Impedance Matching Equation: Developed Using Wheeler s Methodology
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1 Impedce Mtchig Equtio: Developed Usig Wheeler s Methodology IEEE Log Isld Sectio Ates & Propgtio Society Presettio December 4, 03 By Alfred R. Lopez
2 Outlie. Bckgroud Iformtio. The Impedce Mtchig Equtio 3. The Bode d Fo Impedce Mtchig Equtios 4. Wheeler s Sigle- d Double-Tuig Equtios 5. Coversio of Wheeler s Equtios to the Origil Impedce Mtchig Equtio 6. Developmet of the fil form for the Impedce Mtchig Equtio 7. A ote o Triple-Tued Impedce Mtchig
3 Bckgroud Iformtio 940s Wheeler develops impedce mtchig priciples A Wheeler desiged double-tued impedce-mtched IFF te plyed criticl role i WW II Bode d Fo publish their work o impedce mtchig 950 Wheeler publishes Report 48, tutoril o impedce mtchig tht fetures the reflectio chrt s primry tool For sigle- d double-tued impedce mtchig, it presets three equtios tht qutify impedce-mtchig bdwidth limittios relted to specified mximum reflectio mgitude Bsed o the works of Bode d Fo, it qutifies the lw of dimiishig returs for impedce mtchig circuits beyod double tuig 973 Wheeler s three equtios re coverted to the origil Impedce Mtchig Equtio 004 Usig MATCAD to solve Fo s equtios, the fil versio of the Impedce Mtchig Equtio ws developed
4 B Impedce-Mtchig Equtio ( ) b sih B Mximum frctiol impedcemtchig bdwidth B (f H f L )/f 0 f 0 Resot frequecy f H f L Ate (Rtio of rective power to rdited d dissipted power} Mximum reflectio mgitude withi B Number of tued stges i the impedce mtchig circuit l + b l Assumes Lumped-Elemet Circuits Exct for,, d B Error < 0.% for > 0.0 (Mx VSWR >.)
5 Bode Impedce Mtchig Equtio (Hedrik W. Bode) L R 0 Lossless Lumped-Elemet Impedce Mtchig Network C R Geertor Ate B π l ω 0 R L B Theoreticl mximum frctiol bdwidth for specified mximum reflectio mgitude
6 Fo s Impedce Mtchig Equtios (Robert M. Fo) tued stges Alterte - series d prllel All stges tued to f 0 is the tued te th cosh ( b) ( ) ( ) ( ) cosh cosh si sih π th cosh ( ) sih( b) ( b) ( b) B B () NOTE: The Impedce Mtchig Equtio is closed-form pproximte solutio for the Fo Impedce Mtchig Equtios
7 The Bode-Fo Equtio Fo showed tht i the limit cse of B π l
8 We Strted i 973 With Wheeler s Three Equtios for Resot Ate 950 Wheeler Lb Report B t ( φ ) φ t φ Mgitude of impedce phse t edge - bd frequecies (Optimum Sigle Tuig) (Optimum Double Tuig)
9 Wheeler s First Equtio Z Z Z Wheeler s Smll Resot Ate Lumped-Elemet RLC Circuit Exmple: Smll Electric Dipole Cpcitor resoted with series L t f H f R + j ω C 0 f 0 f f H R + j ω0cr f 0 R( + jb) R exp( jφ ) ( φ ) B 0 H f f L 0
10 Wheeler s Optimum Sigle- d Double-Tued Impedce Mtchig (Proof by Ispectio) Optimum Sigle Tuig (Edge-Bd Mtch) t(φ /) Impedce trsformtio c ot reduce jr 0 f H f H Sigle Tuig (Mid-Bd Mtch) t(φ ) B SC R R 0 f H f L OC φ Impedce phse t edge frequecies, f H d f L f L f L Optimum Double Tuig Impedce trsformtio d/or chge i of. secod tuig stge c ot reduce -jr 0
11 Sigle Tuig: Derivtio of t φ From Reflectio Chrt R 0 Z e e jφ e cos cos jφ jφ + + ( φ ) + jsi( φ ) ( φ ) + jsi( φ ) + cos ( φ ) cos( φ ) + + si ( φ ) cos ( φ ) + cos( φ ) + + si ( φ ) cos( φ ) φ t cos( φ )
12 Derivtio of Wheeler Sigle-tued Edge-Bd Mtchig SC L f H OC Similr Trigles f 0 f H & f L Wheeler Double-Tued Mtchig f L. C
13 I 973 we coverted Wheeler s three equtios for resot te to sigle equtio.. 3. B t t ( φ ) ( φ /) φ Impedce (Sigle Tuig) (Double Tuig) phse t edge frequecy SigleTuig : t ( φ) t t ( φ / ) ( φ / ) B - Double Tuig : Wheeler s Equtio: Sigle tuig, Double tuig, B ( ) B -
14 973 Cotiued At this poit we hd explicit expressio tht relted B,,, d for sigle- d double-tued impedce mtchig We were wre of the Bode d Fo results Wheeler clerly defied the lw of dimiishig returs for dded stges beyod double tuig Oe remiig questio ws: How much bdwidth icrese c be chieved with triple tuig over tht of double tuig?
