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IEEETRANSACTIONS ON MICROWAVE THEORYAND TECHNIQUES, VOL. MTT-32, NO. 10, OCTOBER1984 1275
Finite-Element Analysis of
Dielectric-Loaded Waveguides
MITSUO HANO
Abstract —A finite-element analysis in which nonphysical spurious solu- component finite-element methods:
tions do not appearhasbeen established to solvethe electromagnetic field
problem of the closedwavegnide filled with various anisotropic media. This 1) occurrence of the nonphysical spurious solution,
method is basedon the approximate extremization of afunctional, whose 2) restriction on the discontinuity of either permittivity
Euler equation is the three-component cnrlcurl equation derived from the
or permeability of the media.
Maxwell equations, with anewconforming element. Specific examples are
given and the resnfts are compared with those obtained by exact solutions
andIongitmfimd two-component finite-element solutions. Very closeagree- II. VARIATIONAL FORMULATION OFTHE
ment was found and afl nonzero eigenvalues hiwe been proved to have MAXWELL EQUATIONS
one-to-one correspondence to the propagating modesof the wavegnide.
Consider an arbitrarily shaped metal waveguide im-
mersed in several anisotropic and Iossless dielectrics as
1. INTRODUCTION
shown in Fig. 1. This waveguide is assumed to be uniform
A SA RESULT of the broad variety of practical appli- along its longitudinal z axis. ? and ~ denote the tensor
cations of the closed wavegtiide filled with several permittivity and the tensor permeability without off-diag-
kinds of media in microwave and optical frequency re- onal elements, respectively, and they are assumed to be
gions, the development of methods to solve the associated constant in each region.
electromagnetic field problems has attracted the attention Maxwell curl equations for time-harmonic fields are
of many researchers. The finite-element method, which
vXH=jdE (1)
enables one to compute accurately the mode spectrum of a
waveguide with arbitrary cross section, has been widely vXE=–jcofiH (2)
used [1]–[7]. However, the two-component finite-element where the vectors E and H are the dielectric- and the
solutions have been known to include nonphysical spurious
magnetic-field intensity, respectively, and o is an angular
modes [2], [3].
frequency. From (1) and (2), we construct
Konrad has derived a three-component vector varia-
tional expression for electromagnetic field problems [5], E= –(j/a){-lV XH (3)
and has selected a family of functions, as a trial solution, in H= (j/u)p-lv XE. (4)
which each component of the vector field is continuous
By taking the curl of (3) and (4), and then substituting into
along all interelement boundaries [6]. ,Therefore, the
(1) and (2), the following common curlcurl equation is
material parameters are restricted; either permittivity or
obtained:
permeability should be constant in all regions. Spurious
solutions have likewise appeared as the result of this v Xp-yv x v)–arz~v=o (5)
numerical calculation.
We investigated his three-component formulation for the where V denotes either E-or H, and j and Q are the
condition required of trial solutions, and have concluded material tensors as shown in Table I.
that the necessary and sufficient requirement for the trial At the interface of the region, an appropriate boundlary
solution is not so strict as the one in [6], condition must be satisfied by the field vectors. The inter-
In this paper, the functional describing the behavior of face continuity between two contiguous media (say the rth
,.
the electromagnetic fields in anisotropic waveguides is in- and sth) requires that
troduced and the set of trial functions perfectly satisfying nx(v”–v”)=o (6)
the boundary conditions required in the functional, a so-
(j/~) nx(fF1v xV’-j;lvxv’)=o (7)
called conforming element, is derived. This approach has
improved the following two serious problems which are
along their common boundary, where n is a surface normal
inevitable in the previous two-component and three-
unit vector. On the other hand, for .$e electric wall and the
magnetic wall, the boundary condltlon of the electromag-
Manuscript received February 14, 1983; revised May 21, 1984. netic field requires either
The author is with the Department of Electncaf Engineering, Yama-
nxv=o (8)
guchi University, Tofciwadai, Ube 755, Japan.
