Impedance Matching Equation: Developed Using Wheeler’s Methodology IEEE Long Island Section Antennas & Propagation Society Presentation December 4, 2013 By Alfred R. Lopez
Impedance Matching Equation: Developed Using Wheeler’s
Methodology
IEEE Long Island Section Antennas & Propagation Society Presentation
December 4, 2013 By
Alfred R. Lopez
Outline
1. Background Information 2. The Impedance Matching Equation 3. The Bode and Fano Impedance Matching Equations 4. Wheeler’s Single- and Double-Tuning Equations 5. Conversion of Wheeler’s Equations to the Original
Impedance Matching Equation 6. Development of the final form for the Impedance Matching
Equation 7. A note on Triple-Tuned Impedance Matching
Background Information 1940s
Wheeler develops impedance matching principles A Wheeler designed double-tuned impedance-matched IFF antenna played a
critical role in WW II Bode and Fano publish their work on impedance matching
1950 Wheeler publishes Report 418, a tutorial on impedance matching that features
the reflection chart as a primary tool For single- and double-tuned impedance matching, it presents three equations
that quantify impedance-matching bandwidth limitations related to a specified maximum reflection magnitude
Based on the works of Bode and Fano, it quantifies the law of diminishing returns for impedance matching circuits beyond double tuning
1973 Wheeler’s three equations are converted to the original Impedance Matching
Equation 2004
Using MATCAD to solve Fano’s equations, the final version of the Impedance Matching Equation was developed
Impedance-Matching Equation
( )
Γ
−+
Γ
=Γ1ln
nanb11ln
na1sinhnb
1Q1
nB
Bn = Maximum fractional impedance- matching bandwidth Bn = (fH – fL)/f0 f0 = Resonant frequency = Q = Antenna Q (Ratio of reactive power to radiated and dissipated power} Γ = Maximum reflection magnitude within Bn n = Number of tuned stages in the impedance matching circuit
LHff
Assumes Lumped-Element Circuits Exact for n = 1, 2, and ∞ QBn Error < 0.1% for Γ > 0.10 (Max VSWR > 1.2)
Bode Impedance Matching Equation (Hendrik W. Bode)
L
C
R
Lossless Lumped-Element Impedance Matching
Network
R0
Antenna Generator
RLω
Q 1ln
πQ1B 0=
Γ
=
B = Theoretical maximum fractional bandwidth for specified maximum reflection magnitude
Fano’s Impedance Matching Equations (Robert M. Fano)
( )( )
( )( )
( )( )
( ) ( ) QBbsinhasinh
n2πsin2
bcoshnbtanh
acoshnatanh
nacoshnbcosh
=−
=
=ΓΓ n
QBn(Γ)
NOTE: The Impedance Matching Equation is a closed-form approximate solution for the Fano Impedance Matching Equations
n tuned stages Alternate - series and parallel All stages tuned to f0 n = 1 is the tuned antenna
The Bode-Fano Equation
Γ
=∞ 1ln
πQ1B
Fano showed that in the limit case of n = ∞
We Started in 1973 With Wheeler’s Three Equations for a Resonant Antenna
( )
Tuning) Double (Optimum .3
Tuning) Single (Optimum 2
φtan .2
sfrequencie band-edgeat phase impedance of Magnitude φ φtanQB .1
212
EB1
EBEB
Γ=Γ
=Γ
==
1950 Wheeler Lab Report 418
Wheeler’s First Equation
( ) ( )( ) QBφtan
φjexpRjQB1RZ
ff
ff
CRω1j1RZ
ff
ff
Cω1jRZ
EB
EBEB
0
L
0
H
0EB
H
0
0
H
0EB
=⋅=+=
−+=
−+=
Wheeler’s Small Resonant Antenna Lumped-Element RLC Circuit Example: Small Electric Dipole Capacitor resonated with series L
Wheeler’s Optimum Single- and Double-Tuned Impedance Matching (Proof by Inspection)
.
fH
fH
fH
fL
fL
fL
R0 R OC SC
jR0
-jR0
Single Tuning (Mid-Band Match)
tan(φEB) = QB
Optimum Single Tuning
(Edge-Band Match) Γ1 = tan(φEB/2)
Impedance transformation can not reduce Γ1
Optimum Double Tuning
Γ2 = Γ12
Impedance transformation and/or change in Q of second tuning stage can not reduce Γ2
Γ2 Γ1
φEB = Impedance phase at edge frequencies, fH and fL
Single Tuning: Derivation of
( ) ( )( ) ( )
( ) ( ) ( )( ) ( ) ( )
( )( )
=
+
−=Γ
+++
++−=Γ=Γ
++−+
=Γ
+−
=Γ
=
2φtan
φcos1φcos1
φsin1φcos2φcos
φsin1φcos2φcos
1φsinjφcos1φsinjφcos
1e1e
eZ
EB
EB
EB1
EB2
EBEB2
EB2
EBEB2
EB1
EBEB
EBEBEB
φj
φj
EB
φjEB
EB
EB
EBFrom Reflection Chart R0 = 1
=Γ
2φtan EB
EB
Derivation of Γ2 = Γ12
.Wheeler Double-Tuned Matching
Wheeler Single-tuned Edge-Band Matching
OC SC
C
L
2 1
fH
fL
fH & fL f0
212
1
1
2
1
Γ=Γ
Γ=
ΓΓ
Similar Triangles
Γ2 Γ2
Γ1
1
n2
n1
n
1
2Q1)(B
Γ−
Γ=Γ
Wheeler’s Equation: Single tuning, n = 1 Double tuning, n = 2
In 1973 we converted Wheeler’s three equations for a resonant antenna to a single equation
( )( )
Tuning) (Double .3
Tuning) (Single /2φtan .2frequency edgeat phase Impedance φ φtanQB .1
212
EB1
EBEB
Γ=Γ
=Γ==
( ) ( )( )
2
22
21
112
-12
QB :Tuning Double
-12QB
2/φtan12/φtan2φtan :Tuning Single
ΓΓ
=
ΓΓ
=−
=
1973 Continued
• At this point we had an explicit expression that related B, Q, Γ, and n for single- and double-tuned impedance matching • We were aware of the Bode and Fano results • Wheeler clearly defined the law of diminishing returns for added stages beyond double tuning • One remaining question was: How much bandwidth increase can be achieved with triple tuning over that of double tuning?
