sible o ms
o
J( ).
Fo
example,
we
canno ob ain
he
esponse
o an
impulse unc ion and canno e en hope
o
app oxima e
o i
easonably by an
. . .
o economical size.
The same ype
o
commen applies, al hough less o cibly,
o
a
s ep- unc ion inpu . These a e he penal ies o ha ing
an incomple e se
o
da a, and canno be a oided. We can,
howe e , ob ain he esponse o p ac ical inpu s in which he
ise ime
is
ini e, e en hough he p ocess may be expensi e
i he ise ime
is
small. Again, we mus emphasise ha his
is he only known solu ion
o
his ype
o
p oblem
in
which
we equi e he solu ion
o a
di e en ial equa ion we canno
o m.
Fo
a
nonlinea ci cui , we assume ha he nonlinea i ies
a e
in
he lumped po ion
o
he ci cui and a e
o
he ype
ha
may be
ep esen ed
by
ol age-dependen cu en
gene a o s.
I
we
knew he di e en ial ope a o y( ) we could
w i e down
a se o
di e en ial equa ions desc ibing
he
sys em
y( ) ( ) = J( )+ [ ( )]
(2)
whe e
/
is
a
ec o o known nonlinea unc ions. The solu-
ion o eqn.
2 is
gi en by he Vol e a-in eg al equa ion
>(0
=
«W( )+
J
H( -x) [ (x)]dx (3)
As be o e, y( ) is no explici ly de ined; all we know a e he
alues
o
Y(co)
a
speci ic equencies. i0)( )
is
ob ained,
as
in
he linea case, by inding he
i. . . .
o
Z(co)I(co),
and
H( ) is he in e se Fou ie ans o m o
Z(co),
and
is
he e o e
de ined
a
disc e e poin s. Subjec o con e gence, »(() may
hen
be
ound
by a
Pica d i e a ion, o ming successi ely
i0), (1), «(2>,..., using he ecu ence ela ion
(0 = i0)( )+
H( -x) [ U)(x)]dx
.
(4)
whe e, a each s age, he con olu ion in eg al is e alua ed by
o ming
he
. . .
F(co)
o /
[ U)( )]
and,
o
each co,
p e-
mul iplying his ec o
by he
co esponding Z(co)
and,
inally, e ans o ming, ia he
i. . . .,
in o he ime domain.
This ype
o
i e a ion has,
in
ecen yea s, ecei ed de ailed
conside a ion by his w i e and o he s1"4
in
i s applica ion
o inding he s eady-s a e esponse o a nea ly linea sys em.
P esen conside a ions show ha
he
desi able applica ions
o eqn. 3 a e wide han we e o iginally hough , since, o
a
wide class
o
ci cui s,
i
o ms
he
only known me hod
o
analysis. Howe e ,
in
addi ion
o
he di icul ies ha ha e
al eady been obse ed
in
he linea case, he e a e now ad-
di ional di icul ies a ising om he limi ed egion o con e -
gence o
he
Pica d i e a ion. I is shown4 ha , o easonably
well beha ed unc ions/, con e gence o he i e a ion may be
ob ained
in
o he wise noncon e gen cases by he in oduc-
ion o
a
damping ma ix C, modi ying eqn. 3
o
J H( -x) [ (x)]dx
(5)
whe e
U
is he uni ma ix, and, using he i e a ion,
B(1>(0
=
(2U-C)C i0)+C
J
H( -x) [C (0>(x)]dx
(6)
a+1)( ) = C iO>(0+(U-C) U)( )
+ C j H( - x)
[V<J>(T)]
dx
(1)
In p inciple, again o easonably well beha ed/, i
is
always
possible o ind
a
ma ix
C
such ha his i e a ion con e ges,
bu , o g ossly nonlinea ci cui s, his esul is gene ally no
a use ul one
as
he p og ess owa ds
a
solu ion may
be so
slow as o be uneconomical, e en
i
i is no comple ely los
in he ounding e o s o he compu e . Highe -o de i e a i e
solu ions such as he New on-Raphson i e a ion also become
imp ac icably unwieldy
o any bu he
simples sys ems.
ELECTRONICS LETTERS
18 h
Ap il 1974 Vol.10 No.
8
Thus
he
p esen si ua ion may
be
summa ised
by
saying
ha , wi h
he
echniques cu en ly a ailable, nonlinea
ci cui s ha canno be modelled in e ms o lumped elemen s
can
be
analysed
in
he manne desc ibed p o ided ha
he
nonlinea i ies
a e no oo
se e e.
No
o he me hod
o
ob aining such esul s seems
o
be a ailable. No me hod
is
known by which
a
solu ion can be ob ained economically
in
g ossly nonlinea cases, and u he wo k
is
equi ed
in
his
espec .
By
using
he
gene alisa ion
o
nodal analysis
desc ibed
in an
ea lie le e ,5
all
ou ypes
o
dependen
gene a o may be accommoda ed
in
his ype
o
analysis.
T.
