JOURNAL OF RESEARCH of the Nati
onal
Bureau
of
St
andards
- C. E
ngineerin
g a
nd
Instrumentation
Va!.
68C
, N
o.
1, J
anuary
-
March
1964
A
Variable
Impedance
Power
Meter, a nd
Ad
just-
able
Reflection
Co
e
ff
i
cient
Standard
G l
en
n
F.
Eng
en
(Oc tob
er
10, 19G
:3
)
t:
,r
osl mi c
rowav
e
power
m et e rs, such as those of t hc
bolom
ct
ric
and
ca
lorime
t ric
t.v
p
l'S,
com p
il
,tc
ly
ab
s
orb
th
e
pOlY
c r
\\
·
hich
t h
cy
indi
ca
te.
Th
e use of t hcse m
cte
rs is t
hu
s us ua lly
acc
olnp
a ni ed by
th
e re
quir
em cnt t o eit hcr (1) d
ete
rmin
e t hc powe r a t
ano
th e r pla ce in t ill'
Illi c
rowa
ve
circuit
from
t he m
ete
r r
ea
din
g,
or
(2) d
ctc
rmin
e t he
powcr
which
will be d e
li
ve r
ed
to
a
diff
eren t l
oad
whi
ch is to be s ub
st
it u
te
d fo r t he m
ete
r .
Th
e
ev
alua
t ion of t hese rel
at
ion ships pl
ays
an
impor
t
an
t role in t he p o wc r m
eas
u
re-
me n
ts
a rt .
Th
ese
pr
oblems
ar
e all
l'
\·i
atrd
by
t il(' p owe r m
ras
uring d
ev
ice to be d
es
crib
ed.
Thi
s
ins
tru
me
nt, which is ba sed o n rr A
rcLO
ll1
l'
il'r
tcc
hniq
ues, m
ay
bc
u
se
d e it hc r as a fee
d-through
powe r
1n
rte
r
\\
·hi c h
in
d icat('s t il('
pO
W
l'
r d
rl
i
Vl'
r
('
d
to
load s of a rbit
rar
y
imp
e
dan
ce,
or
a s a
te
rmin
a ti ng powl' r n
wt(
' r
\\"
hos
('
in
put
iJllp('d a n
c('
ca
n bl' a
dj
us
te
d
to
a rbi t
rar
y
va
lues
wi
t
hou
t
re
quirin
g r('ca
li
brat
ion.
I n a s
li
g
htl
y m od
ifi
cd
form t hr
in
s
tru
m
ent
a lso giv('s a
ll
indi
cat
ion
of
t he re
fl
ect io n
coeffic ie nt
m
ag
ni tude'
to
w
hi
ch it has b(' c n
adju
stcd .
A com p rclw
ll
si
\"(
'
(,
IT
o r a na l
ys
is pr rmi t s t h
('
acc
ur
ac
y
to
bl'
drt
('rmir1l'd
lIn
d
('J'
gr
ne ra l
op
rrating
co nd it ion s.J
1.
B
ack
gro
und
P O W C
I·
Ill
c ters, I
·o
r
thc
hi
g h,
ultrahi
g h, I
wd
Illi crow,t\·e rre
qu
en
cy
sp
ecL
ruJll, lllay uswtlly be
chlssed
as
cith
cr " r
ee
d-
throu
g h"
or
" le
rmina
L
iJl
g"
Ly pes .
Th
e
form
er
includ
es l
"t
sec
tion
of
I
lppropri
lt
te
\\T
'11·cg
uid
e
and
c
ontain
s
prol
·is
iolJ
fo
rm
e
,1
sul'in g Lhe
pow
er
flow
,
whil
e
'lb
sor
bin
g as
li
tt
le ene r
gy
as
po
ssible.
Th
e sec
ond
ty pc or m
cLe
r te
rmin
a Les
th
e
wav
eg
uid
e a
nd
,
id
ea
ll
y,
,"t
bsol'b s all
of
th e in c
id
e
nt
power.
