«MN NW | W WI H V ._—__— y ‘ W I M : - : -. 1::- .':. f ‘38.! 2"“: £;l~.'v”‘f'*-_é§r"“ :1“ :ka?O: “ilk: Km in:q..?.hhs‘flhf~£§nzz s!‘d‘& f‘d’ :t‘fi‘. - 1 1 . : lurk: f.."fs..“::‘:,' FCR A 'fii’ M: ’3‘" IL: Qi'?LJr.; 51:5 . 294513: Thosis fa! the Gwrm of M. S. MICHIGAN STATE COLLEGE F u. a”. £135? ”1192" a! ME? 3?: E394 This is to certify that the thesis entitled "An Adtsnatic Imuedance Matcher for a lalf Antenna" presented bg Farmon D. Strieter of the requirements for -wave Dipole . has been accepted towards fulfillment ., _ 7 .1 __1'_i._S_:__ degree 1n_-I.;..__ Major proflasscfl Date f//01/5 '7/ ' E T +11V Hblitted to the Sta 8 “allere in partial Degartne l. .1 .311 if: T} (IL: 413:-1‘ by Harmon D. Mtriet ”VT": f. :L inlay J School of Erréuate of rLcult ure and :Jlfill;£. 0” the $11»)ij L3; LITIC 1:; ~D-.L13u 1 ‘1 TT -LL for the de gree of 14$?ET 0* “CIUJCL nt of hie' r‘v-v- -‘ . '7'4‘ VA—‘-L—IA:O 4-r1v: _ u; i.‘.’L I' Studies gpplled reguire_ .L'-‘ 11 of Iichigan Jcicnce eats THESIS’ \ ‘s \‘\ PEETACE 0) PM The purpose of this thesis 1 to :reseht the develop- I ,. (‘1' meat of a method of sutouetic impedance u (‘7 chine for a spe- -J ) L cific tyye of antenn= L The author w'shes to exgress his thanks to the mezbers of the Electrical Engineering Deter ment at Lichigen State College for their rindness and assistance in meeting the re- quirerents for the Easter of science degree. Special thedks are 6J6 Irofessor I. E. Baccus, Irofesscr R. J. Jeffries end Protessor I. O. Ebert for their Kindness and tact during a difficult period; and to Dr. J. A. btrelzoff for his help and advice during the development 09 this thesis. ¥srmon D. Strieter 4 {38-1307 II. IV. v. VI. Outline Introduction 8) Purpose b) Discussion rensmission Lines 8) Input Ingedunce b) hsximun Power “rsnsfer 0) Reflection and standing waves Antenna Couplers a) matching Cransnission Line to untenna b) Coupling the Trans itter to the Line 0) General Considerations Lethod of Autonaticelly Couglin: the Cnrns itter to the Line Detection of error Signs Design Considerations Circuitry Construction GOD‘S? Conclusions Lxgeritentsl Results Restrictions Yecees thensions CD’S) bibliozraphy 'fl ’ ‘ I" "‘ . ~"_. - >-. .L;L:1¢‘ANJ-Lh-J.UKJ-LUJI-'I L‘IA‘ 5.40 “s u M;nns of tr;ns;”rtiug Loner fro. one faint to en- other, t“; s issio' li /“ sle lbbofiid ta. “:eic ,ie sevoivl fs tors “not ;?fect the groper choice o? 1 g,rticuler trans- $155 on 1'“: Cor - ;-v n vurgose. u. 34... -o. lusluz 1.1.9 -2; bf: i' C: ‘4‘ J? 2,.) llzl :il"oo. C) * ”b (—4 L3 ’1 L. P.) r—J ( r.J FJ L-‘T ( L L. ' ‘b (5. L i , >4 > J. S. r... C C H C ct ( u H L L p- } o L) :5 (f‘ t: L H (c "b ,1 S* C C L 1.. ‘. L. (. C l- 1.. (,1 C'.’ (L‘ *t' H a L“ r4 i L: PJ- '._J ("L‘ C c r *3 }_Jo C. *3 £17 cf (D H .t—J LL' H 0 follows: is 07 l J"! 5. M Figure l ’ . ,V - a: o . .0 ' a .' s _— . -' . fi " .1" .-- r~ ' ‘ v‘ - 7, . L1 .- ’- ,- . x . ~ . _ ._ _ ‘1 ,_ ._ - 1 . } 1 ‘ , , f- v _ | . 1 1 . ‘ . ~ L ‘1 ‘ ‘ Ll-AUU "‘f LKA. 14.; .LL.‘ I. txvi ‘I'-..L L D 'J U "“JsknJ , x. .. 0’ .J i- kl. I Li" 1v O_A‘~len tn t.e gbUVC hie.r,n. .e ere GUCel section lb of length Co or Phe shunt ednitcgnee per unit lei th of line is Y unos and thus the admitt Hue o? the element of lige is Yds mhoe. Vhe current dI tout flows ecroee the line or fro: one conuuetor EYds r» F—i I I DJ H II -.'-.. LY Diffareitieting eguqtioss l aid 2 with reageut to s: ——'.3' "— L’a use “as (i, I I I4 C‘; t 4 I I IN (<1 H Solving theoe equations by the ru;ee oi mifferemtietiooz C; C) U H N c4 from ‘i‘JhiCh we Clan write: 3 rats #257 E 2 Ae % Be I‘Z‘ys WE'VE Ce % De ( C»: ,p. I ) To solve for the constants A. B. C. std D, the constants of intergration, we must evaluate the boundary conditions. At 3 = o E z Er L = L 9r Then eguutioxs b and 4 become: @(fi \- \.