**3.3.1 Backward symmetrical CLC**

The proposed backward-wave directional coupler is shown in Fig. 5 (Keshavarz et al., 2011a). It is a coupled-line coupler consisting of an interdigital capacitor with one finger as a CRLH TL in each coupled-line. It is seen that using only one interdigital capacitor to realize the interdigital TLs is more suitable to reach a coupler with better matching and wider bandwidth. As it was mentioned, forω > ω*se* , these interdigital TLs will be operating completely in their RH range for the presented coupler application. So in this coupler, similar to the conventional edge-coupled couplers, the coupling coefficient is (Pozar, 2004):

$$S\_{31} = \frac{jk\sin\theta}{\sqrt{1 - k^2}\cos\theta + j\sin\theta}, \quad k = \frac{Z\_{cc} - Z\_{co}}{Z\_{cc} + Z\_{co}}\tag{6}$$

where, θ*=(2*π*l/λg)* is electrical length and is the length of CLC. Therefore, setting the interdigital capacitor length as *l= λg/*4 or θ*=*π*/2* results in maximum coupling level. On the other hand, selection of *l= λg/*4 preserves the homogeneity condition in CRLH structure (i.e., <sup>4</sup> *<sup>g</sup> p* λ ≤ , where *p* is structural cell size) (Caloz & Itoh, 2005).

The equivalent circuits model of the even and odd modes of Fig. 5 for one cell have been presented in Fig. 6. In this figure, *L* is the inductance for a strip with width *W*′ and , *C Ce o* are the distributed capacitances for the even and odd modes, respectively.

Even and odd mode characteristic impedances ( , *Z Z ce co* ) of the coupled-lines composed of interdigital TLs are obtained from (Caloz & Itoh, 2005) with setting *LL* → ∞ as:

$$Z\_{cc} = \sqrt{\frac{L}{C\_e} - \frac{1}{o\rho^2 C\_e C\_L}} \quad , \quad Z\_{co} = \sqrt{\frac{L}{C\_o} - \frac{1}{o\rho^2 C\_o C\_L}} \tag{7}$$

and

256 Trends in Electromagnetism – From Fundamentals to Applications

theory was used to analyze the phenomenon of complete backward coupling. Then, an asymmetric RH-CRLH coupler was introduced and studied in (Caloz & Itoh, 2004b). It was composed of a conventional right-handed transmission line and a CRLH TL. That coupler showed the advantage of broad bandwidth and tight coupling characteristics, and coupledmode theory based on traveling waves was used to discuss these interesting features. In (Islam & Eleftheriades, 2006), it was shown that the formation of a stop-band and the excitation of complex modes occurred in the case of coupling between a forward wave and a backward-wave mode for a range of frequencies around the tuning frequency. Moreover, authors in (Wang et al., 2007) presented the conditions for tight coupling and detailed

Moreover, some CLCs based on the CRLH TLs with arbitrary coupling levels have been developed, recently (Fouda et al., 2010; Hirota et al., 2009; Hirota et al., 2011; Kawakami et al., 2010; Mocanu et al., 2010). In these couplers, the backward coupling depends on the difference between even and odd modes characteristic impedances and length of the

The interdigital/stub CLCs have been typically adapted to increasing coupling level, but these couplers increase in size (Caloz et al., 2004; Caloz & Itoh, 2004b; Islam & Eleftheriades, 2006), band width of them is narrow (Hirota et al., 2011; Mocanu et al. 2010; Wang et al. 2007) and the multiconductors of the interdigital construction complicate the design

It is considerable that the microstrip CRLH TL structures have been mostly implemented in the form of interdigital capacitors and stub inductors. In the other hand, using shorted stub inductors with large sizes to achieve the required inductances can cause the structure width to be also enlarged. For instance, the length and width of 3-dB microstrip coupled-line coupler proposed in (Caloz et al., 2004) are approximately *λg/*3 and *λg/*6, respectively. Also, bandwidth of the CRLH CLCs which presented in (Mocanu et al., 2010) and (Fouda et al.,

Also, forward coupling level in CRLH coupled line couplers is low (nearly -10 dB in (Fouda

In this section, some of the authors' proposed CLCs based on the CRLH concepts to reach new couplers with better specifications, such as smaller size, broader bandwidth and more simplicity in fabrication are presented. In these new CLCs one has been trying to eliminate some drawbacks and disadvantages of conventional CRLH CLCs mentioned in previous

The proposed backward-wave directional coupler is shown in Fig. 5 (Keshavarz et al., 2011a). It is a coupled-line coupler consisting of an interdigital capacitor with one finger as a CRLH TL in each coupled-line. It is seen that using only one interdigital capacitor to realize the interdigital TLs is more suitable to reach a coupler with better matching and wider

*se* , these interdigital TLs will be operating

ω > ω

formulas were given to define the edges of the coupling range.

coupled lines (Caloz & Itoh, 2005).

procedure (Caloz & Itoh, 2005).

2010) are 25% and 30%, respectively.

**3.3.1 Backward symmetrical CLC** 

bandwidth. As it was mentioned, for

et al., 2010)).

section.

**3.3 Proposed CLCs** 

$$Z'\_{ce} = \sqrt{\frac{L}{C\_e}} \quad , \quad Z'\_{co} = \sqrt{\frac{L}{C\_o}} \tag{8}$$

*Zce*′ and *Zco*′ are even and odd mode characteristic impedances of a conventional microstrip CLC with strips of width *W*′ for each TL, where *W NW* ′( (4 1) ) = − is total width of the interdigital capacitor.

Fig. 5. Structure of the proposed microstrip coupled-line backward coupler on FR4 substrate, = 4.7 *<sup>r</sup>* ε , thickness of 1.6 mm. (a) Structure layout. (b) Fabricated coupler (Keshavarz et al., 2011a).

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 259

mm. Also, the length of the TLs is 12.8 mm (see Fig.5). For better matching and wider bandwidth, we use only one interdigital capacitor, i.e. one cell, in every interdigital TL. Moreover, to reach a large isolation parameter, spacing between the fingers in the lower interdigital capacitor is set larger than the upper one, when ports 1 and 4 are the input and isolated ports, respectively. As shown in the layout of the coupler in Fig. 5, at the all four ports of the structure, tapered microstrip TLs have been used for the impedance matching to 50 Ω, as well as to fit the ports size to the inner conductors of the coaxial-to-microstrip

Fig. 8 presents the full-wave simulated (by using Agilent ADS software) and measured *S*parameters for the coupler of Fig. 5. Excellent agreement can be observed between simulated and experimental results. There is only a small difference between *S*11 parameter of simulated and measurement results. Due to small distance between coupler connectors, we could not connect network analyzer ports to adjacent coupler connectors, directly. Therefore, two interface cables were connected to the coupler connectors and then *S*parameters were measured. This drawback shows its bad effect on *S*<sup>11</sup> parameter more

Using these figures, a amplitude balance of *±*2 dB over a bandwidth of 60% (2.3–4 GHz), a matching (10 dB bandwidth) and an isolation at least *−*20 dB over a bandwidth of 80% (2.2– 4.6 GHz) are observed. Fig. 9 illustrates the phase difference between ports 2 and 3 of the coupler. This phase difference is <sup>0</sup> 90 at design frequency and exhibits a phase-balance

In comparison with the conventional CRLH CLCs, the electrical length of the proposed CLC is more compact than the CRLH CLCs presented in (Islam et al., 2004; Mao & Wu, 2007; Nguyen & Caloz, 2006; Zhang et al., 2008). Moreover, due to the elimination of the stubs in the structure, its width is also smaller. For instance, the width of the coupler is nearly 11 times smaller than CRLH CLC reported in (Caloz et al., 2004) and its coupled-line electrical length is shortened to 60% of the 3-dB CRLH coupler electrical length presented in

′ , b) Spacing between two coupled

*<sup>r</sup>* = 4.7 , thickness of 1.6 mm

ε

(a) (b)

interdigital TLs (*S*), in the proposed coupler on FR4 substrate,

Fig. 7. a) Width of the interdigital TL (*W*) versus *Zce*

(Keshavarz et al., 2011a).

strongly than other *S*-parameters.

(*±*10 <sup>0</sup> ) bandwidth of 1.3 GHz.

transitions.

Fig. 6. (a) Odd and (b) even modes equivalent circuit models of proposed coupler in Fig. 5 (Keshavarz et al., 2011a).

