**4. Numerical characterization of the integral equation**

The linear charge density *ρ* on the metal strip shown in **Figure 4** is related to the distribution potential *ψ* in the cross-section of the coplanar line, and it's zero elsewhere. For that reason, the integral equation Eq. (2) can be written as:

$$\Psi(\mathbf{x},\mathbf{y}) = \frac{1}{\varepsilon} \int\_{-w/2}^{w/2} G(\mathbf{x}, \mathbf{y}|\mathbf{x}\_0, \mathbf{y}\_0) \, \rho(\mathbf{x}\_0, \mathbf{y}\_0) dl\_0 \tag{10}$$

The central strip conductor is considered infinitely thin, that allows us to write:

$$\boldsymbol{\psi}(\boldsymbol{\omega}) = \frac{1}{\varepsilon} \int\_{-w/2}^{w/2} \mathbf{G}(\boldsymbol{\omega}|\boldsymbol{\omega}\_0) \boldsymbol{\rho}(\boldsymbol{\omega}\_0) d\boldsymbol{\omega}\_0 \tag{11}$$

Assuming that the central strip is submitted to a unit potential, so the equation Eq. (11) is convenient to solve numerically using the method of moments. This method is divided into two stages, firstly by developing *ρ x*0, *y*<sup>0</sup> � � in series of N basic functions Eq. (12) in the form of rectangular pulses Eq. (13), secondly by using the Galerkin procedure which allows to write equation Eq. (11) as a linear equations system Eq. (14):

$$\rho\_j(\mathbf{x}\_0) = \sum\_{j=1}^{N} a\_j f\_j(\mathbf{x}\_0) \tag{12}$$

$$\int f\_j(\mathbf{x}\_0) = \begin{cases} 1 & \text{if } \mathbf{x} \in \left[ \mathbf{x}\_j - \frac{h}{2}; \mathbf{x}\_j + \frac{h}{2} \right] \\ 0 & \text{elsewhere} \end{cases} \tag{13}$$

$$\sum\_{i=1}^{N} \psi\_i(\mathbf{x}\_0) = \sum\_{i=1}^{N} \sum\_{j=1}^{N} \frac{a\_j}{\varepsilon\_\varepsilon} \int\_{\mathbf{x}\_j - \frac{h}{2}}^{\mathbf{x}\_j + \frac{h}{2}} G(\mathbf{x}\_i | \mathbf{x}\_j) d\mathbf{x}\_0 \tag{14}$$

So, the linear equations system (14) can be written in the following matrix form:

$$\begin{bmatrix} a\_j \end{bmatrix} = \begin{bmatrix} \mathbf{M}\_{\vec{\eta}} \end{bmatrix}^{-1} \cdot \begin{bmatrix} \boldsymbol{\nu}\_j \end{bmatrix} \tag{15}$$

With *Mij* � � is the matrix of the Green's function obtained by the inversion of the linear equations system Eq. (14), defined on the central strip of the structure

**Figure 4.** *Discretization of the central strip.*

With *a* is the gap between the two infinite metal plate **Figure 2a**, its sum given

*sin* <sup>2</sup> *<sup>π</sup>*

The next step consists of applying a suitable conformal mapping which makes it possible to transform the Green's function into the given structure in **Figure 2a**, it's

� � � *cos* �<sup>1</sup> *<sup>x</sup>*<sup>0</sup>

� � <sup>þ</sup> *cos* �<sup>1</sup> *<sup>x</sup>*<sup>0</sup>

; and *<sup>α</sup>* <sup>¼</sup> *<sup>x</sup>*

So, for a non-homogeneous middle shown in the **Figure 3**, the Green's function

The Green's function Eq. (8) is also a reciprocal function [7], and it's can be expressed as the superposition of two functions for non-homogeneous [8] middle,

� � � � � �

� � � � � �

<sup>2</sup>*<sup>a</sup>* ð Þþ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> *sh*<sup>2</sup> *<sup>π</sup>*

!

<sup>2</sup>*<sup>a</sup>* ð Þþ *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>0</sup> *sh*<sup>2</sup> *<sup>π</sup>*

*b:z*<sup>0</sup>

*b:z*<sup>0</sup>

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

! !

