**1. Introduction**

It is well known that MICs are based on the use of the technology of planar circuits printed partially or entirely, on a flat surface of a dielectric, by an etching operation. The entire circuit can be produced in large numbers at low cost by photolithography.

The technical characteristics of MICs are their small size, their low weight, and their high reliability.

At the end of the 1970s, the advancement of planar circuit technology coupled with the rapid development of microwave semiconductor components, particularly MESFET on gallium arsenide and advances in manufacturing materials technologies, were at the origin of the emergence of microwave monolithic integrated circuit (MMIC) technology. In this technology, the passive and active circuits and their interconnections are produced in large numbers on the same substrate.

In MIC technology, the structure of the planar waveguide consists of block elements according to the development of various functional components or subsystems. The study of planar waveguide structures was an important research subject in the field of MIC circuits. in recent years, the explosive development of commercial microwave applications for the general public, has considerably increased research activities in this field on the one hand, to explore the various new configurations of planar circuits, on the other hand to precisely characterize their electrical performance.

Planar transmission lines are the most essential point in PCM circuits. in the late sixties, with the availability of dielectrics with high dielectric constants of low loss dielectric materials and with the increasing demand for miniaturized microwave circuits for space applications. The intensity of interest in micro-ribbon circuits was renewed. This resulted in the rapid development of the use of micro-stipe lines.

At that time, two other types of planar transmission lines were also invented: they are slotted lines, and coplanar (CPW) respectively proposed by S.B. Cohn and C.P. Wen.

With the growth in operating frequencies, particularly in the millimeter band, the use of the traditional microstrip line becomes problematic because of the increase in losses, the presence of higher order modes and parasitic couplings. In this regard, the interest in uniplanar waveguide structures, using only a one substrate face, was renewed.

Uniplanar transmission structures include the coplanar lines which is modeled in this chapter, the slotted lines, and the two-ribbons coupled lines. These structures have many advantages over microstrip lines, such as easy production of successive parallel connections of passive or active components without the need to resort to metallized holes to a ground plane, low dispersion [1, 2].

With *ρ*ð Þ¼ *x*, *y ρ*ð Þ *x* for *y* ¼ 0 and *x ϵ* strip and *ρ*ð Þ¼ *x*, *y* 0 elsewhere.

*Analysis and Two-Dimensional Modeling of Directional Coupler Based on Two Coplanar Lines*

unit length of the coplanar line, and that will allow us to determine the other

1 *ε* ð

*ψ*ð Þ¼ *x*, *y*

**3. Determination of the Green's function**

created by a unit charge 1ð Þ *C* at the point *x*0, *y*<sup>0</sup>

*∂*2 *∂y*<sup>2</sup> � �*G*<sup>0</sup> *<sup>x</sup>*, *<sup>y</sup>*j*x*0, *<sup>y</sup>*<sup>0</sup>

� � ¼ � <sup>1</sup>

*∂*2 *∂x*<sup>2</sup> þ

*G*<sup>0</sup> *x*, *y*j*x*0, *y*<sup>0</sup>

*G*<sup>0</sup> *x*, *y*j*x*0, *y*<sup>0</sup>

� � ¼ � <sup>1</sup>

4*π*

*<sup>n</sup>*<sup>X</sup> ¼þ∞

*n*¼�∞

following form:

follow:

**83**

characteristic parameters.

*DOI: http://dx.doi.org/10.5772/intechopen.95142*

**Figure 1.** *Micro-coplanar line.*

The determination of *ρ*ð Þ *x*, *y* ¼ 0 makes it possible to evaluate the capacity *C* per

*G x*, *y*j*x*0, *y*<sup>0</sup>

This function used to form the integral equation Eq.(2), thereby it represents the

*∂x*<sup>2</sup> þ

� � ¼ �*δ*ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> *<sup>y</sup>* � *<sup>y</sup>*<sup>0</sup>