In the proposed coupler for given even and odd mode characteristic impedances, according to (7), selection of a small *CL* leads to larger values of *Zce* and *Zco* . This situation is very suitable for elimination of the fabrication restrictions in CLCs with tight coupling level. Consequently, to decrease the value of *CL* in the proposed structure, interdigital capacitors with only one finger (i.e., *N* = 1) are used.

In design procedure, for an indicated coupling-level (*c*) and characteristic impedance ( int *Zc* ), *Zce* and *Zco* can be obtained from conventional expressions as (Pozar, 2004):

$$Z\_{cc} = Z\_c^{\text{int}} \sqrt{\frac{1+c}{1-c}}, \quad Z\_{cc} = Z\_c^{\text{int}} \sqrt{\frac{1-c}{1+c}} \tag{9}$$

With setting *N* = 1, 4 = λ*<sup>g</sup>* and the substrate profile being determined, *CL* and *CR* can be calculated using expressions presented in (Bahl, 2003) and *Zce*′ and *Zco*′ are obtained from (7) and (8). Then, *W*′ and *S* can be determined by using achieved , *Z Z ce co* ′ ′ and relative design graphs for conventional coupled microstrip lines.

For instant, Fig. 7(a) illustrates the required width of the interdigital TL (*W*) in the proposed coupler realized on FR4 substrate, with 4.7 *<sup>r</sup>* ε = and thickness of 1.6 mm, for different values of *Zce*′ . In addition, the necessary spacing between two coupled interdigital TLs (*S*) for the

presented structure versus *Zm* , where <sup>2</sup> *ce co <sup>m</sup> ce co Z Z <sup>Z</sup> Z Z* ′ ′ <sup>=</sup> ′ ′ <sup>−</sup> , has been provided in Fig. 7(b).

As it was mentioned, since *Zce*′ and *Zco*′ would be larger than *Zce* and *Zce* , for constant coupling-level (*c*) and characteristic impedance ( int *Zc* ) in comparison with the conventional CLCs, *W*′ decreases and *S* increases. Therefore in the proposed coupler, the fabrication constrains in conventional edge-coupled couplers to get a tight coupling-level caused very small spacing between two coupled lines (i.e., *S*) can be removed.

To validate the proposed technique, a 3-dB coupled line coupler based on the design procedure and presented expressions has been designed on FR4 substrate with 4.7 *<sup>r</sup>* ε = , thickness of 1.6 mm and *tan δ* = *0.021*. Fig. 5 shows the designed coupler layout and the fabricated structure. A 3-dB coupled line coupler with nearly 60% bandwidth (from 2.3 to 4 GHz) around the design frequency *cf* = 3*.*2 *GHz* is achieved in the measured prototype. The spacing between two TLs (*S*) and width of the interdigital capacitor fingers (*W*) are 0.2 258 Trends in Electromagnetism – From Fundamentals to Applications

(a) (b)

In the proposed coupler for given even and odd mode characteristic impedances, according to (7), selection of a small *CL* leads to larger values of *Zce* and *Zco* . This situation is very suitable for elimination of the fabrication restrictions in CLCs with tight coupling level. Consequently, to decrease the value of *CL* in the proposed structure, interdigital capacitors

In design procedure, for an indicated coupling-level (*c*) and characteristic impedance ( int *Zc* ),

int 1 1 int , 1 1 *ce c co c c c ZZ ZZ*

For instant, Fig. 7(a) illustrates the required width of the interdigital TL (*W*) in the proposed

*Z Z <sup>Z</sup> Z Z*

coupling-level (*c*) and characteristic impedance ( int *Zc* ) in comparison with the conventional CLCs, *W*′ decreases and *S* increases. Therefore in the proposed coupler, the fabrication constrains in conventional edge-coupled couplers to get a tight coupling-level caused very

To validate the proposed technique, a 3-dB coupled line coupler based on the design procedure and presented expressions has been designed on FR4 substrate with 4.7 *<sup>r</sup>*

thickness of 1.6 mm and *tan δ* = *0.021*. Fig. 5 shows the designed coupler layout and the fabricated structure. A 3-dB coupled line coupler with nearly 60% bandwidth (from 2.3 to 4 GHz) around the design frequency *cf* = 3*.*2 *GHz* is achieved in the measured prototype. The spacing between two TLs (*S*) and width of the interdigital capacitor fingers (*W*) are 0.2

′ . In addition, the necessary spacing between two coupled interdigital TLs (*S*) for the

*ce co*

*c c*

*<sup>g</sup>* and the substrate profile being determined, *CL* and *CR* can be

+ − = = − + (9)

′ and *Zco*

= and thickness of 1.6 mm, for different values

′ would be larger than *Zce* and *Zce* , for constant

′ ′ <sup>=</sup> ′ ′ <sup>−</sup> , has been provided in Fig. 7(b).

′ are obtained from (7)

ε= ,

′ ′ and relative design

*Zce* and *Zco* can be obtained from conventional expressions as (Pozar, 2004):

Fig. 6. (a) Odd and (b) even modes equivalent circuit models of proposed coupler in Fig. 5

(Keshavarz et al., 2011a).

With setting *N* = 1, 4 =

As it was mentioned, since *Zce*

of *Zce*

with only one finger (i.e., *N* = 1) are used.

λ

graphs for conventional coupled microstrip lines.

presented structure versus *Zm* , where <sup>2</sup> *ce co <sup>m</sup>*

coupler realized on FR4 substrate, with 4.7 *<sup>r</sup>*

calculated using expressions presented in (Bahl, 2003) and *Zce*

and (8). Then, *W*′ and *S* can be determined by using achieved , *Z Z ce co*

′ and *Zco*

small spacing between two coupled lines (i.e., *S*) can be removed.

ε

Fig. 7. a) Width of the interdigital TL (*W*) versus *Zce* ′ , b) Spacing between two coupled interdigital TLs (*S*), in the proposed coupler on FR4 substrate, ε *<sup>r</sup>* = 4.7 , thickness of 1.6 mm (Keshavarz et al., 2011a).

mm. Also, the length of the TLs is 12.8 mm (see Fig.5). For better matching and wider bandwidth, we use only one interdigital capacitor, i.e. one cell, in every interdigital TL. Moreover, to reach a large isolation parameter, spacing between the fingers in the lower interdigital capacitor is set larger than the upper one, when ports 1 and 4 are the input and isolated ports, respectively. As shown in the layout of the coupler in Fig. 5, at the all four ports of the structure, tapered microstrip TLs have been used for the impedance matching to 50 Ω, as well as to fit the ports size to the inner conductors of the coaxial-to-microstrip transitions.

Fig. 8 presents the full-wave simulated (by using Agilent ADS software) and measured *S*parameters for the coupler of Fig. 5. Excellent agreement can be observed between simulated and experimental results. There is only a small difference between *S*11 parameter of simulated and measurement results. Due to small distance between coupler connectors, we could not connect network analyzer ports to adjacent coupler connectors, directly. Therefore, two interface cables were connected to the coupler connectors and then *S*parameters were measured. This drawback shows its bad effect on *S*<sup>11</sup> parameter more strongly than other *S*-parameters.

Using these figures, a amplitude balance of *±*2 dB over a bandwidth of 60% (2.3–4 GHz), a matching (10 dB bandwidth) and an isolation at least *−*20 dB over a bandwidth of 80% (2.2– 4.6 GHz) are observed. Fig. 9 illustrates the phase difference between ports 2 and 3 of the coupler. This phase difference is <sup>0</sup> 90 at design frequency and exhibits a phase-balance (*±*10 <sup>0</sup> ) bandwidth of 1.3 GHz.

In comparison with the conventional CRLH CLCs, the electrical length of the proposed CLC is more compact than the CRLH CLCs presented in (Islam et al., 2004; Mao & Wu, 2007; Nguyen & Caloz, 2006; Zhang et al., 2008). Moreover, due to the elimination of the stubs in the structure, its width is also smaller. For instance, the width of the coupler is nearly 11 times smaller than CRLH CLC reported in (Caloz et al., 2004) and its coupled-line electrical length is shortened to 60% of the 3-dB CRLH coupler electrical length presented in

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 261

Fig. 9. Measured phase difference between the through port and the coupled port for the

In this section, an asymmetrical coupled-line coupler based on the interdigital TL is presented. Fig. 10 shows layout and circuit model of the interdigital TL and conventional TL which are adjacent to each other as asymmetrical backward coupled-line coupler. As depicted in Fig. 10(b), *Cm* represents the mutual capacitance between interdigital and strip of the microstrip conductors in the absence of the structure ground conductor while *C*1 and *C*2 represent the capacitance between interdigital or microstrip strip conductors and ground, respectively. Moreover, the circuit model includes mutual inductance ( *Lm* ) and self-inductances of interdigital (line 1) and conventional microstrip (line 2) conductors, i.e. *L*<sup>1</sup> and *L*<sup>2</sup> , respectively.*C*int is series interdigital capacitor of line 1. It should be stated that all of parameters in the circuit model are per unit length quantities. Also, Fig. 11 shows the capacitance representation for quasi-TEM mode of cross section of the proposed asymmetrical coupler. For structure analysis, it is assumed that lines 1 and 2 are terminated

proposed coupler of Fig. 5 (Keshavarz et al., 2011a).