15 d, 3 for l l sih B > 973 Cotiued B Wheeler s Equtio: l sih e e B l l l sih e e B l l
16 π l π B Fo Equtio Bode Cotiued??? l B Is 3 : / d ll For > π π π s k k Kew tht,, d π Ref.: L.B.W. Jolley, Summtio of Series, Dover, New York, (40), p. 76, 96
17 973 Impedce-Mtchig Equtio (Origil Equtio) B ( ) sih l Exct for d Approximte for > /3, d > Set letter to Professor Fo skig for help i determiig ccurcy of For /3 B.3 (3% Icrese) B B.65 (65% Icrese) B B3.8 (8% Icrese?) B
18 973 Fo s Reply A sih( ) sih( b) ωc π si cosh( b) cosh( ) th( ) cosh( ) l ρ MAX. th( b) cosh( b) (36) (37) (38) c A ω l Frctiol Bdwidth (Bd-Pss) Coversio R A L A R ω0 ω ω L ω A. ω B c c 0 π B c Fig. 9. Tolerce of mtch for low-pss ldder structure with elemets
19 l 004 Compriso of Fo d Origil Mtchig Equtio Used MATHCAD to solve Fo s equtios Fo 3 B B 3 l π sih l π B sih 3 l > / B. B sih. l sihl B
20 004 Impedce-Mtchig Equtio ( ) + l b l sih b B b coefficiet provides bledig of the sih d l fuctios B 3 /B.4 (4% Icrese)
21 Coclusio Wheeler s developmet of the priciples for double-tued impedce mtchig ws mjor cotributio. Although it ws developed for lumped-elemet circuits it hs broder pplictio Oe c see by ispectio tht his solutios were optimum We hve developed the Impedce-Mtchig Equtio, closed form solutio for the Fo Equtios, which we hope will be helpful d useful to the commuity Wht impressed me the most i ll of this work ws the remrkble fct tht Wheeler s results, usig the reflectio chrt, were ideticl to the results obtied by Fo usig high-level etwork theory
22 Wheeler d Fo Wheeler (Reflectio Chrt), B ( ) B ( ) th cosh cosh cosh Fo (Network Theory),,3. ( ) ( ) ( b) ( ) si sih th cosh π ( ) sih( b) ( b) ( b) B ( ) B ( ) sih sih ( ) sih( b) ( ) sih( b)
23 Triple-Tued Impedce Mtchig
24 Triple-Tued Impedce Mtchig Which circle, A or B, should be used to positio the edge-bd frequecies o the Mx Circle Circle A or Circle B f L Double-Tued Locus Mx Circle / VSWR 3 f H.
25 Triple-Tued Impedce Mtchig Cot d Edge-Bd Frequecies o Horizotl Axis Edge-Bd Frequecies o Verticl Axis f L f H f L f H..
26 Triple-Tued Impedce Mtchig Cot d.
27 Triple-Tued Moopole Ate O Ifiite Groud Ple
28 Triple-Tued Moopole Ate (Cotiued) Triple Tued Double Tued
29 Triple-Tued Moopole Ate (Cotiued)
Impedance Matching Equation: Developed Using Wheeler s Methodology
Impedce Mtchig Equtio: Developed Usig Wheeler s Methodology IEEE Log Isld Sectio Ates & Propgtio Society Presettio December 4, 0 y Alfred R. Lopez Outlie. ckgroud Iformtio. The Impedce Mtchig Equtio. The
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