Th
aL
is, th e power m
ele
r
te
rmin
,1
tes
th
e wav eg
uid
e b y
it
s
ch'
lnt
cteris
ti
c impedallce.
At
mic rowlll'e fre
qu
en cies,
mo
st
pow
er m e
ter
s b e
lon
g
to
thi s
seco
nd
ca
leg
or
y.
1n
Lh
e u
sc
of'
Lh
esc pow or m oters, c
ompari
son th e
or
y
and
Lec
hniqu
es
pla
y ,
"t
n
impor
t
an
t
rol
e.
Thi
s th e
or
y d eals
with
pr
edictillg
lh
e power
delil
·ered to
,t
lo
ad
0
1'
arbitrar
y cha
ra
c leri
st
ics
wh
en th e
pow
er ,
"tb
s
orb
ed by a difl'ere
nt
10
,
tel
is
known
(us
uall
y ,1 tcrlllilH1ting
pow
er m et er).
1n
tIle
laborator
y , this pJ'oble
lll
m
ay
t ak e the I'orm
or
c,
tlibra
t ing on e te
rmin,
1
tin
g
pow
er m
ete
r
in
term
s
of
a
sec
ond
a
nd
s
imibu
Ly pe m eter. In the
fi.
e
ld
it
))l
ay
b e re
quir
ed to d ete
rmin
e
th
e
power
delivered
to
,111
ant
e
nna
by
th
e u
se
of
a m eter
of
thi
s
ty
p e. ·
VVi
t h few ex
ce
ption
s, tlI e
u
se
of a
terminatin
g
typ
e
power
m et er is
motiv
a t ed by
th
e des
ir
e to
pr
e
di
ct
th
e power
whi
c h
will
be
delivered
by a p a
rticular
sig
nal
s
ource
to
another
load
who
se
imp
ed an
ce
difrers
from
th a t
of'
the
power
meter.
Unfortunat
e
ly
,
th
e a
ss
ociated
problem
s
ar
e
not
as
wid
ely
appr
ec
iat
ed
as
tbe
y s
hould
be
,
It
is
probabl
y safe to
sa
y
that
in
man
y c
ase
s
the
accura
cy
realiz
ed in
th
e
c
urr
en t a
rL
is to a s
ub
s
tantial
degre
e
limit
ed
by
the
failur
e
to
appl
y
this
compari.s
on
th
e
or
y.
Th
e vari a ble
imp
eda
nc
e
power
met
er ,
whi
ch is
the
s
ubje
c t
of
thi
s
pap
er ,
provide
s a
poten-
t ial s implifi
ca
tion
ill this
comp
aris
on
theory
and
pro
c
edure
becau
se
it
s
impedance
ma
y b e
'1
dju
s ted to e
qllltllhn
,t 0(' th e
sec
ond
m eLer
orlo
a
el
with
whi
ch
it
is b e
in
g
compared
.
In
a s
li
g
htl
y
modifi
ed
fOl'lll
, this de l·i
ce
al
so
provid
es the m tlg
nilud
e
of
th
e re
fl
ec
tion
co efficient to
whi.
ch
it
h as been
adju
s ted . B ec
au
se
of
the s
tron
g
int
e
rd
ep e
nd
en
ce
be
Lw
een
th
ese
appl
i
ca
tion
s,
th
ey
w
ill
be d e l'e
lop
ed s illlulLall e
ou
sly,
buL
lI
·it h
maj
or e
mpha
sis on th e
pow
er met er th e
or
y. r
rh
e
a
dju
sta ble r eAec tion coeffi c ie
nt
sLandnrd will be
includ
ed as a
Illodif
LCa
Lion of the power m eter
a
ppli
c
ation
.
7
2.
