**2. Composite Right/Left Handed Transmission Lines (CRLH TLs)**

A conventional transmission line (right-handed TL) is represented by a series inductance (*LR*) and a shunt capacitance (*CR*), implying the use of a low pass topology (Pozar, 2004). By interchanging the position of the inductor and capacitor, the resulting structure is referred to as left-handed TL with a high pass configuration (Caloz & Itoh, 2005). In these purely lefthanded transmission lines (PLH TLs), the phase and group velocities are opposite to each other (Pozar, 2004). PLH TLs cannot exist physically because, even if we intentionally provide only series capacitance and shunt inductance, parasitic series inductance (*LL*) and shunt capacitance (*CL*) effects, increasing with increasing frequency, will unavoidably occur due to currents flowing in the metallization and voltage gradients developing between the metal patterns of the trace and the ground plane (Caloz & Itoh, 2002). Thus, the composite right and left handed (CRLH) model represents the most general MTM structure possible. Equivalent circuit model of a CRLH TL for one cell is shown in Fig. 1(a) (Caloz & Itoh, 2004a). In this figure, *LR* and *LL* are right and left handed inductances, respectively, also *CR* and *CL* are right and left handed capacitances, respectively. Many lumped (using SMT chip components) or distributed implementations (microstrip, stripline, CPW, etc.) are possible for CRLH TLs. Interdigital/stub configuration is one of widely used of these implementations (Caloz & Itoh, 2005). A layout of this configuration for one cell is shown in Fig. 1(b). A CRLH TL is constructed of these unit cells connected in series as shown in Fig. 1(c). This structure consists of series interdigital capacitor of capacitance*CL* and parallel short-ended stub working as inductor of inductance *LL*. Moreover, *LR* and *CR* are parasitic elements of interdigital capacitor.

Fig. 1. (a) Equivalent circuit model of a CRLH TL for one cell. (b) Layout of a CRLH TL by using interdigital capacitor and shorted stub inductor for one cell. (c) Microstrip implementation of a CRLH TL.

An interdigital capacitor is a multifinger periodic structure which, as mentioned, can be used as a series capacitor in microstrip transmission lines technology (Bahl, 2003). This capacitor uses the capacitance that occurs across a narrow gap between thin-film conductors. Fig. 2 shows an interdigital capacitor and its equivalent circuit model. As seen in this figure, an interdigital capacitor is made of some gaps. The gap meanders back and forth in a rectangular area forming two sets of fingers that are interdigital. These gaps are essentially very long and folded to use a small amount of area. By using a long gap in a small area, compact single-layer small-value series capacitors can be realized. Typically, its capacitance values range from 0.05 pF to about 0.5 pF. The capacitance can be increased by increasing the number of fingers, or by using a thin layer of high dielectric constant material such as a ferroelectric between the conductors and the substrate (Bahl, 2003).

The value of series capacitance of an interdigital structure can be expressed as (Bahl, 2003):

$$C\_L = \frac{\varepsilon\_{re}'}{18\pi} \left(N - 1\right) \frac{K(\kappa)}{K'(\kappa)} l\_s \qquad \text{ ( $pF$ )}\tag{1}$$

where *re* ε′ is effective permittivity of a strip with width *W* , *N* is the number of fingers and ( ) ( ) *K k K k* ′ is a constant that has been presented in (Bahl, 2003).

As is well-known, the characteristic impedance of a CRLH TL ( *Zc* ) with equivalent circuit model of Fig. 1(a) is given by (Caloz & Itoh, 2005):

$$Z\_c = Z\_L \sqrt{\frac{\left(\frac{\partial}{\partial s\_{\rm se}}\right)^2 - 1}{\left(\frac{\partial}{\partial s\_{\rm sh}}\right)^2 - 1}}\tag{2}$$

