Maxwell, time harmonic, transversetoz

1
Maxwell, time harmonic, transverse-to-z

• collecting all the terms,
• assuming time harmonic solutions
• using Ohms law
• assuming there is no component of either E or H
  in the z direction
• Maxwells equations reduce to
• things to notice
• Ey is connected to Hx via d/dz and w
• Ex is connected to Hy via d/dz and w
• Hy is connected to Ex via d/dz and w
• Hx is connected to Ey via d/dz and w
• Ey and Ex are connected via d/dx and d/dy
• Hy and Hx are connected via d/dx and d/dy

2
Telegraphists equations

• lets consider a long piece of something like
  coax
• i.e., a wire-pair
• one wire carrying a time-varying current out and
  the other carrying the return current
• one wire is at some time varying voltage relative
  to the other
• what might the equivalent circuit look like for a
  short segment of a long wire-pair?
• we would expect something related to current and
  magnetic fields inductance, resistance (if wires
  have finite conductivity)
• we would expect something related to voltage and
  charge capacitance, and conductance if the
  dielectric is leaky

3
Equivalent circuit

• so a reasonable guess for the equivalent circuit
  os a short segment of our wire pair would be
• here R, L, C, and G are per unit length values

4
Circuit response

• in time harmonic form, using phasors, we have

5
The telegraphists equations

• in time harmonic form, using phasors, we have

6
Circuit response

• in time harmonic form, using phasors, we have
• the form of these solutions looks exactly the
  same as what we got from Maxwell for TEM waves!
• the transmission line supports a traveling
  wave!!!
• again, g is called the complex propagation
  constant

7
V and I relationship characteristic impedance

• lets look at the equation for dI/dz
• for the solution in the z direction we use g,
  well designate the voltage as Vo, current as
  Io, and the characteristic impedance Zo is

8
V and I relationship generalized transmission
line

• we can generalize our results by considering the
  transmission line series impedance per unit
  length Z, and the shunt admittance per unit
  length Y
• physically, the most common form for
• Z is the series R-L model
• Y is the shunt G-C model

9
Generalized transmission line impedance

• it is convenient to break the voltage into a
  forward traveling wave V(z) and a backward
  traveling wave V-(z)
• the total voltage is the sum of V(z) and V-(z)
• same notation for current
• the impedance at any point on the line is the
  ratio of TOTAL voltage to TOTAL current
• this is position dependent!

10
Generalized transmission line impedance

• the impedance at any point on the line is the
  ratio of TOTAL voltage to TOTAL current which is
  position dependent
• lets fiddle a bit
• the ratio V-/V is reminiscent of our reflection
  coefficient from plane waves... what are they
  here?

11
Generalized transmission line impedance

• to get any further we need a boundary condition
  to find relationships between the traveling wave
  quantities V- and V
• lets assume that we connect a load with
  impedance ZL at z 0
• what is Z seen at z -l?

12
Generalized transmission line impedance

• now we know what V/I is at z 0
• lets do some algebra

13
Load reflection coefficient

• at z 0 the load voltage reflection
  coefficient is
• more generally, we define a reflection
  coefficient at the location z -l

14
Normalized load impedances

• note that the reflection coefficient equations
  depend on the ratio of the (complex) load
  impedance ZL to the (complex) characteristic
  impedance Zo
• it may make sense to work with a normalized
  load impedance
• where r is the real part of the load impedance
  normalized to Zo
• i.e., r is the normalized load resistance
• and x is the imaginary part of the load
  impedance normalized to Zo
• i.e., x is the normalized load reactance

15
Input impedance

• the impedance Z at the location z -l

16
Input impedance

• a little more fiddling on impedance Z at the
  location z -l

17
Input impedance

• the impedance Z at the location z -l depends on
• the load impedance ZL
• the transmission line characteristic impedance Zo
  
• set mostly by the geometry dielectric constant
  of the wire-pair connected to the load
• the distance l between you and the load
• and the propagation constant g
• which depends on the T-line characteristics AND
  FREQUENCY!

18
Transmission line applets

• various
• http//www.eecg.toronto.edu/bradel/projects/trans
  missionline/
• http//fermi.la.asu.edu/w9cf/tran/
• matching (single stub) http//home3.netcarrier.com
  /chan/EM/PROGRAMS/STUBMATCH/

19
Example L-C transmission line

• physically, the most common form for Z is the
  series R-L model
• if the conductors are perfect then R 0, and
  so
• and for Y is the shunt G-C model
• if the dielectric is lossless then G 0