1. IESE TRANSACTIONS
120
Computer-Aided
Signal
KARL
ON MfCROWAVE
THEORY
AND
TSCHNIQUES,
Determination
Equivalent
Microwave
HARTllIANN,
STUDENT
of the
IEEE,
J. O. STRUTT,
WILLI
KOTYCZKA,
FELLOW,
CIRCUIT
2, FEBRUARY
1972
SmaU-
MEMBER,
IEEE,
AND
IEEE
r-----+
%
C7
c ‘(couector)
II
b’(base)
0
R2
%= !1
EQUIVALENT
NO.
Clj
Abstract—The
actual technology for the production of transistors
is now suitable for bipolar transistors in the 1O-GHZ range. In order
to obtain amplifier circuits in this microwave range, the knowledge
of the exact equivalent circuit is essential. A computer method for
the determination
of the equivalent network is given. The final application and accuracy are shown for a particular microwave transistor.
1. THE
~-20,
Network
of a Bipolar
Transistor
MEMBER,
MAX
VOL.
OF AN INTRINSIC
u~e,
by
I’:Ya&.U~et
0
TRANSISTOR
IN COMMON
CONFIGURATION
EXTRINSIC
O
N THE
tion
imate
equivalent
which
BASIS
of the
only
neglected
RC line
of
direction
the
2). This
our
transistor
of the
[1],
capacitances
CI take
approx-
we
transistor
current
find
Im
into
connection
lows
CT).
’21 e’
at
Co and
CT are
the
in
the
backward
the
resistance
/
.
account
the
internal
base
Fig.
2.
increasing
frequency
Approximation
of the transadmittance
VZ1,’ is complex.
Im L
n
may
Because
4-to-8
GHz),
we
a circle
‘21e’
(Fig.
worse
want
to a defined
[3]
by
becomes
we only
relative
from
Pritchard
be approximated
Y21.’ by
zero
current
the
circle
the
frequency
approximate
the
(in
trans-
3, crossing
the
real
I
Fig.
(see Fig.
1) may
Re
equiv-
range
point.
I’
w
at increas-
to find
of Fig.
Y21.’.
zone,
aDOr
The
(C6,
1)
and
consideration
into
and
approximation
network
at its
elements
m
A
the
(Fig.
density
junction
RI takes
Yzl,’
admittance
axis
and extrinsic
slightly.
circle
case
Winkel
intrinsic
emitter-base
base
frequency.
alent
Te
The
transadmittance
that
Intrinsic
[10].
is doped
shows
1.
determina-
and
small
C6 and
resistance
between
ing
the
[2],
The
the
at
Fig.
experimental
elements
recombination,
elements.
which
the
by
of
valid
extrinsic
The
of
equations
e’(em[tter)
ELEMENTS
intrinsic
circuit
is
EMITTER
AND THE
be expressed
3.
as fol-
Oxtmate
fiiquency
range
Approximation
of the transadmittance
employed
in our computer
program.
rue’
as
:
~’ = Ul),e,” Y,I.I
This
equation
can
work
of Fig.
4. (It
= –
&
be represented
is possible
‘b’”’
=_
1/1
(1]
i- jwL7
by
the
to employ
additional
net-
additional
net-
Manuscript
received
January
18, 1971; revised
April
12, 1971.
The authors
are with
the Department
of Advanced
Electrical
Engineering,
Swiss Federal
Institute
of Technology,
Zurich, Switzerland.
Fig.
4.
Additional
network
for the transadmittance
Y21e/.

2. HAR’CMANN
et al.:
EQUNALF.!NT
NSTWORK
OF MICROWAVS
121
TRANSISTOR
II 1, COMPUTER-AIDED
EQUIVALENT
L,
L2
MEASURED
+
-+J)J=
A.
Relation
DETERMINATION
NETWORK
BASED
SCATTERING
Between
OF THE
ON THE
PARAMETERS
the Scattering
Parameters
awd the
Currents
As
0
Fig.
5. Equivalent
network
of microwave
transistor
package.