**3.3.2 Backward asymmetrical CLC** 

to impedances *Za* and *Zb* , respectively.

(a) (b)

Fig. 10. Proposed asymmetrical coupled-line coupler consisted of interdigital TL and microstrip conventional TL. a) Its layout and b) lumped equivalent circuit model.

(Keshavarz et al., 2011a). Moreover, the bandwidth of the proposed CLC is wider than CRLH CLCs presented in (Islam et al., 2004) and (Nguyen & Caloz, 2006).

In comparison with the conventional planar microstrip CLC realized in the same substrate material and similar spacing between coupled TLs, this CLC achieves higher coupling level. The high coupling level (8 dB or higher) is extremely difficult to achieve in the conventional CLC due to the present limit in fabrication (Pozar, 2004). Also, simulation results show that in the proposed structure if the spacing between the coupled lines increases, the bandwidth increases up to 85% for 7-dB coupling factor. Moreover, this coupler exhibits much higher design simplicity than the existing CRLH CLCs. Due to the wide bandwidth and compact size, the proposed coupler is well suitable for microwave and millimeter-wave integrated circuits, wideband communication systems and many kinds of antenna arrays.

Fig. 8. *S*-parameters of the proposed coupler have shown in Fig.5 (a) Full-wave simulation results. (b) Measurement results (Keshavarz et al., 2011a).

Fig. 9. Measured phase difference between the through port and the coupled port for the proposed coupler of Fig. 5 (Keshavarz et al., 2011a).

#### **3.3.2 Backward asymmetrical CLC**

260 Trends in Electromagnetism – From Fundamentals to Applications

(Keshavarz et al., 2011a). Moreover, the bandwidth of the proposed CLC is wider than

In comparison with the conventional planar microstrip CLC realized in the same substrate material and similar spacing between coupled TLs, this CLC achieves higher coupling level. The high coupling level (8 dB or higher) is extremely difficult to achieve in the conventional CLC due to the present limit in fabrication (Pozar, 2004). Also, simulation results show that in the proposed structure if the spacing between the coupled lines increases, the bandwidth increases up to 85% for 7-dB coupling factor. Moreover, this coupler exhibits much higher design simplicity than the existing CRLH CLCs. Due to the wide bandwidth and compact size, the proposed coupler is well suitable for microwave and millimeter-wave integrated

(a)

(b) Fig. 8. *S*-parameters of the proposed coupler have shown in Fig.5 (a) Full-wave simulation

results. (b) Measurement results (Keshavarz et al., 2011a).

CRLH CLCs presented in (Islam et al., 2004) and (Nguyen & Caloz, 2006).

circuits, wideband communication systems and many kinds of antenna arrays.

In this section, an asymmetrical coupled-line coupler based on the interdigital TL is presented. Fig. 10 shows layout and circuit model of the interdigital TL and conventional TL which are adjacent to each other as asymmetrical backward coupled-line coupler. As depicted in Fig. 10(b), *Cm* represents the mutual capacitance between interdigital and strip of the microstrip conductors in the absence of the structure ground conductor while *C*1 and *C*2 represent the capacitance between interdigital or microstrip strip conductors and ground, respectively. Moreover, the circuit model includes mutual inductance ( *Lm* ) and self-inductances of interdigital (line 1) and conventional microstrip (line 2) conductors, i.e. *L*<sup>1</sup> and *L*<sup>2</sup> , respectively.*C*int is series interdigital capacitor of line 1. It should be stated that all of parameters in the circuit model are per unit length quantities. Also, Fig. 11 shows the capacitance representation for quasi-TEM mode of cross section of the proposed asymmetrical coupler. For structure analysis, it is assumed that lines 1 and 2 are terminated to impedances *Za* and *Zb* , respectively.

Fig. 10. Proposed asymmetrical coupled-line coupler consisted of interdigital TL and microstrip conventional TL. a) Its layout and b) lumped equivalent circuit model.

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 263

*<sup>Z</sup> R R <sup>Z</sup>* =− = π

*C C* = − , <sup>2</sup>

where *Z*1 and *Z*2 are characteristic impedances of uncoupled lines 1 (interdigital TL) and 2

Moreover, the capacitance matrix of the coupled lines (Fig. 10) can be expressed as (Cristal,

21 2 2

*CC CC C*

*C C C CC* + − = = − +

π

*c*

1 int 1

ω

[ ] 1 12 1

*a a Z Z <sup>c</sup>* π

1 0 2

2

= −

2

2

1

1

2

2

*C C*

*m*

*<sup>C</sup>* ′ <sup>=</sup> , <sup>1</sup> 0

with a strip of width *W*′ , where *W*′ (= (2 1) 2 *N S NW* − +′ ) is the total width of the

In coupler design procedure, for an indicated coupling-level (*k*) and impedance ports *Za*

*a ab ZZ Z c c* π

0

 <sup>=</sup> 

*a c*

*<sup>L</sup> <sup>Z</sup>*

= −

*a*

π

*<sup>L</sup> <sup>Z</sup>*

*b c*

*<sup>L</sup> <sup>Z</sup> C*

*b*

*<sup>L</sup> <sup>Z</sup>*

<sup>=</sup> <sup>+</sup>

0

0

π

and *Zb* of lines 1 and 2, respectively, 00 0 , ,

*a c <sup>L</sup> <sup>Z</sup>*

π

1 int 1 1 0 2

<sup>+</sup> <sup>+</sup>

*C C C*

ω

1

1 int 1

( 2) ( 2)

*a*

π

*<sup>L</sup> <sup>Z</sup>*

*C C CC C*

ω

*C*

According to equations (13) and (14), *c* and

) are obtained as (Cristal, 1966):

 

interdigital transmission line ( 0 0 ,

1 2

*<sup>L</sup>* <sup>1</sup> *<sup>Z</sup> C*

where

1966):

( 0 0 , *b b Z Z <sup>c</sup>* π

and

0 *<sup>a</sup> Z <sup>c</sup>* ′ and 0

*<sup>a</sup> Z* π

interdigital capacitor.

′ are *c* and

following equations (Cristal, 1966):

<sup>1</sup>

(conventional microstrip TL), respectively.

2 1

2

*m m m m*

1

<sup>1</sup> ( 2)

mode characteristic impedances of a conventional microstrip TL

 and 0 *<sup>b</sup> Z* π

*C C*

*m*

′ <sup>=</sup> + (16)

*m m*

*<sup>L</sup> <sup>Z</sup>*

2

(12)

*<sup>C</sup>* <sup>=</sup> (13)

(14)

(15)

can be calculated from

mode characteristic impedances of

) and the conventional microstrip transmission line

Fig. 11. Capacitance representation for cross section of the asymmetrical coupler presented in Fig. 10.

Characteristics of the proposed coupled transmission line can be described by a superposition of characteristics of *c* and π modes. A set of two coupled lines can support two fundamental independent modes of propagation (called normal modes). For asymmetrical coupled lines, the two normal modes of propagation are known as *c* and π modes (Mongia et al., 1999). Both *c* and π modes are composed of two traveling waves in the backward and forward directions. The *c* mode is characterized by four parameters: *<sup>c</sup>* γ , *Zc*<sup>1</sup> , *Zc*2 and *Rc* which are the propagation constant of the mode, the characteristic impedances of lines 1 and 2 and the ratio of the voltages on the two lines of the *c* mode, respectively. Similarly, the π mode is also characterized by four parameters: π γ , *Z*π <sup>1</sup> , *Z*π 2 and *R*π which are propagation constant of the mode, characteristic impedances of lines 1 and 2 and the ratio of the voltages on the two lines of theπ mode, respectively (Mongia et al., 1999).

