Next, we study the degree of the influence of this impedance on the coupling of antennas. Figure 4(a) shows the dependence of on the synthesized impedance (Figure 3), normalized relative to the field above an ideal conducting plane, for (solid curve) and (dashed curve). As we can see, a greater attenuation of the field is accompanied by a steeper slope of the impedance alteration, which coincides with the results in the Ref. [18]. The main difference consists in the fact that the impedance obtained in this paper not only has a reactive component but also a resistive component,. In order to measure the degree of influence of the resistive part of the impedance, , on the field attenuation along the impedance structure, we show in Figure 4(b) the De-

(a)(b)

Figure 4. (a) Behavior of the field attenuation for the synthesized impedance: (solid curve) and (dashed curve). (b) Behavior of the field attenuation for the synthesized impedance () with the active component, (solid curve), and without it, (dashed curve).

pendence of for the synthesized impedance (), normalized relative to the field above an ideal conducting plane, with the active component, (solid curve) and without it, , (dashed curve). The calculations show that the presence of the resistive part of the impedance not only doesn’t worsen the level of decoupling between antennas, as stated in Ref. [13], but, in fact, increases it by about an additional 5 dB. These results are probably caused by the different dependence of the impedance obtained in this paper compared with what was analyzed in Ref. [13] (this dependence is called uniform). The results of the design optimization show that greater attenuation of the field is reached with a higher rate of impedance growth (generally of its capacitive part, Figure 3). However, the rate of impedance growth (slope of the curves in Figure 4(b)) cannot be arbitrarily large because of practical limitations on the precision of the production of the structures. Therefore, one way to increase the rate of impedance alteration (increase the decoupling) substitutes the monotonically growing impedance with a periodic variation.

Figure 5 shows the dependence of the initial synthesized impedance with, for the structure with length (, red solid curve and, blue dotted curve) and compressed by a factor of three (, black dashed curve and, purple dash-dotted curve), i.e., with the rate of impedance alteration three times greater. Figure 6 shows the dependence of the field, , normalized relative to the field above an ideal conducting plane, for the initial impedance (synthesized in Figure 5, solid and dotted curves) and periodic impedance (see Figure 5, dashed and dash-dotted curves). Here, the dash-dotted curve shows the case for homogeneous purely reactive

Figure 5. Variation of the initial synthesized impedance with, for the structure with length (, solid curve and, dotted curve) and compressed by a factor of 3, (, dashed curve, and, dash-dotted curve); i.e., the slope of the impedance variation is 3 times larger.

Figure 6. Solid curve: variation of the field, , normalized relative to the field, , above an ideal conducting plane for the initial impedance (synthesized in Figure 5, solid and dotted curves). Dashed curve: periodical impedance (see Figure 5, dashed and dash-dotted curves). Here, the dash-dotted curve shows the case for homogeneous purely reactive impedance,.

impedance. As we can see, the level of decoupling grows not monotonically, but at the end of each period, the growth of the coupling coefficient is sharp near the minimum of the impedance, which is characteristic for the propagation of radio waves above non-uniform spreading surfaces [18]. Nevertheless, from comparison of the curves in Figure 6, we can see that the use of several periods of impedance alteration brings an additional gain in the degree of antennas decoupling, 10~40 dB bigger as compared with the monotonic variation (Figure 5, solid and dotted curves). Definitely, the level of the field behind the impedance decoupling structure is defined primarily through the variation of the impedance. Besides, it is evident that the main role is played here by the rate of impedance increase in immediate proximity to the antenna. The results presented in Figure 6 show that from this point of view, the best results are obtained with constant but large capacitive impedance. Nevertheless, in this case, the field along the structure decreases inversely with distance to the 3/2 power. However, the most important thing is that the constant impedance gives the best results outside the impedance structure, i.e., with, where the placement of the receiving antenna is assumed.

• We next consider the design optimization of the structure for a given field with the following form:

(9)

where is the coefficient defining the degree of reduction of the overall field intensity on the impedance part of the structure.

Figure 7 shows the variation of the synthesized impedance (, solid curve and, dashed curve) for the structure with the parameters: and. The behavior of the impedance shows that the

Figure 7. Variation of the synthesized impedance (, solid curve and, dashed curve). The parameters for the calculation are and.

main load in reduction of the field along the structure is carried by the part at the beginning and at the end of the structure, and this does not depend on the length of the structure itself. For example, the dependence of the synthesized impedance distribution for the structure with length is shown in Figure 8. As the length of the structure increases, the characteristics of the impedance remain the same as in Figure 7. The only difference is that Figure 7 is a compressed version of Figure 8 by a factor of 3 for the length of the structure, with a minor deviation of impedance value.

The real and imaginary values of the impedance are influenced only by the degree of the field reduction on the structure. To demonstrate this fact, we show in Figure 9 the variation of the impedance on the structure with the following parameters: and. When the degree of the field reduction changes from (Figure 7) to (Figure 9), the real and imaginary values of the impedance are reduced approximately 5 times and 2.6 times smaller, respectively.

In order to see the dependence of the field attenuation on the synthesized impedance, we plot as a function of in Figure 10. The solid curve

Figure 8. Variation of the synthesized impedance (, solid curve and, dashed curve) with the length.

Figure 9. Variation of the synthesized impedance (, solid curve and, dashed curve) with the following parameters: and.