L1, and Lo—lead
inductances
between
the reference plane
(scattering
parameters)
and the package. L,, and L,—lead
inductances between the package edge and the gold wire connection. -&, Lf, and L-inductances
of the gold wires of the emitter,
base, and collector lead-in wires to the transistor
pill (L, = O,
because of the actual construction).
Cl, cZ, CS, and C4—capacltances of the package. C~—-capacitance between input and output.
is well
known
parameters
a pure
the
[6],
resistance
pedance,
In
of
our
source
output
terminals
value
Input
1)
Z
to
B. General
%
+
t-!!
n
1 V and
Additional
6.
First
the
are
I
1
I
based,
I
are
for
Relationship
between
s parameters
and
on
by
by
works
the
for
more
validity
The
complicated
of described
R8 and
elements
curves
without
impairing
the
method.)
transistor,
When
L7 are also
found
by
optimiza-
the
tion.
the
gold
including
this
first
complete
the
For
initial
this
proce-
values
said
element
of the
optimization
needed
only
complete
wires
are not
is
values
as initial
network
as,
as long
as in
pill.
optimization
network
con-
optimization
The
are
These
under
package,
The
of the
shorted
first
is
con-
optimization
the
method
pill)
wires.
range
the
The
later.
optimization
example,
this
gold
the
of
consecutive
and collector
short
estimation.
for
for
of the
measured
(without
frequency
which
we find
the
and
elements
error
the
two
base,
5 is used.
in detail
in
alone
obtaining
which
the currents.
sev-
range
the
with
comat
The
until
found
parameters
found
values.
very
the
values
7.
Fig.
is
using
elements,
a digital
measured
frequency
the emitter,
of Fig.
is described
using
are
package
within
used
(5)
limit.
transistor
scattering
– S,,).
are varied
circuit
shorted
are
(4)
calculated
a defined
values
network
dure
(2)
.s12
as compared
[11 ], while
package
50 Q, the
(3)
is found
circuit
the
sideration
the
the
equivalent
measured
of
with
is below
nections
for
to
.s21.
of a specified
parameters,
measured
network
Z. equal
parameters
points
equivalent
steps.
net’#Ork
Complete equivalent
circuit and additional
the complex transadmittance
I?z1,’.
network
compared
The
at the
of Optimization
scattering
frequency
ones,
R3
Fig.
equivalent
calculated
ug~
then
(1 – s,,)
= 0.01.(1
Procedure
The
of the
a voltage
and
Excitation:
1;’
are
“First
are valid.
11” = – 0.01
eral
by
im-
13xcitaiion:
Output
The
ports
reference
7).
12’ = – 0.01
puter.
the
terminals
11’ = 0.01
2)
scattering
at both
called
input
(see Fig.
equations
of the
Z== 20 = 50 ~.
at the
Uol = U02 is equal
If
definition
is terminated
case,
is applied
following
by
transistor
procedure
is taken
into
is terminated,
consideration
(Fig.
6),
11. EQUIVALENT
NETWORK
COMPLETE
In
FOR THE
the
elements
TRANSISTOR
package
For
microwave
influence
of the
transistors
package
parts
of the
leads
bining
Fig.
5 with
equivalent
network
(gold
we
as shown
wires)
Figs.
of the
1 and
have
may
4, we
transistor
to
in Fig.
consider
5. The
be neglected.
find
the
as shown
following
of the
alone
sections
entire
raises
only
the
transistor
optimization
is described
no special
of the
because
the
difficulties.
the
C. Determination
ohmic
Com-
complete
in Fiz.
Many
of the
6.
OJ the Elements
optimization
error
is based —on
function.
the
methods
The
corresponding
of the Transistor
are based
solution
on the gradient
of the
method
of
present
Fletcher
paper
and

3. 122
IEEE TRANSACTIONS
Powell
[4].
Because
culated
frequently,
it
use of computer
as
possible,
their
tance
by
has
to
between
For
four
two
culate
the
[5]
We
input
we
calculate
quency
the
value.
frequency
the sum
has
sum
values
to
of the
errors
be
current
with
of the
the
squared
minimum
errors
is 2n if n is the
number
–IR. @R
‘j@IL
‘+L
juVc#Z
V1~v = Zm. IICV
+lCV
— GICV.
