**5. Solution of equation for current in a slot between two semi-infinite rectangular waveguides**

Now let us solve the second key problem. Let a resonant iris is placed in infinite hollow (*ε*<sup>1</sup> =*μ*<sup>1</sup> =*ε*<sup>2</sup> =*μ*<sup>2</sup> =1) rectangular waveguide so that its slot has arbitrary orientation in the plane of waveguide cross-section and has no contacts with waveguide walls (Figure 4).

**Figure 4.** The problem geometry and notations

A starting point for the analysis is equation (10) written as (index *sl* is omitted)

$$\left(\frac{\mathrm{d}^{2}}{\mathrm{ds}^{2}} + k^{2}\right)\Big|\_{-L}^{L}f(s')\mathbf{4}\frac{e^{-i\mathbf{k}\mathbf{R}(s,s')}}{\mathbf{R}(s,s')}\mathrm{ds}' = -i\alpha H\_{0s}(\mathbf{s}) - \left(\frac{\mathrm{d}^{2}}{\mathrm{ds}^{2}} + \mathrm{k}^{2}\right)\Big|\_{-L}^{L}f(s')\Big[\mathcal{G}\_{0s}^{V\_{1}}(\mathbf{s},s') + \mathcal{G}\_{0s}^{V\_{2}}(\mathbf{s},s')\Big]\mathrm{ds}',\tag{34}$$

where 4 *e* <sup>−</sup>*ikR*(*s*,*<sup>s</sup>* ′ ) *R*(*s*, *s*′ ) is the Green's function of the slot in infinite perfectly conducting plane, *G*0*<sup>s</sup> <sup>V</sup>*1,2(*s*, *s*′ ) are the Green's functions, which takes into account multiple reflection from walls of volumes.

Isolating the logarithmic singularity in the kernel of equation (34) as in (17), we reduce the equation (34) to an integral equation with small parameter

$$\frac{\text{d}^2 f(\text{s})}{\text{d}s^2} + k^2 f(\text{s}) = \alpha \left\{ i a H\_{0s}(\text{s}) + F \left[ \text{s}\_\prime f(\text{s}) \right] + F\_0 \left[ \text{s}\_\prime f(\text{s}) \right] \right\}. \tag{35}$$

Here *<sup>α</sup>* =1 / 8ln *de* / (8*<sup>L</sup>* ) is the natural small parameter of the problem (|*<sup>α</sup>* | <<1), *de* <sup>=</sup>*d e* <sup>−</sup> *πh* <sup>2</sup>*<sup>d</sup>* is equivalent slot width which takes into account a real wall thickness *h* (*h* / *λ* <<1) [3],

Electromagnetic Waves Excitation by Thin Impedance Vibrators and Narrow Slots in Electrodynamic Volumes http://dx.doi.org/10.5772/61188 163

$$\begin{aligned} \left[F\left[s,f(s)\right]\right] &= -4\frac{\mathrm{d}f(s')}{\mathrm{d}s'}\frac{e^{-i\mathrm{d}R(s,s')}}{R(s,s')} + 4\left[\frac{\mathrm{d}^2f(s)}{\mathrm{d}s^2} + k^2f(s)\right]\gamma(s) \\ &+ 4\int\_{-L}^L \left[\frac{\left[\frac{\mathrm{d}^2f(s')}{\mathrm{d}s'^2} + k^2f(s')\right]e^{-i\mathrm{d}R(s,s')} - \left[\frac{\mathrm{d}^2f(s)}{\mathrm{d}s^2} + k^2f(s)\right]}{R(s,s')}\right] \mathrm{d}s' \end{aligned} \tag{36}$$

