**6. Combined vibrator–slot structures**

Now let us consider a problem of electromagnetic waves excitation by a narrow straight transverse slot in the broad wall of rectangular waveguide with a two passive impedance vibrators in it.

are define by the current as

2

( ) <sup>2</sup>

Bez Broja <sup>2</sup>

= + = −α <sup>−</sup> + α i z g

k f kL S Se S

16 cos ( ) 1 , , [1 ( / ) ][cos 2 ( , )] γ ϕ

11 12 12 3 2

ϕ ϕ

and waveguide axis {0 }x are <sup>O</sup> 0 and <sup>O</sup> 30 .

k cos a <sup>ϕ</sup> π

Dummy Text where <sup>0</sup> J is current amplitude, f s( ) is the current distribution function,

= ϕ , ( ,2 ) sa W kd kL <sup>ϕ</sup> <sup>e</sup> is the function of slot self-field, defined by formulas (43).

Reflection and transmission coefficients, <sup>11</sup> S and <sup>12</sup> S for the dominant wave in the slot iris

ϕ ϕ

Figure 5 shows the theoretical and experimental wavelength dependences of power reflec-

iabk k k kL W kd kL (46)

( ) 2cos cos , 1( /) 2( / )

− + <sup>=</sup> <sup>−</sup> <sup>−</sup>

sin cos ( / )cos sin sin 2 2

kL k L k k kL k L kL kL

e

k k k k ϕ ϕ ϕ ϕ ϕ

ϕ ϕ

π ϕ

f kL kL kL

Dummy Text where 2 2 (/) <sup>g</sup> kk a = −π is the propagation constant of H10 wave.

Figure 5. Power reflection coefficient <sup>2</sup> <sup>11</sup> | | S versus wavelength for the iris at a =23.0 mm, b =10.0 mm, 2L =16.0 mm, d =1.5 mm, h =2.0 mm, <sup>0</sup> x a = / 2 , <sup>0</sup> y b = / 2 **Figure 5.** Power reflection coefficient |*S*11|2 versus wavelength for the iris at *<sup>a</sup>* =23.0 mm, *<sup>b</sup>* =10.0 mm, 2*<sup>L</sup>* =16.0 mm, *d* =1.5 mm, *h* =2.0 mm, *x*<sup>0</sup> =*a* / 2, *y*<sup>0</sup> =*b* / 2

**Figure 6.** The geometry of three-element vibrator-slot system and notations

Let a fundamental wave *H*<sup>10</sup> propagates from the area *z* = −*∞* in a hollow infinite rectangular waveguide, the area index is "*Wg*". Two thin nonsymmetrical vibrators (monopoles) with variable surface impedance are located in a waveguide with cross-section {*a* ×*b*}. A narrow transverse slot cut in a broad wall of the waveguide symmetrically relative to its longitudinal axis is radiating into free half-space, the area index is "*Hs*". The vibrators radiuses and lengths are *r*1,2 and *L* 1,2 ((*r*1,2 / *L* 1,2)<<1), the slot width is *d*, the slot length is 2*L* 3 ((*d* / *L* <sup>3</sup>)<<1) and the waveguide wall thickness is *h* . One vibrator is located in the plane {*x*0*y*} and the second vibrator may be shifted along the axis {0*z*} at the distance *z*0 (Figure 6).

For this configuration the system of integral equations relative to electrical currents at the vibrators *J*1,2(*s*1,2) and equivalent magnetic current in the slot *J*3(*s*3) in accordance with (8) may be represented as

Electromagnetic Waves Excitation by Thin Impedance Vibrators and Narrow Slots in Electrodynamic Volumes http://dx.doi.org/10.5772/61188 167

<sup>d</sup> d d d d <sup>d</sup> <sup>d</sup> d 1 2 1 2 1 2 3 3 3 1 2 2 2 2 11 11 1 22 12 2 1 33 13 3 0 1 1111 2 2 2 22 22 2 2 () (,) () (,) () (,) ( ) ( ) ( ), (a) () (,) *L L Wg Wg s s L L L Wg s L s i Wg s L k JsG ss s J sG ss s s ik J s G s s s i E s z sJs k JsG ss s s* w - - - æ öì ü ï ï ç ÷ + +- í ý ¢ ¢¢ ¢ ¢¢ è øï ï î þ - ¢ ¢¢ =- - é ù ë û æ ö ç ÷ <sup>+</sup> ¢ ¢¢ è ø ò ò ò % d <sup>d</sup> <sup>d</sup> d d 2 1 1 2 1 2 3 3 3 3 1 1 3 11 21 1 0 2 2222 2 2 2 1 33 33 33 3 3 11 31 1 0 3 () (,) ( ) ( ) ( ), (b) () (,) (,) () (,) ( ). (c) *L L Wg s L s i L Wg Hs s s L L Wg s s L JsG s s s i E s z sJs k Js G ss G ss s s ik J s G s s s i H s* w w - - ì ü ï ï í ý + = ¢ ¢¢ ï ï î þ =- - é ù ë û æ ö + +- ¢ ¢ ¢¢ é ù ç ÷ ë û è ø - = ¢ ¢¢ ò ò ò ò % (47)

Here *Gs*1,2 *Wg*(*s*1,2, *<sup>s</sup>*′ 1,2) and *Gs*<sup>3</sup> *Wg*,*Hs* (*s*3, *<sup>s</sup>*′ 3) are components of the Green's functions of the rectan‐ gular waveguide and the half-space over the plane [3,4], *G*˜ *s*1 *Wg*(*s*3, *<sup>s</sup>*′ 1)= <sup>∂</sup> <sup>∂</sup> *<sup>z</sup> Gs*<sup>1</sup> *Wg <sup>x</sup>*(*s*3), 0, *<sup>z</sup>*; *<sup>x</sup>* ′ (*s*′ 1), *y* ′ (*s*′ 1), *z*<sup>0</sup> and *G*˜ 3)= <sup>∂</sup> 3), 0, 0 after substitution *<sup>z</sup>* =0 into *G*˜