= Z.,.
%lDV
zii
AC
@rDT-
and
Un-
Origina[
range.
at each
of the
#R= R.CPB
*L =juL
. $L
Q. =ju C. +~
cal-
III-A).
frequency
number
VR=R.
IR
VL =jcoL. IL
Ic =jw C V,
is
we
input
(see Section
over
This
First
Component
of AP
Note; V and 1 are the voltage and current
in the original
network;
~ and @ are the voltage
and current
in the adjoint
network,
of the
to be calculated.
output
integrate
The
only
Sensitivity
(component
of ~)
Branch
Relation in Adjoint
Resistance
Inductance
Capacitance
Current-controlled
voltage source
(see Fig. 11)
transadmit-
1972
I
Branch
Relation
the
However,
are
calculations.
respectively
not
the
on
FEBRUARY
plement
Type
Function
value
and
do
based
elements
TECHNIQUES,
economical
[5].
since
AND
TABLE
be as simple
is
THEORY
be calcu-
limits.
network
excitation,
to
for
Rohrer
pill
parameters
with
output
like
and
The
frequency
scattering
possible
important
calculation
of the Error
every
has
its determination
be extended,
defined
Formation
gradient
Director
Yzle’ is complex.
varied
D.
that
gradient
paper
theory
said
is very
time
Our
excellent
the
ON MICROWAVE
Network
fre-
1
1
individual
of the
variable
elements.
5o11
Sum
E of the squared
errors
‘Y[wl’(.iw) U1’(jh))c
= -+
~
?
Q
k=l
] (~2’(joJk))c
(~2’(j@k))m
–
[2
+
W,’(jhc)
+
~1’’(j%)
I (~1’’(j~k)).
–
(~,’’(j%))m
w,’’(jm)
] (l,’’(jcok))c
–
(l,’’(jok))m
l’]
Fig. 8.
(Li)lOWer
number
~2n;
k’
individual
m
measured
c
of
j~h)
, ~,’(
~1”
( jtih),
jo~)
and
This
of the
a further
second
error
sum
function
network
limit,
sum
current
with
would
output
rapidly
comes
near
if
its
excitation.
limit.
the
this
elements
process,
attain
as the
But
we
the
to E.
physical
limit.
it is very
essential
that
chosen
variation
upper
exact
limits
program
their
the
the
this
definition
elements
are only
at
values
choose
If
useful
limits
corresponding
may
the
subroutine
of EL is not
definition
cannot
yond
zation
(8)
function.
that
be infinite.
W
used,
EL is added
function
input
ET is
sum
argue
minimization
L; be-
limit
of EL is
in the
varied
the
of .E~
optimi-
between
the
Li.
or lower
E. Calculation
ET
respectively.
Error
with
excitation.
increases
elements
current
excitation;
input
ju,)
could
because
output
with
limit;
W is the weight
where
functions;
and
network
EY=E+EL.
value;
weight
input
error
value;
output
Now,
total
value;
We
jak)
1’”(
upper
The
input
j+,),
lower
ith
and adjoint
values
J%+),
~~”(
11’ ( jcw) , 1“(
frequency
frequency
calculated
~1’(
ith
(LJ.PPe,
kl
Original
(6)
where
1,”(
Network
I
IZ
+
I
Adjoint
Y’;k
1’
– U,’(jw))m
of the Error
of the Gradient
Sum Function
EL
(Li)lower
+
1) Gradient
(-w..er
——
.
z
{x,
–
(L,),ower
(L,)..,.,
(7)
tion
}
-xi
is based
network
where
This
Xi
normalized
‘n
number
+ WI’’(joJJ
network
of variable
~(~l’’*(jq))C
element
network
(see
III-E);
With
elements;
– (~1’’”(jtih))~}
lowing
with
method
[5]
and
of the Error
on
the
its mutually
makes
the
Sum
pairing
use
definition
Function
of
adjoint
of
E: The
originally
network
a theorem
calculaspecified
(see Fig.
of Tellegen
of E one can
derive
13).
