Extended Composite Right/Left-Handed Transmission Line and Dual-Band Reactance Transformation

Abstract

An extended composite right/left-handed transmission line is introduced, and its dual-band bandpass filter characteristics are explored. Novel reactance transformations, derived from this transmission line, are formulated to transform a low-pass prototype filter into a dual-band bandpass filter with arbitrary dual pass bands, well-defined in-band attenuation ripples, and high out-of-band rejection. The physical insight into such a dual-band bandpass filter is provided with a dispersion analysis. The transformations are verified by simulated results for dual-band bandpass filters.

1. Introduction

An extended composite right/left-handed (CRLH) transmission line is first introduced. Then, its fundamental properties are investigated. Consequently, a reactance transformation is derived for dual-band filter synthesis.

Although a dual-band bandpass filter design using the reactance transformation stems from the extended CRLH transmission line concept, it is totally different from the approach used by Tseng and Itoh [1]. In their approach, a traditional bandpass filter is first designed using quarterwave short circuited; then the dual-band bandpass filter with arbitrary two pass bands is implemented by replacing the microstrip lines with the CRLH transmission lines; the separation of these two pass bands is determined by the nonlinear phase slope of the CRLH tansmission lines. In our approach, a filter with arbitrary dual pass bands is directly transformed from a low-pass filter prototype; then the dual-band bandpass filter can be realized using traditional or metamaterial transmission lines; the separation of the two pass bands is precisely controlled by the reactance transformation.

In addition, Guan et al. reported dual-band bandpass filters employing two successive frequency transformations [2]. In their approach, the dual-band bandpass filter is also similar to the extended CRLH transmission line. The center frequencies of two pass bands are used to define frequency transformations; that is, the two pass bands are positioned by the center frequencies and their boundaries cannot be accurately specified. Additionally, their frequency transformations can only be used in a narrow band filter. However, our reactance transformation is parameterized by edge frequencies, and the bands are precisely delimited by them. Also, our reactance transformation can be used in an arbitrary band filter.

2. Extended CRLH Transmission Line

Figure 1 shows the equivalent circuit model for an extended CRLH transmission line. It consists of the series resonator comprised of 𝐿₁ and 𝐶₁ (Resonator 1), the series shunt resonator comprised of 𝐿₃ and 𝐶₃ (Resonator 3), the shunt shunt resonator comprised of 𝐿₂ and 𝐶₂ (Resonator 2), and the shunt series resonator comprised of 𝐿₄ and 𝐶₄ (Resonator 4). The elimination of Resonators 3 and 4 reduces the extended CRLH transmission line into a CRLH transmission line [3]. The extended CRLH transmission line becomes a dual CRLH transmission line [4] without Resonators 1 and 2.

Figure 1 

A lumped-element model unit cell of an extended CRLH transmission line. {𝑑 is the length of the unit cell. 𝐿1, 𝐿2, 𝐿3, and 𝐿4 are the inductances in henries/m. 𝐶1, 𝐶2, 𝐶3, and 𝐶4 are the capacitances in farads/m.}

This paper focuses only on an extended CRLH transmission line that satisfies two conditions. The first condition is that the four resonators in Figure 1 have the same resonant frequency

𝜔²₀=1𝐿₁𝐶₁=1𝐿₃𝐶₃=1𝐿₂𝐶₂=1𝐿₄𝐶₄, (1) where 𝜔₀ is the “center frequency.” The series impedance 𝑍 and shunt admittance 𝑌 in Figure 1 are

𝑍=𝑗𝜔𝐿₁+1𝑗𝜔𝐶₁+1𝑗𝜔𝐶₃+1/𝑗𝜔𝐿₃, (2)𝑌=𝑗𝜔𝐶₂+1𝑗𝜔𝐿₂+1𝑗𝜔𝐿₄+1/𝑗𝜔𝐶₄. (3) The second condition requires that the resonant frequencies of 𝑍 and 𝑌 coincide at 𝜔^′₀:

(𝜔^′₀)²=1𝐿₁𝐶₃=1𝐿₄𝐶₂. (4) If either of the two conditions is not satisfied, three or more pass bands can occur.

