**4. Solution of integral equation for current in an impedance vibrator, located in unbounded free space**

Let us use the equation (9) with the approximate kernel (13), being a quasi-one-dimensional analog ofthe exactintegral equation with kernel(11) as starting pointforthe analysis. Note that impedance *zi* (*s*)≡*const*, *ε*<sup>1</sup> =*μ*<sup>1</sup> =1, and index *v* is omitted. Thus, the equation may be written as

$$\left(\frac{\text{d}^2}{\text{ds}^2} + k^2\right)\Big|\_{-\text{L}}^{\text{L}} f(s') \frac{e^{-i\text{l}\mathbb{R}\langle s, s'\rangle}}{R\langle s, s'\rangle} \text{ds}' = -i\alpha \text{E}\_{0s}(\text{s}) + i\alpha \text{z}\_i f(\text{s}),\tag{15}$$

where *R*(*s*, *s*′ )= (*s* −*s*′ ) <sup>2</sup> + *r* <sup>2</sup> . Let us isolate the logarithmic singularity in the kernel of equation (15) by identical transformation

$$\int\_{-L}^{L} f(s') \frac{e^{-i\mathcal{R}(s,s')}}{\mathcal{R}(s,s')} \mathrm{d}s' = \mathcal{Q}(s)f(s) + \int\_{-L}^{L} \frac{f(s')e^{-i\mathcal{R}(s,s')} - f(s)}{\mathcal{R}(s,s')} \mathrm{d}s'.\tag{16}$$

Here

Now we take into account that *n*

d

m

Ohm/m, *E*0*sm*

volumes *V*1 and *V*2.

d 2

*s*

æ ö

axes, *Gsm V*1 (*sm*,*n*, *<sup>s</sup>*′

m

Here *e* → *s* ′ *v* and *e* → *s* ′ *sl*

*s*

2

æ ö

154 Advanced Electromagnetic Waves

è ø

2 2 1

we

æ ö

è ø

1 0

*q*

1

= -

(*sm*) and *H*0*sn*

*<sup>m</sup>*) and *Gsn*

2

*v L*

*v*

*L*


d d

*s s*

1 2

*v*

*V s vv*

*v*

2 2

*m L*

*<sup>M</sup> <sup>L</sup>*

2

å ò <sup>r</sup>

→ *<sup>m</sup>*, *J* → *<sup>m</sup>*(*r* →

d d <sup>d</sup> d d

( ) ( ) ( ),

1 2 1

rot d

+ = ¢ ¢¢ -

*V*1(2)

*k JsG ss s s s*

1 1 () (,)

*s q iq q q q <sup>L</sup> <sup>N</sup> <sup>V</sup>*

*n*

*i E s z sJs*

é ù <sup>=</sup> - - ê ú ë û

*n*

= -

*m*

*m*

å ò

2 2

*q m n L L*

1

*nn s pn n p p n L*

2 2 1

ç ÷ <sup>+</sup> ¢ ¢¢ + + ç ÷

1

*ike J s G s s s i H s*

*V s mm s pm m s p*

*pm p*

Here *zim*(*sm*) are internal lineal impedance of the vibrators (*ZSm*(*r*

(*sm*,*n*, *<sup>s</sup>*′

them, the system (8) splits into two independent equations:

2 2

p

p

*<sup>e</sup> G ss*


1,2

*sl*

*V s sl sl*

*sl L L sl*

*V*

*sl sl*

*L L*


*sl sl*

ç ÷ + ¢ ¢¢ =- +

d d d d

æ ö æ ö ç ÷ <sup>+</sup> ¢ ¢¢ + + ç ÷ ¢ ¢¢ = -

 m

*v v*


<sup>d</sup> <sup>d</sup>

*n*

of linear integral equations relative to the currents in the vibrators and slots

<sup>d</sup> d rot <sup>d</sup>

ç ÷ + - ¢ ¢¢ ¢ ¢¢ ç ÷

() (, ) ( ).

*k J s G s s s ike J s G s s s*

*m n*

*m n*

1 1

å å ò ò

= = - -

*M N L L*

equations (3a) and (3b) on the axes of the vibrators and slots, respectively, and arrive at a system

1 1

( ) (, ) () (,)

2 2 1

æ ö

è ø

*n*

*<sup>L</sup> <sup>N</sup> <sup>V</sup>*

→

*<sup>n</sup>*) are components of the tensor Green's functions in the

d


 j

yj


d

cx x

ç ÷ ¢ ¢¢ ç ÷

*k JsG ss s*

*n*

*n L*

(*sn*) are projections of extraneous sources on the vibrators and slots

 we

= -

å ò

*mq n*

 m

w

For solitary vibrator or slot as well as for the absence of electromagnetic interaction between

<sup>2</sup> <sup>1</sup> 1 0 <sup>1</sup> () (,) ( ) ( ) ( ),

*k J sG s s s i E s i zs J s*

*vv s vv v s v ivvv*

è ø <sup>ò</sup> (9)

*sl sl sl*

*sl sl s sl sl sl sl sl s sl sl sl s sl*

we

*v v*

1 2

*V V*

*v v*


*<sup>e</sup> G ss r r ss r*

*ik s s r*

1,2

*s s*

*d ik s s*

*<sup>d</sup> sl sl*

/ 2 ( ) ()


2 2 / 2 (,) () . ( ) () *sl sl*

1 1 () (,) ( ) ( , ) ( ).

