Electromagnetic field of the ground-based source surrounded...

5. Resonant Frequencies

[26] Now we study some features of the character of the dependence of the functions of influence K_I (12) and K_V (13) on the changing shape and relative thickness of two layers of the plasma coating. It is worth noting that because of the dispersion properties of the plasma one can find a frequency at which the values of |K_I| and |K_V| reach the maximum (resonance) value. At small losses to emission (G 1) and small thermal losses in the plasma (Im e | Ree|) when Ree approx 1- w_p²/w² such resonant frequency in current is determined from the condition of becoming zero of the real part of the denominator (12)

(15)

or the real part of the denominator (13) for the resonance in voltage

(16)

Equations (15) and (16) are biquadratic relative to the resonant frequency. The roots of the each equation form two branches at a change of the spheroid shape Y from strongly prolate to strongly oblate.

[27] The roots of equation (15) (at a>0 ) have the form

(17)

and form two branches depending on the spheroid shape b/a at its fixed filling in by the vacuum a.

[28] At continuous changes in the spheroid shape from strongly prolate one ( Y approx 1, Y^* approx 0 ) to strongly oblate one

Figure 3

the resonant frequency (17) varies from the exit point (the resonance frequency at b/a = 0 for the prolate spheroid) to the coinciding to it entrance point (the resonance frequency at b/a = 0 for the strongly oblate spheroid) equal to w⁽¹⁾ approx

0 and w⁽²⁾ approx

w_p for the low-frequency and high-frequency branches, respectively. It is worth noting that for a small filling in of the plasma spheroid by vacuum ( a³

1 ), the approximate expressions for resonant frequencies (17) for the low-frequency ( w⁽¹⁾ ) and high-frequency branches ( w⁽²⁾ ) for the prolate spheroid

and

and for the oblate spheroid

and

do not describe their values for the oblate shape of the spheroid in the vicinity of Y=1/2 ( a approx

2b ), where the resonant frequencies (17) have an extreme

(18)

These values (18) tend to the entrance (or exit) points of its branches with an increase in the filling in a of the spheroid by vacuum, that is, the curves of the dependence (17) on the spheroid shape b/a become less steep. In more detail for the fixed a, the position of resonance frequencies (17) as a function of b/a for the prolate spheroid and of the inverse parameter a/b for the oblate spheroid is shown in Figure 3a. Solid and dashed curves show the low-frequency and high-frequency branches, respectively, for two values of the relative thickness of the plasma coating: a=0.2 and 0.8. The calculation result illustrate the shift of the resonance frequency to the limiting values w⁽¹⁾=0 and w⁽²⁾=w_p as the vacuum cavity becomes larger in volume ( a =0.8 ). For the small vacuum filling in ( a =0.2 ) resonance frequencies (17) have a well-pronounced extreme for the oblate shape of the spheroid.

[29] In the case of the slot antenna, the roots of equation (16) for the resonant frequencies have a form of complicated radicals, so their simple analytical dependence on the problem parameters may be obtained only in two particular cases: F=1 and F=0.

[30] For F approx 1 we have a two-layer plasma coating of a weakly prolate spheroidal shape with a relatively thick first layer ( a₀³ 1, a₀³ Y ), and equation (16) coincide with resonant equation (15), the roots of which are presented by expression (17). Thus, in this case, the frequencies for the resonances in voltage and current coincide. These resonant frequencies are shown in Figure 3a.

[31] At the absence of the depleted ion layer ( F=0, a₀=1 ), equation (16) is a resonance one for the spheroidal slot antenna with one-layer plasma coating e . The roots of (16) have one frequency branch

to which (as we will see below) the resonant frequencies of the high-frequency branch (16) tend with a decrease of the thickness of the first layer ( a₀

1 ). These frequencies are shown in Figure 3b for two values of the relative thickness of the plasma coating: a=0.2 and 0.8.

[32] For the a₀ values different from zero and unity, the regularities in the behavior of the resonant frequency (18) (related to w_p ) are shown in Figure 3c as a function of the spheroid shape by solid and dashed curves for the low-frequency and high-frequency branches, respectively, for three values of the relative thickness of the first layer a₀. Curves 1 and 2 correspond to a₀ = 0.2, a=0.2 and 0.8; curves 3 and 4 correspond to a₀ =0.8, a=0.2 and 0.8; and curves 5 and 6 correspond to a₀ =0.99, a=0.2 and 0.8.

[33] For the relatively thick first layer ( F approx 1, a₀³ 1 ) the dependence of the resonant frequencies in voltage (curves in Figure 3c) coincide with the corresponding dependence for the resonance in current (curves that correspond to a= 0.2 and 0.8 in Figure 3a).

[34] In the case when the thickness of the first layer becomes too small ( F approx 0, a₀ approx 1 ) the dependence of the resonance frequencies of the high-frequency branch (dashed curves 5 and 6 in Figure 3c) on the spheroid shape coincide with the corresponding dependence for the spheroidal slot antenna with a one-layer plasma coating (curves 0.2 and 0.8 in Figure 3b). The branch of lower resonance frequencies (solid curves 5 and 6 in Figure 3c) shifts to the region w approx 0. Thus with a decrease of the thickness of the first layer of the coating, the branch of higher resonance frequencies transfers into the resonance frequencies for a slot antenna with an one-layer plasma coating, whereas the branch of lower resonance frequencies disappears. Therefore one may take that for the resonance in voltage the high-frequency and low-frequency branches characterize a resonance of the plasma with the outer and inner vacuum, respectively. We will draw the final conclusion after evaluation of the value of the function of influence for the both branches of resonant frequencies.