Introduction
If
two
loads
are
connected
in
tUI'll
to a
constant
voltage,
arbitrary
impedance
,
signal
source
as indi
cate
d in figure 1,
it
can
be
shown
that
the
ratio
of
the
two
powers
is given
by
: [1]1
P
A
[
1-rgrB
I
21
- l
rA
I2
P
B
1
1-rgrA
I
21
- lrBI2
(1)
where
P
A
and
P/3
are
the
powers
absorbed
by
the
loads
of reflection coefficient r
A
and
r
B
,
re-
s
pectively,
and
rg
is
the
reflection coefficient of
the
signal s
ource
. A
major
portion
of
tbe
com-
parison
theory
and
techniques,
referred
to earlier, is
devoted
to
various
methods
of coping
with
the
relationships
and
measurement
problems
imp
l
icit
in
this
equation.
The
existing
techniques
for
dealing
with this
problem
include
the
following.
(1) IV[easurements of r
o
,
r
A
,
and
r/3
,
and
substitution
of their values in
eq
(1).
This
approach
is seldom , if ever, used.
Note
that
both
the
magnitudes
l1nd
phase
angles of
the
reflection coefficien ts
are
required.
Although
a
slotted
line
is
capable
or
providing
both
pieces
of
information
,
the
procedures
associated
with
the
phase
angle
determination
are
somewhat
awkward
and
the
overall
accuracy
leaves
something
to
be
desired.
The
reflectometer
provides
a
more
accurate
measurement
of
the
reflection
coefP~ieJ1t
magnitude,
but
does
not
,
in
its
lIsual
form ,
provide
any
phase
information
.
(2)
Adjust
rg
= o
and
measure
Ir AI
and
I
rB
I.
This
is a useful techn i
que
when
one
is
interested
in
comparing
two
power
meters.
The
techniques
for
matching
a genel'ator
include
the
use
of
attenuators
and
load
isolators such
that
the
equiva
le
nt
generator
impedance
at
the
terminal
surface
of
ioterest
tends
to
become
independent
of
the
actual
source
of energy.
An
alternative
equivalent
to a
matched
generator
is
provided
by
the
use
of a directional
coupler
and
associated
techniques
[2
, 3].
(3)
Use
of a
"reflectorneter
type"
transfer
instrument
[4].
The
cited
reference [4]
de
-
scribes
an
alternative
technique
for
comparing
power
meters
which utilizes a
four
-
arm
wave-
g
uide
junction
together
with
appropriate
power
detectors
and
calibration
procedures.
This
device
may
be
regarded
as a
feed-through
type
wattmeter.
For
example, if a
reflectometer
is
e
mployed
as
the
four-arm
junction
,
it
can
be
shown
that
the
instrument
performs
the
opera-
tions
required
in
item
2
implicit
ly.
It
should
be
noted,
however
,
that
a
reflectometer
is
only
a special
case
of a
more
general
class of
four
-
arm
junctions
which
s'1tisfy
the
criteriq,
outlined
in
[4]
.
(4)
Adjust
one of
the
impedances
so
that
1'.4
= r
B
.
In
this case,
the
ratio
P.1
/
P/3
reduces
to
unity.
Of
the
methods
described,
this
one
is
potentially
the
most
accun1.te,
at
least
in
I
I
~r
LO
AD
B
I B
I
o
FIG
U
RE
1.
Ba
sic circuit jor pow
er
measurements by
comparison.
I Fig
ur
es in brac kcts indicate the liter
at
ur
e references at Ihe end of this paper .
8
----
-
------
TUNING
FIG
li
RE
2. Ci1'cuit f o
l'
1'eco
gn
izing
the equal
ity
oj
two
im
pedances .
r educ
in
g
the
un
ce
rtaint
y
in
tll e
mtio
PA/Pn.
"In
additioll, while the
measurement
of ul
tra
hi
gh
frequ
ency
a
nd
mic
row
ave
imp
e
dan
ce is
~),n
eX
Hcting
art
, the tec
hniqu
es for recognizing the
eq
uuli ty of two im pechtnces are co
111
p
/1
r
at
i vely simple.