2a and ,1a
31for
1ln
aQ1
1lna1sinh
1Q1B
21
n
n
n
==
>Γ
Γ
≈
Γ
=
1973 Continued
n
n
nQB 2
1
1
2
Γ−
Γ=Wheeler’s Equation:
Γ
=
−
=Γ−
Γ
=Γ−Γ
=
Γ
−
Γ
112
12
12
1121
lnsinhee
QBlnln
Γ
=
−
=Γ−
Γ
=Γ−Γ
=
Γ
−
Γ
12112
12
12
1211
212
lnsinhee
QBlnln
πa 1ln
πQ1B
Equation Fano-Bode
=
Γ
= ∞∞
1973 Continued
??? 1ln
aQ1B Is
:3/1 andn all For
nn
Γ
≈
>Γ
π=======
π=∞+
+
+
+
++++
π=∞+
+
++
∞=∑ a ... 2.756a 667.2a 333.2a 2a 1a sa
.........54
32
71
54
32
71
32
51
32
51
31
3111
2.........
54
32
71
32
51
311
54321
n
1kkn
2222
22
Knew that a1 = 1, a2 = 2, and a∞ = π
Ref.: L.B.W. Jolley, “Summation of Series,” Dover, New York, (410), p. 76, 1961
?) Increase (18% 18.1BB
Increase) (65% 65.1BB
Increase) (131% 31.2BB
1/3 For
2
3
2
1
2
=
=
=
=Γ
∞
1973 Impedance-Matching Equation (Original Equation)
Exact for n = 1 and 2 Approximate for Γ > 1/3, and n > 2
Γ
≈Γ1ln
a1sinh
1Q1)(B
n
n
Sent letter to Professor Fano asking for help in determining accuracy of an
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
. QB2
ωA
ωω
LωR2
ωA
LR2A
c
1
c
0
0c
1
1
=
=
=
∞
∞
∞
QBπ1ln =
Γ
1973 Fano’s Reply
n = ∞
n = 1
n = 2
n = 3
n = 4 n = 5 n = 6
Fig. 19. Tolerance of match for a low-pass ladder structure with n elements
c
1
ωA∞
.MAX1ρ1ln
(38) )b cosh()nb tanh(
)a cosh()na tanh(
(37) )na cosh()nb cosh(
(36)
n2πsin
)b sinh()a sinh(ωA
c
1
=
=Γ
−
=∞
Fractional Bandwidth (Band-Pass) Conversion
.
.
0 0.2 0.4 0.6 0.80
0.5
1
1.5
2
2.5
3
3.5
.
2004 – Comparison of Fano and Original Matching Equation
Γ
=1ln
21sinh
QB1
2
π
1ln
QB1
Γ=
∞
Γ
=∞
1lnπ1sinh
QB1
QB1
Γ
=1lnsinh
QB1
1
Γ1ln
Γ
=1ln
a1sinh
QB1
33
Fano
Γ > 1/3
n = 1
n = ∞
n = 3
n = 2
Used MATHCAD to solve Fano’s equations
2004 Impedance-Matching Equation
( )
Γ
−+
Γ
=Γ1ln
nanb11ln
na1sinhnb
1Q1
nB
bn coefficient provides blending of the “sinh” and “ln” functions
B3/B2 = 1.24 (24% Increase)
Conclusion • Wheeler’s development of the principles for double-tuned impedance matching was a major contribution. Although it was developed for lumped-element circuits it has a broader application • One can see by inspection that his solutions were optimum • We have developed the Impedance-Matching Equation, a closed form solution for the Fano Equations, which we hope will be helpful and useful to the community • What impressed me the most in all of this work was the remarkable fact that Wheeler’s results, using the reflection chart, were identical to the results obtained by Fano using high-level network theory
Wheeler and Fano
n2
n1
n
1
2Q1)(B
Γ−
Γ=Γ ( ) ( ) ( )
( )( )
( )( )
( )( ) Γ=
=
−
=Γ
nacoshnbcosh
bcoshnbtanh
acoshnatanh
bsinhasinhn2πsin2
Q1Bn
Wheeler (Reflection Chart) n = 1,2
Fano (Network Theory) n = 1,2,3….∞
1
2Q1)(B 1n 1
Γ−Γ
=Γ= ( ) ( )
( ) ( ) bsinh 1asinh
bsinhasinh2
Q1)(B1
Γ=Γ
=
−=Γ
Triple-Tuned Impedance Matching
.
Triple-Tuned Impedance Matching Which circle, A or B, should be used to position the edge-band frequencies on the
Max Γ Circle
Circle A or Circle B
Max Γ Circle Γ = 1/2 VSWR = 3
fL
fH
Double-Tuned Locus
. .
Triple-Tuned Impedance Matching Cont’d
Edge-Band Frequencies on Horizontal Axis
Edge-Band Frequencies on Vertical Axis
fH fL
fH
fL
.
Triple-Tuned Impedance Matching Cont’d
Triple-Tuned Monopole Antenna On Infinite Ground Plane
Triple-Tuned Monopole Antenna (Continued)
Double Tuned
Triple Tuned
Triple-Tuned Monopole Antenna (Continued)