*s*3 *Wg*(*s*1, *<sup>s</sup>*′ <sup>∂</sup> *<sup>z</sup> Gs*<sup>3</sup> *Wg <sup>x</sup>*(*s*1), *<sup>y</sup>*(*s*1), *<sup>z</sup>*; *<sup>x</sup>* ′ (*s*′ *s*1 *Wg* and *<sup>z</sup>* <sup>=</sup> *<sup>z</sup>*<sup>0</sup> into *G*˜ *s*3 *Wg* after first derivation, *zi*1,2(*s*1,2) is the internal impedance per unit length of the vibrators ([Ohm/m]), *E*0*s*1,2 (*s*1,2) and *H*0*s*<sup>3</sup> (*s*3) are projections of impressed sources fields on the vibrators and the slot axes, *s*<sup>1</sup> = − *L* 1 and *s*<sup>2</sup> = − *L* <sup>2</sup> are end coordinates of mirror vibrator images relative to the lower broad wall of the waveguide [4]

We will seek the solution of equations system (47) by a generalized method of induced EMMF [19,20], using functions *J*1(2) (*s*1(2) )= *J*1(2) <sup>0</sup> *<sup>f</sup>* 1(2) (*s*1(2) ) and *J*3(*s*3)= *J*<sup>3</sup> <sup>0</sup> *<sup>f</sup>* <sup>3</sup>(*s*3) as approximating expres‐ sions for the currents. Here *J*1(2) 0 and *J*<sup>3</sup> <sup>0</sup> are unknown current amplitudes, *<sup>f</sup>* 1(2) (*s*1(2) ) and *f* <sup>3</sup>(*s*3) are predetermined functions of the current distributions. In accordance with (27) and (42) for the vibrator-slot structure excited by the fundamental wave *H*10 we have

$$\begin{aligned} \{f\_{1\langle 2\rangle}(s\_{1\langle 2\rangle}) = \cos \tilde{k}\_{1\langle 2\rangle} s\_{1\langle 2\rangle} - \cos \tilde{k}\_{1\langle 2\rangle} L\_{\; 1\langle 2\rangle} & \qquad \{f\_{\langle 3\rangle}(s\_{\langle 3\rangle}) = \cos k\_3 - \cosh L\_{\; 3} & \qquad \tilde{k}\_{1\langle 2\rangle} = k - \frac{i2\pi z\_{\overline{1}\langle 2\rangle}^w}{Z\_0 \Delta\_{1\langle 2\rangle}}, \\\ z\_{1\langle 2\rangle}^{\text{av}} &= \frac{1}{L\_{\; 1\langle 2\rangle}} \int\_0^{L\_{\; 1\langle 2\rangle}} z\_{\overline{1}\langle 2\rangle}(s\_{\langle 1\rangle}) ds\_{\; 1\langle 2\rangle} \quad \text{are} \qquad \text{average} \qquad \text{values} \quad [4] & \quad \text{of} \qquad \text{internal} \quad \text{impedances}, \end{aligned}$$

Ω1(2)=2ln(2*L* 1(2) /*r*1(2) ).

1

**Figure 6.** The geometry of three-element vibrator-slot system and notations

vibrator may be shifted along the axis {0*z*} at the distance *z*0 (Figure 6).

be represented as

Let a fundamental wave *H*<sup>10</sup> propagates from the area *z* = −*∞* in a hollow infinite rectangular waveguide, the area index is "*Wg*". Two thin nonsymmetrical vibrators (monopoles) with variable surface impedance are located in a waveguide with cross-section {*a* ×*b*}. A narrow transverse slot cut in a broad wall of the waveguide symmetrically relative to its longitudinal axis is radiating into free half-space, the area index is "*Hs*". The vibrators radiuses and lengths are *r*1,2 and *L* 1,2 ((*r*1,2 / *L* 1,2)<<1), the slot width is *d*, the slot length is 2*L* 3 ((*d* / *L* <sup>3</sup>)<<1) and the waveguide wall thickness is *h* . One vibrator is located in the plane {*x*0*y*} and the second

Note that a comparative analysis of the analytical solution of key problems is not limited only by the examples presented above. Thus, the solution for current in the impedance vibrator, located in free space, was preliminary compared with the known approximate analytical solutions of integral equations. The adequacy of the constructed mathematical models to real physical processes and the reliability of simulation results has been also con-

**Figure 5.** Power reflection coefficient |*S*11|2 versus wavelength for the iris at *<sup>a</sup>* =23.0 mm, *<sup>b</sup>* =10.0 mm, 2*<sup>L</sup>* =16.0

Dummy Text where <sup>0</sup> J is current amplitude, f s( ) is the current distribution function,

= ϕ , ( ,2 ) sa W kd kL <sup>ϕ</sup> <sup>e</sup> is the function of slot self-field, defined by formulas (43).

Reflection and transmission coefficients, <sup>11</sup> S and <sup>12</sup> S for the dominant wave in the slot iris

ϕ ϕ

Figure 5 shows the theoretical and experimental wavelength dependences of power reflec-

iabk k k kL W kd kL (46)

( ) 2cos cos , 1( /) 2( / )

− + <sup>=</sup> <sup>−</sup> <sup>−</sup>

sin cos ( / )cos sin sin 2 2

<sup>11</sup> | | S for the iris, which oriented so that the angle between slot axis {0 }s

0.0

0.2

0.4

Reflection coefficient |S11|

2

0.6

0.8

1.0

kL k L k k kL k L kL kL

e

k k k k ϕ ϕ ϕ ϕ ϕ

28 30 32 34 36

 Theory Experimental data ϕ=30<sup>O</sup>

Wavelength λ, mm

<sup>11</sup> | | S versus wavelength for the iris at a =23.0 mm, b =10.0 mm,

ϕ ϕ

π ϕ

f kL kL kL

Dummy Text where 2 2 (/) <sup>g</sup> kk a = −π is the propagation constant of H10 wave.

k cos a <sup>ϕ</sup> π

are define by the current as

2

tion coefficient <sup>2</sup>

166 Advanced Electromagnetic Waves

0.0

0.2

0.4

0.6

Reflection coefficient |S11|

2

0.8

1.0

( ) <sup>2</sup>

Bez Broja <sup>2</sup>

= + = −α <sup>−</sup> + α i z g

k f kL S Se S

16 cos ( ) 1 , , [1 ( / ) ][cos 2 ( , )] γ ϕ

11 12 12 3 2

ϕ ϕ

and waveguide axis {0 }x are <sup>O</sup> 0 and <sup>O</sup> 30 .

