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^-y_g^z

Where y_g = a_g + jb_g

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

a_g is attenuation constant, and

b_g is phase constant.

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

                        e^jwt e^-y_g^z = e ^{(jwt – y}_g^z)

                                e^{(jwt – (a}_g + ^jb_g^)z)

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

                                e^{jwt – j}b_g^z = e ^{j(wt  –} b_g^z)

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.

            E_tan  = 0

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

            H_nor = 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

                          a_x         a_y                            a_z

Ñ x H = d/dx      d/dy            d/dz    = jwε (E_xa_x + E_ya_y + E_za_z)

            H_X           H_Y           H_Z

            a_x    dH_z/dg – dH_y/dz    – a_y    dH_z/dx– dH_x/dz    + a_y    dH_y/dx– dH_x/dg

            = jwεE_xa_x +jwεE_ya_y + jwεE_za_z

Comparing the respective components on both sides, we get

            dH_z/dg  –  dH_y/dz  = jwεE_x

            dH_x/dz  –  dH_z/dx  = jwεE_y

            dH_y/dx  –  dH_x/dg = jwεE_z                                                                      (3)

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

            dE_z/dg –  dE_y/dz  = jwμH_x

            dE_x/dz  –  dE_z/dx  = jwμH_y

            dE_y/dx  –  dE_x/dg  = jwμH_z                                                                                    (4)

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

                        H_x = H_x0 e^-y_g^z

            Thus     dH_x /dz =^-y_gH_x0 e^-y_g^z

            Or        dH_x /dz = -g_gH_x                                                                             (5a)

Similarly          dH_y/dz = -g_gH_y                                                                               (5b)

                        dE_x/dz = -g_gE_x                                                                                                (5c)

and                 dE_y/dz = -g_gE_y                                                                                                (5d)

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

            dH_z/dg –  dH_x/dg = dE_z/dg = dE_x/dg = 0                              (5e)

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

            g_gH_y = jwεE_x                                                                                                    (6a)

            -g_gH_x – dH_z/dx = jwεE_y                                                                              (6b)

            dH_y/dx = jwεE_z                                                                                             (6c)                                      

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

            g_gE_y = – jwμH_x                                                                                                                (7a)

_–g_gE_x-dE_z/dx = -jwμH_y                                                                                                               (7b)

dE_y/dx = – jwμH_z                                                                                                         (7c)

Now use equations 6a and 7b

From equation 6a

Ex = g_gH_y/jwε

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

g²_gH_y/jwε + dEz/dx = jwμH_y

 

            dE_z/dx =( jwμ – g²g/jwε)H_y

jwε (dE_z/dx) = (-w²με – g²g) H_y

 jwε (dE_z/dx) = -H_y(g²g+ H_y w²με)

            = – H_yK²g

            K²g = g²g + w²με

            H_y = – jwε/ K²g   dE_z/dx                                                        (8a)

Again use equations 6a and 7b

From equation

                        H_y = 1/ jwμ ( dE_z/dx  + g_gE_x)

Substituting value of H_y in equation , we have

            g_g/ jwμ ( dE_z/dx  + g²gEx/jwμ )= jwE_x

            g_g/ jwμ ( dE_z/dx ) (jwε- g2g/ jwμ)E_x

            g_g(dE_z/d_x) = (-w² με – g2g) E_x

                _– Y_g(dE_z/d_x) = E_xK²_g

            K²_g = g²g + w² με

            E_x = (gg/ K²_g) dE_z/dx                                                 (8b)

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

            H_x = (-gg/ K²_g) dH_z/dx                                               8c

and     E_y= (jwμ/ K²_g) dH_z/dx                                                           8d

where            K²_g = g²_g + 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 E_z and H_z.

If E_z = 0 and  H_z = 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 E_z or H_z exists :

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