Theory of Doubly-Shielded Coupled Lines for Directional Couplers of Various Directivity Types With Impedance Transformation

Asymmetric coupled lines (CLs) with double shielding in an inhomogeneous dielectric medium are studied. They have specific properties that allow the creation of impedance-transforming directional couplers with a given directivity type. The proposed models and synthesis techniques take into account both electric asymmetry and dielectric inhomogeneity. Numerical simulations and measurements prove the correctness of the general theory, and the possibility of creating co-, contra-, and trans-directional couplers with impedance-transforming and phase-shifting properties. The results will be useful for finding new design solutions for next-generation microwave circuits.


Theory of Doubly-Shielded Coupled Lines for Directional Couplers of Various Directivity Types
With Impedance Transformation
However, until now such CLs remain poorly studied and unreasonably rarely used, although it was correctly noted in [1] that they are required to create serial connections, which are sometimes so necessary for microwaves. The main reasons for the infrequent use of the considered CLs are the difficulty in achieving perfect (double) shielding in designs and the complexity of the technology of their manufacture. However, the main problem is the lack of a general theory of such CLs. The study object in this work will be asymmetric CLs shown in Fig. 1, which will be called "doubly-shielded CLs" There have been many contributions to the theory of DSCLs [1], [2], [3], [4]. However, CLs only with homogeneous dielectrics were studied to create filters in [1], inhomogeneity of the dielectric filling was taken into account only for designing switched-line phase shifters in [2], a theory of asymmetric coupled-line impedance-transforming directional couplers was proposed in [3] and [4], however, with a weak inhomogeneity of the dielectric filling, which allows creating only traditional contra-directional couplers.
The purpose of this research is to supplement the existing theory of DSCLs with a new approach that takes into account both the asymmetry of the CLs and the inhomogeneity of the dielectric filling, on the basis of which to propose new synthesis techniques. This will open another way to build couplers and hybrids of various directivity types with transforming properties and create new microwave devices based on them (e.g., novel 180 • hybrid; reflection-type phase shifter with an increased controllable phase shift, and so on).
The theoretical relationships for the DSCLs have been verified by the numerical simulations of impedance-transforming co-, contra-, and trans-directional couplers based on DSCLs, as well as by comparison with known results.

II. SCHEMES AND DESIGNS OF DSCLS
The generalized construction of the perfect DSCLs is transmission lines in which one conductor (conditionally the first) is surrounded by another hollow conductor (here the second) acting as a perfect screen between the inner and the outer spaces divided by the hollow conductor [1] (see Fig. 1).
The scheme of the loaded section of the DSCLs with the length and the equivalent schemes of the infinitely-short section x → 0 are shown in Fig. 2. Let us begin the modeling of DSCLs by solving the analysis problem.

III. ANALYSIS OF CLS. PRIMARY DISTRIBUTED MATRIX PARAMETERS
The main feature of perfect DSCLs is the complete electrical isolation (shielding) of one conductor, conditionally the first, with another hollow conductor, conditionally the second, from the ground. This suggests that the first conductor does not have any self-capacitance to the ground, i.e., C 01 = 0 [see Fig. 2(c)], which leads to zero self-inductance of the second conductor, i.e., L 02 = 0 [see Fig. 2(d)]. These features and the corresponding assumptions allow us to go from the general case with six parameters to the special case with four parameters when describing DSCLs. Therefore, perfect DSCL as a component of a distributed electrical circuit is fully characterized by a set of four independent parameters. The choice of specific four electric parameters is determined by the problem, which can be both a problem of analysis and synthesis.
The initial data for the quasi-static analysis of DSCLs with an inhomogeneous dielectric filling are the geometric parameters of the structure and the permittivity of the nonmagnetic medium. On their basis, by solving the quasistatic Dirichlet problem for the Laplace equation, electrical parameters, or rather parameters per unit length (PUL) of DSCLs are found. These distributed parameters are further used as coefficients of telegraph equations [11], [18] to find the frequency response of transmission lines and represent a pair of matrices: distributed capacitances C and inductances L, respectively, [see Fig. 2].
where C 11 , C 22 are self-capacitances of the first and second transmission lines; C 01 , C 02 , C 12 are self-partial and mutual capacities, respectively; L 11 , L 22 are self-inductances of the first and second lines; L 01 , L 02 , L 12 are self-partial and mutual inductances, respectively. So, if CLs are perfect DSCLs, then C 01 = 0 and C 11 = C 12 , as well as L 02 = 0 and L 22 = L 12 .