B. M.
NEILL
18 h
Feb ua y
1974
Pos O ice
Resea ch Depa men
Dollis Hill
London
NW2 7DT,
England
Re e ences
1
NEILL,
T. B. M.: 'Imp o ed me hod
o
analysing nonlinea elec ical
ne wo ks', Elec on. Le ., 1969,
5,
pp. 13-15
2
HEYWOOD,
D. R., and
MOORE,
A. D.: 'Commen on an imp o ed
me hod
o
analysing nonlinea elec ical ne wo ks',
ibid.,
1969,
5,
pp.
269-270
3
NEILL,
T.
B. M.: 'Reply
o
commen
on an
imp o ed me hod
o
analysing nonlinea elec ical ne wo ks',
ibid.,
1969,
5,
pp. 270-271
4
NEILL,
T. B. M.: 'Spec al analysis
o
nonlinea ci cui s'
in
'Ne wo k
and signal heo y'. PPL
Con .
Publ. 12, 1972, pp. 122-131
5
NEILL, T. B. M.:
'Gene alisa ion o nodal and mesh analysis', Elec on.
Le ., 1969,
5,
pp. 365-366
SELECTION
OF
ZEROS
OF
REFLECTION
COEFFICIENT
IN
DESIGN
OF
FILTERS
BY
INSERTION LOSS
Indexing e ms: Fil e s, Ne wo k syn hesis, Poles and ze os
I is shown ha
a
cha ac e is ic unc ion ha can be exp essed
as he squa e
o
an odd a ional unc ion
in w,
wi h poles
on
he
j<o
axis and
a
in ini y,
is
always ealisable wi hou using
ans o me s by he selec ion
o
he ze os
o
p(s), and his
is
he key
o a
il e design based
on
2-po syn hesis heo y.
I one wan s
o
inse
an
LC il e be ween
a
uni y-in e nal-
esis ance gene a o and a load esis ance (bo h no malised),
i s ansmission cha ac e is ic may
be
exp essed
by
means
o
i s
inse ion-loss unc ion P20IP2,
o i s
cha ac e is ic
unc ion F((o2), ela ed by
De ining
p(s)p(-s)
= -
=
_**
wi h
= 4 /(l
+
)2; he il e inpu immi ance is1
Z^
o 7(5
= 777
1
+p{s)
and i s ans e unc ion
is
E2
l
+
/(mlm2
—
H(s) =
(1)
(2)
(3)
Fo
a
lowpass il e —o he band con igu a ions a e ob ained
by
a
equency ans o ma ion— he ollowing exp essions2
gi e i s pa ame e ma ix:
o
— Z2i o
-V Y2l
=
—
Z22o Y22
=
—
n2
Z(mlm2-nl n2)(4)
133
I F(co2)
> 0 o all co, and i s
poles
a e
loca ed
on he
jco
axis, aking om
eqn. 1 he
le -hal plane poles
o p(s),
eqn.
2
gi es
an
immi ance ha
is a
eal posi i e unc ion,
and
he
pa ame e s om
eqn.
4
will cons i u e
an odd
eal
posi i e ma ix, which leads
o a
p ac ical ladde il e wi hou
ans o me s, p o ided ha
he
Fialcow-Ge s 3 condi ions
a e
me . A
his poin ,
he
selec ion
o he
ze os
o p(s) is
o g ea es impo ance.
Le
N be an odd and M an
e en polynomial
in s
sa is ying
(a) deg ee
o N >
deg ee
o
M
(b) highes -deg ee- e m coe icien
o
JV
equals
1.
Taking
2)
- - —
s
= jco
as
he
cha ac e is ic unc ion,* hen
Pi
S
=
JCO
M2-N2
S
=
JCO
and, om eqns.
3 and 1,
and
p(s)p(-s)
=
•-«!
n2)
= - -—M
(m2-m1)2-(n2-n1)2
(l- )M2-N:
M2-N2
(5)
(6)
Ob iously, om
eqn.
6, he
immi ance
has a
ze o
o a
pole
a in ini y; he ea e
i
will be supposed
o
ha e a
ze o,
wi hou
loss
o
gene ali y,
and
P(O)
= ±
(7)
To o m
p(s), he
le -hal plane oo s
o
M2-N2
mus
be
ob ained, gi ing Mj+Ni.
I
may
be
easily p o ed ha
(a) deg ee
o
N1
=
deg ee
o
N
(b) deg ee
o
N1
>
deg ee
o Mx
(c) Mx(0)
= M(0)
(d)
Nx
highes -deg ee- e m coe icien equals
1
(e)
he
coe icien
o a
gi en powe
o s in Mi
ponding coe icien
in M
(coe icien s
Mi
M).
The selec ion
o he
ze os
o p(s)
ha gua an ees
he
ul il-
men
o he
Fialcow-Ge s condi ions
by he
ma ix
o
eqn.
4 leads
o
he co es-
coe icien s
p(0)M + N
P(s)
=
whe e
p(0)
is
gi en
in
eqn.
