Mysteries of the Smith Chart
Transmission Lines, Impedance Matching,
and Little Known Facts
Stephen D. Stearns, K6OIK Chief Technologist TRW Firestorm Wireless Communication Products [email protected]
VG Ð 1 Pacificon 2001 PSA-00527 — 10/21/2001 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 Ð 2 Pacificon 2001 PSA-00527—10/21/2001 Part 1: Transmission Line Theory
VG Ð 3 Pacificon 2001 PSA-00527 — 10/21/2001 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 VG Ð 4 Early Smith Chart, published 1939 and 1944 - Smith Pacificon 2001 PSA-00527—10/21/2001 Numbers to Remember!
1.4142135623... 1.7320508075… 1.6180339887... 3.1415926535... 2.718281828459045…
2.54 299,792,458 376.7303134...
VG Ð 5 Pacificon 2001 PSA-00527—10/21/2001 Heaviside’s Vector Formulation of Maxwell’s Theory
∂B ∇×E =− ∂t D∂ ∇×HJ = + t∂ ∇⋅D = ρ ∇⋅B0 = DE= ε BH= µ
“And God said, Let there be light; and there was light.” Genesis 1:3
VG Ð 6 Pacificon 2001 PSA-00527—10/21/2001 Frequency Domain or Phasor Form
∇×EH =−jωµ ∇×HE =()σωε + j ∇⋅E0 = ∇⋅H0 =
VG Ð 7 Pacificon 2001 PSA-00527—10/21/2001 Heaviside’s Telegrapher’s Equations
Uniform transmission line Equivalent circuit of infinitesimal segment
I(x) R∆xL∆x V(x) G∆xC∆x
dV =−()()RjLIx + ω dx dI =−()()GjCVx + ω dx
VG Ð 8 Pacificon 2001 PSA-00527—10/21/2001 Transmission Line Solution TEM Waves
Traveling wave γx Vx()=Vo e Vx() Ix()= Zo Propagation constant
γα=+ βjRjLGjC ω= ω()() + +
Characteristic impedance
RjL+ ω Z = o GjC+ ω
VG Ð 9 Pacificon 2001 PSA-00527—10/21/2001 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) = Velocity factor (dimensionless) f v X = Reactance (Ohms) B = Susceptance (Siemens) s = Standing wave radio (dimensionless)
VG Ð 10 Pacificon 2001 PSA-00527—10/21/2001 Notations (Cont’d) Complex Parameters
Z = R + jX = impedance (Ohms)
ZL = Load impedance (Ohms)
Zi = Input impedance (Ohms)
Z0 = Characteristic impedance (Ohms) z = Z/Z = r + jx = normalized impedance (dimensionless) Y = G + jB0 = admittance (Siemens) y = Y/Y Γ 0 = g + jb = normalized admittance (dimensionless) = Γ γ r + j Γi = complex reflection coefficient (dimensionless) = α + jβ = propagation constant (inverse meters)
VG Ð 11 Pacificon 2001 PSA-00527—10/21/2001 Transmission Line Parameters Physical Dimensions and Material Properties d dielectric dielectric (µ, ε, σ) (µ, ε, σ) a
b a a c Parameter Coax Twinlead 1 1 11+ πδσ R 2πδσ ab a c Ω/m c µ µ b 11 δ −1 d ln ++δ + cosh L H/m 22π aab π22a a 2πσ πσ
b −1 d G S/m ln cosh a 2a 2πε πε C F/m b − d ln cosh 1 a 2a σ =×58. 107 S/m Where skin depthδ For =is copper1 c f 85. mm at 60 Hz πµσc δ= VG Ð 12 6.6 m at 100 MHz Pacificon 2001 PSA-00527—10/21/2001 µ 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
VG Ð 13 Pacificon 2001 PSA-00527—10/21/2001 Comparison of Impedance Formulas Round Open-Wire Line
VG Ð 14 Pacificon 2001 PSA-00527—10/21/2001 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 Ð 15 Pacificon 2001 PSA-00527—10/21/2001 Round vs Square Open-Wire Lines
VG Ð 16 Pacificon 2001 PSA-00527—10/21/2001 Optimal Characteristic Impedances Coax
Ω For minimum loss Zo = 77 Ω For maximum breakdown voltage Zo = 30
Ω For minimum temperature rise Zo = 60
Ω Zo = 50 has no special significance
VG Ð 17 Pacificon 2001 PSA-00527—10/21/2001 Reflection Coefficient and Impedance Relation at a Terminal Plane
Terminal Plane
Ζ Ζ DefinitionL O Γ L ZZ− − Γ= Lo= z 1 + + ZZLoz 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 moves? Γ and z change as the terminal plane
VG Ð 18 Pacificon 2001 PSA-00527—10/21/2001 Relations Between Two Terminal Planes
Input Output Terminal Terminal Plane Plane Impedance relation Ζ Ζ L O Γ L zj+ tan β l = L Ζ zi Γi 1+ jztan β l i L
Cross relations Reflection coefficient relation 1 + Γ e − jl2β = L zi − jl2 β 1 − Γ e − β L ΓΓ= 2jl iL e
1 + Γ e jl2 β z = i L − Γ jl2 β 1 i e
VG Ð 19 Pacificon 2001 PSA-00527—10/21/2001 Velocity Factor
Wavelength
λ = c free space f
v λ = actual f Velocity factor
v λ v == actual f λ c free space
VG Ð 20 Pacificon 2001 PSA-00527—10/21/2001 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 21πfl v = f c −Z cot −1 open Zshort
VG Ð 21 Pacificon 2001 PSA-00527—10/21/2001 Phase Constant
β ==22π f π radians/meter λactualvc f information
❏ Phase constant f give equivalent β and velocity factor v ❏ Both can be calculated from line dimensions and material properties βωω=++Im (RjLGjC )( )
❏ Best to measure!