As it has been shown in (Mongia et al., 1999), the relation between the characteristic impedances, i.e. 12 1 , , *ZZZ c c* π and *Z*π <sup>2</sup> , and also the ratio parameters, i.e. *Rc* and *R*π , are as:

$$\frac{Z\_{c2}}{Z\_{c1}} = \frac{Z\_{\pi 2}}{Z\_{\pi 1}} = -R\_c \ R\_\pi \tag{10}$$

So, a total number of only six quantities, i.e. 1 21 2 *c cc* , , or , or , and π *Z ZZ ZR R* ππ π *c* γ γ are required to characterize asymmetrical coupled lines. For a lossless TEM-mode coupled-line, the propagation constants of both *c* andπ modes are the same, and are given by (Cristal, 1966):

$$
\boldsymbol{\chi}\_c = \boldsymbol{\chi}\_\pi = \mathbf{j} \,\boldsymbol{\beta} \tag{11}
$$

As special case for asymmetrical coupled lines, symmetrical coupled line are completely characterized by four parameters, the even and odd modes characteristic impedances of any lines (as both lines are identical) and even and odd modes propagation constants. In symmetrical coupled lines, *Rc* and *R*π are equal to 1 and -1, respectively (Mongia et al., 1999).

By assuming the quasi-TEM mode for proposed structure and according to equations (10), (11) and (Cristal, 1966) for above asymmetrical coupler (Fig. 10), it is obtained that:

$$R\_c = -R\_\pi = \sqrt{\frac{Z\_2}{Z\_1}}\tag{12}$$

where

262 Trends in Electromagnetism – From Fundamentals to Applications

Fig. 11. Capacitance representation for cross section of the asymmetrical coupler presented

Characteristics of the proposed coupled transmission line can be described by a

two fundamental independent modes of propagation (called normal modes). For asymmetrical coupled lines, the two normal modes of propagation are known as *c* and

the backward and forward directions. The *c* mode is characterized by four parameters: *<sup>c</sup>*

*Zc*<sup>1</sup> , *Zc*2 and *Rc* which are the propagation constant of the mode, the characteristic impedances of lines 1 and 2 and the ratio of the voltages on the two lines of the *c* mode,

As it has been shown in (Mongia et al., 1999), the relation between the characteristic

π

*c*

π

(11) and (Cristal, 1966) for above asymmetrical coupler (Fig. 10), it is obtained that:

*Z Z*

2 2 1 1 *<sup>c</sup> <sup>c</sup>*

So, a total number of only six quantities, i.e. 1 21 2 *c cc* , , or , or , and

π

*Z Z R R*

γ γ

required to characterize asymmetrical coupled lines. For a lossless TEM-mode coupled-line,

*<sup>c</sup> j* π γγ

As special case for asymmetrical coupled lines, symmetrical coupled line are completely characterized by four parameters, the even and odd modes characteristic impedances of any lines (as both lines are identical) and even and odd modes propagation constants. In

By assuming the quasi-TEM mode for proposed structure and according to equations (10),

 β

π

π

mode is also characterized by four parameters:

π

π

π

<sup>2</sup> , and also the ratio parameters, i.e. *Rc* and *R*

= =− (10)

modes are the same, and are given by (Cristal,

= = (11)

are equal to 1 and -1, respectively (Mongia et al.,

 *Z ZZ ZR R* ππ

are

which are propagation constant of the mode, characteristic impedances of lines 1

modes. A set of two coupled lines can support

modes are composed of two traveling waves in

π

γ,

π, are

 π*c*

π γ , *Z*π <sup>1</sup> , *Z*π2

mode, respectively (Mongia et

π

π

in Fig. 10.

and *R*π

as:

1966):

1999).

al., 1999).

superposition of characteristics of *c* and

modes (Mongia et al., 1999). Both *c* and

π

and 2 and the ratio of the voltages on the two lines of the

πand *Z*

respectively. Similarly, the

impedances, i.e. 12 1 , , *ZZZ c c*

the propagation constants of both *c* and

symmetrical coupled lines, *Rc* and *R*

$$Z\_1 = \sqrt{\frac{L\_1}{C\_1} - \frac{1}{a^2 C\_{\text{int}} C\_1}} \quad , \qquad Z\_2 = \sqrt{\frac{L\_2}{C\_2}} \tag{13}$$

where *Z*1 and *Z*2 are characteristic impedances of uncoupled lines 1 (interdigital TL) and 2 (conventional microstrip TL), respectively.

Moreover, the capacitance matrix of the coupled lines (Fig. 10) can be expressed as (Cristal, 1966):

$$\begin{bmatrix} \mathbf{C} \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_1 & \mathbf{C}\_{12} \\ \mathbf{C}\_{21} & \mathbf{C}\_2 \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_1 + \mathbf{C}\_m & -\mathbf{C}\_m \\ -\mathbf{C}\_m & \mathbf{C}\_2 + \mathbf{C}\_m \end{bmatrix} \tag{14}$$

According to equations (13) and (14), *c* and π mode characteristic impedances of interdigital transmission line ( 0 0 , *a a Z Z <sup>c</sup>* π ) and the conventional microstrip transmission line ( 0 0 , *b b Z Z <sup>c</sup>* π) are obtained as (Cristal, 1966):

$$\begin{cases} Z\_{0x^a} = \sqrt{\frac{L\_1}{C\_1} - \frac{1}{\alpha^2 C\_{\text{int}} C\_1}}\\ Z\_{0x^a} = \sqrt{\frac{L\_1}{(C\_1 + 2C\_m)} - \frac{1}{\alpha^2 C\_{\text{int}} (C\_1 + 2C\_m)}}\\ Z\_{0c^b} = \sqrt{\frac{L\_2}{C\_2}}\\ Z\_{0x^b} = \sqrt{\frac{L\_2}{C\_2 + 2C\_m}} \end{cases} \tag{15}$$

and

$$Z\_{0c}^{\prime\prime\,a} = \sqrt{\frac{L\_1}{C\_1}} \qquad \qquad , \quad Z\_{0\pi}^{\prime\prime} = \sqrt{\frac{L\_1}{(C\_1 + 2C\_m)}} \tag{16}$$

0 *<sup>a</sup> Z <sup>c</sup>* ′ and 0 *<sup>a</sup> Z* π′ are *c* and π mode characteristic impedances of a conventional microstrip TL with a strip of width *W*′ , where *W*′ (= (2 1) 2 *N S NW* − +′ ) is the total width of the interdigital capacitor.

In coupler design procedure, for an indicated coupling-level (*k*) and impedance ports *Za* and *Zb* of lines 1 and 2, respectively, 00 0 , , *a ab ZZ Z c c* π and 0 *<sup>b</sup> Z* π can be calculated from following equations (Cristal, 1966):

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 265

(a) (b)

characteristic impedance. (Keshavarz et al., 2011b).

(Keshavarz et al., 2011b).

Fig. 12. Proposed asymmetrical backward coupler based on the interdigital and conventional microstrip coupled TLs. (a) Structure layout. (b) Fabricated coupler.

Fig. 13. Design graph for width of the interdigital TL ( *W*<sup>1</sup> ) on FR-4 substrate versus *c* mode

Fig. 14. Design graph for width of the conventional microstrip TL ( *W*<sup>2</sup> ) on FR-4 substrate

versus *c* mode characteristic impedance. (Keshavarz et al., 2011b).

$$\begin{cases} Z\_{0c}{}^a = \frac{Z\_a Z\_b \sqrt{1 - k^2}}{Z\_b - k \sqrt{Z\_a Z\_b}} \\ Z\_{0x}{}^a = \frac{Z\_a Z\_b \sqrt{1 - k^2}}{Z\_b + k \sqrt{Z\_a Z\_b}} \\ Z\_{0c}{}^b = \frac{Z\_a Z\_b \sqrt{1 - k^2}}{Z\_a - k \sqrt{Z\_a Z\_b}} \\ Z\_{0x}{}^b = \frac{Z\_a Z\_b \sqrt{1 - k^2}}{Z\_a + k \sqrt{Z\_a Z\_b}} \end{cases} \tag{17}$$

In order the values of 0 *<sup>a</sup> Z <sup>c</sup>* and 0 *<sup>b</sup> Z <sup>c</sup>* to be positive, it is necessary that:

$$\frac{1}{k^2} \ge \frac{Z\_a}{Z\_b} \quad \text{and} \quad \frac{1}{k^2} \ge \frac{Z\_b}{Z\_a} \tag{18}$$

where <sup>2</sup> *k* denotes the power coupling coefficient between two coupled lines.

