Back to our slightly more general solution

1
Back to our slightly more general solution

• g is the complex propagation constant
• E and H are perpendicular to each other,
  direction of propagation is given by E x H
• what happens if s is not zero?
• since g2 is complex, so will be the square root
• lets write the real and imaginary parts of g as
• for the z traveling wave the behavior is
  exp-gz exp-azexp-jbz
• so a represents attenuation a is the
  attenuation constant
• b is still the phase (propagation) constant
• applet showing attenuation with finite
  conductivity (try s 0.05)

2
Finite conductivity and loss tangent

• clearly the relative magnitudes of s and we are
  critical in determining whats going to happen
• lets take a quick look back to the original
  Maxwell equation that mixed conductivity,
  dielectric constant, and frequency
• tan q is called the loss tangent tan q
  s/(we)

conduction current
3
Low loss material behavior in a good
dielectric

• lets consider different limits for the loss
  tangent (or equivalently, different limits for s
  compared to we)
• good dielectric
• s ltlt we or equivalently w gtgt s/e , i.e.,
  high frequency
• tan q ltlt 1

4
Low loss material tan q ltlt 1 (w gtgt s/e)

• so the complex propagation constant in the low
  loss limit is
• note that the phase constant in this limit is the
  same as we got for the zero conductivity case
• the attenuation constant is proportional to the
  (frequencyloss tangent) product
• or another way to write the same thing
• the attenuation constant is proportional to the
  conductivity
• since in this limit s ltlt we ? a ltlt b

5
Another way to write loss characteristics

• in the zero conductivity case we had
• but now we have
• what if we try to modify our zero-loss equation
  to make it look like the lossy equation by
  assuming
• now substitute and compare
• so in any thing we have already done
• wherever we saw s, replace it with we
• wherever we saw e, replace it with e
• example the loss tangent tan q s/(we)
  (we)/(we) e/ e

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Complex permittivity formulation

• the complex propagation constant is
• the loss tangent is just
• in the low loss limit
• or
• this keeps things looking a lot like our lossless
  result
• plus the small loss perturbation

7
Wave impedance when s ? 0

• recall we found that the ratio of E/H for the TEM
  wave was a constant
• lets use the solution traveling in the z
  direction, i.e., the minus sign from the

8
Wave impedance for good dielectrics

• so we have for the wave impedance, in the case
  for s ? 0
• if s ltlt we then we are in the good dielectric
  limit
• this is the same as requiring that the operating
  frequency be such that w gtgt s/e
• i.e., you get dielectric behavior so long as the
  operating frequency is HIGHER than the dielectric
  relaxation frequency s/e
• this is the same as small loss tangent tan q
  s/(we) ltlt 1
• the wave impedance for small loss tangent is
• the wave impedance is now complex!
• interpretation were using phasors, so this
  tells us that the phase angle between E and H
  must have changed
• lossless case E and H are in phase
• lossy case they are slightly out of phase

9
Good conductor wave propagation

• lets go back (again) to our exact expression
  for the fields and complex propagation constant
  for a TEM plane wave
• this time, lets consider the case when s gtgt we
  , or equivalently w ltlt s/e (low frequency of
  operation)
• so, what is the square root of j?

10
Good conductor wave propagation

• good conductor s gtgt we , low frequency w ltlt
  s/e
• in the limit s gtgt we well drop the small terms
  inside the bracket, so we have the propagation
  constant in a good conductor
• note that the real part (the attenuation
  constant) and the imaginary part (the phase
  constant) are identical, proportional to the
  square root of (frequencyconductivity)
• pick the sign on gamma to make sure the wave
  DECAYS, not grows!