VG Ð 22 Pacificon 2001 PSA-00527—10/21/2001 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 =× ZZZo open short
❏ Geometric mean of two complex numbers ❏ Calculation is trivial in polar form on Smith Chart
VG Ð 23 Pacificon 2001 PSA-00527—10/21/2001 What Special Lengths of Lossless Line Do
Half wavelength /2 , l = _ = ZZiL
Quarter wavelength, /4 Z 2l = _ = o Zi Z L Eighth wavelength, /8 l = _ = ZZio if Z L and Z o are real (resistive)
VG Ð 24 Pacificon 2001 PSA-00527—10/21/2001 Standing Wave Ratio
Easy to remember from
1 + Γ 1+ Γ s = z = 1 − Γ 1− Γ
− z −1 Γ = s 1 Γ = s +1 z +1
VG Ð 25 Pacificon 2001 PSA-00527—10/21/2001 Part 2: The Smith Chart
VG Ð 26 Pacificon 2001 PSA-00527 — 10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 0 45 5 110 40 70 0.34
0.41 1.0
0.9 0.17 55 1.2
0.08 0.8 35 0.33 2 1.4 60
0.4 120 0.7 60 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 C 0.32 AN 0.43 PT 0.2 1.8 CE 5 30 65 S 0 1 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A 0 C 7 R 0 O 0.4 4
4 ), 0 0.2 1 0.4 o Z 0.05 / 0.3 jX 0 0.45 + 2 ( T
75 N 3.0 E 0.6 N 0.21 O P
30 0.29 0.04 0.3 M 0.46 150 O C > E 0.8 15 — C 80 4.0 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 2 0 0 E 6 E
85 G 1 V I
T 0.8 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3 0.4 0.5
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-2
0 E L
6
0
E C
-85
-1
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
<
S
U
4.0
S
-80 -15
0.8 E
0.21 V
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
N
0.6
I
R
3.0
O -75
,
)
0.2
0
o
-2
Z
/ 0.3 0.05
X
-4
j
0
-
( 4
0 0.45
0.4 T
-1 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44
C
-5
N 0 0.5
2.0
3 A 0
T
-1
C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
-60 0.17
-60 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-7
0 1.2 1 0 0.09 0
-1 0.9
1.0 -4 0
0.34
-5
-45
0.15 0.41
-80 0.1
-100
-90
0.14
0.35
0.11
0.4
0.13 0.12
0.36
0.39
0.37 0.38
VG Ð 27 Pacificon 2001 PSA-00527 — 10/21/2001 Complex Functions
zi wi Complex Number Complex Number y (x, y) v (u, v) z = x + jy w = u + jv ( ) z f w x r u r
❏ Basic types of complex functions ➤ Global Properties Ð Linear Ð lines map to lines Ð Bilinear ÐÊcircles map to circles ➤ Local Properties Ð Conformal ÐÊright angles map to right angles
VG Ð 28 Pacificon 2001 PSA-00527—10/21/2001 Mathematical Basis of the Smith Chart
z −1 Γ= A bilinear conformal + z 1 complex function
(-)+rjx1 ujv+= (+)+rjx1
x v
r u
Right Half Interior Unit Circle Z Plane Γ Plane
VG Ð 29 Pacificon 2001 PSA-00527—10/21/2001 Smith Chart: Impedance Coordinates Γ i
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0 .09 .16 0 45 50
1 110 40 70 0.3 .4 .9 4
0 1.0
0 0.17 55 1.2
0.08 0.8 35 0.33
.7 1.4 60
0.42 120 0 60 Yo) r=0 0.18
jB/ 1.6
.6 (+ 0.07 0 E 30 NC 0.32 TA 0.43 EP 0.2 1.8 65 SC 50 130 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A x=1 0.44 P A C 70 R O 0.4 40
), 0 140 0.4 o .2 .05 Z 0 / 0.3 jX 20 0.45 (+ T
75 N 3.0 E 0.6 N 0.21
O
P 0.29 0.3 30 0.04 M 150 O r=1 0.46 C > E 0.8 15 — C 80 4.0 R N x Series Reactance O A T 0 T .22 0 A C 7 Inductive 1.0 .2 R A
E 8
E 0.4 .0
.2 5 R 1.0 N 0 20 E E
85 G 160 V I
T Reactance 0.8 10 D r=2 C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 (Positive) N T G
G S L
x=1 E L
H
E
10 O T
0.1
.4 O F G 0
170 F N T 0 0
E R R .2 9 .2
L A E 4
.4 6
E F N 0
L V
20 S E A 0.2 M
C
I W T
S
I
S > 0 O 5 I r=0 r=1 r=2
x=0 O N
— 0.25 0.25 N ∞ x=0 C 10 20 50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0
C O 0.0 0.0
180 O
E Γ
±
F
E 0
5 F
F I
F r
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) C
—
I
C < I
E
0.2 I
N
D 20 E
N
0
T A
0
.2
T .2 9
O
I
4 N
L
6 .4
I
N
0
D
D
x=-1 D E
R 0.4
0.1 G E
A
-170 10
G
R
W
E R
O
E E
T
S 0.2 E
0.2
S
S ) 0.6
-90 3 o
H
7
Y
T
0.48
/
G
B j
-10 - N
(
E Capacitive 0.8
r Series Resistance -20
E
L
5
E C
-8
-160 .2 V N
0.22 0 .0
5
A
A 1.0 0.28
T
W
P Reactance 1.0
E
0.47
—
C
<
S
U
4.0
S
-80 -15
0.8
E (Negative)
0.21 V
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
N
0.6 I
R
3.0
O -75
,
)
0.2
o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
0.4 T
-140 0.4
N
E
N
-70 0.19
O
P
M x=-1 0.06
0.31
O
-25 C
E
0.44 C
-50
0.5 N
2.0
A
T
-130
C
0.18
A
0.2 -65 E
R
0.07 E
1.8
V
0.32 I
T
I
-30
.6 C
A
0.43 0 P
A
C
1.6
-60 0.17
-60 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0
.1
-55 -70
6 .09 1.2
0
-110 0.9
1.0 -40
0
-50 .3
-45 4 .41
0.15 0
-80 0.1
-100
-90
0
.1 0.35 4
0.11
0.4
0.13 0.12
0 .3 6
0.39
0.37 0.38
VG Ð 30 Pacificon 2001 PSA-00527—10/21/2001 Smith Chart: Admittance Coordinates Γ i
0.37 0.38 0.36 0.39 0.13 0.12 0.4 0.35 0.14 0.11 -90 -80 -100 0.1 4 0.15 -4 0.4 0.3 5 -50 1 -40 1.0
0.9 -110 0
.16 -70 1.2 - .09 0 0.8 5 5 0.42 -35 0.33 1.4
0.7 0.08 -60 -120 0.17 -60
1.6 CA PA 0.6 0.43 -30 CI 0.32 TI 1.8 VE 0.07 RE 0.18 0.2 A -65 g=0 CT -130
2.0 A -50 N 0.5 CE 0.44 C -25 O 0.31 M 0.06 P 0.19 O N -70 E
N
0.4 -140 T 0.4 0 (- .4 .3 -40 jX 0 / 0 5 Z .0 .2 -20 0 o 5 ), O -75 3.0 R 0.6 I b=-1 N D 0.46
U g=1 C 0.3 -150 0.04 0.29 T -30 I 0.21 V 0.8 E -15 -80 S 4.0 U
S < C Shunt Susceptance — b 0 E 8 1 P .4 .0 W .2 2 T 7
0
1.0 A
.2 5.0 Inductive A 0 0.2 N -160 V -8 C E
g=2 5 E L
-20 0.8 E ( -10 Susceptance - N j B G / 0.48 Y T
H
o -90 0.6 ) S 0.27 S (Negative)
0.23 S E T
E E O
E R W b=1 R
G
10 -170 A E G 0.1 0.4 R E D
D D
N 0
I 6 L .4 4 N
I O .2 T 9 .2 0 T A 0 N
E 20 D N I 0.2
E
I C <
I
— C (G/Yo) COMPONENT CONDUCTANCE OR (R/Zo), COMPONENT RESISTANCE
F
I
F F 50
E g=0 g=1 g=2 F
b=0 ±
E
O
b=0 180 0.0 0.0
O ∞ C 0
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
C N
0.25 0.25
—
N
g O Γ I 5
O 0
>
S
I
S
T
W r I
C
M 0.2 A E
S 20
V
L
0
N
F
E
.4 6
4
E
L A
9 .2
E R
.2 R
0
0
N
T
F
170
0.4 G
b=-1 F
O 0.1
T O 10
E
H
L E
S
L
G
T G N
90 0.48
O
0.6 I
N
N A
W
A
D
0.27
A
U
0.23
R
C
D
10 0.8 T
I
G
V
160
85
Shunt Conductance
