An application of the T‐chart for solving problems associated with terminated finite lossless periodic structures

Abstract

Periodic structures of transmission lines have several applications in microwave technology. Due to complicated equations involved with terminated finite lossless periodic structures, a graphical solution assisting in the analysis and design of these structures is necessary. In this study, the T-chart is employed to solve problems associated with these structures, operating in the passband, readily and effectively, especially for unsymmetrical unit cells associated with periodic structures. Only the periodic structures with passive effective characteristic impedances are considered in this paper. It is found that the T-chart provides accurate solutions, compared to analytical solutions of termi- 594MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. [4]. Using the ABCD matrix technique, it can be shown that the equation of the input impedance at each terminal of terminated finite lossless periodic structures is in the same form as that of CCITLs. Due to complicated equations involved with these structures, a graphical solution assisting in analysis and design of these structures is necessary. In this study, the T-chart developed for CCITLs is employed to solve problems associated with terminated finite lossless periodic structures. It is found that the T-chart depends on the phase angle associated with the effective characteristic impedances of periodically loaded transmission lines. It will be shown that the effective characteristic impedances and the effective propagation constant are dependent on various parameters of each unit cell of the periodic structures in a complicated fashion. Once these effective parameters of these structures are known, the theory of CCITLs and the T-chart can be employed to simplify these problems. This paper is organized as follows. Section 2 presents the theory of terminated finite lossless periodic structures in brief. The theory of CCITLs is discussed in section 3. Section 4 illustrates T-Chart solutions of periodic structures. Finally, conclusions are provided in section 5. THEORY OF TERMINATED FINITE LOSSLESS PERIODIC STRUCTURESThis section provides relevant background information of terminated finite lossless periodic structures for developing its graphical solution. It will be useful when applying the T-chart for solving these problems. Figure 1 illustrates a model of an unsymmetrical unit cell associated with lossless periodic structures. The unit cell consists of a length d of reciprocal lossless uniform transmission line loaded with lossless lumped elements (C 1 , L, C, and L 1 ) across the midpoint of the line. Note that this unit cell is similar to that in [1] (see Fig. 1 in [1]), which exhibits NRI as well as negative group delay. The characteristic impedance and the propagation constant of the unloaded transmission line are defined as Z 1 and k, respectively. Assuming operation in the passband, it is found that at the left terminal of the n th unit cell of terminated finite periodic structures, as shown in Figure 2, the phasor voltage and current can be express...

Equivalent Graphical Solutions of Terminated Conjugately Characteristic-Impedance Transmission Lines With Non-Negative and Corresponding Negative Characteristic Resistances

“…Using the ABCD matrix technique, Z ± 0 can be expressed in terms of the total ABCD parameters of the unit cell of interest as [4] …”

“…As shown in [4], the input impedance Z in,M (see Fig. 1(a)) of the terminated finite periodic TL structure of M unit cells is given by…”

“…One class of transmission lines (TLs), called conjugately characteristicimpedance transmission lines (CCITLs), has been introduced recently in the literature [1][2][3][4][5][6][7]. By definition, a CCITL possesses conjugate characteristic impedances Z ± 0 of waves propagating in the opposite directions along the transmission line.…”

“…Examples of CCITLs are reciprocal lossless uniform TLs, nonreciprocal lossless uniform TLs [8][9][10], exponentially tapered lossless nonuniform TLs [2,11,12] and periodically loaded lossless TLs operated in passband [13][14][15][16][17][18]. Using the ABCD matrix technique, it can be shown that the equation of the input impedance at each terminal of loaded finite lossless periodic structures is in the same form as that of CCITLs [4]. Thus, CCITLs can be conveniently implemented using periodically loaded lossless TLs in practice.…”

Non-Uniqueness of T-Charts for Solving Ccitl Problems With Passive Characteristic Impedances

“…Conjugately characteristic-impedance transmission lines (CCITLs) have found various applications in microwave technology [1][2][3][4][5][6][7]. Examples of CCITLs are reciprocal lossless uniform transmission lines (TLs), nonreciprocal lossless uniform TLs, exponentially tapered lossless nonuniform TLs [3,8] and periodically loaded lossless TLs operated in passband [10][11][12][13][14][15].…”