circuits and systems part ii prof. dr hab. stanisław osowski electrical engineering (b.sc.)...
DESCRIPTION
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)TRANSCRIPT
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