0018-9480/84/1000-1275 $01.00 01984 IEEE
1276 IEEETRANSACTIONS ON MICROWAVE THEORYAND TECHNIQUES, VOL. MTT-32, NO. 10, OCTOBER1984
A
x
,2_v,,r!!rlL4
vy2
7
Vxl v., ~h.
t
Vyl
Perfect conductor xl -------- ~ J
Vzl! - ;V,2
Fig. 1. Configuration of the metal waveguide.
0 Y1 Y* Y
TABLE I
RELATIONS BETWEEN(j, ~) ANDMATERIAL Fig. 2. Rectangular element.
TENSORSAGAINST V.
v Ge trial function is sufficient to satisfy the admissibility re-
E ;: quirement presented by the previous section.
HH :; The vector n X V on the interface of the contiguous
regions can be separated into two components: i.e., V, is a
or longitudinal component and ~ is a tangential component
(j/@) in the x –y plane. The trial solutions are formed by
nx(F1v x J“)=0. (9)
approximating u=as a bilinear form of x and y within
The electromagnetic problem defined by (5) with the each element and OXand Uyas a linear function of x or y.
forced boundary conditions of (6) and (8), is expressed by Within each element, the value of v=is interpolated by the
vertex values of V,, and those of UXand Uyby the side
8F=0 (lo)
values of VXand Vy, respectively, on the element boundary.
in which F is the functional whose variation yields (5) as a Fig. 2 illustrates a rectangular element of which side lines
Euler equation and (7) and (9) as natural boundary condi-
are held parallel to the coordinate axes. The eight nodal
tions. The functional F is determined to have the form points described in the element consist of the four corner
points (corresponding to unknown values of ~) and four
F=+~{(V XV*)”fl-l( VXV)-ti2V*” QV} du (11)
side points (corresponding to unknown values of VX and
~).
where the asterisk denotes the complex conjugate.
Using matrix notation, the approximate vector func-
In this paper, traveling waves of the form
tional form of o is expressed as
V= (ixVx+iy~y + izVz) e-~fl’ (12)
V={ UX7.7YUZ} (14)
are treated, where ~ is the propagation constant. By sub-
where
stituting (12) into (11), the particular functional is given by
Ux= {Vx}q(px]
(15)
Uy={m%yl
02= {VZ}T[’BI 1
and
+p2(px;ll~12 +Py;11L12)
[ffxlT=[&f21
)
–@2(9xxlvx1+2~yylvy12+qzzlvz12~)~. (13) (17)
[9YIT=[L {21
The surface integral in (13) is to be evaluated over the cross [vzlT = [{ICI (,$2 {2’$, {2’$21 /
section of the waveguide.
(,= (-%- x)/kx,
{2= (x -x,)/hx
III. FINITE-ELEMENT METHOD
&=( Y2-Y)/~y, $2= (y - .YJ/hy ’18)
)
The selection of a family of trial solutions for the
Rayleigh–Ritz technique is facilitated if the cross section and T denotes transverse. Equation (14) can be rewritten as
of the waveguide is represented by a series of finite ele-
ments. If we consider the subdomain as one region, the 0= {V}qfl] (19)
EMW: FINITE-ELEMENTANALYSISOFWAVEGUIDES 1277
where dimension of v,, that is four. The matrix [S] is the matrix
representation of the curl operator on the space having [Q]
(20)
as a basis.
MCfxoo Substituting (19) and (25) into (13) and performing the
indicated integrations, the contribution of the particular
[Q]= o 9, 0 (21)
element “e” to the total values of F is obtained. The
o 0 %
resulting expression with respect to the parameters gives
and [0] is a zero matrix.