11
Skin depth in a good conductor

• the plane wave field inside a good conductor is
  exponentially decaying
• with attenuation coefficient a (units 1/length)
  a given by
• or equivalently, with a 1/e decay distance of
• d is the skin depth for a plane wave in a good
  conductor
• the plane wave field is still propagating with
  phase constant b
• recall that the wavelength l is 2p/b, so here

12
Heavisides analogy for the skin effect

• what happens when a pulse of current is sent down
  a wire of radius r?
• how far into the wire does the current extend?
• Heaviside used the following analogy
• imagine a pipe full of water that is initially at
  rest, then suddenly jerked into uniform motion
  (in a direction lined up with the length of the
  pipe)
• initially only the water right next to the pipes
  wall will move, and then slowly the rest of the
  water will also begin to move with the pipe
• what would happen if you reversed the direction
  of the pipes motion before all the water had a
  chance to catch up?
• the water near the surface would follow the
  pipes motion, but the water in the middle
  wouldnt get a chance to move at all!
• now reverse again, and again only the water
  adjacent to the surface would move
• for a single frequency ac problem the depth of
  current flow is about one skin depth!
• the part of the wire in the middle (deeper than
  the skin depth) never actually gets the chance
  to carry any current!
• you might as well have used a hollow pipe for
  your wire!
• clearly the ac resistance of the wire could be
  bigger than the dc resistance!

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Wave impedance in a conductor

• the wave impedance is (use the sign for g for
  propagation in the z direction)
• in the case of s gtgt we we then have
• so finally we have

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Typical numbers for real materials

• recall the dielectric relaxation frequency s/e
  
• good dielectric HIGH FREQUENCY limit w gtgt
  s/e
• good conductor LOW FREQUENCY limit w ltlt s/e
• skin depth calculators
• http//www.microwaves101.com/encyclopedia/calsdept
  h.cfm

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Regardless of whether we call it a conductor
(w ltlt s/e) or a dielectric (w gtgt s/e)

• g is the complex propagation constant
• E and H are perpendicular to each other,
  direction of propagation is given by E x H
• what happens if s is not zero compare w to s/e
• since g2 is complex, so will the square root
• lets write the real and imaginary parts of g as
• so a represents attenuation a is the
  attenuation constant
• b is still the phase (propagation) constant
• applet showing attenuation with finite
  conductivity (try s 0.05)

16
Dispersion

• lets consider the simple case of a uniform
  plane wave in a medium with zero conductivity
• what would happen if we had two waves
  propagating, each at a slightly different
  frequency, and e a function of w?
• assume the two frequencies are w dw with
  corresponding phase constants b db

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Dispersion

• if we have two waves propagating, each at a
  slightly different frequency, where the two
  frequencies are w dw with corresponding phase
  constants b db
• then the solution is proportional to
• in words, this looks just like a wave at the
  frequency w and associated phase constant b, with
  phase velocity w/b
• but multiplied with an amplitude modulation
  function
• the velocity of a phase front for this
  modulation envelope is dw/db

18
Group velocity

• if we have two waves propagating, each at a
  slightly different frequency, the solution
  behaves like a wave at the frequency w with
  associated phase constant b, traveling at the
  phase velocity w/b
• but it is multiplied by an amplitude modulation
  function traveling at the velocity dw/db
• in the limit of infinitesimal variation we obtain
  the group velocity vg
• omega-beta diagrams
• plot the frequency vs beta
• slope from origin to a point on the curve is the
  phase velocity
• slope of tangent is the group velocity

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Group velocity and the speed of light

• if we have two waves propagating at slightly
  different frequencies w dw, the solution
  behaves like a wave at the frequency w with
  associated phase constant b, traveling at the
  phase velocity vp w/b , but it is multiplied by
  an amplitude modulation function traveling with
  group velocity vg
• special relativity says
• the velocity of information cannot be greater
  than c, the speed of light
• since packets carry information, and group
  velocity is usually (but not always) related to
  packet velocity, vgroup is normally less than c

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Summary of electromagnetics Maxwells equations

• summarizing everything we have so far, valid even
  if things are changing in time
• plus material properties

• where do we go from here? What about power flow
  carried by a wave?

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a for the good dielectric case

• so we have
• lets fiddle
• but b 2p/l so
• hence the when you travel a distance of one
  wavelength the amplitude decreases by exp(-ptanq)

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Dispersion example
23
omega-betas
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Surface impedance of a conductor

• recall we looked at a dc problem that consisted
  of a sheet of uniform conducting material

• if the width and length are the same (i.e., its
  a square) we defined a sheet resistance Rs

• for the ac problem of a good conductor we got for
  the wave impedance

• this looks similar to the dc sheet resistance of
  a slab that was d thick
• since the wave decays rapidly with distance,
  almost all the ac current occurs in a thickness
  of roughly a skin depth d