2.1. Auxiliary CRLH Transmission Line

Here, an auxiliary CRLH transmission line is introduced to simplify the analysis of the extended CRLH transmission line. The auxiliary CRLH transmission line is comprised of the elements 𝐿₁, 𝐶₃, 𝐿₄, and 𝐶₂, as shown in the inset of Figure 1. From (4), it is known that the series and shunt resonant frequencies coincide at 𝜔^′₀. The high-pass cutoff frequency 𝜔^′_UPPER and low-pass cutoff frequency 𝜔^′_LOWER are expressed as follows [5]:

(𝜔^′_UPPER)²=12{2(𝜔^′₀)²+(𝜔^′_𝑅)²+[2(𝜔^′₀)²+(𝜔^′_𝑅)²]²−4(𝜔^′₀)⁴}=22/(𝜔^′₀)²+1/(𝜔^′_𝐿)²−[2/(𝜔^′₀)²+1/(𝜔^′_𝐿)²]²−4/(𝜔^′₀)⁴,(𝜔^′_LOWER)²=12{2(𝜔^′₀)²+(𝜔^′_𝑅)²+[2(𝜔^′₀)²+(𝜔^′_𝑅)²]²−4(𝜔^′₀)⁴}=22/(𝜔^′₀)²+1/(𝜔^′_𝐿)²+[2/(𝜔^′₀)²+1/(𝜔^′_𝐿)²]²−4/(𝜔^′₀)⁴, (5) where (𝜔^′_𝑅)²=4/𝐿₁𝐶₂ and (𝜔^′_𝐿)²=1/4𝐿₄𝐶₃. From (5), three frequency relationships are obtained as follows:

𝜔^′_𝑅=𝜔^′_UPPER−𝜔^′_LOWER, (6)1𝜔^′_𝐿=1𝜔^′_LOWER−1𝜔^′_UPPER, (7)𝜔^′_UPPER𝜔^′_LOWER=𝜔^′_L𝜔^′_R=(𝜔^′₀)². (8)

2.2. Dispersion of the Extended CRLH Transmission Line

The dispersion analysis for the extended CRLH transmission line is presented here. Using the 𝐴𝐵𝐶𝐷 matrix of the two-port network described in Figure 1 and the Bloch-Floquet theorem, the propagation constant 𝛾 is determined by

1𝛾=𝛼=𝑑arccos(𝑍𝑌+22),if−4<𝑍𝑌<0, (9)𝑗𝛾=𝑗𝛽=𝑑arccosh(𝑍𝑌+22),if𝑍𝑌<−4or𝑍𝑌>0, (10) where 𝑍 and 𝑌 are defined in (2) and (3), respectively. Equation (9) corresponds to an evanescent wave whereas (10) represents a propagating wave. 𝑍𝑌 can be written as the follows: 𝑤𝑍𝑌=−4(𝜔^′_𝑅−𝜔^′_𝐿𝑤)², (11) where 𝑤=𝜔₀(𝜔𝜔₀−𝜔₀𝜔). (12)𝜔^′_𝑅 and 𝜔^′_𝐿 are defined in (6) and (7), respectively. 𝑍𝑌=0 gives two resonant frequencies: 𝜔_ResUpper=12[4𝜔²₀+(𝜔^′₀)²+𝜔^′₀𝜔],_ResLower=12[4𝜔²₀+(𝜔^′₀)²−𝜔^′₀]. (13)𝑍𝑌=−4 yields four cutoff frequencies: 𝜔_𝑈𝑈=12[4𝜔⁴₀+(𝜔^′_UPPER)²+𝜔^′_UPPER𝜔],_𝑈𝐿=12[4𝜔⁴₀+(𝜔^′_LOWER)²+𝜔^′_LOWER𝜔],_𝐿𝑈=12[4𝜔⁴₀+(𝜔^′_LOWER)²−𝜔^′_LOWER𝜔],_𝐿𝐿=12[4𝜔⁴₀+(𝜔^′_UPPER)²−𝜔^′_UPPER]. (14) These frequencies are related as follows: 𝜔²₀=𝜔_ResUpper𝜔_ResLower=𝜔_𝑈𝑈𝜔_𝐿𝐿=𝜔_𝑈𝐿𝜔_𝐿𝑈. (15) Equations (5) and (8) can be used to show that 𝜔^′_UPPER>𝜔^′₀>𝜔^′_LOWER. Thus, it follows from (13) and (14) that 𝜔_𝑈𝑈>𝜔_ResUpper>𝜔_𝑈𝐿>𝜔₀>𝜔_𝐿𝑈>𝜔_ResLower>𝜔_𝐿𝐿. (16)