F’=*({v}f[K]{v} -J{v}~[M]{v}) (30)
The partial derivatives of [TX], [qY], and [9,] of (17) with
respect to x and y are given by in which
+.1 = [%1[901! &[8J=[~y][%l [KI=[SI*(JJITIP-’[ WWY)[SI’
(22)
+[%l=[ffzl[%l> J8%Y % 1=[%1[9,1
I (31)
where
[kf]=fJ[Q]@[Q]=dxdy
[%1=+[-:]> [4=+-[-;]
Y
–1 !1‘Bz-]1=*[-iA411 o 0
o ~ o A’& o (32)
[AZ]=;
x 1 0 0 M3J
0 [
and
(23)
[KJ=P2PY;’[QJ+J I
and
[K221=P2ZZ;[Q21 +Z’[Q51
[901 =[11. (24)
[KSSl=P.;’IQCl+ PY;’[Q~l
From (22), (23), and (24), the v x v is derived as follows: (33)
VXV={V}T[S][*] [K,zl= [K2JY= -P.; ’[Q81
(25)
[K,,] = [K,’]f= -jPp;~[Qg]
where
[W1]=9..[Q1]
“]”[’! :: T] ‘2’) [~221=~yy[Q21 (34)
00
[JLs]=%z[Qs] }
[q?]= ? Ipx o (27)
where ~denotes the complex conjugate and transverse, and
[1o 0 90
the matrices [Qj] (i= 1- 10) are given in the Appendix. By
and [1] is a unit matrix. applying the Silvester’s inequality to (31), the rank of [K]
On the other hand, from the commutativity of the dif- will become equal to that of [S].
ferential operators d/ax and 8/ dy, the following relation Summing the contribution of all elements over the cross
is obtained: section of the waveguide yields
[B.][AY] ‘[ A.][BX]. (28)
Using the relation of (28), it is derived that the rank of
=;({P}’[k]{P}-J{ P} ’[ Aq{P}) (35)
the 8X 5 matrix [S] of (26) becomes four. This factor can
be explained as follows. From (19) and (25), the curl where
operator v x is a linear operator from the space having
[k]=~[K] (36)
[@] as a basis to the space having [~] as a basis. Therefore,
e
the operator is a degenerate operator with a kernel, which
[M]=z[il’1]
(37)
is the subspace satisfying the following relation:
e
vlvZ + jbor = O (29)
where {~} is an ordered array of the three-component
where vl is a transverse operator and q is a transverse nodal variables. The matrices [~] and [~] are an adjoint
component of o. The nullity of the operator is equal to the matrix. Hence, the variation of F in (35) gives the follow-
1278 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.MTT-32,NO.10,OCTOBER1984
2,0 I I
1.0
0,5
‘Perfect ccnductor
Fig. 3. Cross section ofhalfdielectric-loaded metal wavegnide; (l=(.,
PI=Po, ~2=4co, P2=I.L0.
0 2,0 4,0 6,0 8,0
k,H
2,0 Fig. 5. Comparison of exact solution and present three-component
finite-elementanalysisresults.
1,5
1,0 )E, j HZ
0.5
0 2,0 4.0 6.0 8,0
koH
Fig. 4. Dispersion characteristicsfrom two-component finite-element E.
anrdysis.
Fig. 6. Plots of field intensity of LSM1l mode for (a) E- and (b)
ing algebraic eigenvalue problem: H-presentationat/lH = 5.0.
[R]{ P}-L7[M]{F}=0. (38)
The matrix [~] has components proportional to the ~“, /31, the two-component finite-element analysis. In Fig. 4, the
and B2. The solution of this eigenvalue problem will pro- occurrence of the spurious modes and the difficulties at
vide the required results on the angular frequency of /3/k0 = 1 can be found. Fig, 5 shows the dispersion char-
various modes on a particular waveguide. From the anal- acteristics obtained from the present finite-element analysis
ogy between the space of the element and the space of the for the E-formulation and from the exact solutions. On
cross section of the waveguide, the rank of [~] is equal to comparing the results of Fig. 4, the spurious modes have
NX+ NY, where NX and NY are the number of unknown not occurred at all in Fig. 5. And then, it is confirmed from
values of {VX} and {VY}, respectively. Therefore, the alge- the numerical experiment that the algebraic system of (38)
braic system of (38) has N, zero eigenvalues where N, is has the implicit zero eigenvalues, of which the number is
the number of unknown values of {V,}. Other field compo- equal to that of the longitudinal nodal points. All nonzero
nents can be derived from the eigenvector of (38) by (3) or eigenvalues were found to have one-to-one correspondence
(4). to the propagation modes from its field distribution.