As it was mentioned, for indicated coupling level ( *k* ) and ports impedance ( , *Z Za b* ) in the proposed coupler, the *c* and π characteristic impedances, i.e. 0 0 , *a a Z Z <sup>c</sup>* π , <sup>0</sup> *<sup>b</sup> Z <sup>c</sup>* and <sup>0</sup> *<sup>b</sup> Z* π , can be determined using (17). It is clear from (15) that selecting a small *C*int in the introduced coupler, increases values of 0 *<sup>a</sup> Z <sup>c</sup>* ′ and 0 *<sup>a</sup> Z* π′ which can lead to smaller value for*Cm* . It means that in this situation, the required spacing between two coupled-lines can be increased in comparison with the conventional microstrip coupled-lines. It is due to the inverse relationship between mutual capacitance value and spacing between coupled lines. Therefore, it is suitable for realizing high coupling-level coupled-line couplers with relatively larger spacing between two lines than conventional coupled-line couplers.

Fig. 12 illustrates the layout and fabrication of the proposed asymmetrical coupler that above considerations have been considered in its design (Keshavarz et al., 2011b).

For an asymmetric coupled microstrip line of the type shown in Fig. 12, the design graphs presented in Figs. 13, 14 and 15 can be used to determine the necessary interdigital and microstrip strip widths and spacing for a given set of characteristic impedances, <sup>0</sup> , and *a b ZZ Z oc c m* on FR-4 substrate with 4.6 *<sup>r</sup>* ε = and thickness of 1.6 mm. In Fig. 15, *Zm* is defined as:

$$Z\_m = \frac{2Z\_{0c}{}^b Z\_{0\pi}{}^b}{Z\_{0c}{}^b - Z\_{0\pi}{}^b} \tag{19}$$

The asymmetrical coupled line coupler presented in this study is a 3-dB coupler at center frequency of 3 GHz which is simulated on FR-4 substrate with 1.6 mm substrate thickness and a dielectric constant of 4.6. Impedances of all four ports have been considered equal to 50 50 Ω ==Ω ( ) *Z Z a b* . The final structure of designed coupler has been presented in Fig. 12(b) with 1 2 *W mm W mm* = = 0.6 , 1 and the spacing between two coupled lines (*s*) is 0.2 mm.

264 Trends in Electromagnetism – From Fundamentals to Applications

*<sup>a</sup> a b <sup>c</sup>*

<sup>−</sup> <sup>=</sup> <sup>−</sup>

*ZZ k <sup>Z</sup> Z k ZZ*

*a a b*

 <sup>−</sup> <sup>=</sup> +

*ZZ k <sup>Z</sup> Z k ZZ*

*ZZ k <sup>Z</sup> Z k ZZ*

*<sup>b</sup> a b <sup>c</sup>*

<sup>−</sup> <sup>=</sup> <sup>−</sup>

*b a b*

<sup>−</sup> <sup>=</sup> <sup>+</sup>

2 2 1 1 and *a b b a Z Z k k Z Z*

As it was mentioned, for indicated coupling level ( *k* ) and ports impedance ( , *Z Za b* ) in the

be determined using (17). It is clear from (15) that selecting a small *C*int in the introduced

that in this situation, the required spacing between two coupled-lines can be increased in comparison with the conventional microstrip coupled-lines. It is due to the inverse relationship between mutual capacitance value and spacing between coupled lines. Therefore, it is suitable for realizing high coupling-level coupled-line couplers with

Fig. 12 illustrates the layout and fabrication of the proposed asymmetrical coupler that

For an asymmetric coupled microstrip line of the type shown in Fig. 12, the design graphs presented in Figs. 13, 14 and 15 can be used to determine the necessary interdigital and microstrip strip widths and spacing for a given set of characteristic impedances,

> 0 0 0 0 2 *b b <sup>c</sup> <sup>m</sup> b b c Z Z <sup>Z</sup> Z Z*

<sup>=</sup> <sup>−</sup>

The asymmetrical coupled line coupler presented in this study is a 3-dB coupler at center frequency of 3 GHz which is simulated on FR-4 substrate with 1.6 mm substrate thickness and a dielectric constant of 4.6. Impedances of all four ports have been considered equal to 50 50 Ω ==Ω ( ) *Z Z a b* . The final structure of designed coupler has been presented in Fig. 12(b) with 1 2 *W mm W mm* = = 0.6 , 1 and the spacing between two coupled lines (*s*) is 0.2

π

π

characteristic impedances, i.e. 0 0 ,

*ZZ k <sup>Z</sup> Z k ZZ*

0

0

*<sup>a</sup> Z <sup>c</sup>* and 0

π

<sup>0</sup> , and *a b ZZ Z oc c m* on FR-4 substrate with 4.6 *<sup>r</sup>*

*<sup>a</sup> Z <sup>c</sup>* ′ and 0

In order the values of 0

proposed coupler, the *c* and

coupler, increases values of 0

defined as:

mm.

π

0

0

where <sup>2</sup> *k* denotes the power coupling coefficient between two coupled lines.

*<sup>a</sup> Z* π

relatively larger spacing between two lines than conventional coupled-line couplers.

ε

above considerations have been considered in its design (Keshavarz et al., 2011b).

π

2

1

*b ab*

1

1

*a ab*

*b ab*

1

*a ab*

*<sup>b</sup> Z <sup>c</sup>* to be positive, it is necessary that:

2

2

(17)

2

≥ ≥ (18)

*a a Z Z <sup>c</sup>* π, <sup>0</sup>

′ which can lead to smaller value for*Cm* . It means

= and thickness of 1.6 mm. In Fig. 15, *Zm* is

(19)

*<sup>b</sup> Z <sup>c</sup>* and <sup>0</sup>

*<sup>b</sup> Z* π, can

Fig. 12. Proposed asymmetrical backward coupler based on the interdigital and conventional microstrip coupled TLs. (a) Structure layout. (b) Fabricated coupler. (Keshavarz et al., 2011b).

Fig. 13. Design graph for width of the interdigital TL ( *W*<sup>1</sup> ) on FR-4 substrate versus *c* mode characteristic impedance. (Keshavarz et al., 2011b).

Fig. 14. Design graph for width of the conventional microstrip TL ( *W*<sup>2</sup> ) on FR-4 substrate versus *c* mode characteristic impedance. (Keshavarz et al., 2011b).

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 267

(b)

Fig. 16. Magnitude of the S-parameters for the proposed coupler obtained by full-wave simulation, equivalent circuit model and measurement results. (a) | |,| | *S S* 12 14 (b) | |,| | *S S* 11 13

Fig. 17. Phase difference between the through and the coupled ports for the proposed

11

 = 

*S*

*S*

0

0

12 13

<sup>=</sup>

14

The scattering parameters of an ideal forward-wave directional coupler, as shown in Fig. 3,

( ) 2

*j l*

<sup>−</sup> = −

β β

− +

*e o*

*<sup>l</sup> S je*

( ) cos[ ] <sup>2</sup>

β β

*e o*

(20)

( ) sin[ ] <sup>2</sup>

β β

*e o*

( ) 2

*j l*

β β

<sup>−</sup> = −

− +

*e o*

*<sup>l</sup> S je*

(Keshavarz et al., 2011b).

coupler of Fig. 12. (Keshavarz et al., 2011b).

**3.3.3 Forward symmetrical CLC** 

are given by (Mongia et al., 1999):

Fig. 15. Design graph for two lines separation ( *S* ) on FR-4 substrate versus *Zm* . (Keshavarz et al., 2011b).

So, this coupler is more compact than CRLH coupled line couplers reported in (Abdelaziz et al., 2009; Garcia-Perez et al., 2010; Joon-Boom et al., 2001), due to the elimination of the stubs. The structure coupled-line length ( ) is equal to 12 mm, which is approximately λg/4 at center frequency of 3 GHz and is smaller than the CRLH microstrip CLC with the coupled line length around λg/3 (Caloz et al., 2004).

