Mysteries of the Smith Chart
Transmission Lines, Impedance Matching,
and Little Known Facts
Stephen D. Stearns, K6OIK
Chief Technologist
TRW Firestorm Wireless Communication Products
stearns@ieee.org
VG – 1
PSA-00527 — 10/21/2001
Pacificon 2001
Outline
❏ Transmission Line Theory
➤ Historical development
➤ Heaviside’s rewrite of Maxwell’s theory, Telegrapher’s equations,
➤ Impedance, reflection coefficient, SWR, phase constant, and velocity
factor
➤ Special facts for λ/2, λ/4, and λ/8 lossless lines
❏ The Smith Chart
➤ Bilinear complex functions
➤ Impedance and admittance coordinates (circles, circles, and more
circles)
❏ Impedance Matching
➤ Why match? Impedance matching vs. conjugate impedance
matching
➤ Single frequency matching
➤ Multiple-frequency and broadband matching
VG – 2
PSA-00527—10/21/2001
Pacificon 2001
Part 1: Transmission Line Theory
VG – 3
PSA-00527 — 10/21/2001
Pacificon 2001
Key Dates in Electrical Transmission
1830s Magnetic telegraphs - Gauss, Henry
1839
Electromagnetic telegraph - Wheatstone & Cook
1844
Telegraph in America - Morse
1850s Thousands of miles of telegraph line U.S. and Europe
1851
40-mile cable under English Channel
1855
Distributed analysis of transmission line - Lord Kelvin
1858
Transatlantic cable, project delayed by civil war
1873
Theory of electrodynamics - Maxwell
1876
Invention of telephone - Bell
1880s Vectors, vector calculus, reformulation of Maxwell’s
theory, transmission line theory - Heaviside
1886
Experimental confirmation of Maxwell’s Theory - Hertz
1937
Early Smith Chart, published 1939 and 1944 - Smith
VG – 4
PSA-00527—10/21/2001
Pacificon 2001
Numbers to Remember!
1.4142135623...
1.7320508075…
1.6180339887...
3.1415926535...
2.718281828459045…
2.54
299,792,458
376.7303134...
VG – 5
PSA-00527—10/21/2001
Pacificon 2001
Heaviside’s Vector Formulation of Maxwell’s Theory
∂B
∇×E = −
∂t
∂D
∇×H = J+
∂t
∇⋅D = ρ
∇⋅B = 0
D = εE
B = µH
“And God said, Let there be light; and there was light.” Genesis 1:3
VG – 6
PSA-00527—10/21/2001
Pacificon 2001
Frequency Domain or Phasor Form
∇ × E = − jωµH
∇ × H = (σ + jωε )E
∇⋅E = 0
∇⋅H = 0
VG – 7
PSA-00527—10/21/2001
Pacificon 2001
Heaviside’s Telegrapher’s Equations
Uniform transmission line
Equivalent circuit of
infinitesimal segment
I(x)
R∆x
V(x)
L∆x
G∆x
C∆x
dV
= −( R + jωL) I ( x )
dx
dI
= −(G + jωC )V ( x )
dx
VG – 8
PSA-00527—10/21/2001
Pacificon 2001
Transmission Line Solution
TEM Waves
Traveling wave
V ( x ) = Vo eγx
V ( x)
I( x) =
Zo
Propagation constant
γ = α + jβ = ( R + jωL)(G + jωC )
Characteristic impedance
Zo =
VG – 9
PSA-00527—10/21/2001
R + jωL
G + jωC
Pacificon 2001
Notations
Real Parameters
R=
Series resistance per unit length (Ohms/meter)
L=
Series inductance per unit length (Henries/meter)
G=
Shunt conductance per unit length (Siemens/meter)
C=
Shunt capacitance per unit length (Farads/meter)
α=
Attenuation constant (nepers/meter)
β = Phase constant (radians/meter)
λ=
Wavelength (meters)
vf =
Velocity factor (dimensionless)
X=
Reactance (Ohms)
B=
Susceptance (Siemens)
s=
Standing wave radio (dimensionless)
VG – 10
PSA-00527—10/21/2001
Pacificon 2001
Notations (Cont’d)
Complex Parameters
Z = R + jX = impedance (Ohms)
ZL = Load impedance (Ohms)
Zi = Input impedance (Ohms)
Z0 = Characteristic impedance (Ohms)
z=
Z/Z0 = r + jx = normalized impedance (dimensionless)
Y = G + jB = admittance (Siemens)
y = Y/Y0 = g + jb = normalized admittance (dimensionless)
Γ = Γr + j Γi = complex reflection coefficient (dimensionless)
γ=
VG – 11
PSA-00527—10/21/2001
α + jβ = propagation constant (inverse meters)
Pacificon 2001
Transmission Line Parameters
Physical Dimensions and Material Properties
d
dielectric
(µ, ε, σ)
a
dielectric
(µ, ε, σ)
b
a
a
c
Parameter
Coax
µ
2π
G S/m
C F/m
Where skin depth is
VG – 12
PSA-00527—10/21/2001
1
πaδσ c
1  1 1
+
2πδσ c  a b 
R Ω/m
L H/m
Twinlead
 b δ  1 1 
ln a + 2  a + b  
µδ
−1 d 
cosh
+
2 a 
π  2 a
2πσ
b
ln
a
2πε
b
ln
a
1
δ=
πfµσ c
πσ
cosh −1
d
2a
πε
cosh −1
For copper
d
2a
σ c = 5.8 × 10 7 S/m
 8.5 mm at 60 Hz
δ =
6.6 µm at 100 MHz
Pacificon 2001
Round Open-Wire Transmission Line Formulas
d
s
❏ Approximate formula
➤ Widely published by ARRL and others
➤ Accurate only for large spacings: s/d > 3
or large impedances: Z0 > several hundred
❏ Exact formula
➤ Accurate for all spacings and impedances
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Pacificon 2001
Comparison of Impedance Formulas
Round Open-Wire Line
VG – 14
PSA-00527—10/21/2001
Pacificon 2001
K6OIK Square Open-Wire Transmission Line Formula
w
s
❏ Excellent approximation in the range of practical interest
➤ Accurate for small spacings: 1 < s/w < 3
or small impedances: 0 < Z0 < several hundred
VG – 15
PSA-00527—10/21/2001
Pacificon 2001
Round vs Square Open-Wire Lines
VG – 16
PSA-00527—10/21/2001
Pacificon 2001
Optimal Characteristic Impedances
Coax
For minimum loss
Zo = 77Ω
For maximum breakdown voltage
Zo = 30Ω
For minimum temperature rise
Zo = 60Ω
Zo = 50Ω has no special significance
VG – 17
PSA-00527—10/21/2001
Pacificon 2001
Reflection Coefficient and Impedance Relation
at a Terminal Plane
Terminal
Plane
ΖO
Definition
ΖL
ΓL
Z L − Zo z − 1
Γ=
=
Z L + Zo z + 1
Inverse
1+ Γ
z=
1− Γ
❏ For every terminal plane, the complex load impedance and
complex reflection coefficient seen to the right give the same
information for that terminal plane
❏ Question: How do Γ and z change as the terminal plane
moves?
VG – 18
PSA-00527—10/21/2001
Pacificon 2001
Relations Between Two Terminal Planes
Input
Terminal
Plane
Output
Terminal
Plane
ΖL
ΓL
ΖO
Ζi
Γi
Impedance relation
z L + j tan βl
zi =
1 + jz L tan βl
Cross relations
− j 2 βl
1 + ΓL e
zi =
1 − ΓL e − j 2 βl
Reflection coefficient relation
Γi = ΓL e − j 2 βl
1 + Γi e j 2 βl
zL =
1 − Γi e j 2 βl
VG – 19
PSA-00527—10/21/2001
Pacificon 2001
Velocity Factor
Wavelength
λ free space =
λ actual =
c
f
v
f
Velocity factor
λ actual
v
vf = =
c λ free space
VG – 20
PSA-00527—10/21/2001
Pacificon 2001
How To Measure Velocity Factor of a Line
(One Way To Do It)
Known length
Antenna
Analyzer
open
circuit
Same length
Antenna
Analyzer
short
circuit
2πfl
vf =
c
VG – 21
PSA-00527—10/21/2001
cot
−1
1
− Z open
Z short
Pacificon 2001
Phase Constant
β=
2π
2πf
=
λ actual v f c
radians/meter
❏ Phase constant β and velocity factor vf give equivalent
information
❏ Both can be calculated from line dimensions and material
properties
β = Im ( R + jωL)(G + jωC )
❏ Best to measure!
VG – 22
PSA-00527—10/21/2001
Pacificon 2001
How to Measure Complex Zo of A Line
(One Way to Do It)
Unknown or
arbitrary length
Antenna
Analyzer
open
circuit
Same length
Antenna
Analyzer
short
circuit
Zo =
Z open × Z short
❏ Geometric mean of two complex numbers
❏ Calculation is trivial in polar form on Smith Chart
VG – 23
PSA-00527—10/21/2001
Pacificon 2001
What Special Lengths of Lossless Line Do
Half wavelength, l = _/2
Zi = Z L
Quarter wavelength, l = _/4
Z o2
Zi =
ZL
Eighth wavelength, l = _/8
Zi = Z o
VG – 24
PSA-00527—10/21/2001
if Z L and Zo are real (resistive)
Pacificon 2001
Standing Wave Ratio
Easy to remember from
VG – 25
PSA-00527—10/21/2001
1+ Γ
s=
1− Γ
1+ Γ
z=
1− Γ
s −1
Γ =
s +1
z −1
Γ=
z +1
Pacificon 2001
Part 2: The Smith Chart
VG – 26
PSA-00527 — 10/21/2001
Pacificon 2001
0.4
0.39
0.38
0.37
0.36
0.12
0.13
5
-4
0.9
1.2
0.0
8
3
0.6
0.4
1.6
-60
0.7
1.4
0.8
-55
1.0
0
-5
0.4
1
0.1
0.11
-100
-90
0.15
0.14
-80
0.35
0.4
2
0.0
9
-65
0.5
1.8
06
0.