where

252 Trends in Electromagnetism – From Fundamentals to Applications

A conventional transmission line (right-handed TL) is represented by a series inductance (*LR*) and a shunt capacitance (*CR*), implying the use of a low pass topology (Pozar, 2004). By interchanging the position of the inductor and capacitor, the resulting structure is referred to as left-handed TL with a high pass configuration (Caloz & Itoh, 2005). In these purely lefthanded transmission lines (PLH TLs), the phase and group velocities are opposite to each other (Pozar, 2004). PLH TLs cannot exist physically because, even if we intentionally provide only series capacitance and shunt inductance, parasitic series inductance (*LL*) and shunt capacitance (*CL*) effects, increasing with increasing frequency, will unavoidably occur due to currents flowing in the metallization and voltage gradients developing between the metal patterns of the trace and the ground plane (Caloz & Itoh, 2002). Thus, the composite right and left handed (CRLH) model represents the most general MTM structure possible. Equivalent circuit model of a CRLH TL for one cell is shown in Fig. 1(a) (Caloz & Itoh, 2004a). In this figure, *LR* and *LL* are right and left handed inductances, respectively, also *CR* and *CL* are right and left handed capacitances, respectively. Many lumped (using SMT chip components) or distributed implementations (microstrip, stripline, CPW, etc.) are possible for CRLH TLs. Interdigital/stub configuration is one of widely used of these implementations (Caloz & Itoh, 2005). A layout of this configuration for one cell is shown in Fig. 1(b). A CRLH TL is constructed of these unit cells connected in series as shown in Fig. 1(c). This structure consists of series interdigital capacitor of capacitance*CL* and parallel short-ended stub working as inductor of inductance *LL*. Moreover, *LR* and *CR* are parasitic

(a) (b)

(c) Fig. 1. (a) Equivalent circuit model of a CRLH TL for one cell. (b) Layout of a CRLH TL by

An interdigital capacitor is a multifinger periodic structure which, as mentioned, can be used as a series capacitor in microstrip transmission lines technology (Bahl, 2003). This capacitor uses the capacitance that occurs across a narrow gap between thin-film conductors. Fig. 2 shows an interdigital capacitor and its equivalent circuit model. As seen

using interdigital capacitor and shorted stub inductor for one cell. (c) Microstrip

**2. Composite Right/Left Handed Transmission Lines (CRLH TLs)** 

elements of interdigital capacitor.

implementation of a CRLH TL.

Fig. 2. (a) Interdigital capacitor. (b) Its equivalent circuit model.

According to Fig. 2, the equivalent circuit model of an interdigital capacitor is similar to the equivalent circuit model of one cell of CRLH TL when *LL* → ∞. Inserting *LL* → ∞ into (2) results the characteristic impedance ( int *Zc* ) of a TL consists of cascaded interdigital capacitors as:

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 255

Symmetrical coupled lines represent a very useful but restricted class of couplers. In many practical cases, it might be more useful or even necessary to design components using asymmetrical coupled lines. For example, in some situations, the terminal impedance of one of the coupled lines may be different from those of the other. It may then be more useful to choose two coupled lines with different characteristic impedances. Also, an asymmetrical coupled-line coupler has usually broader bandwidth than symmetrical one (Mongia et al.,

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

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

Fig. 4. Prototype of a CRLH edge-coupled directional coupler constituted of two

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

Fig. 3. Typical structure of a coupled-line coupler (CLC).

1999).

**3.2 CRLH CLCs** 

coupling-level.

constraints (Mongia et al., 1999).

interdigital/stub CRLH TLs.

$$\left| Z\_c^{\text{int}} = \sqrt{\frac{\left(\frac{\partial}{\partial \mathbf{o}\_{sc}}\right)^2 - 1}{\alpha \rho^2 \mathbf{C}\_L \mathbf{C}\_R}} = \sqrt{\frac{L\_R}{\mathbf{C}\_R} - \frac{1}{\alpha \rho^2 \mathbf{C}\_L \mathbf{C}\_R}} \tag{4}$$

It is seen from above equation that int *Zc* is real forω ω<sup>&</sup>gt; *se* . From TL theory, it is clear that *R R L <sup>C</sup>* is the characteristic impedance of a microstrip TL consists of a strip with width *W NW* ′ = − (4 1) .

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

$$\mathcal{J}^{\text{int}} = \sqrt{\mathcal{o}^2 L\_R C\_R - \frac{C\_R}{C\_L}} \tag{5}$$

So, in a transmission line composed of interdigital capacitors which can be named ''interdigital transmission line'', forω ω> *se* , the propagation constant int ( ) β is real and positive. It means that in this frequency interval, the interdigital transmission line operates in the right-handed (RH) band.
