ACCELERATOR DEPARTMENT (AGS)

Internal Report

BROOIMAVEN NATIONAL LABORATORY Associated Universities, Inc,

Upton, L.I . , N.Y.

s- BQ?=- WHICH CAN Q

J. L. Walters

May 9, 1962

That the Smith chart is useful i n studying stub tuners i s well known, and

what follows here i s e i ther similar t o or a s l i g h t amplification of material ap-

1 pearing i n Ragan .

The following properties w i l l be shown.

1. The wavelength-spaced double stub tuner cannot tune a load which appears

a t stub 1 (the one nearer the load) with a conductive component greater than

the character is t ic admittance. (Among those excluded here a r e a l l r e s i s t ive

loads of l e s s than 50 ohms)

2. For the 318 wavelength spacing admittances cannot be tuned which appear a t

stub 1 with a conductive component greater than twice Yo.

3. A 5/12 wavelength spacing tuner w i l l match any admittance which appears

a t stuh 1 with a real p a r t l e s s than 4 timesYo. The r e s i s t ive loads which

cannot be: tuned are those of l e s s than 12% ohms (for a 50 ohm l ine) .

4. Any double stub tuner can match any f i n i t e load i f the distance to the

load can be changed by .jC wave length.

5. A t r i p l e stub tuner of any equal spacing can match any f i n i t e load.

6. The .j; wavelength spaced double stub tuner cannot be tuned by making

successive moves which reduce VSWR. For any other spacing the process i s

convergent .

' - leWean. G. L. ''Microwave Transmission Circuits" McGraw-Hill 1948 pp. 472-478 -7 r- . ?? ,-, .>-+..by ' -7 ,

L

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

- 2- JLW-4

7 , I f VSr!4R i s measurable, i t should be poss ib le t o a d j u s t a double s tub

tuner with 3 or 4 moves and a t r i p l e s tub tuner with 4 or 5 moves.

To any complex admittance (or impec!ance) with p o s i t i v e r e a l p a r t there

corresponds a point on the Smith char t . The r i g h t ha l f plane i s s o mapped

i n t o a c i r c l e t h a t the transformation,

Y Y* = Yo

1 + j t an 9

i s a r o t a t i o n about the cen te r of the c i r c l e of 28. This t ransformation

~ i v e s the admittance Y p , which appears when a l o s s i e s s t ransmission l i n e

of c h a r a c t e r i s t i c admittance Yo i s terminated by an admittance Y1 a t an

e l e c t r i c a l d i s t ance 0.

Figure 1 shows a Smith c h a r t on which some c i r c l e s of cons tant VSWR have

been drawn. A typ ica l po in t , 0.2 + 50.7 has been labeled Y1. Adding

negative susceptance t o Y w i l l change the combined admittance i n the 1

d i r e c t i o n of arrow a along the G = 0.2 c i r c l e . I f the measuring point

i s moved along the l i n e toward the generator , Y w i l l be transformed t o 1

values ly ing on the VSiJR = 7 c i r c l e i n d i r e c t i o n b. Adding pure conductance

w i l l change the t o t a l admittance along the B = 0.7 c i r c l e a s indica ted by

arrow c . The s tub tuning problem becomes t h a t of moving from an admit-

tance point such a s Y t o the cen te r of the c h a r t v i a paths li!ce a and b 1

only.

Consider a s the f i r s t example the double s tub tuner with 118 wave length

spacing. Assume t h a t both s tubs a r e s e t a t t h e i r wave pos i t ions so t h a t

they a r e adding no susceptance; point 1 i s the admittance seen a t s t u b . 1 .

-3- JLW-4 \

I f shor t 1 i s moved i n from i t s wave posi t ion, the negative susceptance

which i s thereby added t o Y l ' w i l l produce a new combined admittance lying on

the G = 0.2 c i r c i e i n d i rec t ion a. It can be seen t ha t moving the shor t

180 - 110 = 70 Smith char t degrees, (which i s 35 e l e c t r i c a l degrees, or

0.098 wavelengths a s read from the outer sca le ) , w i l l put the combined

admittance on the r ea l ax i s minimizing the VSWR t o a value of 5. This i s

shown on Figure 2 as point 2.

This combined admittance, Y2, a t s tub 1, w i l l appear a s Y a t s tub 2. To 3

f ind Y follow the VSWR = 5 c i r c l e fo r 90' cw, which i s equivalent t o 3 '

going 1/8 wavelength toward the source. Y3 = 0.42 + j0.93. The shor t i n

stub 2 may be moved t o minimize VSWR. An inward movement of 43' ( e l ec t r i c a l )

w i l l add 0.93 of negative susceptance t o produce Y on the r e a l ax i s . A t 4

stub 1 Y w i l l be t rans formei to Y5 by a 90' CCW path along the VSWR 2 . 5 4

c i r c l e . The shor t a t s tub 1 can be moved out t o supply e n o u ~ h pos i t ive

susceptance to move back t o a lower VSWP. a t point 6 on the r e a l ax i s . Y6

i s transformed t o Y a t s tub 2 where the sho r t can be moved i n to.produce 7

Y on the r e a l ax i s and so on. By moving one shor t and then the other i n 8

d i rec t ions t o always reduce VSWR, the center of the char t can be approached.