-70
2.0
R
O
),
Zo
X/
E
IV
CT
DU
IN
-75
0.8
40
-1
0.4
0.0
7
-1
30
0.2
-110
44
0.
0.6
(-j
CO
M
PO
N
EN
T
-12
0
CAP
AC
ITI
V
E
RE
AC
TA
NC
E
-35
-4
0
-70
4
0.3
6
0.1
3
1.0
)
/Yo
(-jB
CE
AN
T
EP
SC
SU
1.0
5
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
80
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
4.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.2
9
0.0
0.8
1.0
5.0
0.28
5
0.2
20
10
0.0
4
5
14
0
0.0
5
(+
jX
/Z
0.4
20
0.22
0.4
0.6
0.4
6
15
0
0.6
0.3
-60
0.1
0.4
85
70
0.
44
0.5
0.
06
25
-20
0.3
0.2
1
-30
0.
31
7
8
0.2
20
-85
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
0.28
4
0.0
0
-15 -80
50
10
20
50
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN
D
E
G
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
6
0.4
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
W
ERA
O
T
0.48
± 180
HS
TO
T
0.47
G
17
0
R—
-170
EN
L
E
0.47
>
AV
W
160
<—
-90
90
-160
30
0.22
0.48
0.1
0.
19
0.3
0.3
0.2
-4
0
-30
0.2
0.1
60
2
35
0.3
)
/Yo
(+jB
0.3
4
1
0.2
9
0.2
30
0.
70
0.2
0.2
40
0.3
-20
0.
4
0.4
40
0.1
0.35
8
0.1
0
-5
-25
80
31
0.
4
R
,O
o)
CE
AN
PT
CE
US
S
VE
TI
CI
PA
A
C
3.0
3
0.4
0
13
0
0.36
19
0.
VG – 27
PSA-00527 — 10/21/2001
0.
7
12
90
-15
2
0.4
110
0.37
4.0
0.0
1
0.4
0.13
0.38
5.0
8
0.0
0.4
0.12
-10
9
0.0
0.39
100
50
0.11
0.
0.1
0.14
0.15
0.1
6
3
7
2
0.1
8
3.0
15
10
Pacificon 2001
Complex Functions
zi
wi
(x, y)
y
x
Complex Number
Complex Number
z = x + jy
w = u + jv
(u, v)
v
f( )
zr
u
wr
❏ Basic types of complex functions
➤ Global Properties
– Linear – lines map to lines
– Bilinear – circles map to circles
➤ Local Properties
– Conformal – right angles map to right angles
VG – 28
PSA-00527—10/21/2001
Pacificon 2001
Mathematical Basis of the Smith Chart
Γ=
u + jv =
z −1
z +1
A bilinear conformal
complex function
(r - 1) + jx
(r + 1) + jx
v
x
u
r
Right Half
Z Plane
VG – 29
PSA-00527—10/21/2001
Interior Unit Circle
Γ Plane
Pacificon 2001
-90
5
-4
-5
0.9
-60
1.2
0.7
1.4
0.8
1.0
0
-70
6
0.1
0.15
4
0.3
0.39
0.4
0.38
0.37
0.11
-100
0.12
0.13
0.36
-55
-35
0.14
-80
-4
0
0.35
0.6
-65
44
0.
1.6
1.8
2.0
0.5
0.
31
7
06
0.
-70
-1
E
IV
CT
DU
IN
-75
R
,O
o)
/Z
jX
40
5
0.0
5
x=
-1
9
0.4
)
/Yo
(-jB
CE
AN
PT
CE
S
SU
1.0
8
0.
1
0.2
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
85
10
6
150
0.0
0.4
4
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
80
1.0
4.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
0.2
0.1
0.4
1
-110
0.0
9
0.4
2
0.0
1
8
2
0
CAP
0.4
AC
ITI
3
VE
0.0
R
EA
7
CT
-1
AN
30
CE
CO
M
PO
N
EN
T
(-
-85
20
6
r=1
0.2
0.4
4
0.0
0
-15 -80
0.8
5.0
0.28
0.4
(+
jX
/Z
14
0.0
5
0.4
5
20
0.22
0.
19
-60
0.1
0.4
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
ERA
0.48
± 180
S TOW
H
TO
T
170
NG
R—
-170
ELE
V
0.47
>
WA
160
—
<
-90
90
-160
0
70
0.
44
0.5
0.
06
25
0.22
0.48
0.4
-20
0.6
0.47
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
10
20
50
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0
-30
0.3
r Series Resistance
50
0.3
-20
3.0
0.2
r=2
20
0.8
4
r=2
30
0.28
1.0
0.
r=1
0.1
0.2
-4
3
0.3
0.3
60
0.1
r=0
2
1.0
4
0.3
0.6
0.1
8
0.1
0
-5
-25
0.2
0.3
9
4
35
1
0.3
-30
0.
70
0.2
Capacitive
Reactance
(Negative)
0.35
0.2
0.1
40
30
Inductive
Reactance
(Positive)
4.0
)
/Yo
(+jB
CE
AN
PT
E
SC
SU
0.36
0.3
0.2
x=0
80
0.2
x=-1
0
0.13
40
VG – 30
PSA-00527—10/21/2001
x=0
VE
TI
CI
PA
CA
0.37
-15
7
0.0
90
31
0.
x=1
0
12
0.38
19
0.
x Series Reactance
R
,O
o)
2
1
110
0.12
5.0
0.4
0.4
0.4
0.39
100
-10
0.11
50
3
0.4
0
13
8
9
0.2
0.0
0.6
0.1
0.
8
0.0
1
x=
r=0
Smith Chart: Impedance Coordinates
Γi
0.14
0.15
6
3
7
2
0.1
8
3.0
15
10
∞
Γr
Pacificon 2001
70
0.
44
0.
06
0.0
7
R
,O
o)
3
0.4
0
13
0.5
65
55
50
45
0.11
0.9
1.0
1.2
0.39
100
1.4
0.7
0.6 60
1.6
0.1
0.4
90
80
0.37
0.38
0.13
0.12
40
0.36
40
0.2
30
9
20
8
0.6
20
0.2
25
0.35
70
30
35
60
0.3
0.1
4
0.1
50
0.3
0.3
0.1
1.8
0.8
0.4
9
0.0
1
110
0.4
8
0.0
2
0.4
120
Yo)
jB/
E (+
NC
TA
EP
SC
SU
VE
TI
CI
PA
CA
0.
5.0
0.8
1.0
4.0
0.28
0.6
0.4
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
-20
0.28
0.22
0.3
1
0.2
9
0.2
1
-30
4
0.
31
0.
0
1
2.0
19
0.
0.0
5
0.4
5
14
(+
jX
/Z
0.2
150
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.48
0.0
D <—
0.49
GEN
RD LOA
E
TOWA
0.48
R
± 180
A
TO
THS
0.47
17
70
0
R
-1
ENG
—>
VEL
0.47
WA
0.0
160
<—
4
-90
90
-160
0.4
85
-85
6
80
0.2
4
0.0
0
1.0
b=
RE
AC
TA
75
N
C
EC
OM
PO
N
EN
T
0.3
6
0.4
IND
UCT
IVE
8
-15
0.1
0.2
20
50
10
0.25
0.26
0.24
0.27
0.25
0.24
0.26
0.23
O
C
E
N
F
F
O
I
I
C
T
I
C
E
E
0.27
N
L
T
F
E
IN DEG
R
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
5
0.0
-75
o)
-80
jB/Y
E (NC
TA
EP
SC
SU
E
IV
CT
DU
IN
0.
5
0.4
40
-1
0.23
-70
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
1.4
1.6
0.22
06
0.
R
,O
o)
/Z
jX
10
10
Shunt Conductance
0.3
0
-4
0.
19
b=
-1
0.2
44
0.
0
-65 .5
g=2
0.6
1.0
1.8
2.0
3.0
4.0
5.0
10
20
50
0.6
0.2
Capacitive
Susceptance
(Positive)
4
0.
Γr
0
g=1
-60
0.
31
0.7
0.4
0.2
-5 5
0.8
0.9
1.0
0
-5
5
-4
1.2
1.4
1.6
1.8
-10
2.0
0.1
0.4
b=-1
1.0
g=2
5.0
0.2
1
0.4
Pacificon 2001
VG – 31
PSA-00527—10/21/2001
20
∞
g
4.0
-15
0
0.6
g=1
-20
3.0
0
8
0.1
0
-5
-70
.16
Inductive
Susceptance
(Negative)
b=1
0.8
b Shunt Susceptance
-25
g=
-30
2
0.3
-110
0.0
9
0.4
2
0.0
-12
8
0
CAP
0.4
AC
ITI
3
VE
0.0
RE
7
AC
-1
TA
30
NC
EC
OM
PO
N
EN
T
(7
-60
0.1
-35
0.3
3
4
0.3
0.39
0.38
0.1
0.4
0.11
-100
-90
0.15
0.37
0.36
0.12
0.13
0.14
-80
-4
0
0.35
b=0
50
b=0
g=0
Smith Chart: Admittance Coordinates
Γi
15
3.0
0.3
3
2
0.1
8
6
7
0.14
0.15
0.1
-90
0.4
0.38
0.39
0.37
0.11
-100
0.12
0.36
0.13
0.14
-80
0.35
0.4
2
0.0
9
-5
0.9
1.2
1.0
0
0.7
1.4
0.8
-55
-110
0.4
1
-4
5
0
0.0
8
-35
-4
-70
6
0.1
0.15
4
0.3
0.0
7
30
0.4
3
0.6
1.6
1.8
2.0
31
-6 0
0.
-1
-12
0
CAP
AC
ITI
VE
R
EA
CT
AN
CE
0.5
-20
(-j
CO
M
PO
N
EN
T
19
-65
1.0
3
0.3
0.
-60
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
0.2
20
10
80
0.0
4
0.4
6
1 50
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
1.0
5.0
4
0.2
0.4
0.6
8
0.
