Lehner G. Electromagnetic Field Theory for Engineers and Physicists

Подождите немного. Документ загружается.

484 Time Dependent Problems II (Electromagnetic Waves)

united, while they are spatially separated for the LC oscillator, namely in the two

distinct components. Nevertheless, the transition is gradual. It is possible to start

from an LC oscillator and make a smooth transition into a resonant cavity

(Fig. 7.32).

Notice that for the standing waves of this section as described by eqs. (7.384),

(7.385), (7.387), (7.388) (contrary to the previous sections) it is not permissible to

replace the differential operator by . The reason is that the standing wave is

created by superposition of the two functions and , which,

in the said equation., resulted in the sine and cosine functions, respectively. This

had prevented us to obtain the fields of eq. (7.387) and (7.388) from eqs. (7.347)

and (7.353), rather we had to go back to the more generally applicable eqs. (7.238),

(7.239), (7.248), and (7.249).

It shall also be pointed out that eqs. (7.389) (7.390), (7.391) – just as

analogous equations for mechanical oscillations and waves – have a plausible

Fig. 7.31 Energy in a resonant cavity

mag.

()W

el..

()

mag.

Fig. 7.32 Transition from LC oscillator to resonant cavity

–

– z[]exp +ik

z[]exp

7.10 Circular Wave Guide 485

explanation. Consider, for example, a string, mounted at both ends. The relation for

this vibrating string is

where a is the length of the string and n is a whole number (Fig. 7.33). For a wave

guide, this applies for waves in x and y direction, while it applies to all three

directions in case of the resonant cavity. The waves in the x and y direction of a

wave guide are standing waves, while the wave in z direction is travelling. In case

of the resonant cavity, all three directions carry standing waves. Intuitively, one

may picture a standing wave by superposition of an incident wave and waves,

reflected at the boundary (the reader shall thereby be reminded of sects. 7.1.5 and

7.1.6).

7.10 Circular Wave Guide

7.10.1 Separation of Variables

For circular cylindrical wave guides, whether simply or multiply connected

(Fig. 7.34), the best approach is to use cylindrical coordinates. According to

eq. (3.33), the two-dimensional Laplacian in the x-y-plane takes the form

(7.395)

The formulas applicable to and are (7.344) and (7.350),

respectively, where N is given by (7.338). Therefore we obtain

(7.396)

Separating like in Sect. 3.7 gives

---

Fig. 7.33 Oscillating string

∆

---

r∂

∂

r∂

∂ 1

-----

∂

∂ϕ

---------

---

r∂

∂

r∂

∂ 1

-----

∂

∂ϕ

---------

N++





486 Time Dependent Problems II (Electromagnetic Waves)

(7.397)

We could have picked a sine instead of the cosine, but this does not constitute a

significant difference. is a general cylinder function:

(7.398)

The specific form of given in (7.397) allows to solve an abundance of

problems. Notice, that in general, it has to be , if the area under

consideration includes the z-axis, since diverges for . However, if the

area does not include the z-axis, then the complete solution (7.398) has to be used.

Fig. 7.35 shows a number of arrangements, which can be solved using

(7.397). First, there is the “normal” cylindrical wave guide (Fig. 7.35a),

characterized by an ideal insulator surrounded by an ideal conductor. From this, the

coaxial cable (Fig. 7.35b) emerges when another ideal conductor is inserted inside

the insulator. Fig. 7.35c shows the so-called Sommerfeld conductor, where a

medium of finite conductivity is surrounded by an ideal insulator. The Harms-

Goubau conductor of Fig. 7.35d has a conductor of finite conductivity at its center

and is surrounded by two layers of dielectrics with different permittivity. Another

interesting case is the so-called light conductor or light guide, shown in Fig. 7.35e,

an arrangement important in optics and has gained significance in

telecommunications as well. Subsequently, we will limit our discussion to ordinary

wave guides and the coaxial cable. The limit of ideal insulators and media with

infinite conductivity can – with the exception of super conductors – not be realized

at all or only approximately. Using these limits, however, simplifies the theoretical

analysis significantly, without misrepresenting the material points of the results.