-’ IP 0711) ( Differentiating equations 5 end 4 so toot we may use the some set of boundary coalitions, we get: - rfys -/:fys 5-: - We - Bfl’ye dI _ /Z}$ -12y8 a: Dme I C) m ‘4. (D 1 -5... Using egustions l uno a, we rat that: to II 0 :5. (D <4 (.0 I Tm (D I E 0) 6112830 H H I I k r I m I4 A \1 v #51 II 0 l U ‘<:IL\ E the v,lueo for the const;nts of intergrLtion is fullQAS: - 3‘ Ir z “ 797/ 2 y B 2 5;. - l£,.£ 2 2 y 02.13.171er 2 2 U I I I I .e mi; tnen rewrite equetions 5 and 4: z n Z - S . _ Er { Ir y; r5§° % Er - Ir/E’a /E§ (9} 2 V 3 V s T r I333 14/"' -/§§ I _ (Ir : nn/EIS / Ir : L ‘ée (10) 6 m I dinoe Lr 2 Irér eni by definition 20 =V§t equations 9 and 10 Gun be rewritten as follows: 3 ‘1 S Y?- ;{ by u _ " - fly ‘ : gfléagsfnegl. err. / .r to e (11) T. L‘J l t‘ H L O ' 9 -_‘I [:33 o Ir f :01 e _ Ar - 40 b 20 Ar 40 [Ya A {V L C" (D H (j, H I By further manipul tion, eguutio s 11 one i? renged in terms of hyicrboiio functions: 3 = chph /:§ 3 / Irno sigh/Z} s (is) I 3 Iroph]/:§ s / 5% sinh yfiy s (14) 1.; These equations fully define the volt 3e and current at H' I J H H. [3 (I) 0 any goint on the general trngsnies 0. on. (I (1 L. OJ L4 C.‘ (‘1‘ U” Iv H D I C) (1‘ <2 (L H O H D CL H (/3 Cf irimsry importenoe to the the intut ingedsnoe of ‘ M 33 - Lr cosh / Ir’o sinh 31 I53 1' - —— - “h“n ‘5 Is nr cosh {’lr a o = ,0 3r cosh.Yl Lo sinh Jl “ Zo cosh (l {for sinh Xl A pertioulir otse thet is of great interest is when the load imgedenoe Zr is eguel to the oherecteristio impedonoe of the transmission line Lo. Then, - 20 no oosh.JlAf';o sinh Xl : 70 Lo cosh.Jl.fr40 sinh Kl From the irootioul stendpoint, tie grorer termination of the trinsnission line is rost eosily get when the chersoteris~ tic ingednnoe is e reel eonstsnt. This hill make the trans- nitted energy q maximum. . Consider q source with in internul ifi}€CfiflC€ oi feeding ‘ '3 ' ‘-,-.— ‘ ~ ,' .. ,i‘ ». f ‘n ,— ‘ 3-. ,. . -. n 1. A .. ,, ' a i. H; «4&4 .2. v u o 4:10.: $01: 94;. ugh; .L i ' 9‘ V L; . ' lo a 1 9 fine r - I t 4 VOL r4 o t1~ ~Jurne u it The imgedsnces will be of the form Li Zr : hi x in = Br ,1 jXr She power delivered to the loss ingedsnoe mey be formulated on the besis of tne simple circuit shown in Figure 3: Zn 9 3 Br ~»\ ,.. r — I E . - . if; Firur ’ r - i t - s - v .s 4 8 Q P I T ai bf Ir (n f nr) f (X ¥ Ar) :ri II fl 0" if . 'I" ‘i‘ ’ n +' . ‘ ’\ " - ‘I '- ‘ . I' 4 r‘\ '\ ‘r‘ -" v'; 1'24' "x’, ‘ u .Lhe f1;:Al-flU.Il V.._'.lUC‘ Oi ufllb Chilesoloi‘i 4..) Cu L16 unverinlfledo Differentiating this expression with respect to Kr and Br: £155 __ fin? (X£+XR) . JXR .. [(35 +301 (Xw Xx)?2 fl :: (fiifflfl)+(x£+XR): Zflxflififlg) JR“ [(R£+Hn)+(X£+Xe)J Setting these pertiel uifferentietions to zero, we obtsin: 9135 _ I O aXR if 1&1 Z -AR L” = 0 if 111 = r; a 'riI' C (.1 C‘I' .- C This gives exgre sion for the conditions recesssry for maximum power trsnsfer. Thus, if the tno imrednnces ore pure resistunces, the conoition for nexinun 0 er trons fer is simgly thet they sh-ill be e‘uel. ghen they have reactince components hose or, the letter should be equal in megnitude and of opposite Sign. This is equiVslent to u resonsnce con- cition. In the trsismissi n line, the ch cteristic ii£-c nce A0 is the one irted'nce one the ter instion of the other. Since in communication m3 tens, ti; gOWLr evsileble is rela- tively low, it is essentiel tnnt the s stem be designed to de- liver the largest gsrt possible of the input power. Later on we slsll see tint tze condition for mexinum power transfer to be realized sinultn eouslv nith no reflec- tion occurs when the “txiootci stic ingessnce Lo reduces to s (3.02.574 n7: Reflection Referring to equations ll end 13 for the voltsges and currefl1 s on the trsnsti;11o line, .. r- s (ernole IZ§S¥ ;r - 30's [2- (ll) r213 