E E
N
20 0.2
1.0 R 5.0
0
E
E
.4 8
R
Capacitive A
1
7 .2 2
A
.0 C
0
.2
T
T
0
O A
N R
4.0 Susceptance
80
C
—
0.8
E 15
>
C
0.46
O 150
(Positive) M 0.04
30 0.3
0.29 P
O
0.21 N
0.6
E
3 7
N
.0
5
T
b=1 ( +
0
20
j .4
X
.3
5 /
0 0
Z
.0
140
o .2
0.4 5 )
0 ,
0.4 40 O
R
70
C
A
P 0.44
A
C
25
I 0.31 0.06 T
I
V 0.5
E 2.0
0.19
S 130 U
S
65
C
50
E
0.43
P 0.2 1.8
T
A
N
0.32
C
0.07 30
E 0.6
(
+
j
B
1.6 / 0.18
Y
o
)
60
120
0.7 0.42 60
1.4 0.33
35
0.8 0.08
1.2 5
0.17
0 5
1.0 0
.4
4 .9
110
40 1 .3
0 70
50
4
0 5
.0 6
9 .1
0.4 0
100 80
0.35
90
0
.39 6
.3
0.1 0
0.15 0.38 0.37
0 .1 4 1 .1
0
0.12 0.13
VG Ð 31 Pacificon 2001 PSA-00527—10/21/2001 Smith Chart: Constant SWR Circles
Γi
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 0 45 5 110 40 70 0.34
0.41 1.0
0.9 0.17 55 1.2
0.08 0.8 35 0.33
1.4 60 0 0.7 0.42 12 0 6 ) Yo 0.18 jB/ 1.6
(+ 0 0.07 0.6 E 3 NC 0.32 TA 0.43 EP 0.2 1.8 65 SC 50 130 SU
E 2.0 0 6 0.5 .1 .0 IV 9 0 IT 0 4 C 25 .3 .4 A 1 0 P A C 70 R O 0.4 4
40 ), 0 0.2 1 0.4 o Z 0.05 / 0.3 jX 20 0.45 (+ T .0 75 N 3 E 0.6 N 0.21
O
P 0.29 0.3 3 0.04 0 5 M 0 0.46 1 O C > E 0.8 15 — C .0 80 4 R N O A 0.22 T T C .0 0.28 A 1 R A E
E 0.47 5.0 R 1.0 N 0.2 20 E E
85 G 160 V I
T D 0.8 10 C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L
H
E
10 O T
0.1
.4 O F G 0
170 F N T 0
0
E R R .2 9 .2
L A E 4
.4 6
E F N 0
L V
20 S E A 0.2 M
C
I W T
S
I
S > 0 O 5 I
O
N — Γ 0.25 0.25 N r
C
0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3 0.4
C O 0.0 0.0
180
O
E
±
F
E 0
5 F F
1.5 234 I
F
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) C
—
I
C < I
E
0.2 I
N
D 20 E
N
0
T A
0
.2
T .2 9
O
I
4 N
L
6 .4
I
N
0
D
D
D E
R 0.4
0.1 G E
A
-170 10
G
R
W
E R
O
E E
T
S 0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-20
E L
E C
-85
-160
V N
0.22 0.2 .0
5
A
A 1.0 0.28
T
W
.0
P 1
E
0.47
—
C
<
S
U
4.0
S
-80 -15
0.8 E
0.21 V
I -3
0
T
5 0.29
0
1 C
0.04
- 0.3 U
0.46 D
N 0.6 I
.0 R
3
O -75
,
)
0.2
o
-20
Z
/ 0.3
0.05
X -40
j
-
(
0.45
0.4 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25 C
E
0.44 C
-50
0.5 N
2.0
A
T
-130
C
0.18
A
0.2
-65 E
R
.8
0.07
E
1
V
0.32 I
0 T
I
-3 C
A
0.43
0.6 P
A
.6 C
1 -6 0
0
0.17
2 0
6
- -1
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55 -70
1.2
0.09
-110 0.9
1.0 -40
0.34
-50
-45
0.15 0.41
-80 0.1
-100
-90
0.14
0.35
0.11
0.4
0.13 0.12
0.36
0.39
0.37 0.38
VG Ð 32 Pacificon 2001 PSA-00527—10/21/2001 Smith Chart: Constant Impedance Magnitude Circles
Γ i
0.12 0.13 0.11 0.14 0.38 0.37 0.1 0.1 0.39 0.36 5 90 0.4 100 80 0.35 0.1 0.09 1 6 45 50 70 0.3 110 40 4
0.41 1.0
0.9 0.17 55 .8 1.2
0.08 0 35 0.33 1.4 6
0.42 120 0.7 0 60 ) Yo 0.18 jB/ 1.6
(+ 0 0.07 0.6 E 3 NC 0.32 A .8 T .2 1 2 0 0.430.5 EP C 50 30 65 S 1 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R O 0.4 40
), 0.2 140 0.4 o Z 0.05 / 0.3 jX 20 0.45 (+ T 5 7 N 3.0 E N 0.6 x 0.21
O
P 0.29 0.3 30 0.04 M 0.46 150 O C > E 0.8 15 — C .0 80 4 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 20 E E
85 G 160 V I
T D 0.8 10 C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N 2 T G
G S L
E L H
E
10 O T
0.1 O F G 0.4
170 F N T 0.24 0.26 1 R E R
A L E
E F N 0.49
L V
20 S E A 0.2 M
C
I W T
S
I
S >
O
50 I
0.5 O
N
— 0.25 0.25
N
C
10 20 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0
C O 0.0 0.0
180 O
E Γ
±
F E 50 F
r F I
F r
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) C
—
I
C < I
E
0.2 I
N
D 20 E
N
0.24
T A
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.4
0.1 G E
A
-170 10
G
R
W
E R
O
E E
T
S 0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-20
E L
E C
-85
-160
V N
0.2 0.22
5.0
A
A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
<
S
U
4.0
S
-80 -15
0.8 E
0.21 V
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
N
0.6
I
5 R
3.0
O -7
,
)
0.2
o
-20
Z
/ 0.3
0.05
X -40
j
-
(
0.45
.4 0.4 T
-140 0
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25 C
E
0.44 C
-5
0.5 N
2.0
A
0
T
-130
C
0.18
A
.2
0
-65 E
R
.8
0.07 E
1
V
0.32 I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6 -6
0.17
20 0
-60 -1
0.08 0.7
0.33
1.4 -35
0.42 .8
0 0.16
-55
.9 -70 1.2
0.09
-110 0
1.0 -40
0.34
-50
-45
0
0.41
.1
-80 .1 5 0
-100
-90
0
.1 1
0 4 .1
0
.3 .4
5 0
0.13 0.12
0 .3 9 6 .3
0
0.37 0.38
VG Ð 33 Pacificon 2001 PSA-00527—10/21/2001 Smith Chart: Constant Impedance Phase Angle Circles
Γi
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45 50
110 40 70 0.34 .9
0.41 1.0
0 0.17 55 1.2
0.08 0.8 35 0.33
.7 1.4 60
0.42 120 0 60 ) /Yo 0.18
(+jB 1.6 0.07 0.6 E 30 NC 0.32 TA 0.43 EP 0.2 1.8 65 SC 50 130 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R
O 0.4 40
.4 ), 0.2 140 0 o Z 0.05 / 0.3 jX 20 0.45 (+ T
75 N 3.0 E 0.6 N 0.21
O
P 0.29 0.3 30 0.04 M 0.46 150 O C > x E 0.8 15 — C .0 80 ° 4 R N 45 O A T 0 T .22 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 20 E E
85
G 160 V
I
° T .8 D 0 10 C R 0.23 U A 0.2 D A W
.48 N N A 7 90 I 0.6 O 0 135 90 N ° T G G S L
E L
H
0 E
1 O T .1
° 0 O F 45 G 0.4
170 F N T 0.24 0.26
E R R 9
A L E
E F N 0.4
L V
20 S E .2 A 0 M
C
I W T
S
I
S > 0 O 5 I
O
N — Γ
° 0.25 0.25 ° N r
° C 10 20 50
180 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0
C O 0.0 0.0
180
O
E
±
F
E 0
5 F
F
I
F
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) C
—
I
C < I
.2 E 0 I
0 N
D 2 E
N
0.24
T A
0.26
T
O
I
N
L
I
N
0.49 D
° 315 D
D E
R .1 0.4
0 G E
A 0
-170 1
G
R
W
E R
° ° O
E E
T S 0.23
225 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
270 (
E 0.8
-20
E
L
5
E C
-8
-160
V N
0.2 0.22
5.0
A
A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
<
S
U
4.0
S °
-80 -15
0.8
E 315
0.21 V
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
N
0.6 I
R
3.0
O -75
,
)
0.2
o
-20
Z
/ 0.3
0.05
X -40
j
-
(
0.45
0.4 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25 C
E
0.44 C
-50
0.5 N
2.0
A
T
-130
C
0.18
A
0.2 -65 E
R
0.07 E
1.8
V
