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
1950
Bode and Fano publish their work on impedance matching
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

B n
Impedance-Matching Equation

1
Q b n sinh


1 a n
1 ln


1






1
 b n a n ln


1



Assumes Lumped-Element Circuits Exact for n = 1, 2, and ∞
QB n
Error < 0.1% for Γ > 0.10 (Max VSWR > 1.2)
B n
= Maximum fractional impedance-
matching bandwidth
B n
= (f
H
– f
L
)/f
0 f
0
= Resonant frequency = f
H f
L
Q = Antenna Q (Ratio of reactive power
to radiated and dissipated power}
Γ = Maximum reflection magnitude
within B n n = Number of tuned stages in the
impedance matching circuit

Bode Impedance Matching Equation
(Hendrik W. Bode)
R
0
Lossless Lumped-Element
Impedance Matching
Network
L
C
R
Generator Antenna
B

1
Q ln
π
1

Q

ω
0
L
R
B = Theoretical maximum fractional bandwidth for specified maximum reflection magnitude

Fano’s Impedance Matching Equations
(Robert M. Fano) n tuned stages
Alternate - series and parallel
All stages tuned to f
0 n = 1 is the tuned antenna
Γ n
  cosh cosh
 
  tanh cosh
 
 
 tanh cosh
 
  sinh
2
 sin

π
2 n
 
 sinh
 

QB QB n
(Γ)
NOTE: The Impedance Matching Equation is a closed-form approximate solution for the Fano Impedance Matching Equations

The Bode-Fano Equation
Fano showed that in the limit case of n = ∞
B


1
Q ln
π
1


We Started in 1973 With Wheeler’s Three
Equations for a Resonant Antenna
1950 Wheeler Lab Report 418
1 .
QB
 tan
 
EB
φ
EB

Magnitude of impedance phase at edge band frequencie s
2 .

3 .

2
1
 tan


φ
EB
2



(Optimum Single Tuning)
 
1
2
(Optimum Double Tuning)

Wheeler’s First Equation
Wheeler’s Small Resonant Antenna
Lumped-Element RLC Circuit
Example: Small Electric Dipole
Capacitor resonated with series L
Z
EB

R

1 j
ω
0
C

 f
H f
0
 f f
H
0


Z
Z
EB
EB tan


EB
R
R
 


1
1


 j
QB
ω jQB
0

1
CR

R


 f
H f
0
 exp f f
L



0 j φ
EB


Wheeler’s Optimum Single- and Double-Tuned
Impedance Matching (Proof by Inspection) jR
0
Optimum
Single Tuning
(Edge-Band Match)
Γ
1
= tan(φ
EB
/2)
Impedance transformation can not reduce Γ
1 f
H
Γ
1
Γ
2 f
H
Single Tuning
(Mid-Band Match) tan( φ
EB
) = QB
SC R R
0 f
H f
L
OC
φ
EB
= Impedance phase
at edge frequencies,
f
H
and f
L f
L f
L
Optimum
Double Tuning
Γ
2
= Γ
1
2
Impedance transformation
.
second tuning stage can not reduce Γ
2
-jR
0

From
Reflection
Chart
R
0
= 1
Single Tuning: Derivation of 
EB
 tan


φ
EB
2


Z
EB
 e j
φ
EB

EB

EB

1


 e j
φ
EB

EB


1 e j
φ cos cos
EB

 
φ
EB
EB
1

 j j sin sin
 
EB
 
EB cos
2 cos
2
 
EB
EB



1

1
2
2 cos
 
EB cos
 
EB

1

1

1
 cos cos
 
EB
 
EB
 tan
φ
EB
2

1


1
 sin sin
2
2
EB
EB

Wheeler
Single-tuned
Edge-Band
Matching
Wheeler
Double-Tuned
Matching
SC
Derivation of Γ
2
= Γ
1
2 f
0
L f
L
Γ
1
1 f
H f
H
& f
L
OC
Similar
Triangles


2
1


1
1

2
 
1
2
1
2
.
C

In 1973 we converted Wheeler’s three equations for a resonant antenna to a single equation
1 .
QB
2 .

1

 tan tan

 
φ
EB
EB
/2

φ
EB

Impedance
(Single Tuning) phase at edge frequency
3 .

2
 
1
2
(Double Tuning)
Single Tuning : tan

1

2 tan tan
2

φ

/
φ
2
/

2
 QB
1

1
2

1
-

1
2
Double Tuning : QB
2
Wheeler’s Equation:
Single tuning, n = 1
Double tuning, n = 2
B n
(

)

1 2
 n
1
Q
1
  n
2

2
1 -

2

2

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?