0.28
0.6
o)
jB/Y
E (NC
TA
EP
4
C
4
-75
US
0.
40
ES
-1
06
IV
0.
CT
DU
IN
R
-70
O
),
Zo
X/
5
4.0
0.2
9
0.0
0.8
0.22
0.8
85
5
(+
jX
/Z
14
0
0.0
5
0.4
20
-20
0.3
0.2
1
-30
0
-30
-85
0.6
0.2
-4
0.2
4
0.0
0
-15 -80
0.6
0.2
6
0.4
0.
8
5
0.4
70
0.
44
65
0.5
0.
06
25
0.28
2
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
30
10
20
50
0.1
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.4
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
—
0.49
GEN
D LOAD <
ERA
OWAR
T
0.48
±
S
180
TO
GTH
170
70
N
R—
-1
E
VEL
0.47
>
A
W
1
0
6
<—
6
0
-90
90
-1
0.2
3
0.3
60
7
0.1
0.3
4
2
0.3
35
0.22
20
0.48
0.3
9
0.2
30
1.5
70
1
0.2
0.2
40
0.2
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.35
0.3
1.0
0.
4
0.4
40
0.1
0.36
-25
80
8
0.1
0
-5
90
31
0.
VG – 32
PSA-00527—10/21/2001
4
0.
R
,O
o)
0.4
3
0.4
0
13
Yo)
jB/
E (+
NC
TA
EP
C
S
SU
VE
TI
CI
PA
A
C
0
12
3.0
2
0.4
110
0.37
4.0
7
0.0
1
0.4
0.13
0.38
-15
8
0.0
0.12
5.0
9
0.0
0.4
0.39
100
-10
0.11
50
0.1
19
0.
0.47
Smith Chart: Constant SWR Circles
Γi
0.14
0.15
0.1
6
0.3
3
0.1
7
0.3
2
50
0.1
8
3.0
15
10
Γr
Pacificon 2001
0.4
0.38
0.37
0.39
0.12
0.13
0.36
-
0.0
9
45
0.35
0.9
1.2
0.0
8
0.0
7
30
0.6
0.4
3
1.6
-60
0.7
1.4
0.8
-55
1.0
0
-5
0.4
1
0.1
0.11
-100
-110
-70
0.14
-80
-4
0
-90
0.4
2
6
0.1
0.15
4
0.3
-1
1.8
2.0
-12
0
CAP
AC
IT
I
V
ER
EA
CT
AN
CE
0.
31
0.2
(-
CO
M
PO
N
EN
T
0.
19
-35
1.0
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
80
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
4.0
0.3
0
-65 .5
0.6
)
/Yo
(-jB
CE
AN
PT
CE
-75
US
0
S
4
E
-1
06
IV
0.
CT
DU
IN
R
-70
,O
)
o
/Z
jX
5
0.0
0.8
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.
0.2
9
44
0.
0.2
20
10
0.0
4
5
0
14
0.0
5
(+
jX
/Z
0.4
20
1.0
5.0
0.28
0.3
0.2
1
-30
-20
0.4
6
150
0.6
0.22
0
-60
0.1
0.4
85
70
0.
44
65
0.5
0.
06
25
0.2
-4
-30
IND
UCT
IVE
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
2
-20
0.8
0.2
-85
30
10
20
50
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.6
4
0.0
0
-15 -80
0.
8
6
0.4
0.2
0.22
20
5
0.4
60
7
0.1
35
3
0.3
0.3
4
-25
70
8
0.1
0
-5
40
0.28
1.0
0.
4
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.48
0.0
0.49
AD <—
GEN
ARD LO
ERA
0.48
± 180
S TOW
TO
GTH
170
0
N
R—
-17
E
VEL
0.47
>
A
W
160
<—
-90
90
-160
0.3
9
0.2
30
0.2
0.35
2
0.3
0.36
1
0.2
0.1
1
80
0.2
4
0.
0.13
0.3
VG – 33
PSA-00527—10/21/2001
r
0.4
40
1
0.5
2
R
,O
o)
0.37
31
0.
x
0.5
0.4
3
0.4
0
13
Yo)
jB/
E (+
NC
TA
EP
SC
U
S
VE
TI
CI
PA
CA
120
90
3.0
2
0.4
110
0.38
4.0
7
0.0
1
0.4
0.12
-15
8
0.0
0.39
100
5.0
0.11
-10
9
0.0
0.4
50
0.1
19
0.
0.47
Smith Chart: Constant Impedance Magnitude Circles
Γi
0.14
0.15
0.1
6
0.3
3
0.1
7
50
0.3
2
0.1
8
3.0
15
10
Γr
Pacificon 2001
-90
0.39
0.4
0.38
0.11
-100
0.12
0.13
0.37
0.36
-55
-4
5
0.9
1.2
1.0
0
-5
-60
0.7
1.4
0.8
-35
0.14
-80
-4
0
-70
6
0.1
0.15
4
0.3
0.6
1.6
1.8
2.0
0
-65 .5
31
0.35
-85
1.0
o)
jB/Y
5
E (0.0
NC
TA
EP
4
C
4
-75
US
0.
40
ES
-1
06
IV
0.
CT
DU
IN
R
-70
O
),
Zo
X/
0.
0.2
4
0.0
0
-15 -80
0.2
20
10
0.0
4
0.4
6
150
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
80
4.0
315°
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
85
IND
UCT
IVE
0.8
1.0
5.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.
0.28
0
.1
0
.41
-110
0.0
9
0.4
2
0.0
-1
8
2
0
C
A
P
0.4
AC
ITI
3
VE
0.0
R
E
7
AC
-1
TA
30
NC
EC
OM
PO
N
EN
T
(-j
0.2
6
0.4
0.6
5
0.4
0.4
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
D <—
0.49
GEN
RD LOA
ERA
TOWA
0.48
±
S
180
H
TO
170
70
NGT
R—
-1
E
L
E
0.47
>
AV
W
160
<—
-90
90
-160
0.
8
0.48
45°
0.3
9
0.
1
-60
0.47
(+
jX
/Z
14
0
5
0.0
5
0.4
20
0.2
9
-4
0
0.4
°
0.6
0.22
0.3
0.2
1
-30
0.6
45
70
0.
44
0.
2.0
65
0.5
06
25
-20
-20
90°
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
30
0.2
-30
0.1
270°
0.2
3
0.3
60
7
0.1
35
-25
0.3
4
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
4
0.
70
2
0.3
0.2
40
0.22
20
0°
0.35
0.28
0.8
0.
4
5°
0.3
1
0.2
9
0.2
30
0.1
0.36
8
0.1
0
-5
80
0.2
5°
31
VG – 34
PSA-00527—10/21/2001
0.13
0.3
0°
0.4
40
180°
0.37
31
0.
x
R
,O
o)
1.0
3
0.4
0
13
Yo)
jB/
E (+
NC
TA
EP
SC
U
S
VE
TI
CI
PA
CA
120
3.0
2
0.4
110
90
4.0
7
0.0
1
0.4
0.38
-15
8
0.0
0.4
0.12
5.0
9
0.0
0.39
100
-10
0.11
50
0.1
19
0.
5°
13
22
Smith Chart: Constant Impedance Phase Angle Circles
Γi
0.14
0.15
0.1
6
0.3
3
0.1
7
0.3
2
50
0.1
8
3.0
15
10
Γr
Pacificon 2001
Smith Chart: Multiplication, Division, Squares, and
Square Roots
Unary Operators
squares
square roots
tangents
cotangents
inverse tangents
Inverse cotangents
a2
a
tan θ
cot θ
tan-1 a
cot-1 a
Binary Operators
multiplication X
division ÷
geometric mean
VG – 35
PSA-00527—10/21/2001
a•b
c/a
ab
Pacificon 2001
0.4
0.38
0.37
0.39
0.12
0.14
-80
-90
0.11
-100
0.1
0.4
1
5
-4
0.35
2
0.9
1.2
0.7
1.4
0.8
-55
1.0
0
-5
0
0.13
0.36
-110
0.0
9
0.4
-35
-4
-70
6
0.1
3
8
0.0
7
30
0.6
1.8
(-j
3
0.4
1.6
-60
7
0.15
4
0.3
0.0
0.3
-1
2.0
-12
0
CAP
AC
ITI
VE
R
EA
CT
AN
CE
-60
1.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.
0
0.5
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1.0
a=4
1.0
5.0
0.28
-65
4.0
0.3
31
0.2
0.2
0.8
0.22
CO
M
PO
N
EN
T
0
(+
jX
/Z
14
0.0
5
0.4
5
20
.29
0.6
15
3.0
-20
0.3
1
-30
-20
0.1
0.6
80
0.3
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
0.6
0.2
0.
-30
IND
UCT
IVE
0
70
44
0.
0.5
06
0.
25
10
20
50
0.1
0.4
0.2
20
10
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.8
o)
jB/Y
E (NC
TA
EP
5
SC
44
-7
.
U
0
40
ES
-1
06
IV
0.
CT
DU
IN
R
-70
O
),
Zo
X/
5
0.0
0.
8
5
0.4
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.1
0.28
6
0.3
0.22
20
4
0.0
0
-15 -80
50
1
9
0.2
30
0.4
30
0.
19
0.1
0.2
2
60
0.3
35
0.2
-4
0
-√ 2
0.3
4
0.2
c=2
70
-25
40
0.2
0.2
√ c=1.414
0.35
0.3
1.0
0.
4
0.4
40
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
D <—
0.49
GEN
RD LOA
A
W
ERA
O
T
0.48
± 180
TO
THS
0.47
G
17
70
N
0
R—
-1
ELE
V
0.47
>
WA
0.0
1
—
0
60
<
4
-90
90
-16
0.4
85
-85
6
31
0.1
0.36
80
0.
b=0.5
0.13
19
VG – 36
PSA-00527—10/21/2001
4
0.