1.11" (.10 . m 1 _ 1 s Ir(Zr idol 8/29: _ é; - zo VZ§ (12) she sr ; so e we observe that in the genersl Cage in thich the losd impe- p: H I e LY ;snce Zr is ct ecual to the characteristic ilbeiun e of t line Lo, es ch equation consis b5 01 tfiO terns, cxe of which varies exponentially with glue 3, the other with minus 8. Toking s to be csitive as measured from tie receiving end, the wave thst travels from the SELCihg son to the rece ivin5 end csn be identified as the cosponent Veryi;5 with eufs’ re- prese ts a wave of voltage ac current pregressing fro:1 t1e re- Cin: end. This “eve is called the .. ’-.J cciving end tower; the se. reflected wove. n reflected move will always be present un- a-t- ~. 4-” - 1s ee1ol to the 0.3130U6r10t10 impe- (U , ,.‘ r‘ .A 1 . .1'~ -'. ‘. less the lOdQ 115eo1ic- ounce. hell-cteo waves Will be tress t in Varglflg degiees ce- on the unou t of fliSmUtCh presett, with the reflected 1 I ‘v'\ y’ Y‘ I‘ v" :~l tJCx.LAllJJ Wave being o maximum under the two extra es conditions of o- pen or snort circuitec terrinetions. The ratio of oaglituaes of the reflect o and incident voltage moves st the receivimr end of the line is called the "reflec tion coefficient." From eguation ll, with s : 0, we get that the reflec- ’) *1 le cted voltese st lend 1613th VUlUUgse at 103d F43": tion coefficient K C -7- hr Lo Then equations ll std 12 may be rewritten using the re- flection coefficient as follows: 1'- :-' ' -' (5 E = Jorge»? )1 4-40) [‘8‘ :58 J (1-5) ZI'J 391 15- tie '85] ( 14) H H '1 again we see that if Zr = 23, the expression: for the volt- age and cirrent at any point on the line sirglify to n ‘ o = Epe vs (15) t I: Ipe 5’3 (16) Since we are concerned primarily with high frequency lines, and since the trans issioh lines that we ere concerned (1' (D with will be reietively short, the attenuation rss ray be neglected. In general X’=¢(fi/A?Nhere a( is the attenuation constant and ,A? is the ghase COASEQJt. Thus, for our lines of negligible attenuationCK = O, equatiOUs 15 en; 14 may be rewritten. 6’ ‘335 E : J3"]:‘(gir7z LOJZ— j S lie ] (1-7) 5 -st I : Ir( Zr )1 201E 3 S -1:e (18) £7 hub 383 he tern varying with e hes :r; viou 513 1‘21 C“ ( l (C Li) H 551 ( ..) c i- H' ’ "b H (D Q.- 11‘ U; (1" wave progressing fror the source toward the load, and the . . - '83 - A , term 1nvolv1ng e 3 as the refiectea Wave Moving frog the 10: d bec { to the source. The magnituoe of the reflected teve is dege;rwe1t on the value K, the efiection coefficient. The actuzl vulture at any point on the tre nsmission line ,. u is the sum of the incident and refl~cted W~ e vo that point. The resultant tote? volteg“ stead still on the line oscillati g in magnitude with re- sgect to time but having fixed positiOns of maxima end mini- ma. This is due to the phase of the reflecte; and incident weves canceling or adding to the emplituoe of each other. For an exemtle which will help to clarify whet has been soid previonsly, let us tohe J typical Case of 5 transmission line 2200 electrical length and terninete; in a resistance of one-half the char eteristi \ (i f. O sistehce of the line. flat—7' %Zfl=ge 0 ‘__ '— A better understanding of the Conditions Lresent any be had if we tronsfor; equotiOLS l7 and 13 to another form. i jEs .-st st ~jBSW Lr -~ “‘ 23 0 Pi H ('3’) jeS 6-333 333 _o-jes W I : %§[ZC€ 1: 71 :1ng _. " recognizing that 0 is 00323 = ex—iw‘ ‘ t .- ; ' 1:. - ‘2 ' ' .-., " -. :3 ”- end sinus - 9. L is no-1nec ea .43 A .-D I-‘Il 1:. .-. Lr(cos:7‘— ,1 32:1 cm3 ) (:21) I l I H H C. C\ U. :4“ m ‘k (—1. ”I O H ‘J V to {\3 Solvinj these equations with our 4. ind Jfld glotting and voltage ogves versie 0 the emglitudee of the resulte tygicel eXengle in U 3., *— LL U current distance, we arrive at the following H Mas .1} (in: f} +270° The glotting of the ., H 1 r'. ' I :_. {:3 D - -‘w‘J—v this L ceec clearly 11 “n .hese coves are know; LCD .fl- 4 ., ~ .1. - i x V i:;1;X.Lml.11 VOlUJ’SS 01' 4. age or current (node) is TI M Emex — ." r .. h J O : -‘--—-.