[7].
the
fol-
equations:
~(~,’’(j~k))c
wl’’(jow))c
+
13p
the
Wz’’(jbk)
[ (~2’’*(j@k))c
–
(~2’’*(j@k))m]
ap
1}
(9)

4. et al.:
HARTMANN
---B ----Eir_3
EQUIVALENT
NETWORK
oF
MIC3Z01VAVFi
TRANSISTOR
c,
-----
RL
T
I’
I
q
Original
Fig.
12,
network
Relation
Adjoint
between
original
and
current-contl-oiled
voltage
L4
network
adjoint
sour,:e.
network
for
a
-----
----
Fig. 9.
r————
I_’_’_- ‘—l
ic..??.2u!2??_o
(?2
C*
123
Transformed
network
corresponding
to Fig. 10.
CT
—---
RL
C*
R2
C3
I
I’
LA
-----
-.-—
Fig.
10.
Part
I’
of the
original
network
of
a broken line.
Fig.
6 as delimited
by
C7
L
1
AdJolnt
Additmr@{
netwc,rk
Fig. 13.
Complete
addltiona(
adjoint
[network
network
and adjoint
additional
network,
------------Fig. 11.
Transformed
network
corresponding
to Fig.
9.
where
Pth
P
network
Because
element;
for
=
ap
Re
~
(lo)
[s’+s”];
first-order
sensitivity
with
input
s“
Re
first-order
sensitivity
with
output
real
[5]
the
Now
[5],
directions
we must
Output
voltages
lated.
to
input
and
of Fig.
change
the
currents
the
measured
values,
8;
value,
the
UO1= 1 V,
U02 = O
necessary
branch
Fig.
original
network
output
currents
have
to
is not
work
and
we
line
network
of the
input
analyses,
excitation,
(see Fig.
of the
network
insert
are calcumust
the
have
very
8 are
sign
essential
of the
rela-
of the
been
may
Adjoint
In~at
sources
of Fig.
8.
part
deterorig-
by an interrupted
use of an additional
Y210’. The
to the
ad joint
original
9, its current
a further
The
of the
source
net-
network,
source
having
transformation
11.
source
voltage
voltage
to Fig.
UW., as a current-controlled
Ul, U2 as current-controlled
consider
voltage
of the
is equal
With
Fig.
is completely
network:
of the
is marked
because
part
transplaced.
We
following
network
6 which
10 is equivalent
w-e obtain
procedure
transadmittance
other
~’ is
one.
of the adjoint
simple
sensitivity
I.
This
previous
ad joint
the
The
to Table
in Fig.
for
+0.01.&.
excitation:
to the
inal
network
12’=
c) Determination
derived:
two
output
frequency
Because
I are
perform
excitation
every
and
of Table
have
Input
At
analogous
excitation;
i.e.,
according
mination
relations
with
a)
excitation;
part.
we
namely,
error
current
of 111-A;
b)
s’
the
tions
calculated
k=l
V.):
the
theory
k=n
dE
In
this
and
sources,
1’ depends
I)
(by
(by
l’)
on 1:
Network
Brarwh :
According
*elk’ =
((~1’’(jw))m
–
(Id’’),)
k
“ w’,’(jc.-%),
=
frequency
. point.
.
are
(11) .
.
treated
In
the
to
[5]
the
as shown
adjoint
current-controlled
in Fig.
network
voltage
sources
12.
