S-plane

Im[s]

Re[s]

1

1RC 1

( )1

CH ss

( )H j

3dB

0

t

1

( )h t

0 1

R

1

C

1( ) ( ) th t u t e

C

0

1

1X

S-plane

Im[s]

Re[s]

1

0.5RC 1

( )2

CH ss

2

( )H j

3dB

0

t

1

( )h t

0 1

R

1

C

2

0.5

21( ) ( ) th t u t e

C

Example 6 RLC Circuit

2 2 2

1 1( )

( / ) 1/ 2

s sH s

C s G C s LC C s s

1( ) ( ) cos( )t

dh t u t e tC

For 0.3 1d Fig 3.2

For 0.1 1d Fig 3.3

See section 2 Chapter 5 for derivation of h(t)

Physical Interpretation of Poles and Zeros

( )( )

( )LV s

H sI s

1

1

( )( )

( ) ( )

( )

m

iin

jj

s zP s

H s KQ s

s p

Poles

Any pole of a network function is a natural frequency of the corresponding (output) network variable.

n1 i

i=1

1

( )K

( ) =( )

m

iin

ij

j

s zH s K

s ps p

Using partial-fraction expansion

Residue at pi

1

1

( ) [ ( )] 0i

np t

L ii

v t H s K e t

L

For input current = ( )t

For i = 1

11( ) 0p t

Lv t K e t Natural frequency p1

If a particular input waveform is chosen over the interval [0,T] then for t > T

1( ) p tLv t Ke t T

Any pole of a network function is a natural frequency of the corresponding (output) network variable,

Summary

but any natural frequency of a network variable need not be a pole of a given network function which has this network variable as output.