—-— - L) H fishil D W1 -"~ l; r m {‘3 —————I AJX = D" Pt. Lain Ltending weve ratio wire trons ission O h"! (D slotted line ‘4 Sufi. the greetel the rcf The stigdifig wave r currem' must be used. Tigure 5 result; t voltoje ofil current for DHOES tne maxigi “ll ninihi present. .. 1-.- . A ' .- . r .-. n r. ‘« 1- ' a +-‘. .3 VJ {4111‘s ‘t'u'llV’db 0.1..‘4 tile 1"}th 0.1. Lille (3 ti-node) t the minimuh volt- 4. U culled the stelf' ratio. TN an bi reedily mede on ( coexiul lines, o The greeter the hignftuge of the fhctor ehd hism tch yreeent. 4,- U etio be defined in erms of the rv . - 'r I l t) ".1494 \l \J eflection coefficient es / l - 7A7 This way be reurrengec as - - 1 - [m x/ - '/—:.:in: M - 52-7—1 ’ / f mm of ecu tion 23 that germits cbtnining :4 5) Dre H O C' Figure 5 is the mignitude of K from a Knowleeu' f the stont‘ in; WJVE ra- LU (‘ O tic. Irecticelly steihiig, it is olnost imgossible to obtain a t;niin3 wove ratio of unity. Uozever, this ideal can be U2 closely egproiineted on; by careful en of 1"1euince latchcrs, he Lower lost one to reflection can be Kett very s ell. h discussion of Litcfiifig the trensxis- sion line to the loed or redigtor will be yresehted illustre- tion some 0? the LethOds and circuits thereby the standing wave ratio any be Keit a mini mum. -11- intennn I i“ewncw :AtChiflfl Jet1orKs Lvery antenna system. no matter what its lhy 3105 1 form, will have 3 definite v lue of impedence at the goint where the line is to be connected.‘ fine :roblem is to transform this antenna in; 1; the line. ?he con itions existi1“ :t t standing wive ratio on the line. There are Verious tyges of hetcninfi networis thet are eVei able to notch the antenna ingut imLedunce to the trons- mission line. In the followix3 section a few methods of im- pedance matching will be piv n, thou h for u more thorough end rigorous treothent of this these, the reader is referred to the references :5 given in the hptendix. the Quarter— eve Transformer \ h\r,.' :‘1:-1;D-. - ‘W- . . :7"? .‘r “ ’ «.1. ' .‘ r ‘1.” |‘- ‘1 -4; ehfr:s ion for the inlet inreddnce of A loseiess line i‘ .3 nay be obtained for ecuetions tl and LL. Dividing eguition 31 by re, we obtain .. '2 . 73 ' , n“ _ :r(ccs:i:s f 3 2.51néfis) uD '- fife) 9 Jr ccsf‘i-I—s 71 j Ho oinLEIV-s 9 ; Ho coszfize f j Zr Sin-i-s _ h Lr % 3 Bo ten éfize (n Bo / er ten g}: “COS-axis 71 3— sin 1’10 This Loy be furt he r Ie r11: e as Ar '3 tn 32?; 713° rn A 71 jar -12.. If the line is made a quarter-wave long, that is, if s : 1h/4 then equation 24 degenerates to 33 2 §%3 This states that the input impedance of the line is e- qual to the square of the cnaract:ristic resistance of the line divided by the load impedance. Thus we can thinK of the quarter-wave line as a transformer to match a load of Zr to a source of 33 ohms. This match can be obtained if the char- acteristic impedance R0' of the matching quarter-wave sec- tion of line is properly chosen so that RO' : ZsZr (35) To couple a transmission line to an antenna of impedance Zr, equation 24 may be used. The antenna impedance is then transformed to a value equal to the characteristic impedance of the transmission line. This value of Ro', the characteris- tic impedance of the matching section, is just the value re- quired to achieve critical coupling and maximum power trans- fer and consequently, the sta ding wave ratio will be essen- tially unity. The half Wave Line as a Transformer From equation 23, when a length of line having 3 a AVB is used r {jPo tan - a _ Z “S - “0 Bo / JLr tan ‘ Zr A half-wave l ngth