Download Definition of transfer function

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
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
Y ( s)
H ( s) 
X ( s)
Sometimes transfer function is denoted also by T(s)
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Definition of transfer function (cont.)
Voltage transfer function
U 2 ( s)
H u (s) 
U1 (s)
Current transfer function
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I 2 (s)
H i ( s) 
I1 ( s )
Definition of transfer function (cont.)
Voltage-to-current transfer function
H ui ( s) 
U 2 ( s)
I1 ( s )
Current-to-voltage transfer function
H iu ( s ) 
I 2 (s)
U1 ( s)
Special case of transfer function is the input impedance
U1 ( s )
Z we ( s) 
I1 ( s )
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Transfer function of RLC circuits
Each RLC element has its operator description
Element
Operator description
Resistance R
ZR  R
Inductance L
Z L  sL
Mutual inductance  M
Z M   sM
Capacitance C
ZC 
1
sC
General form of transfer function
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L(s) bm s m  bm1s m1  ...  b1s  b0
H ( s) 
 n
M ( s)
s  an1s n1  ...  a1s  a0
Impulse and step responses
Impulse response is the time response of the circuit for Dirac
impulse excitation at zero initial conditions
H ( s) 




Y ( s) Y ( s)
1
1

 Y ( s)  H ( s) y(t )  L Y ( s)  L H ( s)  h(t )
X ( s)
1
Step response is the time response of the circuit for unity
Heaviside excitation at zero initial conditions
Y ( s) Y ( s)
1
H ( s) 

 Y ( s)  H ( s)
X ( s) 1 / s
s
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1

y (t )  L Y ( s )  L  H ( s )
s

1
1
Example
Transfer function of the circuit is given in the form
H ( s) 
1
s  1s  5
Impulse response


1
1 st
1 st 1 t 1 5t
h(t )  L 
 lim s 1
e  lim s 5
e  e  e

s5
s 1
4
4
 s  1s  5
1
Step response
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

1
1
st
y (t )  L1 

lim
e

s 0

s  1s  5
 s s  1s  5 
1
1
 lim s 1
e st  lim s 5
e st  0,2  0,25e t  0,05e 5t
ss  5
ss  1
Example (cont.)
Impulse response
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Step response
Stability of linear circuits
Stability BIBO (Bounded Input – Bounded Output): the
circuit is stable if at bounded input excitation the output signal is
also bounded at any time t.
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Dependence of stability on the placement of poles
Impulse response of 2nd order
transfer function
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Frequency 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.
H ( j )  H ( s) s  j
• Magnitude characteristics (magnitude of spectral function)
H ( j )
• Phase characteristics (phase of spectral function)
 ( )  arg( H ( j )
• Logarithmic magnitude characteristics
20 log 10  H ( j) 
Example
Transfer function is given in the form
0.003s 4  0.082 s 2  0.287
H ( s)  4
s  0,945s 3  1,487 s 2  0,778s  0,322
Magnitude characteristics
0.003 4  0.082 2  0.287
H ( j )  4
  1,487 2  0,322  j  0,945 3  0,778 
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Linear and logarithmic form of magnitude characteristics
First order transfer functions
1) Integrator
k
H (s) 
s
Frequency characteristics
k
k  j 90
H ( j ) 
 e
j 
Magnitude and phase characteristics
H ( j ) 
k

 ( )  90
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,
First order transfer functions (cont.)
2) Differentiator
H ( s )  ks
Frequency characteristics
H ( j)  kj  ke
Magnitude and phase characteristics
H ( j )  k,
 ( )  90
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j 90
First order transfer functions (cont.)
3) Phase shifter
sa
H (s) 
sa
Frequency characteristics
 j  a
 2  a 2 e  j  
 
 j 2  
H ( j ) 



1
e
,

(

)

arctg
 
j  
2
2
j  a
a
 a e
Magnitude and phase characteristics
H ( j )  1,
 
 ( )  2 arctg  
a
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Frequency characteristics of nth order
transfer function
General form
bm s m  bm1s m1  ...  b1s  b0
H ( s) 
an s n  an 1s n 1  ...  a1s  a0
Frequency characteristics
b  j   bm1  j   ...  b1 j  b0
H ( j )  m
 A( )  jB( )
n
n 1
an  j   an 1  j   ...  a1 j  a0
m 1
m
Magnitude and phase characteristics
 B( ) 

H ( j )  A ( )  B ( ) ,  ( )  arctg 
 A( ) 
2
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2
Example
Determine the voltage transfer function of the circuit. Assume:
R=1, L=2H, C=1F
Solution:
Operator form of the circuit
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Example (cont.)
Current I(s)
I (s) 
U1 ( s )
sC
 2
U1 ( s )
R  sL  1 / sC s LC  sRC  1
Output voltage
1
1
U 2 ( s) 
I (s)  2
U1 ( s )
sC
s LC  sRC  1
Voltage transfer function
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1
U 2 (s)
LC
H u ( s) 

,
U1 ( s) s 2  s R  1
L LC
0,5
H u ( s)  2
s  0,5s  0,5