**3.2 CRLH CLCs**

254 Trends in Electromagnetism – From Fundamentals to Applications

1 *se* <sup>1</sup> *<sup>R</sup> <sup>c</sup>*

2 2

*<sup>C</sup>* is the characteristic impedance of a microstrip TL consists of a strip with width

int 2 *R R R*

So, in a transmission line composed of interdigital capacitors which can be named

positive. It means that in this frequency interval, the interdigital transmission line operates

Coupled line couplers are indispensable components in radio frequency (RF)/microwave communication systems. In these structures two unshielded transmission lines are close together, as indicated typically in Fig. 3, and power can be coupled between the lines. Such lines are referred to as coupled transmission lines (Mongia et al., 1999). The coupler is frequently utilized in a variety of circuits including modulators, balanced amplifiers, balanced mixers, and phase shifters. Rapidly expanding applications such as modern wireless technology continue to challenge couplers with extremely stringent requirements—

In general, two types of CLCs have been proposed; backward and forward CLCs. When the coupled port is located on the same side of the structure as the input port and power is subsequently coupled backward to the direction of the source, this coupler is conventionally called a backward coupler and otherwise the CLC is called forward coupler (Mongia et al.,

On the other hand, two types of edge-coupled backward CLCs have been presented. The first is the symmetrical coupler. When the two lines constituting a CLC are the same, the structure is called symmetric. In the symmetric structures, coupling mechanism is based on the difference between the characteristic impedances of the even and odd modes. The second one is the asymmetrical coupler. This coupler is asymmetrical as it is constituted of two different transmission lines. In this case, decomposition in even and odd modes is not possible anymore. The analysis becomes more difficult and the even/odd modes have to be replaced by the more general c and π modes, which are two fundamental independent

 ω *<sup>C</sup> L C C*

*L R R L R*

ω ω

 ω

*L*

> *se* , the propagation constant int ( )

= =− (4)

= − (5)

<sup>&</sup>gt; *se* . From TL theory, it is clear that

β

is real and

*C C C C C*

2

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

Similarly, the propagation constant for this TL is obtained as (Caloz & Itoh, 2005):

β

ω ω

high performance, broad bandwidth, and small size (Pozar, 2004).

ω

 −

ω

ω

int

It is seen from above equation that int *Zc* is real for

''interdigital transmission line'', for

**3. Coupled-Line Couplers (CLCs)** 

modes, as described in (Mongia et al., 1999).

in the right-handed (RH) band.

**3.1 Conventional CLCs** 

*R R L*

1999).

*W NW* ′ = − (4 1) .

The conventional CLC has several intrinsic drawbacks. First, their operating bandwidths are usually limited. Second, to raise the coupling level of a coupler, a very small space between the coupled lines is required and it is usually difficult to obtain due to fabrication constraints (Mongia et al., 1999).

As mentioned, in the past few years there has been a great interest in the field of metamaterials, especially composite right/left-handed structures (e.g. interdigital/stub configurations), and the microwave circuits based on the unusual properties of them (Caloz & Itoh, 2005). By closely placing two identical CRLH lines in parallel, such as configuration shown in Fig. 4, a strong contrast exists between the impedances of two fundamental modes of propagation (i.e. the even and odd mode impedances), which would result in high coupling-level.

Fig. 4. Prototype of a CRLH edge-coupled directional coupler constituted of two interdigital/stub CRLH TLs.

For the first time, a novel composite right/left-handed coupled-line directional coupler composed of two CRLH TLs was proposed in (Caloz et al., 2004) and an even/odd-mode

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 257

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):

sin ,

<sup>−</sup> = = <sup>+</sup> − +

other hand, selection of *l= λg/*4 preserves the homogeneity condition in CRLH structure

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*

Even and odd mode characteristic impedances ( , *Z Z ce co* ) of the coupled-lines composed of

1 1 , *ce co*

, *ce co*

(a) (b)

, thickness of 1.6 mm. (a) Structure layout. (b) Fabricated coupler

Fig. 5. Structure of the proposed microstrip coupled-line backward coupler on FR4

*L L Z Z*

*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

2 2

*e o e L o L*

*e o*

 ω*C C C C* = − = − (7)

*C C* ′ ′ = = (8)

*k j Z Z*

*l/λg)* is electrical length and is the length of CLC. Therefore, setting the

 θ

*jk Z Z S k*

θ

*ce co ce co*

*/2* results in maximum coupling level. On the

(6)

1 cos sin

θ

θ*=*π

<sup>31</sup> <sup>2</sup>

≤ , where *p* is structural cell size) (Caloz & Itoh, 2005).

are the distributed capacitances for the even and odd modes, respectively.

ω

interdigital TLs are obtained from (Caloz & Itoh, 2005) with setting *LL* → ∞ as:

*L L Z Z C C*

where,

(i.e., <sup>4</sup> *<sup>g</sup> p* λ

and

interdigital capacitor.

substrate, = 4.7 *<sup>r</sup>* ε

(Keshavarz et al., 2011a).

θ*=(2*π

interdigital capacitor length as *l= λg/*4 or

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

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 coupled lines (Caloz & Itoh, 2005).

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 procedure (Caloz & Itoh, 2005).

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., 2010) are 25% and 30%, respectively.

Also, forward coupling level in CRLH coupled line couplers is low (nearly -10 dB in (Fouda et al., 2010)).