The matrix of distributed capacities has two design types: with real C and with air C(1) dielectric filling, respectively, In this case, it is assumed that the permittivity of the entire medium in the construction of CLs (see Fig. 1) is equal to one (ε r = 1) because it is very convenient for searching the inductance matrix L by the well-known matrix formula [11], [18] where ε 0 = 8.54 · 10 −12 (F/m); μ 0 = 0.4π · 10 −6 (H/m) are electric and magnetic constants. Taking into account the zero values of the two distributed parameters C 01 = 0 and L 02 = 0, it is possible to determine such parameters of the DSCLs as the self-impedances of the first and second transmission lines, calculated by the formulas [8], [11] Z 1 = L 11 /C 11 = (L 01 + L 12 )/C 12 Next, one can write the coefficients of capacitive and inductive coupling, respectively, [11] as well as the coefficient of their imbalance [12] Moreover, based on the conditions of physical realizability, the values of the coefficients should lie in the following limits 0 < (k C , k L ) < 1; −1 < k LC < 1.
The initial data for the search for modal parameters are matrices of distributed capacitances C and inductances L, the product of which is founded at the first step For this product, the spectral decomposition is determined by solving the problem of eigenvalues where c is the speed of light in free space; v is a vector composed of modal velocities: in-phase v c and out-of-phase v π ; ε r is the modal effective dielectric permittivities of the lines during in-phase ε rc and out-of-phase ε rπ excitations, respectively, where U is the normalized modal voltage matrix composed of the eigenvectors of the product matrix LC, is written [14], [15], [16] where R c , R π are modal numbers characterizing the relation of voltage on the lines, which take the values R c = 1 and R π = 0, respectively. This can be described as the "congruent case" proposed by Speciale [16] in the "zero-limit form," where R π = 0. Moreover, from the condition of physical realizability, the following inequality R π < R c must be satisfied. The found modal permittivities ε rc and ε rπ , as well as the modal velocities v c and v π allow us to determine the modalphase ratio m Based on the obtained parameters, the modal current matrix J is calculated [14], [15], [16], the elements of which, as a result of the corresponding normalization, acquire units of admittance where Z c1 is the first line impedance during in-phase excitation; Z π 1 is the impedance of the first line during out-ofphase excitation; Z c2 is the impedance of the second line during in-phase excitation; Z π 2 is the impedance of the second line during out-of-phase excitation. In this case, two modal characteristic impedances are calculated by the following formulas: (14) and the other two impedances reach their limits Note that these modal characteristic impedances Z c1 , Z π 1 , Z c2 , Z π 2 and modal numbers R c and R π , in general, are related in a known manner [14] − Since DSCLs do operate in the limit mode, then ratio (16) becomes zero Further, using the matrices of modal voltages U and modal currents J, the matrices of characteristic impedances Z and characteristic admittances Y are found [7], [17] where Y 11 , Y 22 , and Y 12 are the self and mutual characteristic admittances of the CLs, while Z m = 1/Y 12 is the mutual impedance. Z 11 , Z 22 , and Z 12 are self and mutual characteristic impedances DSCLs calculated by the formulas The voltage transformation ratio n of the DSCLs is written as the square root of the impedance transformation ratio The coefficient of impedance coupling k of the DSCL is given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