 Theory Experimental data ϕ=0<sup>Ο</sup>

28 30 32 34 36

Wavelength λ, mm

2L =16.0 mm, d =1.5 mm, h =2.0 mm, <sup>0</sup> x a = / 2 , <sup>0</sup> y b = / 2

Figure 5. Power reflection coefficient <sup>2</sup>

mm, *d* =1.5 mm, *h* =2.0 mm, *x*<sup>0</sup> =*a* / 2, *y*<sup>0</sup> =*b* / 2

For this configuration the system of integral equations relative to electrical currents at the vibrators *J*1,2(*s*1,2) and equivalent magnetic current in the slot *J*3(*s*3) in accordance with (8) may

In accordance with the generalized method of induced EMMF, we multiply equation (47a) by the function *f* 1(*s*1), equation (47b) by the function *f* <sup>2</sup>(*s*2), and the equation (47c) by the function

*f* <sup>3</sup>(*s*3) and integrate the equations (47a) and (47b) over the length of the vibrators, and the equation (47c) over the length of the slot. As a result, we obtain a system of linear algebraic equations relative to the current amplitudes *J*1,2,3 0

$$\begin{split} &f\_1^0 Z\_{11}^\pm + f\_2^0 Z\_{12} + f\_3^0 Z\_{13} = -\frac{i\alpha}{2k} \int\_{-l\_1}^{l\_1} f\_1(\mathbf{s}\_1) E\_{0\varsigma\_1}(\mathbf{s}\_1) d\varsigma\_1, \\ &f\_2^0 Z\_{22}^\pm + f\_1^0 Z\_{21} = -\frac{i\alpha}{2k} \int\_{-l\_2}^{l\_2} f\_2(\mathbf{s}\_2) E\_{0\varsigma\_2}(\mathbf{s}\_2) d\varsigma\_2, \\ &f\_3^0 Z\_{33}^\pm + f\_1^0 Z\_{31} = -\frac{i\alpha}{2k} \int\_{-l\_3}^{l\_3} f\_3(\mathbf{s}\_3) H\_{0\varsigma\_3}(\mathbf{s}\_3) d\varsigma\_3. \end{split} \tag{48}$$

Here

$$\begin{split} Z\_{11(22)} &= \frac{4\pi}{ab} \sum\_{n=1}^{\alpha} \sum\_{\imath=0}^{\alpha} \left\{ \frac{\varepsilon\_{\imath} (k^2 - k\_y^2) \tilde{k}\_{\imath(2)}^2}{k k\_z (\tilde{k}\_{1(2)}^2 - k\_y^2)^2} e^{-k\_z \varepsilon\_{\imath(2)}} \sin^2 k\_x \mathbf{x}\_{01(02)} \\ &\times \left[ \sin \tilde{k}\_{1(2)} L\_{\mathfrak{u}(2)} \cos k\_y L\_{\mathfrak{u}(2)} - \frac{\tilde{k}\_{\mathfrak{u}(2)}}{k\_y} \cos \tilde{k}\_{\mathfrak{u}(2)} L\_{\mathfrak{u}(2)} \sin k\_y L\_{\mathfrak{u}(2)} \right]^2 \right\}, \end{split} \tag{49}$$

$$F\_{1(2)}^z = -\frac{i}{r\_{1(2)}} \int\_0^{L\_{1(2)}} f\_{1(2)}^2(s\_{1(2)}) \overline{Z}\_{\text{S1(2)}}(s\_{1(2)}) \text{ds}\_{1(2)} \tag{50}$$

$$\begin{split} Z\_{12} = Z\_{21} &= \frac{4\pi}{ab} \sum\_{n=1}^{\alpha} \sum\_{n=0}^{\alpha} \left\{ \frac{\tilde{\kappa}\_{n} (k^{2} - k\_{y}^{2}) \tilde{k}\_{1} \tilde{k}\_{2} e^{-k\_{z}z\_{0}}}{k k\_{z} (\tilde{k}\_{1}^{2} - k\_{y}^{2}) (\tilde{k}\_{2}^{2} - k\_{y}^{2})} \sin k\_{x} x\_{01} \\ &\times \sin k\_{x} x\_{02} \left[ \sin \tilde{k}\_{1} L\_{1} \cos k\_{y} L\_{1} - (\tilde{k}\_{1} \,/\, k\_{y}) \cos \tilde{k}\_{1} L\_{1} \sin k\_{y} L\_{1} \right] \\ &\times \left[ \sin \tilde{k}\_{2} L\_{2} \cos k\_{y} L\_{2} - (\tilde{k}\_{2} \,/\, k\_{y}) \cos \tilde{k}\_{2} L\_{2} \sin k\_{y} L\_{2} \right] \end{split} \tag{51}$$

$$\begin{aligned} Z\_{33}^{\text{Hls}} &= \text{Si}4kL\_3 - i\text{Ci}\text{in}4kL\_3 - 2\cos kL\_3 \left[ 2\left(\sin kL\_3 - kL\_3\cos kL\_3\right) \right. \\ &\times \left( \ln \frac{16L\_3}{d\_c} - \text{Ci}\text{in}2kL\_3 - i\text{Si}2kL\_3 \right) + \sin 2kL\_3 e^{-ikL\_3} \right] \end{aligned} \tag{52}$$

$$\mathcal{Z}\_{33}^{W\_3} = \frac{8\pi}{ab} \sum\_{m=1,3\dots n=0,1}^{n} \left\{ \frac{\mathcal{E}\_\mathbf{x}k}{k\_z(k^2 - k\_x^2)} e^{-\frac{k\_z\frac{d}{4}}{4}} \left[ \sin kL\_3 \cos k\_x L\_3 - (k \;/\ k\_x) \cos kL\_3 \sin k\_x L\_3 \right]^2 \right\},\tag{53}$$

