Waves between parallel planes
Let us discuss how waves propagate through parallel planes and derive the necessary relation of transverse electric and magnetic waves:
(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
= jwεExax +jwεEyay + jwεEzaz
Comparing the respective components on both sides, we get
dHz/dg - dHy/dz = jwεEx
dHx/dz - dHz/dx = jwεEy
dHy/dx - dHx/dg = jwεEz (3)
Similarly expanding equation (2) and equating respective components on both sides, we get
dEz/dg - dEy/dz = jwμHx
dEx/dz - dEz/dx = jwμHy
dEy/dx - dEx/dg = jwμHz (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 = jwεEx (6a)
-ggHx – dHz/dx = jwεEy (6b)
dHy/dx = jwεEz (6c)
Similarly by substituting equations 5c, d and e in equation 4, we have
ggEy = - jwμHx (7a)
-ggEx-dEz/dx = -jwμHy (7b)
dEy/dx = - jwμHz (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
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 :
- Transverse Electric (TE) Waves or H Waves (Ez = 0, Hz ¹ 0)
- Transverse Magnetic (TM) Waves or E Waves (Hz = 0, Ez ¹ 0)