From this analysis for the first time, it is found that in order to achieve a 3 dB coupling level at the center of the operating frequency band, different types of directional couplers must provide different levels of impedance coupling corresponding to them, in particular In this case, the double inequality always holds Next, one can find the main modal parameter of DSCLs, which is characteristic impedance Z 0 [14] where modal impedances Z c and Z π are determined by the formulas [18] In addition, one can write the normalized self ρ and mutual r impedances found through modal impedances and elements of the impedance matrix according to Feldstein and Yavich [11] from where one can get back to modal impedances and get following forms of characteristic impedance matrix Z and again determine the coefficient of impedance coupling according to Feldstein and Yavich [11] Relationships (26)-(32) for finding the normalized modal impedances Z c /Z 0 and Z π /Z 0 , as well as the self ρ and mutual r normalized impedances, can be compactly represented in a form of a geometric drawing (Fig. 3) with three half -circles for each of the three types of 3-dB hybrids based on DSCL: 1) codirectional (COD); 2) contra-directional (CTD); and 3) trans-directional (TRD). From the drawing (see Fig. 3) the following numerical values of the self ρ and the mutual r normalized impedances are clearly determined Based on the modal parameters (9), (14) and using the wellknown definitions of a single line impedance Z 0 and modal velocity v, through distributed capacitance C and inductance L, we can write From here it is possible to determine the coefficients of capacitive and inductive coupling (5) through modal-phase m (12) and modal-impedance Z c2 /Z π 1 ratios, as well as coupling impedance r (33), in a novel form The calculation according to formulas (36), (37) gives the following numerical values of the capacitive and inductive coupling coefficients in the DSCL for the matched 3-dB  Fig. 4(a)] is very clearly replaced by the equivalent circuit shown in Fig. 4(b), which consists of two isolated sections of transmission lines connected in series at the near end and has characteristic terminal loads at the far end as in [1].
In addition, there are πand T-shaped equivalent circuits for DSCL [ Fig. 4(c) and (d)], which are completely matched and non-reflective, i.e., they are characteristic terminal loads [13]. Moreover, for perfect DSCLs, taking into account the limiting values of the known parameters (15), the structures of the πand T-shaped equivalent circuits [ Fig. 4(c) and (d)] are simplified and reduced to a single characteristic terminal load circuit in form of two series-connected resistors [ Fig. 4(e)], the values of which are equal to the modal characteristic impedances Z 1π and Z 2c . From Fig. 4(e), we can see that the impedance between the 1st and 2nd lines is Z 1π = Z m , and the ones between the 2nd line and the ground are Z c2 = Z 12 . Therefore, all values of the characteristic load elements are expressed through the corresponding modal impedances (14)-(22), and there is no need to introduce additional designations for the load parameters. In addition, relation (26) for the characteristic impedance Z 0 is also satisfied here.
And finally, an equivalent circuit in the form of two separate load resistors Z 01 and Z 02 , which are connected between each of the lines and ground and matched with a DSCL section is shown in Fig. 4(f). The calculated values of the matched resistances for hybrids of various types of directivities are determined by the formulas given in Table I, and their characteristic impedance for any m is calculated as

VI. SYNTHESIS OF TRANSFORMING HYBRIDS BASED ON DSCL
There are three types of directivities of the couplers based on CLs: 1) codirectional (COD); 2) contra-directional (CTD); and 3) trans-directional (TRD). The main characteristics of the three types of matched 3-dB couplers (i.e., hybrids) based on the DSCL sections, in which an in-phase (here fast) mode wavelength θ c is equal to 90 • , are given in Table II.
Each of these DSCL hybrids is completely described by a set of four independent parameters (two impedance and two phase), for example, Z π 1 , Z c2 , ε rc , ε rπ or Z 0 , k, ε rc , ε rπ . The main relationships for couplers based on DSCL sections are presented in Table III Fig. 4(e)] and the specification of the modalphase ratio corresponding to the type of directivity. In a contradirectional hybrid, it is necessary to provide a modal-phase ratio (12) equal to one (m = 1), and in co-and trans-directional hybrids that ratio must be equal to three (m = 3 or 1/3), i.e., m = 3 ±1 .
