general approach to wave guide analysis

7/28/2019 General Approach to Wave Guide Analysis

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General Approach to Wave Guide Analysis

In case of parallel wave guide the modalpropagation was visualized as super position of

multiply reflected plane wave from the twoconducting sheets. This approach althoughprovides better physical understanding of themodal propagation, becomes algebraicallyunmanagable for complicated waveguide instructure.

In the following we develop a general frameworkfor analyzing the wave guide structure likea Rectangular Wave Guide.

A rectangular wave guide is a hollow metallic pipewith rectangular cross section. The electromagnetic energy propagates along the length ofpipe. The net wave propagation therefore is alongthe length of the pipe.

A direction along the length of the pipe is calledthe Longitudinal direction. Whereas any directionperpendicular to the wave propagation is calledthe TRANSVERSE DIRECTION.

Let us choose the co-ordinate system such that

the - axis is along the longitudinal direction.

You can note that in general there are six fieldcomponents, three for electric field nad three formagnetic field which are related throughMaxwell's equations. All the six componentstherefore cannot be independent. We can selectto field components as an independent

components and the remaining four componentscan be obtained from the Maxwell's equation.

Since the - direction is a specialdirection(direction of the net wave propagation)we choose the longitudinal electric and magnetic

field components ( , ) as independent


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components and derive the transversecomponents in terms of the longitudinalcomponents using the Maxwell's equation.

Since the net wave is travelling in a - direction

any field component in the - direction will be of

the type , where is the phase constant

of the net wave propagating along the -direction.

If we define a parameter (transversepropagation constant) as

we can write transverse electric and magneticfield in term of the longitudinal field components

( , ) as

-----------(6.28

)

-----------(6.29)

Here transverse is defined by

-----------(6.30)

in Cartesians co-ordinate system.


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From the above equations we can make someimportant observations as follows :

(1) Transverse fields can exist provided at least one

of the longitudinal components ( or ) is non-

zero, except when = 0. That is, in general thereis no transverse electromagnetic wave

propagation except when .

(2)For we get implying that thetransverse electromagnetic wave can exist in awaveguide if its propagation constant is same asthat of the unbound medium filling the waveguide.

(3) The fields corresponding to , have electricfield transverse to the direction of wavepropagation and hence represent the Transverseelectric (TE) wave.

(4) The fields corresponding to =0 have magneticfield transverse to the direction of wavepropagation and hence represent Transversemagnetic (TM) wave.

(5) For TM or TE case, or respectively is to benon zero, and has to be non zero. Otherwisethe transverse fields would become infinite. Inother words, the TE and TM modes can not havethe phase constant same as that of the unboundmedium. The TE and TM modes then essentiallyhave to be dispersive modes i.e., their phasevelocity should vary as a function of frequency.

In Cartesian co-ordinate system we can explicitly

write the transverse field component in terms ofthe longitudinal components as

--------


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---(6.31)

-----------(6.32)

-----------(6.33)

-----------(6.34)

So, in the analysis of a wave guide first we

obtain or which is consistent with theboundary conditions and then

subsequently obtain the transverse componentusing the above equation.

Transverse Magnetic ModeA rectangular waveguide with cross-


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section is shown in the figure below

The walls of the waveguide are made of idealconductior and the medium filling the waveguideis ideal dielectric.

As a convention and the -axis is oriented alongthe broader dimension of the wave guide, and

the -axis is oriented along the shorter dimension

of the waveguide. The -axis is oriented along thelength of the waveguide and the waveguide isassumed to be infinite length.

For transverse magnetic mode, we haveand . The transverse fields are therefore

represented in terms of components only.

The wave equation is to be solved for withappropriate boundary conditions. In Cartesian co-

ordinates the wave equation for can be writtenas

-------- (6.35)

The equation can be solved by the separation of


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variables i.e by assuming that is given as

--------(6.36)

The fields are assumed to be of sinosoidal naturewith an angular frequency .

Substituting for from 6.36 into 6.35, we get

-------- (6.37)

Note that in equation 6.37 the first term is a

function of , the second term is a function of

only, the third term is a function of only andfourth term is a constant. Since the equation is to

be satisfied for every value of each term inequation 6.37 must be constant i.e

----------(6.38)

--------

--(6.39)

--------
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--(6.40)

and are real constants.

From the physical understanding of reflection ofwaves from parallel conducting boundaries we

expect a standing wave kind of behaviour in

and directions and a travelling wave kind of

behaviour in direction. In any case, we expect a

wave phenomenon in direction which can beproperly represented by putting a negative sign

infront of the constant , and . Insteadof negative sign if the positive sign was used thesolutions will have real exponential functionswhich would not represent the wavephenomenon.

The equations 6.38, 6.39 and 6.40 can be re-written as

----------(6.41)

----

------(6.42)

----
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------(6.43)

These equations are identical to the transmissionline equations.