*k J sG s s s k J sG s s s iH s*

are unit vectors of vibrator and slot axes at the sources, and

2 2 <sup>1</sup> ( ) [2 sin( / 2)] 2 2 (,) (, ) , ( ) [2 sin( / 2)]

j

j

2 2

x

x

è ø è ø ò ò (10)

2 2 1 20

0

*V V mm s qm m s nn s qn n*

r

*<sup>m</sup>*) ≪1 according to inequalities (4) and project the

d

→

w

*<sup>m</sup>*)) measured in

(8)

2

*n*

() (,)

*<sup>m</sup>*)=2*πrmzim*(*r*

*nn s pn n*

$$
\Omega(s) = \int\_{-L}^{L} \frac{\text{ds}'}{\sqrt{\left(s - s'\right)^2 + r^2}} = \Omega + \gamma(s)\_{\prime} \tag{17}
$$

*<sup>γ</sup>*(*s*)=ln (*<sup>L</sup>* <sup>+</sup> *<sup>s</sup>*) <sup>+</sup> (*<sup>L</sup>* <sup>+</sup> *<sup>s</sup>*) <sup>2</sup> + *r* <sup>2</sup> (*L* −*s*) + (*L* −*s*) <sup>2</sup> + *r* <sup>2</sup> <sup>4</sup>*<sup>L</sup>* <sup>2</sup> is a function, equal to zero at the vibrator center and reaching maximal value at its ends where the current in accordance with boundary condition (7) is equal to zero, Ω=2ln <sup>2</sup>*<sup>L</sup> <sup>r</sup>* is a large parameter. Then, equation (15) in view of (16) is transformed to integral equation with a small parameter

$$\frac{\text{d}^2 \text{J}(\text{s})}{\text{ds}^2} + k^2 \text{J}(\text{s}) = \alpha \left\{ \text{i}o\nu \mathbb{E}\_{0\text{s}}(\text{s}) + F[\text{s}\_\text{} \text{J}(\text{s})] - \text{i}o\nu \mathbb{Z}\_{\text{l}} \text{J}(\text{s}) \right\}. \tag{18}$$

Here *<sup>α</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> <sup>Ω</sup> <sup>=</sup> <sup>1</sup> 2ln *<sup>r</sup>* / (2*<sup>L</sup>* ) is a natural small parameter of the problem (|*<sup>α</sup>* | <<1),

$$\begin{split} \left[F\{s,f(s)\}\right] &= -\frac{\mathrm{d}f(s')}{\mathrm{d}s'} \frac{e^{-i\mathrm{d}\mathcal{R}(s,s')}}{\mathrm{R}\langle s,s'\rangle} \Big|\_{-L}^{L} + \left[\frac{\mathrm{d}^{2}f(s)}{\mathrm{d}s^{2}} + k^{2}f(s)\right] \Big\gamma(s) \\ &+ \int\_{-L}^{L} \frac{\left[\mathrm{d}^{2}f(s')}{\mathrm{d}s'^{2}} + k^{2}f(s')\right] e^{-i\mathrm{d}\mathcal{R}(s,s')} - \left[\frac{\mathrm{d}^{2}f(s)}{\mathrm{d}s^{2}} + k^{2}f(s)\right]}{\mathrm{R}\langle s,s'\rangle} ds' \end{split} \tag{19}$$

is the vibrator self-field in free space.

To find the approximate analytic solution of equation (18) we will use the asymptotic averaging method. The basic principles of the method are presented in [3,4]. To reduce the equation (18) to a standard equation system with a small parameter in compliance with the method of variation of constants we will change variables

$$\begin{aligned} \mathbf{J}(\mathbf{s}) &= A(\mathbf{s}) \cos k\mathbf{s} + B(\mathbf{s}) \sin k\mathbf{s}, \\ \frac{\mathbf{d}J(\mathbf{s})}{\mathbf{d}s} &= -A(\mathbf{s}) k \sin k\mathbf{s} + B(\mathbf{s}) k \cos k\mathbf{s}, \\ \frac{\mathbf{d}^2 J(\mathbf{s})}{\mathbf{d}s^2} + k^2 I(\mathbf{s}) &= -\frac{\mathbf{d}A(\mathbf{s})}{\mathbf{d}s} \sin k\mathbf{s} + \frac{\mathbf{d}B(\mathbf{s})}{\mathbf{d}s} \cos k\mathbf{s}, \end{aligned} \tag{20}$$

where *A*(*s*) and *B*(*s*) are new unknown functions. Then the equation (18) reduces to a system of integral equations