To form a contra-directional quadrature coupler (3 dB) with perfect matching and directivity, one more condition must be realized For a trans-directional quadrature coupler necessary condition is the following: (see Table III). This shows that both the contra-and transdirectional 3-dB couplers based on the DSCL sections have the property of double impedance transformation Z 02 /Z 01 = 2 ±1 , and their characteristic impedance Z 0 is calculated using the same formula (40). Unlike the two previous couplers, the codirectional matched coupler based on the DSCL section is not quadrature, but it is in-phase/out-of-phase (i.e., sum-difference). And in despite of the side-by-side asymmetry of the coupled-line coupler, it does not provide a line-to-line transformation. However, it can provide an end-to-end impedance transformation between the near Z in and the far Z out ends of the DSCL section (see Table III) with a characteristic impedance In the absence of transformation, when Z 0 = Z in = Z out such hybrid becomes the most broadband.
So, the procedure for calculating (synthesizing) a singlesection matched transforming 3-dB coupler based on a DSCL section with a given type of directivity is as follows: 1) Set the numbering of CLs: the first is internal; the second is external (shielding) according to Figs. 1, 2, and 4. If the numbering is inverse, then the subsequent relations will need to be renumbered, and in the matrix forms, rearrange both the rows and columns. 2) Based on the given terminating impedances Z 01 , Z 02 or Z in , Z out , find the value of the characteristic impedance of the Z 0 system according to (40) or (43). 3) Based on the found characteristic impedance Z 0 and the given type of (co-, contra-, trans-) directivity, find the values of modal-characteristic impedances Z π 1 = Z 0 /r and Z c1 = Z 0 r , where the mutual normalized impedance r is determined from (32). 4) Given the type of directivity, according to Table III, determine the value of the modal-phase ratio m. 5) Using the found ratio m and the given value of one of the modal permittivities (for example ε rc ), find the value of the second modal permittivity (in this example ε rπ ) from (12). Thus, as a result of the synthesis, we obtain a complete electrical description of the coupler based on the DSCL section with a given directivity type in the form of a set of four modal parameters: two impedance and two phase, namely Z π 1 , Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. Z c2 , ε rc , ε rπ . Further, this allows for choosing a design and determining its physical parameters (see Fig. 1) [19].

VII. ANALYSIS OF THE CHARACTERISTIC IMPEDANCE MATRIX
In some cases, the mathematical spectral decomposition of the characteristic impedance matrix Z of CLs is of interest [17], [18]. We perform it for all types of couplers (co-, contra-, and trans-directional) based on DSCLs and present in a general form where z (c,π ) are the eigenvalues of the impedance matrix Z; V is a matrix composed of eigenvectors of the matrix Z, written as where r c , r π are the modal numbers from the matrix V, satisfy the following values of the "metallic means" [20]: "silver" 1 + √ 2 = 2.414 in the case of a codirectional coupler; "gold" (1 + √ 5)/2 = 1.618 in the case of a contra-directional coupler, and another, so far nameless, but for which we offer the term "platinum" (1 + √ 17)/4 = 1.281 in the case of a trans-directional coupler. Moreover, the product of modal numbers for all types of couplers are always the same and equal to minus one, i.e., r c r π = −1, hence The main formulas obtained from the above-written relations (44)-(46) are given in Table III.

VIII. NUMERICAL RESULTS AND THEIR DISCUSSION
For a numerical and graphical illustration of the obtained design relationships, take three directional couplers with the design parameters given in Table IV.
These circuits are quarter-wave DSCL impedancetransforming terminated sections, which perform the functions of co-, contra-, and trans-directional matched 3-dB couplers (hybrids) shown in Fig. 5.
The couplers were analyzed in the frequency domain, and their calculated frequency dependences of the modulus s i j = 20 log(|S i j |) (dB) and phase shift ϕ i j = arg(S i j )(deg) of the main coefficients S i j (i , j = 1, . . . , 4) of the scattering matrix S are shown in Fig. 6. The electric length of all couplers is the same and is θ c = 90 • at a frequency of f = 1 GHz. In this case, the electric length of the DSCL sections θ c is determined by the geometric length l and permittivity ε rc for the fastest (here in-phase) mode. Thus, here c is the speed of light in free space; l is the geometric length of the DSCL section. From the above-presented dependences and preliminary calculations, it was found that a co-directional coupler terminated at the input and output by 50 and 25 , respectively, (i.e., with double impedance ratio), has an operating frequency band at matching level S 22 = 15 dB and coupling at the far end S 41 = (3-3.2) dB only 10%. However, if the same impedances do terminate all ports (e.g., 50 ), and there is no end-to-end impedance transformation (i.e., there is symmetry between the near and far ends of the DSCL section), then with the same level of matching S 11 = S 22 = 15 dB and coupling at the far end of S 41 = (3-3.3) dB, the operating frequency band expands up to 16% [ Fig. 6(a)].