<sup>2</sup> *ch*�<sup>1</sup>

; *<sup>T</sup>*� <sup>¼</sup> <sup>1</sup>

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

(7)

ð Þ *z*<sup>0</sup>

1 A

ð Þ *z*<sup>0</sup>

.

(8)

(9)

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

<sup>2</sup> *ch*�<sup>1</sup>

<sup>2</sup> *ch*�<sup>1</sup>

; *<sup>β</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> � <sup>4</sup> *<sup>x</sup>*

<sup>þ</sup> *<sup>R</sup>: ln sin* <sup>2</sup> *<sup>R</sup>*� ð Þþ *sh*<sup>2</sup> *<sup>T</sup>*<sup>þ</sup> ð Þ

ð Þ� *<sup>z</sup> ch*�<sup>1</sup>

� �

*sin* <sup>2</sup> *<sup>R</sup>*<sup>þ</sup> ð Þþ *sh*<sup>2</sup> *<sup>T</sup>*<sup>þ</sup> ð Þ

ð Þ *z*0*:*

ð Þ� *<sup>z</sup> ch*�<sup>1</sup>

ð Þ� *<sup>z</sup> ch*�<sup>1</sup>

*b* � �<sup>2</sup> � �

� � � �

� � � �

<sup>þ</sup> *sh*<sup>2</sup> <sup>1</sup>

<sup>þ</sup> *sh*<sup>2</sup> <sup>1</sup>

<sup>4</sup>*<sup>π</sup> ln sin* <sup>2</sup> *<sup>π</sup>*

*(a) Micro-coplanar line without central conductor; (b) structure with two infinite ground planes.*

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

by [6] as:

**Figure 2.**

given as:

*G x*, *y*j*x*0, *y*<sup>0</sup> � � ¼ � <sup>1</sup>

With: *<sup>z</sup>* <sup>¼</sup> *<sup>α</sup>*þ*<sup>β</sup>*

is written as [3]:

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

� � ¼ � <sup>1</sup>

With: *<sup>R</sup>*� <sup>¼</sup> <sup>1</sup>

And: *<sup>R</sup>* <sup>¼</sup> *<sup>ε</sup>*1�*ε*<sup>2</sup>

**Figure 3.**

**84**

*ε*1þ*ε*<sup>2</sup> .

*Micro-coplanar (non-homogeneous middle).*

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

<sup>4</sup>*<sup>π</sup> ln*

2 � �<sup>1</sup>*=*<sup>2</sup>

� � ¼ � <sup>1</sup>

*sin* <sup>2</sup> <sup>1</sup>

0 @

*sin* <sup>2</sup> <sup>1</sup>

; *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *<sup>α</sup>*0þ*β*<sup>0</sup> 2 � �<sup>1</sup>*=*<sup>2</sup>

and for isotropic or anisotropic [9–10] middles.

<sup>2</sup> *cos* �<sup>1</sup> *<sup>x</sup>*

*b:z*

<sup>2</sup> *cos* �<sup>1</sup> *<sup>x</sup> b:z*

<sup>2</sup> *cos* �<sup>1</sup> *<sup>x</sup> b:z*

<sup>4</sup>*<sup>π</sup> ln sin* <sup>2</sup> *<sup>R</sup>*� ð Þþ *sh*<sup>2</sup> *<sup>T</sup>*� ð Þ

� � � *cos* �<sup>1</sup> *<sup>x</sup>*<sup>0</sup>

� � � �

*sin* <sup>2</sup> *<sup>R</sup>*<sup>þ</sup> ð Þþ *sh*<sup>2</sup> *<sup>T</sup>*� ð Þ

*b:z*<sup>0</sup>

!

**Figure 5.** *Variation of the charge density on the metal strip.*

studied show in **Figure 4**. And **Figure 5** shows the variation of the charge density *ρ j* ð Þ *x*<sup>0</sup> (solution of the integral equation) as a function of subdivisions *N* of the central strip of with *w*.

