Waves between parallel planes

Let us discuss how waves propagate through parallel planes and derive the necessary relation of transverse electric and magnetic waves:

Assumptions :

(a)  Pair of parallel planes are perfectly conducting.

(b)  Separation between the planes is ‘a’ meter in x – direction.

(c)  Space between planes is perfect dielectric (σ = 0) of permittivity ε and permeability μ.

(d)  Planes are of infinite extent in the y and z direction.

(e)  As the plane is extended to infinity in the y – direction there are no boundary conditions to be met in this direction, therefore field is uniform in the y- direction  i.e. derivative with respect to y is zero (d/dy =0)

(f)     Direction of propagation of wave is along z-direction, therefore the variation of all the field component in the z-direction is expressed as e-ygz

Where yg = ag + jbg

Here ygis  propagation constant and it is not equal to y(yg ¹ y). In special case of uniform plane waves, yg reduces to y.

ag is attenuation constant, and

bg is phase constant.

(g)  In time varying form, the field variation is expressed as

ejwt e-ygz = e (jwt – ygz)

e(jwt – (ag + jbg)z)

If there is no attenuation, ag = 0 then field variation is expressed as

ejwt – jbgz = e j(wt  – bgz)

Boundary Conditions :

In order to determine the electromagnetic field configuration between parallel planes, Maxwell’s field equation are solved with the following boundary condition :

(I)                 Electric field must terminate normally on the conductor, that is, tangential component of electric field must be zero.

Etan  = 0

(II)              Magnetic field must lie tangentially along the wall surface, that is, the normal component of magnetic field must be zero.

Hnor = 0

Derivation of field equations :

In general, Maxwell’s equations (Modified Ampere’s Circuital law and Faraday’s law of em induction) in non-conducting region (σ = 0) between the planes are

Ñ x H = jwεE   (Modified Ampere’s circuital law)           (1)

Ñ x E = jwμH  (Faraday’s law of em iduction)                 (2)

Expanding equation (1) in rectangular coordinates, we get

ax         ay                            az

Ñ x H = d/dx      d/dy            d/dz    = jwε (Exax + Eyay + Ezaz)

HX           HY           HZ

ax    dHz/dg – dHy/dz    – ay    dHz/dx– dHx/dz    + ay    dHy/dx– dHx/dg

= jExax +jEyay + jEzaz

Comparing the respective components on both sides, we get

dHz/dg  –  dHy/dz  = jEx

dHx/dz  –  dHz/dx  = jEy

dHy/dx  –  dHx/dg = jEz                                                                      (3)

Similarly expanding equation (2) and equating respective components on both sides, we get

dEz/dg –  dEy/dz  = jHx

dEx/dz  –  dEz/dx  = jHy

dEy/dx  –  dEx/dg  = jHz                                                                                    (4)

From assumption (f), as the direction of propagation is along z-direction, the variation of field components can be expressed as

Hx = Hx0 e-ygz

Thus     dHx /dz =-ygHx0 e-ygz

Or        dHx /dz = -ggHx                                                                             (5a)

Similarly          dHy/dz = -ggHy                                                                               (5b)

dEx/dz = -ggEx                                                                                                (5c)

and                 dEy/dz = -ggEy                                                                                                (5d)

also from assumption (e), d/dy = 0

dHz/dg –  dHx/dg = dEz/dg = dEx/dg = 0                              (5e)

By substituting equations 5a, b and e,  we have

ggHy = jEx                                                                                                    (6a)

-ggHx – dHz/dx = jEy                                                                              (6b)

dHy/dx = jEz                                                                                             (6c)

Similarly by substituting equations 5c, d and e in equation 4, we have

ggEy = – jHx                                                                                                                (7a)

ggEx-dEz/dx = -jwμHy                                                                                                               (7b)

dEy/dx = – jHz                                                                                                         (7c)

Now use equations 6a and 7b

From equation 6a

Ex = ggHy/jwε

Putting value of Ex in equation 7(b), we have

g2gHy/jwε + dEz/dx = jwμHy

dEz/dx =( jwμ – g2g/jwε)Hy

jwε (dEz/dx) = (-w2με – g2g) Hy

jwε (dEz/dx) = -Hy(g2g+ Hy w2με)

= – HyK2g

K2g = g2g + w2με

Hy = – jwε/ K2g   dEz/dx                                                        (8a)

Again use equations 6a and 7b

From equation

Hy = 1/ jwμ ( dEz/dx  + ggEx)

Substituting value of Hy in equation , we have

gg/ jwμ ( dEz/dx  + g2gEx/jwμ )= jwEx

gg/ jwμ ( dEz/dx ) (jwε- g2g/ jwμ)Ex

gg(dEz/dx) = (-w2 με – g2g) Ex

– Yg(dEz/dx) = ExK2g

K2g = g2g + w2 με

Ex = (gg/ K2g) dEz/dx                                                 (8b)

Similarly by using and solving equations 6b and 7a, we get

Hx = (-gg/ K2g) dHz/dx                                               8c

and     Ey= (jwμ/ K2g) dHz/dx                                                           8d

where            K2g = g2g + w2με

Equations 8(a,b,c and d) represent the equations of plane waves propagating in +z direction varying sinusoidally between the infinite parallel planes.

In equation, the components of electric and magnetic fields strengths are expressed in terms of Ez and Hz.

If Ez = 0 and  Hz = 0, all the components will vanish, therefore it is observed that there must be a z component of either E or H i.e. comonent along the direction of  propagation.

Therefore, the propagating waves in parallel plane guide are classified into following types according to whether Ez or Hz exists :

1. Transverse Electric (TE) Waves or H Waves (Ez = 0, Hz ¹ 0)
2. Transverse Magnetic (TM) Waves or E Waves (Hz = 0, Ez ¹ 0)
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