Coupled Strip Lines with Highly Unbalanced Electromagnetic Coupling and their Use

 Abstract — The coupled lines being considered are with horizontally and vertically positioned strip conductors. Constructions like this help gain vastly different phase velocities of synphase and antiphase wave types. The task of calculating the electric field in the per unit parameters’ cross section was being solved using the grid method. The capacities per unit were being defined as a sum of the partial volumes in the chosen subareas of the strip structure, in which the accumulated energy of the electric field was calculated. The inverse problem of finding the relative permittivity using the set phase velocities relation of synphase and antiphase waves was solved. There are given synthesis examples of the constructions using the set phase velocities relation. Simulation and experimental research of several coupled lines sections with different relation of phase velocities was complete. It was shown that having a substantial difference in the phase velocities, resonant fluctuations appear in the sections with scheme in which one of the unloaded on ports 2 and 4 lines is under the floating induced potential by the second line. That line in such scheme is included between ports 1 and 3 for passage. Herewith depending on the phase velocities relation, pass-bands of differing width and multiplicity are observed.

for the identical lines, were reviewed in article [1]. In this work the introduced coefficients are: coefficient of line coupling by the voltage In general, coefficients U K and I K may not be equal. In this case the inequality points out the unbalance of electromagnetic coupling between the lines. Using U K and I K we have four values of the spread coefficients of synphase (index "c") and antiphase (index "π") waves: The inequality с     was not used in article [1] for the analysis of the transfer matrix and devices based on coupled lines, because it was relied on that in coupled lines with TEM waves at any frequency the waves move at the same velocity.
In work [2] the matrix ABCD was found, Z , Yparameters of the identical coupled lines in a heterogeneous dielectric environment. The heterogeneity of the dielectric environment was being taken into account by introducing an electric length inequality of the coupled lines at their synphase (index "e") and antiphase (index "o") excitement. Since in [2] were reviewed coupled lines with the identical parameters, for naming synphase and antiphase types of excitement were used the terms "even" (index "e") and "odd" (index "o") wave and electric wave lengths accordingly 0 2 ее e f l f where 0e f and 0o f are frequencies, at which the coupled lines have a quarter length of a wave at even and odd excitement.
The found transfer matrice in [2], impedances and conductivities matrices Z , Y were used for analyzing a number of equivalent schemes reviewed earlier Jones and Boljaln [3]. Zysman and Johnson showed [2] that the frequency characteristics of the schemes change vastly due to the inequality of electric lengths еo    . Further progress in researching of the coupled lines with unbalanced electromagnetic coupling is associated with articles [4][5][6][7]. In those works, using insignificantly different ways the task of calculating matrix parameters of coupled synphase and antiphase modes. This allowed us to avoid the interference of waves in coupled lines, which reduces decoupling and is accompanied by resonant phenomenons [8 -10].
The other direction consists of the search for an optimal ratio of the inequality of e v and o v for solving the problems of improving frequency selective characteristics of devices based on CL [11,12], and creating short impulse equipment defense devices using modal filters [13,14], and the designing of trans directional couplers [15,16].
Next, we will be reviewing the general case of similar coupled lines, in which the voltages and currents of normal waves in the lines during synphase and antiphase excitement are not equal. Thus, we will hold on to naming synphase waves using the index «c», and antiphase using the index « π », as in article [5].
The purpose of this work consists of researching the influence of the phase velocities inequality