*Zc* ¼

ffiffiffi *L C* r

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

P*<sup>N</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>ρ</sup> <sup>j</sup> ψ*

*paC* <sup>¼</sup> <sup>1</sup>

With *vp* is the propagation velocity in the coplanar line and *vpa* is the propagation

A coupled coplanar line configuration consists of two transmission lines placed parallel to each other and in proximity as shown in **Figure 8**. In such a configuration there is a continuous coupling between the electromagnetic fields of the two lines. Coupled lines are utilized extensively as basic elements for coplanar directional coupler which is the subject of this study, filters, amplifiers, and a variety of other

Because of the coupling of electromagnetic fields, a pair of coupled lines can support two different modes of propagation. These modes have different characteristic impedances *Zco* for odd mode, and *Zce* for even mode. The general theory of coupled lines (telegraphist equations) is used as method of analysis to determine those impedances for each mode of propagation, and to calculate the coupling coefficient *K*.

*<sup>ε</sup>eff* <sup>¼</sup> *vp vpa* � �<sup>2</sup>

velocity in the air, while *L* and *C* is the is the inductance and capacity per unit

length of the micro-coplanar line. And *ψ* is the unit potential.

**6. Analysis of a directional coupler in CPW technology**

useful circuits.

**87**

**Figure 7.**

*Variations of the effective permittivity.*

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

*C* ¼

*<sup>L</sup>* <sup>¼</sup> <sup>1</sup> *v*2

<sup>¼</sup> *vpL* <sup>¼</sup> <sup>1</sup>

*vpC* (16)

*<sup>ε</sup>*0*μ*0*<sup>C</sup>* (18)

(17)

(19)

#### **5. Characteristics of the coplanar line**

In this part we present the numerical results of the variation of the characteristic impedance *Zc* of the transmission line shown in **Figure 6** and of the variation of the effective dielectric permittivity *εeff* as a function of the ratio *w=b* shown in **Figure 7**. The line is assumed to be lossless ð Þ *R* ¼ *G* ¼ 0 , then its characteristic impedance is given by:

**Figure 6.** *Variations of the characteristic impedance for different εeff .*

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

**Figure 7.** *Variations of the effective permittivity.*

studied show in **Figure 4**. And **Figure 5** shows the variation of the charge density

ð Þ *x*<sup>0</sup> (solution of the integral equation) as a function of subdivisions *N* of the

In this part we present the numerical results of the variation of the characteristic impedance *Zc* of the transmission line shown in **Figure 6** and of the variation of the effective dielectric permittivity *εeff* as a function of the ratio *w=b* shown in **Figure 7**. The line is assumed to be lossless ð Þ *R* ¼ *G* ¼ 0 , then its characteristic impedance is

*ρ j*

**Figure 5.**

given by:

**Figure 6.**

**86**

central strip of with *w*.

**5. Characteristics of the coplanar line**

*Variations of the characteristic impedance for different εeff .*

*Variation of the charge density on the metal strip.*

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

$$Z\_c = \sqrt{\frac{L}{C}} = v\_p L = \frac{1}{v\_p C} \tag{16}$$

$$\mathbf{C} = \frac{\sum\_{j=1}^{N} \rho\_j}{\boldsymbol{\nu}} \tag{17}$$

$$L = \frac{1}{v\_{pa}^2 \mathcal{C}} = \frac{1}{\varepsilon\_0 \mu\_0 \mathcal{C}}\tag{18}$$

$$
\varepsilon\_{\sharp \overline{\mathcal{J}}} = \left(\frac{\upsilon\_p}{\upsilon\_{pa}}\right)^2 \tag{19}
$$

With *vp* is the propagation velocity in the coplanar line and *vpa* is the propagation velocity in the air, while *L* and *C* is the is the inductance and capacity per unit length of the micro-coplanar line. And *ψ* is the unit potential.

#### **6. Analysis of a directional coupler in CPW technology**

A coupled coplanar line configuration consists of two transmission lines placed parallel to each other and in proximity as shown in **Figure 8**. In such a configuration there is a continuous coupling between the electromagnetic fields of the two lines. Coupled lines are utilized extensively as basic elements for coplanar directional coupler which is the subject of this study, filters, amplifiers, and a variety of other useful circuits.

