§ 15-8 The transfer function(or network function) H(s)
We define transfer function H(s) as a ratio of the Laplace
transform of system output (or response) Vo(s) to the Laplace
transform of the input (or forcing function) Vi(s) when all initial
conditions are zero,then
)(
)()(
sV
sVsH
i
O? )()()( sVsHsVor io ?
)()(1)( sHsVsVw h e n oi ??
)()(1)( ttsVa n d ii ?? ??
The inverse function corresponding to the transfer
function H(s) is the unit-impulse response of the circuit.
)()]([)]([)( 11 tsVLsHLth oO ???? ??
H(s) usually is a ratio of two polynomes containing s,by
means of partial-fraction-expansion obtain
?
? ?
?? n
i i
i
ps
k
sD
sNsH
1)(
)()( ??
??
?? ?
???
n
i
tp
i
n
i i
i iek
ps
kLsHLth
11
11 ][)]([)(
§ 15-9 The complex-frequency plane
)),,, ()((
)),,, ()((
)(
)()(
21
21
0
n
m
pspsps
zszszsH
sD
sNsH
???
?????
H0 real number
z1,z2,… zm zeros
p1,p2,… pn poles
The complex frequency ?? js ??
?
?j
?
?j
0
)( p la n es ?
Example 2,Find,Z(s)=?
)51)(51(
2)(
jsjs
sksZ
????
??
262
2)(
2 ??
??
ss
sksZor
262
213)(
2 ??
???
ss
ssZ 13
26
21,1)0(,????? kkZIf
(a) (b)
(a) The pole-zero constellation of some impedance Z(s);
(b) A portion of the rubber-sheet model of the magnitude
of Z(s).
?j
?
2?
5j1??
5j1??
planes ?j
?
5j1??