This
I1w
y be
done
Hs indi
C/1te
d ill
fi
g
ur
e
2.
For
definiteness, it will b e ass umed
that
r
..
t
is
va
riabl
e while r n is fixedi'1
va
lu
e.
Th
e proce
dur
e consi
sts
of
te
rminatin
g a
rm
2 of t he
co
upler
by
r
B
and
ad jus tin g t he
tra
nsform er f
or
a dete
cto
r null. L
on,d
B is
the
n
replaced
by
load A.
If
r
A
is
1I0W
ad jus ted so
tlll1
t a
det
ec
tor
lIulli
s
aga
iJl
r e/1lized ,
rA
= r
n
as
required.
Tn
t he d i
sc
ussion to follow , it will b e assum ed
that
load A is
/1
te
rll1il1l
1ting
type
p
ower
meter
of re
fl
ect
iOlI
coe fficiellt f A and which has becn
ca
libra
ted
by so me m
ef)'
ns s uch
that
it
indi
cates
L1l
e
},bso
l'b
ed powe
r.
It w
ill
fr eq ue
ntl
y be refe
rr
ed
to
<1S
t he
"s
tand
a rd
."
Lo
ad
B,
for
which
the
power is required , m
ay
be
}mo
th
cr
powcr meter or
other
ter
min
l1t
i
oll.
such
as
an
a
nt
e
llllH.
Th
e tecJlIliqu e o utlined /1bovc of
In
'1k
in
g
f'
A = f'1i
11<
1s
found
but
li
tt
le u
se
bec}IU
Se
of th e
implici t requ irem
ent
for
}1
t
Ull
ill
g tp1lls fo rm er, and
the
ull ce r
ta
il1
ty
it
C}
1n
introdu
ce.
Fo
r
e:
\ all1pl
e,
if t
hi
s transform er is
<l
.
dcl
ecl
ft
h
e<1C
1 of or inclu ded as p}lrt of th e
stan
dard
,
an
un
-
cer tain
ty
is
introduc
e
cl
into its
c<l
lib
mt
i
on
unl ess
the
tran
sfo
rJll
er loss is
known
.
Alternatively
if t h e
tra
nsform er is
in
troduc
ed
in
co
njun
ct
ion with l
oa
d B , it mu
st
eit her
become
p ar t of t
hi
s
lo
ad
or
a simib
11'
en o
l'
will b e
in
t roduced . A
lth
oug
h
the
attemlll.t ion in <1
typ
i
ca
l waveguid e
transforme
r is probtLb ly n
ot
larg
e,
the uncc
rt
'1
in
t.v
in
t
hi
s loss us ually
precludes
t he u
se
of
trans
form ers
ill
L1li
s ma
nn
er.
Thi
s tec hnique of
adjusting
for r A=
f/j
has found, Jlowever, a
limi
ted
use
at
th
e
Boulder
L
n,
boratori
es of th e
l\ational
Bureau
of StaJ1(hu-ds.
Herc
t he
il11
-
pedf1l1ce
of
a.
bolometer
moun
t of special design is ad justed , before cl
dibmtioll
,
to
eq
un,l
tha
t
of
the
load
of in ter est.
The
design of t he
NBS
bolomet
er
1ll
0
Ull
t
pe
rmi ts its
ca
libnt
t ion
in
t he
microcalorimeter
[5]
and
thi s
ca
libra
tion
procedure
is ind epende'1t of
the
bol
ometer
mo
un
t
impedance
.
The
propo
sed power m eter, which is th e
prim
ary subj
ect
of t his p
ape
r , is or
the
te
rmin
at
in
g
type
and
includes a
means
of
ad
ju
st
ing
its
input
imp
edance
without
requiring
r eca
libration
.
In
com
mon
with
other
ter
min
at
ing
meters
, it
indi
cates
the n
et
power
ab
sorbe
d .