Because of the coupling of electromagnetic fields, a pair of coupled lines can support two different modes of propagation. These modes have different characteristic impedances *Zco* for odd mode, and *Zce* for even mode. The general theory of coupled lines (telegraphist equations) is used as method of analysis to determine those impedances for each mode of propagation, and to calculate the coupling coefficient *K*.

**Figure 8.** *Directional coupler.*

We suppose that the propagation is along the axis *OZ*, the telegraphist equation is written [10]:

$$\begin{cases} -\frac{d}{dz}[V] = [Z].[I] \\\\ -\frac{d}{dz}[I] = [Y].[I] \end{cases} \tag{20}$$

*Re*,*<sup>o</sup>* <sup>¼</sup> <sup>ð</sup>*L*2*C*<sup>2</sup> � *<sup>L</sup>*1*C*1Þ �

odd mode are:

**Figure 9.**

**Figure 10.**

**89**

*Variations of the coupling coefficient.*

q

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

*Zce* <sup>¼</sup> *<sup>ω</sup> βe*

*Variations of the characteristic impedance for even and odd mode.*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>L</sup>*2*C*<sup>2</sup> � *<sup>L</sup>*1*C*<sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>4</sup>ð Þ *LmC*<sup>2</sup> � *<sup>L</sup>*1*Cm* ð Þ *LmC*<sup>1</sup> � *<sup>L</sup>*2*Cm*

> 1 *C*<sup>1</sup> � *ReCm*

� � (28)

(27)

2ð Þ *LmC*<sup>2</sup> � *L*1*Cm*

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

*<sup>L</sup>*<sup>1</sup> � *Lm Ro* � �

Therefore, the characteristic impedances of the two coupled lines for even and

¼ *βe ω*

With ½ � *Z* is the impedance matrix and ½ � *Y* represents the admittance matrix of the directional coupler. This system can be written as:

$$\begin{cases} \begin{aligned} -\frac{dv\_1}{dz} &= Z\_1 i\_1 + Z\_m i\_2 \\\\ -\frac{dv\_2}{dz} &= Z\_m i\_1 + Z\_2 i\_2 \\\\ -\frac{di\_1}{dz} &= Y\_1 v\_1 + Y\_m v\_2 \\\\ -\frac{di\_2}{dz} &= Y\_m v\_1 + Y\_m v\_2 \end{aligned} \end{cases} \tag{21}$$

With *Z*1, *Z*<sup>2</sup> are the impedances of the coupled lines per unit length, and *Y*1, *Y*<sup>2</sup> their admittance, where *Zm*, *Ym* are the mutual impedance per unit length and the mutual admittance per unit length respectively.

The system Eq. (21) is a system of homogeneous first-order differential equations with constant coefficients. The resolution of this system gives the voltage and the current for even and odd propagated modes is as follow:

$$\nu\_1 = \left(A\_1 e^{-\gamma\_r Z} + A\_2 e^{-\gamma\_r Z}\right) + \left(A\_3 e^{-\gamma\_o Z} + A\_4 e^{-\gamma\_o Z}\right) \tag{22}$$

$$w\_2 = R\_\epsilon \left( A\_1 e^{-\gamma\_\epsilon Z} + A\_2 e^{-\gamma\_\sigma Z} \right) + R\_o \left( A\_3 e^{-\gamma\_\sigma Z} + A\_4 e^{-\gamma\_\sigma Z} \right) \tag{23}$$

$$i\_1 = Y\_{\epsilon1} \left( A\_1 e^{-\gamma\_r Z} - A\_2 e^{-\gamma\_r Z} \right) + Y\_{\sigma1} \left( A\_3 e^{-\gamma\_\sigma Z} - A\_4 e^{-\gamma\_\sigma Z} \right) \tag{24}$$

$$i\_2 = Y\_{\epsilon2} R\_{\epsilon} \left( A\_1 e^{-\gamma\_r Z} - A\_2 e^{-\gamma\_r Z} \right) + Y\_{o2} R\_{\sigma} \left( A\_3 e^{-\gamma\_\sigma Z} - A\_4 e^{-\gamma\_\sigma Z} \right) \tag{25}$$