R
,O
o)
0
0.37
8
0.1
0
-5
3
0.4
0
13
)
/Yo
(+jB
CE
AN
PT
E
C
US
ES
IV
IT
AC
P
CA
12
0.4
2
3.0
0.4
110
90
4.0
7
0.0
1
0.4
0.38
-15
8
0.0
0.4
0.12
5.0
9
0.0
0.39
100
-10
0.11
50
0.1
0.
0.48
Smith Chart: A Nomogram for Math Calculations
Γi
0.14
0.15
0.1
6
0.3
3
0.1
7
2
8
15
10
Γr
Pacificon 2001
Part 3: Impedance Matching
VG – 37
PSA-00527 — 10/21/2001
Pacificon 2001
Impedance Matching Categories
❏ Single frequency matching
➤ Manual synthesis using Smith Chart
➤ Eight canonical networks
➤ Lumped elements
➤ Series and parallel stubs
➤ Transmission line sections
❏ Multiple frequency matching
➤ Ladder networks
➤ Multiple stubs
➤ Multiple line sections
❏ Broadband matching
➤ Maximize SWR bandwidth
➤ Software for computer-aided manual design
➤ Software for network optimization
➤ Smith Chart used for visualization only
VG – 38
PSA-00527—10/21/2001
Pacificon 2001
Bandwidth Classifications
Fractional Bandwidth
VG – 39
PSA-00527—10/21/2001
Narrowband
< 10%
Moderate band
10% to 50%
Broadband
> 50%
Pacificon 2001
Two Kinds of Matching
❏ Conjugate Matching
❏ Load Matching
ΖS
+
∼
ΖL
ΖO
ΖL
-
Zs = ZL*
ZL = ZO
❏ Best use at source
(transmitter)
❏ Best used at ends of transmission
lines
❏ Maximizes power delivery to
the load
❏ Minimizes reflections
❏ Does not minimize reflections
unless Zs is real
❏ Does not maximize delivered
power unless Z0 is real
❏ Normally done by the
transmitter manufacturer at the
circuit design level
❏ Ideally Zs (ext) = 50 Ω
VG – 40
PSA-00527—10/21/2001
Pacificon 2001
Where Should Matching Network Go?
❏ Poor Setup
High to
Low SWR
High SWR
Antenna
Tuner
Tx
Long Transmission Line
❏ Good Setup
Perfect 1:1
Tx
❏ Best Setup
Low SWR
Antenna
Tuner
VG – 41
PSA-00527—10/21/2001
Very Low-Loss Line
Matching
Network
Insertion Loss x dB
Low SWR
Tx
Low SWR
Low SWR
Matching
Network
Pacificon 2001
Necessary Test Equipment
❏ Antenna analyzer
➤ Autek
➤ CIA
➤ MFJ
❏ Noise bridge (less accurate)
❏ Network analyzer (more accurate)
➤ Hewlett-Packard
VG – 42
PSA-00527—10/21/2001
Pacificon 2001
Matching Network Design Recipe
1. Measure transmission line parameters Zo, vf
2. Measure antenna feedpoint impedance across band(s) of interest
3. Measure or calculate impedance across band(s) of interest at network
insertion point
4. Narrowband match:
➤ Select appropriate lossless L network, 2 or 4 choices
➤ Select lumped elements vs stubs
➤ Calculate component values
➤ Calculate SWR and SWR bandwidth
➤ Build and test
5. Broadband match:
➤ Use design software - winSMITH or equivalent
➤ Design n-stage lossless ladder network
➤ Select lumped elements vs stubs
➤ Calculate component values
➤ Calculate SWR and SWR bandwidth
VG – 43
PSA-00527—10/21/2001
Pacificon 2001
How to Measure Antenna Feedpoint Impedance
❏ Measure impedance through known line
❏ Divide measure impedance by Zo
❏ Plot impedance point on Smith Chart
❏ Move counter clockwise on chart by electrical length of line
❏ Read coordinate values from chart
❏ Multiply result by Zo
VG – 44
PSA-00527—10/21/2001
Pacificon 2001
-90
0.4
0.38
0.39
0.12
-55
5
-4
0.9
1.2
1.0
0
-5
-60
0.7
1.4
0.8
6
0.1
0.15
4
0.3
-70
0.14
-80
-4
0
0.13
0.11
-100
0.6
1.6
1.8
2.0
0.5
0.1
0.4
1
-110
0.0
9
0.4
2
0.0
1
8
2
0
CAP
0.4
AC
I
3
T
IVE
0.0
RE
7
AC
-1
TA
30
NC
EC
O
M
PO
N
EN
T
(-j
-35
0.37
0.36
-65
9
0.6
1.0
o)
jB/Y
5
E (0.0
NC
TA
EP
5
SC
44
-7
.
U
0
40
ES
-1
06
IV
0.
CT
DU
IN
R
-70
O
),
Zo
X/
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
>
10
4
0.0
6
15
0
0.4
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
80
4.0
+b
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.
0.2
0.
31
0.35
0
0.8
85
+x
1.0
5.0
0.28
0.2
-85
20
<
0.8
0.3
0.3
0.2
6
5
14
0
0.0
5
(+
jX
/Z
0.4
20
0.22
0.2
0.
19
-60
4
0.0
0
-15 -80
0.6
-20
0.3
0.2
1
-30
-4
0
-30
0.1
-x
0.6
0.4
0.3
.45
70
0.
44
65
0.5
0.
06
25
10
20
50
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.4
> WAVELEN
GTHS
TOW
ARD
0.49
GEN
ERA
0.48
TO
170
R
0.47
160
90
0.
8
0.2
20
0.0
-b
0.28
0.0
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
0.22
± 180
50
9
0.49
30
1
D<
RD LOA
TOWA
THS
-170
ENG
VEL
WA
-90
-160
0.2
0.2
0.48
60
7
35
3
0.3
4
0.2
30
4
0.
70
0.3
0.2
40
0.2
1.0
0.
4
0.4
40
0.1
0.35
0.1
)
/Yo
(+jB
CE
AN
PT
E
SC
SU
0.36
2
80
0.3
VE
TI
CI
PA
CA
0.13
31
0.
VG – 45
PSA-00527—10/21/2001
R
,O
o)
120
0.4
2
0.37
8
0.1
0
-5
-25
0.4
-20
3.0
7
0.0
110
90
4.0
3
0.4
0
13
1
0.4
0.38
-15
8
0.0
0.4
0.12
5.0
9
0.0
0.39
100
-10
0.11
50
0.1
19
0.
0.47
Smith Chart: Effect of Adding Series Reactances or
Shunt Susceptances
Γi
0.14
0.15
0.1
6
0.3
3
7
0.1
2
0.1
8
3.0
15
10
Γr
Pacificon 2001
0
0.1
-90
0.4
9
2
0.9
1.2
8
0.7
1.4
0.8
-55
1.0
0
-5
5
-4
0.15
4
0.39
0.38
0.37
0.36
0.12
0.13
0.11
-100
0.4
.41
0.14
-80
0.35
0.0
-110
0
-70
0.0
-35
-4
0.3
0.4
0.6
1.6
1.8
0.2
-60
2.0
31
6
3
0.0
7
30
-1
-12
0
CAP
AC
ITI
VE
R
EA
CT
AN
CE
0.5
0.6
(-j
CO
M
PO
N
EN
T
0.
-65
1.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
0.
8
0.2
19
0.1
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.2
20
10
80
0.0
4
150
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
1.0
5.0
0.28
0.3
0.
-60
IND
UCT
IVE
4.0
0.22
9
)
/Yo
(-jB
CE
AN
PT
E
C
44
-75
US
0.
40
ES
-1
06
IV
0.
CT
U
D
IN
R
-70
O
),
Zo
/
X
5
0.0
0.8
-20
0.3
1
-30
-20
0.4
6
(+
jX
/Z
14
0.0
5
5
0.4
20
0.2
0.2
-4
0
-30
0.1
0.4
85
0.6
10
20
50
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DE
G
R
LE OF
E
ES
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.6
5
0.4
8
-85
0.
0.2
20
6
0
70
0.
44
06
0.5
0.
25
0.28
4
0.0
0
-15 -80
65
2.0
1 .8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
30
0.22
0.4
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
GEN
LOAD <—
D
R
A
ERA
TOW
0.48
S
7
180
±
H
.4
TO
T
0
170
70
NG
R—
-1
E
L
VE
0.47
>
A
W
1
0
6
<—
6
0
-90
90
-1
0.2
3
0.3
60
7
0.1
35
1
0.2
9
0.2
30
0.2
4
0.2
Forbidden Area
0.1
2
0.3
)
/Yo
(+jB
0.3
0.3
0.8
0.
4
0.4
40
0.48
31
0.1
70
0.
(a)
R
,O
o)
E
IV
IT
AC
P
CA
E
NC
TA
EP
SC
SU
40
19
VG – 46
PSA-00527 — 10/21/2001
Z
3
0.4
0
13
0.35
-25
7
80
0.
Z0
4
0.
1.0
0.0
120
0.4
2
90
0.36
8
0.1
0
-5
0
3.0
0.4
110
0.37
-15
1
0.4
0.13
0.38
4.0
.08
0.4
0.12
5.0
9
0.0
0.39
100
-10
0.11
50
0.1
0.14
0.15
6
0.3
3
0.1
7
0.3
2
0.1
8
50
3.0
15
10
Pacificon 2001
5
-4
0.9
1.2
1.0
0
-5
0
-4
0.39
0.6
1.4
0.7
1.6
1.8
0.2
0.4
0.38
0.37
0.11
-100
-90
0.8
-35
0.14
-80
0.15
4
0.3
-70
6
0.1
0.12
-55
-60
7
0.1
0.13
-60
2.0
31
3
0.3
0
-65 .5
0.
0.35
44
0.
06
19
0.36
-70
),
Zo
X/
40
-1
0.1
0.4
1
-110
0.0
9
0.4
2
0.0
-12
8
0
CAP
0.4
AC
IT
3
I
V
ER
0.0
EA
7
C
-1
T
AN
30
CE
CO
M
PO
N
EN
T
(-j
0.
0.6
1.0
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
0.1
0.4
0.2
20
10
80
15
0
6
4
0.0
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
1.0
5.0
0.2
0.4
0.6
0.