Electromagnetic Waves Excitation by Thin Impedance Vibrators and Narrow Slots in Electrodynamic Volumes http://dx.doi.org/10.5772/61188 157

$$\begin{split} \frac{\mathrm{d}A(\mathrm{s})}{\mathrm{d}\mathbf{s}} &= -\frac{\alpha}{k} \Big[ iaoE\_{\mathrm{bs}}(\mathrm{s}) + F\Big[ s, A(\mathrm{s}), \frac{\mathrm{d}A(\mathrm{s})}{\mathrm{d}\mathbf{s}}, B(\mathrm{s}), \frac{\mathrm{dB}(\mathrm{s})}{\mathrm{d}\mathbf{s}} \Big] - io\mathrm{z}\_{\mathrm{l}}[A(\mathrm{s})\cos k\mathbf{s} + B(\mathrm{s})\sin k\mathbf{s}] \Big] \sin k\mathbf{s}, \\ \frac{\mathrm{d}B(\mathrm{s})}{\mathrm{d}\mathbf{s}} &= +\frac{\alpha}{k} \Big[ io\mathrm{z}\_{\mathrm{lo}}(\mathrm{s}) + F\Big[ s, A(\mathrm{s}), \frac{\mathrm{dA}(\mathrm{s})}{\mathrm{d}\mathbf{s}}, B(\mathrm{s}), \frac{\mathrm{dB}(\mathrm{s})}{\mathrm{d}\mathbf{s}} \Big] - io\mathrm{z}\_{\mathrm{l}}[A(\mathrm{s})\cos k\mathbf{s} + B(\mathrm{s})\sin k\mathbf{s}] \Big] \cos k\mathbf{s}. \end{split} \tag{21}$$

This system is equivalent to the equation (18) and represents the standard equations system unsolvable with respect to derivatives. The right-hand sides of the equations are proportional to small parameter *α*, therefore, the functions *A*(*s*) and *B*(*s*) in the left-hand sides of the equations system (21) are slowly varying functions and the system can be solved by the asymptotic averaging method. Then, we replace the system (21) by the simplified system wherein assume *dA*(*s*) *ds* =0 and *dB*(*s*) *ds* =0 in rigth-hand members and carry out partial averaging over the explicit variable *s* to obtain the equations of first approximation. The term *partial averaging* means that averaging operator acts on all terms, but containing *E*0*s*(*s*) and it may be done for the system (21). The averaged system can be written as

$$\begin{aligned} \frac{\mathbf{d}A(s)}{\mathbf{ds}} &= -\alpha \left\{ \frac{i\alpha}{k} E\_{0s}(s) + \overline{F}[s, \overline{A}(s), \overline{B}(s)] \right\} \sin ks + \chi \overline{B}(s), \\\frac{\mathbf{d}\overline{B}(s)}{\mathbf{ds}} &= +\alpha \left\{ \frac{i\alpha}{k} E\_{0s}(s) + \overline{F}[s, \overline{A}(s), \overline{B}(s)] \right\} \cos ks - \chi \overline{A}(s), \end{aligned} \tag{22}$$

where *χ* =*α iω* <sup>2</sup>*<sup>k</sup> zi* ,

d 2 2 ( ) ( ), ( )

¢ W = =W+

*s s s ss r*

g

<sup>2</sup> + *r* <sup>2</sup>

vibrator center and reaching maximal value at its ends where the current in accordance with

{ } <sup>d</sup>

+= + -

d d d d

2 2

2 2

*s s*

¢ é ù =- + + ê ú ¢ ¢ ë û

(, ) 2

*L*


( ) ( ) [ , ( )] () () (, )

*Js e J s FsJs kJs s s Rss <sup>s</sup>*

( ) ( ) ( ) [ , ( )] ( ) . *s i J s k Js i E s Fs Js i zJs*

2ln *<sup>r</sup>* / (2*<sup>L</sup>* ) is a natural small parameter of the problem (|*<sup>α</sup>* | <<1),

2

*ikR s s*


é ùé ù ¢ + -+ ¢ ê úê ú ¢ ë ûë û <sup>+</sup> ¢ ¢ <sup>ò</sup>

2 (, ) 2

( ) ( ) ( ) ( )

*J s J s kJs e kJs*

(, )

*Rss*

To find the approximate analytic solution of equation (18) we will use the asymptotic averaging method. The basic principles of the method are presented in [3,4]. To reduce the equation (18) to a standard equation system with a small parameter in compliance with the method of

( ) () () ( ) sin ( ) cos , cos sin 0 ,

=- + ç ÷ + =

where *A*(*s*) and *B*(*s*) are new unknown functions. Then the equation (18) reduces to a system

*J s As Bs A s k ks B s k ks ks ks s s s*

2


*<sup>r</sup>* is a large parameter. Then, equation (15) in

d

*s*

(18)

(19)

(20)

<sup>4</sup>*<sup>L</sup>* <sup>2</sup> is a function, equal to zero at the

w

g

æ ö

è ø

*L*

*L*

<sup>2</sup> + *r* <sup>2</sup> (*L* −*s*) + (*L* −*s*)

view of (16) is transformed to integral equation with a small parameter

a w

d d d d


d d d d d d

( ) () () ( ) sin cos ,

*J s As Bs kJs ks ks s s s*

*<sup>L</sup> ikR s s*


boundary condition (7) is equal to zero, Ω=2ln <sup>2</sup>*<sup>L</sup>*

2 2 0

d 2

<sup>Ω</sup> <sup>=</sup> <sup>1</sup>

*s*

*L*

*L*


variation of constants we will change variables

( ) ( )cos ( )sin ,

*J s A s ks B s ks*

= +

d dd d d d

+ =- +

2

is the vibrator self-field in free space.