Where *Re*,*<sup>o</sup>* and *Ye*1,2; *<sup>o</sup>*1,2 are functions depending on the impedance and admittance of the coupled line. As a result, the propagation constants of the two modes are expressed as a function of linear capacitances and inductances:

$$\begin{split} \gamma\_{\epsilon,\rho} &= j\beta\_{\epsilon,\rho} \\ &= j\frac{\alpha}{\sqrt{2}} \sqrt{(L\_1\mathbf{C}\_1 + L\_2\mathbf{C}\_2 - 2L\_m\mathbf{C}\_m) \pm \sqrt{(L\_2\mathbf{C}\_2 - L\_1\mathbf{C}\_1)^2 + 4(L\_m\mathbf{C}\_1 - L\_2\mathbf{C}\_m)(L\_m\mathbf{C}\_2 - L\_1\mathbf{C}\_m)^2} \end{split} \tag{26}$$

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

$$R\_{\epsilon,\rho} = \frac{(L\_2\mathbf{C}\_2 - L\_1\mathbf{C}\_1) \pm \sqrt{(L\_2\mathbf{C}\_2 - L\_1\mathbf{C}\_1)^2 + 4(L\_m\mathbf{C}\_2 - L\_1\mathbf{C}\_m)(L\_m\mathbf{C}\_1 - L\_2\mathbf{C}\_m)}}{2(L\_m\mathbf{C}\_2 - L\_1\mathbf{C}\_m)}\tag{27}$$

Therefore, the characteristic impedances of the two coupled lines for even and odd mode are:

$$Z\_{c\varepsilon} = \frac{\alpha}{\beta\_{\varepsilon}} \left( L\_1 - \frac{L\_m}{R\_o} \right) = \frac{\beta\_{\varepsilon}}{\alpha} \left( \frac{1}{C\_1 - R\_{\varepsilon} C\_m} \right) \tag{28}$$

**Figure 9.** *Variations of the characteristic impedance for even and odd mode.*

**Figure 10.** *Variations of the coupling coefficient.*

We suppose that the propagation is along the axis *OZ*, the telegraphist equation

*dz* ½ �¼ *<sup>V</sup>* ½ � *<sup>Z</sup> :*½ �*<sup>I</sup>*

(20)

(21)

(26)

*dz* ½�¼*<sup>I</sup>* ½ � *<sup>Y</sup> :*½ �*<sup>I</sup>*

*dz* <sup>¼</sup> *<sup>Z</sup>*1*i*<sup>1</sup> <sup>þ</sup> *Zmi*<sup>2</sup>

*dz* <sup>¼</sup> *Zmi*<sup>1</sup> <sup>þ</sup> *<sup>Z</sup>*2*i*<sup>2</sup>

*dz* <sup>¼</sup> *<sup>Y</sup>*1*v*<sup>1</sup> <sup>þ</sup> *Ymv*<sup>2</sup>

*dz* <sup>¼</sup> *Ymv*<sup>1</sup> <sup>þ</sup> *Ymv*<sup>2</sup>

With *Z*1, *Z*<sup>2</sup> are the impedances of the coupled lines per unit length, and *Y*1, *Y*<sup>2</sup> their admittance, where *Zm*, *Ym* are the mutual impedance per unit length and the

The system Eq. (21) is a system of homogeneous first-order differential equations with constant coefficients. The resolution of this system gives the voltage and

�*γoZ* <sup>þ</sup> *<sup>A</sup>*4*<sup>e</sup>*

�*γoZ* <sup>þ</sup> *<sup>A</sup>*4*<sup>e</sup>*

�*γoZ* � *<sup>A</sup>*4*<sup>e</sup>*

�*γoZ* � *<sup>A</sup>*4*<sup>e</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>L</sup>*2*C*<sup>2</sup> � *<sup>L</sup>*1*C*<sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>4</sup>ð Þ *LmC*<sup>1</sup> � *<sup>L</sup>*2*Cm* ð Þ *LmC*<sup>2</sup> � *<sup>L</sup>*1*Cm*