Electromagnetic Waves Excitation by Thin Impedance Vibrators and Narrow Slots in Electrodynamic Volumes http://dx.doi.org/10.5772/61188 169

$$\begin{aligned} Z\_{13} &= -Z\_{31} = \\ Z\_{2} &= \frac{4\pi}{ab} \sum\_{n=1}^{n} \Biggl[ \frac{\varepsilon\_{n}k\tilde{k}\_{1}e^{-k\_{z}z\_{0}}}{i(\tilde{k}\_{1}^{2} - k\_{y}^{2})(k^{2} - k\_{z}^{2})} \sin k\_{x}x\_{01} \sin \frac{k\_{y}a}{2} \Bigg[ \sin \tilde{k}\_{1}L\_{1}\cos k\_{y}L\_{1} - \frac{\tilde{k}\_{1}}{k\_{y}}\cos \tilde{k}\_{1}L\_{1}\sin k\_{y}L\_{1} \\ & \times \Bigg[ \sin kL\_{3}\cos k\_{x}L\_{3} - \frac{k}{k\_{x}}\cos kL\_{3}\sin k\_{z}L\_{3} \Bigg] \Bigg, \\ Z\_{11(22)}^{2} = Z\_{11(22)} + F\_{12}^{z}, Z\_{33}^{2} = Z\_{33}^{\mathrm{Hz}} + Z\_{33}^{\mathrm{Wg}}, \end{aligned} \tag{54}$$

where *ε<sup>n</sup>* ={ 1, *n* =0 2, *n* ≠0 , *kx*(*y*)<sup>=</sup> *<sup>m</sup>*(*n*)*<sup>π</sup> <sup>a</sup>*(*b*) , *kz* <sup>=</sup> *kx* <sup>2</sup> <sup>+</sup> *ky* <sup>2</sup> −*k* <sup>2</sup> , *m*, *n* are integers; *Si* and *Cin* are integral sine and cosine.

The energy characteristics of the vibrator-slot system: the reflection and transmission coeffi‐ cients, *S*11 and *S*12, and power radiation coefficient |*S*Σ|<sup>2</sup> , are defined by the expressions

$$S\_{11} = \frac{4\pi i}{abkk\_{\mathcal{g}}} \left[ J\_3 \frac{2k\_{\mathcal{g}}^2}{k^2} f(kL\_3) - J\_1 \frac{k\_{\mathcal{g}}}{\tilde{k}\_1} \sin\left(\frac{\pi x\_{01}}{a}\right) f(\tilde{k}\_1 L\_1) e^{-ik\_{\mathcal{g}}z\_0} - J\_2 \frac{k\_{\mathcal{g}}}{\tilde{k}\_2} \sin\left(\frac{\pi x\_{02}}{a}\right) f(\tilde{k}\_2 L\_2) \right] e^{2ik\_{\mathcal{g}}z},\tag{55}$$

$$S\_{12} = 1 + \frac{4\pi i}{abk\_{\text{s}}} \left| f\_{\text{s}} \frac{2k\_{\text{g}}^2}{k^2} f(kL\_{\text{s}}) + f\_1 \frac{k\_{\text{g}}}{\tilde{k}\_1} \sin\left(\frac{\pi x\_{01}}{a}\right) f(\tilde{k}\_1 L\_1) e^{\tilde{k}\_{\text{g}} \tau\_0} + f\_2 \frac{k\_{\text{g}}}{\tilde{k}\_2} \sin\left(\frac{\pi x\_{02}}{a}\right) f(\tilde{k}\_2 L\_2) \right|,\tag{56}$$

$$\|\|\mathbf{S}\_{\mathbf{z}}\|\|^2 = \mathbf{1} - \|\mathbf{S}\_{\mathbf{z}1}\|^2 - \|\mathbf{S}\_{\mathbf{z}2}\|^2 \tag{57}$$

In expressions (55)-(57)

*f* <sup>3</sup>(*s*3) and integrate the equations (47a) and (47b) over the length of the vibrators, and the equation (47c) over the length of the slot. As a result, we obtain a system of linear algebraic

0

1

*L*

w

1

*L*

ò


() () , <sup>2</sup>

() () . <sup>2</sup>

2

*s*

3

1(2)

*x*

d

*<sup>z</sup> k z*

*<sup>r</sup>* = - ò (50)

}

*ikL*

<sup>2</sup> <sup>4</sup>

*xx x*


*x*

2

þ

1(2) 2

() () ,

*S*

2 2 <sup>0</sup> 1 2

*k k kke*

% %

( )( )

*s*

2

*L*

2 22 1 21 22 0 2 2


*<sup>i</sup> J Z J Z f s E s ds <sup>k</sup>*

w

+ + =-

3 33 1 31 33 0 3 3


*<sup>i</sup> J Z J Z f s H s ds <sup>k</sup>*

w

1 11 2 12 3 13 11 0 1 1

*<sup>i</sup> J Z J Z J Z f s E s ds <sup>k</sup>*

2 3

*L L*

ò

3

2 22


*<sup>z</sup> n y k r*

11(22) 2 22 01(02)

*k k L kL k L kL k*

*<sup>i</sup> F fsZ s s*

12 21 2 22 2 01 1 0 1 2

<sup>4</sup> ( ) sin

*n y*

*m n zy y*

*x yy y*

´ - é ù

*kx kL kL k k kL kL*

% %%

*yy y*

Si Cin ( )

<sup>8</sup> sin cos ( / )cos sin , ( )

ìï <sup>=</sup> <sup>í</sup> é ù - ë û <sup>ï</sup> - <sup>î</sup> å å (53)

