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Composite right-/left-handed (CRLH) transmission lines have gained great interest in the microwave community. In practical applications, such CRLH sections realized by series and shunt resonators have a finite length. Starting from the observation that a high-order Chebyshev filter also exhibits a periodic central section of very similar structure, the relations between finite length CRHL transmission lines and Chebyshev filters are discussed in this paper. It is shown that a finite length CRLH transmission line in the balanced case is equivalent to the central part of a low-ripple high-order Chebyshev band-pass filter, and a dual-CRLH transmission line in the balanced case is equivalent to a low-ripple high-order Chebyshev band-stop filter. The nonperiodic end sections of a Chebyshev filter can be regarded as matching sections, thus leading to an even better amplitude and phase response. It is also shown that, equally to a CRHL transmission line, a Chebyshev filter exhibits negative phase velocity in part of its passband. As a consequence, an improved behavior of finite length CRLH transmission lines may be achieved adding matching sections based on filter theory; this is demonstrated by a simulation example.

In the past years, metamaterials [

Figure

Equivalent circuit models of two finite length negative refractive index transmission lines. (a) CRLH transmission line. (b) Dual-CRLH transmission line.

In a dual-CRLH transmission line, the series-branch and the shunt-branch resonators of a CRLH transmission line are interchanged resulting in a different type of negative refractive index transmission line [

A conventional filter, on the other hand, is generally not a uniform structure. However, looking at the filter coefficients

The starting points of our considerations were on the one hand, the periodic nature of the central part of Chebyshev filters, and on the other hand, practical problems of implementing finite length CRLH transmission lines into a (typically) 50 Ohm environment. Therefore, we wanted to find the relations between CRLH transmission lines and Chebyshev filters to design better CRLH transmission line sections with the help of classical filter theory.

Typically, CRLH transmission lines are considered as tools for dispersion engineering—this is in some way equivalent to frequency dependent phase control. Amplitude plays a minor role, although the practical implementation of a finite length section of a CRLH transmission line (and only this is feasible in reality) leads to problems with impedance match. On the other hand, filters are normally considered as components which control the amplitude performance of a signal while maintaining a good impedance match. Looking at many system applications, however, group delay (and therefore phase behavior as a function of frequency) plays an important role as well—there are even special filter design procedures strongly concentrating on group delay/phase behavior (e.g., Bessel or linear phase filters). This indicates closer relations between the two concepts than usually perceived.

In this paper, we deepen and extend the discussion in [

The equivalent circuit of a CRLH transmission line is a cascaded network as shown in Figure

Characteristic impedance of a CRLH transmission line in the balanced/unbalanced case.

Element values of 41st-order prototype Chebyshev lowpass filters with different passband ripples.

The balanced condition of a CRLH transmission line is

In the balanced case, the characteristic impedance of the CRLH transmission line is

There are four independent parameters, namely,

An

From (

Since the mapping from a prototype lowpass to a band-pass filter is a generic formula, it can be applied to any type of prototype lowpass filter. Thus, the balanced case of a CRLH transmission line is automatically realized in band-pass filters from

In most prototype lowpass filters, the element values

The element values of a 41st-order Chebyshev prototype lowpass filters with several different values of passband ripple are shown in Figure

When the element value is

In a high-order Chebyshev prototype lowpass filter with larger passband ripple, element values oscillate around

Once two cut-off frequencies and the system impedance are given, a uniform CRLH transmission line in the balanced case and a high-order Chebyshev band-pass filter with low passband ripple can be uniquely implemented with

We first consider a CRLH transmission line in the balanced case. The characteristic impedance is dependent on frequency as in (

When the passband ripple of a Chebyshev prototype lowpass filter is high, the element values of the central part are alternatively

Therefore, the central part of a high-ripple high-order Chebyshev band-pass filter is equivalent to a CRLH transmission line with mismatched impedance to the source and load.

When the external and the design impedances of a CRLH transmission line are different, matching circuits have to be applied. Even if both impedances are equal, some matching may be required to improve the overall passband performance. Since a CRLH transmission line is equivalent to the central part of a Chebyshev band-pass filter, the impedance matching circuits can be constructed from the corresponding band-pass filter design. This procedure is equivalent to the design of an entire filter and automatically results in good matching performance.

As an example, we have designed a CRLH transmission line of finite length in the balanced case, for example, with 21 series resonators and 20 shunt resonators according to the circuit model in Figure

Comparison of 10 dB bandwidths between CRLH transmission lines. (a) Conventional CRLH transmission line with 20 symmetric unit elements; (b) CRLH transmission line from 41st-order Chebyshev band-pass filter with 0.01 dB passband ripple; (c) CRLH transmission line with three resonators exchanged at either end.