Agreement between the finite-element solutions and the
IV. EXAMPLESAND CONSIDERATION~
exact solutions is excellent. Fig. 6 shows the. field intensity
To demonstrate the excellent quality and the accuracy of configuration of the LSMII mode taken at ~H = 5.0. These
the finite-element analysis of the previous section, the field configurations are almost identical with those ob-
solutions for sample problems are given and are Gompared tained by the exact solution so that the values of HX over
with the conventional two-component finite-element solu- all cross sections of the waveguide are equal to zero.
tions [2], [3] due to insufficient data of the three-compo- Second, a problem consisting of a rectangular metal
nent one [6]. In our program, all the eigenvalues of (38) are waveguide with rnicrostrip of finite thickness in the center,
obtained. as shown in Fig. 7, is treated. This waveguide geometry is
First, the problem consisting of a rectangular metal given in [2] and the spurious modes were shown to be
waveguide half-filled with dielectric, as shown in Fig. 3, is mixed with physical modes in the solution of the two-
treated. The propagation modes in this waveguide are component finite-element method. Fig. 8 shows the disper-
classified into LSM, LSE, and TE modes, asis well known. sion characteristics obtained from our method where the
Fig. 4 shows the dispersion characteristics obtained from spurious modes have not occurred at all and the number of
HANO:FINITE-ELEMENTANALYSISOFWAVKNJIDBS 1279
[Qd=[%][%][%]T (A4)
i_f
#l Air
[Q,] =[~y][%o][~y]T (A5)
(A6)
[QJ=[BZl[~y][BJT
b
[QT]=[4][%][4]T (A7)
(A8)
[Qg]=[%l[%JIAy]T
1 [’Qg]=
[CJJB.IT (A~) ,
\Perfect conductor
; [Qm]=[L][4]T (AlID)
Fig. 7. Hrdfcrosssectionofclosedmicrostrip; a= 2b = 2W = 4H.
where
[uxx]=J’’Jx*[qx] [qx]”dxdy=~[: ;] (All)
_o_. _._.—
13- #-0-”-” y~ x,
./.-
12- ,.0-.
O,*. ‘--O--- DaIY
[qy] =Jy2Jx2[9y][9yl’dxdY= y[; ;] (Al~)
11-
. 10- -“OOF”Z--- —.— Present analysis y~ xl
-. 9- E/EO=16 [42211
?
5? 8. 1 , 1
0.05 0.1 ----0-’15 ((WC) 2 [qz]=Jy’Jx’[qz] [9z]’dxdy=# ; ; } ;
7.5- y~ xl
.x*-”-”-”””-”
7.0- #“-”- 1224
6.5- -w”’ (A13)
-...--”/
[um]=Jy2Jx2[qo] [qo]~dxdy=fixhy [l]. (A14)
E/E@=9
:::. /p4-
y, xl
ACKNOWLEDGMENT
The author wishes to thank Prof. H. Kayano for his
helpful discussions and advice, and Prof. H. Matsumolto
for-his helpful advice.
2.5 c/Eo= 4
IV3FERENCES
0.1 0.2 0.3 0.4 0.5 0.6 (mH/c) 2
[1] P.Silvester,“A generafhigh-orderfinite-elementwaveguideanalysis
Fig. 8. Comparison of thetwo-component tmdpresent three-component program; L%&?Trans. Microwave Theo~ Tech., vol. MTT-17, pp.
finite-element ar2alysi8results. 204-210,Apr. 1969.
[2] P. Daly, “Hybrid-mode analysisof microstrip by finite-element
methods,” IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp.
the zero eigenvalues were confirmed to be equal to that of 19-25, Jan. 1971.
[3] M. Ikeuchi,H. Sawrdri,andH. Niki, “Analysis of open-typedielec-
the longitudinal nodal points, aswell.
tric waveguides bythefinite-elementiterativemethod; IEEE Trans.