8
0.28
0.3
0.
-30
)
/Yo
(-jB
CE
AN
T
EP
SC
-75
SU
VE
TI
C
DU
IN
R
O
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.22
5
4.0
0.2
9
0.0
0.8
-20
0.3
0.2
1
-30
5
8
0.4
5
14
0
0.0
5
(+
jX
/Z
0.4
20
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.2
4
0.26
0.23
0.27
LECTION COEFFICIEN
T
F
E
I
N
R
D
E
GREE
LE OF
S
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.4
0.
0.2
85
0.6
0.28
-85
70
0.
44
06
2.0
0.5
0.
25
0.22
20
6
65
1.8
50
1
9
4
0.0
0
-15 -80
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
0.2
0.2
30
0.4
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
W
ERA
O
T
0.48
± 180
HS
TO
T
0.47
G
17
0
R—
-170
EN
L
E
0.47
>
AV
W
160
<—
-90
90
-160
30
0.2
-4
0
8
0.2
0.1
60
2
35
0.3
0.3
4
0.2
0.2
70
0.3
4
0.
-20
0.
4
0.4
40
0.48
31
0.1
40
0.
Forbidden Area
0.35
19
VG – 47
PSA-00527 — 10/21/2001
Z
R
,O
o)
0
80
0.
Z0
3
0.4
0
13
0.8
0
1.0
.07
Yo)
jB/
E (+
NC
A
T
EP
SC
SU
VE
TI
I
C
PA
CA
12
0.36
0
-5
-25
2
0.4
0.0
3.0
0.4
110
90
4 .0
1
0.4
0.37
-15
8
0.4
0.13
0.38
5.0
9
0.6
0.0
0.39
100
0.12
-10
0.11
50
0.1
0.14
0.15
0.1
6
0.3
3
7
0.1
2
0.1
8
3.0
15
10
(b)
Pacificon 2001
0.39
0.4
0.38
0.11
-100
0.37
-4
5
0.9
1.2
0.7
1.4
0.8
1.0
0
-5
0
-4
0.12
-55
-35
-90
0.15
4
0.3
-70
6
0.1
0.14
-80
0.35
-60
1.6
-60
7
0.1
0.13
0.36
0.6
44
06
0.
-70
),
Zo
X/
40
-1
1.8
2.0
0
-65 .5
0.1
0.4
1
-110
0.0
9
0.4
2
0.0
-12
8
0
CAP
0.4
AC
IT
3
I
V
ER
0.0
EA
7
C
-1
T
AN
30
CE
CO
M
PO
N
EN
T
(-j
0.
-20
0.3
0.2
5
1.0
)
/Yo
(-jB
CE
AN
T
EP
SC
-75
SU
VE
TI
C
DU
IN
R
O
0.6
0.0
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
80
150
6
4
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
1.0
5.0
Forbidden Area
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.
0.28
5
4.0
0.22
0.4
0.8
0.2
9
3
0.1
0.4
0.2
20
10
0.0
5
14
0
0.0
5
(+
jX
/Z
0.4
20
-20
0.3
0.2
1
-30
31
0.
0.3
0.6
0.4
0.6
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.2
4
0.26
0.23
0.27
FLECTION COEFFICIENT
E
I
N
R
D
E
GREE
LE OF
S
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
19
0.
-30
0.
8
0.2
85
70
44
0.
2.0
0.5
06
0.
25
0.28
-85
65
1.8
50
0.22
20
6
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.1
1
9
4
0.0
0
-15 -80
0.3
0.2
0.2
30
0.4
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
D <—
0.49
GEN
RD LOA
A
W
ERA
O
0.48
± 180
HS T
TO
T
0.47
G
17
0
R—
-170
EN
L
E
0.47
>
AV
W
160
<—
-90
90
-160
30
0.2
-4
0
8
0.2
0.1
60
2
35
0.2
4
0.
0.3
4
0.3
0.8
0.
4
0.4
40
0.48
31
0.2
70
0.
0.1
40
19
VG – 48
PSA-00527 — 10/21/2001
Z
R
,O
o)
0.35
0.
Z0
3
0.4
0
13
1.0
0
.07
Yo)
jB/
E (+
NC
A
T
EP
SC
SU
VE
TI
I
C
PA
CA
120
80
0.3
2
0.36
0
-5
-25
0.0
0.4
0.4
110
3.0
1
0.4
90
4.0
8
0.4
0.37
-15
9
0.13
0.38
5.0
0.0
0.39
100
0.12
-10
0.11
50
0.1
0.14
0.15
0.1
6
0.3
3
7
0.1
2
8
3.0
15
10
(c)
Pacificon 2001
0.4
0.39
0.38
0.37
0.36
5
-4
0.4
9
2
0.9
1.2
8
0.6
0.4
1.6
-60
0.7
1.4
0.8
-55
1.0
0
-5
1
0.4
0.1
-90
0.11
-100
0.12
0.13
0.35
0.14
-80
0.15
0.3
4
0.0
-110
0.0
-35
-4
0
-70
6
0.1
3
3
0
-65 .5
0.0
7
-1
30
1.8
2.0
-12
0
CAP
AC
ITI
V
E
RE
AC
TA
NC
E
0.2
-70
R
O
),
Zo
/
X
E
IV
CT
DU
IN
-75
0.8
40
1.0
-1
(-j
CO
M
PO
N
EN
T
06
0.
0.6
44
9
0.
1.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.6
Forbidden Area
0.
8
0.2
1
0.
3
0.3
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
80
15
0
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
1.0
5.0
0.28
0.
19
-60
IND
UCT
IVE
4.0
0.3
-30
)
/Yo
(-jB
CE
AN
T
EP
SC
SU
0.8
0.22
5
0.6
0.1
0.4
0.2
20
0.0
4
(+
jX
/Z
14
5
0.0
5
0.4
20
-20
0.3
1
-30
0.0
8
0.4
6
0.3
85
0.6
0.2
5
0.4
0.
0.2
20
-85
0
70
44
0.
0.5
06
0.
25
0.28
6
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
0.22
4
0.0
0
-15 -80
50
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
EFLECTION COEFFICIENT IN
R
D
E
G
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.4
30
0.2
0
-4
7
0.2
0.1
60
2
0.3
35
1
0.2
9
0.2
30
0.4
4
0.2
0.2
70
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
W
ERA
O
T
0.48
± 180
HS
TO
T
0.47
G
17
N
0
R—
-170
ELE
V
0.47
>
A
W
160
<—
-90
90
-160
0.4
40
0.48
31
0.1
40
0.
0.
4
R
,O
o)
0.35
19
VG – 49
PSA-00527 — 10/21/2001
Z
3
0.4
0
13
80
8
0.1
0
-5
-25
7
)
/Yo
(+jB
CE
AN
T
EP
SC
SU
VE
TI
I
C
PA
CA
120
0.36
0.
Z0
4
0.
-20
0.0
0.4
2
110
3.0
0.4
100
90
4.0
1
0.4
0.37
-15
8
0.0
0.4
0.13
0.38
5.0
9
0.12
-10
0.0
0.39
10
0.11
50
0.1
0.14
0.15
0.3
0.1
6
0.3
3
7
0.1
2
0.1
8
3.0
15
10
(d)
Pacificon 2001
0.4
0.39
0.38
0.37
0.36
5
-4
0.4
9
2
0.9
1.2
8
0.6
0.4
1.6
-60
0.7
1.4
0.8
-55
1.0
0
-5
1
0.4
0.1
-90
0.11
-100
0.12
0.13
0.35
0.14
-80
0.15
0.3
4
0.0
-110
0.0
-35
-4
0
-70
6
0.1
0
-65 .5
1.8
2.0
1
0.
3
0.3
3
0.0
7
-1
30
-12
0
CAP
AC
ITI
V
E
RE
AC
TA
NC
E
0.2
-70
R
O
),
Zo
/
X
(-j
CO
M
PO
N
EN
T
E
IV
CT
DU
IN
-75
0.8
40
1.0
-1
0.6
06
0.
9
44
1.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
0.
8
0.2
0.
Forbidden Area
0.3
0.
19
-60
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
80
15
0
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
4.0
1.0
5.0
0.28
0.2
0
-4
-30
)
/Yo
(-jB
CE
AN
T
EP
SC
SU
0.8
0.22
5
(+
jX
/Z
14
5
0.0
5
0.4
20
-20
0.3
1
-30
0.0
0.6
0.1
0.4
0.2
20
10
0.0
4
0.6
0.2
5
0.4
8
0.2
20
0.
0.4
6
0.3
85
0
70
44
0.
0.5
06
0.
25
0.28
-85
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
EFLECTION COEFFICIENT IN
R
D
E
G
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
6
50
0.22
4
0.0
0
-15 -80
30
1
0.2
9
0.2
30
0.4
0.2
3
60
7
35
0.1
4
0.2
0.2
70
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
W
ERA
O
T
0.48
± 180
HS
TO
T
0.47
G
17
N
0
R—
-170
ELE
V
0.47
>
A
W
160
<—
-90
90
-160
0.4
40
0.48
31
0.1
40
0.
0.
4
R
,O
o)
0.35
19
VG – 50
PSA-00527 — 10/21/2001
Z
3
0.4
0
13
80
2
0.3
7
0.36
0.
Z0
4
0.
-20
0.0
)
/Yo
(+jB
CE
AN
T
EP
SC
SU
VE
TI
I
C
PA
CA
120
110
90
8
0.1
0
-5
-25
2
0.4
0.4
100
3.0
1
0.4
0.37
4.0
8
0.0
0.4
0.13
0.38
-15
9
0.12
5.0
0.0
0.39
-10
0.11
50
0.1
0.14
0.15
0.3
0.1
6
0.3
3
7
0.1
2
0.1
8
3.0
15
10
(e)
Pacificon 2001
0.4
0.39
0.38
0.37
0.36
5
-4
0.4
9
2
0.9
1.2
8
0.6
0.4
1.6
-60
0.7
1.4
0.8
-55
1.0
0
-5
1
0.4
0.1
-90
0.11
-100
0.12
0.13
0.35
0.14
-80
0.15
0.3
4
0.0
-110
0.0
-35
-4
0
-70
6
0.1
3
0
-65 .5
0.0
7
-1
30
1.8
2.0
-12
0
CAP
AC
ITI
V
E
RE
AC
TA
NC
E
0.2
-70
R
O
),
Zo
/
X
E
IV
CT
DU
IN
-75
0.8
40
1.0
-1
(-j
CO
M
PO
N
EN
T
06
0.
