An improved analytical model for broadside coupled transmission line used on planar circuit

Broadside coupled transmission lines have the advantage of more compact size and higher coupling factor compared to the traditional coupled microstrips in the realization of planar circuits such as substrate integrated baluns, directional couplers, and filters on printed circuit boards. A broadside coupled transmission line is a structure with interesting properties composed of two signal lines with strong coupling placed on the top and bottom of a dielectric substrate, surrounded by coplanar ground. This configuration allows the realization of the practical circuit with several hundred ohms impedance for even mode excitation, with the help of lower even mode capacitance. This leads to a high coupling factor and benefits to a high impedance transformation ratio balun or a wide bandwidth filter design. In this paper, an analytical model describing the symmetrical pair of conductors for broadside coupled transmission lines is presented. The analytical model based on the physical geometries of the coupled lines is verified by electromagnetic (EM) simulations and measurements. The results are in very good agreement with each other.


Introduction
When two unshielded transmission lines are in close proximity, power can be coupled from one line to the other due to the interaction of electromagnetic fields.The power can be coupled from one line to the other due to the interaction of electromagnetic fields.Such lines are called coupled transmission lines [1].Coupled transmission lines can support two distinct propagating modes, and this feature can be used to implement a variety of practical directional couplers, hybrids, and filters.Generally, coupled transmission lines can be implemented with microstrip, stripline, or even coaxial lines.
The cross-sectional view of the well-studied traditional coupled microstrip line is illustrated in Fig. 1(a).The two signal lines are located on the same side, i.e., the top side, while the ground plane (GND) is located on the opposite side.The electrical characteristics of such coupled microstrip lines can be completely determined from the effective capacitances between the lines.The related formula analysis is fully investigated in [1].Such structure is successfully used in filter and power divider design [2][3][4][5][6], etc.Moreover, [7] presents a method to estimate the near-end and far-end cross-talk of a coupled transmission lines based on eigenvector decomposition.While the cross-sectional view of the broadside coupled transmission line is depicted in Fig. 1 (b).It can be seen as two coplanar waveguides (CPWs) transmission lines [8] mounted back-to-back, without ground combined together.It can also be seen as a broadside coupled stripe line [8] without cover but with the suspended ground.Due to its simplicity and compactness, this structure can easily be realized on printed circuit boards (PCB).
In practical applications, such as for wider bandwidth filters, balun with high impedance transformation ratio, etc.They generally require very tightly coupled lines to pursue the high coupling factor, which is difficult to fabricate for the traditional coupled microstrip lines.But this feature can be easily implemented by broadside coupled lines because of its relatively smaller equivalent capacitance under even-mode excitation.This broadside coupled structure is successfully used for the design of planar balun as output matching network in [9,10].However, to the best of the author's knowledge, the circuit design based on this structure completely relies on EM simulations, and there is no systematic investigation of this structure and no analytic model yet for the designer to refer to as to guide for engineering the coupled lines [11][12][13].Therefore, such a model is developed in this paper using the same methods as for coplanar waveguides [14] and adapting the model for broadside coupled transmission lines to extract the even and odd characteristic impedances.Analytical formula derivation based on the even and odd impedance analysis is presented in Section 2. Section 3 presents a comparison among EM simulations using the commercial software ANSYS Electronics HFSS and measurements is adopted to verify the derived analytical model.Very good agreements are obtained.

Theoretical analysis
The configuration under investigation is sketched in Fig. 1(b).It is a structure composed of two signal traces (Signal) placed on top and bottom of a dielectric substrate, surrounded by ground and realizing a microwave coupled transmission line.The height of the dielectric substrate is H, and the relative dielectric constant is ∈ r .The signal trace is located on the middle of two separated ground traces (GND).The distance or gap between Signal and GND is defined as S, and the width of the signal trace is defined as W. Generally, for PCB applications, the signal and ground trace is copper, and the thickness t is normally 1 oz (1.35 mil), for simplicity, the copper thickness was ignored in the following analytical derivation.
In this paper, start with the analytical model based on coplanar waveguide [8] and broadside coupled stripline [15] to calculate the C (o,e)2 in dielectric layer that depicted in Fig. 1(b).Then improved formulas for C (o,e)1 are derived to applied to the air layer with infinity height.Importantly, a coefficient α (o,e) is employed to correct the C (o,e)1 since the changed boundary conditions.Additionally, to simplify the analytical process, separate the structure into two symmetrical planes by a magnetic wall or electric wall with even-mode or odd-mode excitation.
Based on this strategy, the characteristic impedance derivations are derived as follows.

