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RADIO SC IEN CE Journal of Research, NBS/USNC- URSI Vol. 68D No.4, April 1964
Some Basic Microwave Phase Shift Equations
Robert W. Beatty
Contribution from the Radio Standards Laboratory , Natio n al Bureau of Standard s, Boulder , Colo.
(R ecei ved November 14,1963 )
The phase di fferences between terminal var i ab les (v oltag e, cu rr ent , ar t r ave li ng wave types) at the output with resp ect to s imilar ones at th e in put of a 2-port are exp r essed in te rm s of the scat t er in g coefficients of t he 2 -p ort and of the r eflection coefficients of t he system into w h ich it is in se rted. A va ri ety of phase s hift s may be defined for a give n 2-port, depending upon wh i ch of the term m al va n a bles a rc co nSIdered, whether or not generator :Lnd l oad r e fl ect ion coefficients are assumed to va nish , and in wh ich dir ection the 2- port has bccn inserted into the syste m. It is s hown that a reaso na ble e hoice for one of the two " ch aracte ri st ic" pha se' s h ifts of a 2-po r t is lf 2J, the ar 'gu me nt of S 2J, one of the s cattering coeffici ents. It [ollo \l 's that t he other "cha r acte ri s tic" ph a c s hift is lfr2. Th e co rT es pondin g change in characteristic pha se s hift for va ri ab le pha se shi fter::> is t he change in eithe r "' 21 or h 2 ( \ 'hi ch eve r is appropr i ate ) from th e i nitia l to th e fina l set ting.
. Ideal pha se s hift ers a r ~ di sc ussed , a nd ex pr essio ns for t Il(' ch ange in output l eve l of va r-r a bl e pha se s hlfk rs a re gIven. Th e Importance of us ing a llonrefl ect inrr syste m in ph ase s hift meas lIrem ents i::> e mph asi~ed. "
1. Introduction
Th e m eas ur e lll cllL of Lhe p hasc s hi ft in g pl'ope rLi es of microwave d evices llfLS received relatively li tt le fL Ltentioll, but is becoilllng more important. On e canno t fiJld a refcreJice to microwave phase shilL m easu rements in an y of the well-knowll te xts on microwave mea s urement s that h ave be en pub lished to dat e. How ever , ev id ences of in cr eas in g inl erest include the r ecent devel opment by commercial organi zat i.on s of severa l differen t s wep t J'requell cy systems for microw ave phase shift mea s uremenl s. In addition, paper s on this su bj ect have b een sur- veyed in a rece n t article by R. A. S parks [1963]. It seems appropriate at this time t.o examine co ncepts of microwave phase s hift in order to clefLr ly
1
b rG r l^ a
i (^5 ) VI 52 1
°
understand what ty pes of phfL sc s hift may occur ",nd how they may be descri bed in a q u anLiLaL ivc WfLy. Spec ifically, it is p 1"O pO cd in t hi s pap er to presen t expressio ns re l at in g Lil e phase s hifL s associ ated with fL !2- port to Lhe sCfLt Lcl'in g coefficients of thfLt 2-port fLnd the re fl ecLion cocffLcients of Lhc system in which Lhe 2-port is COl1n ected.
2. Ge n era l
Several Lypes of phAse s hift of 2-ports may be cons idered , as wi ll be sh own wiLh reference to fi gure l. The usefuln ess of thc diil'erent types of phase shift consider ed will depen d upon what types of
2
r -~^ ° L - b
i
(^5 ) V (^5 )
°
1 2 FIGURE 1. Representation of a 2-port, showing t wo se ts of terminal vw·ia bi es.