Characteristics of insertion loss of the 1st and 2nd lines coincide with each other S 31 = S 42 ; all matching and isolation curves S 11 = S 22 = S 21 also coincide. In addition, when the 1st port is excited, then the phase difference between the output 3 and 4 ports is and when the 2nd port is excited, then the phase difference between the same 3 and 4 ports is already at the center of the operating frequency band [ Fig. 6(b)]. Wherein ϕ 2 − ϕ 1 = 180 • over the entire operating frequency band, i.e., the coupler is in-phase/out-of-phase (i.e., sum/difference), which is similar to the case of an asymmetric co-directional coupler based on microstrip lines described in [18, p.153] and [21].
The contra-directional coupler [ Fig. 6(c) and (d)] is perfectly matched at all ports and at all frequencies, so only the characteristics of the coupling S 21 and the insertion loss S 31 are displayed in the graph field, but the characteristics of return Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. i.e., the coupler is quadrature. The operating frequency band at the coupling level of S 21 = (2.53-3.5) dB is very wide and reaches 60%.
Frequency characteristics of a trans-directional coupler based on a DSCL section during excitation of the near end of the external line, i.e., 2nd port, shown in Fig. 6, e where the greatest number of different curves are observed (there are six of them). These characteristics are selected and denoted as follows (see Table II): S 11 , S 22 are frequency responses of return loss for the 1st and 2nd ports (red and black dashed lines), respectively. S 12 is the frequency response of the near-end coupling between the 2nd and 1st ports (blue dashed line).
S 32 is the frequency response of the insertion loss, i.e., farend coupling between the 2nd and 3rd ports (black dash-dotted line).
S 31 , S 42 are frequency responses of insertion loss in the 1st and 2nd lines, i.e., here the internal and external isolations (red and black solid lines), respectively. It can be seen from the graphs [ Fig. 6(e) and (f)] that for a trans-directional coupler with an exciting 2nd port, at the level of external isolation, S 42 = 15 dB (almost coinciding with the level of the return loss of the 1st port S 11 ) and at the level of near-end coupling S 12 = (2.8-3.5) dB, the operating frequency band reaches 14%. This is quite acceptable for a large number of  applications. Also, note that the phase difference between the output 1 and 3 ports in the working frequency band is with an error of not more than ±1 • , i.e., the trans-directional coupler is quadrature with high precision.

IX. EXPERIMENTAL VALIDATION
As shown above, the presented general theory of the DSCLs falls into three special cases and corresponds to three types of directional couplers: COD, CTD, and TRD. To date, the case of the impedance-transforming CTD couplers has already been proposed and verified in [4]. Here we present an experimental validation of the theory for the cases of COD and TRD couplers.
For the first time, using theoretical model and design relationships a COD coupler (3 dB 180 • hybrid) based on DSCLs is proposed and implemented with the following physical parameters: (d w h 1 h 2 t ) = (1 2.4 0.4 2 0.035 40) mm; ε r = 16, as shown in Fig. 7.
Simulated and measured S-parameters are presented in Fig. 11. Simulated results show that the center of the operating  Further, using the theoretical model and synthesis technique novel 3-dB 50/25-impedance-transforming TRD coupler is proposed and implemented with the following physical parameters: (w 1 w 2 h 1 h 2 t ) = (2 4 1 1.15 0.035 33) mm; ε r1 = 16, as shown in Fig. 12. This TRD coupler based on imperfect DSCLs is fabricated using 1-mm thick dielectric substrates. The material of the main substrate is FLAN-10 (relative dielectric permittivity is ε r = 10), and DSCLs are implemented on FLAN-16 (ε r1 = 16), both manufactured by CJSC "Factory "Moldavizolit" (Tiraspol, Transdnistrian Moldavian Republic).