ë

4 4 2cos 2 sin cos

22 2 2 22 2

Cin Si <sup>3</sup>

33 2 2 3 3 3 3

*<sup>k</sup> <sup>Z</sup> e kL k L k k kL k L*

33 3

33 3 3 3 33 3

*Z kL i kL kL kL kL kL*

sin cos ( / )cos sin ,

*kL kL k k kL kL*

´ - é ù ë û

<sup>16</sup> ln 2 2 sin 2 ,

=- - - é

*<sup>L</sup> kL i kL kL e*

*e z*

æ ö ù ´--+ ç ÷ ú è ø úû

% %%

sin sin cos ( / )cos sin

*Z Z k x ab kk k k k k*

åå % %

e- ¥ ¥

ì <sup>ï</sup> - = = <sup>í</sup> - - ïî

02 1 1 1 1 11 1

ë û

<sup>ü</sup> é ù <sup>ï</sup> ´ - ê ú <sup>ý</sup> ë û ï

<sup>4</sup> ( ) sin ( )

*k kk Z e kx ab kk k k*

> 1(2) 1(2) 1(2) 1(2) 1(2) 1(2) 1(2)

sin cos cos sin ,

*y y y*

1 0 1(2)

% % %

1(2) 2 1(2) 1(2) 1(2) 1(2) 1(2) 1(2)

*L*

1(2) 0

= =

p e

ì <sup>ï</sup> - <sup>=</sup> <sup>í</sup> - ïî

*m n z y*

åå % %

*L*

ò

1

(48)

(49)

(51)

(52)

}

*s*

() () , <sup>2</sup>

equations relative to the current amplitudes *J*1,2,3

168 Advanced Electromagnetic Waves

S

000

+ =-

+ =-

0 0

S

0 0

p¥ ¥

*z*

3

*e*

*<sup>d</sup> <sup>k</sup> Wg <sup>n</sup>*

*ab kk k*

*m n z x*

e¥ ¥ -

*d*

1,3.. 0,1..

= =

p *Hs*

= =

S

Here

$$\begin{split} J\_{1} &= \mathbb{I} \left\{ \left( Z\_{11}^{\Sigma} Z\_{22}^{\Sigma} Z\_{33}^{\Sigma} - Z\_{21} Z\_{12} Z\_{33}^{\Sigma} - Z\_{31} Z\_{13} Z\_{22}^{\Sigma} \right) \\ & \times \left[ \frac{k^{2}}{k\_{g} \tilde{k}\_{1}} \sin \frac{\pi \chi\_{01}}{a} f\_{1}(\tilde{k}\_{1} L\_{1}) e^{-ik\_{g} \tilde{\tau}\_{0}} Z\_{22}^{\Sigma} Z\_{33}^{\Sigma} - \frac{k^{2}}{k\_{g} \tilde{k}\_{2}} \sin \frac{\pi \chi\_{02}}{a} f\_{2}(\tilde{k}\_{2} L\_{2}) Z\_{12} Z\_{33}^{\Sigma} - f\_{3}(\mathcal{k} L\_{3}) Z\_{13} Z\_{22}^{\Sigma} \right] \end{split}$$

$$\begin{split} \mathcal{J}\_{1} &= \mathbb{E}\left\{ \left( \mathbf{Z}\_{11}^{\Sigma} \mathbf{Z}\_{22}^{\Sigma} \mathbf{Z}\_{33}^{\Sigma} - \mathbf{Z}\_{21} \mathbf{Z}\_{12} \mathbf{Z}\_{33}^{\Sigma} - \mathbf{Z}\_{31} \mathbf{Z}\_{13} \mathbf{Z}\_{22}^{\Sigma} \right) \right. \\ &\times \left[ \frac{k^{2}}{k\_{\mathcal{g}} \tilde{k}\_{2}} \sin \frac{\pi \mathbf{x}\_{02}}{a} f\_{2}(\tilde{k}\_{2} \mathbf{L}\_{2}) (\mathbf{Z}\_{11}^{\Sigma} \mathbf{Z}\_{33}^{\Sigma} - \mathbf{Z}\_{31} \mathbf{Z}\_{13}) - \frac{k^{2}}{k\_{\mathcal{g}} \tilde{k}\_{1}} \sin \frac{\pi \mathbf{x}\_{01}}{a} f\_{1}(\tilde{k}\_{1} \mathbf{L}\_{1}) e^{-\tilde{k}\_{1} z\_{0}} Z\_{21} \mathbf{Z}\_{33}^{\Sigma} + f\_{3}(\mathbf{k} \mathbf{L}\_{3}) \mathbf{Z}\_{13} Z\_{21} \right] . \end{split}$$

$$\begin{split} \mathcal{J}\_{3} &= \left\{ f \left( f\_{21}^{\Sigma} Z\_{12}^{\Sigma} Z\_{33}^{\Sigma} - Z\_{21} Z\_{12} Z\_{33}^{\Sigma} - Z\_{31} Z\_{13} Z\_{22}^{\Sigma} \right) \right. \\\\ &\times \left[ f\_{3} (k L\_{3}) (Z\_{11}^{\Sigma} Z\_{22}^{\Sigma} - Z\_{21} Z\_{12}) + \frac{k^{2}}{k\_{\mathcal{g}} \bar{k}\_{2}} \sin \frac{\pi \chi\_{02}}{a} \, f\_{2} (\tilde{k}\_{2} L\_{2}) Z\_{12} Z\_{31} - \frac{k^{2}}{k\_{\mathcal{g}} \bar{k}\_{1}} \sin \frac{\pi \chi\_{01}}{a} \, f\_{1} (\tilde{k}\_{1} L\_{1}) e^{-ik\_{\mathcal{g}} z\_{2}} Z\_{22} \right] \, . \end{split}$$
 
$$f\_{1(2)} (\tilde{k}\_{1(2)} L\_{1(2)}) = \sin \tilde{k}\_{1(2)} L\_{1(2)} - \tilde{k}\_{1(2)} L\_{1(2)} \cos \tilde{k}\_{1(2)} L\_{1(2)},$$
 
$$f\_{3} (k L\_{3}) = \frac{\sin k L\_{3} \cos \{\pi L\_{3} \, /\.a\} - (ka / \pi) \cos k L\_{3} \sin \pi (\pi L\_{3} / a)}{1 - [\pi / \left< \{ka\} \right>^{2}}.$$