Microwave Theory Tech.,vol. MTT-29, pp.234-239,Mar. 1981.
V. CONCLUSION [4] C. Yeh, S.B.Dong, andW. Oliver, “Arbitrarily shapedinhomoge-
neousopticaffiber orintegratedopticalwaveguide~J. Appl. Phys.,
In this paper, the finite-element method for solving the vol. 46,pp.2125-2129, May 1975.
dielectric-loaded waveguide problems was presented in [5] A. Konrad, “Vector variational formulation of electro-magnetic fields
which the nonphysical spurious solutions included in the in anisotropic media; IEEE Trunk. Microwave Theory Tech., vol.
M’IT-24, pp.553-559,Sep.1976.
solution of the two-component finite-element method do [6] A. Konrad, “High-order triangular finite elementsfor electromag-
not appear. This program has a specific number of zero neticwavesin anisotropicmedia,” IEEE Trans. Microwave Theory
Tech.,vol. MTT-25, pp.353-360,May 1977.
eigenvalues. The element used in our formulation is re-
[7] N. Mabaya, P. E. Lagasse, and P. Vandenbulke, “Finite element
stricted to the rectangle, so that the arbitrary cross section
analysis of optical waveguides~ IEEE Trans. Microwave Theo~
of the waveguide must be divided into the small rectangu- Tech.,vol. MTT-29, pp.600-605,June 1981.
lar region.
I m
Future problems in the present finite-element analysis .
Mitsuo Hano wasborn in Yamagucbi, Japan, in
will be the formulation with the triangular element and the
1951. He received the B.S. and M.S. degreesin
treatment of needless zero eigenvalues. electrical engineering from Yamaguchi U2river-
sity, in 1974and 1976, respectively.
APPENDIX From 1976 to 1979, he was a member of the
Faculty of Science,Yamaguchi University. Since
The [Qj] matrices in (33) and (34) are given by 1979, he has been a member of the electrical
engineetig faculty atYamaguchi University. He
[Q,]= [Uxx] (Al) hasbeen engagedin researchof fight modulation
using the magnetooptic effect and electromag-
[Q,] =[q,] “(A2)
netic propagation.
Mr. Hano is amember of the Institute of Electrical Engineers of Japan
[Q31= [%1 (A3) and the Institute of Electronics and Communication Engineers of Japan.
1280 IEEETRANSACTIONS ONMICROWAVE THEORYAND TECHNIQUES, VOL. MTT-32, NO. 10, OCTOBER1984
Phase Shifts in Single- and Dual-Gate GaAs
MESFET’S for 2–4-GHz Quadrature
Phase Shifters
JYOTI P. MONDAL, ARTHUR G. MILNES, FELLOW, IEEE, JAMES G. OAKES, MEMBER, IEEE,
AND SHING-KUO WANG, MEMBER, IEEE
Abstract —The variation of transmission phase for single- anddual-gate peared a need to realize a phase shifter which can be easily
GaAs MESFET’S with biaschangeanditsprobable effects ontheperform- integrated with the rest of the circuitry using the same
anceof anactive phaseshifter have been studied for the frequency range 2 technology. The p-i-n and ferrite approaches are not con-
to 4 GHz. From measured S-parameter vahtes for single- and dtud-gate
venient for monolithic integration. In this paper, a phase
transistors, the element values of the equivalent circuits were fitted by
shifter using dual-gate MESFET’S reported recently by
using the computer-aided design program SUPER COMPACT.