In addition to the equivalent circuit model which is used to simulate the designed coupler, a full-wave electromagnetic simulator (ADS) is also used to examine the structure. Fig. 16 illustrates the full-wave and equivalent circuit model analysis results of the proposed asymmetric backward coupler along with its measured S-parameters. Excellent agreement can be observed between full-wave simulated and experimental results. The elements of the equivalent circuit model are obtained using equations (13) and (15) and for this example are equal to 1 *L nH* = 7.33 , <sup>1</sup> int <sup>2</sup> <sup>2</sup> *C pF C pF L nH C pF* = == = 0.7 , 1.82 , 6.18 , 0.86 . Using this figure, performance of the introduced 3-dB edge-coupled coupled-line coupler can be stated as the following: the power which is coupled to port 3 is approximately -3 dB, the return loss is less than -14 dB and the isolation is better than -13 dB over the bandwidth of 66% from 2.2 GHz to 4.2 GHz. Moreover, Fig. 17 shows the phase difference between the ports 2 and 3 of the coupler. As it is seen, this difference is equal to 90 ± 10° for a frequency range from 2.2 GHz to 3.5 GHz. Proposed asymmetrical backward coupler exhibits reachable dimension, broad bandwidth and smaller size than the conventional and CRLH couplers.

Fig. 16. Magnitude of the S-parameters for the proposed coupler obtained by full-wave simulation, equivalent circuit model and measurement results. (a) | |,| | *S S* 12 14 (b) | |,| | *S S* 11 13 (Keshavarz et al., 2011b).

Fig. 17. Phase difference between the through and the coupled ports for the proposed coupler of Fig. 12. (Keshavarz et al., 2011b).

#### **3.3.3 Forward symmetrical CLC**

266 Trends in Electromagnetism – From Fundamentals to Applications

Fig. 15. Design graph for two lines separation ( *S* ) on FR-4 substrate versus *Zm* . (Keshavarz

So, this coupler is more compact than CRLH coupled line couplers reported in (Abdelaziz et al., 2009; Garcia-Perez et al., 2010; Joon-Boom et al., 2001), due to the elimination of the stubs. The structure coupled-line length ( ) is equal to 12 mm, which is approximately λg/4 at center frequency of 3 GHz and is smaller than the CRLH microstrip CLC with the coupled

In addition to the equivalent circuit model which is used to simulate the designed coupler, a full-wave electromagnetic simulator (ADS) is also used to examine the structure. Fig. 16 illustrates the full-wave and equivalent circuit model analysis results of the proposed asymmetric backward coupler along with its measured S-parameters. Excellent agreement can be observed between full-wave simulated and experimental results. The elements of the equivalent circuit model are obtained using equations (13) and (15) and for this example are equal to 1 *L nH* = 7.33 , <sup>1</sup> int <sup>2</sup> <sup>2</sup> *C pF C pF L nH C pF* = == = 0.7 , 1.82 , 6.18 , 0.86 . Using this figure, performance of the introduced 3-dB edge-coupled coupled-line coupler can be stated as the following: the power which is coupled to port 3 is approximately -3 dB, the return loss is less than -14 dB and the isolation is better than -13 dB over the bandwidth of 66% from 2.2 GHz to 4.2 GHz. Moreover, Fig. 17 shows the phase difference between the ports 2 and 3 of the coupler. As it is seen, this difference is equal to 90 ± 10° for a frequency range from 2.2 GHz to 3.5 GHz. Proposed asymmetrical backward coupler exhibits reachable dimension,

(a)

broad bandwidth and smaller size than the conventional and CRLH couplers.

et al., 2011b).

line length around λg/3 (Caloz et al., 2004).

The scattering parameters of an ideal forward-wave directional coupler, as shown in Fig. 3, are given by (Mongia et al., 1999):

$$\begin{cases} S\_{11} = 0\\ S\_{12} = -je^{\frac{-j(\beta\_c + \beta\_o)l}{2}} \cos[\frac{(\beta\_c - \beta\_o)l}{2}]\\ S\_{13} = 0\\ S\_{14} = -je^{\frac{-j(\beta\_c + \beta\_o)l}{2}} \sin[\frac{(\beta\_c - \beta\_o)l}{2}] \end{cases} \tag{20}$$

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 269

*W* , *Z* is characteristic impedance of the transmission line and *c* is the speed of light. Also, *C*12 represents the capacitance between the two coupled lines without stubs and ground conductor. *C*int is capacitance per unit length of the interdigital capacitor formed between

(a)

(b) (c)

Some extra distributed shunt capacitance and inductance per unit length are added to the equivalent circuit models for the even and odd modes, respectively, which are given based

<sup>1</sup> ( ) ( tan ( )) 2 2

β

> ω

(24)

β

> ω

*<sup>s</sup>* represent characteristic impedance and phase constant of the shunt stubs,

*<sup>s</sup> <sup>s</sup> s ss a s s s*

β

ω

ω

*<sup>Z</sup> l s Z ls <sup>L</sup> d d ls ls <sup>C</sup> d Z Z d*

<sup>+</sup> <sup>+</sup> = ≈

*s ss a s*

β

+ + = ≈

Series impedance and shunt admittance of these equivalent circuit models in even and odd

1 1 ( ) ( tan ( )) 2 2

*s s s s*

Fig. 18. (a) Proposed forward-wave coupled-line coupler with periodic stubs. (b) Even mode, and (c) odd mode equivalent circuit models of each coupled line for one period

is effective permittivity of a microstrip transmission line with a strip with width

where *re* ε

the two coupled lines.

(Keshavarz et al., 2010).

where *Zs* and

modes are given by:

respectively.

on the TL theory as (Pozar, 2004):

β

where β *<sup>e</sup>* and β*<sup>o</sup>* are even and odd mode propagation constants of coupled lines, respectively. Also, *l* is length of the coupled line. As it was mentioned, forward-wave directional couplers cannot be realized using TEM mode transmission lines such as coaxial lines. It is due to this fact that for the TEM mode, the propagation constant of the even and odd modes are equal, and as shown in (20), there is no coupling between ports 1 and 4. Therefore, forward-wave coupling mechanism can only be appeared in non-TEM coupled TLs such as metallic waveguides, fin lines, dielectric waveguides and also quasi-TEM mode TLs like microstrip lines at high operating frequencies. In these transmission line structures, in general, the phase velocities of the even and odd modes are not equal (Mongia et al., 1999).

From (20), it is clear that complete power can be transferred between lines if the length *l* of the coupled line is chosen as:

$$d = \frac{\pi}{|\mathcal{B}\_e - \mathcal{B}\_o|}\tag{21}$$

Above result is significant in the sense that even for arbitrarily small values of difference in the propagation constants of even and odd modes, complete power can be transformed between the lines if the length of the coupler is chosen according to (21). In this situation, the directivity and isolation of the coupler are thus infinite. Also, the phase difference between ports 1 and 4 ( *S*<sup>41</sup> and *S*<sup>21</sup> ) is 90º. However, in general, situation (21) cannot be completely satisfied. Hence, some finite amount of backward-wave coupling always exists between coupled lines.

Our proposed forward-wave coupled-line coupler is shown in Fig. 18(a), where the coupledlines have the same width of *W* and periodic stubs have been loaded between these coupled-lines (Keshavarz et al., 2010). In this structure, *Ws* and *<sup>s</sup>* are the width and length of the periodic stubs, respectively, and *<sup>s</sup> d* is a period of the stubs. The mid plane (red line in Fig. 18(a)) between the coupled-lines remains two different equivalent circuits for the even and odd modes. The even and odd modes are associated with a magnetic wall (open-circuit) and an electric wall (short-circuit), respectively. These two equivalent circuit models have been presented in Figs. 18(b) and 18(c) for one period. In these circuits, *Ce* and *Co* are even and odd mode capacitances per unit length, respectively, and *L* is inductance per unit length of the coupled-lines. *Ce* and*Co* are equal to:

$$\mathbf{C}\_{\varepsilon} = \mathbf{C}\_{11} = \mathbf{C}\_{22}, \quad \mathbf{C}\_{o} = \mathbf{C}\_{11} + 2\mathbf{C}\_{12} + \mathbf{C}\_{\text{int}} \tag{22}$$

where*C*11 and *C*22 represent the capacitance between one strip conductor and ground in absence of the other strip conductor, in planar structures. Because of the strip conductors of the coupled lines are identical in size and location relative to the ground conductor, *C*11 will be equal to *C*22 or *C C* 11 22 = . From transmission line theory, it is well known that the value of *C*11 is (Pozar, 2004):

$$C\_{11} = \frac{\sqrt{\varepsilon\_{re}} \, Z}{c} \tag{23}$$

268 Trends in Electromagnetism – From Fundamentals to Applications

respectively. Also, *l* is length of the coupled line. As it was mentioned, forward-wave directional couplers cannot be realized using TEM mode transmission lines such as coaxial lines. It is due to this fact that for the TEM mode, the propagation constant of the even and odd modes are equal, and as shown in (20), there is no coupling between ports 1 and 4. Therefore, forward-wave coupling mechanism can only be appeared in non-TEM coupled TLs such as metallic waveguides, fin lines, dielectric waveguides and also quasi-TEM mode TLs like microstrip lines at high operating frequencies. In these transmission line structures, in general, the phase velocities of the even and odd modes are not equal (Mongia et al.,