detectors are used in a measurement of phase difference a nd whether they respond t o wave am - plitudes or gen eralized vo lt age an d c urr ent. (An elect ri c fie ld pro be in a sl otted line would r esp on d to v, for example.) In figure 1, t he termina l surfaces 1- 1, and 2- 2, in the waveguide leads of the 2-port, are the places where the complex amplitudes a and b of the incident and emergent voltage waves, a nd v and i, the generaliz ed voltage and curr ent, [Kerns, 1949] are to be considered. Th e assumptions inherent in this representation of a 2- port , such as single-mode propagation in the lossless waveguide leads have been set forth by D. M. Kerns [1949 and 1951]. The rel at ion sh ip between the two sets of termina l va ri ab les is as fo llows
v= a + b }
Zoi = a - b
(1)
Th e amp litud e be of the ge n erator wave, a], b ],
and r e are rel ated by the followin g eq uation:
(2)
One can cons id er the phase shi ft of the 2-port to be the difference in phase between Vz and V I, between iz and ii' between bz and aJ, or between b 2 and be, for exampl e. "The ph ase shift of a 2-port" may, therefore, be a misleading and amb iguous expression , since it seems to imply either that there is only one, or that all phase shifts of a given 2-port are the same. In the fo llowing, eq uations will be given to show how these va rious phase shifts will differ in general, and w1der what conditions some of them may be the same. The equations will be given for the case of a linear 2- port (which may be nonreciprocal) inserted into a system in such a direction that a generator feeds arm 1 and the lo ad terminates a rm 2. For simpli c- ity , the symbo ls such as t/; v for phase shift do not in dicate that this direction has been chose ll. One can easily obtain the corresponding phase shift for the opposite direction of energy flow thru the 2-port by interchan ging subscripts 1 a nd 2 in the equations. If the phase shifter is a reciprocal one, the condi-
tion Z 01 8 2 1 = Z 02 81 2 will hold, where Z 01 and Z 02 ar e
th e characteristic impedances of the w:weguide
leads 1 an d 2 of the 2- port. Very often , Z 01 = Z02, an d
the re ciprocity condition is written 8 2]= 8 12 , Thi s
condition may be s ub sti tuted into the pha se shift equations in order to reduce the numb er of variables by one for a reciprocal 2-port.
3. Phase Shift Equations
The derivation of the following expressions ror phase shift will not be given , as th ey follow from s traightforward algebraic manipulation iof eqs (1 ) and (2) and the scattering equations of the 2-por t.
bl= 811al + 812a2 ~.
b 2 = 8 z1 al + 8 2Zaz j
3.1. Phase Shift of v
The ph ase shift of v, denoted by t/; ., may be written as fo ll ows
This phase shift depends upon the reflection coefficient of the lo ad as well as upon the character is- ti cs of the 2-port. When the load is nonrefiecting,
(5)
If one emp loys the impedance matrix instead of the scattering matrix of the 2-port, the equations cor- responding to (4) an d (5) are
(6)
and
(7)
where normalized impedances are used. For an open-circuited phase shifter,
(8)
3.2. Phase Sh ift of i
Proceeding in a similar way,
When th e load is nonreflecting,
(10)
If one employs th e admitlance matrix in s tead of the scatterin g m airix of th e 2-port, the equations corresponding to (9) a nd (10) are
and
(12)
where norma liz ed adm.ittances are use d.
J '> I
The differential phase shift is
i11/; -. [1S 21 • (l-iSllrG)(I - iS22rL) _i SI2iS21rGrL].
- aIg iS 21 (1-1s11rG)(I-1S22rrJ-1S1lS21rGrL (27) If only t.he generator is nonreflecting,
(28)
If only the load is nonreflecting,
(29)
If both generator and load are nonreflecting,
5. Cha racte ristic Phase Shift
The foregoing equations have served t.o illustrate that the pha se differ ences considered will in genernl depen d not only upon the characte ristics of t.he 2-port, but nlso upon t he characteristics of the load, and in some cases also upon the characteristics of the generator. If Ol1e is intereste d in a phase difference which depends only upon characterist ics of the 2-port, then (5), (10) , (15), and (20) can be considered. Of the se, (15) and (20 ) are simplest. Thu s it would seem desirable to select 1/; 21 as one of the characteristic phase shifts of a 2-port. (The other would be 1/; 12') It would be defuled 2 as the pha se difference between b 2 and bG when nom-eflecting ge nerator and load fire connected to arms 1 and 2, respectively, of the 2-port. The differential phfise shift of a 2-port in a non- reflecting syste m as given by (30) is the n sim ply the differen tial characteristic pha se differen ce, as defllJed above. "\Then phas e differences of 2-ports are mensured lllder different source or load conditions. different r esu lts will be expected. The discrepande s can be called mismat ch errors, which can be evaluated by reference to t he foregoing equations.
