# Referring to Problem 4.3 and the geometry described therein, this problem concerns calculating the..

Referring to Problem 4.3 and the geometry described therein,
this problem concerns calculating the real time-averaged power,

where _ = 2_/T, T is the temporal period
and  represent
the real, time varying EM fields of the system.

(a) Using the general eigensolutions of Problem 4.3, give a
general expression for the real time-averaged power that is transmitted in the  direction.

(b) Using the formula of (a), numerically calculate SAV
for the numerical case used in your Problem 4.3e solution (this was the
lossless case) in the regions exterior to the multilayer system.
»

Referring to Problem 4.3 and the geometry described therein,
this problem concerns calculating the real time-averaged power,

where _ = 2_/T, T is the temporal period
and  represent
the real, time varying EM fields of the system.

(a) Using the general eigensolutions of Problem 4.3, give a
general expression for the real time-averaged power that is transmitted in the  direction.

(b) Using the formula of (a), numerically calculate SAV
for the numerical case used in your Problem 4.3e solution (this was the
lossless case) in the regions exterior to the multilayer system. Verify
numerically that the incident power equals the sum of the power reflected and
transmitted from the multilayer system, thus verifying that conservation of
power holds for the system.

Problem 4.3

A plane wave of peak amplitude E0 is
normally incident on a four-layer system located between free space on the
incident side and an infinite dielectric region on the transmit side.

(a) Using general layer thicknesses (i.e., s_,
= 1, 2, 3, 4) and material
parameters (i.e., relative permeability __ and permittivity, __,
= 1, 2, 3, 4), using the state
variable method, and using coordinate system of Figure 5.19, find the general
form of the EM fields in each region, expressed in terms of forward and
backward traveling plane waves of unknown amplitude.

(b) By matching EM boundary conditions at all material
interfaces, formulate matrix equations from which the EM fields in all regions
of the system may be found.

(c) Form a ladder matrix equation, which relates the EM
field amplitudes of the plane waves of (b) in the layer that is adjacent to the
transmit side (call this layer Region 4, assumed to have width s4)
to the EM field amplitudes of the plane waves in the layer that is adjacent to
the incident side (call this layer Region 1, assumed to have width s1).

(d) Using the ladder matrix found in (c), matching EM
boundary conditions, formulate a reduced matrix equation from which the EM
fields in all regions of the system may be found.

(e) Picking specific layer thicknesses (i.e., s_,
= 1, 2, 3, 4) and lossless
layer material parameters (i.e., relative permeability __
and permittivity, __,  = 1, 2, 3, 4) of your choice, solve numerically
all equations formulated in (a)(d). Present and compare numerical results of
(b) and (d).

(f) Repeat (e) using lossy layer material parameters.

(g) Comment on the advantages and disadvantages of the
solutions as determined by the methods described in (b) and with the solution
described in (d) and (e).

»

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