ð Þ *x* � *x*<sup>0</sup>

inverse of Laplacian operator Eq.(3), where the point ð Þ *x*, *y* is said field point

� � <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

With *δ* is the Dirac function, so the Green's function *G*<sup>0</sup> is written in the

q

To calculate the Green's function corresponds to the coplanar line without the central strip shown in **Figure 2a**, first we calculate the Green's function of the electrical potential created by a distribution of charges between two infinite ground planes shown in **Figure 2b** using the multiple image method [2], given by [3] as

*ln <sup>y</sup>* � *<sup>y</sup>*<sup>0</sup>

*y* � *y*<sup>0</sup>

<sup>2</sup>*<sup>π</sup> ln*

*G*<sup>0</sup> *x*, *y*j*x*0, *y*<sup>0</sup>

� �*ρ*ð Þ *<sup>x</sup> dl* (2)

� � said source point.

(3)

� � (4)

þ *cte* (5)

2

2

(6)

*∂*2 *∂y*<sup>2</sup> � ��<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> *<sup>y</sup>* � *<sup>y</sup>*<sup>0</sup> � �<sup>2</sup>

� �<sup>2</sup> <sup>þ</sup> ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> � <sup>2</sup>*na*

!

� �<sup>2</sup> <sup>þ</sup> ð Þ *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>0</sup> � <sup>2</sup>*na*

To solve the poison's equation, we inverting the Laplacian operator using the Green's function to the integral operator Eq. (2) to form the integral equation:

The determination of the Green's function *G* corresponding to the studied problem constitute the most delicate step. Once this function is obtained, the second step is to solve numerically the integral equation by the method of MOMs.

The physical modeling and then the numerical characterization of these planar transmission lines have been an important axis in the field of scientific research in recent years. Many techniques and numerical methods have been developed and used for the numerical characterization of uniplanar structures. In general, the numerical methods for the study of MMICs can be classify into two groups: the first includes the integral methods like MOMs, NEWTON-COTES FORMULAS, … and the other group is derivative like the FDTD, TLM ... DF method.

The analysis in this chapter is devoted to modeling a directional coupler in CPW technology by modeling one and two micro-coplanar lines on a dielectric substrate, the problem starts with solving the poison's equation which is transformed to the integral equation in which the unknown is the charge density *ρ* on the metal strip, by using the method of the Green's function [3] and the conformal mapping [4] technique. Subsequently we apply the numerical method (MOMs) [5] to solve the integral equation for obtaining the unknown function which is used to determine the variation of capacitance *C*, the value of the characteristic impedance *Zc*, and the effective permittivity *εeff*.

The second part of this analysis is to search the expression of the characteristic impedance for the odd *Zco* and even *Zce* modes as a function of the geometric dimensions of the directional coupler in CPW technology and the coupling coefficient *K* as a function of the gap ? between the two strip lines by using the general theory of coupled lines (telegraphist equations) and the results of the first part.

### **2. Statement of the problem in quasi-TEM mode**

In this part we focused to the formulation of the problem studied by determining the different characteristics of one and two micro-coplanar lines in the quasi-TEM approximation, using the integral equation method using the Green's function technique and the conformal mapping. The integral equation method is suitable for planar structures and it's most used to solve the electromagnetic problems.

The problem starts with solving the poison's equation Eq. (1) to obtaining the linear charge density *ρ* on the central metal strip shown in **Figure 1**.

$$
\left(\frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial \mathbf{y}^2}\right)\boldsymbol{\upmu}(\mathbf{x}, \mathbf{y}) = -\frac{1}{\varepsilon}\boldsymbol{\uprho}(\mathbf{x}, \mathbf{y})\tag{1}
$$

*Analysis and Two-Dimensional Modeling of Directional Coupler Based on Two Coplanar Lines DOI: http://dx.doi.org/10.5772/intechopen.95142*