For the normal full-gate voltage range Oto – 2 V at VD~= 4 V, the Kumar et al. [1] will be studied. This phase shifter uses the
single-gate MESFET varies in transmission phase from 142° to 149° at operating principle shown in Fig. 1. Two signals, 90°
2 GHz, and from 109” to 119° at 4 GHz. However, with drain voltage out-of-phase, are presented to the two channels, namely, x
varied from 0.3 to 4 V and aconstant gate-voltage bias of OV, the phase
and y. The output of each channel is controlled by a
shifts are much larger, 105° to 145° at 2 GHz and 78” to 112° at 4 GHz.
variable gain amplifier using a dual-gate MESFET. These
This suggests that large phase shifts may be expected in adual-gate device
andthis isfound to be SO.With V~~= 4V and VG~l= –1.0V,Vfiation of two signals are then combined by an in-phase combiner to
control (second) gate bias from O to – 1.75 V for the NE463 GaAs produce a resultant vector, as shown in Fig. l(b). The
MESFET produces a transmission phase variation from 95° to 132° at 2 vector amplitude as well as the angle of rotation can easily
GHz and 41” to S8° at 4 GHz.
be controlled by adjusting the individual x and y compo-
Such phase shifts cause both mnpfitode and phase errors in phase-sfdfter
nents. In this way, one can achieve a phase shift of 00 to
circuits of the kind where signsdsfrom two FET channels are combkred in
quadrature with their gate voltages controlled to provide O“ to 90” phase 90° and, with four dual-gate FET’s, Kumar et al. [1] have
control with constant mnpfitade. For the single-gate FET examined, the shown how one may have a full 00 -to-360° phase shift.
expected ampfitude and phase errors are 0.30 dB and 6° at 2 GHz, and This type of phase-shifting technique is different from
0.36 dB and 10° at 4 GHz. If dual-gate FET’s are used in similar circuits,
that studied by Tsironis and Harrop [2], where the intrinsic
the distribution of errors is different. For NFA63 devices, the correspond-
circuit elements are changed by changing one of the gate
ing figures are 0.56 dB and 2° at 2 GHz and 1.2 dB and 3° at 4 GHz. The
advantage of the dual-gate configuration is that the input impedance biases, and this in turn changes the transmission phase.
conditions are more constant than for the single-gate configuration. They obtained a gain of 4 dB with 120° continuous phase
shift at 12 GHz. This is suitable for narrow-band applica-
I INTRODUCTION
tions. The advantages of a dual-gate MESFET phase shifter
PHASE-SHIFT CIRCUITS are needed in phased-array are stated in [1], to which may be added the advantage of
antennas to steer the radiation direction by varying monolithic integration. Such shifters are limited to a low-
the phase across the array elements. The type of phase power stage and are followed by amplification before the
shifters to be used is decided by specific requirements like signal is fed to the antenna elements. The phase-shifted
low VSWR, power-handling capability, insertion loss, amplitude reported in [1] shows a fluctuation of +2.5 dB;
switching speed, and bandwidth, together with cost, size, this kind of amplitude variation would produce unaccept-
weight, and other mechanical considerations. Switched able beam control in a phased-array antenna. One cause
transmission-line phase shifters using p-i-n diodes and for such a variation becomes apparent if one considers the
ferrite phase shifters are among the technologies used. way the phase shift is being carried out. For obtaining a
With the rapid development of microwave integrated 450 phase shift, both the channels are switched on (i.e., 0.0
circuits on semi-insulating GRAS substrates, there has ap- V on the control gate); then, keeping one of the channels
fixed, the other control gate voltage is ramped linearly
Manuscnpt received February 23, 1983; revised April 30, 1984. This from OV to pinchoff. This rotates the vector resultant from
work was supported in part by the Westinghouse R & D Center and by 45° to 0°, as in Fig. l(c). With the channel action inter-
Carnegie-Mellon University.
changed, the vector is rotated from 450 to 90°. If we
J. P. Mondal and A. G. Milnes are with Carne~e-Mellon University,
Pittsburgh, PA 15213. assume each channel has constant transmission phase, the
J. G. Oakes was with the Westinghouse R & D Center, Pittsburgh, PA resultant amplitude will vary from (fiA ) at 450 to A at 0°