0.32 I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
-60 0.17
-60 -120
0.08 .7
0
0.33
1.4 -35
0.42
0.8 0.16
-55 -70
1.2
0.09
-110 0.9
1.0 -40
0.34
-50
-45
0 0.41
.15
-80 .1 0
-100
-90
0.14
0
0.11
.35 .4
0
0.13 0.12
0.36
0.39
0.37 0.38
VG Ð 34 Pacificon 2001 PSA-00527—10/21/2001 Smith Chart: Multiplication, Division, Squares, and Square Roots
Unary Operators squares a 2 square roots a tangents tan Inversecotangents cotcotangents cot inverse tangents tan -1θ a -1 aθ Binary Operators multiplication a X ¥ b division ÷ geometric mean ab c/a
VG Ð 35 Pacificon 2001 PSA-00527—10/21/2001 Smith Chart: A Nomogram for Math Calculations Γ i
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0 .16 0.09 45 50 √ 7 0 110 40 c=1.4140 .34 0.41 1.0
5 0.9 0.17 5 1.2
0.08 0.8 35 0.33 1.4 60
0.42 120 0.7 60 ) Yo .6 0.18 7 jB/ 1
.0 (+ 0 0.6 CE 30 3 N 0.32 .4 TA 0 EP 0.2 1.8 65 SC 50 130 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R O 0.4 40
), 0 140 0.4 o .2 0.05 Z 0 / .3 jX 20 0.45 (+ T
75 N 3.0 E 0.6 N 0.21 O
P 0.29 0.3 a=43 0.04 M 0 0.46 150 O C > E 0.8 15 — C 80 4.0 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 20 E E
85 G 160 V I
T D 0.8 10 C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
0 E 1 O T
0.1 O F G 0.4
170 F N T 0.24 0.26
E R R 9
A L E
.4
E F N 0
L V
20 S E .2 A 0 M
C
I W T
S
I
S >
O
50 I
O
N
— 0.25 0.25 c=2 N
C
0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3 0.4 0.5
C O 0.0 0.0
180 O
E Γ
±
F E
50 F
F I
F r
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) C
—
I
C < I
.2 E
0 I
N
D 20 E
N
0.24
T A
0.26
T 9
O
I
N
L
.4
I
N
0
D
D
D E
R .1 0.4
0 G E
A 0
-170 1
G
R
W
E R
O
E E
T
S 0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-20
E L
E C
-85
-160
V N
0.2 0
5.0
.2
A
A 1.0 0
2 .2 T
W
P
1.0 8
E
0.47
—
C
<
S
U
4.0
S
-80 -15
0.8 E
0.21 V
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
N
0.6 I
R
3.0
O -75
,
)
0
o
.2 -20
Z
/ 0
0.05
.3 X -40
j
-
(
0.45
0.4 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25 C
E
0.44 C
-50
0.5 N
2.0
A
T
-130
C
0.18
A
7 0.2 -65 E
R
.0
0 E
1.8
V
0.32
I
T
3 I
-30 C
.4
A
0 0.6 P
A
C
1.6
-60 0.17
-60 -120
b=0.50.08
0.7
0.33
1.4 -35
0.42
0.8 0
.1
-55 -70
1.2 6
0.09
-110 0.9
1.0 -40
0
-50 .3
-45 4
0.15 0.41
-80
0.1
-100 √ -90
0.14
- 0.35
0.11 2
0.4
0.13 0.12
0.36
0.39
0.37 0.38
VG Ð 36 Pacificon 2001 PSA-00527—10/21/2001 Part 3: Impedance Matching
VG Ð 37 Pacificon 2001 PSA-00527 — 10/21/2001 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 Ð 38 Pacificon 2001 PSA-00527—10/21/2001 Bandwidth Classifications
Fractional Bandwidth Narrowband < 10% Moderate band 10% to 50% Broadband > 50%
VG Ð 39 Pacificon 2001 PSA-00527—10/21/2001 Two Kinds of Matching
❏ Conjugate Matching ❏ Load Matching
Ζ S
+ Ζ Ζ Ζ ∼ L O L -
Zs = ZL*ZL = ZO ❏ Best use at source ❏ Best used at ends of transmission (transmitter) lines ❏ Maximizes power delivery to ❏ Minimizes reflections the load ❏ Does not maximize delivered ❏ Does not minimize reflections power unless Z0 is real unless Zs is real ❏ Normally done by the transmitter manufacturer at the circuit design level ❏ Ideally Z (ext) = 50 s Ω VG Ð 40 Pacificon 2001 PSA-00527—10/21/2001 Where Should Matching Network Go?
❏ Poor Setup High to Low SWR High SWR
Antenna Tx Long Transmission Line Tuner
❏ Good Setup
Perfect Low1:1 SWR Low SWR Antenna Matching Tx Tuner Network ❏ Insertion Loss x dB Best Setup Very Low-Loss Line
Low SWR Low SWR Matching Tx Network
VG Ð 41 Pacificon 2001 PSA-00527—10/21/2001 Necessary Test Equipment
❏ Antenna analyzer ➤ Autek ➤ CIA ➤ MFJ ❏ Noise bridge (less accurate) ❏ Network analyzer (more accurate) ➤ Hewlett-Packard
VG Ð 42 Pacificon 2001 PSA-00527—10/21/2001 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 Ð 43 Pacificon 2001 PSA-00527—10/21/2001 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 Ð 44 Pacificon 2001 PSA-00527—10/21/2001 Smith Chart: Effect of Adding Series Reactances or Shunt Susceptances
Γi
0.12 0.13 0.11 0.14 0.38 0.37 0.1 0.1 0.39 0.36 5 90 0.4 100 80 0.35 0.16 5 0.09 4 50
110 40 70 0.34 .9
0.41 1.0 5 .2 5 0 0.17 8 1
0.0 0.8 35 0.33 1.4 60
0.42 120 0 0.7 6 ) /Yo 0.18
+jB 1.6
.6 ( 0.07 0 E 30 NC 0.32 TA .2 0.43 EP 0 1.8 65 SC 50 130 U S 0
6 E 2.0 .1 .0 0.5 IV 9 0 IT 5 0 4 C 2 .3 .4 A 1 0 P A -b C 70 R O 0.4 40
), 0.2 140 0.4 o Z 0.05 / 0.3 jX 0.45 + 20 ( T
75 N 3.0 E 0.6 N 0.21 O
P
3 0.29 0.04 0 0.3 5 M +x 0 0.46 1 O C > E 0.8 15 0 C 8 4.0 R N O A T 0 T .2 0 A C 7 1.0 .2 2 R A .4 E 8
E 0 5.0 R 1.0 N 0.2 20
E 5 E
8 G 160 V I
T 0.8 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0 0.1
O F G -x0.4
17 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W +b T
S
I
S > 50 O I
O N Γ 0.25 0.25
N
C
0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3 0.4 0.5
C O r 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D
E R
70 0.1 0.4 G
E
A
-1 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-20
E
L
5
E C
-8
-160
V N
0.2 0
5.0
.2
A
A 1.0 0
2 7 .2 T
W
P
.4 8 1.0
E 0
C
<
S
U
0 4.0
S
-8 -15
0.8 E
0.21 V
I -
0 3
T
5 0.29
0
1 C
0.04
- 0.3 U
0.46 D
N
0.6
I
R
3.0
O -75
,
)
0.2
o
-20
Z
/ 0.3 0.05
5
X -40
j
-
(
0.4
0.4 T
-140 0.4
N
E
N
-70
0 O
6 .1
P
.0 9
M
0
5
0
O
-2
4 .3
C
.4 1 E
0 C
-50
0.5 N
2.0
A
T
-130
C
0.18
A
.2
0 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30
.6 C
A
0.43 0 P
A
C
1.6
-60 0 0.17
-6 -120
0.08 0.7
0.33
1.4 -35
0.42 5
0.8 .2 0.16
-5
.9 -70 1 0.09
-110 0
1.0 -40
5
0.34
-50
-4
0
0.41
.1
-80 .1 5 0
-100
-90
0.14
0
0.11
.3 .4
5 0
0.13 0.12
0.36
0.39
0.37 0.38
VG Ð 45 Pacificon 2001 PSA-00527—10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45 50 110 40 70 0.34
0.41 1.0
0.9 0.17 55 1.2
0.08 0.8 35 0.33 1.4 60
0.42 120 0.7 60 ) Yo 0 B/ 1.6
(+j .18 0.07 0.6 E 30 C 0 3 AN .32 0.4 PT 0.2 1.8 CE 50 65 US
130 S .0
.5 E 2 0.19 0 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R O 0.4 40
