Engineering of Distributed Systems Introduction to Electromagnetic Waves

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Engineering of Distributed SystemsIntroduction
to Electromagnetic Waves
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The mythical equipotential wire
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But every wire has parasitics
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Why do wires act like transmission lines?
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Signals take time to propagate Propagating
Signals must have energy All conductors have
inductance and capacitance Inductance and
Capacitance Stores Energy
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Discrete Approximation to a Lossless Transmission
Line

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Fundamental Equations of Continuous Transmission
Lines

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From Dual State Equations to the Wave Equation
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From Dual State Equations to the Wave Equation
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Wave Equation Intuition

• Same equation for V and I Both propagate
• Does not specify relationship of V and I
• The larger l c , the larger the wavelength in
  space, i.e. the slower the propagation in time.

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Transmission Line Math
Lets try a sinusoidal solution for V and I
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Transmission Line Algebra
Divide Equations, then simplify
Simplify by substitution method
Propagation Velocity
Characteristic Impedance
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Why do wires act like transmission lines?
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Signals take time to propagate Propagating
Signals must have energy Inductance and
Capacitance Stores Energy An Infinite
transmission line looks like a
Resistor
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Why do wires act like transmission lines?
Signals take time to propagate Propagating
Signals must have energy Inductance and
Capacitance Stores Energy Without termination,
energy reaching the end of a finite
transmission line has nowhere to go - so it
_______________________
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Echoes
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Modeling Reflections as Reverse Propagation
Direction

• Right Hand Boundary Constraint
• Open I 0
• Reverse Waveform
• Positive Voltage
• Negative Current
• Short V 0
• Reverse Waveform
• Negative Voltage
• Positive Current
• Left Hand Boundary Constraint?

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How do waves know which way to go?

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Answer They dont !!!
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Bidirectional Propagation Demo

• Electrical String Nodes

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Waves can Propagate in All Directions
Simultaneously

• Linear Superposition
• Waves traveling both directions do NOT interact
• How can this work when waves traveling in
  opposite direction collide?
• If current 0, voltage is not zero
• If voltage 0, current is not zero
• Phase of voltage and current (-pi/2, pi/2) at
  various spatial locations remembers which
  direction waves are traveling

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Standing Waves

• Waves reflect off both ends.
• If boundary conditions identical
• period N round trip time
• N harmonic
• If boundary conditions opposite
• ½ period N round trip time
• N harmonic
• Envelope of Waveform in Space Nodes Peaks
• Nodes are NOT zero signal points
• Zero in one variable
• Maximum in other

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Virgil Fox
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Terminations

• Ideal Termination Characteristic Impedance of
  Transmission Line
• Parallel termination
• Absorb signal at end
• Series termination
• Absorb signal at source (after reflection at end)

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Parallel Termination
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Series Termination
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When is a wire a transmission line?
Rule of Thumb
Transmission Line
Equipotential Line
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Making Transmission LinesOn Circuit Boards
Insulating Dielectric
Copper Trace
w
t
h
Voltage Plane
h / (w sqrt(e r ) )
e r w/h
1/sqrt(e r )
h/w
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Actual Formulas
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A Typical Circuit Board
1 Ounce Copper
G-10 Fiberglass-Epoxy
Speed of light in vacuum approx. 30 cm (a.k.a.
one foot) / ns Z0 of free space approx. 377
ohms
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Impedance Mis-Match

• Some of wave energy is reflected
• How much?
• Part of this weeks lab
• General principle
• Impedance Match to get best energy transfer
• Similar to impedance match idea of compartment
  systems

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Linear Model of Lance Armstrong

• Thevenin
• Velocity Source max. speed at no load
• Series Damper torque / speed relationship
• Norton
• Torque Source max. torque at stall
• Parallel Damper torque / speed relationship

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Is Lance Impedance Matched with Terrain?

• A Depends on Terrain
• If Terrain High Torque / Low Velocity, we need
• If Terrain Low Torque / High Velocity, we need

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Impedance Matching of Transmission Lines

• Can use transformer
• Transformer can be continuous