we
must
insert
the
voltage

5. IEEE TRANSACTIONS
124
Q
ON MICROWAVE
THEORY
AND
TSCHNIQUJ=,
FEBRUARY
1972
d,
Read:
parameter
3) Initia(
values
b) Excitation-Response
situation
at thespeclfied
Apply
analysis
Cait
Uo, lvolt
=
store
-
program and
the branch
voltages
Form error excitation
and currents necessary
for gradierrt calculation
u~~=ovo(t
frequency points
{k )m ,(l;k )m,(l;k)m,(l;k)m
k=
r
the input excitation
L
~~1, ‘t’~2
1,.. ....n
!I
Calculate and store
part ial gradient
Form theerror
E;
relative to input excitation
components
excitation
the
Call analysis program
to calculate
the branch
—
for input
(grad’)
voltages
Appty error
excitation
to ad joint
and currents
V& ,$$2
network
!I
Cal[ anatysis
program and
store the branch vo(tage$
App[y the output excitation
u~, = o Vo(t
U02 = 1 volt
and currents
for
gradient
Apply error excitation %$,.’i’~2
Form error
necessary
excitation
‘i& ,4’~2
to ad joint network
calculation
1
,
r
Ca(culate
Calcu(ate the total
gradient
Form the error E;
relative to output excitation
partial
(grad’ + grad”)
and store
the
Cal 1 ana(ysis
partial gradient
components for output
excitation
(grad”)
—
-
program
to calcu(ate the branch
voltages
and currents
t
Yes
END
Adjust
network
parameters
according to
F(etcher - PoweN method
Fig.
source
(+)
into
source
of the
obtain
the
Except
I.
In
ad joint
for
gradient
the
controlling
controlled
network
case
must
“n
diagrams
The
be zero.
voltage
Thus
we
elements,
calculated
gradient
Go to r
for computer
the
according
component
partial
to Table
numerical
trast
would
Re
.
[S’+
Re
[
variable
have
verified
been
2)
Gradient
be determined
3) Normalization
For the
Elements:
OR1)’
–
(14)
(~Rl”@RI)”]O
gram
it
(in
the
our
sidered.
sensitivity
case
Thus
of the
I&),
the
the
controlling
controlled
relationship
branch
current
of (14)
has
has
elements
to
be con-
we
to unity
need
The
the
total
{ –
(~RiI[@R2
(1R2[@R2
+
+
method
leads
relationships
@R2’])’
@m’1)”}.
(15)
of this
EL:
Function
by a simple
analytical
of the Gradient
proper
that
(initial
normalized
normalized
axi
Re
by
the
con-
number
to
an
chapter
numerically.
dET
—
gradient
(n being
the
network
value
gradient
X~O) at
8E
—“t’”:
dxi
This
gradient
calculation.
and
optimization
of the Network
subroutine
elements
the
probe nor-
start.
Hence
components.
gradient
of the
dE
—+—
= ax{
ilEL
to be changed
as follows:
.
the
this
The
the Error
is essential
malized
For
Thus
saving.
Of
of
2n calculations
elements).
time
becomes
S”]
–(lR1o
determination
require
important
can
dlz
—.
8R1
optimization.
The
of
13.
branch
are
of RI the
Flow
of Fig.
the controlling
components
the
14.
branch.
branch
1
6’X;
error
function
is

6. HARTMANN
et Q1.:
EQUIVALENT
NETwORK
OF MICROWAVE
TRANSISTOR
125
1
—
78
-90”
Fig.
16.
Scattering
parameter
s,, (transistor
AT- IO IA:
Ic, 3 mA). A measured;
u calculated.
-1
Fig.
15.
AT-101A:
Scattering
parameters
su and siz (transistor
Avantek
UcE, 10V; lc, 3 mA).
A measured;
U calculated.
Ld
where
S,2
p’
initial
value
of the
P
actual
value
of the
This
computer
wave
transistor
Without
in
the
have
The
flow
in
Fig.
14.
the
Control
Data
Type
of Technology
bias
to
much
by
more
computer
used
for
which
of the
are
Collector—Base
Current
0165
0.023
.4
0.005
0.158
0,006
5
Im(SIZ)
laborious
17.
used
0 1s6
-00!5
6
-0026
0166
-0037
7
-0035
0162
-0050
8
—
Scattering
in
summarized
these
programs
is one
of the
computers
Swiss
Federal
Institute
computed
AT-101
BY
values
A)
is
This
final
model
C,=o. oll
C,=o. ool
C,= O.013
C,= O.005
C,= O.152
C,= O.316
C,=4.011
in the
frequency
AT-101A:
range
from
As a comparison,
and
those
shown
the
calculated
in
measured
with
Figs.
the
scattering
final
parameters,
computed
model,
15–17.
ACKNOWLEDGMENT
of a microwave
at
the
following
The
authors
Laboratory,
wish
Swiss
to
Federal
of the
above
thank
Institute
U.