), 0.2 140 0.4 o Z 0.05 / 0.3 jX 0.45 + 20 ( T
75 N 3.0 E 0.6 N 0.21 O P
30 0.29 0.04 0.3 M 0.46 150 O C > E 0.8 15 — C .0 80 4 R N O A 0.22 T T 0.28 A C 1.0 R A
E E
0.47 .0 5.0 R 1 N 0.2 20 E E
85 G 160 V I
T 0 0.8 1 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y Forbidden Area
T
0.48
/
G
B
j
0
-1 - N
(
E 0.8
-20
E L
E C
-85
-160
V N 0.2 0.22
.0 5.0
A A 1 0.28
T
W
P 1.0
E
0.47
—
C
< S
.0 U
4
S
-80 -15
0.8 E
0.21 V
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
N
0.6
I
R
3.0
O -75
,
)
0.2
o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
0.4 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44
.5 C
.0
-50
0 N
2
A
T
-130
C
0
A
.18 7
0.2 -65 E
R
.0
0 E
1.8
V 0
I
T
.32
I
-30 C
A
0.43 0.6 P
A
C
1.6
-60 0.17
-60 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-70
1.2
0.09
-110 0.9
1.0 -40
0.34
-50
-45
0.15
0.41
-8 0 0 0 0.1
-1
-90
0.14
0.35 0.11
0.4
0.13
Z 0 Z 0.12
0.36
0.39
0.37 0.38
VG Ð 46 (a) Pacificon 2001 PSA-00527 — 10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.1 0.1 0.39 0.36 5 90 0.4 100 80 0.35 0.16 0.09 0 45 5 110 40 70 0.34
0.41 1.0
0.9 0.17 55 1.2
0.08 0.8 35 0.33 1.4 60
0.42 120 0.7 60 ) Yo .6 0.18 B/ 1
(+j 0.07 0.6 E 30 NC 0.32 TA 0.43 EP 0.2 1.8 65 SC 50 130 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R 0 O 0.4 4
4 ), 0 0.2 1 0.4 o Z 0.05 / 0.3 jX 0.45 + 20 ( T
75 N 3.0 E 0.6 N 0.21 O P
30 0.29 0.04 0.3 M 0.46 150 O C > E 0.8 15 — 0 C .0 8 4 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 20 E E
85 G 160 V I
T 0.8 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S Forbidden Area L E L H
10 E T O
0.1
O F G 70 0.4
1 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-20
E L
E C
-85
-160
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
< S
.0 U
0 4
S
-8 -15
0.8 E
0.21 V
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
N
0.6
I
R
3.0
O -75
,
)
0.2
o
-20
Z
/ 0.3 0.05
X
-4
j
0
-
( 4
0 0.45
0.4 T
-1 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44 C
-50
0.5 N
2.0
A
T
-130
C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
.6 C
1
-60 0.17
-60 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-70
1.2 0.09
-110 0.9
1.0 -40 0
0.34
-5
-45
0
0.41
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-80 .1 5 0
-100
-90
0.14
0
0.11
.3 .4 5
0
0.13
Z 0 Z 0.12
0.36
0.39
0.37 0.38
VG Ð 47 (b) Pacificon 2001 PSA-00527 — 10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.1 0.1 0.39 0.36 5 90 0.4 100 80 0.35 0.16 0.09 45 50 110 40 70 0.34
0.41 1.0
0.9 0.17 55 1.2
0.08 0.8 35 0.33 1.4 60
0.42 120 0.7 60 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 NC 0.32 TA 0.43 EP 0.2 1.8 65 SC 50 130 U S 0
6 E 2.0 .1 .0 0.5 IV 9 0 IT 0 4 C 25 .3 .4 A 1 0 P A C 70 R O 0.4 40
), 0.2 140 0.4 o Z 0.05 / 0.3 jX 0.45 + 20 ( T
75 N 3.0 E 0.6 N 0.21 O P
30 0.29 0.04 0.3 M 0.46 150 O C > E 0.8 15 — C 80 4.0 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 20 E E
85 G 160 V I
T 0.8 10 D C R 0.23 U A 0.27 D A
W Forbidden Area N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-20
E L
E C
-85
-160
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
<
S
U
4.0
S
-80 -15
0.8 E
0.21 V
I -30
T 0.29
C
0.04
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0.46 D
N
0.6
I
R
3.0
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)
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o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
0.4 T
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N
E
N
-70
0 O
6 .1
P
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0
0
O
-25
4 .3
C
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0 C
-50
0.5 N
2.0
A
T
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C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
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-60 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-70
1.2
0.09
-110 0.9
1.0 -40
0.34
-50
-45
0
0.41
.1
-80 .1 5 0
-100
-90
0.14
0
0.11
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5 0
0.13
Z 0 Z 0.12
0.36
0.39
0.37 0.38
VG Ð 48 (c) Pacificon 2001 PSA-00527 — 10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45 50 110 40 70 0.34
0.41 1.0
55 0.9 0.17 8 1.2
0.0 0.8 35 0.33
1.4 60 .42 0.7 0 120 0 6 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 C 0.32 AN 0.43 PT 0.2 1.8 CE 5 30 65 S 0 1 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R
O 0.4 40
.4 ), 0.2 140 0 o Z 0.05 / 0.3 jX 0.45 + 20 ( T 5 7 N 3.0 E 0.6 N 0.21 O
P
3 0.29 0.04 0 0.3 5 M 0 0.46 1 O C > E 0.8 15 — 0 C .0 8 4 R N O A 0.22 T T 0.28 A C 1.0 R A .47 E
E 0 5.0 R 1.0 N 0.2 20 E E
85
G 160 V
I
T .8 0 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W E
Forbidden Area R O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B
j
0
-1 - N
( .8
E 0
-20
E L
E C
-85
-160
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
< S
.0 U
4
S
-80 -15
0.8 E
0.21 V
I -3
0
T
5 0.29 0
C
0.04
-1 0.3 U
0.46 D
N
0.6 I
5 R
3.0
O -7
,
)
0.2
o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
.4
0 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44
C
-5
N 0 0.5
2.0
3 A 0
T
-1
C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
-60 0 0.17
-6 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-70
1.2
0.09 0
-110 0.9
1.0 -4
0.34
-50
-45
0.15
0.41
-8 0 0 0 0.1
-1
-90
0.14
0.35 0.11
0.4
0.13
Z 0 Z 0.12
0.36
0.39
0.37 0.38
VG Ð 49 (d) Pacificon 2001 PSA-00527 — 10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45 50 110 40 70 0.34
0.41 1.0
55 0.9 0.17 8 1.2
0.0 0.8 35 0.33
1.4 60 .42 0.7 0 120 0 6 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 C 0.32 AN 0.43 PT 0.2 1.8 CE 5 30 65 S 0 1 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R
O 0.4 40
.4 ), 0.2 140 0 o Z 0.05 / 0.3 jX 0.45 + 20 ( T 5 7 N 3.0 E 0.6 N 0.21 O
P