Based upon (9) and (10), the dispersion of the extended CRLH transmission line is plotted in Figure 2(a). It is noted that the four cutoff frequencies define key features of the dual passband behavior. 𝜔_𝐿𝐿 and 𝜔_𝐿𝑈 are the lower and upper edges of the lower pass band, respectively. 𝜔_𝑈𝐿 and 𝜔_𝑈𝑈 are the lower and upper edges of the upper pass band, respectively. A wave exhibits phase-backward propagation when the operating frequency is between 𝜔_𝐿𝐿 and 𝜔_ResLower, or between 𝜔_𝑈𝐿 and 𝜔_ResUpper. However, a wave exhibits phase-forward propagation when the operating frequency is between 𝜔_ResLower and 𝜔_𝐿𝑈, or between 𝜔_ResUpper and 𝜔_𝑈𝑈. Otherwise, the wave is evanescent. It is noteworthy that 𝜔_ResLower and 𝜔_ResUpper are transition points between forward and backward waves. The performance between 𝜔_ResLower and 𝜔_ResUpper is similar to that of a dual CRLH transmission line [4] whereas the performance below 𝜔_ResLower or greater than 𝜔_ResUpper is similar to that of a CRLH transmission line [3, 5].

Figure 2 

The dispersion (a) and equivalent circuits (b) of the extended CRLH transmission line (𝛽 and 𝛼 are defined in (9) and (10).)

Figure 2(b) represents the equivalent circuit behavior of the extended CRLH transmission line at different frequencies. When the operating frequency is less than 𝜔_ResLower, the extended CRLH transmission line is equivalent to a pure left-handed transmission line. Thus, the high-pass cutoff frequency 𝜔_𝐿𝐿 appears. When the operating frequency is between 𝜔_ResLower and 𝜔₀, the extended CRLH transmission line is equivalent to a pure right-handed transmission line with the low-pass cutoff frequency 𝜔_𝐿𝑈. When the operating frequency is between 𝜔₀ and 𝜔_ResUpper, the extended CRLH transmission line is equivalent to a pure left-handed transmission line with the high-pass cutoff frequency 𝜔_𝑈𝐿. When the operating frequency is greater than 𝜔_ResUpper, the extended CRLH transmission line is equivalent to a pure right-handed transmission line with the low-pass cutoff frequency 𝜔_𝑈𝑈.

2.3. Important Frequency Expressions

Using (6), (7), and (14), 𝜔^′_𝑅 and 𝜔^′_𝐿 are rewritten by

𝜔^′_𝑅=(𝜔_𝑈𝑈−𝜔_𝐿𝐿)−(𝜔_𝑈𝐿−𝜔_𝐿𝑈), (17)1𝜔^′_𝐿=1(𝜔_𝑈𝐿−𝜔_𝐿𝑈)−1(𝜔_𝑈𝑈−𝜔_𝐿𝐿). (18) Equations (17), (18), and (15) are important expressions that are used in Section 3 to transform frequency and elements.

It should be noted that all formulations presented in Section 2 are derived from the infinite extended CRLH transmission line. In practice, no transmission line is infinite. However, it has been verified in [3] that the infinite-structure approximation provides reasonable accuracy when a sufficiently large number (>3) cells are used.

3. Dual-Band Reactance Transformation

In this section, the dual-band frequency and element transformations are derived. The frequency transformation is a reactance function and converts a low-pass prototype filter with well-defined insertion loss [6] to a dual-band bandpass filter. Simulation results are presented to verify the transformations.

