Electromagnetic waves and Applications (Part III)

1
Lecture 2
2
Review lecture 1
R, L, G, C ?
3
?1.4 Field analysis of transmission lines

• Derive the transmission line parameters (R, L, G,
  C) in terms of the electromagnetic fields
• Retrieve the telegrapher equations using these
  parameters

Example
Voltage V0e?j?z Current I0e?j?z
4
Work (W) and power (P)
H Multiplies the two sides of the first Maxells
equation
E Multiplies the two sides of the conjugated
second Maxells equation
5
(Time averaged)
Poynting law
6
Calculate magnetic energy
Calculate the time-average stored magnetic energy
in an isotropic medium ( the results are valid
for any media )
7
Transmission line parameter L

• The time-average stored magnetic energy for 1 m
  long transmission line is

And circuit line gives .
Hence the self inductance could be identified as
8
Transmission line parameter C

• Similarly, the time-average stored electric
  energy per unit length can be found as

Circuit theory gives ,
resulting in the following expression for the
capacitance per unit length
9
Transmission line parameter R

• The power loss per unit length due to the finite
  conductivity of the metallic conductors is

(Rs 1/?? is the surface resistance and H is the
tangential field)
The circuit theory gives ,
so the series resistance R per unit length of
line is
10
Transmission line parameter, G

• The time-average power dissipated per unit
  length in a lossy dielectric is

Circuit theory gives , so
the shunt conductance per unit length can be
written as
11
Homework
1. The fields of a traveling TEM wave inside the
coaxial line shown left can be expressed
as where ? is the propagation constant of the
line. The conductors are assumed to have a
surface resistivity Rs, and the material filling
the space between the conductors is assumed to
have a complex permittivity ? ? - j?" and a
permeability µ µ0µr. Determine the transmission
line parameters (L,C,R,G).
2. For the parallel plate line shown left, derive
the R, L, G, and C parameters. Assume w gtgt d.
12
Transmission line parameters for different line
types
13
(No Transcript)
14
Derive Telegrapher Equations from Field Analysis

• The fields inside the coaxial line will satisfy
  Maxwell's curl equations

Expanding the above equations in cylindrical
coordinates and then gives the following vector
equations (TEM waves,Ez Hz 0, no ?-dependence)
15
Using the above equations, we obtain
The same telegraphic equations as derived from
distributed theory.
16
Waves in lossless coaxial waveguides
17
Waves in lossless coaxial waveguides
Voltage between the two conductors at z 0
18
Propagation Constant, impedance and Power
Flow for the Lossless coaxial Line

• Characteristic impedance

• Power flow (computed from the Poynting vector)

(Match the circuit theory)
19
Surface resistance and surface current of metal
Energy entering a conductor
metal
dielectric
Evanescent
The contribution to the integral from the surface
S can be made zero by proper selection of this
surface (Snell law --gt refraction angle ? 0).
Therefore,
From vector identity, we have
The energy absorbed by a conductor
20
Grad, Div and Curl in Cylindrical Coordinates