He-Xiu Xu, Guang-Ming Wang, and He-Ping An

Metamaterials have shown great promise as substrates for compact RF and microwave filters. By forming composite right/left-handed (CRLH) transmission lines (TLs) on such materials, it is possible to take advantage of the dual-band properties of these structures by merit of their unique hyperbolic- linear relationship. By applying CRLH TLs in a Hilbert fractal-shaped geometry, it was possible to design a diplexer operating at 0.96 and 1.69 GHz and measuring just 33 x 33 mm. It features output-to-input isolation of better than 20.89 dB and insertion loss of less than 0.31 dB.

When left-handed (LH) bulk metamaterials were first fabricated in 2000,¹ their large dimensions, complexity, and high transmission losses did not appear promising for microwave applications. However, several researchers proposed different methods based on transmission-line theory for analyzing² and designing³ structures on LH metamaterials. Furthermore, researchers in ref. 4 proposed some general characteristics for CRLH TLs, including their capability for miniaturization and their dual-band, broadband, zero-order resonance nature.

Fractal theory has been attractive for the design of microwave components especially where miniature wideband requirements were critical. Hilbert fractal curves have been used in the miniaturization of superconducting filters,⁵ with a square Sierpinski fractal geometry employed in the design of complementary split ring resonators (CSRRs) to enhance the frequency selectivity to some degree.⁶

Conventional diplexers, which are used for connecting a receiver and transmitter to a common antenna, consist of bandpass and bandstop filters. They are typically large. A hybrid configuration is often used for classical diplexers, although it is also large, with high insertion loss and limited isolation. Diplexers based on CRLH TLs have been fabricated in multilayer configurations and have shown great promise in achieving miniaturization but without sacrificing performance.^7,8 In this report, the authors have developed a compact diplexer based on CRLH TLs using Hilbert fractal curves; the diplexer was formed by combining a pair of three-port networks operating at an arbitrary pair of frequencies (at 0.96 and 1.69 GHz).

A Hilbert fractal curve can be generated in an iterative fashion by using collinear transformations, as outlined in Fig. 1. This approach consists of forming a continuous line by connecting the centers of a uniform background grid. The fractal curve is fit in a square section defined with an external side, s. By increasing the iteration level of the curve, the space between lines diminishes accordingly and the length of the curve increases according to Eq. 1:

However, there is a tradeoff between the line spacing and miniaturization, as inadequate line spacing may result in reciprocal coupling between adjacent TLs. To avoid this, the diplexer detailed here was formed by uniting four Hilbert fractal curves of first iteration order (Fig. 2).

A lossless reciprocal three-port network has two key characteristics. First, its three ports cannot be matched simultaneously. Second, any two ports can be matched if the third port is allowed to be completely mismatched. In this work, the authors designed a diplexer at 0.96 and 1.69 GHz. The lower and higher operating frequencies of the diplexer are denoted as ω_L and ω_H, respectively. At ω_L, ports 1 and 2 are impedance matched, while port 3 is mismatched completely. Simultaneously, at ω_H, ports 1 and 3 are impedance matched, while port 2 is mismatched completely. For this circuit, the ideal scattering matrices, [S]_L and [S]_H can be described as shown in Eqs. 2 and 3, respectively:

Since a CRLH TL has a dual-band nature, the characteristics described above can be realized by one three-port CRLH TL network. For a balanced configuration, the CRLH TL can be decoupled into LH and right-handed (RH) subcircuits. The LH part can be realized by adding lumped elements, and the RH portion by adjusting microstrip lines. The total phase shift of the CRLH TL, f^CRLH, is the phase shift of the LH TL, f^LH and that of the RH TL, f^RH, described in Eq. 4, where N is the number of cells and L_R, L_L, C_R, and C_L are RH and LH equivalent lumped-element values.

The characteristic impedance of the CRLH TL, Z, should be determined from two times the termination impedance, Z₀, which yields Eq. 5:

The electrical dimensions (f_L, f_H) of the three transmission-line sections for a conventional three-port diplexer operating at ω_L and ω_H, respectively, have been calculated for comparison with the novel diplexer design. The novel CRLH TL-based diplexer should have exactly the same phase response and characteristic impedance as a conventional microstrip diplexer operating at ω_L and ω_H, respectively, as shown by Eqs. 6a and 6b:

Inserting Eq. 4 into Eq. 6 together with Eq. 5, the lumped-element component values needed for the diplexer are given by Eqs. 7a, 7b, 7c, and 7d.

With the values of L_R and C_R, the lengths of the RH microstrip lines can be calculated by Eq. 8:

However, L_R and C_R should have positive values, thus f_L and f_H should satisfy the following constraint condition, or f_L should be added to 2p and then inserted into Eq. 8 in order to calculate f^RH by means of Eq. 9:

The above processes can be repeated to find the values of the lumped elements for the other two TL sections. Since there is no need to load LH lumped elements into the TL between ports 2 and 3, the LH inductor and capacitor values for this TL section are zero (see table). A T-type circuit has been adopted here for the compact diplexer with the aim of more conveniently matching the input and output ports, so capacitor C_L should be multiplied by 2, as shown in Fig. 3.

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