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Ahmed1968-FEM-Waveguides.pdf
region directly. If the sample is of uniform resistivity, the
calculation of the voltage drop across the illuminated region
is trivial, and is given by
RTxlL)IT
Vj = (Rj -
where Rj = sample resistance when length L — x is illuminated
RT — zero-illumination resistance
L = sample length
IT = threshold current.
The deduced threshold voltage as a function of length for a
uniform sample is shown in Fig. 2.
average resistivity over the length 0^-y is the quantity
required for the calculation of the threshold voltage, but, if
the point-to-point resistivity p is required, it can be obtained
from the following relationship:
- ,
dp
A plot of resistance down the sample length (= py/A,
A — sample area) is shown in Fig. 3 for a very poor sample,
350
300
measured by laser
250
measured by probing
centre of sample
200
150,
cc 100
50
0
50
100
150 200 250 300 350 400 450 50O 55O
distance from anode y
Fig. 3 Comparison of sample resistance determined by laser
experiment and potential probing
100
150
200
effective sample length,
50
250
300
Fig. 2 Variation of threshold voltage against effective sample
length for a 300/nm long sample
However, if the material resistivity is not uniform along the
sample length, as is the case for a few samples of poorer
material, the simple calculation gives erroneous results, and
it is first necessary to determine the resistivity profile. This
can be done by potential probing,2 but it is convenient to
deduce the resistivity from a measurement of the terminal
resistance of the sample as increased lengths are illuminated.
If the carrier mobility and the illumination intensity of the
laser image are assumed to be constant, the average sample
resistivity over the illuminated region can be determined in
the following way:
If [n0 = average zero-illumination carrier concentration
nL = laser-injected carrier concentration
then
nT = nQ + nL
or
O~T
= <70 + aL
0)
For convenience of calculation, it is simpler to deal in terms
of resistivity; therefore eqn. 1 becomes
H
and is compared with the value of py/A obtained by potential
probing down one track in the centre of the sample.
If the mobility or the laser illumination is nonuniform, the
resistivity can still be obtained, but now measurements of
resistance for two illumination levels are required. If the
ratio of the two levels is known, the resistivity can be calculated
in a similar manner to above.
The authors wish to thank J. Sarma for his very valuable
assistance in this work.
F. A. MYERS
J. McSTAY
13th August 1968
Department of Electrical & Electronic Engineering
University of Leeds
Leeds 2, England
B. C. TAYLOR
Royal Radar Establishment
Malvern, Worcs., England
References
1 HAYDL, w. H.: 'A wide range variable-frequency Gunn oscillator',
Appl. Phys. Letters, 1968, 12, pp. 357-359
2 THIN, H. w.: 'Potential distribution and field dependence of electron
velocity in bulk GaAs measured with a point contact probe',
Electronics Letters, 1966, 2, pp. 403-405
Po + PL
where pT = l/cr r , p~0 = l/a 0 and pL = \\aL.
Now, if a length y of material is illuminated, the overall
measured resistivity, calculated from the terminal resistance
and the sample dimensions, is
where p = material resistivity
p = average material resistivity from 0
We also have, in the absence of illumination,
\ p
From eqns. 2 and 3, the final equation for p is
p2y + pL(Pm - p0) + pLlipm
- p0) = 0
Now pm, pQ and pL can be determined from the sample
terminal resistance, and hence p can be determined. The
ELECTRONICS LETTERS 6th September 1968 Vol. 4 No. 18
FINITE-ELEMENT METHOD FOR
WAVEGUIDE PROBLEMS
A numerical method is presented from which it is possible to
calculate the propagation coefficients of waveguides with
arbitrary boundaries and dielectric fillings.
The difficulties in obtaining analytic solutions in compact
form to problems involving waveguides with arbitrary
boundaries and dielectric fillings lead to the use of more
direct numerical methods. The standardfinite-differencetechnique1'2 could be usefully applied to many problems of this
type, but difficulties arise when the waveguide contains
singular points and the dielectric surfaces do not conform to
certain simple geometric shapes. Further difficulties arise
when the permittivity is not sectionally constant. The
387
finite-element method proposed here is applicable to waveguides containing arbitrary isotropic media. This method3
has, as its starting point, a variational integral equation rather
than the differential-operator equation employed in the finitedifference method.