Fig. 7.34 Circular cylindric wave guides

z z

rN()mϕcos i ωtk

z–()[]exp=

r 0→

7.10 Circular Wave Guide 487

7.10.2 TM Waves in a Circular Cylindrical Wave Guide

For the ordinary wave guide, we have to let in eq. (7.398). Here, we are

interested in TM waves and thus with (7.397) find

(7.399)

According to (7.349) we also have to ensure that

(7.400)

It follows that may not assume every value. If, for instance, is the radius of

the wave guide, then it must be

(7.401)

We have already found the zeros of in Sect. 3.7.3.3.:

(7.402)

If we only consider ideal insulators ( ), then with (7.338) we obtain

(7.403)

The associated wave is called the wave of the wave guide. Its fields are

derived from eqs. (7.238) and (7.239):

Fig. 7.35

a) b) c)

d) e)

κ 0=

κ∞=

κ 0=

κ∞=

κ finite=

κ 0=

κ finite=

κ 0=

≠

κ 0=

εε

rN()mϕcos i ωtk

z–()[]exp=

()

boundary

N()0=

κ 0=

εµω

–

----------=

488 Time Dependent Problems II (Electromagnetic Waves)

(7.404)

Notice that has to assume one of the values allowed by

eq. (7.402). It becomes clear from inspection that with this restriction, , , and

vanish when , as it must be. is the derivative of the Bessel function,

where the subject of derivative is its entire argument. waves exist for

, (i.e., N>0). n=0, N=0 would result in a TEM wave, which is not

possible in the current case, and even more so, according to our general

conclusions regarding TEM waves.

Similar to the rectangular wave guide, we find the phase velocity from the

dispersion relation, here eq. (7.403)

(7.405)

and the group velocity is

(7.406)

The wave length in z-direction, the propagation direction in the wave guide is

(7.407)

is the related free space wave length. The largest possible free space wave length

of the wave is the critical wave length

(7.408)

A few values of are listed in Tab. 7.1. Notice that the quantities with the

double index represent the zeros of the Bessel functions. They are

dimensionless and should not be confused with the wave length , , , etc.

' rN()mϕcos–=

-----------

rN()msin ϕ=

+ NC

rN()mϕcos=

iωεm

-------------

rN()msin ϕ–=

iωε NC

' rN()mϕcos–=











i ωtk

z–()[]exp⋅

N εµω

–=

= J

m 0≥ n 0>

----

εµω

----------–

----------------------------------==

dω c

--------==

2π

------

2π

------

---------





–

-----------------------------------

------

2πr

-----------





–

--------------------------------------–==

2πr

-----------=

λλ

7.10 Circular Wave Guide 489

7.10.3 TE Waves in a Circular Cylindrical Wave Guide

For TE waves we have

(7.409)

where

, (7.410)

which requires that

(7.411)

If one addresses the zeros of in their order by , then one obtains the

dispersion relation

(7.412)

and for it becomes

(7.413)

In analogy to (7.405) and (7.406), one finds here again

(7.414)

The related fields of the wave are determined by (7.248) and (7.249)

Table 7.1 The values of

1 2 3 4 5

0 2.40483 5.52008 8.65373 11.79153 14.93092

1 3.83171 7.01559 10.17347 13.32369 16.47063

2 5.13562 8.41724 11.61984 14.79595 17.95982

3 6.38016 9.76102 13.01520 16.22347 19.40942

4 7.58834 11.06471 14.37254 17.61597 20.82693

5 8.77148 12.33860 15.70017 18.98013 22.21780

6 9.93611 13.58929 17.00382 20.32079 23.58608

()0=

rN()mϕcos i ωtk

z–()[]exp=

r∂

∂Π





' r

N()0=

' µ

N µ

κ 0=

εµω

–

----------=

⋅ c

490 Time Dependent Problems II (Electromagnetic Waves)

(7.415)

N has to assume a value permitted by eq. (7.412). Eq. (7.407) applies to , if

is replaced by . The critical wave length is found in analogy to (7.408)

(7.416)

A number of values for is given in Tab. 7.2. This table and the one for

make it particularly evident that the largest critical wave length of all TM and TE

waves belongs to the wave, with a value of

(7.417)

Finally, we will present a few graphs, illustrating the field of some TM and

TE waves, possible in a circular cylindric wave guide (Fig. 7.36, Fig. 7.37).