of line may then be considered as a one to one transformer. This has a very useful application -13- if the antenna impedance equals th characteristic impedance of the transnission line and the line itself matches the im- pedance of the transmitter. However, these conditions are the exception rather than the usual Case. The exponential line for impedance transformation (com- monly called the "Delta" match); the "T" matching section, the "Gamma" match, are all methods which may be used to match the antenna input impedance to the transmission line. These methods are infrequently used and are more QifinUlt to con- struct and adjust then the quarter-wave matching transformer. There are many other variations of Circuitry that en- able an effective match between the antenna and transmission line, but the subject itself would be a volumnetric topic and be much too lengthy for the purposes of this paper. Refer- ence may be made to several periodicals as listed in the hp- pendix for further information. Coupling the fransmitter to the Line As has been mentioned previously, the tranSLitter, or radio freque.cy amplifier, requires a definite value of load resistance if the desired power output is to be obtained. Since the input impedance is will seldom be the sa.e as the load impedance required by the transmitting tube or tubes, the impedance must be transformed. The principle of impedance transformation is illustrated in the following figure: -14- :~ :3 5.1.4 gnu/173 C ? fgflISFOKfl/I; AN—TWW" ZS ‘9 dann/A/ll/ 0.1/1}, 2 FLA: Kzlarwwvis I? Figure 4 Figure 4 shows a networh connecting a load impedance Zr which may be thought of as the input impedance of the trans- mission line and radiator, to the two input terminals 1-2. If this connecting network is made of pure reactances, any power delivered to the input terminals 1-2 must in turn be transferred to the load 5r. Honever, the resistive and re- active components of the input or driving point impedance at 1—2 will, in general, be different from the impedance fir con- nected to the output terminals 3-4. Hence, the reactance networK may be considered to be and "impedance-transforming" networ: changinq the impedance Zr into an impedance Ls. If the networh contains resistive components, then the output power would be less than the input power and this po- wer loss would be undesirable. Therefore, the networks are made of reactances with the lowest possible resistance. Two reactance of opposite sign mry be arranged as in figure 5 to transform at one frequency a load resistance Pr to provide a desi ed load Rin for the generator, vhere Hr1<:§in. :413‘ ’77mT————-—~'e A 1 ‘éBw--9- ‘32-]? l/ae ‘ E 45: ‘0’ t Figure 5 Operation of this circuit may be readily understood if it is noted that the circuit to the right of the terminals a,b constitutes a parallel circuit that at anti—resonance ap- pears as a resistance load on the generator. The value of this resistance load is a function of the L to C ratio chosen for the reactance matching section. Thus one can select the values resulting in an antiresonant load Bin for matching to D “g. Simultaneous conditions to be realized are that the cir- cuit to the right of a,b be in anti-resonance and have an an— ti-resonant impedance equal to Bin. - l - Rrg ; W“ \IZ'C' L2 (3L) - - _ L Rantiresonance - Rln “5§; (37) from which L = RinRrC Inserting this expression for L in equation 26 and solv- ing for C, we obtain 1 Rin whim RI 83 C - l ( Likewise. from equation 27, L C:'T‘—.—.