Gysel,
Microwave
of Technology,
for
work.
REFERENCES
10 V
3 mA
C,= O.085
is valid
su (transistor
3 mA).
4 to 8 GHz.
his discussions
L,=o. zl
L2=0.20
parameter
10V; Ic,
APPLICATION
given:
Voltage
-0013
METHOD
Capacitances
(pF)
.
0032
UCE,
been
OBTAINED
Inductances
(nH)
L,= O.09
L,=O.31
L,3=0.36
L,=O.42
(S12)
OIL6
are
Avantek
L,=O.16
R?
0 IL5
computer
Fig.
programs
have
PRESENTED
the
(Type
conditions
Im(s12)
0146
8
the
PROGRAMS
RESULTS
THE
an example,
Reverse
Collector
4
of Zurich.
OF
As
6500,
Center
ACTUAL
transistor
from
calculation
network
procedure
computer
Computer
SOME
of the
above
The
of the
T7.
range
been
COMPUTER
diagrams
out
[GHZ]
0136
a micro.
[9]).
lV.
carrying
to
gradient
equivalent
would
and
applied
frequency
described
f
R@ (5,2)
element.
been
CALCULATED
0.162
of
optimization
[8]
the
the
determination
has
~,2
—
element;
network
program
GHz.
(see
network
MEAsuRED
[1]
Resistors
(Q)
[2]
R,=
14.8
R,=279.8
R,=
7.7
R,=
0.3
[3]
[4]
[5]
[6]
J. Te Winkel,
“Drift
transistor
simplified
electrical
characterization, ” Electron.
Radio Eng., vol. 36, 1959, pp. 28@-288.
W. Baechtold,
“The small-signal
and noise properties
of bipolar
transistors
in the frequency
range 0.6 to 4 GHz/s”
(in German),
thesis 4207, Swiss Fed. Inst. Tech., Zurich,
Switzerland.
R. L. Pritchard,
Electrical
Characteristics
qf Transistors.
New
York:
McGraw-Hill,
1967.
R. Fletcher
and M. J. D. Powell,
“A rapidly
convergent
descent
method
for minimization,”
Cow@.d. ~., vol. 6, no. 2, 1963, pp.
. .
163-168.
S. IV, Director
and R, A. Rohrer,
“Automated
network
design—
The frequency-domain
case, ” IEEE
Trans. Givcwit Theory, vol.
CT-16,
Aug. 1969, pp. 330--337.
Hewlett-Packard,
“S-parameters-Circuit
al]alysis
and
de-

7. IEEE TRANSACTIONS
126
ON MICROWAVE
sign, ” Application
Note 95, Sept. 1968.
[7] C. A. Desoer and E. S. Kuh, Basic Ciwait
Theory, vol. 2.
New
York:
McGraw-Hill,
1967.
[8] W. Thommen
and M. J. 0. Strutt,
“Noise figure of UHF Transisters, ” IEEE
T?am. Electron Devices, vol. ED-12,
Sept. 1965,
pp. 499–500.
[9] W. Thommen,
“Contribution
to the signal- and noise equivalent
circuit
of UHF
bipolar
transistors”
(in German),
thesis 3658,
Cavity
Perturbation
of the
ISMAIL
MEMBER,
IEEE,
AND
Swiss Fed. Inst. Tech., 1965, Zurich,
Switzerland.
W. Beachtcld
and M. J. O. Strutt,
“Noise in microwave
tranTrans.
-kfic~owave Theo~y Tech., vol. MTT-16,
sisters, ” IEEE
Sept. 1968, pp. 578-585.
K. Hartmann,
W. Kotyczka,
and M. J. 0. Strutt,
“~quivalent
networks
for three different
microwave
bipolar
transwtor
packages in the 2-10 GHz range, ” E1.ctYon. Lett., no. 18, Sept. 1971,
pp. 51@511.
[10]
[11]
for
C(IM
PLEX
GEORGE
semiconductor
T
two
ductor
slab
been
or
fills
to
reported
by
recently
achieved
with
the
this
that
is
the
for
used
Nag
is very
section
lems.