3 0.29 0.04 0 0.3 5 M 0 0.46 1 O C > E 0.8 15 — 0 C .0 8 4 R N O A 0.22 T T 0.28 A C 1.0 R A .47 E
E 0 5.0 R 1.0 N 0.2 20 E E
85
G 160 V
I
T .8 0 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B
j
0
-1 - N
( .8
E 0
-20
E L
E C
-85
-160
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
< S
.0 U
4
S
-80 -15
0.8 E
0.21 V
I
Forbidden Area -3
0
T
5 0.29 0
C
0.04
-1 0.3 U
0.46 D
N
0.6 I
5 R
3.0
O -7
,
)
0.2
o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
.4
0 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44
C
-5
N 0 0.5
2.0
3 A 0
T
-1
C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
-60 0 0.17
-6 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-70
1.2
0.09 0
-110 0.9
1.0 -4
0.34
-50
-45
0.15
0.41
-8 0 0 0 0.1
-1
-90
0.14
0.35 0.11
0.4
0.13
Z 0 Z 0.12
0.36
0.39
0.37 0.38
VG Ð 50 (e) Pacificon 2001 PSA-00527 — 10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45 50 110 40 70 0.34
0.41 1.0
55 0.9 0.17 8 1.2
0.0 0.8 35 0.33
1.4 60 .42 0.7 0 120 0 6 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 C 0.32 AN 0.43 PT 0.2 1.8 CE 5 30 65 S 0 1 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R
O 0.4 40
.4 ), 0.2 140 0 o Z 0.05 / 0.3 jX 0.45 + 20 ( T 5 7 N 3.0 E 0.6 N 0.21 O
P
3 0.29 0.04 0 0.3 5 M 0 0.46 1 O C > E 0.8 15 0 Forbidden Area — C .0 8 4 R N O A 0.22 T T 0.28 A C 1.0 R A .47 E
E 0 5.0 R 1.0 N 0.2 20 E E
85
G 160 V
I
T .8 0 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B
j
0
-1 - N
( .8
E 0
-20
E L
E C
-85
-160
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
< S
.0 U
4
S
-80 -15
0.8 E
0.21 V
I -3
0
T
5 0.29 0
C
0.04
-1 0.3 U
0.46 D
N
0.6 I
5 R
3.0
O -7
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)
0.2
o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
.4
0 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44
C
-5
N 0 0.5
2.0
3 A 0
T
-1
C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
-60 0 0.17
-6 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-70
1.2
0.09 0
-110 0.9
1.0 -4
0.34
-50
-45
0.15
0.41
-8 0 0 0 0.1
-1
-90
0.14
0.35 0.11
0.4
0.13
Z 0 Z 0.12
0.36
0.39
0.37 0.38
VG Ð 51 (f) Pacificon 2001 PSA-00527 — 10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45 50 110 40 70 0.34
0.41 1.0
55 0.9 0.17 8 1.2
0.0 0.8 35 0.33
1.4 60 .42 0.7 0 120 0 6 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 C 0.32 AN 0.43 PT 0.2 1.8 CE 5 30 65 S 0 1 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R
O 0.4 40
.4 ), 0.2 140 0 o Z 0.05 / 0.3 jX 0.45 + 20 ( T 5 7 N 3.0 E 0.6 N 0.21 O
P
3 0.29 0.04 0 0.3 5 M 0 0.46 1 O C > E 0.8 15 — 0 C .0 8 4 R N O A 0.22 T T 0.28 A C 1.0 R A .47 E
E 0 5.0 R 1.0 N 0.2 20 E E
85
G 160 V
I
T .8 0 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B
j
0
-1 - N
( .8
E 0
-20
E L
E C
-85
-160
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
< S
.0 U
4
S
-80 -15
0.8 E
0.21 V
I
Forbidden Area -3
0
T
5 0.29 0
C
0.04
-1 0.3 U
0.46 D
N
0.6 I
5 R
3.0
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)
0.2
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Z
/ 0.3 0.05
X -40
j
-
(
0.45
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0 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44
C
-5
N 0 0.5
2.0
3 A 0
T
-1
C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
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0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-70
1.2
0.09 0
-110 0.9
1.0 -4
0.34
-50
-45
0.15
0.41
-8 0 0 0 0.1
-1
-90
0.14
0.35 0.11
0.4
0.13
Z 0 Z 0.12
0.36
0.39
0.37 0.38
VG Ð 52 (g) Pacificon 2001 PSA-00527 — 10/21/2001 0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45 50 110 40 70 0.34
0.41 1.0
55 0.9 0.17 8 1.2
0.0 0.8 35 0.33
1.4 60 .42 0.7 0 120 0 6 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 C 0.32 AN 0.43 PT 0.2 1.8 CE 5 30 65 S 0 1 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R
O 0.4 40
.4 ), 0.2 140 0 o Z 0.05 / 0.3 jX 0.45 + 20 ( T 5 7 N 3.0 E 0.6 N 0.21 O
P
3 0.29 0.04 0 0.3 5 M 0 0.46 1 O C > E 0.8 15 0 Forbidden Area — C .0 8 4 R N O A 0.22 T T 0.28 A C 1.0 R A .47 E
E 0 5.0 R 1.0 N 0.2 20 E E
85
G 160 V
I
T .8 0 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O I
O
N
— 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
—
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B
j
0
-1 - N
( .8
E 0
-20
E L
E C
-85
-160
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
< S
.0 U
4
S
-80 -15
0.8 E
0.21 V
I -3
0
T
5 0.29 0
C
0.04
-1 0.3 U
0.46 D
N
0.6 I
5 R
3.0
O -7
,
)
0.2
o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
.4
0 T
-140 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44
C
-5
N 0 0.5
2.0
3 A 0
T
-1
C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
-60 0 0.17
-6 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 0.16
-55
-70
1.2
0.09 0
-110 0.9
1.0 -4
0.34
-50
-45
0.15
0.41
-8 0 0 0 0.1
-1
-90
0.14
0.35 0.11
0.4
0.13
Z 0 Z 0.12
0.36
0.39
0.37 0.38
VG Ð 53 (h) Pacificon 2001 PSA-00527 — 10/21/2001 Matching: Four L-Networks Using Lumped Elements
Γi
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 5 Z 4 Z 50 L 110 40 70 0.34 L
0.41 1.0 .2 0.9 0.17 55 1
0.08 0.8 35 0.33 1.4 60
0.42 120 0 0.7 6 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 NC 0.32 TA 0.43 EP 0.2 1.8 65 SC 50 130 SU
E 2.0 0.19 0.5 V Start I 0.06 IT 0.31 C 25 A 0.44 P A 0 C 7 R O 0.4 40
), 0.2 140 0.4 o Z 0.05 / 0.3 jX 0.45 + 20 ( T
75 N 3.0 E 0.6 N 0.21 O
P
3 0.29 0.04 0 0.3 5 M 0 0.46 1 O C > E 0.8 15 C .0 80 4 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 20 0 E 5 E
8 G 16 V I
T 0.8 10 D C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0.24 0.26 E R R
A L E
E F N 0.49
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O Finish I
O N Γ 0.25 0.25
N
C
0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3 0.4