How
eve
r ,
the
probl
em
of
tra
nsform er losses is
a\
'
oided
and
,
within
cer
ta
in
limits, t his
meter
may
be
r
egar
ded as the eq
ui
va
len t of a c
onv
e
ntion
al
terminating
ty
pe power
meter
pr
eced
ed
by a
lossless
and
a
dju
sta
bl
e
tuning
tra
nsform er.
Thi
s device is in
appearan
ce, a t
un
ed
re
fl
ectometer
, a
nd
in f
act
a n
alternati\
"e
u
se
fo
r
th
c
"po
wer trans rer in s
trument
" descr i
bed
in
[4
].
Fo
r a simplified expl
anatio
n, co ns
id
er
fl
g ure 3.
By
u
se
or t
un
ers
1
~
aJICi
Ty it is possible to co mpens
ate
for
imp
erf
eet
c
oupler
dire
c
ti\
'
ity
,
etc
.,
o
t h
at
t he
power
s
indi
cf1ted
by
detector
s 3 a
nd
4
are
respect
i\'ely proporLionill to
the
powers
associ
ated
wit
h the in c
id
ent
f1Jld
em er
ge
nt
W~L
\
'
es
on a
rm
2.
Th
ese
proportionality
factor
s
may
be
determin
ed
by
a s
ui
tab
le
cl),
libra
t io n.
Jf
arm
1 is co
nn
ected to the
signa
l so urce , the device
may
be
regarded
as a fee
d-
t
hrou
gh
wattmeter
since
the
n
et
power
l
eav
in
g
arm
2 is
the
differen ce
between
the
emerge
n t
and
incident
powers. Alte1'l1lLtively,
with
a
rm
2 co
nn
ecte
d to
the
signal s
our
ce,
and
arm 1
te£'-
712
- 3 78- (;{- 2
9
POWER OETECTORS
b
l
+-:-+al
-'Io"NoIV-....-...----"
FIGum
3.
Ba
sic f o
ur
arm
j nnction used
in
pow
er
and
impedanc
e mea
su
n:ng applications.
2
m
in
ated
by
a
pa
ss
i\
'e
10fLd
,
the
dm'ice
be
c
ome
s a terminaLing pow er m
ete
r with th e n
et
power
de
liyered to
fLrm
2 given
by
the difference
between
the
incide
nt
fLncl
e
mergent
powers.2
Th
e
imp
edance obse
ned
at
arm
2
under
these conditions depe
nds
, to s
ome
extent,
upon
tbe
imp
e
dan
ce
terminating
fLrm
l.
On
the
other
hand,
th
e
proportionality
factors (
betw
een
tbe
in c
id
e
nt
or
emerge
nt
Wayes
at
fLrm
2
and
the
detector
s
on
arms
3
and
4) a
re
ind
epe
ndent
of
t he impe
dan
ce which
terminat
es
ann
l.
Thi
s mak
es
possibl e the d escribed
op
e
ration.
In
the
sect
ions to follow
the
pertinent
theory will
be
de\
-eloped.
For
th
e
sa
ke
of c
omplet
eness,
a nd
in
ord
er to
bring
out
those fea
tur
es which are pertinen t to the
pre
se
nt
appli cation
s,
the
earlier th e
ory
[3,
4, 6] will be re\-iewed
in
s
ome
d
eta
il.
3.
Review of
Prier
Ar
t
C
on
sider fir
st
a four
arm
wa\
'eguide
jun
ction
<lssemb
led
fr
om
a pt
lir
of dir
ect
ional co uplers
an
d
tun
in
g
tran
sform ers as shown in figure
3.
Anns
3
and
4
fLre
term
in
ated
by c
on
stfL
nt
im-
p e
dan
ce power detectors. Since th e subse
quent
opemtion
does
not
re
quir
e access to
the
te
r-
m i
nfL
l s
urfa
ces between
the
jun
ct
ion s
and
these detectors, it is
not
r
eq
uired that
the
te
rminal
s urfaces employed in
the
ana
l
ys
is coincide
with
tho
se
of t he
mating~wn\
'eg
uid
e
flfLng
es.