From (20), it is clear that complete power can be transferred between lines if the length *l* of


Above result is significant in the sense that even for arbitrarily small values of difference in the propagation constants of even and odd modes, complete power can be transformed between the lines if the length of the coupler is chosen according to (21). In this situation, the directivity and isolation of the coupler are thus infinite. Also, the phase difference between ports 1 and 4 ( *S*<sup>41</sup> and *S*<sup>21</sup> ) is 90º. However, in general, situation (21) cannot be completely satisfied. Hence, some finite amount of backward-wave coupling always exists between

Our proposed forward-wave coupled-line coupler is shown in Fig. 18(a), where the coupledlines have the same width of *W* and periodic stubs have been loaded between these coupled-lines (Keshavarz et al., 2010). In this structure, *Ws* and *<sup>s</sup>* are the width and length of the periodic stubs, respectively, and *<sup>s</sup> d* is a period of the stubs. The mid plane (red line in Fig. 18(a)) between the coupled-lines remains two different equivalent circuits for the even and odd modes. The even and odd modes are associated with a magnetic wall (open-circuit) and an electric wall (short-circuit), respectively. These two equivalent circuit models have been presented in Figs. 18(b) and 18(c) for one period. In these circuits, *Ce* and *Co* are even and odd mode capacitances per unit length, respectively, and *L* is inductance per unit

where*C*11 and *C*22 represent the capacitance between one strip conductor and ground in absence of the other strip conductor, in planar structures. Because of the strip conductors of the coupled lines are identical in size and location relative to the ground conductor, *C*11 will be equal to *C*22 or *C C* 11 22 = . From transmission line theory, it is well known that the value

*re Z*

*c* ε

11

*C*

β β

π

*l*

*<sup>o</sup>* are even and odd mode propagation constants of coupled lines,

<sup>=</sup> <sup>−</sup> (21)

11 22 11 12 int , 2 *CC C CC C C e o* == =+ + (22)

= (23)

where

1999).

coupled lines.

of *C*11 is (Pozar, 2004):

β*<sup>e</sup>* and β

the coupled line is chosen as:

length of the coupled-lines. *Ce* and*Co* are equal to:

where *re* ε is effective permittivity of a microstrip transmission line with a strip with width *W* , *Z* is characteristic impedance of the transmission line and *c* is the speed of light. Also, *C*12 represents the capacitance between the two coupled lines without stubs and ground conductor. *C*int is capacitance per unit length of the interdigital capacitor formed between the two coupled lines.

Fig. 18. (a) Proposed forward-wave coupled-line coupler with periodic stubs. (b) Even mode, and (c) odd mode equivalent circuit models of each coupled line for one period (Keshavarz et al., 2010).

Some extra distributed shunt capacitance and inductance per unit length are added to the equivalent circuit models for the even and odd modes, respectively, which are given based on the TL theory as (Pozar, 2004):

$$\begin{split} L\_a &= \frac{1}{d\_s} (\frac{Z\_s}{a\nu} \tan \beta\_s (\frac{l\_s + s}{2})) = \frac{Z\_s \,\beta\_s (l\_s + s)}{2 \,\alpha d\_s} \\ C\_a &= \frac{1}{d\_s} (\frac{1}{a\nu Z\_s} \tan \beta\_s (\frac{l\_s + s}{2})) = \frac{\beta\_s (l\_s + s)}{2 \,\alpha Z\_s d\_s} \end{split} \tag{24}$$

where *Zs* and β*<sup>s</sup>* represent characteristic impedance and phase constant of the shunt stubs, respectively.

Series impedance and shunt admittance of these equivalent circuit models in even and odd modes are given by:

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 271

Fig. 20 presents some curves for selecting dimension of the proposed coupler for three coupling-levels (0-dB, 3-dB and 6-dB) with 0.2 , 0.6 *W mm d mm s s* = = and *S mm* = 0.2 on

coupling-level, dimension of the coupler increase. But, it is interesting to note that for a fixed coupling-level, the area of the coupler (product of the stub length by the structure length)

The proposed structure of the forward-wave CLC in this section is fabricated on FR4 substrate with 1.6 mm thickness and dielectric constant of 4.6, as shown in Fig. 21. The fullwave simulator Agilent Technologies Advanced Design System (ADS) is used to examine the structure. For good matching, the width of the microstrip transmission lines for 50Ω port impedances is selected equal to 1 mm (i.e. *W mm* = 1 ). To have a coupling level of 0-dB, according to the derived relations and Fig. 20, the length ( *l* ) and width ( 2 *sl W* + ) of the structure in Fig. 18 have been chosen equal to 26 mm and 4 mm, which are approximately

Fig. 20. Data for designing dimension of the proposed coupler on FR-4 substrate

Therefore, the proposed CLC is more compact than the microstrip coupler with the coupledline length around 0.75λg presented in (Fujii & Ohta, 2005). Also, the width ( *Ws* ) and period distance ( *<sup>s</sup> d* ) of the stubs are considered as: 0.2 , 0.6 *W mm d mm s s* = = and the space between

The measured and simulated S-parameters of the proposed coupler are shown in Fig. 22. This figure shows the measured amplitude balance of ±2 dB over a bandwidth of 66% (2-4 GHz). In this figure, full-wave simulation and equivalent circuit model results have also been presented for verification. A good agreement between measurement, full-wave simulation and equivalent circuit model results is obtained and thus the usefulness of the presented equivalent circuit model is validated. The element values of the equivalent circuit model (Fig. 18) for the layout are: 1.8 , 3.2 , 0.1 , *L nH L nH C pF* === *a a* 0.2 *C pF <sup>e</sup>* =

= = *h mm* ). These curves illustrate that with increasing the

FR-4 substrate ( 4.6, 1.6 *<sup>r</sup>* ε

will remain constant, approximately.

(ε

and 1.8 *C pF <sup>o</sup>* = .

λg/2 and λg/13 at center frequency of 3 GH, respectively.

*<sup>r</sup>* = 4.6, *h* = 1.6 *mm* ) (Keshavarz et al., 2010).

the stubs and transmission lines is 0.2 mm (i.e. *S mm* = 0.2 ).

$$\begin{aligned} Z\_e &= j \text{odL}\_\prime & Y\_e &= j \text{od}(\mathbf{C}\_e + \mathbf{C}\_a) \\ Z\_o &= j \text{odL}\_\prime & Y\_o &= j \text{od}\mathbf{C}\_o + 1/j \text{odL}\_a \end{aligned} \tag{25}$$

According to the TL theory, the propagation constants and the characteristic impedances of the transmission coupled-lines in even and odd modes are:

$$\begin{aligned} \mathcal{Y}\_e &= \sqrt{\mathcal{Z}\_e \mathcal{Y}\_e} = j\alpha \sqrt{\mathcal{L}(\mathcal{C}\_e + \mathcal{C}\_a)} = j\mathcal{B}\_e\\ \mathcal{Y}\_o &= \sqrt{\mathcal{Z}\_o \mathcal{Y}\_o} = j\alpha \sqrt{\mathcal{L}(\mathcal{C}\_o - 1/\alpha^2 \mathcal{L}\_a)} = j\mathcal{B}\_o \end{aligned} \tag{26}$$

and

$$\begin{aligned} Z\_{cc} &= \sqrt{\frac{Z\_c}{Y\_c}} = \sqrt{\frac{j\alpha \mathcal{L}}{j\alpha (\mathcal{C}\_c + \mathcal{C}\_a)}} = \sqrt{\frac{L}{(\mathcal{C}\_c + \mathcal{C}\_a)}}\\ Z\_{co} &= \sqrt{\frac{Z\_o}{Y\_o}} = \sqrt{\frac{j\alpha \mathcal{L}}{j\alpha (\mathcal{C}\_o - 1/\alpha^2 \mathcal{L}\_a)}} = \sqrt{\frac{L}{(\mathcal{C}\_o - 1/\alpha^2 \mathcal{L}\_a)}} \end{aligned} \tag{27}$$

Since, the length of the stubs is relatively large, the value of *C*12 would be very smaller than *C*<sup>11</sup> and *C*int . So, (22) can be approximated as:

$$\mathbf{C}\_o \equiv \mathbf{C}\_{11} + \mathbf{2}\mathbf{C}\_{\text{int}} \tag{28}$$

As it is seen in (26), the difference between β *<sup>e</sup>* and β*<sup>o</sup>* in proposed structure becomes larger than conventional structures without stubs in coupled line couplers. Moreover, this difference can be controlled by stub length, so that for a fixed coupling-level, increasing length of stubs ( *<sup>s</sup>* ) results reduction of structure length (Fig. 19).