Even mode
Follow the general idea to formulate even/odd mode characteristic impedance, the analytic expression for even/odd mode characteristic impedance is given by: where c v is the velocity of propagation in free space, c v =2.998 × 10 8 m/s.C e is the even-mode capacitance per unit length, C a e is the capacitance when the substrate's dielectric is replaced by air.To express the C e effectively, the structure in Fig. 1 can be modeled as two symmetrical half-planes which are separated by magnetic walls (dashed line in Fig. 1, its corresponding E field distribution is illustrated in Fig. 2(a)).Due to the symmetrical distribution of the E field, the total even-mode capacitance C e can be considered as the sum of two components, C e1 and C e2 , representing the capacitance in air and dielectric, respectively.Then the equations with physical dimensions can be written as follows: k e1 and k e2 are two intermediate parameters involved in the derivation of the two capacitors above.Observably, k e1 is the case of k e2 when the height h goes to infinity high.To be accurate, a coefficient α e is employed to illustrate the fact that no cover and suspended ground in the air layer.They are expressed as follows: where K(k) and K(k ′ ) are the complete elliptic integrals of the first kind

Odd mode
A same analysis process with even mode, but take the magnetic wall under even mode as the electrical wall under odd mode in Fig. 1(b).Its corresponding E field distribution is illustrated in Fig. 2(b).Then the total capacitance of odd mode can be described as C o = C o1 + C o2 , then all equations are given by:

Analytical model verification
To verify the formulas derived in Section 2, coupled structures with different dimensions were simulated and fabricated.Practically, a matrix for W/H and S/H dimensions equal to 2, 3, 5 and 8 were fabricated on a commercial material: 30 mil Rogers 4350B with dielectric constant ∈ r = 3.66.The responding fabricated circuits are shown in Fig. 3.The line length is 28.8 mm for all dimensions.
The corresponding schematic model with the specified port numbers is indicated in Fig. 4. The features of impedance matrix Z can be written as below for its symmetrical characteristic [1].
where Z 0e and Z 0o are the even and odd impedance separately, θ is the electrical length.Due to its symmetrical features, only Z 11 , Z 12 , Z 13 and Z 14 are   required to fully describe this structure.For simplicity, a factor k f is introduced in this work, Z 0e and Z 0o then can be calculated as: In practise, the scattering matrix S is first simulated and measured, then convert them into Z matrix to calculate Z 0e and Z 0o .
For the S parameter measurements of these DUTs, for more clarity, the transparent layout of the test fixture without DUT shown in Fig. 5 was designed.Observably, two sides of PCB are used, which may increase little but not too much cost compared to conventional microstrip structure.While the photograph for the test fixture with DUT is shown in Fig. 6.
First, in measurements, the S-parameters of the whole board was measured, then de-embedding the test fixture to obtain the desirable broadside coupled line structure's S-parameters.The comparison of calculation, simulation and measurements of one special case when S = 3 mm and W = 3 mm are shown in Fig. 7.The calculated results matched well with simulations and measurements.
Finally, the calculated, simulated and measured characteristic impedances of the even and the odd modes are shown in Fig. 8 It can be observed that Z 0o is not sensitive with the gap space S between signal and ground trace that indicated in Fig. 8(a).This is also in agreement with predictions since the main capacitance of the even mode is from the parallel plate capacitor between the two signal traces.Meanwhile, it can be noticed that the even mode impedance Z 0e can reach up to more than 200 Ω, which means the coupling factor (C F = (Z 0e − Z 0o )/(Z 0e + Z 0o )) is very high compared to the traditional coupled microstip structure.

Conclusions
Broadside coupled transmission lines are adequate to be used in planar baluns, directional couplers, and filters for their easy realization on PCB and have the advantage of compact features.This paper derives an analytical model that can be used to describe this complex structure as an aid in the design phase of other passive components based on these coupled transmission lines.Moreover, a method is introduced to calculate the even/odd characteristic impedance from measured scattering parameters.The analytical derivations are verified by EM simulations and measurements.The model shows very good agreement with the simulated and measured results for a large span of geometrical shapes of broadside coupled transmission line structures.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .Fig. 2 .
Fig. 1.Cross-sectional view of (a) conventional coupled micrstrip line; and (b) broadside coupled transmission line.Where the dashed line represents the Ewall or H-wall.

Fig. 3 .
Fig. 3. Fabricated designs with different dimensions.The layout at the top and the bottom side are symmetrical same.They will be used as DUT in measurement.All DUT has the same length 28.8 mm.

Fig. 4 .
Fig. 4. Corresponding schematic model for the broadside coupled structure shown in Fig. 3 with specified port numbers.
(a) and Fig. 8 (b) respectively as a function of W/H and S/H.The analytical model match very well with the simulated and the measured results.