2

of integral equations

2

*<sup>γ</sup>*(*s*)=ln (*<sup>L</sup>* <sup>+</sup> *<sup>s</sup>*) <sup>+</sup> (*<sup>L</sup>* <sup>+</sup> *<sup>s</sup>*)

156 Advanced Electromagnetic Waves

Here *<sup>α</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup>

$$\left[\overline{F}\text{[s, }\overline{A}(s), \overline{B}(s)] = \left[\overline{A}(s')\sin k s' - \overline{B}(s')\cos k s'\right] \frac{e^{-ikR(s, s')}}{R(s, s')}\right]\_{-L}^{L} \tag{23}$$

is self-field of the vibrator (19), averaged over its length.

We will seek the solution of the equations system (22) in the form

$$\begin{aligned} \overline{A}(\mathbf{s}) &= \mathbb{C}\_1(\mathbf{s}) \cos \chi \mathbf{s} + \mathbb{C}\_2(\mathbf{s}) \sin \chi \mathbf{s}, \\ \overline{B}(\mathbf{s}) &= -\mathbb{C}\_1(\mathbf{s}) \sin \chi \mathbf{s} + \mathbb{C}\_2(\mathbf{s}) \cos \chi \mathbf{s}. \end{aligned} \tag{24}$$

Then, substitution (24) into (22) gives

$$\begin{split} \frac{\mathrm{dC}\_{1}\mathrm{(s)}}{\mathrm{ds}} &= -\alpha \left\{ \frac{i\alpha}{k} E\_{0s}(\mathrm{s}) + \overline{F}[\mathrm{s}, \mathrm{C}\_{1}, \mathrm{C}\_{2}] \right\} \mathrm{sin}(\mathrm{k} + \mathrm{\mathcal{K}}) \mathrm{s}, \\ \frac{\mathrm{dC}\_{2}\mathrm{(s)}}{\mathrm{ds}} &= +\alpha \left\{ \frac{i\alpha}{k} E\_{0s}(\mathrm{s}) + \overline{F}[\mathrm{s}, \mathrm{C}\_{1}, \mathrm{C}\_{2}] \right\} \mathrm{cos}(\mathrm{k} + \mathrm{\mathcal{K}}) \mathrm{s}. \end{split} \tag{25}$$

Then we find *C*1(*s*) and *C*2(*s*) by solving system (25), determine *A*¯(*<sup>s</sup>*) и *B*¯(*<sup>s</sup>*) from (24), and substitute them as approximating functions for the current into (20). Thus, the general asymptotic expression in parameter *α* for the current in a thin impedance vibrator under arbitrary excitation may be presented as

$$\mathbf{J}(\mathbf{s}) = \overline{A}(-\mathbf{L})\cos(k\mathbf{s} + \chi\mathbf{L}) + \overline{B}(-\mathbf{L})\sin(k\mathbf{s} + \chi\mathbf{L}) + \alpha \int\_{-\mathbf{L}}^{\circ} \left\{ \frac{i\alpha}{k} E\_{0s}(s') + \overline{F}[\mathbf{s}', \overline{A}, \overline{B}] \right\} \sin\tilde{k}(\mathbf{s} - \mathbf{s}')d\mathbf{s}',\tag{26}$$

where *k* ˜ <sup>=</sup>*<sup>k</sup>* <sup>+</sup> *<sup>χ</sup>* <sup>=</sup>*<sup>k</sup>* <sup>+</sup> *<sup>i</sup>*(*<sup>α</sup>* /*r*)*<sup>Z</sup>*¯ *S* , *Z*¯ *S* =*R*¯ *<sup>S</sup>* <sup>+</sup> *iX*¯ *<sup>S</sup>* is the normalized complex surface impedance: *Z*¯ *<sup>S</sup>* =2*πrzi* / *Z*0.

For electrically thin vibrators (|(*k εμr*) 2 *ln*(*k εμri* )| <<1, *ri* is the radius of the inner conductor) with the parameters of material *ε*, *μ*, *σ*, from which they are made, the formulas of the distributed surface impedance *Z*¯ *<sup>S</sup>* are presented in Table 1.