is self-field of the slot in infinite perfectly conducting plane,

$${}^{\circ}F\_{0}\left[\begin{matrix}\mathbf{s},f(\mathbf{s})\end{matrix}\right] = -\frac{\mathbf{d}f(\mathbf{s}')}{\mathbf{d}\mathbf{s}'}\left[\mathbf{G}\_{0\mathbf{s}}^{V\_{1}}(\mathbf{s},\mathbf{s}') + \mathbf{G}\_{0\mathbf{s}}^{V\_{2}}(\mathbf{s},\mathbf{s}')\right]\bigg|\_{-\mathsf{L}}^{\mathsf{L}} + \int\limits\_{-\mathsf{L}}^{\mathsf{L}}\frac{\mathbf{d}^{2}f(\mathbf{s}')}{\mathbf{d}\mathbf{s}'^{2}} + k^{2}f(\mathbf{s}')\bigg[\mathbf{G}\_{0\mathbf{s}}^{V\_{1}}(\mathbf{s},\mathbf{s}') + \mathbf{G}\_{0\mathbf{s}}^{V\_{2}}(\mathbf{s},\mathbf{s}')\bigg] \mathbf{ds}' \tag{37}$$

is self-field of the slot, which takes into account multiple reflection from walls of volumes.

To solve the equation (35) by averaging method we change the variable according to (20) and obtain the standard system of integral equations relative to new unknown functions *A*(*s*) and *B*(*s*) which is equivalent to initial equation (35)

$$\begin{aligned} \frac{\mathrm{d}A(s)}{\mathrm{ds}} &= -\frac{\alpha}{k} \{iooH\_{0s}(s) + F\_N\left[s, A(s), \frac{\mathrm{d}A(s)}{\mathrm{ds}}, B(s), \frac{\mathrm{dB}(s)}{\mathrm{ds}}\right] \} \sin ks, \\ \frac{\mathrm{d}B(s)}{\mathrm{ds}} &= +\frac{\alpha}{k} \{iooH\_{0s}(s) + F\_N\left[s, A(s), \frac{\mathrm{d}A(s)}{\mathrm{ds}}, B(s), \frac{\mathrm{dB}(s)}{\mathrm{ds}}\right] \} \cos ks, \end{aligned} \tag{38}$$

where *FN* = *F* + *F*0 is the total self-field of the slot.

Assuming, as in Section 4, *dA*(*s*) *ds* =0 and *dB*(*s*) *ds* =0 in the right-hand members of equations (38) and making partial averaging over the variable *s*, we derive the equations of the first approx‐ imation by averaging method

$$\begin{aligned} \frac{\mathbf{d}\overline{A}(s)}{\mathbf{ds}} &= -\alpha \left\{ \frac{i\alpha}{k} H\_{0s}(s) + \overline{F}\_N[s, \overline{A}, \overline{B}] \right\} \sin ks \,\, \\\frac{\mathbf{d}\overline{B}(s)}{\mathbf{ds}} &= +\alpha \left\{ \frac{i\alpha}{k} H\_{0s}(s) + \overline{F}\_N[s, \overline{A}, \overline{B}] \right\} \cos ks \,\, \end{aligned} \tag{39}$$

where

**5. Solution of equation for current in a slot between two semi-infinite**

of waveguide cross-section and has no contacts with waveguide walls (Figure 4).

A starting point for the analysis is equation (10) written as (index *sl* is omitted)

d d d d

2 2 <sup>0</sup> 0 0 ( )4 ( ) ( ) (, ) (, ) , (, )

*<sup>e</sup> k Js s i H s k Js G ss G ss s*

æ ö æ ö ç ÷ <sup>+</sup> ¢ ¢ =- - + ç ÷ ¢ ¢ ¢¢ é ù <sup>+</sup> ë û ¢ è ø è ø ò ò (34)

Isolating the logarithmic singularity in the kernel of equation (34) as in (17), we reduce the

{ } <sup>d</sup>

( ) () () ,() ,() . *<sup>s</sup> J s kJs i H s F sJs F sJs*

Here *<sup>α</sup>* =1 / 8ln *de* / (8*<sup>L</sup>* ) is the natural small parameter of the problem (|*<sup>α</sup>* | <<1), *de* <sup>=</sup>*d e* <sup>−</sup>

equivalent slot width which takes into account a real wall thickness *h* (*h* / *λ* <<1) [3],

2 0 0

a w

+= + +

*s s s*

is the Green's function of the slot in infinite perfectly conducting plane,

) are the Green's functions, which takes into account multiple reflection from walls

é ùé ù

1 2

ë ûë û (35)

*πh* <sup>2</sup>*<sup>d</sup>* is

*V V*

Now let us solve the second key problem. Let a resonant iris is placed in infinite hollow (*ε*<sup>1</sup> =*μ*<sup>1</sup> =*ε*<sup>2</sup> =*μ*<sup>2</sup> =1) rectangular waveguide so that its slot has arbitrary orientation in the plane

**rectangular waveguides**

162 Advanced Electromagnetic Waves

**Figure 4.** The problem geometry and notations

where 4

of volumes.