0.6
44
9
0.
1.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
0.
8
0.2
1
0.
3
0.3
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
80
15
0
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
1.0
5.0
0.28
0.3
0.
19
-60
)
/Yo
(-jB
CE
AN
T
EP
SC
SU
4.0
0.22
5
0.8
-20
0.3
1
-30
0.0
0.6
0.1
0.4
0.2
20
10
0.0
4
(+
jX
/Z
14
5
0.0
5
0.4
20
0.2
5
0.4
8
0.2
20
0.
0.4
6
0.6
0.28
85
0
70
44
0.
0.5
06
0.
25
0.22
-85
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
EFLECTION COEFFICIENT IN
R
D
E
G
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
6
50
1
0.2
4
0.0
0
-15 -80
30
0.2
0
-4
-30
0.3
Forbidden Area
9
0.2
30
0.4
0.2
3
60
7
35
0.1
4
0.2
0.2
70
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
W
ERA
O
T
0.48
± 180
HS
TO
T
0.47
G
17
N
0
R—
-170
ELE
V
0.47
>
A
W
160
<—
-90
90
-160
0.4
40
0.48
31
0.1
40
0.
0.
4
R
,O
o)
0.35
19
VG – 51
PSA-00527 — 10/21/2001
Z
3
0.4
0
13
80
2
0.3
7
0.36
0.
Z0
4
0.
-20
0.0
)
/Yo
(+jB
CE
AN
T
EP
SC
SU
VE
TI
I
C
PA
CA
120
110
90
8
0.1
0
-5
-25
2
0.4
0.4
100
3.0
1
0.4
0.37
4.0
8
0.0
0.4
0.13
0.38
-15
9
0.12
5.0
0.0
0.39
-10
0.11
50
0.1
0.14
0.15
0.3
0.1
6
0.3
3
7
0.1
2
0.1
8
3.0
15
10
(f)
Pacificon 2001
0.4
0.39
0.38
0.37
0.36
5
-4
0.4
9
2
0.9
1.2
8
0.6
0.4
1.6
-60
0.7
1.4
0.8
-55
1.0
0
-5
1
0.4
0.1
-90
0.11
-100
0.12
0.13
0.35
0.14
-80
0.15
0.3
4
0.0
-110
0.0
-35
-4
0
-70
6
0.1
0
-65 .5
1.8
2.0
1
0.
3
0.3
3
0.0
7
-1
30
-12
0
CAP
AC
ITI
V
E
RE
AC
TA
NC
E
0.2
-70
R
O
),
Zo
/
X
(-j
CO
M
PO
N
EN
T
E
IV
CT
DU
IN
-75
0.8
40
1.0
-1
0.6
06
0.
9
44
1.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
0.
8
0.2
0.
Forbidden Area
0.3
0.
19
-60
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
80
15
0
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
4.0
1.0
5.0
0.28
0.2
0
-4
-30
)
/Yo
(-jB
CE
AN
T
EP
SC
SU
0.8
0.22
5
(+
jX
/Z
14
5
0.0
5
0.4
20
-20
0.3
1
-30
0.0
0.6
0.1
0.4
0.2
20
10
0.0
4
0.6
0.2
5
0.4
8
0.2
20
0.
0.4
6
0.3
85
0
70
44
0.
0.5
06
0.
25
0.28
-85
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
EFLECTION COEFFICIENT IN
R
D
E
G
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
6
50
0.22
4
0.0
0
-15 -80
30
1
0.2
9
0.2
30
0.4
0.2
3
60
7
35
0.1
4
0.2
0.2
70
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
W
ERA
O
T
0.48
± 180
HS
TO
T
0.47
G
17
N
0
R—
-170
ELE
V
0.47
>
A
W
160
<—
-90
90
-160
0.4
40
0.48
31
0.1
40
0.
0.
4
R
,O
o)
0.35
19
VG – 52
PSA-00527 — 10/21/2001
Z
3
0.4
0
13
80
2
0.3
7
0.36
0.
Z0
4
0.
-20
0.0
)
/Yo
(+jB
CE
AN
T
EP
SC
SU
VE
TI
I
C
PA
CA
120
110
90
8
0.1
0
-5
-25
2
0.4
0.4
100
3.0
1
0.4
0.37
4.0
8
0.0
0.4
0.13
0.38
-15
9
0.12
5.0
0.0
0.39
-10
0.11
50
0.1
0.14
0.15
0.3
0.1
6
0.3
3
7
0.1
2
0.1
8
3.0
15
10
(g)
Pacificon 2001
0.4
0.39
0.38
0.37
0.36
5
-4
0.4
9
2
0.9
1.2
8
0.6
0.4
1.6
-60
0.7
1.4
0.8
-55
1.0
0
-5
1
0.4
0.1
-90
0.11
-100
0.12
0.13
0.35
0.14
-80
0.15
0.3
4
0.0
-110
0.0
-35
-4
0
-70
6
0.1
3
0
-65 .5
0.0
7
-1
30
1.8
2.0
-12
0
CAP
AC
ITI
V
E
RE
AC
TA
NC
E
1
0.
3
0.3
-70
R
O
),
Zo
/
X
E
IV
CT
DU
IN
-75
0.8
40
1.0
-1
(-j
CO
M
PO
N
EN
T
06
0.
0.6
44
9
0.
1.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
0.
8
0.2
0.
19
-60
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
80
15
0
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
1.0
5.0
0.28
0.3
-30
)
/Yo
(-jB
CE
AN
T
EP
SC
SU
4.0
0.22
5
0.8
-20
0.3
1
-30
0.0
0.6
0.1
0.4
0.2
20
10
0.0
4
(+
jX
/Z
14
5
0.0
5
0.4
20
0.2
5
0.4
8
0.2
20
0.
0.4
6
0.6
0.28
85
0
70
44
0.
0.5
06
0.
25
0.22
-85
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
EFLECTION COEFFICIENT IN
R
D
E
G
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
6
50
1
0.2
4
0.0
0
-15 -80
30
0.2
0
-4
0.2
0.3
Forbidden Area
9
0.2
30
0.4
0.2
3
60
7
35
0.1
4
0.2
0.2
70
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
0.49
AD <—
GEN
ARD LO
W
ERA
O
T
0.48
± 180
HS
TO
T
0.47
G
17
N
0
R—
-170
ELE
V
0.47
>
A
W
160
<—
-90
90
-160
0.4
40
0.48
31
0.1
40
0.
0.
4
R
,O
o)
0.35
19
VG – 53
PSA-00527 — 10/21/2001
Z
3
0.4
0
13
80
2
0.3
7
0.36
0.
Z0
4
0.
-20
0.0
)
/Yo
(+jB
CE
AN
T
EP
SC
SU
VE
TI
I
C
PA
CA
120
110
90
8
0.1
0
-5
-25
2
0.4
0.4
100
3.0
1
0.4
0.37
4.0
8
0.0
0.4
0.13
0.38
-15
9
0.12
5.0
0.0
0.39
-10
0.11
50
0.1
0.14
0.15
0.3
0.1
6
0.3
3
7
0.1
2
0.1
8
3.0
15
10
(h)
Pacificon 2001
-90
0.4
0.38
0.39
0.12
0.13
0.37
0.36
0.35
-4
0.0
2
0.4
9
0.9
1.2
0.0
7
30
0.6
0.4
3
1.6
-60
0.7
1.4
0.8
-55
1.0
50
5
-
1
0.4
0.1
0.11
-100
0.14
-80
-110
0
-4
-70
6
0.1
0.15
4
0.3
8
0.0
-35
0
-65 .5
1.8
5
-85
2.0
0.0
1.0
10
<
o)
jB/Y
E (NC
TA
EP
C
4
4
-75
US
0.
40
ES
-1
06
IV
0.
CT
DU
IN
R
-70
O
),
Zo
X/
5
0.6
-1
0.4
0.8
(-j
CO
M
PO
N
EN
T
-12
0
CAP
AC
I
T
IVE
RE
AC
TA
NC
E
0.2
4
0.0
0
-15 -80
20
6
0.4
80
>
0.0
4
0.4
6
15
0
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
4.0
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
85
0.8
0.3
-60
0.2
> WAVELEN
GTHS
TOW
ARD
0.49
GEN
ERA
0.48
TO
170
R
0.47
160
90
(+
jX
/Z
14
5
0.0
5
0.4
20
Finish
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.2
9
0.
31
7
0.1
0.4
0.0
0.6
0.2
0.0
0.6
1.0
5.0
0.28
0.3
0.2
1
-30
-30
8
± 180
0
70
0.4
0.22
1.0
4
0.
0.49
0.
44
65
0.5
0.
06
25
0.
19
3
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.3
0.28
D<
RD LOA
TOWA
THS
-170
ENG
VEL
WA
-90
-160
30
-20
0
0.1
60
0.3
Start
0.1
10
20
50
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.
4
0.22
20
0.48
0.3
9
0.2
30
0.2
35
2
0.3
0.2
70
1
0.2
0.1
40
8
0.1
0
-5
-25
)
/Yo
(+jB
CE
AN
PT
CE
S
SU
0.35
0.2
-4
0.4
VE
TI
CI
PA
CA
80
0.2
4
0
0.36
0.3
0.
R
,O
o)
12
-20
3.0
3
0.4
0
13
2
0.13
4.0
0.4
0.37
-15
7
0.0
110
90
5.0
1
0.4
-10
8
0.0
0.4
0.38
40
VG – 54
PSA-00527—10/21/2001
9
0.12
31
0.