Thi
s
p er
mit
s n su b
sLa
nti
al simplifi
cat
ion in
the
trea tm en t by use of t he following artifice.
For
convenience
it
will be ass
um
ed t
hat
the
power d
etect
or
s are of
the
bolom
et
ric
type
.
In
general
th e
bolomet
er
mount
re
fle
cts
a
certa
in fra
ct
ion of
th
e incident e
ner
gy
and
abso
rb
s fLnother
p ortion.
Th
e r e
mainder
is d eli \'ered to the bolom eter or power se ns
in
g ele
ment.
Jt
is
thus
poss
ible to c
on
s
tru
ct
an
equi\-al
ent
circ
uit
of
the
bolom eter
mount
as s hown in
fi
gure
4.
Thi
s
circuit co nsists of a
lo
ssless transformer, a matched
attenuato
r,
and
a n " ideal " d
etect
or
co
n-
nected
in
tandem
fLS
shown.
Th
e ch
amcter
istics of
the
transform er H,nd
aU
e
nufL
t
or
are
ad-
j u
ste
d
HS
re
quir
ed to
duplicate
the
power reflected
~tl1d
absorbed 3
by
the
actu
al bolometer
mount.
Th
e
"idea
l" d
etect
or by definition includes a
mat
ched a
ndlo
ssless wa\'eguide
st
ru
ct
ure
,,
-hich d eliyers all
oJ
th e
in
cide
nt
power to th e bolom
ete
r ele
ment.
Th
e power det
ect
or
s shown
ill
fi
g
ur
e 3
are
now re
pla
ced by t hese eq ui
\'
a
lent
s and th e term inal s
urfa
ces be tween the
foUl'
a
nn
jun
ct
ion
fLnd
detector c
ho
sen as indi
cate
d in
fi
gure
5.
Thi
s
pro
ce
du
re, in e
ff
ect, shifts
th e nonicleal be
ha\
'ior of th e detectors to th e jun
ct
ion ,
and
in
co
rpomt
es it as
par
t of th e
j
un
ction
behfL\
'ior.
Thi
s s
tmta
gem
permit
s
an
anal
ys
is or the gene
ntl
behtlyior or th e de\'ice
LOSSLESS
TRANSFORMER
MAT
CHED
ATTENUATOR
"IDEAL"
DETE
CTOR
FIG
U
HE
4.
Equi
va
le
nt
circuit
of
bolomet
er
mo
unt
(pow
er
de
tector) .
Z
Th
e f
en
ei
er lll
ay
\\
-o
nder
if
th
e power
inpu
t a t
ar
m 1 could a lso be
obta
ined fro lll
th
e readin
gs
on a
rm
s 3 a
nd
4.
Tn
order to achieve
thi
s re·
suIL
th
e
jun
c
tion
wo
uld
h
ft
y c
to
s
ati
s
fy
ce
rt
a
in
add
iti
onal cond
iti
on
s,
t h e r
ea
li
zation
of
wh
ich
would
pr
oyc
diITi
cu
lt
in
t he
ex
is
tin
g a
rt.
3 Strict ly spea kin g th e a
tL
enu ator is adju sted so
th
at
it
s
effi
ciency
equ
als th e effective
effi
c
ien
cy of
the
bolometer
mo
un
t.
Th
es
e conce
pt
s are
explain ed in
[5J.
10
"IDEAL"
DETECTORS
F , G
L:
HE
5.
I
ll
ustrating the choice
of
termi
na
l
swj
ac
e
emplo yed in the an al ysis.
2
un d er th e Silll plify ill
g"
ass
um
pt
ion of
Jl
hltc
hed
<t
ll
d id oal d e t
ec
to
rs on a
rm
s 3 a
nd
4.