*G*0*<sup>s</sup> <sup>V</sup>*1,2(*s*, *s*′

*e* <sup>−</sup>*ikR*(*s*,*<sup>s</sup>* ′ )

*R*(*s*, *s*′ )

d d


*s s Rss*

2 (, ) 2

2 2

equation (34) to an integral equation with small parameter

2

d 2

*s*

*L L ikR s s*

*L L*


w

$$\begin{aligned} \overline{\boldsymbol{F}}\_{N}[\mathbf{s}, \overline{\boldsymbol{A}}, \overline{\boldsymbol{B}}] &= [\overline{\boldsymbol{A}}(\mathbf{s}')\sin k\mathbf{s}' - \overline{\boldsymbol{B}}(\mathbf{s}')\cos k\mathbf{s}'] \mathbf{G}\_{\boldsymbol{s}}^{\mathbb{Z}}\langle\mathbf{s}, \mathbf{s}'\rangle \Big|\_{-\mathbf{L}}^{\mathbb{L}} \\ \mathbf{G}\_{\boldsymbol{s}}^{\mathbb{Z}}\langle\mathbf{s}, \mathbf{s}'\rangle &= \mathbf{G}\_{\boldsymbol{s}}^{\mathbb{V}\_{1}}\langle\mathbf{s}, \mathbf{s}'\rangle + \mathbf{G}\_{\boldsymbol{s}}^{\mathbb{V}\_{2}}\langle\mathbf{s}, \mathbf{s}'\rangle \end{aligned} \tag{40}$$

is the slot total self-field, averaged over the slot length.

Solving the system (39), we obtain the general asymptotic expression for the current in narrow slot, located in arbitrary position relative to the walls of coupling volumes

$$f(\mathbf{s}) = \overline{A}(-L)\cos k\mathbf{s} + \overline{B}(-L)\sin k\mathbf{s} + a \int\_{-l}^{s} \left\{ \frac{i\alpha}{k} H\_{0s}(s') + \overline{F}\_N[s', \overline{A}, \overline{B}] \right\} \sin k(\mathbf{s} - \mathbf{s}')d\mathbf{s}'.\tag{41}$$

To determine constants *A*¯( <sup>±</sup> *<sup>L</sup>* ) and *B*¯( <sup>±</sup> *<sup>L</sup>* ) we will use the boundary conditions (7) and the symmetry conditions, uniquely related both to slot excitation method and its position in waveguide. Then, in terms of symmetric and antisymmetric magnetic current components, marked by indexes *s* and *a*, respectively, for arbitrary slot excitation by *H*0*s*(*s*)=*H*0*<sup>s</sup> <sup>s</sup>* (*s*) <sup>+</sup> *<sup>H</sup>*0*<sup>s</sup> <sup>a</sup>* (*s*) with an accuracy of order *α* <sup>2</sup> we have

$$\begin{split} f(\mathbf{s}) &= f^{\circ}(\mathbf{s}) + f^{\circ}(\mathbf{s}) = a \frac{i\alpha}{k} \Bigg\{ \Big\}\_{-\mathsf{L}}^{\circ} H\_{0\varsigma}(\mathbf{s}') \sin k(\mathbf{s} - \mathbf{s}') \mathrm{ds}' \\ &- \frac{\sin k(\mathsf{L} + \mathsf{s}) \Big\{}^{\mathsf{L}} H\_{0\varsigma}^{\varsigma}(\mathbf{s}') \sin k(\mathbf{L} - \mathbf{s}') \mathrm{ds}' \bigg{} - \frac{\sin k(\mathsf{L} + \mathsf{s}) \int\_{0}^{L} H\_{0\varsigma}^{\varsigma}(\mathbf{s}') \sin k(\mathbf{L} - \mathbf{s}') \mathrm{ds}' \bigg{} \Big{} \\ &- \frac{\sin 2kL + aN^{\prime}(\mathsf{k}d\_{\varsigma}, 2\mathsf{k}L)}{\sin 2kL + aN^{\prime}(\mathsf{k}d\_{\varsigma}, 2\mathsf{k}L)} \Bigg{]}, \end{split} \tag{42}$$