7,
and,
ese ing
he
symbol
+ o
ob ain
Z(s) and he
symbol
— o Y(s), is
equi alen
o
aking
Z(0)
= and 7(0) = 1/ .
F om eqns.
2, 4 and
5,
he
p oo ha his choice
o
p(s)
is
co ec educes
o
e i ying ha
coe icien s—- (
ML
—
M
I
^
coe icien s
M
2
1
+
}
1
+
and
coe icien s-—-
{M
+
MI ^
coe icien s
M
2
1
+
J
1
+
whe e
he
ac o
1/2 has
been in oduced,
so
ha
(8)
n2
=
has
i s
highes -deg ee coe icien equal
o
uni y,
and o
p es-
e e consis ency wi h
eqn. 6;
e i ica ion
o
eqn.
8 is im-
media e once
i has
been es ablished ha coe icien s
M 5s
coe icien s
M.
* Ob iously,
we
suppose
he
ze os
o M o be on
he
jco
axis
134
I
is
in e es ing
o
men ion ha , when
M is a
cons an ,
all
il e ansmission ze os
a
in ini y, since
he
ze o-deg ee
nume a o e m
is
independen
o he
choice
o he
ze os
o
p(s),
he
ul ilmen
o
he
Fialcow-Ge s condi ions will
no
depend
on
ha selec ion.
22 nd Feb ua y
1974J.
B.
MARINO
ETS Jngenie os
de
Telecomunicacion
Uni e sidad Poli ecnica
de
Ba celona
331 A enida
del
Caudillo
Ba celona, Spain
Re e ences
1
TUTTLE,
D.
F.: 'Redes elec icas' (Dossa , Mad id, 1964), chap.
13
2
VLACH,
j.: 'Compu e ized app oxima ion
and
syn hesis
o
linea
ne wo ks' (Wiley, 1969), chap.
4
3
FIALCOW,
A. D.,
and
CERST,
I.: 'The
ans e unc ion
o
gene al
wo
e minal-pai
RC
ne wo ks', Qua . Appl. Ma h.,
1952,
10, pp.
113-127
OBLIQUITY FACTOR
FOR
RADIATION
FROM SOLID-STATE LASER
Indexing e ms: Lase beams, Solid lase s
The adia ed ield consis s
o
o wa d adia ion
a an
angle
0
and backwa d adia ion
a
n
—
0 e lec ed om
he
dielec ic-
ai in e ace in o
an
addi ional componen
o
ield
a
angle
0.
The o al adia ion con ains
a
ac o ha simpli ies in o
an
exp ession ecen ly ob ained
by
mo e sophis ica ed me hods,
and which
is
he e in e p e ed
as
he
Huygens obliqui y ac o
o
he
a angemen .
A
small co ec ion
o he
exp ession
is also ob ained.
Recen ly,1
by
means
o
ma hema ical echniques in ol ing
Fou ie in eg als
and
saddle-poin asymp o ic e alua ions,
exp essions ha e been p oduced
o
some p ope ies
o
solid-
s a e-lase adia ion, including
he
damped in e e ence wa es
a
he
dielec ic-ai bounda y,
and an
exp ession leading
o
he obse ed sha pening
o
he
adia ion-pa e n.
The
la e
ac o
can
be
w i en
g(0)
=
whe e
2cos0
{cos0+V(£2-sin20)}
{P/ko+V(e2-sin26)} (0
g(6)
=
adia ion-pa e n ac o addi ional
o
main
adia ion-pa e n exp ession
6
=
angle o -axis
a
which
he
ield
is
obse ed
e2
=
ela i e pe mi i i y
o
lase ma e ial
/?
=
axial p opaga ion ac o
in
lase
kQ =
p opaga ion ac o
in ai
I
is
in e es ing
o
no e ha eqn.
1
can
be
ob ained by elemen-
a y means,
and
ha
i
ep esen s,
o he
dielec ic-ai
con-
igu a ion, he o m aken
by he
Huygens obliqui y ac o .*
Fig.
1
shows
he
a angemen , wi h
an
inciden su ace
wa e
o
he o m1
Ey
=
Ei{x) exp
(-j iz)
z < 0
(2)
whe e E,(x)
is
he
ans e se- ield dis ibu ion
o
he inciden
su ace-wa e mode.
In
egion
3
(ai ),
he
ield
can
be
ound
om
he
o m
o
Huygens's in eg al
in
cylind ical co-o di-
na es:'
00
y/
=
(8; £o)~*exp(-77 /4)
dy/Q exp(-jko )
8n
V
~¥oIn-
V? ]dX (3)
*
The
Huygens obliqui y ac o
is he
adia ion pa e n
o an
elemen
o a
adia ing ape u e.
I
has
he
alue
cos
0+cos
0
o
adia ion
a an
angle
0 o
inciden adia ion
a an
angle
< >.
I is
slowly a ying,
and
is
o en igno ed when
calcula ing adia ion pa e ns om la ge ape u es
ELECTRONICS LETTERS
18 h
Ap il 1974 Vol.10 No.
8