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)