7.10.4 The Coaxial Cable

An in depth discussion of the coaxial cable requires one to start with the solutions

(7.397), (7.398). However, we will not present the coaxial cable in its full

Table 7.2 The values of

1 2 3 4 5

0 3.8317 7.0156 10.1735 13.3237 16.4706

1 1.8412 5.3314 8.5363 11.7060 14.8636

2 3.0542 6.7061 9.9695 13.1704 16.3475

3 4.2012 8.0152 11.3459 14.5859 17.7888

4 5.3175 9.2824 12.6819 15.9641 19.1960

5 6.4156 10.5199 13.9872 17.3128 20.5755

6 7.5013 11.7349 15.2682 18.6374 21.9318

iωµm

--------------

rN()msin ϕ=

iωµ NC

' rN()mϕcos=

' rN()mϕcos–=

-----------

rN()msin ϕ=

rN()mϕcos=











i ωtk

z–()[]exp⋅

2πr

-----------=

' µ

()0=

()

2πr

-----------

2πr

1.84

----------- 3 . 4 1 r

===

7.10 Circular Wave Guide 491

generality, which would provide all of its TE and TM waves. We will limit

ourselves to the case , i.e., its TEM wave, which requires to solve

eq. (7.396) in its form

(7.418)

The Ansatz

(7.419)

yields an equation for :

(7.420)

Its general solution is

(7.421)

Fig. 7.36 TM modes in cylindric wave guide

magnetic fieldelectric field

Fig. 7.37 TW modes in cylindric wave guide

magnetic fieldelectric field

N 0=

---

r∂

∂

r∂

∂ 1

-----

∂

∂ϕ

---------





Rr() mϕcos i ωtk

z–()[]exp=

Rr()

---

r∂

∂

r∂

∂ m

-------

–





Rr() 0=

Rr() C

⁄ln+= for m 0=

492 Time Dependent Problems II (Electromagnetic Waves)

(7.422)

which can also be obtained from the cylindrical functions when taking a limit. We

have met eq. (7.420) and its solutions (7.421) and (7.422) before in Sect. 3.7.3.5,

eqs. (3.262) through (3.264). The result for the Hertz vector is

(7.423)

and

(7.424)

Now one has the freedom to take as either the z-component of the electric or

the magnetic Hertz vector and calculate the fields from (7.345), (7.346) or (7.351),

(7.352), respectively. Either way, after considering the boundary conditions (7.363)

and (7.364) at the outer conductor ( ) and at the inner conductor ( ) of

the coaxial cable, it turns out that there is only one case where not all fields are

identically zero (Fig. 7.38). This case requires that and if one regards

as , (7.345), (7.346) yield the fields (for ):

(7.425)

Rr() C

m–

+=for m 0>

----ln+





i ωtk

z–()[]exp for m 0==

m–

+()mϕcos i ωtk

z–()[]exp for m 0>=

= rr

Fig. 7.38 Coaxial cable

m 0= Π

κ 0=

------

i ωtk

z–()[]exp–=

iωε

---------

i ωtk

z–()[]exp–=











7.10 Circular Wave Guide 493

This describes an electric field which is purely radial and a magnetic field that is

purely azimuthal. Fig. 7.38 illustrates this. Since

(7.426)

one finds

(7.427)

i.e., the wave behaves in this respect like a plane wave. There is no difficulty in

satisfying the boundary conditions, because there are neither components of E

parallel to the boundary, nor perpendicular components of H. must decrease as

so that E remains source-free, while needs to decrease as so that H

remains irrotational.

For the sake of completeness, it shall be noted that the just derived TEM wave

can also be obtained from

For

is the most general solution of (7.418). If we take for , then it has to be

because of (7.363), which in conjunction with provides the

fields given in (7.425). If we take for , then it has to be because

of (7.355), which in conjunction with yields as given above. With

the appropriate choice of , this provides the fields (7.425).

Wave types are also called modes. The just found TEM mode of the coaxial

cable is very similar to the TEM wave between two parallel plates. The two

converge for . We could think of Fig. 7.29 as being a piece of an annular

region with a very large radius.

According to (7.426), the TEM wave propagates with the speed of light for

that particular dielectric:

(7.428)

The just found wave is also used in transmission theory, where the telegrapher's

equation is used instead of the wave equation. To better illustrate the relation, we

shall now derive the telegrapher's equation and briefly discuss it.

7.10.5 Telegrapher's Equation

To derive the telegrapher's equation, one imagines the transmission line as being a

continuous analog of a network circuit, where a transmission line is nothing more

than a multiply connected wave guide of arbitrary cross section. The coaxial cable

N εµω

–0==

-----=

1 r⁄ H

1 r⁄

ϕ+()i ωtk

z–()[]exp=

m 0=

----ln+





ϕ+()i ωtk

z–()[]exp=

0= D

1= Π

«–

----

εµ

---------- c== =