**Table 1.** The formulas of the distributed surface impedance *<sup>Z</sup>*¯ *S*

Then we find *C*1(*s*) and *C*2(*s*) by solving system (25), determine *A*¯(*<sup>s</sup>*) и *B*¯(*<sup>s</sup>*) from (24), and substitute them as approximating functions for the current into (20). Thus, the general asymptotic expression in parameter *α* for the current in a thin impedance vibrator under

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

= - + +- + + + í ý ¢ ¢ - ¢ ¢

*ln*(*k εμri*

with the parameters of material *ε*, *μ*, *σ*, from which they are made, the formulas of the

*<sup>S</sup>* are presented in Table 1.

 a

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

c

*<sup>S</sup>* <sup>+</sup> *iX*¯

2

**№ Design type of vibrator Breadboard view of vibrator Formula for impedance**

*s*

*L*


*s*

î þ <sup>ò</sup> % % % (26)

ì ü

*<sup>S</sup>* is the normalized complex surface impedance:

*Z*¯ *<sup>S</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>i</sup> <sup>Z</sup>*0*σ*Δ<sup>0</sup>

*Z*¯

*Z*¯ *<sup>S</sup>* = −*i*

*Z*¯

*Z*¯

*Z*¯ *<sup>S</sup>* (*s*)=*<sup>R</sup>*¯

*<sup>S</sup>* <sup>=</sup> <sup>1</sup>

*Z*0*σh <sup>R</sup>* + *ikr*(*ε* −1)/ 2

*Z*0*σh <sup>R</sup>* −*i* / *krμ*ln(*r* /*ri*

)

*<sup>S</sup>* (*s*)

*<sup>S</sup>* (*s*) <sup>+</sup> *iX*¯

)

2 *krε*

*L* <sup>2</sup> *L* <sup>1</sup> + *L* <sup>2</sup>

*<sup>S</sup>* <sup>=</sup> <sup>1</sup>

*<sup>S</sup>* =*ikrμ*ln(*r* /*ri*

is the radius of the inner conductor)

w

)| <<1, *ri*

arbitrary excitation may be presented as

c

> *S* , *Z*¯ *S* =*R*¯

˜ <sup>=</sup>*<sup>k</sup>* <sup>+</sup> *<sup>χ</sup>* <sup>=</sup>*<sup>k</sup>* <sup>+</sup> *<sup>i</sup>*(*<sup>α</sup>* /*r*)*<sup>Z</sup>*¯

For electrically thin vibrators (|(*k εμr*)

distributed surface impedance *Z*¯

1 The solid metallic cylinder of the

2 The dielectrical metalized cylinder with covering, made of the metal of

thickness

3 The metal-dielectrical cylinder (*L1* is the thickness of the metal disk, *L2* is the thickness of the dielectric disc)

4 The magnetodielectrical metalized cylinder with the inner conducting cylinder with the radius *ri*

5 The metallic cylinder with covering, made of magnetodielectric of the *r-ri* thickness, or the corrugated

skin-layer thickness

radius, Δ<sup>0</sup> =*ω* / *k* 2*πσωμ* is the

where *k*

*<sup>S</sup>* =2*πrzi* / *Z*0.

158 Advanced Electromagnetic Waves

*r*Δ<sup>0</sup>

the *hR*Δ<sup>0</sup>

cylinder

*Z*¯

The formulas have been obtained in the frames of impedance conception [4], and they are just for thin cylinders both of infinite and finite extension, located in free space. It is necessary to introduce the multiplier *μ*<sup>1</sup> / *ε*1 in all formulas for the vibrator in the material medium with the *ε*1 and *μ*1 parameters. We note, that most of the formulas for impedances include the parameters *ε* and *μ*, smooth change of which (in the case of their dependence from the static electrical and magnetic fields) and the characteristics of radiation of the system, correspond‐ ingly, (at its fixed geometrical sizes) can be made, for example, by external field effects.

The constants *A*¯( ± *<sup>L</sup>* ) and *B*¯( ± *<sup>L</sup>* ) can be found employing the boundary conditions (7) and the symmetry conditions [5], unambiguously related to a method of vibrator excitation; if *E*0*s*(*s*)= *E*0*<sup>s</sup> <sup>s</sup>* (*s*), *J*(*s*)= *J*(−*s*)= *J <sup>s</sup>* (*s*) and *A*¯(<sup>−</sup> *<sup>L</sup>* )= *<sup>A</sup>*¯( + *<sup>L</sup>* ) , *B*¯(<sup>−</sup> *<sup>L</sup>* )= <sup>−</sup> *<sup>B</sup>*¯( + *<sup>L</sup>* ) ; if *E*0*s*(*s*)= *<sup>E</sup>*0*<sup>s</sup> <sup>a</sup>* (*s*), *J*(*s*)= − *J*(−*s*)= *J <sup>a</sup>* (*s*) and *A*¯(<sup>−</sup> *<sup>L</sup>* )= <sup>−</sup> *<sup>A</sup>*¯( + *<sup>L</sup>* ) , *B*¯(<sup>−</sup> *<sup>L</sup>* )= *<sup>B</sup>*¯( + *<sup>L</sup>* ). Then, in terms of symmetric and antisymmetric current components, marked by indexes *s* and *a*, respectively, for arbitrary vibrator excitation by *E*0*s*(*s*)= *E*0*<sup>s</sup> <sup>s</sup>* (*s*) + *<sup>E</sup>*0*<sup>s</sup> <sup>a</sup>* (*s*) it is not difficult to show that

