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 – 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
µδ −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
VG – 13 PSA-00527—10/21/2001
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 AV W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 0.49 AD 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 VEL 0.47 WA 0.0 160 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 WAVELE 0.49 NGTH S TOW ARD 0.48 0.0 0.49 AD A W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 D AV W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 D 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
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 A W 1 0 6 WAVELE 0.49 NGTH S TOW ARD 0.0 0.49 AD AV W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 D AV W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 0.49 AD A W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 0.49 AD A W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 0.49 AD A W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 0.49 AD A W 160 WAVELE 0.49 NGTH S TOW ARD 0.0 0.49 AD A W 160 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
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 WAVELE 0.49 NGTH S TOW ARD 0.0 D A W 1 0 6 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
VG – 62 PSA-00527—10/21/2001
Pacificon 2001
ARRL Radio Designer
VG – 63 PSA-00527—10/21/2001
Pacificon 2001
ARRL Radio Designer
VG – 64 PSA-00527—10/21/2001
Pacificon 2001
Caron’s Example 11: Long Wire Receiving Antenna
VG – 65 PSA-00527—10/21/2001
Pacificon 2001
Caron’s Solution
Max VSWR=6.2
VG – 66 PSA-00527—10/21/2001
Pacificon 2001
Caron’s Solution Max VSWR=6.2
VG – 67 PSA-00527—10/21/2001
Pacificon 2001
Eagleware’s Solution 1
Max VSWR=5.05
VG – 68 PSA-00527—10/21/2001
Pacificon 2001
Eagleware’s Solution 2
Max VSWR=4.77
VG – 69 PSA-00527—10/21/2001
Pacificon 2001
Eagleware’s Solution 3
Max VSWR=6.59
VG – 70 PSA-00527—10/21/2001
Pacificon 2001
Eagleware’s Solution 4: Fano Limit
Max VSWR=3.42
VG – 71 PSA-00527—10/21/2001
Pacificon 2001
GM3HAT 4-Band Dipole of Delight
VG – 72 PSA-00527—10/21/2001
Pacificon 2001
GM3HAT Feedpoint Impedance At Five Harmonic Frequencies
VSWR=11.02
VSWR=1.62
VG – 73 PSA-00527—10/21/2001
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
VG – 75 PSA-00527—10/21/2001
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
VG – 76 PSA-00527—10/21/2001
Pacificon 2001
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.
VG – 77 PSA-00527—10/21/2001
Pacificon 2001
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 PSA-00527—10/21/2001
Pacificon 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.
❏ 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. VG – 79 PSA-00527—10/21/2001
Pacificon 2001
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. VG – 80 PSA-00527—10/21/2001
Pacificon 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 – 81 PSA-00527—10/21/2001
Pacificon 2001