II. CONSTRUCTION AND THE MODEL OF COUPLED LINES
For researching the dependence of coupled lines parameters from the ratio c vv  we have taken the construction of coupled strip lines (CSL) with the cross section shown on Fig. 1. This construction represents the modification of coupled lines with a vertically positioned substrate (VIP), which were suggested and researched in articles [17,18]. Having the gap lets us reduce own capacities of horizontally positioned strips and thus increase the characteristic impedance during synphase excitement of conductors. Besides, there is an emerging possibility of varying electromagnetic coupling's degree of unbalance.
The modification of VIP with a gap in the ground plane was used for building the C-section with an unbalanced electromagnetic coupling in the correctors of group delay [19,20].
In articles [17,18,21] the calculation of primary and secondary parameters CSL with a vertical substrate using various methods with set geometrical sizes and substrate properties was studied.
But in practice it's often needed to solve a inverse problem, that is to find the geometrical sizes and substrate parameters using the set matrices of the primary parameters in the form of С capacities' matrix and L inductances' matrix. The initial data may also be secondary parameters in the form of characteristic impedances of synphase 0c Z and antiphase 0 Z  excitement, and coefficients of the capacitive C k and inductive L k coupling [7], which define the ratio of synphase and antiphase phase velocities in accordance with (4) [22]. The being examined coupled lines with a cross section according to Fig. 1 own the following distinctive features.
1) The constituents of the strips' per unit capacities, which are executed on horizontally and vertically positioned substrates are to varying extents dependent from sizes  . Calculation of the primary parameters was done using the grid method [23]. Herewith the goal was to independently from the chosen method (as a tool) justify the algorithm of the sizes' synthesis using the set primary parameters, which lets us define the being physically implemented construction of the 3 CSL's cross section with unequal phase velocities of normal waves.
The calculation of the electric field during synphase and antiphase excitement of coupled strips ( Fig. 1) was being done by transitioning from the Laplace's differential equation to finite-difference approximation and solving using the iterative procedure on a PC [23] with a deviation of 6 10  with net size 125×95. The results of the calculation in the form of equipotential lines during synphase and antiphase excitement of coupled lines are shown on Fig. 2 and Fig. 3.  Herewith we took the following cross section sizes of strip lines 1 4 6 h  mm, 10 a  mm, 1.0 d  mm and relative permittivity 14 1.0     , 23 2.68     . As a result of solving finite-difference equations for potentials , ij U , in the grid's couplings the electric fields' projections x E и y E on the axes x and y were being calculated. Then, for the synphase and antiphase excitement the overall energy was being found (5), which was stored in the electric field [23]  where the electric field's sum of tension vectors' projection squares on the axes x and y is calculated as follows indexes , ij are taken from within their alternation range limits for the corresponding subareas. In expressions (7.1) -(7.6) sum limits contain the 16 ,...,  subareas' coordinates of limits, which are gained after the strip structure's cross section sampling (see Fig. 4). Formulas (7.1) -(7.6) are used while calculating the matrices of per unit capacities and coupled lines' inductances. For this the current-carrying strips potential is set to +1 at synphase excitement and -1 at antiphase excitement. Then the Laplace's equation in the finite-difference form is solved and the dependence  The per unit capacity of a strip for the synphase modes having the identical coupled lines' sizes filled with 14 ,...,  dielectrics is defined as follows in (9)-(12) While being filled with air the per unit capacity at synphase excitement is found the following way The per unit capacities at antiphase excitement are found similarly: Formulas (9) -(12) let us represent per unit capacities с С , The coefficients of capacitances' matrix while being filled with air are written down based on (10) and (12) 66 11 22 11 (1) (1) 0.5 Having the matrix of per unit capacities while being filled with air written down, let's find the matrix of per unit inductances (1) Next let's define the effective permittivities at synphase excitement and for antiphase excitement   . Wherein the straight and inverse problems were solved. The straight problem -the calculation of primary and secondary parameters, and the inverse problem -defining 23 ,  using formula (21). As we can see, the full matching of 23 ,  were being defined using formula (21) and the calculation of coupled lines' primary and secondary parameters was being done using the grid method by the program NETEPSILON (Table 1). Knowing the matrices of per unit capacities, inductances and effective permittivities lets us define the characteristic impedances of the synphase and antiphase wave [22] It is obvious that if changing   Fig. 5 and Fig. 6 the dependencies of  The analysis of graphs on Fig. 5-8 shows that the dominant contribution to the total capacity of antiphase fluctuations type is the capacity between vertically positioned strips (coefficient 3 WE  ). Functions which approximate dependencies     .
Addressed to expression (22) Constructions with different ratios of synphase and antiphase waves phase velocities are synthesized. As a basis we took a construction with sizes and relative permittivities 1 0 w  , 10 a  mm, 4 6 h  mm, 1 2 14 1.0     . During the synthesis, as usual, two iterations were being done. Their purpose came down to on the first step determine by using formulas (21) and (23)    Ohms.
Example 5. As a base construction we took the strip structure with The objective is to based on this construction, get the ratio π c vv 2.5, using the estimations of expressions (21) and (23), and conduct an experimental researches of coupled lines' sections with synthesized parameters and the strip switch on scheme as on fig. 9. Wherein the condition of  parameters was made, and its frequency characteristics were experimentally measured using the scheme on Fig. 9. The results of calculating the coefficients of the scattering matrix 31 S and 11 S the section of the coupled lines showed a significant difference in their frequency dependence from the experimental one. One of the reasons, according to our assumption, was to deviate 3  the material of the vertical substrate from the specified nominal value. A change of up to 15 resulted in a perfectly satisfactory correspondence between the calculated and experimental dependencies 31 () Sf , 11 () Sf as shown in the following subsection.