Let us consider several distribution functions for the surface impedance along the vibrator, namely: 1) *ϕ*0(*s*1(2) )=1, the constant distribution, 2) *ϕ*1(*s*1(2) )=2 1−(*s*1(2) / *L* 1(2) ) , the triangular distribution linear decreasing to the vibrator end, and 3) *ϕ*2(*s*1(2) )=2(*s*1(2) / *L* 1(2) ), the triangular linear increasing distribution. All distribution have equal average values *ϕ*0,1,2(*s*1(2) ¯ )=1. The expression for *F*1(2) *<sup>z</sup>*<sup>0</sup> with the distribution function 1), in accordance with (50), can be presented as

$$\begin{split} F\_{\mathbf{1}(2)}^{\pm 0} &= -\frac{2i(\overline{R}\_{\mathrm{S1}(2)} + i\overline{\mathbf{X}}\_{\mathrm{S1}(2)})}{\tilde{k}\_{\mathrm{1}(2)}^{2}L\_{\mathrm{1}(2)}r\_{\mathrm{1}(2)}} \left[ \left( \frac{\tilde{k}\_{\mathrm{1}(2)}L\_{\mathrm{1}(2)}}{2} \right)^{2} \left( 2 + \cos 2\tilde{k}\_{\mathrm{1}(2)}L\_{\mathrm{1}(2)} \right) - \frac{3}{8} \tilde{k}\_{\mathrm{1}(2)}L\_{\mathrm{1}(2)} \sin 2\tilde{k}\_{\mathrm{1}(2)}L\_{\mathrm{1}(2)} \right] \\ &= \tilde{F}\_{\mathrm{1}(2)}^{\mathrm{s}} \left( \overline{R}\_{\mathrm{S1}(2)} + i\overline{\mathbf{X}}\_{\mathrm{S1}(2)} \right) \Phi\_{\mathrm{1}(2)} \end{split} \tag{58}$$

with the distribution function 2) as

$$\begin{aligned} &F\_{1(2)}^{z1} = \tilde{\tilde{\mathbf{F}}}\_{1(2)}^{z2} \\ &\times \left\{ \overline{\mathbf{R}}\_{\text{S1(2)}} \boldsymbol{\Phi}\_{1(2)} + i \overline{\mathbf{X}}\_{\text{S1(2)}} \left[ \left( \frac{\tilde{\boldsymbol{k}}\_{1(2)} \boldsymbol{L}\_{1(2)}}{2} \right)^2 \left( 2 + \cos 2\tilde{\boldsymbol{k}}\_{1(2)} \boldsymbol{L}\_{1(2)} \right) - \frac{7}{4} \sin^2 \tilde{\boldsymbol{k}}\_{1(2)} \boldsymbol{L}\_{1(2)} - 2 \left( \cos \tilde{\boldsymbol{k}}\_{1(2)} \boldsymbol{L}\_{1(2)} - 1 \right) \right] \right\} \end{aligned} \tag{59}$$

and with the distribution function 3) as

$$\begin{aligned} F\_{\mathbf{1}(2)}^{\pm} &= \tilde{F}\_{\mathbf{1}(2)}^{\pm} \left( \tilde{\mathbf{R}}\_{\mathbf{3}\mathbf{1}(2)} \boldsymbol{\Phi}\_{\mathbf{1}(2)} \\ + i \tilde{\mathbf{X}}\_{\text{S1}(2)} \left[ \left( \frac{\tilde{\mathbf{k}}\_{\mathbf{1}(2)} L\_{\mathbf{1}(2)}}{2} \right)^{2} \left( 2 + \cos 2 \tilde{\mathbf{k}}\_{\mathbf{1}(2)} L\_{\mathbf{1}(2)} \right) + \frac{7}{4} \sin^{2} \tilde{\mathbf{k}}\_{\mathbf{1}(2)} L\_{\mathbf{1}(2)} \\ - \frac{3}{4} \tilde{\mathbf{k}}\_{\mathbf{1}(2)} L\_{\mathbf{1}(2)} \sin 2 \tilde{\mathbf{k}}\_{\mathbf{1}(2)} L\_{\mathbf{1}(2)} + 2 \left( \cos \tilde{\mathbf{k}}\_{\mathbf{1}(2)} L\_{\mathbf{1}(2)} - 1 \right) \right] \end{aligned} \tag{60}$$

Since the formulas for *F*1(2) *z*0,1,2 differ from one another, in spite of equal average values of functions *ϕ*0,1,2(*s*1(2) ) and identical functional dependences in formulas for currents, the cur‐ rent amplitudes and, hence, energy characteristics will be substantially different.

Figures 7, 8 shows the wavelength dependences of the radiation coefficient, modules of the reflection and transmission coefficients in the wavelength range of the waveguide single-mode regime, obtained using the following common parameters: *a* =58.0 mm, *b* =25.0 mm, *h* =0.5 mm, *<sup>r</sup>*1,2 =2.0 mm, *<sup>L</sup>* 1,2 =15.0 mm, *R*¯ *<sup>S</sup>* 1(2)=0, *x*<sup>01</sup> =*a* / 8, *x*<sup>02</sup> =7*a* / 8, *d* =4.0 mm and 2*L* <sup>3</sup> =40.0 mm.