A variational expression for a waveguide filled with an
isotropic medium, whose relative permittivity er and permeability \xr may be functions of the transverse co-ordinates,
may be written in Euler density form as
&&R\V,tf>\2ds
all M vertices,
<f>2, . . . < £ „ , 0 , ,
, <f>2, . . . 4>N)
(7)
The optimum set of <j> and iff for a certain functional form
of NK(x, y) may be obtained by minimising the functionals
given in eqns. 5, 6 and 7 with respect to each of <f>m and i/fm,
i.e. equating
fcr-,-1.2,...*
This way of minimisation of /,
eigenvalue problem of the form
-Koj>j>(^<t>2+-Tl(j2)ds
• - ••
(1)
where 0 and if* are the axial components of the magnetic field
and the normalised electric field and
= propagation coefficient through waveguide
=
W=
TLJLr
\
(^
CO
and the region of integration is the entire guide cross-section.
For inhomogeneously filled guides, P, r, W and R are
sectionally constant and can be taken outside the integration;
for homogeneously filled guides, the above extremisation
function splits up into two extremum functionals of the form
\vt(f>\2ds-
K2
(2)
A6
= K2Bd
-
K
2
<t> < b ifj ds
. . .
(3)
corresponding to the existence of separate TE and TM modes.
It is shown4-5 that the right-hand side of eqns. 1, 2
and 3 are variational for K2. Therefore, among the set of
trial functions <j>(x, y) and ift(x, y), the correct values will be
the set which yields the smallest possible values for the
extremum functional /.
Instead of approximating the true <f> and iff by a set of
normal-mode functions of the waveguide defined over the
entire cross-section, the finite-element method employs a set
of algebraic functions defined over a subsection of the whole
guide cross-section. These subsections or subregions may be
either triangular or regularly polygonal in shape and are called
elements. Thus, in the finite-element method, the entire
domain, over which the differential operator equations are
defined, is divided into a finite number of elements on each
of which the actual mode function is approximated by a set
of algebraic functions, which are only defined over the
element under consideration and are zero outside. Furthermore, these algebraic functions must satisfy the condition
that they are linearly dependent on the values of <j> and if/
at the vertices of the element. Hence, if an element has
n vertices, the potentials (f> and ifj within it may be approximated by
(8)
and J2 results in a matrix
. . . .
(9)
where A and B are square sparse symmetric and diagonally
dominant matrices, and, furthermore, matrix B will always be
positive definite. The matrices A and B are of order N for
homogeneous media and of order 2N for inhomogeneous and
for generalised media. The components of the column matrix 6
are the vertex values of <f> and 0.
To illustrate the method, we consider a homogeneous waveguide whose cross-section is divided into a finite number
(say p) of triangular elements and examine a typical element
(the eth) which is described by the vertices /, j and m in
cyclic order. If the values of <f> at these vertices are denoted
by <f>h <j>j and <f>m, the potential cf> over the entire cross-section
can be approximated as
P
ZJ
.
ZJ
e=l K=i,j,m
.
.
. (10)
where the superscript V indicates that all values are confined
to the eth element. Now, for a triangular element, Nt(x, y) is
found to be
Nt(x,
2
(6)
and
-0
+ 2
(5)
y) = — (a, + b,x + c,y)
(11)
2A
where at = Xjym - yjxm, bt = y} - ym, ctt = xmm - x;; and
Ae — area of the triangular element.
The values of the other parameters a, b and c can be
obtained by the cyclic rotation of the co-ordinates (xh yt),
(xj, y}) and (xm, ym) of the vertices /, j and m of the triangular
element.
If we now consider the minimisation of the functional / t
as given in eqn. 2 with respect to potential <\>t (at the /th
vertex), we obtain
. . . .
(12)
Substituting the functional form of <f> as given in eqn. 10 into
eqn. 12, we obtain
^
ly
NeKN^Kdxdy . (13)
2
j
, y) = 2 NK(x, y)<f>K\<fiK
(4)
K=l
where (f>K and ifjK are the values of <j> and ijj at the vertex K,
and NK(x, y) is a predetermined function of (x, y), which is
uniquely defined and differentiable over the element and
reduces to zero outside the element.