�*γoZ* � � (22)

�*γoZ* � � (23)

�*γoZ* � � (24)

�*γoZ* � � (25)

With ½ � *Z* is the impedance matrix and ½ � *Y* represents the admittance matrix of

� *d*

8 >><

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

>>:

the directional coupler. This system can be written as:

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

the current for even and odd propagated modes is as follow:

�*γeZ* <sup>þ</sup> *<sup>A</sup>*2*<sup>e</sup>* �*γeZ* � � <sup>þ</sup> *<sup>A</sup>*3*<sup>e</sup>*

�*γeZ* <sup>þ</sup> *<sup>A</sup>*2*<sup>e</sup>*

�*γeZ* � *<sup>A</sup>*2*<sup>e</sup>*

�*γeZ* � *<sup>A</sup>*2*<sup>e</sup>*

are expressed as a function of linear capacitances and inductances:

�*γeZ* � � <sup>þ</sup> *Ro <sup>A</sup>*3*<sup>e</sup>*

�*γeZ* � � <sup>þ</sup> *Yo*<sup>1</sup> *<sup>A</sup>*3*<sup>e</sup>*

�*γeZ* � � <sup>þ</sup> *Yo*2*Ro <sup>A</sup>*3*<sup>e</sup>*

Where *Re*,*<sup>o</sup>* and *Ye*1,2; *<sup>o</sup>*1,2 are functions depending on the impedance and admittance of the coupled line. As a result, the propagation constants of the two modes

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mutual admittance per unit length respectively.

*v*<sup>1</sup> ¼ *A*1*e*

*v*<sup>2</sup> ¼ *Re A*1*e*

*i*<sup>1</sup> ¼ *Ye*<sup>1</sup> *A*1*e*

ð*L*1*C*<sup>1</sup> þ *L*2*C*<sup>2</sup> � 2*LmCm*Þ �

r q

*i*<sup>2</sup> ¼ *Ye*2*Re A*1*e*

*γ<sup>e</sup>*,*<sup>o</sup>* ¼ *jβ<sup>e</sup>*,*<sup>o</sup>*

**88**

<sup>¼</sup> *<sup>j</sup> <sup>ω</sup>* ffiffi 2 p

� *d*

� *dv*<sup>1</sup>

� *dv*<sup>2</sup>

� *di*<sup>1</sup>

� *di*<sup>2</sup>

is written [10]:

**Figure 8.** *Directional coupler.*

$$Z\_{co} = \frac{\alpha}{\beta\_o} \left( L\_1 - \frac{L\_m}{R\_\epsilon} \right) = \frac{\beta\_o}{\alpha} \left( \frac{1}{C\_1 - R\_o C\_m} \right) \tag{29}$$

For the coupling coefficient *K*, it is given by the following formula:

$$K = \frac{Z\_{\alpha\varepsilon} - Z\_{co}}{Z\_{\alpha\varepsilon} + Z\_{co}} \tag{30}$$

**References**

4200-6145-1.

September1989.

of Lausanne 1983.

9480(96)00469–3.

119–126.

**91**

[1] K.C. GUPTA, RAMESH GARG AND INDER. BAHL AND MAURIZIO BOZZI. Microstrip lines and slot-lines,

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

anisotropic electrostatic field problems, IEEE TRANS. On MTT, vol. MTT-26,

[10] ROBERT E. COLLIN, Field Theory of Guided Waves. Edition Mc Graw Hill,

No 7, July 1978, pp 510–512.

New York, 1962. ISBN: 978–0–

879-42237-0.

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

[2] R.W. JAKSON, Consideration in the

[3] WALTON C. GIBSON, The method of moments in electromagnetics. Chapman & hall/CRC. ISBN-13: 978–1–

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**Figures 9** and **10** show the variations of *Zco*, and *Zce* as a function of the ratio W / S, and the variation of *K* the coupling coefficient as a function of the gap S between the coupled lines respectively.