The solution to equations 6.41, 6.42 and 6.43 canbe appropriately written as

--------

--(6.44)----------(6.45)

----------(6.46)

where are arbitrary constants whichare to be evaluated by boundary conditions.

If we assume that waveguide is of infinite length,

we can take only one travelling wave in -direction. We can

then choose . Substituting forfrom 6.44, 6.45 and 6.46 into equation 6.36 the
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general solution for can be written as----------

(6.47)

Boundary conditions for thelongitudinal electric fieldcomponent

The component is parallel to all the four walls of the

waveguide. Since the tangential field of the electric field is zero atthe conducting boundary, we get the boundary condition for .

---------- (6.48)

The boundary conditions for = and = walls give =

0, = 0 and the boundary conditions for , and give

---------- (6.49)

---------- (6.50)

where and are integers.

Substituting for and , we finally get the solution for as

---------- (6.51)

Constant is essentially a combination of , , whichgives the amplitude of the longitudinal component of the electricfield.


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Transverse field componentfor TM mode

Substituting for from equation 6.51 and inequations 6.31, 6.32, 6.33 and 6.34 we get the transverse electricand magnetic field components as

---------- (6.52)

---------- (6.53)

---------- (6.54)

---------- (6.55)

The integers and define the order of the mode and the mode

is designated as

We can make the following observations regarding mode :

(1) Similar to that of the parallel plane waveguide the fields exists inthe discrete electric and magnetic field pattern called modes ofwaveguide.

(2) All field components are sinosoidally in and directions.

(3) All transverse fields go to zero if either or is zero. In other

words, both the indices and have to be non-zero for

existence of the TM mode. That is, and modes can notexist. Consequently, the lowest order mode which can exist

is mode.

Substituting from 6.51 into 6.35, we get what is called thedispersion relation for the mode as

---------- (6.56)
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The dispersion relation suggest that the phase constant for the

mode is different for different modes (for different values of

and ) and is no more proportional to .

Implication of this would be discussed later along with thecharacteristics of the TE mode.

The analysis of TE mode in a rectangular waveguide can becarried out on the line similar to that of the the TM mode.

For TE mode we have

---------- (6.57 )

The wave equation is solved for in this case.

In the case of TM mode the wave equation was solved forwhich was tangential to all the four walls of the waveguides. We

therefore had boundary conditions on .

In the TE case however the independent component istangential two the walls of the waveguide which do not impose

any boundary conditions on . The tangential component ofmagnetic field is balanced by the appropriate surface currents onthe walls of the waveguides.

The analysis procedure for TE mode therefore is slightly differentthan that of the TM mode. We have seen in the case of parallel

plane waveguide that the tangential component of the magneticfield is maximum at the waveguide walls. Also in Cartesian co-ordinate system the solution to the wave equation are sinosoidalin nature.

One can note that for = , (vertical walls) and for

, (horizontal walls) the tangential component of magnetic field


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is maximum.

Substituting for from

---------- (6.58)

and = 0 in 6.31, 6.32, 6.33 and 6.34, we get the transversefield components as

---------- (6.59)

---------- (6.60)

---------- (6.61)

---------- (6.62)

In this case also we get

---------- (6.63)

Following observations can be made regarding the TE mode :

(1) The fields for the TE modes have similar behaviour to the fields ofthe TM modes i.e they exist in the form of discrete pattern, they

have sinosoidal variations in and directions, indices and

represent number of half cycles of the field amplitudes in anddirection respectively and so on.

(2) Unlike TM mode both indices and need not be non-zero forthe existence of the TE mode. However, of both the indices zero

makes the magnetic field independent of space and thereforecannot exist. In other words, mode cannot exist but

and modes can exist.

(3)The lowest order mode for the TE case therefore would be
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and .

Cutt-off Frequecy of TE and TM mode

For both and modes the phase constant is given by

--------- (6.64 )

For the mode to be travelling has to be a real quantity. Ifbecomes imaginary then the fields no more remain travelling but

become exponentially decaying

The frequency at which changes from real to imaginary iscalled the cut-off frequency of the mode. At cut-off frequency

therefore giving

--------- (6.65 )

--------- (6.66 )

The cut-off frequencies for lowest TM and TE modes

i.e can be obtained from eqn. 6.70 as

--------- (6.67 )

--------- (6.68 )

--------- (6.69)


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Since by definition we have we get the frequencies as

--------- (6.70)

We can make an important observation that if at all the electromagnetic energy travels on a rectangular waveguide itsfrequency has to be more than the lowest cut-off frequency

i.e .

As the order of the mode increases the cut-off frequency alsoincreases i.e with increasing frequency there is possibiltyof existence of higher order mode.

The very first mode that propagates on the rectangular waveguide

is mode and therefore this mode is called thedominant mode of the rectangular waveguide. The cut-offfrequency for the dominant mode is

--------- (6.71)

The equation suggest that for propagation of an electro magnetic

wave inside a rectangular waveguide the width of awaveguide should be greater than half the wave length of thewave.

general approach to wave guide analysis

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