When the functional form of <f> and ip as given in eqn. 4
is substituted into the right-hand side of the extremum
functions J, Jx and J2 and the relevant integrations are
carried out, / , Jx and J2 will be functions of the set of variables
<f>K and ipK at the vertices of the elements; i.e., if there are in
388
Now considering the contribution to :—r- by the eth element
dcpi
only, we get, after substituting the values of N(x, y) as given
in eqn. 11 and integrating over the domain described by the
element e,
1
{b bbj
''
ELECTRONICS LETTERS 6</J September 1968 Vol. 4 No. 18
which can be written in a compact form, by using matrix
notation, as
f>e)
•
•
which access may be made at two or more ports, as indicated
in the conventional symbol shown in Fig. 1A. In lossless
• (15)
where Se and Fe are 1 X N row matrices whose elements are
given by
SeiK = (bibK + CiCK)l4Ae (if / and K belong to same element)
= 0 (if vertices i and K belong to different elements)
Ae
e
F iK = — (if i and K belong to the same element but / ^ K)
Fig. L A 3-port circulator
Ae
= — (if i and K belong to the same element and i = K)
6
= 0 (if i and K belong to different elements).
Therefore, when all the elements are considered, minimisation of Jx with respect to </>,- will generate an equation of the
form
p
- K2[F*] x [«£]) = 0
s
•»— internal circulation loop
I
1
(16)
e = l 16, it is clear that minimisation of Jx with respect
From eqn.
to all N values of <j>} results in a matrix eigenvalue equation
= K2Fcf>
(17)
where S and F are square symmetric matrices of order N.
Eqn. 17 can be solved either for all N values of K2 by
matrix inversion or for the lowest value of K2 by an iterative
technique, using successive overrelaxations.
s. AHMED
9th August 1968
0
[ access port
Fig. I B General form of operational-amplifier
lumped circulator
stage of proposed
n such stages are needed for an n-port circulator
Department of Electrical & Electronic Engineering
University of Leeds
Leeds 2, England
References
1
COLLINS, J. H., and DALY, P . : 'Calculation for guided electromagnetic
waves using finite-difference methods', / . Electronics Control, 1963,
14, p. 361
2 HANNAFORD, c. D. : 'The finite-difference method for waveguides',
thesis, Department of Electrical & Electronic Engineering, University of Leeds, 1967
3 ZIENKIEWICZ, o. c.: 'Finite-element in the solution of field problems',
The Engineers, 24th Sept. 1965
4
MORSE, p. M., and FESHBACK, H. : 'Methods of theoretical physics'
5
(McGraw-Hill, 1953), pt. 11
AHMED, s.: 'Wave equations and variational formulation for
generalised isotropic medium in wave-guides', research report
LU/EE/68/01.
REALISATION OF THE CIRCULATOR
CONCEPT USING DIFFERENTIAL-INPUT
OPERATIONAL AMPLIFIERS
A 3-port unit that satisfies the requirements of the circulator
concept is realised with a differential-input operational amplifier. It provides, between two of its ports, a direct measure of
reflection coefficient at the third port. One unit may be
employed as a hybrid transformer; two or more units may be
connected in a ring to form an n-port circulator.
The usefulness of the circulator as a basic circuit component
in coaxial, stripline and waveguide systems operating in the
v.h.f. and higher-frequency bands suggests that lumped
realisations of the circulator concept may be equally valuable
at lower frequencies, particularly if realisable in microelectronic form. The need for a degree of precision in the
realisation of this concept suggests the use of operational
amplifiers to supply the required nonreciprocal directivity,
and such units are now readily available in compact solidstate form, with bandwidths of the order of 10 MHz at low
gain, so that there is now the possibility of building circulators
from zero frequency up to the h.f. band (3-30MHz).
Essentially the circulator is a closed transmission loop, into
ELECTRONICS LETTERS
6th September 1968 Vol. 4 No. 18
Fig. 1C Circuit configuration for form shown in Fig. 1B
Fig. I D Port-to-port circuit diagram corresponding to Fig. 1c
distributed structures, the transmission path (line, guide) has
a fixed characteristic impedance that is purely resistive (i?0),
and each access port is normally terminated with this value
of impedance. Ideally, when an e.m.f. is introduced in series
with any one port, termination energy is transmitted through
the circulator in only one direction round the loop and is
entirely absorbed in the termination on the next port, provided
that this matches Ro. Otherwise there is some reflected energy
which will be transmitted to the next port, and so on. In
practice, such ideal operation can be closely approximated
only over a restricted frequency band.
In order to realise a lumped circulator, preferably with
grounded ports, using active units we would, on the basis of
known operational-amplifier circuit technique, expect a stage
having the configuration shown in Fig. 1B to be sufficiently
general for the required realisation. 1 The problem is to
determine a network which is aperiodic in action and, when
loaded with a specified characteristic resistance RQ at the
port indicated, will produce zero output (v2 = 0). We have
389
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