First,
enough
problem
between
the
in
attenuators
sample
should
the
due
and
re-
ple
at
other
phase
has
has
precise,
since
the
20 dB)
hand,
and
when
the
ac-
commercial
shifters.
the
The
fact
transverse
[10].
quency
shift
in the
interaction
sample
surement
of
placed
Manuscript
received
January
28, 1971; revised
May
28, 1971.
This work was supported
by the National
Aeronautics
and Space
Administration
under Grant NGL 23-005-183.
I. I. Eldumiati
is with Sensors, Inc., Ann Arbor,
Mich.
G. I. Haddad
is with the Electron
Physics Laboratory,
Department of Electrical
Engineering,
university
of Michigan,
Ann Arbor,
Mich. 48104.
under
a high
that
Q-factor,
the
er than
that
of
the
the
energy
the
and
energy
perturbing
the
the
If
the
sample
in the
material
can
the
meaas
sample
cavity
is chosen
is much
it can
parameters
be related
the
properties
sample
cavity,
samstrong
and
the
of a reentrant
the
defre-
the
The
for
size of the sample
within
by
cavity
material
stored
in
in
in the
post
con-
resonant
field,
suitable
the
to
techniques
inserting
perturbations.
stored
change
by
very
central
sam-
difficult
microwave
the
electric
fields
changes
external
guide
are measured
and
cavity
method
of small
a
result
the
change
the
this
of the
of the
perturbation
of maximum
between
Imakes
mode
be
is
sam-
InSb.
of a resonant
region
may
parameters
Q-factor
to the
surface
like
cavity
material
the
idea
problem
TEIO
measuring
for
mode
of a circular
scheme
material
with
erroneous
a
is normal
mode
a
gap
” Helm’s
contact
to the
This
air
even
to give
the
a TEIO
using
a small
suggested
the
converting
into
this
is
effect.
field
is tangential
by
The
termining
ple
by
method
is
that
large
sample
walls,
“gap
electric
a brittle
second
ductivity
[7]-
fact
boundary.
with
The
been
accuracy
mode,
fill
and
field
the
realize
the
guide
electric
very
to available
completely
surface,
rectangular
the
The
(nearly
a transmission
when
method
[4].
of the
prob-
get
the
shown
[6]
this
to
section;
is always
Helm
serious
cross
when
was
IEEE
hard
waveguide
effect
overcome
advantage
ple’s
serious
Recently,
to
two
is very
waveguide
and
this
MEMBER,
poses
it
there
sample
[5].
complex
the
is not
high
using
more
and
important
the
Second,
fitting,
results
cases
to fill
crystal.
SENIOR
waveguide
many
becomes
single
tight
Material
of the
in
samples
and
materials,
On
a
a semiconsection
This
method
small,
further
one
[1 ]– [3 ] and
and
reflection
is very
is degraded
standards
authors
Datta
to be measured
method
determine
of
measured
first
a waveguide
high-conductivity
phase-angle
curacy
been
the
coefficients.
many
by
the
with
VSWR
made
transmission
reviewed
especially
In
are
conductivity
has
methods.
completely
measurements
flection
material
different
Dielectric
I. HAD DAD ,
cross
takes
microwave
Measurement
and
transducer
INTIIODUCTION
MTT-20, NO. 2, FEBRUARY 1972
VOL.
Semiconductor
Abstract—Cavity perturbation
techniques offer a very sensitive
and highly versatile means for studying the complex microwave
conductivity
of a bulk material. A knowledge of the cavity coupling
factor in the absence of perturbation,
together with the change in the
reflected power and the cavity resonance frequency shift, are adequate for the determination
of the material properties. Thk eliminates the need to determine the @factor change with perturbation
which may lead to appreciable error, especially in the presence of
mismatch loss. The measurement
accuracy can also be improved by
a proper choice of the cavity coupling factor prior to the perturbation.
HE
TECHNIQUES,
Conductivity
of a Bulk
1, ELDUMIATI,
I.
AND
Techniques
Microwave
Constant
THEORY
is
with
such
small-
be shown
as a result
to
the
cavity