CIRCUITS and SYSTEMS – part II

Prof. dr hab. Stanisław Osowski

Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 12

Transfer function concept

3

Definition of transfer function

Transfer function is defined as the ratio of Laplace transform of output signal Y(s) and input signal X(s) at zero initial conditions

Sometimes transfer function is denoted also by T(s)

)()()(sXsYsH

4

Definition of transfer function (cont.)

Voltage transfer function

Current transfer function

)()()(

1

2

sUsUsH u

)()()(

1

2

sIsIsH i

5

Definition of transfer function (cont.)Voltage-to-current transfer function

Current-to-voltage transfer function

Special case of transfer function is the input impedance

)()()(

1

2

sIsUsH ui

)()()(

1

2

sUsIsH iu

)()()(

1

1

sIsUsZwe

6

Transfer function of RLC circuits

Each RLC element has its operator description

General form of transfer function

011

1

011

1

......

)()()(

asasasbsbsbsb

sMsLsH

nn

n

mm

mm

Element Operator description

Resistance R RZR

Inductance L sLZL

Mutual inductance M sMZM

Capacitance C sC

ZC1

7

Impulse and step responses

Impulse response is the time response of the circuit for Dirac impulse excitation at zero initial conditions

Step response is the time response of the circuit for unity Heaviside excitation at zero initial conditions

)()(1

)()()()( sHsYsYsXsYsH )()()()( 11 thsHLsYLty

)(1)(/1

)()()()( sH

ssY

ssY

sXsYsH

)(1)()( 11 sHs

LsYLty

8

Example

Impulse response

511)(

sssH

Step response

Transfer function of the circuit is given in the form

ttst

sst

s eees

esss

Lth 551

1

41

41

11lim

51lim

511)(

ttst

sst

s

sts

eeess

ess

esssss

Lty

551

01

05,025,02,01

1lim5

1lim

511lim

511)(

9

Example (cont.)

Impulse response Step response

10

Stability of linear circuitsStability BIBO (Bounded Input – Bounded Output): the circuit is stable if at bounded input excitation the output signal is also bounded at any time t.

Dependence of stability on the placement of poles

11

Impulse response of 2nd order transfer function

12

Frequency characteristics

• Magnitude characteristics (magnitude of spectral function)

• Phase characteristics (phase of spectral function)

• Logarithmic magnitude characteristics

Spectral transfer function is the frequency characteristics of the circuit. It represents the dependence of output signal on the frequncy at the sinusoidal input signal of unity magnitude and changing frequency.

jssHjH )()(

)( jH

)(log20 10 jH

)(arg()( jH

13

ExampleTransfer function is given in the form

Magnitude characteristics

322,0778,0487,1945,0287.0082.0003.0)(

234

24

ssss

sssH

778,0945,0322,0487,1287.0082.0003.0)(

324

24

j

jH

Linear and logarithmic form of magnitude characteristics

14

First order transfer functions 1) Integrator

Frequency characteristics

Magnitude and phase characteristics

sksH )(

90)( jekjkjH

90)(

,)(

kjH

15

First order transfer functions (cont.)2) Differentiator

Frequency characteristics

Magnitude and phase characteristics

kssH )(

90)( jekkjjH

90)(

,)(

kjH

16

First order transfer functions (cont.)

3) Phase shifter

Frequency characteristics

Magnitude and phase characteristics

asassH

)(

ae

ee

aa

ajajjH j

j

j

arctg)( ,1)( 2

22

22

a

jH

arctg2)(

,1)(

17

Frequency characteristics of nth order transfer function

General form

Frequency characteristics

Magnitude and phase characteristics

011

1

011

1

......)(

asasasabsbsbsbsH

nn

nn

mm

mm

)()(

......)(

011

1

011

1 jBAajajajabjbjbjbjH n

nn

n

mm

mm

)()()( ,)()()( 22

ABarctgBAjH

18

ExampleDetermine the voltage transfer function of the circuit. Assume: R=1, L=2H, C=1F

Solution:Operator form of the circuit

19

Example (cont.)Current I(s)

Voltage transfer function

)(1/1

)()( 121 sU

sRCLCssC

sCsLRsUsI

Output voltage

)(1

1)(1)( 122 sUsRCLCs

sIsC

sU

5,05,05,0)(

,1

1

)()()(

2

21

2

sssH

LCLRss

LCsUsUsH

u

u