—_" kinhr -15- Inserting irtO equation 26, leads to L : (29) fie note that L will be imaginary if En:>'Rin. Therefore, if Rn:>’Hin, the matching circuits components of C and L should be interchanged. Ye will then obtain through a simi- lar process 1 - 1 Fr - ,fi L - WRrv‘Rin l (3“) _ Rin Hr , L - T Lin ' 1 (51) For matched conditions Bin will be equal to Rg and thus maximum power will be transferred. The above circuitry is often used for matching a radio frequency amplifier to the transmission line. These are quite useful and their simplicity is such that they are easily con- structed and adjusted. Generally, inductive coupling, or coupling by means of the magnetic field, is used to transfer the radio frequency power to the antenna coupler, transmission line and thus to the radiator. a modification of inductive coutling, called link coupling particularly adapts itself to good constructional procedure and has found a wide acceptance in the design of an- tenna coupling impedance matching. Link coupling gives the ef- fect of inductive coupling between two coils that have no mu- tual inductance. he link coils usually have a small number of turns compared with the tank coil of the final raiio fre- quency amplifier. The number of turns is not greatly important -17- because the coefficient of coupling is relatively independent of the number of turns on either coil. It is more important that both link coils should have about the same number of turns. Usually the length of the coil is small compared with the wavelength and therefore, the length is not critical if this is so. However, if the construction of the link neces- sitates that the length becomes a a;preciable fraction of a wavelength, transmission line methods must be used so that the input impedance to the radio frequency amplifier is of the proper value for maximum power transfer. The following diagram, showing a rsdio frequency ampli- fier coupled by means of line coupling to an impedance match- ing network clarifies what is meant by link coupling. _|_ _. .— ..-- 0MPFDWW€£. ' ‘1'" 13 fl» Tahiti 0&177W’7 fa}; b 3 Akflwanf a e :F 0 5* Figure 6 The pi impedance matching network, illustrated in Figure 7 is much more general in its application than the network given above. The pi network has the advantage of being able to match a wider range of impedances. :fi : «gm 1‘ 4: a. 7 £6 EC 0 Rad/3731? : 3 v‘ “1 Figure 7 -15- Tor the design consideration, let us take the general "T" section network of pure reactances. figure 8 shows a generator of internal resistance Bi connected to a load R2 through the T section network. 4_a. A a h ,______w4pl ea-A————v X. X2 8.: 25&~-—9- ‘xs ‘-mz}”v ’2; ,Ah :4 Figure 8 ”or the generator to be matches to the load, it is ne- cessary tn;t the impedance ziin at the terminals a,b be equal to Bi. Kith the load E2 connected, the impedance Ziin must N H H D I if} H H LJ. M H ‘k ,2 3115 : -X1X2 -Kit5 - .2Kafjrine {_jts?2 F2 ¥J(X2 fKal mhis can be simplified to .L slag g JF5R1 / ngh1 = -X1K2 - X1X3 - rftg / jzlgg / Equating the real terms, R192 : -X1X2 - X1X3 - XgXB (65) Since E1 a.d R2 are real and positive, one or more of ‘I "K the terms on the right side 0 this equation must be positive. This can be accomplished Owl by havi_: one of the reactances L1" to be Opposite in sign to the oth r. Tlus one reactive arm of the T section must be opposite in sign to the other two. -19- That is, the T section must be composed 0? one capacity end two inductances or two capacities and one inductance. By equating the reactive ter 3 of Equation 32: E1(Xg / i3) : 32(K1 X5) ' (34) 3:1 71:53 - $3112. (K2 7‘ is) (35) Equation 53 hey be rewritten as 21 33 : — (K1 / X ) (X3 % Kg) - X3 (56) Equations 55 and 56 may be combined in one of two ways, civinvz ‘ -I J 9 X2 7! X:- : 2(- 3%?- (X3“-RlR2 ) v : f _z.