C O r 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
I
C <
I
E
0.2 I
N
D 20 E
N
0.24
A T
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0.23 E
0.27
S
S ) 0.6
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-20
E
L
5
E C
-8
-160
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E 0.47
C
< S
.0 U
4
S
5
-80 -1
0.8 E
0.21 V
I -3
0
T
5 0.29 0
C
0.04
-1 0.3 U
0.46 D
.6
N
0
I
R
3.0
O -75
,
)
0.2
o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
.4 0.4 T
-140 0
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25
C
E
0.44 C
-50
0.5 N
2.0
A
T
-130
C
0.18
A
0.2 -65
E
R
0.07 E
1.8
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6
-60 0 0.17
-6 -120
0.08 0.7
0.33
1.4 -35
0.42
0.8 .2 0.16
-55
-70
1 0.09 -110 0.9
1.0
Z -40 Z
5
0.34
-50
4
-
0.15
0.41
L -80 L 0.1
-100
-90
0.14
0.35
0.11
0.4
0.13 0.12
0.36
0.39
0.37 0.38
VG Ð 54 Pacificon 2001 PSA-00527—10/21/2001 Matching: Four L-Networks Using Stubs
Γi
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 80 0.35 9 0.4 100 0.1 0.0 6 Z 45 Z 50 0 70 0 L .41 110 4 .34 L
0 1.0 5 .2 5 0.9 0 8 1 .1
0.0 0.8 5 7 3 0
2 1.4 60 .3 .4 0.7 3 0 120 60 ) Yo 0.18 B/ 1.6
(+j 0.07 0.6 E 30 NC 0.32 TA .8 0.43 EP 0.2 1 0 65 SC 50 13 U S 0
6 E 2.0 .1 .0 0.5 IV 9 0 T Start 5 0 I 4 C 2 .3 .4 A 1 0 P A 0 C 7 R O 0.4 40
), 0.2 140 0.4 o Z 0.05 / 0.3 jX 0.45 + 20 ( T
75 N 3.0 E 0.6 N 0.21 O P
30 0.29 0.04 0.3 M 0.46 150 O C > E 0.8 15 C 80 4.0 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 2 0 0 E 6 E
85 G 1 V I
T 0.8 10 D C R 0 U A 0 .2 D 8 A .2 W 3
.4 N N A 7 I 0.6 O 0 90 N T G
G S L
E L H
10 E T O
0.1
O F G 0.4
170 F N T 0 0 E R R .2 9 .2
L A E 4 .4 6
E F N 0
L V
20 S E
A M 0.2 C
I W T
S
I
S > 50 O Finish I
O N Γ 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O r 0.0 0.0
180
O
E
±
F E
50 F
F I
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) F C
I
C <
I
E
0.2 I
N
D 20 E
N
0
A T
0
.2
T 9 .2
O
I
4 N
L
.4
6
I
N
0
D
D
D E
R 0.1 0.4 G
E
A
-170 10
G
R
W
E R
O
E E
T
S
0 E
.2 0
S
S
) 0.6 8 .2
-90 3 o
H
.4 7
Y
T
0
/
G
B j
-10 - N
(
E 0.8
-2
0 E L
6
0
E C
-85
-1
V N
0.2 0.22
5.0
A A 1.0 0.28
T
W
P 1.0
E 0.47
C
<
S
U
4.0
S
-80 -15
0.8 E
0.21 V
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
.6
N
0
I
R
3.0
O -75
,
)
0.2
o
-20
Z
/ 0.3 0.05
X -40
j
-
(
0.45
0.4 T
-140 0.4
N
E 0
N
-7
0 O
6 .1
P
.0 9 M
0
0
O
-25
4 .3
C
.4 1 E
0
C
-5
N 0 0.5
2.0
3 A 0
T
-1
C
0.18
A
0.2
-65 E
R
.8
0.07 E
1
V
0.32
I
T
I
-30 C
A
0.43 0.6 P
A
C
1.6 -60
0
8 .1
-60
-120 .0 7
0 0.7 5
0
1.4 2 .3 -3
.4 3
0 .8
0 0
9 .1
-55
-70
1.2 .0 6 0 0 -110 0.9
1.0
Z -4 Z
0.3
-50
1
-45
.4 4 0.15
0
L -80 L 0.1
-100
-90
0
.1 1 0.35 .1 4
0
0.4
0 .1 2 3 .1 0
0 9 .3 .3 6
0
0 .3 8 7 .3 0
VG Ð 55 Pacificon 2001 PSA-00527—10/21/2001 Reactance & Susceptance of Lumped Elements
Inductors
−1 Xf= 2π L B = L L 2πfL Capacitors
−1 X = BfC= 2π C 2πfC C
VG Ð 56 Pacificon 2001 PSA-00527—10/21/2001 Reactance & Susceptance of Stubs
❏ Assuming Zo is real, then ❏ Shorted Stubs
− 2 πlf = 1 XZ= tan Bshorted shorted o π vcf 2 lf Zo tan vcf
❏ Open Stubs
−Z 12 πlf X = o B = tan open open 2πlf Zo vcf tan vcf
VG Ð 57 Pacificon 2001 PSA-00527—10/21/2001 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 Ð 58 Pacificon 2001 PSA-00527—10/21/2001 ∏-Resonant and T-Resonant Network For Moderate and Broadband Matching
Γ i
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 80 0.35 Z Z 0.4 100 0.16 0.09 45 L L 50
110 40 70 0.34 .9
0.41 1.0
.2 5 0 5 0 8 1 .1
0.0 0.8 7 35 0
2 1.4 60 .3 .4 0.7 3 0 120 60 ) /Yo 0.18
(+jB 1.6 0.07 0.6 E 30 C 0.32 AN 0.43 PT 0.2 1.8 5 CE 5 30 6 S 0 1 SU
E 2.0 0 6 0.5 .1 .0 IV 9 0 IT f 25 0 4 C .3 .4 A 1 0 P A 0 l C 7 R O 0.4 40
), 0.2 140 0.4 o Z 0.05 / 0.3 jX 20 0.45 (+ T .0 75 N 3 E 0.6
N 0.21
O
.3 P 0.29 0 30 0.04 M 0.46 150 O C > E 0.8 15 — C 80 4.0 R N O A Z T 0 T .2 0 A C 7 1.0 .2 2 Z R A L .4 8 E
E 0 5.0 R 1.0 N 0.2 20 0 L E E
85 G 16 V I
T D 0.8 10 C R 0.23 U A SWR = 1.5 0.27
D A
W
N .6 N A I 0 O 0.48 90 N T G
G S L
E L H
E
10 O T
0.1 O F G 0.4
170 F N T 0.24 0.26 R E f R A L E
E F N 0.49
L V
mid 20 S E A 0.2 M A B C I W T
S
I
S >
O
50 I
O
N — Γ 0.25 0.25
N
C
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3
C O 0.0 0.0
180
O
E r
±
F E
50 F
F
I
F
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) C
—
I
C < I
E
0.2 I
0 N
D 2 E
N
0.24
T A
0.26
T
O
I
N
L
I
N
0.49
D
D
D E
R 0.4
0.1 G E
A
-170 10
G
R
W
E R
O
E E
T
S 0.23 E
0.27
S
S .6 ) 0
-90 o
H
Y
T
0.48
/
G
B j
-10 - N
(
E 0.8
-20
E L
60
E C
-85
-1
V N
0.2 0
5.0
.2
A
A 1.0 0
2 .2 7 T
W .0
P
1 8 .4
E
0
—
C
<
Z S Z
U f
4.0
S
5 .8
-80 -1
L 0
E L
0.21 V u
I -30
T 0.29
C
0.04
-150 0.3 U
0.46 D
.6
N 0 I
.0 R
3
O -75
,
)
0.2
o
-20
Z
/ 0.3
0.05
X -40
j
-
(
0.45
.4 0.4 T
-140 0
N
E 0
N
-7
0 O
6 .1
P
.0 9 M
0
0
O
-25
4 .3
C
.4 1 E
0
C
-5
N 0 0.5
2.0
3 A
0
T
5 -1
C
0
A
.1 7 0.2 -6 E
R
8 .0
0 E
1.8
V
0
I
T
.3 3 I
-30 C
2 .4
A
0 0.6 P
A
C
1.6 -60
0
8 .1
-60
-120 .0 7
0 0.7
0
1.4 2 .3 -35
.4 3
0
0.8 .2 0.16
-55 -70
1
0.09 0
-110 0.9
1.0 -4
0.34
-50
-45
0.15 0.41
-80 0.1
-100
-90
0
.1 1 0.35 .1 4
0
0.4
0.13 0.12
0
9 .3 .3 6 Z 0 0.37
L 0.38 ZL
VG Ð 59 Pacificon 2001 PSA-00527—10/21/2001 Caron’s Example 10: VHF Folded Blade Antenna
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 0 45 5 110 40 70 0.34
0.41 1.0
0.9 0.17 55 1.2
0.08 0.8 35 0.33 1.4 60
0.42 120 0.7 60 ) /Yo 0.18
(+jB 1.6 0.07 0.6 E 30 C 0.32 AN 0.43 PT 0.2 1.8 5 CE 5 30 6 S 0 1 SU
E 2.0 0.19 0.5 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R 0 O 0.4 4
4 ), 0 0 5 1 0.4 o .2 0.0 Z 0 5 / .3 jX 20 0.4 (+ T
75 N 3.0 E 0.6 N 0.21
O
P 0.29 0.3 30 0.04 M 0.46 150 O C > E 0.8 15 — C 80 4.0 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 20
E 5 E
8 G 160 V I
T D 0.8 10 C R 0 U A 0 .2 D 8 A .2 W 3
.4 N N A 7 I 0.6 O 0 90 N T G
G S L
E L
H
0 E
1 O T .1
0 O F G 0.4
170 F N T 0.24
0.26
E R R
A L E
E F N 0.49