It
mu
st
be
re
m e
mb
er ed , ho wc \'
l'
r, t h
aL
t ho n ow
tc
nnill
,.l s
urf
ac
os 3 a nd 4 m'e,
in
g en er al, n
ot
ava
il
ab
l e f
or
su
ppl
em en
ta
r y
ob
sor
nlt
ion s,
.I
n a d
di
tion , cer ta in
of
t h e d et ec
to
r Ch
'"l'H
cte
ri
s ti cs h a \"o been
ab
s
orb
ed w
ith
in
th
e
ju
n ct io n ,
Th
ese Jllu
st
b e ta k en
in
to
n
cc
o u
nL
ill
,)
n
,v
a t t em
pt
to
c
on
s t
ru
ct
th e
jun
c t ion behll\'io r from t he c h
nm
cteris ti cs
of
its in
cl
i\' i
cl
ual co m po nen Ls.
Th
e
sys
t em bc
ll<
l\
'io r will b e d esc ribed i
ll
Lel'lll S o f a p
pr
op
ri
at
ely nO
I'Il
l<
tii zcd i
ll
cid c
ll
t
li
ll
d
em er
ge
n t " \' oI
Lll
ge"
Wi
lye a mp
li
t ud es lit the d i
ff
ere
ll
L Lel'llli
na
l s
Ul'
faccs ,
Th
es
E'
wi
ll
be
cl
e3
-
ig n a
te
d
b
~
'
am a nd b
IN
resp
ect
iYol
y,
wJ
lO
r
l'
t h e s u bscrip
ts
tHk
e t ho \·alu es 1 . , , 4 co r res po ndill g
to t he differe
nL
te
l'l
ll
illll ls
of
in te res t.
Th
e
pow
e rs , P3 a
ll
d i
\,
d e
li
\'ered Lo
cl
etec to r
s:)
a l
lc/
4 ,tr e gi
ve
n b y
I
Z
~
3
!
2
a lld
I
Z
~4
1
"
r esp ec
tiv
el
y,
':....1
0 uo
(N o
te
t hHt t
il
l'
a'
s a lld b
's
ar
o i
ll
ge
n
er
al co
mp
lex, ) T he n et
pow
e r em erg
in
g f
rom
a
rm
2 is
g iven by :
(2)
III
acc
ord
a n
ce
wi t h fr e
qu
e
ntl
y u sed mi c r
owav
e
pr
act
i
ce
,
th
e nO
l'
l
ll
aiiilation is I
Il
'ld e
ill
s uch a W<ty
th.tt
t h e Chll
l'<t
c te ris t ic illl pe
c/
n
ll
C'
e, Zo, r e
du
ces
to
u ni ty .
Thu
s , t hi s fa
ct
or
d
oE'
S n
ot
a
ppear
in th e eq ua t io ns w hic h foll
ow
.
Th
e r ela t io ns hi ps im
po
se
d by t he j
Ull
ct
ion O
il
t he
a's
<
lIlcl
b' s a re g i\Te
ll
h
y:
(3)
wh
er e t ho S",.
II
ar
e t he sca
tter
i
ng
coe
ffi
cien ts of th e foul'
Il
l'lll jUll
eL
io n (
n,
fo
ur
th e
qu
at
ion 11l-
\' ol vill g
b,
co uld be wri
tte
n b u t is
no
t r
eq
uir
ed ill t he
an
al
ys
is ).
Th
o ltd v
an
tage
of
th e choice
o
i'
r efere n
C'
e
pl
,w es o u tlin ed ea rli er is now ev id e
ll
L.
O
rdin
a ril y eq s (3 ) w
ould
includ
e
te
r
ms
in
a3 a nd a
4
, S ix
te
rm
s
11Il\
'e b
ee
n e
limin
at
ed by choos
in
g t h e t e
rmin
al s
urf
ac
es so tha t a3 nnd
a
4
"/t
il
ish ,
So
h -
ill
g eqs (3 ) for a 2 Iw d b
2
ill
Lel'lll S of b
3
ft
ll
d b
1
y ields :
(4)
11