Calculated using the conformal mapping technique [19], distributed parameters of the DSCL are (L 11     not complete, we need all six parameters because four is not enough. For measurement purposes, additional single-section impedance transformers are connected to the near-end (1) and the far-end (3) coupled ports of the TRD coupler, as shown in Fig. 13. Electrical parameters of the transformers are   Fig. 15.
Simulated results show that the center of the operating frequency band is 2.06 GHz. The coupling is (3.5 ± 0.6) dB in the frequency band 1.85-2.26 GHz (20%) at return loss S 11 and isolation S 42 levels of −10 dB. Measured results show that the center of the operating frequency band is Simulated and measured differential phase shift between 1 and 3 ports of the impedance-transforming TRD coupler based on DSCLs.
1.85 GHz. The coupling is (3.7 ± 0.7) dB in the frequency band 1.68-2.02 GHz (18%) at return loss S 11 and isolation S 42 levels of −10 dB. Thus, the errors in calculating the center frequency and the operating frequency band do not exceed 11%. Fig. 16 shows that the phase difference between the output signals of the 1 and 3 ports of the coupler at 50-reference impedance is close to 90 • in the operating frequency band, i.e., impedance-transforming TRD coupler is a real quadrature hybrid.
Thus, a measurement-based validation of the general theory of DSCLs showed good agreement between the simulated and experimental results, sufficient for most practical applications.
X. CONCLUSION 1) Using of DSCLs having an inhomogeneous dielectric filling allows the creation of directional couplers and hybrids with any specified type of directivity: codirectional; contra-directional; trans-directional. 2) Perfect DSCLs are completely described by a set of four independent parameters: pair impedance ones, and pair phase ones. Those may be Z π 1 , Z c2 , ε rc , ε rπ , or Z 0 , k, ε rc , m, etc. 3) In a homogeneous dielectric medium, when a modalphase ratio equals one (m = 1), only the contradirectional operation mode of the coupler is possible. 4) In an inhomogeneous dielectric medium, when a modalphase ratio is equal to 3 or 1/3 (m = 3 ±1 ), it is possible to achieve either co-or trans-directional operation modes by changing the modal-impedance ratio (i.e., impedance coupling k). 5) Although three different directivity types of couplers can be formed on the basis of the DSCL section, the codirectional coupler is in-phase/out-of-phase (i.e., not quadrature), and the contra-directional and transdirectional these are quadrature. 6) A matched codirectional 3-dB coupler based on the DSCL section with the same terminating loads Z 01 = Z 02 at all ports has a modal-impedance ratio of Z c2 /Z π 1 = 1/2. 7) A matched contra-directional 3-dB coupler based on the DSCL section has a terminating impedance ratio Fig. 17. Geometric drawing, which illustrates the relationship between the impedance parameters of the CLs, where (Z 11 Z 22 ) 1/2 , Z 12 , and Z 0 are self, mutual, and characteristic impedances, respectively.
of Z 02 /Z 01 = 1/2 and the same modal characteristic impedances Z π 1 = Z c2 . 8) A matched trans-directional 3-dB coupler based on the DSCL section has a double ratio of both terminating impedances (at each of both ends of section Z 02 /Z 01 = 2) and modal impedances Z c2 /Z π 1 = 2. 9) A matched codirectional coupler based on the DSCL section has no property of line-to-line transformation, but it provides the impedance transformation between the near and far ends of the section. At the same time, it becomes as broadband as possible if the transformation is not carried out (i.e., it is terminated with the same impedances at all ports). 10) To obtain equal power division (3 dB coupling at the center of the operating frequency band) between the output ports of the couplers of different directivity types, different values of the impedance coupling k are required. At the same time, a codirectional coupler needs "weak" coupling √ 1/3 = 0.577, a contra-directional coupler needs "average" coupling √ 1/2 = 0.707, and a trans-directional coupler needs "tight" coupling √ 2/3 = 0.816, which was revealed in this study for the first time.