IV. FREQUENCY CHARACTERISTICS -DEGRADATION OF ALL-PASS SECTION PROPERTIES
It is known [3], that the coupled lines' sections with their switch on scheme as on Fig. 9 are all-pass while the electromagnetic couplings are balanced. Next it is shown, that while the synphase and antiphase velocities are unequal and even with the identical lines there are resonant fluctuations in the sections, which lead to losses and reflections in the frequency range with the observed differing periodicity of pass and reflect strips. The calculation of frequency characteristics was done using articles [5][6][7], which gave identical results.  Fig. 10 shows the frequency dependencies of matrix dispersion coefficients 31 S and 11 S for coupled lines with lengths l=0.1 m, synthesized with the parameter π 08 c v v .  (example 2). In frequencies up to 4 GHz we can see the presence of resonates, which repeat from the first one to the second after 764 MHz, but from the third one to the fourth already after 884 MHz. The periodicity's disturbance is caused by the synphase and antiphase waves interference particularities, which were studied in article [24]. In this case the phase velocities ratio π 08 c v v .  conditions the specified resonants' repetency. Having the multiple ratio of π 08 c v v .  , not only a "breakdown" of resonants is possible, but also the fusion of neighboring pass bands. This will further be shown using the section example with π 2 c vv . The construction's dispersion matrix coefficients' frequency dependence, which is characterized by π 10 c v v .  (example 3), confirmed the absence of resonants (Fig. 11). From the shown figure, a small development periodicity of the being implemented decay S 31 while with fluctuations of the input inductance, which can be observed by increasing the return losses 11 S . Wherein the phase-frequency characteristic is close to the linear one. The frequency characteristics are shown on Fig. 15-17. We can observe the alternation of wide and narrow pass bands. Those are formed, as it seems, at the expense of resonant fluctuations' on frequencies 1.5199 GHz and 3.0399 GHz disappearance while putting together constitutive waves with close but being inverted phases with +0 on 0  . Fig. 15. The simulative and experimental frequency dependencies of the section transfer coefficient (Fig. 9) while π 2 636 On Fig. 15 we can observe an unusual dependency of return losses on the frequency, which is uncharacteristic for coupled lines with not a big phase velocities difference of normal waves. The primary parameters' analysis, based on the numerical method's usage for the electric's field and accumulated energy in the selected coupled lines' cross section subareas calculation, can be applied to other coupled lines' types. This will allow us go from the heuristic search of optimal solutions while building modal filters [13,14] to using analytic expressions for finding the relative dielectric permittivities of substrates.

VI. CONCLUSION
The suggested approach, which's point consists of solving inverse finding dielectric permittivities and strip sizes problems based on the Laplace's numerical equation and the selected subareas' electric field's accumulated energy defining can be used for other coupled lines' types. The substrates' values of the relative dielectric permittivities being found cannot correspond to the being manufactured foiled materials' permittivities. In example 2, the necessity of getting the relative dielectric permittivities 2 17.2  , 3 2.42  is shown this way. Materials with those parameters can be manufactured with the help of additive technologies of multicomponent printing using different dielectrics with specified in advance components' contents in percent [25]. Another possible way -use multilayered substrates made from different available dielectrics with different thickness and dielectric permittivities, including the ones made with technologies of printing. In this case the presented way of solving the set phase velocities ratio getting problem with a limit for the other parameters is also usable. This way, the conducted in the study CSL synthesis' possibilities by the criteria of the set ratio of π c vv make the new problem of dielectric materials with a set dielectric permittivity manufacturing using additive printing methods technological process's development expedient.