HW8_Solution

HW8_Solution - ECE 201A Homework 8 solution 1. (a) Maxwells...

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ECE 201A Homework 8 solution 1. (a) Maxwell’s equations in a uniform, source free and isotropic dielectric medium are 0 E jH ωμ ∇× =− GG HjE ωε = 0 H ∇= G 0 E ∇• = G N () () 2 00 0 E EE j H j j E ω μω μ ε ⎛⎞ ∇×∇× =∇ ∇• −∇ ⎜⎟ ⎝⎠ G G G 22 0 0 ωμε ∇+ = (b) ˆ Tz a z ∇=∇ + , where ˆˆ Tx y aa x y ∂∂ + . Hence 2 2 T z ∇=∇ •∇=∇+ . If ( ) , j z x y e β = , () 2 2 2 ,, TT T jz E Exye E Exy e E E zz ββ −− + = = ∇− G G G G Substituting this result in to the wave equation 2 22 0 T E = ∇− + = G , or ( ) 2 0 0 T ωμε β +− = G G . (c) For a TEM wave T E E = and T HH = , i.e., fields have only x and y components. Then 0 E becomes 0 ˆ T T aE j H z × =− 0 z directed x and y directed x and y directed ˆ T z T E E aj H z ∇× + × G ±²³²´ ±²³ ²´ Hence for the equality to hold the z component of this vector equation should vanish. Therefore z directed 0 E ∇× = G . Hence T E G can be taken as the gradient of a scalar function since ( ) 0 φ ∇ ×∇ . Then
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() , j z TT Ex y e β φ =−∇ G (d) Since 0 T E ∇• = G in each source free region ( ) 2 ,0 T jz T y e e ββ φφ −− =−∇ •∇ = −∇ = G requires ( ) 2 T xy ∇= (e) Using the wave equation for E G 22 2 0 0 T EE ωμε β ∇+ = GG . Since T E E = for a TEM wave 222 0 0 T = But N 2 0 0 T T E ⎛⎞ ⎜⎟ = ⎝⎠ G or Hence for a non trivial TEM wave 0 or 0 ωμε = . Since has to be constant everywhere within the cross section of the waveguide dielectric properties of the medium should be uniform. In other words ε should be constant throughout the medium. The last equality ( ∇∇ = ∇∇ ) can also be proven writing it explicitly, i.e., ˆˆ Tx y aa x y ∂∂ + 2 2 2 2 2 2 ˆ ˆ ˆˆˆ y x y x y xyx a a x yx y x x y y x y aaa xx yy x x + = + + = + + + ∂ ∂ =+ ++ = + 2 2 2 2 2 2 2 ˆ T yT T a y y x y ⎛⎞⎛⎞ = + ⎜⎟⎜⎟ ⎝⎠⎝⎠ =∇ ∇ To summarize for a TEM mode to exist the dielectric inside the waveguide should be uniform and it should be possible to set up a DC field profile within the waveguide so that Laplace’s equation can be solved and a DC potential distribution is found. This requires at least two independent conductors so that a DC potential can be applied between them to set up a DC field. For a time varying excitation this DC electric profile within the waveguide propagates with the velocity which is the velocity of light in the medium filling the waveguide.
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(a) This waveguide can support TE to x, TM to x and TE to z, TM to z modes. Principle of guiding depends on total internal reflection at the air dielectric interface and reflection at the dielectric perfect conductor interface. Using a plane wave model it is seen that a TE polarized plane wave can be trapped within the
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This note was uploaded on 12/02/2009 for the course ECE 000 taught by Professor O during the Spring '09 term at UCSB.

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HW8_Solution - ECE 201A Homework 8 solution 1. (a) Maxwells...

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