$$\begin{split} f(\mathbf{s}) &= f^\circ(\mathbf{s}) + f^\circ(\mathbf{s}) = a \frac{i\alpha}{k} \left\{ \int\_{-L}^{\bar{s}} E\_{0s}(s') \sin \tilde{k}(\mathbf{s} - s') \, \mathrm{d}s' \\ &- \frac{\sin \tilde{k}(L+\mathbf{s}) + \alpha P^\circ[kr, \tilde{k}(L+\mathbf{s})]}{\sin 2\tilde{k} + \alpha P^\circ(kr, 2\bar{k}L)} \int\_{-L}^{L} E\_{0s}^\circ(s') \sin \tilde{k}(L-s') \, \mathrm{d}s' \\ &- \frac{\sin \tilde{k}(L+\mathbf{s}) + \alpha P^\circ[kr, \tilde{k}(L+\mathbf{s})]}{\sin 2\tilde{k} + \alpha P^\circ(kr, 2\bar{k}L)} \int\_{-L}^{L} E\_{0s}^\circ(s') \sin \tilde{k}(L-s') \, \mathrm{d}s' \right\} \, \end{split} \tag{27}$$

where *P <sup>s</sup>* and *P <sup>a</sup>* are the functions of vibrator self-fields equal to

$$P^{\prime}[kr,\tilde{k}(L+s)] = \int\_{-\mathbb{L}}^{\circ} \left[ \frac{e^{-i\mathbb{R}\mathcal{S}(s^{\prime},-L)}}{R(s^{\prime},-L)} + \frac{e^{-i\mathbb{R}\mathcal{S}(s^{\prime},L)}}{R(s^{\prime},L)} \right] \sin\tilde{k}(s-s^{\prime}) \,\mathrm{d}s^{\prime} \bigg|\_{s=L} = P^{\prime}\{kr,2\tilde{k}L\}, \tag{a}$$
 
$$P^{\prime}[kr,\tilde{k}(L+s)] = \int\_{-\mathbb{L}}^{\circ} \left[ \frac{e^{-i\mathbb{R}\mathcal{S}(s^{\prime},-L)}}{R(s^{\prime},-L)} - \frac{e^{-i\mathbb{R}\mathcal{S}(s^{\prime},L)}}{R(s^{\prime},L)} \right] \sin\tilde{k}(s-s^{\prime}) \,\mathrm{d}s^{\prime} \bigg|\_{s=L} = P^{\prime}\{kr,2\tilde{k}L\}. \tag{b}$$

It is evident that if an impedance vibrator is located in restricted volume *V* , the expression for the current coincides with (27), but the functions of vibrator self-field (28) must contain components of electric Green's function for corresponding electrodynamic volume.

Let us consider a problem of vibrator excitation at its geometric center by a lumped EMF with amplitude *V*0. The mathematical model of excitation is presented as

$$E\_{0s}(\mathbf{s}) = E\_{0s}^s(\mathbf{s}) = V\_0 \delta(\mathbf{s} - \mathbf{0}),\tag{29}$$

where *δ*(*s* −0)=*δ*(*s*) is Dirac delta-function. Then the expression for the current (27) is

$$J(\mathbf{s}) = -\alpha V\_0 \left(\frac{\mathbf{i}\alpha}{2\tilde{k}}\right) \frac{\sin\tilde{k}(L-\|\mathbf{s}\|) + \alpha P\_\delta^\*(kr, \tilde{k}\mathbf{s})}{\cos\tilde{k}L + \alpha P\_\mathcal{L}^\*(kr, \tilde{k}L)}.\tag{30}$$

Here *P<sup>δ</sup> s* (*kr*, *k* ˜*s*)=*P <sup>s</sup> kr*, *k* ˜(*L* + *s*) −(sin*k* ˜*s* + sin*k* ˜ <sup>|</sup>*<sup>s</sup>* |) *PL s* (*kr*, *k* ˜ *L* ) and *P <sup>s</sup> kr*, *k* ˜(*L* + *s*) are defined by the formula (28a). Explicit expressions for *P<sup>δ</sup> s* (*kr*, *k* ˜*s*) and *PL s* (*kr*, *k* ˜ *L* ) can be expressed explicitly in terms of generalized integral functions [4,5]. Thus, *PL s* (*kr*, *k* ˜ *L* ) which will be needed below may be presented as

