**7. Conclusion**

For the arbitrary vibrator-slot structures and coupled electrodynamic volumes expressions for

The choice of slot dimensions was stipulated by its natural resonance at the average wave-

of the currents with respect to the vibrator (*sv* =0) and slot (*ssl* =0) centers, respectively), in accordance with the results, presented in Sections 4 and 5 (see formulas (27) and (42)), can be

have been selected so that their resonant wavelength was within the waveguide operating range. Here we present the results only for vibrators with inductive impedances ( X<sup>S</sup>1( 2) >0),

> , 0

*v*

% %

ò

*s s*

ò

*sl*

*L*

*sl*

*s*

ò

% %


*s s a*


sin ( ) ( )sin ( )

*v v v*

> , 0

*s a*

*v v sl*

( ) , (a) sin ( ) ( )sin ( )

*kL s E s kL s s*

*kL s H s kL s s*

ì ü ï ï í ý ï ï + - ¢ ¢¢ î þ <sup>ò</sup>

*kL s E s kL s s*

ì ü ï ï - + ¢ ¢¢ í ý ï ï + + ¢ ¢¢ î þ

*v v sv v v v*

*v v sv v v v*

*sl sl s sl sl sl sl*


d ,

) ( )sin ( )

*s H s kL s s*

*l sl s sl sl sl sl*

, 0

*s a*

0

*s a*

*sl*

*sl sl*

sin ( ) ( )sin ( )

) (the subscripts *s*, *a* denote the symmetric and antisymmetric components

d

d

d

, (b)

65 70 75 80 85 90 95 100 105

ZS1=ikr<sup>1</sup> ln(5.5)

ZS2=ikr<sup>2</sup> ln(5.5)

VSWR=1.5

 |S<sup>Σ</sup> | 2 |S11| |S12| Slot (|S<sup>Σ</sup> | 2 )

> |S<sup>Σ</sup> | 2 |S11| |S12|

z0 =64.0 mm

Wavelength λ, mm

60 65 70 75 80 85 90 95 100 105

Wavelength λ, mm

65 70 75 80 85 90 95 100 105 0.0

res λ =86.0 mm. The dimensions of the vibrators

Wavelength λ, mm

res res λ ≠λ v sl , Z Z S S 1 2 ≠ , experimental data are

*<sup>S</sup>* 2, experimental data are marked by

(61)

*f v s*,*a*

circles

(*sv*) and *f sl*

marked by circles

*s*,*a* (*ssl*

obtained from the following relations

,

*f s*

,

*f s*

*sl sl*

( )

:

sin (

*s*

*k L*

*L s a*

+

*L s a v v L*

65 70 75 80 85 90 95 100 105

ZS1=ikr<sup>1</sup> ln(5.5) ZS2=ikr<sup>2</sup> ln(5.5)

VSWR=1.5

 |S<sup>Σ</sup> | 2 |S11| |S12| Slot (|S<sup>Σ</sup> | 2 )

VSWR=1.5

 |S<sup>Σ</sup> | 2 |S11| |S12| 0.0

0.0

0.2

*res* <sup>≠</sup>*λsl res* , *Z*¯ *<sup>S</sup>* <sup>1</sup> <sup>≠</sup>*<sup>Z</sup>*¯

0.4

0.6

ZS1=ikr<sup>1</sup>

ZS2=ikr<sup>2</sup> ln(4.0)

VSWR=1.5

z0 =64 mm

ln(4.0)φ<sup>1</sup> (s1 )

Energy characteristics

0.8

1.0

0.2

0.4

Energy characteristics

0.6

0.8

ZS1=ikr<sup>1</sup>

ZS2=ikr<sup>2</sup> ln(4.0)

VSWR=1.5

z0 =64.0mm

1.0

0.2

0.4

0.6

Energy characteristics

0.8

 |S<sup>Σ</sup> | 2 |S11| |S12| Slot (|S<sup>Σ</sup> | 2 )

res res λ =λ v sl , Z Z S S 1 2 =

ln(4.0)φ<sup>2</sup> (s1 )

1.0

z0 =32.0 mm

Wavelength λ, mm

60 65 70 75 80 85 90 95 100 105

Wavelength λ, mm

65 70 75 80 85 90 95 100 105 0.0

**Figure 8.** The energy characteristics versus wavelength at *λv*1,2

length of the waveguide frequency range <sup>3</sup>

Wavelength λ, mm

Figure 8. The energy characteristics versus wavelength at 1,2

Figure 7. The energy characteristics versus wavelength at 1,2

0.0

0.0

0.2

0.4

0.6

Energy characteristics

0.8

ZS1=ikr<sup>1</sup> ln(4.0)

ZS2=ikr<sup>2</sup>

VSWR=1.5

z0 =64 mm

ln(4.0)φ<sup>1</sup> (s2 )

1.0

0.2

0.4

Energy characteristics

0.6

ZS1=ikr<sup>1</sup> ln(4.0)

172 Advanced Electromagnetic Waves

ZS2=0 z0 =64.0mm

0.8

1.0

0.2

0.4

0.6

Energy characteristics

0.8

 |S<sup>Σ</sup> | 2 |S11| |S12| Slot (|S<sup>Σ</sup> | 2 )

1.0

:

This chapter presents the methodological basis for application of the generalized method of induced EMMF for the analysis of electrodynamic characteristics of the combined vibratorslot structures. Characteristic feature of the generalization to a new class of approximating functions consists in using them as a function of the current distributions along the impedance vibrator and slot elements; these distributions are derived as the asymptotic solution of integral equations for the current (key problems) by the method of averaging. Comparison of theoret‐ ical and experimental curves indicates that the solution of integral equations for combined vibrator-slot structures by the generalized method of induced EMMF with approximating functions for the currents in the impedance vibrator and the slot, obtained by averaging method is quite legitimate. It should be noted that for simple structures similar to that considered in the model problem, the proposed approach yields an analytic solution of the electrodynamic problem. For more complex structures, the method may be used to design effective numerical-analytical algorithms for their analyses.

The demonstrative simulation (the comparative analysis of all electrodynamic characteristics in the operating frequencies range) has confirmed the validity of the proposed generalized method of induced EMMF for analysis of vibrator-slot systems with rather arbitrary structure (within accepted assumptions). Here, as examples, some fragments of this comparative analysis were presented. This method retains all benefits of analytical methods as compared with direct numerical methods and allows to expand significantly the boundaries of numerical and analytical studies of practically important problems, concerning the application of single impedance vibrator, including irregular vibrator, the systems of such vibrators and narrow slots. And this is a natural step in the further development of the general fundamental theory of linear radiators of electric and magnetic types.