where *N <sup>s</sup>* (*kde*, <sup>2</sup>*kL* ) and *<sup>N</sup> <sup>a</sup>* (*kde*, 2*kL* ) are the functions of self-field which are equal

$$\begin{aligned} N^{\varepsilon}(kd\_{\varepsilon}, 2kL) &= \int\_{-L}^{L} [\mathbf{G}\_{s}^{\varepsilon}(\mathbf{s}, -L) + \mathbf{G}\_{s}^{\varepsilon}(\mathbf{s}, L)] \sin k(L-s) \mathrm{d}s \,, \\ N^{\varepsilon}(kd\_{\varepsilon}, 2kL) &= \int\_{-L}^{L} [\mathbf{G}\_{s}^{\varepsilon}(\mathbf{s}, -L) - \mathbf{G}\_{s}^{\varepsilon}(\mathbf{s}, L)] \sin k(L-s) \mathrm{d}s \, \end{aligned} \tag{43}$$

which are completely defined by the Green's functions of the coupling volumes.

Supposing that dominant wave *H*10 with amplitude *H*<sup>0</sup> is propagated from the region *z* = −*∞*, we have

$$H\_{0s}(\mathbf{s}) = 2H\_0 \cos \phi \left[ \sin \frac{\pi \mathbf{x}\_0}{a} \cos \frac{\pi (s \cos \phi)}{a} + \cos \frac{\pi \mathbf{x}\_0}{a} \sin \frac{\pi (s \cos \phi)}{a} \right]. \tag{44}$$

The symmetric and antisymmetric components of the slot current, relative to the slot center *s* =0, become equal

$$\begin{split} f(s) &= f\_o f(s) = -\alpha 2H\_o \cos \phi \frac{2io \,/\, k^2}{[1 - (k\_\rho / k)^2][\sin 2kL + \alpha 2W\_\rho^{\rm u}(kd\_\rho, 2kL)]} \\ &\times \left\{ \sin \frac{\pi \mathbf{x}\_o}{a} \sin kL (\cos ks \cos k\_\rho L - \cos kL \cos k\_\rho s) + \cos \frac{\pi \mathbf{x}\_o}{a} \cos kL (\sin ks \sin k\_\rho L - \sin kL \sin k\_\rho s) \right\}, \end{split} \tag{45}$$

where *J*0 is current amplitude, *<sup>f</sup>* (*s*) is the current distribution function, *k<sup>φ</sup>* <sup>=</sup> *<sup>π</sup> <sup>a</sup>* cos*φ*, *W<sup>φ</sup> sa* (*kde*, 2*kL* ) is the function of slot self-field, defined by formulas (43).

Reflection and transmission coefficients, *S*11 and *S*<sup>12</sup> for the dominant wave in the slot iris are define by the current as

$$S\_{11} = \left(1 + S\_{12}\right)e^{2j\varphi},\ S\_{12} = -\alpha \frac{16\pi k\_{\varphi}\cos^{2}\phi\, f(k\_{\varphi}L)}{\dot{\alpha}bk^{3}[1 - \left(k\_{\varphi}\,/\, k\right)^{2}][\cos kL + \alpha 2\mathcal{W}\_{\phi}\{\mathrm{kd}\_{\varepsilon}, kL\}]},\tag{46}$$

$$f(k\_{\boldsymbol{\phi}}\boldsymbol{L}) = 2\cos k\_{\boldsymbol{\phi}}L\frac{\sin kL\cos k\_{\boldsymbol{\phi}}L - (k\_{\boldsymbol{\phi}}/k)\cos kL\sin k\_{\boldsymbol{\phi}}L}{1 - (k\_{\boldsymbol{\phi}}/k)^2} - \cos kL\frac{\sin 2k\_{\boldsymbol{\phi}}L + 2k\_{\boldsymbol{\phi}}L}{2(k\_{\boldsymbol{\phi}}/k)}J$$

where *kg* <sup>=</sup> *<sup>k</sup>* <sup>2</sup> <sup>−</sup>(*<sup>π</sup>* / *<sup>a</sup>*) 2 is the propagation constant of *H*10 wave.