6. An Ideal Phase Shifter
The co ncept of an ideal phase sh ifter is useful for comparison purposes in evaluatin g the performance of actual phas e shifters. Such a phase shifter would be nonreflecting (Sll = S2 2= 0) , and lo ss less, bu t could be nonreciprocal. Consequently
2 This definition is in harm ony with thut given in IRE Standa r ds on Antennas and Waveguid es: Wavegu ide and waveguide component measur ements , 1959, Proc. IRE 47, No.4, 568 - 582.
One notes that the phase shifts of an ideal phase shifter more closely approach 1/;21, than one which is not ideal, even if generator and load reflections are present. For example,
(31)
and (34)
The differential phase shift of an idcal phase shifter IS
(35)
It is seen that the use of an ideal phase shifter does not obviat e the need for a nonreflecting system, if the characteristic phase shift is to be produced. In an ideal situation, the output level is not chf) .nged when the pha se shifter is adjusted. How- ever, it can be seen by application of (26) that the change in level of either V2 or b 2 expre..<;sed in decibels is
t:.L = 20 10g lO [^16 ib2[ 2
= 20 10g lo [i S^21. (l -f Sllra)(I - fS22 rL ) _ 1S1lS21r Gr L[.
1S 21 (1- ' Sllr G) (1 -' S22rL) -'SI2'S21r Gr L
(36)
Even with an ideal phase shirter, a change in level will occur, given by
i1L (for ideal pha se s hift er ) = 20 log lo
1
1 -rG rL ei (1 11'"1 +1f I2 J[. (3 7)
1- r Gr LCi (' f2!+'fI2 )
One can see from the above that there would be no change in level from an ideal phase shifter in a non- reflecting system (r G = r L = 0), but that such a change in level would generally occlli' if the phase shifter were not idea l.
7. Conclusions
It hn,s been shown that one can reasonably choose either 1/; 21, the argument of S21, or ""1 2, the argument. of S12, as the ehamcteristic phase shift of a 2-port. For a variable pha se shifter, the characteristic differential pha se shift i11/; equals the change in 1/; 21 between initial and ma l settings. In measuring these quantities, it is important to insert the phase shifter into nonreflecting systems
-;'
(rC= rL = o), a nd in us in g the phase shifter there-
after, it is no less important to duplicate Lh ese co ndition s. AllY deviation from these condition s will result in a mismatch error , and these h fwe becn analyzed [Scharer, 1960] for some types or measUl'C- Jll cn t systems. The use or an ideal phase shiJter which is lossless alld nonreHecting does not obviate the need for a nonr e fl ect in g system in pha se shift measurements, al though the mismatch errors will usually be sm aller ir thc phase shifter is ideal. It was sh ow11 that there will be no change in the output level of an ideal variable phase shifter when operated in a nonreflect- ing sys tem, but some change can be expected if the system has reflections.
Constr uctive criticism an d suggest ion s for im- proving the pre sentati on were given by David M.
K el'll s of NBS and Helmut M. Altschuler of the Polytechnic Institute of Brooklyn. The encourage-
m ent given by G. E. Schafer of NBS is also gratefully
ack ll owledgcd.
8. References
Kern s, David M. C\I[ay 1949), Ba sis of t he appli cat ion of n etwo rk e quations to wav eg uid e problem s, J. Re s. NBS 42, No.5, 515- 540, RP. Kern s, David M. ( Apr. 1951), Analys is of sy mm et rical wave- g uid e junc tions, J. R es. NBS 46 , No.4, 267- 28 2, RP2195. Sc haf er, George E. (Nov. 1960 ), Mi s match e rror s in micro- wave phase s hift meas ur eme nt s, IRE Tran s. on MTT 8, No.6,6 17- 622. Sparks, R i chard A. (Jan. 1963 ), Microwave pha se m eas ure- ment s, Microwaves 2, N o.1 , 14- 25.
( Paper 68D4 -3 52)
