Coupled Lines
1) Equations differentiate them that they describe two coupled lines:
If we have two placed side by side microstrips, line 1
characterized from an inductance series L1dz and from one ability parallel C1dz while line 2 characterized from an inductance series Ldzand from one 2
abilities parallel C2dz,
in virtue of the vicinity we will have one then mutual Mdz inductance
and one mutual ability Cmdz,
this by means of the theorem of Miller pu² to be subdivided on the
single lines, in particular the ability parallel total of line 1
becomes 
while that one on line two becomes 
. The equation to the nodes to line 1 gives
to us 
while that one to the nodes to line 2 door
to 
. In analogous way from the equations to the
meshes ottengono 
e 
.
2) Disaccoppiamento of the equations differentiates them that they describe two coupled lines:
The hypotheses that are made are the two following:
to) the two lines 1 = C 2 coupled is equal that is is using onesymmetrical structure thereforeL1 = L 2 eC
b) diagonalizziamo the system
imposing to every section the variable change of 
, 
, 
,
where arbitrarily the line over is the 1
and that one under the 2.
Replacing b) nella the and
nella and adding the two obtained equations 
while embezzling is obtained 
,
analogous for the tensions is gained then has 
e 
. Therefore we have obtained a
system of 4 equations differentiates them disaccoppiate that it
concurs us of dealing the lines like if they were separate, relative
to the way even with characteristic and 
constant
stiffness of 
phase and an other relative one to
the way odd with characteristic and 
constant
stiffness of phase 
, however in the
continuation will be supposed band = bor .
3) Conditions of closing standard for a divisor to coupled lines:
We consider 2 coupled lines in which to door 1 it is
applied to a generator of having tension largeness impressa 2V and
inner stiffness Z0 while all the
other doors are sluices on one stiffness Z0. The conditions of closing to the section of income
1 are 
e 
in which
replacing the expressions for the disaccoppiamento 
, 
, 
, 
obtain 
e 
. Analogous the
conditions of closing to the escape section are 
e 
in which replacing the expressions
for the disaccoppiamento 
, 
, 
, 
obtain 
e 
.
These disaccoppiate equations being concur to us of separately dealing the Even line and linens to Odd.
4) Equations of plan for a divisor to lines coupled in closing conditions standard:
We consider the single Even line in how much the closing
conditions standard taxes on the divisor have concurred of
disaccoppiare the equations, draft of a log of line with transmission
matrix 
, fixing hour Jand = Jor = /2 J =p the matrix
is simplified remarkablly and obtains the two equations 
, 
that replaced in the condition draft from the closing to section 1 give
back 
to having place 
.
is had 
then, 
e 
. It turns out you for the Odd
line are obtained replacing and with or in the found equations.
5) Condition of adaptation for directional couplers and expressions of the tensions to the doors according to Z0and, Z0or:
Adaptation is had if 
, replacing e and being , 
e , , 
is reached the 
relation that concurs to write the largenesses to the lines
Even and Odd in simpler shape, in particular is had:
;
;
; 
;
;
;
.
Replacing in the expressions of the tensions 
, 
is had that it is
logical inasmuch as we had a generator from 2V and we have tax the
adaptation condition, 
where C is the
connection factor and usually comes expressed in dB, 
that is to door 2 marks them exits sfasato of 90°.
6) Parameters of scattering of 2 lines coupled in closing conditions standard and realization coupler to 3dB:
It is necessary to observe that all the doors are adapted
therefore S11 = S22 = S33 = S44 = 0 in how
much the reflected wave are null, must then be had S12 = S21 = S34 = S43 = 
, moreover S23 = S32 = S41 = S14 = 
,
finally ha
S31 = S13 = S24 = S42 =
. In order to obtain a coupler to 3dB it is simply necessary
to impose 
.
7) Meant of the ways Even and Odd in the case of lines coupled Stripline type:
Two plans of mass are had separate to you from a dielectric in which they are dipped two conductors, we then choose a system of reference x,y regarding which the structure is symmetrical, we separately analyze to the Even way and the Odd way.
For the way Even we place Vor = 0 of it derive that V1 = V2 = Vand therefore in the symmetry point you
must be a maximum or a minimum of upgrades them pertanto 
follows some that the symmetry plan is equivalent to
one ideal magnetic wall. In analogous way for the Odd way we
place Vand = 0, we obtain V1 = Vor e V2 = - Vor therefore in
the symmetry plan upgrade them must be cancelled with to its
derivative therefore the symmetry plan are equivalent to one wall
ideal electrical worker.
8) Coupler of Lange:
Much width is a coupler characterized from a band, approximately eighth, is constituted from finger of various length, in particular those courts is l/4 to the higher frequencies while those long ones are l/4 to the lower frequencies. One characteristic important is that the model that has been developed works perfectly however is of the difficulties to realize couplers to 3dB with this geometry.
9) Matrix of transmission for lines sluices coupled on an open:
In order to calculate the transmission matrix we pass for the Z matrix, in particular we consider two coupled lines of transmission, with marks them that the 1 while door 4 is sluice on an open, other enters to the door with marks them that enters to door 2 while door 3 is sluice also it on an open.
Imposing the condition of 
closing to
income section 0 is obtained 
that while imposing
the condition of 
closing to escape the J section it obtains that 
. Replacing 
is had
and 
.
From these 
e 
can be obtained.
10) Matrix of transmission of the commensurato filter:
In order to calculate the transmission matrix we pass for the Z matrix, in particular we consider two coupled lines of transmission, with it marks them that she enters to door 1 and she exits from door 2 while the other to door 3 is sluice on a short circuit while to door 4 it is sluice on an open, evidently the behavior will be of type lowpass .
Imposing the condition of 
closing to
income section 0 is obtained 
that while imposing
the condition of 
closing to escape the J section it obtains that 
. Replacing 
is had
and 
. From these , 
can be obtained
while in order to gain B we leave from 
therefore 
.