Figure 7. The energy characteristics versus wavelength at 1,2 res res λ =λ v sl , Z Z S S 1 2 = **Figure 7.** The energy characteristics versus wavelength at *λv*1,2 *res* <sup>=</sup>*λsl res* , *Z*¯ *<sup>S</sup>* <sup>1</sup> <sup>=</sup>*<sup>Z</sup>*¯ *S* 2

1.0

( )

SSS S S

2 2

é ù ´ -+ <sup>ê</sup> - <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

p

3 3 2

)=1, the constant distribution, 2) *ϕ*1(*s*1(2)

2

1(2) 2 1(2) 1(2) 1(2) 1(2) 1(2) 1(2)

ê ú <sup>=</sup> - ç ÷ + - ç ÷

% %%% %

2 ( ) <sup>3</sup> 2 cos2 sin 2

*F kL kL kL*

1(2) 1(2) 2 1(2) 1(2) 1(2) 1(2) 1(2) 1(2) 1(2) 1(2) 1(2)

*R iX kL kL kL*

<sup>ì</sup> <sup>é</sup> <sup>ï</sup> æ ö ùüï <sup>ê</sup> ´ F+ <sup>í</sup> ç ÷ +- - - úý êç ÷ <sup>ï</sup> è ø ûïþ <sup>î</sup> êë

( )

<sup>7</sup> 2 cos2 sin

1(2) 1(2) 2 1(2) 1(2) 1(2) 1(2) 1(2)

% % %

2 4

*iX kL kL*

( )

2 4

linear increasing distribution. All distribution have equal average values *ϕ*0,1,2(*s*1(2) ¯

distribution linear decreasing to the vibrator end, and 3) *ϕ*2(*s*1(2)

2

2

ê + ++ ç ÷

1(2) 1(2) 1(2) 1(2) 1(2) 1(2)

*kL kL kL*

%% %

<sup>3</sup> sin 2 2 cos 1

ùüï - +- úý

*k L*

{

= F

1(2) 1(2) 1(2) 1(2)

*S*

éæ ö

êç ÷ êè ø ë

*k L*


p

*k k x x f kL Z Z Z Z f kL Z Z f kL e Z Z k k a a k k*

1(2) 1(2) 1(2) 1(2) 1(2) 1(2) 1(2) 1(2) 1(2) *f kL kL kL kL* ( ) sin <sup>=</sup> - cos , % %% %

sin cos( / ) ( / )cos sin( / ) ( ) . 1 [ / ( )] *kL L a ka kL L a f kL ka*

Let us consider several distribution functions for the surface impedance along the vibrator,

p

( )

é ù <sup>+</sup> æ ö

2 8

3 3 11 22 21 12 2 2 2 12 31 1 11 31 22 2 1

( )( ) sin ( ) sin ( ) , *<sup>g</sup> ik z g g*

S S - S

3 3 3 3

*<sup>z</sup>*<sup>0</sup> with the distribution function 1), in accordance with (50), can be presented

è ø ë û

( ) ( )

<sup>7</sup> 2 cos2 sin 2 cos 1

% % %% (59)

ûïþ

pp

02 01

% % % %

 p

)=2 1−(*s*1(2) / *L* 1(2)

)=2(*s*1(2) / *L* 1(2)

3 11 22 33 21 12 33 31 13 22

0 1(2) 1(2) 1(2) 1(2)

*i R iX k L*

1(2) 1(2) 1(2)

*kLr*

1(2) 1(2) 1(2) 1(2)

with the distribution function 2) as

and with the distribution function 3) as

2

*z z*

*F FR*

%

*S*

4

( )

*S S*

= +F

*S S*

*F R iX*

*z S S*

*J ZZZ ZZZ ZZZ*

= --

1

170 Advanced Electromagnetic Waves

namely: 1) *ϕ*0(*s*1(2)

expression for *F*1(2)

*z*

%

1 1(2) 1(2)

*z z*

=

%

*F F*

as

0

) , the triangular

), the triangular

)=1. The

(58)

(60)

60 65 70 75 80 85 90 95 100 105 0.0 0.2 0.4 0.6 0.8 Energy characteristics ZS1=ikr<sup>1</sup> ln(4.0) ZS2=0 z0 =64.0mm |S<sup>Σ</sup> | 2 |S11| |S12| Slot (|S<sup>Σ</sup> | 2 ) VSWR=1.5 60 65 70 75 80 85 90 95 100 105 0.0 0.2 0.4 0.6 0.8 Energy characteristics |S<sup>Σ</sup> | 2 |S11| |S12| Slot (|S<sup>Σ</sup> | 2 ) ZS1=ikr<sup>1</sup> ln(4.0)φ<sup>2</sup> (s1 ) ZS2=ikr<sup>2</sup> ln(4.0) z0 =64.0mm VSWR=1.5 The choice of slot dimensions was stipulated by its natural resonance at the average wave‐ length of the waveguide frequency range *λ*<sup>3</sup> *res* =86.0 mm. The dimensions of the vibrators 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*¯ *S* 1(2) >0), known to increase the vibrator electrical length, i.e. to increase *λ*1,2 *res* as compared to case *Z*¯ *<sup>S</sup>* 1(2)=0, without decreasing a distance between the vibrators ends and the upper broad wall of the waveguide. This is very important for increasing the breakdown power for waveguide device as a whole.