15235. He is now with Raytheon, Northborough, MA 01532.
(or 900), where A is the maximum amplitude in any
S. K. Wang was with the Westinghouse R & D Center, Pittsburgh, PA
15235. He ISnow with Hughes Aircraft, Torrence, CA 90509. channel with the control gate bias at OV. This will cause a
0018 -9480/84/1000-1280$01 .00 01984 IEEE
MONDAL et a[.: PHASESH3FTSIN SINGLE- AND DUAL-GATE MESFET’S 1281
X.Channel ,
‘“’”m”
I 3dBW’ ;Two;hannels ~In-Phase~
I Coupler 1w!ihAmplifiers[CombinerI 1 ‘in
(a)
———— ———. —
Y R
SP
--
‘ .\ @=ian-l (AylAx)
L
G \ R2= AX2+Ay2 SP
\ Source
\ Where,ITXI =Ax
T \ 1~1 =Ay ‘Y
\ (a)
H
h x
G
Intrinsic Elements:Cg~=t42PF
(b)
Rin=6 n
Y Cdq=0.031pF
————--—— gmo=U.8mmho
A7 lncusofRa, Ay-O T= lZ8pS
&- Rd$=293Q
~ Ifilmax= IXI ~ax =A Mrinsk Elements: Cd~=0.126PF
T Lgp=0,1nH
Iv
kL.._.&x ‘s’ =ao’ n“
>:;;;”
9P
Ax
%P=0“4 o
(c)
‘d’= Z**
Fig. 1. Quadrature phase shifter operation: (a) Schematic diagram of (b)
active phase shifter showing input and output couplers and x- and
y-channel amplifiers. (b) Resultant output vector composed of x- and
Fig. 2. Equivalent circuit model of the single-gate FET. (a) The circuit
y-channel components. (c) Phase\arrplitude pattern used by Kumar
model and elements. Intrinsic elements inside the dotted line may
et al. [1].
change with bias. (b) Typical element values for V~~ = 4.0 V, VG~= O
V, and 1~~= 34mA.
maximum deviation of amplitude at 450, with respect to
the amplitude at phase shifts of 0° and 90°. This accounts
for 3-dB variation from the minimum value, occurring at
II. TRANSMISSION PHASECHARACTERISTICSIN A
N x 90° phase shifts, N being an integer. During the
SINGLE-GATE FET
subsequent analysis, we will point out that phase variation
with gate voltage will add to the amplitude and phase A single-gate GaAs MESFET (LN1-5 # 2B) fabricated
fluctuation. at Westinghouse was chosen for this investigation. The gate
Before considering phase shifters using dual-gate length was 1 ~m and the gate width was 4X 75 pm. The
MESFET’S, it is interesting to examine the probable per- source–gate distance was 1pm and the gate–drain distance
formance of a phase shifter using single-gate FET’s in the was 1 pm. The channel doping was 1.1X 1017cm –3and the
two channels, instead of dual-gate MESFET’S, The gate pinchoff voltage was just under – 2 V on the gate. The
voltages of the single-gate FETs will be varied to change S-parameters of this transistor were measured from 2 to 4
the amplitudes in the two channels. GHz at different gate-bias points with the drain voltage
Section II describes the variation of the single-gate FET fixed. They were then used to determine the equivalent
intrinsic elements with gate-bias change and their effect on circuit model of Fig. 2. The typical element values after
the transmission phase characteristics. Tha change in the computer fitting using SUPER COMPACT are given in
intrinsic elements with drain-bias change in a single-gate the caption. The bias-voltage variation affects only the
FET and with control gate-bias change in a dual-gate FET intrinsic elements of the equivalent circuit. The parasitic
are discussed with their effects on the transmission phase due to bonding wires were, therefore, removed from the
in Section III. Section IV shows the overall effects on the model. The resulting circuit has a transmission phase given ‘
performance of single- and dual-gate FET phase shifters by
@=
and discusses a possible correction for the amplitude error. @fl4-1)–(tan-l (3) (1)
While variable gain amplifier-shifters using single- or
Y
dual-gate FET’s must include matching networks, it is
where
believed that the variation of transmission phase with bias
in the FET itself is the primary source of phase errors. This
tiT2 +sin~r
study, therefore, focusses on the FET intrinsic phase re- A=
sponse. Cos6.)’7+’ (/.?7172
1282 IEEETRANSACTIONS ONMICROWAVE THEORYAND TECHNIQUES, VOL. MTT-32, NO. 10, OCTOBER1984
Gd,+ GL G, GL + Gd, 1.51 1 I 1 1
X=(I) T3 + ll-~ Single Gate FEI
( gmo gmo Gcis 1.4’ LN1-5,
+— G, Cdg + cd, ——cdg +Costi’r 1.3
1.2i
gmo Gds G(i. )
G~ + Gd~ 1.1
cd,
—14)3T1T2— — u2r1r2 sin a7 ~.