3.1. Dual-Band Bandpass Frequency Transformations

The desired frequency transformation is obtained by replacing the reactance and the susceptance in the prototype filter by series and shunt resonators such that

𝑗Ω𝐿=𝑍, (19)𝑗Ω𝐶=𝑌, (20)

where Ω, 𝐿 and 𝐶 are the normalized frequency, inductance, and capacitance of the low-pass prototype filter. 𝑍 and 𝑌 are defined in (2) and (3), respectively. Multiplying the right and left sides of (19) and (20), respectively, the dual-band bandpass frequency transformation is obtained as follows: Ω²𝑤=(𝜔^′_𝑅−𝜔^′_𝐿𝑤)², (21) where the definitions of 𝑤, 𝜔^′_𝑅 and 𝜔^′_𝐿 are given in (12), (17) and (18), respectively. The normalized low-pass cutoff frequency, Ω²_𝑅=4/𝐿𝐶=1, of the pure right-handed transmission line is used for the derivation of (21). With the substitution of 𝑤, the explicit expression of 𝑤/𝜔^′_𝑅 shows it is a low-pass to bandpass transformation with the passband bandwidth of 𝜔^′_𝑅 [6]. 𝜔^′_𝐿/𝑤 is a low-pass to bandstop transformation with the stopband bandwidth of 𝜔^′_𝐿 [6]. The frequency mappings of (21) are depicted in Figure 3. It is observed that the subtraction of the low-pass to bandstop mapping from the low-pass to bandpass mapping gives the low-pass to dual-band bandpass mapping.

Figure 3 

Mapping from normalized frequency of a prototype low-pass filter into actual filter frequency. (The solid line, plotted by Ω=𝑤/𝜔𝑅−𝜔𝐿/𝑤, is for the dual-band bandpass filter. The circle and dot lines describe the bandstop transformation with minus sign, Ω=−𝜔𝐿/𝑤, and the bandpass transformation, Ω=𝑤/𝜔𝑅, resp. Here, 𝑤=𝜔0(𝜔/𝜔0−𝜔0/𝜔).)

3.2. The Element Transformations

Substituting Ω with (21) in (19) and (20), and using (1), the element transformations are obtained as follows:

𝐿₁=𝐿𝜔^′_𝑅,𝐶₁=1𝐿₁𝜔²₀,𝐶₂=𝐶𝜔^′_𝑅,𝐿₂=1𝐶₂𝜔²₀,𝐶₃=1𝐿𝜔^′_𝐿,𝐿₃=1𝐶₃𝜔²₀,𝐿₄=1𝐶𝜔^′_𝐿,𝐶₄=1𝐿₄𝜔²₀. (22) It is observed that 𝐿₁, 𝐶₁, 𝐿₂, and 𝐶₂ constitute a bandpass filter with the center frequency 𝜔₀ and passband bandwidth of 𝜔^′_𝑅. 𝐿₃,𝐶₃,𝐿₄, and 𝐶₄ form a band stop filter with center frequency 𝜔₀ and a stop band bandwidth of 𝜔^′_𝐿.

3.3. Simulations

Dual-band bandpass filter, specified by the band edge frequencies of 0.9 GHz, 1.1 GHz, 1.718 GHz, and 2.1 GHz, was transformed from a Chebyshev low-pass filter with 1-dB passband ripple using (22). Figure 4 shows simulated attenuation responses of lumped-element dual-band bandpass filters of orders 3, 4, and 5. The lower and upper pass bands have the same ripple magnitude. The attenuation tends to infinity at the center frequency 𝜔₀=1.375GHz. The observed pass band center frequencies, 1.01GHz and 1.871 GHz, agree with the computed 𝜔_ResLower and 𝜔_ResUpper, respectively.

Figure 4 

Simulated attenuation responses of the dual-band bandpass filters transformed from 1-dB Chebyshev low-pass filters for filter orders of 3, 4, and 5. (The response of the lower and upper pass bands in the upper subplot is amplified in the left and right bottom subplots, resp.)

4. Conclusion

The extended CRLH transmission line has been introduced and characterized. Novel dual-band bandpass reactance transformations were derived. A dual structure of the extended CRLH transmission line can be explored using an approach similar to that described here. A dual-band bandstop frequency transformation can be obtained by replacing Ω by −1/Ω.