In the coupled-line couplers, input matching condition for termination of impedance ( ) *ZZ Z c in c* = is achieved under condition which is given by (12).

Fig. 19.| | β β*e o* − for three lengths of the stubs ( 2,4 6 *sl and mm* = ) (Keshavarz et al., 2010).

270 Trends in Electromagnetism – From Fundamentals to Applications

*Z jL Y j C C Z j L Y j C j L*

= =+

*e ee ea e*

= = +=

ω

*<sup>Z</sup> j L <sup>L</sup> <sup>Z</sup>*

ω

ω

== = + +

*ZY j LC C j*

== − =

*e ea ea*

*Y jC C C C*

ω

= = <sup>=</sup> − −

*<sup>Z</sup> j L <sup>L</sup> <sup>Z</sup>*

ωω

ω

ω

γω

the transmission coupled-lines in even and odd modes are:

γ

*<sup>e</sup> ce*

*<sup>o</sup> co*

length of stubs ( *<sup>s</sup>* ) results reduction of structure length (Fig. 19).

( ) *ZZ Z c in c* = is achieved under condition which is given by (12).

*e o* − for three lengths of the stubs ( 2,4 6 *<sup>s</sup>*

*C*<sup>11</sup> and *C*int . So, (22) can be approximated as:

As it is seen in (26), the difference between

and

Fig. 19.| | β β

, () , 1 *e e ea o oo a*

( )

( )( )

*o o ao a*

*Y j C LC L*

Since, the length of the stubs is relatively large, the value of *C*12 would be very smaller than

β*<sup>e</sup>* and

than conventional structures without stubs in coupled line couplers. Moreover, this difference can be controlled by stub length, so that for a fixed coupling-level, increasing

In the coupled-line couplers, input matching condition for termination of impedance

ωω

2

 β

2 2

( 1 )( 1 )

β

(1 )

 ω

= =+ (25)

β

 ω

11 int 2 *CC C <sup>o</sup>* ≅ + (28)

*l and mm* = ) (Keshavarz et al., 2010).

*<sup>o</sup>* in proposed structure becomes larger

(26)

(27)

 ω

According to the TL theory, the propagation constants and the characteristic impedances of

*o oo o ao*

*ZY j LC L j*

Fig. 20 presents some curves for selecting dimension of the proposed coupler for three coupling-levels (0-dB, 3-dB and 6-dB) with 0.2 , 0.6 *W mm d mm s s* = = and *S mm* = 0.2 on FR-4 substrate ( 4.6, 1.6 *<sup>r</sup>* ε = = *h mm* ). These curves illustrate that with increasing the coupling-level, dimension of the coupler increase. But, it is interesting to note that for a fixed coupling-level, the area of the coupler (product of the stub length by the structure length) will remain constant, approximately.

The proposed structure of the forward-wave CLC in this section is fabricated on FR4 substrate with 1.6 mm thickness and dielectric constant of 4.6, as shown in Fig. 21. The fullwave simulator Agilent Technologies Advanced Design System (ADS) is used to examine the structure. For good matching, the width of the microstrip transmission lines for 50Ω port impedances is selected equal to 1 mm (i.e. *W mm* = 1 ). To have a coupling level of 0-dB, according to the derived relations and Fig. 20, the length ( *l* ) and width ( 2 *<sup>s</sup> l W* + ) of the structure in Fig. 18 have been chosen equal to 26 mm and 4 mm, which are approximately λg/2 and λg/13 at center frequency of 3 GH, respectively.

Fig. 20. Data for designing dimension of the proposed coupler on FR-4 substrate (ε*<sup>r</sup>* = 4.6, *h* = 1.6 *mm* ) (Keshavarz et al., 2010).

Therefore, the proposed CLC is more compact than the microstrip coupler with the coupledline length around 0.75λg presented in (Fujii & Ohta, 2005). Also, the width ( *Ws* ) and period distance ( *<sup>s</sup> d* ) of the stubs are considered as: 0.2 , 0.6 *W mm d mm s s* = = and the space between the stubs and transmission lines is 0.2 mm (i.e. *S mm* = 0.2 ).

The measured and simulated S-parameters of the proposed coupler are shown in Fig. 22. This figure shows the measured amplitude balance of ±2 dB over a bandwidth of 66% (2-4 GHz). In this figure, full-wave simulation and equivalent circuit model results have also been presented for verification. A good agreement between measurement, full-wave simulation and equivalent circuit model results is obtained and thus the usefulness of the presented equivalent circuit model is validated. The element values of the equivalent circuit model (Fig. 18) for the layout are: 1.8 , 3.2 , 0.1 , *L nH L nH C pF* === *a a* 0.2 *C pF <sup>e</sup>* = and 1.8 *C pF <sup>o</sup>* = .

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 273

(b) Fig. 22. Magnitude of the S-parameters, (a) 11 12 *S S*, (b) 13 14 *S S*, for the proposed coupler in Fig. 21 obtained by the full-wave simulation, equivalent circuit model and measurement

Fig. 23. Even and odd modes characteristic impedances of the coupler presented in Fig. 21

In this chapter, some new techniques for realizing compact and tight coupling microstrip backward and forward CLCs with obtainable dimension, broad bandwidth and smaller size than the most conventional microstrip and CRLH couplers have been introduced. We presented three CLCs based on the concept of CRLH CLCs; a symmetrical backward CLC,

New symmetrical backward coupler structure consists of only one interdigital capacitor in each coupled TL without shorted stubs as the CRLH TL. Designed and fabricated 3-dB microstrip coupler at center frequency about = 3.2 *cf* GHz exhibits a matching (10-dB) bandwidth of over 2 GHz, a phase-balance (*±*10º) bandwidth of 1.3 GHz and at least 20-dB isolation between adjacent ports. The coupled-line length and the width of the proposed

an asymmetrical backward CLC and a symmetrical forward CLC.

results (Keshavarz et al., 2010).

(Keshavarz et al., 2010).

**4. Conclusion** 

In comparison with the conventional forward CLCs, the electrical length of the proposed CLC is more compact than CLCs presented in (Chang et al., 2001; Deng et al., 2002; Lauro et al., 2009). For instance, the coupled-line electrical length of the coupler is shortened to 50% of the conventional CLC electrical length reported in (Deng et al., 2002). Moreover, the bandwidth of the proposed CLC is wider than forward CLCs presented in (Deng et al., 2002; Lauro et al., 2009; Chang et al., 2001; Sen-Kuei & Tzong-Lin, 2010). For example, compared with the forward couplers reported in (Deng et al., 2002) and (Lauro et al., 2009), the proposed structure is capable of producing 65% bandwidth enhancement for the amplitude and a 0-dB coupling level with a smaller coupled-line length. Moreover, the proposed structure exhibits broader bandwidth than couplers presented in (Chang et al., 2001; Huang & Chu, 2010; Ikalainen & Matthaei, 1987; Sen-Kuei & Tzong-Lin, 2010; Lauro et al., 2009).

Fig. 23 shows the even- and odd-mode characteristic impedances computed using full-wave simulation. This result indicates that the proposed structure is matched to 50 Ω port impedance over the operating bandwidth, such that the additional tapered structure at each port for impedance matching can be eliminated. Hence, the proposed forward coupler would be more compact in size. As it was mentioned, for the proposed forward CLC, the coupler area is approximately constant. It means that reduction of the structure length results width increasing, proportionally (Fig. 20).

Fig. 21. Proposed forward symmetrical coupler which realized on FR-4 substrate (ε*<sup>r</sup>* = 4.6, *h* = 1.6 *mm* ) (Keshavarz et al., 2010).

Fig. 22. Magnitude of the S-parameters, (a) 11 12 *S S*, (b) 13 14 *S S*, for the proposed coupler in Fig. 21 obtained by the full-wave simulation, equivalent circuit model and measurement results (Keshavarz et al., 2010).

Fig. 23. Even and odd modes characteristic impedances of the coupler presented in Fig. 21 (Keshavarz et al., 2010).