Gds () Gds 1.0’
z
G~ i- Gd, U2 : 0.9
y=G, - = (’~) cdgG;,cds s
( Gdsgmo ) z: 0.8 _-/4
GL+ Gd, ~o~~~ 0.7
—6)2T1T2
() Gd, 0.6
6J2 c 0.5
—— c ~+Cd~r3 +ti#sinu7
gmo ( gs Gd$ ) ds 0.4
in which 0.3 I I I I
-2.0 -1,5 -1.0 -0.5
GateVoltage
r = transit time in the gate region,
Fig. 3. Amplitude and angle of S21are shown against the gate voltage at
~1= RinCg~, 2, 3, and 4 GHz. The values are calculated from the equivalent circuit
derived from the measured S-parameters.
Cdg
~2. —
“~
i3m0
r 1 [ I 1 1 I
73 = (:dg + cgs)/Gds~ I
6.0
G,= ~ and G~ = l/R~.
0.45
s
-& 5.0 0.40
2
In the above equation, we have identified three im- c = 0.35
&
portant time constants rl, Tz,and r~, which are dependent a- 4.0 Q0,30
I
on elements that change with bias.
The variations of transmission phase and amplitude with
gate bias are given in Fig. 3 for 2, 3, and 4 GHz. The phase
variation near maximum amplitude (i.e., near V~= O V) -Cn=.0.05 I~cdq r I 1 I
increases as the frequency increases. In a quadrature phase- u o I I I I
shift circuit, this will effectively increase the amplitude EzzzzI
variation of the resultant vector tip. A figure of merit can
be defined as the ratio of the slope of the magnitude of Szl
versus gate bias to the slope of the phase of S21with gate
bias, i.e.,
-2.0 -1.5 -1.0 -0.5 0
(lAS211/AV~) GateVoltage
Figure of merit=
(A <S,l/AV~) “ Fig. 4. The variation of the elements in the equwalent circuit of a
single-gate FET with change in gate-to-source voltage. The values are
The higher the figure of merit is, the less the amplitude obtained through the fitting of the measured S-parameters.
fluctuations with phase change, as will be seen from the
discussion in Section IV. The variation of the FET ele-
ments with the gate bias is shown in Fig. 4, As the FET The fall in Fig. 4 of transconductance g~O when ap-
approaches pinchoff, the depletion region under the gate proaching pinchoff is expected. The channel resistance R~$
increases. This results in a decrease in both R,n and Cg,, as shows a small decrease near %~ = OV and then steadily
shown in the upper curves of Fig. 4. The transit time ~ in increases as the channel is pinched off. As these changes
the gate region also follows the same pattern. occur, the time constants of the device also change.
As shown by Engelmann and Liechti [3], the pinchoff of The effect of each of three previously identified element
the channel reduces the dipole region. We believe this groups or time constants [RznC~~, Cd~/gmo, and (Cg, +
reduction of charge in the dipole modifies the electric field C~g)/Gd,l On the transmission phase of the FET is shown
and so reduces the transit time. The variation of Cdg is in Table I. These time constants have been varied as the
small. Others who have considered the bias dependence of gate-voltage changes. All FET elements were fixed at the
the elements for the large-signal case include Willing et al. Vg~= – 1.0 V values, except those in the time constant
[4] and Tajima et al. [5]. In our analysis, we have ap- being examined. As the time constant was varied, the phase
proximately maintained R,nC@,,proportional to transit time of Szl was recorded and the maximum and minimum
[6]. values at 2 and 4 GHz are noted in Table I. The R,nC~,
