The Smith Chart and S-Parameters
1
Reflection Coefficient
1 1 1
1
nl nl
o l o
l
o l
o l
z z z
z z
z z
z
z
z
Constant Resistance
3
Proof:
assume: Z
0= R
0+ JX
0, Z
L=R
L+ JX
L z
nl=r
l+ jx
l= Z
L ⁄ Z01 1 1
1
nl nl
o l o
l
o l
o l
z z z
z z
z z
z
z
z
5
2 2 2 2
2 2
) 1
(
2 )
1 (
1
i r
i i
r
i r
l
l
jx j
r
i r
i r
l
l
j
jx j
r
1
1
2 2
2 2
1 2
1
i r
r
i r
r
l
2 2
2
( )
)
( x a y b r
2
2 21 0 1
1
l i
l l
r
r r
r
Constant Reactance
7
2 2
2
( )
)
( x a y b r
2
2
21
2
i r
r
i
x
l
2 2
2
1 )
( 1 )
( )
1
(
r ix
x
2 2 2 2
2 2
) 1
(
2 )
1 (
1
i r
i i
r
i r
l
l
jx j
r
9
A Z Smith Chart
11
13
An Admittance or Y Smith Chart
A ZY Smith Chart
15
Impedance Matching
17
Eight Possible Impedance-Matching
Networks
19
Example#1
Cont’d
21
Match process of the first circuit:
Example#2
Low-pass structure
Cont’d
23
Example#3
Two-step matching (a,b’,d,d’,c)
Cont’d
25
Cont’d
27
Circuit Model for Transmission Line
) m / Ω R: (
L: (H/m)
C: (F/m)
G: (S/m)
v(z,t)-RΔz i(z,t)- LΔz ( , )– v(z+Δz,t) = 0 KVL:
i(z,t)-GΔz v(z+Δz,t)-CΔz ( , ) – i(z+Δz,t)=0 KCL:
( , )
= -Ri(z,t) - L ( , )
( , )
= -Gv(z,t) - C ( , )
( ) = -(R+jωL)I(z)
( ) = -(G+jωC)V(z) Sinusoidal steady-State Analysis:
Cont’d
( ) = -(R+jωL)I(z)
( ) = -(G+jωC)V(z)
² ( )
² - γ²V(z) = 0
² ( )
² - γ²I(z) = 0
γ= α+jβ = ( + )( + )
V(z) = +
I(z) = + or V(z) = +
I(z) = +
■
Z-Parameters
29
S-Parameters
= , = , = , =
S-Parameters
31