ZL
0.0
0.39
100
50
0.11
0.
ZL
0.1
19
0.
0.47
Matching: Four L-Networks Using Lumped Elements
Γi
0.14
0.15
0.3
0.1
6
7
ZL
3
0.3
2
50
0.1
8
3.0
15
10
Γr
ZL
Pacificon 2001
0.4
0.38
0.39
0.12
0.13
0.37
0.36
0.35
0
0.0
9
5
-4
0.9
1.2
1.6
-60
0.7
1.4
0.8
-55
1.0
0
-5
0.4
1
0.1
0.11
-100
-90
0.14
-80
-110
-4
0
-70
6
0.1
0.15
.42
3
0.6
0.4
0
-65 .5
1.8
2.0
0.0
5
1.0
)
/Yo
(-jB
CE
AN
PT
CE
-75
US
0
0.
S
4
E
-1
06
IV
0.
CT
DU
IN
R
-70
,O
)
Zo
X/
-12
0
CAP
A
C
ITI
VE
RE
AC
TA
NC
E
31
4
0.3
8
44
0.
0.0
(-j
CO
M
PO
N
EN
T
19
-35
20
10
80
4
0.0
6
15
0
0.4
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
4.0
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
0.8
0.3
0.0
7
-1
30
5
0.4
9
0.6
>
0
(+
jX
/Z
14
0.0
5
0.4
5
20
Finish
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.2
-20
85
0.6
1.0
5.0
0.28
0.3
1
-30
0.2
-85
70
0.
44
65
0.5
0.
06
25
0.2
0.
-60
0.2
<
0.4
0.22
0.8
0.2
6
0.6
4
0.0
0
-15 -80
8
0.4
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
0.1
-20
0
-30
0.1
0.4
> WAVELEN
GTHS
TOW
ARD
0.49
GEN
ERA
0.48
TO
170
R
0.47
160
90
0.
0.0
0.3
10
20
50
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
1.0
4
0.
0.0
50
0.22
20
± 180
30
0.28
0.49
0.3
9
0.48
D<
RD LOA
TOWA
THS
-170
ENG
VEL
A
W
-90
-160
0.1
1
-4
7
Start
0.1
0.2
0.3
3
0.3
4
0.3
60
2
35
0.3
70
0.2
0.
0.35
0.2
30
0.2
40
0.2
0.1
0.36
0.2
0.4
0
80
8
0.1
0
-5
-25
Yo)
jB/
E (+
NC
TA
EP
C
S
SU
VE
TI
CI
PA
CA
12
0.13
0.3
4
R
,O
o)
2
3.0
0.4
0.37
4.0
0
13
3
0.4
40
VG – 55
PSA-00527—10/21/2001
0.
7
0.4
90
-15
0.0
8
110
0.38
5.0
0.0
1
0.4
0.12
31
0.
ZL
9
0.0
0.39
100
-10
0.11
50
ZL
0.1
19
0.
0.47
Matching: Four L-Networks Using Stubs
Γi
0.14
0.15
0.1
6
7
ZL
3
2
8
3.0
15
10
Γr
ZL
Pacificon 2001
Reactance & Susceptance of Lumped Elements
Inductors
X L = 2πfL
−1
BL =
2πfL
−1
XC =
2πfC
BC = 2πfC
Capacitors
VG – 56
PSA-00527—10/21/2001
Pacificon 2001
Reactance & Susceptance of Stubs
❏ Assuming Zo is real, then
❏ Shorted Stubs
Xshorted
 2πlf 
= Zo tan

 vf c 
Bshorted
−1
=
 2πlf 
Zo tan

v
c
 f 
❏ Open Stubs
Xopen
VG – 57
PSA-00527—10/21/2001
− Zo
=
 2πlf 
tan

v
c
 f 
Bopen
 2πlf 
1
=
tan

Zo
 vf c 
Pacificon 2001
Broadband Matching Network Design Recipe
Using 4-Element π-Resonant and T-Resonant Networks
❏ Putting network insertion point close to load (antenna) gives
greatest SWR bandwidth
❏ Step 1: Using an L-network, move the midband impedance
point to prime point A or B. Bandwidth will be maximized if
the minimum reactance or susceptance L-network is chosen
in this step.
❏ Step 2: Wrap the impedance locus into the SWR circle by
adding a series or parallel resonant circuit as required to
complete the π-resonant or T-resonant network
VG – 58
PSA-00527—10/21/2001
Pacificon 2001
∏-Resonant
and T-Resonant Network
For Moderate and Broadband Matching
Γi
0.
0
44
70
0.
0
4.0
15
45
1.0
50
0.9
55
1.0
5.0
SWR = 1.5
10
8
0.6
fmid
0.2
0.1
0.4
B
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
A
10
20
0.2
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Γr
50
0.2
50
0.1
ZL
20
IND
UCT
IVE
0.28
1.0
0.
80
0.3
0.8
0.22
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
0.4
5
(+
jX
/Z
5
14
4
0.0
0.
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
20
0.4
10
1.0
E
IV
CT
DU
IN
1.0
0.8
2.0
1.8
1.6
0.35
1.2
1.4
0.15
-4
0
0.14
-80
0.36
1.0
-70
0.9
6
5
-4
4
0
0.1
-5
-35
0.3
0.7
-60
0.8
3
7
-90
0.12
0.13
0.38
0.37
0.11
-100
0.39
-55
0.1
0.3
0.6
2
-110
0.1
0.4
0.0
1
-75
6
-70
-1
40
0
0.
0.0
7
30
3
8
0
9
0.4
-1
0.4
0.0
5
R
O
),
Zo
X/
-12
0
CAP
AC
I
T
IVE
RE
AC
TA
NC
E
0.2
-30
-60
0.3
(-j
0.5
1
-65
0.
3
4
9
4
0.
CO
M
PO
N
EN
T
0.4
0.
1
8
0.1
0
-5
-25
0.0
0.6
-20
-85
E
NC
TA
EP
SC
SU
4.0
3.0
0.3
0
5.0
4
0.0
0
-15 -80
0.2
9
0.2
1
-30
0.2
4
0
0.
ZL
0.28
fu
0.3
-4
.45
8
0.22
0.47
0.
-20
0.2
-15
)
/Yo
(-jB
0.6
-10
0.0
4
20
3.0
0.6
9
0.2
30
0.4
6
15
0
25
1
0.2
85
fl
0.4
0.2
6
0.4
0.1
8
0.3
2
50
0.3
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.48
0.0
—
0.49
GEN
D LOAD <
ERA
OWAR
0.48
± 180
HS T
TO
170
0
NGT
R—
-17
E
L
E
0.47
>
AV
W
160
<—
-90
90
-160
7
3
40
.42
VG – 59
PSA-00527—10/21/2001
0.3
30
1
ZL
60
0.2
0.1
ZL
35
3
0.
VE
TI
CI
PA
CA
0.1
9
ZL
R
,O
o)
6
4
1
0.
65
0.5
0
6
13
Yo)
jB/
E (+
NC
TA
EP
SC
SU
0.3
2.0
0
0.1
70
1.6
12
ZL
0.15
0.35
40
1.8
2
0.14
0.36
80
1.4
0.4
110
0.8
.08
0.6 60
0
1
0.4
7
0.0
3
0.4
9
90
0.7
0.0
0.37
0.38
0.39
100
0.4
1.2
0.1
0.13
0.12
0.11
ZL
ZL
Pacificon 2001
-4
5
0.9
1.2
1.0
0
-5
-4
0
0.4
0.38
0.37
0.39
0.12
0.11
-100
-90
1.4
0.8
-35
0.14
-80
0.15
4
0.3
0.7
1.6
-60
-70
6
0.1
3
0.3
1.8
2.0
0.5
31
0.13
-55
0.6
-65
0.
0.36
44
0.
06
0.
19
0.35
-60
-30
-70
E
IV
CT
DU
IN
-75
R
O
),
Zo
X/
40
-1
)
/Yo
(-jB
CE
AN
PT
CE
S
SU
1.0
105
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
IND
UCT
IVE
0.2
20
10
80
0.0
4
1.0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
4.0
0.3
0.2
0.1
0.4
1
-110
0.0
9
0.4
2
0.0
-12
8
0
CAP
0.4
AC
I
3
T
IVE
0.0
RE
7
A
C
-1
TA
30
NC
EC
OM
PO
N
EN
T
(-j
0.4
5
0.0
0.8
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.4
0.6
8
0.
0.2
9
100
0.2
0.
7
0.1
0.6
0.4
6
150
(+
jX
/Z
14
0
0.0
5
0.4
5
20
1.0
5.0
0.28
110
-4
0
2
1.0
85
0.6
0.22
5
0.4
0.6
0.2
-85
0.
8
4
0.0
0
-15 -80
70
44
0.
2.0
65
0.5
06
0.
25
-20
6
0.4
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
30
0.3
0.2
8
0.1
0
-5
-25
60
0.28
4
0.
0.2
1
-30
120
35
10
20
50
0.1
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
130
0.3
4
0.22
20
0.4
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
D <—
0.49
GEN
RD LOA
ERA
TOWA
0.48
±
S
180
H
TO
170
70
NGT
R—
-1
E
L
VE
0.47
>
A
W
1
0
6
<—
6
0
-90
90
-1
0.3
9
0.2
30
0.3
70
1
0.2
140
40
0.2
0.2
0.35
0.3
150
3.0
0.
4
0.4
40
0.48
31
160
0.36
0.
0.1
80
19
VG – 60
PSA-00527—10/21/2001
R
,O
o)
0
90
-20
3
0.4
0
13
)
/Yo
(+jB
CE
AN
T
P
CE
US
ES
IV
IT
C
PA
CA
12
0.8
2
0.4
110
0.37
4.0
7
0.0
1
0.4
0.13
0.38
-15
8
0.0
0.4
0.12
5.0
9
0.0
0.39
100
-10
0.11
50
0.1
0.