$$\begin{aligned} P\_{\perp}^{\circ}(\text{kr}, \tilde{\text{k}L}) &= \cos \tilde{\text{k}L} [2\ln 2 - \gamma(\text{L}) - (1/2)] \text{Cin} (2\tilde{\text{k}L} + 2\text{kL}) + \text{Cin} (2\tilde{\text{k}L} - 2\text{kL})] \\ &- (\text{i } / \text{2}) [\text{Si}(2\tilde{\text{k}L} + 2\text{kL}) - \text{Si}(2\tilde{\text{k}L} - 2\text{kL})] \\ &+ \sin \tilde{\text{k}L} \{ (1/2) [\text{Si}(2\tilde{\text{k}L} + 2\text{kL}) + \text{Si}(2\tilde{\text{k}L} - 2\text{kL})] - (\text{i } / \text{2}) [\text{Cin}(2\tilde{\text{k}L} + 2\text{kL}) - \text{Cin}(2\tilde{\text{k}L} - 2\text{kL})] \}. \end{aligned} \tag{31}$$

where *Si*(*x*) and *Cin*(*x*) are sine and cosine integrals of complex argument.

Since the current distribution (30) is now known we can calculate electrodynamic character‐ istics of an impedance vibrator. Thus, an input impedance *Zin* =*Rin* + *i Xin* of vibrator in a feed point is equal

$$Z\_{in} \text{[Ohm]} = \frac{V\_0}{J(0)} = \left(\frac{60\tilde{k}}{ak}\right) \frac{\cos\tilde{k}L + aP\_L^\*(kr, \tilde{k}L)}{\sin\tilde{k}L + aP\_{\delta L}(kr, \tilde{k}L)},\tag{32}$$

where

$$\begin{split} &P\_{\vartheta \downarrow}(kr, \tilde{k}L) = \Big[\int\_{-\ell}^{\ell} \frac{e^{-ikR(s,l)}}{R(s,l)} \sin \tilde{k} \, \mathrm{l} \, \mathrm{s} \, \mathrm{l}s \\ &= \sin \tilde{k}L \{-\gamma(\mathrm{L}) + (1/2)[\mathrm{Cinc}(2\tilde{k}L + 2k\mathrm{L}) - \mathrm{Cinc}(2\tilde{k}L - 2k\mathrm{L})] - \mathrm{Cinc}(\tilde{k}L + k\mathrm{L}) + \mathrm{Cinc}(\tilde{k}L - k\mathrm{L}) \\ &+ (i/2)[\mathrm{Si}(2\tilde{k}L + 2k\mathrm{L}) - \mathrm{Si}(2\tilde{k}L - 2k\mathrm{L})] - \mathrm{I}[\mathrm{Si}(\tilde{k}L + k\mathrm{L}) - \mathrm{Si}(\tilde{k}L - k\mathrm{L})] \, \mathrm{l} \\ &+ \cos \tilde{k}L[\mathrm{(1/2)}] \mathrm{Si}(2\tilde{k}L + 2k\mathrm{L}) + \mathrm{Si}(2\tilde{k}L - 2k\mathrm{L})] - \mathrm{Si}(\tilde{k}L + k\mathrm{L}) - \mathrm{Si}(\tilde{k}L - k\mathrm{L}) \\ &- (i/2)[\mathrm{Cinc}(2\tilde{k}L + 2k\mathrm{L}) + \mathrm{Cinc}(2\tilde{k}L - 2k\mathrm{L})] + \mathrm{I}[\mathrm{Cinc}(\tilde{k}L + k\mathrm{L}) + \mathrm{Cinc}(\tilde{k}L - k\mathrm{L})] \, \mathrm{l} . \end{split} \tag{33}$$

Note, that an input admittance *Yin* =*Gin* + *iBin* can be calculated as *Yin* =1 / *Zin*.

was determined with some error.

was determined with some error.

rectangular waveguides

rectangular waveguides

Let us consider a problem of vibrator excitation at its geometric center by a lumped EMF with

d

*s*

%% % (30)

˜ *L* ) and *P <sup>s</sup> kr*, *k*

*s* (*kr*, *k*

> *s* (*kr*, *k*

˜*s*) and *PL*

*s L*

% %

a

*s* (*kr*, *k*

˜ <sup>|</sup>*<sup>s</sup>* |) *PL s* (*kr*, *k*

Cin Cin Si Si


Since the current distribution (30) is now known we can calculate electrodynamic character‐ istics of an impedance vibrator. Thus, an input impedance *Zin* =*Rin* + *i Xin* of vibrator in a feed

<sup>0</sup> <sup>60</sup> cos ( , ) [ ] , (0) sin ( , )

sin { ( ) (1 / 2)[ (2 2 ) (2 2 )] ( ) ( )

= - + + - - - ++ -

% % % %%

*kL L kL kL kL kL kL kL kL kL*

è ø +

*J k kL P kr kL*

Cin Cin Cin Cin

% %%

(2 2 )] ( ) ( )

*kL kL kL kL kL kL*


*<sup>V</sup> ik kL P kr kL <sup>Z</sup>*

æ ö <sup>+</sup> = = ç ÷

a

d

Si Si Si Si

+ + - - - +- -

( / 2)[ (2 2 ) (2 2 )] [ ( ) ( )]}


% % %% % % Si Si Si Cin Cin Cin Cin

( / 2)[ (2 2 ) (2 2 )] [ ( ) ( )]}.