**Figure 1.** *Micro-coplanar line.*

Planar transmission lines are the most essential point in PCM circuits. in the late sixties, with the availability of dielectrics with high dielectric constants of low loss dielectric materials and with the increasing demand for miniaturized microwave circuits for space applications. The intensity of interest in micro-ribbon circuits was renewed. This resulted in the rapid development of the use of micro-stipe lines. At that time, two other types of planar transmission lines were also invented: they are slotted lines, and coplanar (CPW) respectively proposed by S.B. Cohn and C.P. Wen. With the growth in operating frequencies, particularly in the millimeter band,

Uniplanar transmission structures include the coplanar lines which is modeled in this chapter, the slotted lines, and the two-ribbons coupled lines. These structures have many advantages over microstrip lines, such as easy production of successive parallel connections of passive or active components without the need to resort to

The physical modeling and then the numerical characterization of these planar transmission lines have been an important axis in the field of scientific research in recent years. Many techniques and numerical methods have been developed and used for the numerical characterization of uniplanar structures. In general, the numerical methods for the study of MMICs can be classify into two groups: the first includes the integral methods like MOMs, NEWTON-COTES FORMULAS, … and

The analysis in this chapter is devoted to modeling a directional coupler in CPW technology by modeling one and two micro-coplanar lines on a dielectric substrate, the problem starts with solving the poison's equation which is transformed to the integral equation in which the unknown is the charge density *ρ* on the metal strip, by using the method of the Green's function [3] and the conformal mapping [4] technique. Subsequently we apply the numerical method (MOMs) [5] to solve the integral equation for obtaining the unknown function which is used to determine the variation of capacitance *C*, the value of the characteristic impedance *Zc*, and the

The second part of this analysis is to search the expression of the characteristic

In this part we focused to the formulation of the problem studied by determining the different characteristics of one and two micro-coplanar lines in the quasi-TEM approximation, using the integral equation method using the Green's function technique and the conformal mapping. The integral equation method is suitable for

The problem starts with solving the poison's equation Eq. (1) to obtaining the

*ψ*ð Þ¼� *x*, *y*

1 *ε*

*ρ*ð Þ *x*, *y* (1)

planar structures and it's most used to solve the electromagnetic problems.

linear charge density *ρ* on the central metal strip shown in **Figure 1**.

*∂*2 *∂y*<sup>2</sup> 

*∂*2 *∂x*<sup>2</sup> þ

impedance for the odd *Zco* and even *Zce* modes as a function of the geometric dimensions of the directional coupler in CPW technology and the coupling coefficient *K* as a function of the gap ? between the two strip lines by using the general theory of coupled lines (telegraphist equations) and the results of the first part.

the use of the traditional microstrip line becomes problematic because of the increase in losses, the presence of higher order modes and parasitic couplings. In this regard, the interest in uniplanar waveguide structures, using only a one

metallized holes to a ground plane, low dispersion [1, 2].

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

**2. Statement of the problem in quasi-TEM mode**

the other group is derivative like the FDTD, TLM ... DF method.

substrate face, was renewed.

effective permittivity *εeff*.

**82**

With *ρ*ð Þ¼ *x*, *y ρ*ð Þ *x* for *y* ¼ 0 and *x ϵ* strip and *ρ*ð Þ¼ *x*, *y* 0 elsewhere.

The determination of *ρ*ð Þ *x*, *y* ¼ 0 makes it possible to evaluate the capacity *C* per unit length of the coplanar line, and that will allow us to determine the other characteristic parameters.

To solve the poison's equation, we inverting the Laplacian operator using the Green's function to the integral operator Eq. (2) to form the integral equation:

$$\Psi(\mathbf{x}, \mathbf{y}) = \frac{1}{\varepsilon} \int G(\mathbf{x}, \mathbf{y} | \mathbf{x}\_0, \mathbf{y}\_0) \rho(\mathbf{x}) dl \tag{2}$$

The determination of the Green's function *G* corresponding to the studied problem constitute the most delicate step. Once this function is obtained, the second step is to solve numerically the integral equation by the method of MOMs.