Figure 10. Behavior of the field attenuation relative to Figure 7 (solid curve) and with a purely reactive impedance, , (dashed curve).

corresponds to the impedance distribution (both and, shown in Figure 7) and the dashed curve with the purely reactive impedance. This result indicates that the prevailing role in the reduction of the level of the complete field behind the impedance structure is played by the reactive part of impedance. The presence of an active component leads to an additional decrease of the field behind the structure, making it 3~5 dB smaller.

We notice from the behavior of the curves in Figures 7-9 that the main load falls on the initial and final parts of the structure, (the so-called “take off” and “landing” grounds of the structure [25]). To confirm this, we design a structure with zero impedance on its initial and final parts, shown in Figure 11, where the parameters are used for and. Figure 12 shows the dependence of on the impedance distribution for a structure with zero impedance on its initial part (at the distances and, solid curve), and with zero impedance on its final part (at the distances and, dashed curve). For the comparison, the dash-dotted curve corresponds to the pure impedance structure. The calculations show that the main role in providing decoupling between antennas belongs to the initial part (the “take off” stripe), directly

Figure 11. Synthesis of structures with zero impedance on their initial and final sections. The parameters for the calculation are and.

Figure 12. Behavior of the field attenuation for the variation of the synthesized impedance structures with zero impedance on its initial section (at the distances and, solid curve) and with zero impedance on its final part (at the distances and , dashed curve). For comparison, the dash-dotted curve corresponds with the pure impedance structure.

touching the opening of the transmitting antenna. The impact of the final part in providing decoupling is much less. In addition, the “take off” stripe defines not only the level of the complete field above the impedance structure but also behind it, where the placement of the receiving antenna is assumed. At the same time, the presence of the ideal conducting part, even if it is very small, leads to sharp growth in the value of the complex impedance of the structure. This, in turn, makes its practical realization more difficult. This statement applies also for any length of the structure. We note that increasing the length of the impedance structure causes a significant reduction of the field only on the structure itself, but this reduction is much smaller behind it.

Finally, we consider the method of pointwise synthesis. The distributions of reactive impedance and the variation of for the designed structure are shown in Figures 13 and 14, respectively, with the same parameters as in Figure 7, where the active part of the impedance is taken to be equal to zero (solid curves) and the structure with reactance is calculated by Equation (7)

Figure 13. Variation of the reactive impedance relative to the initial impedance (solid curve) and the optimized impedance (dashed curve) in accordance with Equation (7). The parameters are the same as in Figure 7.

Figure 14. Behavior of the field attenuation for the structure, in which after the solution of the problem of synthesis, the active part of the impedance is set equal to zero (solid curves) and for the structure with reactance calculated by Equation (7) (dashed curves). The reactance optimized in accordance with Equation (7) (Figure 13, dashed curve) completely differs from the initial calculation (Figure 13, solid curve) because in it, the capacitive impedance along with the inductive impedance has emerged. The parameters are the same as in Figure 7.

(dashed curves). The impedance of the reactance optimized in accordance with Equation (7) (Figure 13, dashed curve) differs completely from the initial calculation (Figure 13, solid curve), because in it, the capacitive impedance along with the inductive impedance has emerged. We obtain more detail in the behavior of the complete field on the impedance part of the structure. In Figure 14, behavior of the field attenuation for the structure, in which after the solution of the problem of synthesis, the active part of the impedance is set equal to zero (solid curves) and for the structure with reactance calculated by Equation (7) (dashed curves). The alternating character of the impedance distribution suggests that a structure which realizes it (for example, a corrugated one) will turn out to be more broadband. Common to all calculations was the large negative reactance near the opening of the transmitting antenna.

The reactance given in Equation (7) has a dependence similar to the function,. A similar form of the impedance (reactance) distribution was obtained in Ref. [26], where the authors considered the design optimization of an impedance plane, transforming the cylindrical front of the wave of a linear source into the front of the heterogeneous flat wave reflected in a given direction. This similarity encourages the use of the results of this paper in the interest of decoupling antennas.

5. Conclusions

In summary, we have solved the problems of the optimization of the design of a complex passive surface impedance for a given electromagnetic field distribution and by means of pointwise synthesis, with the purpose of thereby creating optimized decoupling structures. We present different variations of the field along the impedance structure. We have shown that a greater attenuation of the field is accompanied with a greater degree of impedance variation and that the use of several periods of impedance variation brings an additional gain in the amount of antenna decoupling. We also calculate the degree of influence of the resistive part of the impedance on the rate of the field attenuation along the impedance structure. Finally, we notice that the most efficient way of reducing mutual coupling between antennas located on the same plane is the placement of the impedance structure with a large value of reactive impedance near the transmitting antenna. However, during the solution of the design optimization, the interaction between antennas was not taken into account. It is evident that the presence of the receiving aperture antenna will lead to undesirable reduction of antenna decoupling by means of the heterogeneous character created by the aperture of the receiving antenna. Future research will investigate this question as well.

The derived impedance variation can be used as an independent solution of the problem of providing electromagnetic compatibility, as well as the first step in further optimization of the structure with the help of non-linear programming methods.

6. Acknowledgements

One of the authors (J.-F. D. Essiben) wishes to thank Professor Yu. V. Yukhanov from the Taganrog Institute of Technology at the Southern Federal University in Russia for helpful discussions.

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