1.0

Wavelength λ, mm Wavelength λ, mm 65 70 75 80 85 90 95 100 105 0.0 0.2 0.4 0.6 0.8 1.0 |S<sup>Σ</sup> | 2 |S11| |S12| Energy characteristics Wavelength λ, mm ZS1=ikr<sup>1</sup> ln(4.0) ZS2=ikr<sup>2</sup> ln(4.0)φ<sup>1</sup> (s2 ) z0 =64 mm VSWR=1.5 65 70 75 80 85 90 95 100 105 0.0 0.2 0.4 0.6 0.8 1.0 |S<sup>Σ</sup> | 2 |S11| |S12| Energy characteristics Wavelength λ, mm ZS1=ikr<sup>1</sup> ln(4.0)φ<sup>1</sup> (s1 ) ZS2=ikr<sup>2</sup> ln(4.0) z0 =64 mm VSWR=1.5 Figure 8. The energy characteristics versus wavelength at 1,2 res res λ ≠λ v sl , Z Z S S 1 2 ≠ , experimental data are marked by circles As might be expected from physical considerations, displacement of the impedance vibrator along the longitudinal axis of the waveguide at a distance *z*<sup>0</sup> from the centre of the slot, where the maximum mutual influence between elements of the structure is observed, are multiple of *λ<sup>G</sup>* / 4 (Fig. 7: *z*<sup>0</sup> =*λ<sup>G</sup>* / 4 =32.0 mm and *z*<sup>0</sup> =*λ<sup>G</sup>* / 2 =64.0 mm). Here *λ<sup>G</sup>* =2*π* / (2*π* / *λsl res* ) <sup>2</sup> −(*π* / *a*) <sup>2</sup> is resonant wavelength of the slot in the waveguide, and *λsl res* is the resonant wavelength of the slot in the free half-space over the plane. As seen from Figure 7, an acceptable reflection coefficient |*S*<sup>11</sup> | and high level of radiation could not be achieved if the monopoles have the equal distributed impedances *Z*¯ *<sup>S</sup>* <sup>1</sup> <sup>=</sup>*<sup>Z</sup>*¯ *<sup>S</sup>* 2. The maximum of radiation coefficient |*S*Σ|<sup>2</sup> and almost perfect agreement with the feed line, as well as tuning to other resonant wavelengths can be achieved by changing the distribution functions of impedance along the monopoles axes (Figure 8). Fig. 8 also shows that the results of mathematical modeling are confirmed by the experimental data. Experimental models have been made in the form of corrugated brass rods (see photo in Figure 8).

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

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),

res λ =86.0 mm. The dimensions of the vibrators

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

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

0.0

0.2

0.4

0.6

Energy characteristics

0.8

1.0

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

VSWR=1.5

0.0

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 =

1.0

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

z0 =64.0 mm

Wavelength λ, mm

z0 =32.0 mm

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

Figure 8. The energy characteristics versus wavelength at 1,2 res res λ ≠λ v sl , Z Z S S 1 2 ≠ , experimental data are marked by circles **Figure 8.** The energy characteristics versus wavelength at *λv*1,2 *res* <sup>≠</sup>*λsl res* , *Z*¯ *<sup>S</sup>* <sup>1</sup> <sup>≠</sup>*<sup>Z</sup>*¯ *<sup>S</sup>* 2, experimental data are marked by circles

For the arbitrary vibrator-slot structures and coupled electrodynamic volumes expressions for *f v s*,*a* (*sv*) and *f sl s*,*a* (*ssl* ) (the subscripts *s*, *a* denote the symmetric and antisymmetric components 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 obtained from the following relations The choice of slot dimensions was stipulated by its natural resonance at the average wavelength of the waveguide frequency range <sup>3</sup> res λ =86.0 mm. The dimensions of the vibrators 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),

$$\begin{aligned} \boldsymbol{f}\_{v}^{s,s}(\boldsymbol{s}\_{v}) & \coloneqq \begin{bmatrix} \sin\tilde{\kappa}(L\_{v}-s\_{v}) \int\_{-L\_{v}}^{s\_{v}} E\_{0s\_{v}}^{s,s}(s\_{v}') \sin\tilde{k}(L\_{v}+s\_{v}') \mathrm{d}s\_{v}' \\ \quad \phantom{\rm L}\_{L} \\ + \sin\tilde{\kappa}(L\_{v}+s\_{v}) \int\_{-s\_{v}}^{s\_{v}} E\_{0s\_{v}}^{s,s}(s\_{v}') \sin\tilde{k}(L\_{v}-s\_{v}') \mathrm{d}s\_{v}' \\ \quad \phantom{\rm L}\_{v} \\ \int\_{-L\_{d}}^{s\_{d}} \left( \boldsymbol{L}\_{d} - s\_{d} \right) \int\_{-L\_{d}}^{s\_{d}} H\_{0s\_{d}}^{s,s}(s\_{d}') \sin\boldsymbol{k}(L\_{d}+s\_{d}') \mathrm{d}s\_{d}' \\ \quad \phantom{\rm L}\_{L} \\ + \sin\boldsymbol{k}(L\_{d}+s\_{d}) \int\_{-s\_{d}}^{L\_{d}} H\_{0s\_{d}}^{s,s}(s\_{d}') \sin\boldsymbol{k}(L\_{d}-s\_{d}') \mathrm{d}s\_{d}' \end{bmatrix}, \end{aligned} \tag{61}$$

where *E*0*sv s*,*a* (*sv*) and *H*0*ssl s*,*a* (*ssl* ) are projections of symmetric and antisymmetric components of impressed sources on the vibrator and the slot axes. Here the sign ~ means that after integration in expressions (61) only multipliers, depending upon coordinates *sv* and *ssl* , are left.

Note once more that for arbitrary orientations of the vibrator, or the slot relative to the waveguide walls, or for another impressed field sources, the expressions (61) should be used to determine the distribution functions of electric and magnetic currents in the vibrator and slot. For example, for the longitudinal slot in the broad wall of waveguide, i.e. if axes {0*ssl*} and {0*z*} coincide, we obtain

$$\begin{aligned} f\_{sl}^{s}(\mathbf{s}\_{sl}) &= \cos k \mathbf{s}\_{sl} \cos k\_{\clubsuit} \mathbf{L}\_{sl} - \cos k \mathbf{L}\_{sl} \cos k\_{\clubsuit} \mathbf{s}\_{sl}, \\\ f\_{sl}^{s}(\mathbf{s}\_{sl}) &= \sin k \mathbf{s}\_{sl} \sin k\_{\clubsuit} \mathbf{L}\_{sl} - \sin k \mathbf{L}\_{sl} \sin k\_{\clubsuit} \mathbf{s}\_{sl}. \end{aligned} \tag{62}$$

If vibrator is excited at its base by voltage *δ* -generator as in a waveguide-to-coaxial adapter we have

$$f\_v(\mathbf{s}\_v) = \sin \bar{k} (L\_v - \mathbf{s}\_v). \tag{63}$$