(x 2 - R n.) — x3 (37) 4&2 — R1 3 l- ‘0 The second combination yields: ‘xr 1r - R 2 X : R 2 - - ~ .l 1 f E: (x3 FIRE) is _ (53) Equations 37 and 38 supply the values for X1 and X2 arms of the T section in terms of the third arm X3. Two values of each are possible depending on the choice 0? the plus signs or the minus signs of the radicals. Since there are three unknowns and only two equations, it is necessary to assume a value for one of the reactance arms; after which the other two are readily deternined. By using the T to Z7 transformations, the T sections developed by the equations may be readily tragsferred tOlfir- sections. The transformation formulas are readily computed from the equivalent input 'npedance with the terminals open circuited or shot circuited. network sec- cf J; (D *3 d- O f he transformation equations for tion are given as follows: a ’ 59 Zb 2 2133 f 3123'“: 71 Zo_3 “5 30 : 3152 % 3133 % Zeus A further advantage of section networks, in addition to its impedance matching capabilities, is its action as a low pass filter and the filtering action reduces unwanted harmonics materially. This type of network is a convenient method of matching an end fed ?ertz (or random lengtm antenna. nutomatic Impedance-Ketchinz automatic impedance matching devices are desirable in‘ hatching the input impedance of a given antenna to the char- acteristic impedance of a given feeder line, such that the power to the antenna is at all times, within a given frequen- cy ranee, the optimum available in the feed r. The re ainier of this paper presents a solution to the s ecific problen of auto atically matching the input impedance of a "doublet " antenna to a 75 ohm transmission line at fre- q;encies from 6.6 to 7.5 mega“ycles. The limits of acceptable impedance matching must be such that the standing wave ratio will not exceed 1-1.35 at any frequency within this range. The general block diagram of the circuit is as follows: 1 the :s ——: =e 4: ‘—- aufiflwr Afirwmmm MW?" 72/1519 17734 ’ If r” otk TR : V fl 0-— Vigure 9 e power factor should be x l Sensing Unit ’ m This factor lends itself admirably for purely resistive with the a device that will between 7"‘or optimum power transfer, use as a "sensing " device for, with unity power factor, is hevin; '3318 1" "4'7 L‘J 7- l the antenna tile enable the designer to use the ~D L essentially unity. input ingedance of reactive components balanced out. measure the re ctive: mifferences o ghase a- this vvill "error-sensing" network. ‘OStcf- jeely discriminator circuit, developed for voltage and current, phase angle differences as the fiyzmzflmf frequency modulation detection, with a few minor variations The “‘ adapts itself adhirably for use as a phase angle detector. The basic circuit of the detector is shown in Wigure lO. IV 21/” 1;. a *3. I‘ our/”7‘33“ BeTécJ'ax Tigure lO -23- The significant cistinction is that the circuit of the phase-angle detector i3 untuxzd, so that it is phase-conscious, rather than frequency-conscious as is the more familiar cir- cuit of the Foster-Seeley freque cy discriminator. Qualitatively, the operation of the phase-angle detector can best be described by reference to the vector diagrams of *igure ll and 18. 54 If, .7 1 £2 4 15 g’, ;; 'fi .> V‘ I2 V; 12 V 56 4‘3 Unity P.?. Leading P.F. Lapping P.?. (A) (B) (C) Figure ll The voltage across capacitor 02 is aLWEys in phase with the voltage, V1, across the transmission line. Similarly, the voltage developed across the incuctor L2 is always dif- ferent in phase from the line current, I by 90 degrees. L, Using the baseof the vector diagram as the voltage a- cross CZ (E02, and with the coil center tapped, then the volt- age Ea, across La, leads the line current by 90 degrees. The voltage Eb, across Lb lags the line current by 90 degrees. Thus, with the coil cen er—tapped, Ea equals Eb and with re- spect to the waltage and current in the transmission line, the vector diagrams of Vigure ll correctly describe the conditions present. Referring to