0.47
Caron’s Example 10: VHF Folded Blade Antenna
0.14
0.15
0.1
6
0.3
3
0.1
7
0.3
2
50
0.1
8
3.0
15
10
Pacificon 2001
-5
5
-4
0.15
0.39
0.4
0.38
0.37
0.36
-90
0.11
-100
0.12
0.13
-55
0.9
-60
1.2
0.7
1.4
0.8
1.0
0
-70
4
0.3
6
0.1
3
0.3
-35
7
0.1
0.14
-80
-4
0
0.35
0.6
-65
1.6
1.8
2.0
0.5
0.
31
-60
-70
-1
E
IV
CT
DU
IN
-75
R
O
),
Zo
X/
40
1.0
o)
jB/Y
E (NC
TA
P
E
SC
SU
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1.0
4.0
50
80
15
0
RE
AC
TA
75
NC
EC
OM
PO
N
EN
T
0.8
20
10
5.0
4.0
3.0
2.0
1.8
IND
UCT
IVE
0
(+
jX
/Z
14
0.0
5
0.4
5
20
0.3
0.
19
0.2
0.1
0.4
1
-110
0.0
9
0.4
2
0
.
-12
0
8
0
CAP
0.4
AC
I
3
T
IVE
0.0
RE
7
AC
-1
T
A
30
NC
EC
OM
PO
N
EN
T
(-j
06
0.
0.8
44
0.
0.2
0
-4
0.4
-80
1.0
5
0.0
8
4
0.0
0
0.6
1.0
5.0
0.1
0.4
120
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
140
0.6
0.
8
0.28
-15
70
0.
44
0.5
0.
06
25
0.2
9
5
0.4
65
2.0
1.8
1.6
55
1.2
45
50
1.0
0.9
0.8
1.4
0.7
0.6 60
30
0.22
10
6
0.4
0.
0.2
0.0 —> WAVELE
0.49
NGTH
S TOW
ARD
0.0
—
0.49
GEN
D LOAD <
ERA
OWAR
T
0.48
±
S
180
TO
TH
G
17
70
N
0
R—
-1
ELE
V
0.47
>
WA
0.0
1
—
60
<
4
-90
90
-160
0.4
85
-85
6
0.3
0.48
0.1
7
-20
0.3
0.2
1
-30
4
0.
0.3
3
10
20
50
0.25
0.2
6
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
EES
ANG
0.2
130
-30
0.2
2
0.3
0.4
60
-25
0.1
35
0.28
150
0.3
4
0.22
20
105
70
1
0.2
9
0.2
30
100
40
0.2
160
0.35
0.3
0.6
0.
4
0.4
40
110
0.36
8
0.1
0
-5
)
/Yo
(+jB
CE
AN
PT
E
SC
SU
-20
3.0
E
IV
IT
AC
P
CA
80
-15
7
0.0
90
4.0
0
0.37
31
0.
VG – 61
PSA-00527—10/21/2001
R
,O
o)
12
0.13
0.38
5.0
2
0.4
110
0.12
-10
3
0.4
0
13
1
0.4
20
8
0.0
0.4
0.2
9
0.0
0.39
100
50
0.11
0.6
0.1
19
0.
0.47
Caron’s Matching Solution
0.14
0.15
0.1
6
0.1
8
Max VSWR=1.67
0.3
2
50
3.0
15
10
Pacificon 2001
WinSmith Display
Max VSWR=1.59
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ARRL Radio Designer
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ARRL Radio Designer
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Caron’s Example 11: Long Wire Receiving Antenna
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Pacificon 2001
Caron’s Solution
Max VSWR=6.2
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Caron’s Solution
Max VSWR=6.2
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Eagleware’s Solution 1
Max VSWR=5.05
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Eagleware’s Solution 2
Max VSWR=4.77
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Eagleware’s Solution 3
Max VSWR=6.59
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Eagleware’s Solution 4: Fano Limit
Max VSWR=3.42
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GM3HAT 4-Band Dipole of Delight
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GM3HAT Feedpoint Impedance
At Five Harmonic Frequencies
VSWR=11.02
VSWR=1.62
VG – 73
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Pacificon 2001
K6OIK 80m Suboctave Band Matching Network
150Ω
89°@ 3.5 MHz
5.9 µH
25Ω
91°@ 3.5 MHz
VG – 74
PSA-00527—10/21/2001
18.8 µH
Zant
Pacificon 2001
K6OIK’s Multiband Match Performance
VSWR=1.46
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Pacificon 2001
Software for Smith Charting and Network Design
❏ Smith Chart Analysis & Display
➤ MicroSmith 2.3, ARRL, 1992, $39
A primitive DOS program.
➤ winSMITH 2.0, Noble Publishing, 1995, $79
Written by Eagleware. Easy to use. Restricted to ladder networks.
Doesn’t have series stubs. Lacks an optimizer.
❏ Matching Network Optimization & Synthesis
➤ ARRL Radio Designer 1.5, ARRL, 1995, $150
No longer sold. ARD’s optimizer works with Serenade netlists and
handles more variables than Serenade SV’s optimizer.
➤ Serenade SV (student version), Ansoft, 2000, $0 (free)
Download (about 20 Mbytes) from:
http://www.ansoft.com/about/academics/sersv/index.cfm
➤ Advanced Automated Smith Chart 3.0, Artech House, 1998, $395
➤ =MATCH=, Eagleware, $699 (requires GENESYS Basic, $1997)
➤ Harmonica Linear Design Suite, Ansoft, 2000, $6900
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References: Articles
❏ Robert L. Thomas, “Broadband Impedance Matching in High-Q
Networks,” EDN, pp. 62–69, December 20, 1973.
❏ Neal C. Silence, “The Smith Chart and Its Usage in RF Design,”
RF Design, pp. 85–88, April 1992.
❏ Thomas R. Cuthbert, Jr., “Broadband Impedance Matching Methods,”
RF Design, pp. 64–91, August 1994.
❏ Thomas R. Cuthbert, Jr., “Broadband Impedance Matching - Fast and
Simple,” RF Design, pp. 38–50, November 1994.
❏ William E. Sabin, “Broadband HF Matching with ARRL Radio
Designer,” QST, pp. 33–36, August 1995.
❏ William E. Sabin, “ARRL Radio Designer and the Circles Utility, Part 1:
Smith Chart Basics,” QEX, pp. 3–9, Sept/Oct 1998.
❏ William E. Sabin, “ARRL Radio Designer and the Circles Utility, Part 2:
Small-Signal Amplifier Design,” QEX, pp. 3–11, Nov/Dec 1998.
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References: Articles (Cont’d)
❏ Steve Sparks, “A Practical Amateur Application of the Smith Chart,”
Communications Quarterly, pp. 102–106, Summer 1999.
➤ This article contains serious Smith charting errors. See the comments by Garry
Shapiro, NI6T, Communications Quarterly, p. 3, Fall 1999.
❏ K.C. Chan and A. Harter, “Impedance Matching and the Smith Chart –
The Fundamentals,” RF Design, pp. 52–66, July 2000.
VG – 78
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References: Books
❏ Robert A. Chipman, Transmission Lines, Schaum Outline Series,
McGraw-Hill, 1968.
➤ Basic, mathematical. A classic, but out-of-print.
❏ Robert L. Thomas, A Practical Introduction to Impedance Matching,
Artech House, 1976, ISBN 0890060509.
➤ Intermediate, mathematical. A nice treatment of 4-element networks for wideband
matching.
❏ Pieter L. D. Abrie, The Design of Impedance-Matching Networks for
Radio-Frequency and Microwave Amplifiers, Artech House, 1985,
ISBN 0890061726.
➤ Advanced, mathematical.
❏ Wilfred Caron, Antenna Impedance Matching, ARRL, 1989,
ISBN 0872592200.
➤ Intermediate, non-mathematical. Caron illustrates manual Smith chart methods at
their best, but which nonetheless have been completely replaced by computeraided network design.
❏ Brian C. Wadell, Transmission Line Design Handbook, Artech House,
1991, ISBN 0890064369.
➤ Analysis and formulas for many transmission lines. Comparable to M.A.R. Gunston
or K.C. Gupta, et al., below.
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References: Books (Cont’d)
❏ Phillip H. Smith, Electronic Applications of the Smith Chart in
Waveguide Circuit, and Component Analysis, 2nd edition, Noble
Publishing, 1995, ISBN 1884932398.
➤ Originally published by McGraw-Hill, 1969, and reprinted by Krieger, 1983.
Intermediate, mathematical.
❏ K.C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and
Slotlines, 2nd ed., Artech House, 1996, ISBN 089006766X.
➤ Analysis and formulas for many transmission lines.
❏ M.A.R. Gunston, Microwave Transmission-Line Impedance Data,
Noble Publishing, 1997, ISBN 1884932576.
➤ Originally published by Van Nostrand Reinhold, 1972. Analysis and formulas
for many transmission lines. Comparable to B. Wadell above.
❏ ARRL Antenna Book, 18th edition, chapters 24-28, ARRL, 1997,
ISBN 0872596133.
➤ Elementary, non-mathematical.
❏ M. Walter Maxwell, W2DU, Reflections II: Transmission Lines and
Antennas, 2nd ed., Worldradio Books, 2001, ISBN 0970520603.
➤ A non-mathematical treatment of transmission lines and matching that
examines and corrects common misunderstandings.
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References: Application Notes, Videos, and Web Sites
❏ Application Notes
➤ Times Microwave’s Complete Coaxial Cable Catalog & Handbook
Download from:
http://www.timesmicrowave.com/products/military/TL14/TL14.htm
❏ Videos
➤ Glenn Parker, Introduction to the Smith Chart, Noble Publishing,
1996. 50 minutes, $99.
❏ Useful Web Sites
➤ http://www.timesmicrowave.com/calculators/index.htm
➤ http://www.sss-mag.com/smith.html
➤ http://www.ee.surrey.ac.uk/Personal/D.Jefferies/smith.html
➤ http://www.scott-inc.com/html/smith.htm
❏ This Presentation is at
➤ http://www.fars.k6ya.org/docs/smithchart.html
VG – 81
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