This mechod of adjustment i s qui te useful when working a t wavelengths of 10 cm

or l e s s ; a 3/8 inch coax double stub tuner f i t s i n the palm; the short ing

plungers; resemble hypodermic syringes and a r e ea s i l y moved. The stub tuners

which a r e used i n the 200 MC, 3 218 inch coax system of the LINAC a r e

adjusted by means of a handwheel. The screw i s % x 13 and the t~avelength i s

1.5 meters. I n the example j u s t given, t o t a l motions of the shor t s was 279'.

This i s equal t o (150)(13)(279)/(2.54)(360) = 595 turns of the handwheel.

I f a VSWR i s measureable, the Smith char t can be used t o move t o a match by

a more d i r e c t route a s shown i n Figure 3. Notice the 314 c i r c l e ; t h i s i s

pa r t of the G = l c i r c l e which has been rota ted about the o r i g in 118 wavelength.

I f the admittance can be moved t o t h i s c i r c l e , i t w i l l appear a t s tub 2 on the

G=1 c i r c l e where adding susceptance w i l l br ing i t d i r e c t l y t o the or ig in .

To move from Y = 0.2 + j0.7 t o IT = 1.0 d i r e c t l y requires a t o t a l motion of

the shor ts of 76'. An exploratory t r i p t o the r e a l ax i s must be made t o

f ind the locat ion of Y1. This can be done a s follows s t a r t i n g with both

shor t s a t the wave posi t ions:

1. Measure VSWR. The posi t ion of Y1 i s somewhere on the c i r c l e of t ha t VSWR, f o r example VSWR = 7 .3

2. Move the shor t a t s tub 1 enough t o change the VSbJR; i f moving the sho r t

i n (towards the main l i n e ) decreases VSWR, Y1 m v ~ s t l i e on the upper, o r +jB

semicircle, and conversely.

3. Move the ~ h 0 r t t o minimize VSWX; the motion of the shor t required t o do

t h i s gives; d e l t a B so t ha t Y l i e s a t one of the in te r sec t ions of the 1

appropriate B curve and the VSWR c i r c l e . The s i z e of d e l t a vswr can be used

t o determine which of the two in te r sec t ions . With the locat ion of Y1 known,

s tub 1 can be used t o move t o the rota ted G = l c i r c l e and then d i r e c t l y t o

the center .

Figure 5 shows the d i r ec t path fo r Y l y in s ins ide the G e l c i r c l e . 1

Figure 6 shows the G = 1 c i r c l e , the ro ta ted G = 1 c i r c l e and some of the

other G c i r c l e s f o r G g rea te r than 1. I f Y1 l i e s ins ide the hatched area ,

adding susceptance w i l l only move Y around ins ide the hatched a rea , and i t

w i l l be impossible t o g e t t o ro ta ted G = 1. The hatched area i s enclosed

by the G c i r c l e which i s tangent t o the ro ta ted G = 1 c i r c l e ; f o r the 1/8

wave length case t h i s i s G = 2.0. Figure 7 shows how these inaccess ible ad-

mittances a r e transformed when viewed a t s tub 2. Add5ng 9 j B a t s tub 2 w i l l

move Y onto the ro ta ted G = 1 c i r c l e and a t h i rd stub could be used t o go t o

the center along G = 1. Figure 8 shows a t r i p l e s tub tuner d i r e c t path

where stub 1 has been used t o minimize VSWR.

Similar reasoning appl ies t o other s tub spacings. Figure 9 shows the c i r c l e

of inaccessible admittances fo r 5/12 wave.length spacing; a l s o shown a r e the

rota ted G = 1 c i r c l e and a d i r e c t path t o the center. Figure 10 shows the

path of successive reduction of VSWR. The numbers near the G c i r c l e s ind ica te

the stubs which a r e being changed. The ne t motions of the shor ts a t s tubs 1

and 2 should add up t o those shown i n Figure 9 , i . e . path 1 , 2 e q ~ a l s the t o t a l

motion of s tub 1;path 3,4 is the t o t a l movement cf s tub 2. Comparing Figures

9 and 3, i t can be noted t ha t i n performing the same match a l a rger VSWR i n

the l i n e between stubs has resul ted than i n the case of the 1/8 wave length

spacing. This e f f e c t becomes more pronounced-as the spacing i s decreased;

ac tua l ly t h i s means a s the spacing approaches + wave. A t wave spacing the

rota ted G = 1 c i r c l e coincides with the unrotated G = 1 c i r c l e , the c i r c l e

of inaccessible admittances has shrunk t o zero, but i n f i n i t e VSlJR i s required

i n the in te r s tub l i n e fo r any matching. Figure 11 shows the rota ted G = 1

A c i r c l e f o r the - spacing, the region of inaccess ible admittances, and the 4

timing path. Figure 1 2 shows that: ~r~oving stub 1 t o minimize VSWR puts the

transformed admittance a t stub 2 ins ide the shaded area and on the hor izontal

a x i s so t ha t adding any susceptance w i l l increase VSWR. Figure 13 shows the

h s i t ua t i on f o r a spacing jus t s l i g h t l y l e s s than -. Both the d i r e c t path 4

1,2,3,4 and the path obtained by successively reducing VSWR a re shown. It

can be seen how slowly t h i s process converges.

JL!?:mo0n 5/9/62 Dis t r : AGS Linac Group

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