the secondary side of the phase-angle detec— tor of Figure 12, the operation may be described as follavs: The voltage across either D1 or D2, is the sum of the voltage across C2 and the voltage across the center-tapped 0011. La or Lb. With a change in phase angle between the , voltage across 02 (which voltage is always in phase with the transmission line voltage VL and the voltage across the coil), auses the vector sum of these voltages to change and the rectified voltages developed across the diode load resistors, El and R2 will change. [a6 64 5., £36 . [a 5;» Eu .- £2 £7 £7 5 [01» id; “ f4, Leading P. F. 'Lagging P.F. Unity P.V. (A) (B) (C) Figure 12 Referring to Figure 12, it is seen that when the line cur- rent leads the line voltag., the voltage Edl will be less than E and the output voltage will be positive. Likewise, when d2 the line current lags the lire voltage, wigure 123, Edl Will be greater than Ed2 and the output voltage will be negative. at unity power facto , the rectified voltages are equal and the output voltage is less than zero. Thus, the two prime requisites, zero output at unity power-factor (resistive load), and the sign of the error volt- age being dependent on the sign of the phase angle, are met so that this device is suitable to control a servo-system to cor- rect the phase angle to zero. 31th the sensing unit now available, the choice of the coupling network must be decided. Vita the design criterion in mini, the small range of freguencies envolved makes it logical that the following network would be well suited. 23W again, from practic;l exPsrience, the reactance of he double antenna that is ”cut" for the mid-frequency of this range of frequencies, will be fairly Shall such that the com- ponent of L and C will not be too critical. It was decided that the coupling network in use by the author would be a- ‘apted for automatic control of the impenan e matching of the transmission line to the antenna. This was built up several years ago from surplus components then available. An ordinary D.C. amplifier amplifies the output from the "Sensing“ circuit and controls the reversible motor which in turn, controls the variable condenser such that the Optimum output is fed into the radiating system at all times. The following schematic was subsequently built up for obtain- ing the experimental results. \\ \\\§n. Vcksax wake? \o wauKva ‘10 &0)-——-- (577111 —fl|' \\D l / 4. “e \ Cl 3., 3 C) 4 {1 JP. U 9 010 011 D1, R1 R2, ‘Q “4 a5, F*7 -36A- Parts List for T ner: 200 mmf 05, Cg 100 mmf .Ol mfd 06, 07, C8 .015 mfd .05 mfd, 600 volts 40 mfd, 200 volts Dz, 1334 Rectifier 502, % watt Rs 0.1 megohm, l watt 50K potendometer s6 47K, % watt 0.1 megohm potentiometer 20K, i watt 50 ohms, g watt 1 turn coil, center tapped Ryz S.p.s.t. relay, 5000 ohm RFC 3.5 mho, P.F. chohe 2 Selenium rectifier . 5‘ no can the the prOper d surement of for details on the operati'n fiers, a concise ex The fol as set up in Frequency in Nos. 0 o o o o o o o O n / QQQQQQO‘JO‘O‘C‘DO} developn various planation of this be seen from the schematic, the crucial point is ‘0 ent of this diagram is the 'sensing" unit. after etermination of.a suitable circuit for the mea- error, straightforward circuitry results. of the direct current ampli- ref re given in the appendix that give '._ ""5 CI I 0112,84 (‘3 simple circuit. Experimental Pesults: lowing data was obtained from the prototype model the laboratory. Standing Jave Patio 0 o o o o o o o 0 012302610 l—‘l-JE‘OC): :43 25 01 7‘ (II HHHHHHHHHHH This information can be conveniently graphed (refer to figure 15) and shows that our final results are guite accept- able. 7ithin the range of frequencies dasired, the standing wave ratio is well within tolerable limits. 6.?! .3530; :00 15...... c a: z