The trade-off is a hybrid design, in which the periodic structure of the CRLH transmission line is applied, but several resonators near either end are exchanged with those from the corresponding band-pass filter, for example, a 41st-order Chebyshev band-pass filter with 0.01 dB passband ripple. The detailed procedure for our example is as follows: (a) design a 41st-order Chebyshev filter with 0.01 dB passband ripple and a CRLH transmission line with 41 periodic unit cells with the same cut-off frequencies and characteristic impedance, respectively; (b) choose the first and last three LC resonators from the Chebyshev filter, and place them into the CRLH transmission line at the corresponding positions. The simulated results of the hybrid design with three resonators exchanged at either end are shown in Figure

If the dual-CRLH transmission line is taken into consideration, a similar procedure can be applied. The passband ripple of a high-order Chebyshev band-stop filter corresponds to the mismatch between the characteristic impedance of a dual-CRLH transmission line and the system impedance.

In the case that even order Chebyshev filters are concerned, the load impedance, which is only dependent on the passband ripple and independent on the filter order, is always different from the source impedance. However, when the passband ripple is small, the load impedance is close to the source impedance. Therefore, there is not much difference between filter element values of even and odd order for low passband ripple and high order of the filter. On the other hand, when the passband ripple is high, the difference between even order and odd filters appears only in a limited number of elements near the load; these can be understood as an impedance matching circuit. Thus, with respect to even order Chebyshev filters, the analysis in this paper is also suitable.

Another point that we would like to stress here is the design flexibility of a CRLH transmission line. Between CRLH transmission lines and Chebyshev band-pass filters, there are three common parameters, that is, two cut-off frequencies and the matching impedance. Besides these parameters, there is one more parameter in the design of band-pass filters, that is, the passband ripple. It reflects the impedance mismatch between CRLH transmission lines and source/load. Once some passband ripple variation is allowed in a limited range, it gives more flexibility in the design of a CRLH transmission line.

Left-handed transmission lines support electromagnetic waves with phase and group velocities that are antiparallel to each other. The apparent backward wave propagation is one key characteristic of CRLH transmission lines. We will show that this does exist with Chebyshev band-pass filters in part of the frequency band due to a similar phase performance compared to CRLH transmission lines.

Group delay, which is an important parameter of a filter, is defined as

Group delay of a finite length CRLH transmission line with 20 symmetric unit elements, the correspondent 41st-order Chebyshev band-pass filter, and the modified CRLH transmission line with three resonators from the Chebyshev band-pass filter at either end.

When a band-pass filter is built from lumped/quasilumped elements, the dimensions of each element are much smaller than the wavelength in the passband of the filter. The phase shift along the filter can be calculated based on its equivalent

Phase shifts along a 41st-order Chebyshev band-pass filter from the lower cut-off frequency

This phase analysis shows the equivalence between a finite length transmission line and a Chebyshev filter also with respect to approaching a negative phase velocity, where the filter has a superior amplitude and group delay performance compared to conventional finite length CRLH transmission lines.

The characteristics of finite length CRLH transmission lines and high-order Chebyshev band-pass filters are analyzed, and synthesis formulas based on the matching impedance and cut-off frequencies are shown. From the analysis of element values in Chebyshev prototype lowpass filter, close relations between a CRLH transmission line and a Chebyshev band-pass filter are revealed. It is proven that a CRLH transmission line—in the balanced case—can be considered as the central part of a high-order low passband ripple Chebyshev band-pass filter with identical matching impedance and cut-off frequencies.

The meaning of the passband ripple in a Chebyshev band-pass, which has no obvious counterpart in a CRLH transmission line, corresponds to an impedance mismatch between the CRLH transmission line and the source/load. The formula to compute the passband ripple from the impedance mismatch is presented as well. In addition, impedance matching circuits in finite length CRLH transmission line applications can be design based on the classical filter theory, achieving much better performance.

Phase analysis, including group delay and phase distribution, shows that Chebyshev filters have similar and partly even better phase responses compared to conventional finite length CRLH transmission lines. Negative phase velocity is supported by Chebyshev filters in the same way as with CRHL transmission lines. This confirms the relation between Chebyshev filters and CRLH transmission lines.

Similar results can be obtained for dual-CRLH transmission lines (another kind of negative refractive index transmission lines) and Chebyshev band-stop filters.

Summarizing, this means that a finite length CRLH/dual-CRLH transmission line can be designed from a Chebyshev band-pass/band-stop filter. By allowing a reasonable passband ripple, there is even more design freedom. Based on impedance matching, the design from classical filter theory can achieve smooth broadband responses.

First examples for such an approach have already been presented in [

Finally, we would like to state that we do not at all want to replace the CRLH transmission line concept by filter theory. The CRLH transmission line concept is very useful to get new insight and to arrive at new design concepts, mainly by phase/dispersion engineering which is generally not at all evident from filter theory, but filter theory may help to improve its performance in the case of finite CRHL transmission line length.

An

The element values are

In the beginning part of the prototype lowpass filter (

The element values can be approximated by the infinite products of the sine and cosine functions and the Wallis' series. Then the element values are

Therefore, for a high-order low-ripple (

On the other hand, when the passband ripple is high, that is,

For a high-order Chebyshev prototype lowpass filter, that is,

Thus the sequence of element values keeps

For example, we show (Figure