% % %%

*i kL kL kL kL i kL kL kL kL*

*i kL kL kL kL i kL kL kL kL*

( / 2)[ (2 2 ) (2 2 )]}

% %

*s L*

*L*

% % Ohm (32)

d

a

% % %

a

*i kL kL kL kL*

Si Si Cin Cin

sin {(1 / 2)[ (2 2 ) (2 2 )] ( / 2)[ (2 2 ) (2 2 )]},

*kL kL kL kL kL i kL kL kL kL*

a d

(29)

˜(*L* + *s*) are defined

˜ *L* ) which will be

(31)

(33)

˜ *L* ) can be expressed

<sup>000</sup> ( ) ( ) ( 0), *<sup>s</sup> s s Es Es V s* = = -

where *δ*(*s* −0)=*δ*(*s*) is Dirac delta-function. Then the expression for the current (27) is

sin ( | |) ( , ) ( ) . 2 cos ( , )

*<sup>i</sup> k L s P kr ks Js V k kL P kr kL*

˜*s* + sin*k*

æ ö - + = - ç ÷ è ø +

amplitude *V*0. The mathematical model of excitation is presented as

0

a

˜(*L* + *s*) −(sin*k*

by the formula (28a). Explicit expressions for *P<sup>δ</sup>*

g

*in*

(, ) (, ) sin | | (, )

*L ikR s L*

ò % %


Si

*kL kL kL*

cos {(1 / 2)[ (2 2 )

+ + +

*L <sup>e</sup> P kr kL ks s RsL*


g

˜*s*)=*P <sup>s</sup> kr*, *k*

needed below may be presented as

Here *P<sup>δ</sup> s* (*kr*, *k*

160 Advanced Electromagnetic Waves

*s*

point is equal

where

*L*

d

w

explicitly in terms of generalized integral functions [4,5]. Thus, *PL*

( , ) cos {2ln 2 ( ) (1 / 2)[ (2 2 ) (2 2 )]

where *Si*(*x*) and *Cin*(*x*) are sine and cosine integrals of complex argument.

= -- ++ -

+ ++ - - +- -

%% % % %

*<sup>L</sup> P kr kL kL L kL kL kL kL*

% % % %

To confirm the validity of the above analytical formulas we present the results of a comparative analysis of calculated and experimental data available in the literature. Figure 2 and Figure 3 show the graphs of the input admittance for two realizations of surface impedance: 1) metal wire (radius *ri* =0.3175 cm), covered by dielectric (*ε* =9.0) shell (radius *r* =0.635 cm), the experi‐ mental data [21] at Figure 2 and 2) metal wire (*ri* =0.5175 cm), covered with ferrite (*μ* =4.7) shell (*r* =0.6 cm), the experimental data from [22] at Figure 3. The plots show that trends of the theoretical curves coincide with that of the experimental curves, especially near the resonance for *Bin* =0, though in absolute values some difference is observed. In our opinion, the discrep‐ ancy of theoretical curves, obtained by solving the integral equation for the current by averaging method, and the experimental curves may be caused by evident fact that vibrator self-field (19) was averaged and the current amplitude was determined with some error. integral equation for the current by averaging method, and the experimental curves may be caused by evident fact that vibrator self-field (19) was averaged and the current amplitude integral equation for the current by averaging method, and the experimental curves may be caused by evident fact that vibrator self-field (19) was averaged and the current amplitude

Figure 2. The input admittance of metal wire (radius ir =0.3175 cm), covered by dielectric shell ( ε =9.0, radius r =0.635 cm) versus its electrical length at the frequency f =600 MHz **Figure 2.** The input admittance of metal wire (radius *ri* =0.3175 cm), covered by dielectric shell (*ε* =9.0, radius *r* =0.635 cm) versus its electrical length at the frequency *f* =600 MHz Figure 2. The input admittance of metal wire (radius i<sup>r</sup> =0.3175 cm), covered by dielectric shell ( <sup>ε</sup> =9.0, radius <sup>r</sup> =0.635 cm) versus its electrical length at the frequency <sup>f</sup> =600 MHz

Figure 3. The input admittance of metal wire (radius ir =5175 cm), covered with ferrite shell ( µ =4.7, r =0.6 cm) versus frequency for L =30.0 cm Figure 3. The input admittance of metal wire (radius ir =5175 cm), covered with ferrite shell ( µ =4.7, r =0.6 cm) versus frequency for L =30.0 cm **Figure 3.** The input admittance of metal wire (radius *ri* =5175 cm), covered with ferrite shell (*μ* =4.7, *r* =0.6 cm) versus frequency for *L* =30.0 cm

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

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

Now let us solve the second key problem. Let a resonant iris is placed in infinite hollow (ε =µ =ε =µ = 1 12 2 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).

Now let us solve the second key problem. Let a resonant iris is placed in infinite hollow (ε =µ =ε =µ = 1 12 2 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).
