Dividing and couplers
1) Geometry of the Wilkinson divisor:
A line with characteristic stiffness ZC is connected through one l/4 with stiffness Z2 to
a cargo ZL2 and through l/4 with Z stiffness3 to a cargo ZL3 between these two lines it is then mails a R resistance
that has the task of disaccoppiare. In the case in which the
cargo stiffness it is Z0= 50W then is necessary two adapters
of stiffness in l/4. The
scope of the structure is to realize one partition of 
power while ovviamente 
.
2) Equations for the plan of the Wilkinson divisor:
to) replacing 
and 
in the present and
becoming simpler holding also that so that does not slide current in R
it is necessary that V are had1 =
V2 are obtained 
.
b) replacing 
in the
condition of adaptation to door 1 of the 
divisor and being it is
obtained 
.
c) the relation between the stiffnesses
of the two features of line to l/4 is obtained replacing in the the
expressions of the powers sended in income to the 2 /4 lines tol in particular 
and 
in particular has 
, one relation similar to that one between stiffnesses the cargo.
d) the value of the R resistance must
be such that it mails in parallel to - Y23 gives a null equivalent admittance that is must be
had 
. Y23 comes calculated taking advantage of one of the equations
that alloy the coefficients of the Y matrix to the coefficients of
matrix ABCD in particular 
where 
stiffness Z 2, one net c on admittance
having 0 Z and /4 is that one ofthe obtained matrix multiplying matrices ABCD ofthe having cascade of a long line l /4 a longline l stiffness Z3 , replacing has 
.
3) progettuali Options for the Wilkinson divisor:
Being of usual ZC = Z0 = 50W , has only a freedom degree and therefore enough to impose a value in order to find the others:
to) condition of adaptation to all the doors Z2 = ZL2 from which drift Z3 = ZL3 , in the case but in which it is demanded that the power comes fairly shared on the two doors this progettuale option lead to Z2 = 100W that is much neighbor to the realizable maximum value in microstrip.
b) 
c) placing ZL2 = ZL3 = Z0 I
do not have need of second l/4 that they adapt to me towards the cargos Z0 , otherwise they are necessary and
their characteristic stiffnesses are 
e
.
4) Branch-Line Divisor:
It is a structure 4 doors constituted from two separated
horizontal lines through two long lines ciascuna l/4 and spaced between they of l/4, being doubly symmetrical we have that the S
parameters from 16 are reduced to 4 and can be gain imagining to
separately apply to the structure an equal feeding to you
(… are applied to the 4 door 1 and to the same tension 
of it derives that does not slide current in the
cross-sectional lines that can therefore be considered of the stub
opened) and an uneven feeding (… apply to the door the 1
tension while to 4 door the tension 
derives some the cross-sectional lines can be
considered of the stub in short).
5) S Parameters of theBranch-Line divisor:
The S parameters of the structure can be gain to you in
function of the S parameters of the model Open Circuit and of the
model Short Circuit in particular adding the two contributions 
is had, in analogous way 
is
had and taking advantage of the relations of symmetry 
, 
, 
, 
they obtain also 
e 
.
and 
come gain to you to leave
from the matrix of transmission that obtains from the product of the
matrices of the first one stub in opened along l/8, of the line of long transmission l/4 and of the 2° stub in opened also
it along l/8, 
that in virtue of the relations between the S matrix and
matrix ABCD it concurs to obtain 
e 
, from these expressions is
obtained can be obtained those for 
and 
simply replacing Y2 with â?"Y2 .
6) progettuali Options of theBranch-Line divisor:
The condition of adaptation to door 1 can be obtained
uguagliando to zero one of the two moltiplicandi of 
the , has the cases:
to) 
from which it follows that 0S41 = and therefore door
4 is disaccoppiate from the 1 while marks them to door 2 sfasato of
90° and that one to door 3 of 180°. A divisor to 3dB can be
obtained imposing |S21|= |S31| that gives back the 
condition where Z1 is the stiffness of the horizontal features of line.
b) 
from which it follows that 0S21 = and therefore door
2 is disaccoppiate from the 1 while marks them to door 3 sfasato of
180° and that one to door 4 of 90°. A divisor to 3dB can be
obtained imposing |S31|= |S41| that gives back the 
condition where Z2 is the stiffness of the vertical features of line.
7) Rat-Race Divisor:
It is a structure to 4 doors realized in microstrip in which they are had of lthe /4 between doors 1 and 2, 2 and 3, 3 and 4 while one has 3l/4 between the door 1 and door 4, it achieves some that the structure possesses a solo vertical symmetry slowly and therefore the S parameters necessary to characterize it are 6 instead that 16. As in the case of Branch-Line the S parameters come gain imagining to apply an equal feeding to you (…applying Vto to door 1 and to door 4 and Vb to door 2 and door 3 of it it derives that the current is null on the symmetry plan and therefore the line features that to it reach can be consider you of stub aperti) and one feeding dispari (…applying Vto to door 1, â?"Vto to door 4, Vb to the door 2 e â?"Vb to door 3 of it it derives that the tension is null on the symmetry plan and therefore the line features that to it reach can be consider you of the stub in corto).
8) S Parameters of the Rat-Race divisor:
The S parameters of the structure can be gain to you in
function of the S parameters of the model Open Circuit and of the
model Short Circuit in particular adding the two contributions is had, in analogous way has
and taking advantage of the relations of symmetry , , , they obtain also e ,
moreover in adding regarding Branch-Line they have 
and 
, and 
come gain to you to leave from the transmission matrix that
obtains from the product of the matrices of the first one stub in
open, the line of long transmission l/4 and of the 2° stub in open, obtain 
that in virtue of the relations between the S matrix and
matrix ABCD it concurs to obtain 
,
e 
, from
these expressions can be simply obtained those for , and 
replacing Y2 with â?"Y2 .
9) progettuali Options of the Rat-Race divisor:
The condition of adaptation to door 1 
can be obtained uguagliando to zero 
,
replacing it with to the expressions found for the model Open Circuit
and for the model Short Circuit in the expressions of the parameters
of scattering of the Rat-Race, we have that it marks them that enters
from door 1 exits sfasato of 90° from door 2 and from door 4 while
does not exit from door 3 while it marks them in income from door 3 it
arrives sfasato of 90° to door 2.
In order to obtain a divisor to 3dB it is necessary to
impose |S21| = |S41|, it is obtained
.
