### I. Introduction

^{th}-mode resonance frequency and its BW of N unit cell CRLH TL has been conducted as far as we know.

^{th}mode resonance frequency to show the correlation between the BW and the dispersion curve slope, this study provides the guideline for easily designing a resonance antenna.

### II. Resonance Mechanism of the Negative/Positive n^{th} Mode of the N Unit Cell CRLH TL

^{th}mode for N cells with an open boundary condition at both ends. For the n

^{th}mode, because the total electrical length is a multiple of

*π*with an open-ended boundary, the

*n*

*π*/

*N*phase delay is expected to occur per unit cell when

*n*= 0, ±1, ⋯, ±

*N*–1, and the current distribution has n time variations of a standing wave pattern for both the positive and negative modes. The length of one variation corresponds to a half wavelength of each mode. Therefore, both the n

^{th}positive and negative modes have the same wavelength (or the same magnitude of a propagation constant) even though the corresponding frequencies are different because of different dispersion branches. However, the phase difference between the positive and the negative mode is 180° because of the opposite direction of the propagation direction between the two modes. The phase difference of 180° is easy to understand from the current standing wave pattern with an open boundary condition at both ends given by

*N*unit cell CRLH TL with an open boundary condition is simulated with the Advanced Design System (Fig. 2). If a lossless structure with no resistance exists, only the magnitude of the reactance will distribute the amplitude of the current and the voltage. For

*N*= 3, the voltages and the currents of each node in Fig. 2 are measured with probes labeled

*P*

_{0},

_{1},

_{2}, ⋯,

*N*and compared (Fig. 3).

^{th}mode.

^{th}mode, no current flows and the voltage operates without a phase delay because of the zero value of

*Y*

*. The 0*

_{p}^{th}-order resonance with an open-ended boundary condition occurs at the shunt resonance (

*Y*

*= 0). Moreover, the number of half wavelength variations increases as the mode number of*

_{p}*n*increases. When the distributions of the positive and negative modes are compared, note that the current distribution changes to 180° in the phase as discussed previously, and the voltage distribution is the same as that in the open boundary condition.

*P*

_{0}probe is fixed at +1 Vdc at the input node. Therefore, its amplitude is +1 V regardless of the mode. For the positive mode,

*Y*

*/2 has a positive value because*

_{p}*C*

*is predominant over*

_{R}*L*

*; for the negative mode,*

_{L}*Y*

*/2 has a negative value because*

_{p}*L*

*is predominant over*

_{L}*C*

*. Therefore, the direction of the current changes in the negative mode. The reason for the existence of negative modes is that they are another combination that makes the total input reactance zero. In other words,*

_{R}*Z*

*and*

_{s}*Y*

*of the negative mode have the same ratio as*

_{p}*Z*

*and*

_{s}*Y*

*of the positive mode. The only difference is that the signs of Im(*

_{p}*Z*

*) and Im(*

_{s}*Y*

*) of the negative mode are negative and those of Im(*

_{p}*Z*

*) and Im(*

_{s}*Y*

*) of the positive mode are positive. Fig. 3 satisfies the condition of*

_{p}^{th}-order mode angular frequency is given by the series resonance of

### III. Derivation of Resonance Frequency

*T*circuit as shown in Fig. 4.

*θ*=

*n*

*π*/

*N*) between two terminals:

*N*is the number of total unit cells, and

*n*is the number of arbitrary unit cells.

*AD*–

*BC*= 1), we can summarize both as Eq. (5):

*ω*. By substitution into a quadratic equation, the following equation can be obtained:

*C*

*,*

_{R}*L*

*,*

_{R}*C*

*, and*

_{L}*L*

*) and their values. The structure in Fig. 6 is obtained from [13] and simulated by High Frequency Structure Simulator (HFSS). Table 1 compares the resonance frequencies of the three-unit cell CRLH TL obtained from Eq. (6), HFSS, and Ansoft Designer, respectively.*

_{L}*C*

*,*

_{R}*L*

*,*

_{R}*C*

*, and*

_{L}*L*

*) from the real structure. The method of parameter extraction from the dispersion curve inevitably cannot produce perfect matched results in all modes. However, the overall results are in good agreement.*

_{L}### IV. Correlation between Slope and BW

*df*/

*dθ*(

*θ*= electrical length per unit cell) is a measure of how much the frequency changes as the electrical length changes. The higher the slope is, the greater the frequency changes even with a change in electrical length. In other words, a large value of

*S*

_{11}< –10 dB, and the slope is calculated by differentiating the dispersion curve with the equation of

*df*/

*dθ*. Tables 2 and 3 show the correlation between slope and BW for various cells and modes according to an unbalanced or a balanced condition [5]. The correlation coefficient is defined as

##### (7)

^{th}mode of the unbalanced condition, Tables 2 and 3 show that the slope is always closely related to the BW. Fig. 7 illustrates that the BW, which is correlated with the dispersion curve slope, is significantly improved when the balanced condition is satisfied [14]. In other words, the overall BW improves because the average slope from 0° to 180° increases from 5.3 MHz/° to 25.3 MHz/°. This finding can be confirmed by comparing Tables 2 and 3.

### V. Conclusion

^{th}-mode resonance frequency of an N unit cell CRLH transmission line. The resonance mechanism of the n

^{th}positive/negative mode is investigated by the current distribution of an N unit cell CRLH transmission line. Both the positive and the negative n

^{th}resonance modes have n times current variation, but their phase difference is 180°. The slope of the CRLH dispersion curve is closely correlated with the BW. When the CRLH TL is designed with a balanced condition, the BW increases from the slope.