1973 Continued
Wheeler’s Equation:
QB n
2
 n
1

1
  n
2
QB
1

2

1
  2

2
1

 
 e ln



1



2
 e
 ln



1




1 sinh

 ln


1





QB
2

2

1
 

2
1

 

2 e
1
2 ln


1


  e

1
2 ln



1




1 sinh


1 ln
2
1



B n

1
Q sinh


1 a n a
1
1 ln
1




1 , and a
2


2
1
Q ln a n
1
 for
 
1
3

1973 Continued
Bode Fano Equation
B


1
Q ln
π
1
 a

 π
For all n and
 
1 / 3 :
Is B n

1
Q ln a n
1

???
Knew that a
1
= 1, a
2
= 2, and a
∞
= π
Ref.: L.B.W. Jolley, “Summation of Series,” Dover, New York, (410), p. 76, 1961
1

1
3

1
5


2
3


2

1
7


2
3
4
5


2

.........
 

2
1

1

1
3

1
3

1
5


2
3


2

1
5


2
3


2

1
7


2
3
4
5


2

1
7


2
3
4
5


2

.........
   a n
 k n 

1 s k a
1

1 a
2

2 a
3

2 .
333 a
4

2 .
667 a
5

2.756
...
a

 

1973 Impedance-Matching Equation
(Original Equation)
B n
(

)

1
Q sinh


1 a n
1 ln
1



Exact for n = 1 and 2
Approximate for Γ > 1/3, and n > 2
For
 
1/3
B
2
B
1

2 .
31 (131% Increase)
B

B
B
B
3
2
2

1 .
65 (65% Increase)

1 .
18 (18% Increase ?)
Sent letter to Professor Fano asking for help in determining accuracy of a n

1973 Fano’s Reply
3.2
2.8
2.4
2
A
1

ω c
 sinh( a sin
)




2 sinh( b )
(36)
π n 


  cosh( nb ) cosh( na )
(37) tanh( na ) cosh( a )
 tanh( nb )
(38) cosh( b )
1.6
1.2
0.8
1 ln
ρ
1
MAX .
0.4
0
0 0.2
0.4
0.6
0.8
1 1.2
1.4
n = ∞ ln
1


π
QB n = 6 n = 5 n = 4 n = 3
A
1

ω c
1.6
1.8
n = 2 n = 1
.
Fractional Bandwidth
(Band-Pass) Conversion
A

1

2 R
L
A
1

ω c
A
1

ω c


2
ω
2
0
R
QB
L
ω
ω
0 c
2
Fig. 19. Tolerance of match for a low-pass ladder structure with n elements

ln
1

3.5
2004 – Comparison of Fano and Original
Matching Equation
3
2.5
2
Used MATHCAD to solve Fano’s equations
Fano n = 1 n = 2 n = ∞ n = 3
1
QB

1
QB


 sinh

 ln
1
π
π
1
 ln
1
QB
3
1

 sinh



1 a
3 ln



1
 







1.5
1
1
QB
2
 sinh



1
2
.
ln



1
 





.
1
QB
1
 sinh

 ln
1



Γ > 1/3
0.5
.
0
0 0.2
0.4
1
QB
0.6
0.8

2004 Impedance-Matching Equation
B n

1
Q b n sinh




1 a n
1 ln



1
 

 




1
 b n a n ln



1
 

 b n
coefficient provides blending of the
“sinh” and “ln” functions
B
3
/B
2
= 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
Wheeler (Reflection Chart) n = 1,2
B n
(

)

1 2
 n
1
Q
1
  n
2
Fano (Network Theory) n = 1,2,3….∞
B n cosh cosh

1 tanh cosh
 
 
 
 
Q


π
2 sin cosh nb
 
2 n sinh tanh
 

  sinh
 
 n

1 B
1
(

)

1
Q 1

2
 
B
1
(

)
 sinh
1
Q sinh
 
2
 sinh
 

1
 sinh
 
 

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 f
L
Double-Tuned Locus
Max Γ Circle
Γ = 1/2
VSWR = 3 f
H
.

f
H f
L
Triple-Tuned Impedance Matching Cont’d
Edge-Band Frequencies on Horizontal Axis
Edge-Band Frequencies on Vertical Axis f
L f
H
.
.

Triple-Tuned Impedance Matching Cont’d
.

Triple-Tuned Monopole Antenna
On Infinite Ground Plane

Triple-Tuned Monopole Antenna (Continued)
Triple Tuned
Double Tuned

Triple-Tuned Monopole Antenna (Continued)