L V
20 S E .2 A 0 M
C
I W T
S
I
S >
O
50 I
O
N
— 0.25 0.25
N
C
0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3 0.4
C O 0.0 0.0
180
O
E
±
F E
50 F
F
I
F
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) C
—
I
C < I
E
0.2 I
N
D 20 E
N
0.24
T A
0.26
T
O
I
N
L
I
N
0.49 D
D 160
D E
R .1 0.4
0 G E
A 0
-170 1
G
R
W
E R
O
E E
T
S 0 E
.2 0
S
S .6
) 150
0 .2 8
-90 3 o
H
7 .4
Y
T
0
/
G
B j
0
-1 - N
(
E 0.8
-20
E L
E C
-85
-160
V N 0.2 0.22
140 5.0
A
A 1.0 0.28
T
W
P 1.0
E
0.47
—
C
< S
.0
U
105 4
S
-80 -15
0.8 E
0.21 V
I -30
T
130 0.29
C
0.04
-150 0.3 U
0.46 D
N
0.6
I 100
R
3.0
O -75
,
)
120 0.2
o
-20
Z
/ 0.3 0.05
X
-4
j
0 -
(
0
0.45 110
0.4 T
-14 0.4
N
E
N
-70 0.19 O
P
M
0.06
0.31
O
-25 C
E
0.44
C
-5
N 0 0.5
2.0
3 A
0
T
5 -1
C
0
A
.1 7 0.2 -6 E
R
8 .0
0 E
1.8
V
0
I
T
.3 3 I
-30 C
2 .4
A
0 0.6 P
A
C
1.6
-60 0.17
-60 -120
0.08 .7
0
0.33
1.4 -35
0.42
0.8 0.16
-55 -70
1.2
0.09 0
-110 0.9
1.0 -4
0.34
-50
-45
0.15
0.41
-8 0 0 0 0.1
-1
-90
0.14
0.35
0.11
0.4
0.13 0.12
0.36
0.39
0.37 0.38
VG Ð 60 Pacificon 2001 PSA-00527—10/21/2001 Caron’s Matching Solution
0.12 0.13 0.11 0.14 0.38 0.37 0.15 0.1 0.39 0.36 90 0.4 100 80 0.35 0.16 0.09 45 50 110 40 70 0.34
0.41 1.0 .2 0.9 0.17 55 1 0.08 0.8 35 Max VSWR=1.67 0.33 1.4 60
0.42 120 0.7 60 ) /Yo 0.18
(+jB 1.6 0.07 0.6 E 30 NC 0.32 TA 0.43 EP 0.2 1.8 65 SC 50
130 SU
.5 E 2.0 0.19 0 IV 0.06 IT 0.31 C 25 A 0.44 P A C 70 R O 0.4 40
), 0.2 140 0.4 o Z 0.05 / 0.3 jX 20 0.45 (+ T 5 7 N 3.0 E 0.6 N 0.21
O
P 0.29 0.3 30 0.04 M 0.46 150 O C > E 0.8 15 — 0 C 8 4.0 R N O A 0.22 T T 0.28 A C 1.0 R A E
E 0.47 5.0 R 1.0 N 0.2 20 E E
85 G 160 V I
T D 0.8 10 C R 0.23 U A 0.27 D A W
N N A I 0.6 O 0.48 90 N T G
G S L
E L H
E
10 O T 110
0 0.1
O F G 7 0.4
1 F N T 0.24
0.26 R E 120 R
A L E
E F N 0.49
L V
20 S E A 0.2 M
C
I W T 160 S I
S >
O
50 I
O
N
— 0.25 0.25
100 N
C
0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 10 20 50 0.1 0.2 0.3 0.4
C O 0.0 0.0 180
130 O
E
±
F E
50 F
F
I
F
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo) C
—
I
C < I
E
0.2 I
N
D 20 E
N
0
T A
0
.2
T .2 9 O
105 I 4 N
L 6
150 I
N
0.4
D
D 140
D E
R 0.4
0.1 G E
A
-170 10
G
R
W
E R
O
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VG Ð 61 0.38 Pacificon 2001 PSA-00527—10/21/2001 WinSmith Display
Max VSWR=1.59
VG Ð 62 Pacificon 2001 PSA-00527—10/21/2001 ARRL Radio Designer
VG Ð 63 Pacificon 2001 PSA-00527—10/21/2001 ARRL Radio Designer
VG Ð 64 Pacificon 2001 PSA-00527—10/21/2001 Caron’s Example 11: Long Wire Receiving Antenna
VG Ð 65 Pacificon 2001 PSA-00527—10/21/2001 Caron’s Solution
Max VSWR=6.2
VG Ð 66 Pacificon 2001 PSA-00527—10/21/2001 Caron’s Solution
Max VSWR=6.2
VG Ð 67 Pacificon 2001 PSA-00527—10/21/2001 Eagleware’s Solution 1
Max VSWR=5.05
VG Ð 68 Pacificon 2001 PSA-00527—10/21/2001 Eagleware’s Solution 2
Max VSWR=4.77
VG Ð 69 Pacificon 2001 PSA-00527—10/21/2001 Eagleware’s Solution 3
Max VSWR=6.59
VG Ð 70 Pacificon 2001 PSA-00527—10/21/2001 Eagleware’s Solution 4: Fano Limit
Max VSWR=3.42
VG Ð 71 Pacificon 2001 PSA-00527—10/21/2001 GM3HAT 4-Band Dipole of Delight
VG Ð 72 Pacificon 2001 PSA-00527—10/21/2001 GM3HAT Feedpoint Impedance At Five Harmonic Frequencies
VSWR=11.02
VSWR=1.62
VG Ð 73 Pacificon 2001 PSA-00527—10/21/2001 K6OIK 80m Suboctave Band Matching Network
150Ω 89°@ 3.5 MHz
5.9 µH 18.8 µH Ω 25 Zant 91°@ 3.5 MHz
VG Ð 74 Pacificon 2001 PSA-00527—10/21/2001 K6OIK’s Multiband Match Performance
VSWR=1.46
VG Ð 75 Pacificon 2001 PSA-00527—10/21/2001 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
VG Ð 76 Pacificon 2001 PSA-00527—10/21/2001 References: Articles
❏ Robert L. Thomas, “Broadband Impedance Matching in High-Q Networks,” EDN, pp. 62Ð69, December 20, 1973. ❏ Neal C. Silence, “The Smith Chart and Its Usage in RF Design,” RF Design, pp. 85Ð88, April 1992. ❏ Thomas R. Cuthbert, Jr., “Broadband Impedance Matching Methods,” RF Design, pp. 64Ð91, August 1994. ❏ Thomas R. Cuthbert, Jr., “Broadband Impedance Matching - Fast and Simple,” RF Design, pp. 38Ð50, November 1994. ❏ William E. Sabin, “Broadband HF Matching with ARRL Radio Designer,” QST, pp. 33Ð36, August 1995. ❏ William E. Sabin, “ARRL Radio Designer and the Circles Utility, Part 1: Smith Chart Basics,” QEX, pp. 3Ð9, Sept/Oct 1998. ❏ William E. Sabin, “ARRL Radio Designer and the Circles Utility, Part 2: Small-Signal Amplifier Design,” QEX, pp. 3Ð11, Nov/Dec 1998.
VG Ð 77 Pacificon 2001 PSA-00527—10/21/2001 References: Articles (Cont’d)
❏ Steve Sparks, “A Practical Amateur Application of the Smith Chart,” Communications Quarterly, pp. 102Ð106, 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 Ð The Fundamentals,” RF Design, pp. 52Ð66, July 2000.
VG Ð 78 Pacificon 2001 PSA-00527—10/21/2001 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. Radio-Frequency and Microwave Amplifiers, ❏ Pieter L. D. Abrie, The Design of Impedance-Matching Networks for 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 computer- aided 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. VG Ð 79 Pacificon 2001 PSA-00527—10/21/2001 References: Books (Cont’d)
Waveguide Circuit, and Component Analysis ❏ Phillip H. Smith, Electronic Applications of the Smith Chart in , 2nd edition, Noble Publishing, 1995, ISBN 1884932398. Slotlines ➤ 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 , 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. Antennas➤ Elementary, non-mathematical. ❏ M. Walter Maxwell, W2DU, Reflections II: Transmission Lines and , 2nd ed., Worldradio Books, 2001, ISBN 0970520603. ➤ A non-mathematical treatment of transmission lines and matching that examines and corrects common misunderstandings.
VG Ð 80 Pacificon 2001 PSA-00527—10/21/2001 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 Ð 81 Pacificon 2001 PSA-00527—10/21/2001