1 2

slot, located in arbitrary position relative to the walls of coupling volumes

*V V*

¢¢¢ = +

(, ) (, ) (, )

*ss s*

is the slot total self-field, averaged over the slot length.

S

164 Advanced Electromagnetic Waves

with an accuracy of order *α* <sup>2</sup>

*s a*

(*kde*, <sup>2</sup>*kL* ) and *<sup>N</sup> <sup>a</sup>*

*s*

*a*

0 0

where *N <sup>s</sup>*

we have

*G ss G ss G ss*

[ , , ] [ ( )sin ( )cos ] ( , ) ,

= ¢ ¢¢ ¢ ¢ -

Solving the system (39), we obtain the general asymptotic expression for the current in narrow

*<sup>N</sup> <sup>s</sup>*

ì ü

d <sup>0</sup> ( ) ( )cos ( )sin ( ) [ , , ] sin ( ) . *s*

= - +- + í ý ¢ ¢ + - ¢ ¢

To determine constants *A*¯( <sup>±</sup> *<sup>L</sup>* ) and *B*¯( <sup>±</sup> *<sup>L</sup>* ) we will use the boundary conditions (7) and the symmetry conditions, uniquely related both to slot excitation method and its position in waveguide. Then, in terms of symmetric and antisymmetric magnetic current components,

d

, sin 2 ( ,2 ) sin 2 ( ,2 )

+ + ï

(*kde*, 2*kL* ) are the functions of self-field which are equal

+ -+ - ¢ ¢¢ ¢ ¢¢ï

*s a e e*

*kL N kd kL kL N kd kL*

*kL s H s kL s s kL s H s kL s s*

0 0

( ,2 ) [ ( , ) ( , )]sin ( ) ,

= -+ -

= -- -

Supposing that dominant wave *H*10 with amplitude *H*<sup>0</sup> is propagated from the region *z* = −*∞*,

0 0

*aa aa*

é ù

 p

( cos ) ( cos ) ( ) 2 cos sin cos cos sin . *<sup>s</sup>*

pj

= + ê ú

*N kd kL G s L G s L k L s s*

S S

*N kd kL G s L G s L k L s s*

which are completely defined by the Green's functions of the coupling volumes.

*x x s s Hs H*

p

j

S S

( ,2 ) [ ( , ) ( , )]sin ( ) ,

sin ( ) ( )sin ( ) sin ( ) ( )sin ( )

<sup>ï</sup> - - <sup>ý</sup>

d d

ò ò (42)

 a

d

d

 pj

ë û (44)

*L <sup>i</sup> J s A L ks B L ks H s F s A B k s s s <sup>k</sup>* w

marked by indexes *s* and *a*, respectively, for arbitrary slot excitation by *H*0*s*(*s*)=*H*0*<sup>s</sup>*

*L L s a s s L L*


we have

0

*L*

ò



ò

*e ss L L*

*e ss L*

*s L*

() () () ( )sin ( )

a

w

*<sup>i</sup> Js J s J s H s ks s s <sup>k</sup>*

a

*s*

ò


ìï =+= <sup>í</sup> ¢ ¢¢ ïî


a

*<sup>N</sup> <sup>s</sup> <sup>L</sup>*

*F s A B A s ks B s ks G s s*

*L*


î þ ò (41)

(40)

*<sup>s</sup>* (*s*) <sup>+</sup> *<sup>H</sup>*0*<sup>s</sup>*

ü

ï þ *<sup>a</sup>* (*s*)

(43)

S

Figure 5 shows the theoretical and experimental wavelength dependences of power reflection coefficient |*S*11|2 for the iris, which oriented so that the angle between slot axis {0*s*} and waveguide axis {0*x*} are 0*<sup>O</sup>* and 30*<sup>O</sup>*.

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 confirmed by comparative calculations, obtained by the numerical method of moments and other methods, in particular, by the finite element method implemented in the software package *Ansoft HFSS*.
