Download WavesFinal

Waves
The Scatterometer and Other Curious Circuits
Peter O. Brackett, Ph.D.
[email protected]
©Copyright 2003 Peter O. Brackett
-Fields versus CircuitsUnderstanding Field-Theoretic Device
Electrodynamics

Circulators
 Isolators
 Gyrators
 Filters
The operation of “wave devices”
is often explained in terms of
field theory and material properties
using the“wave variables” a and b.
In this presentation we examine their
operations in terms of circuit theory
and the “electrical variables” v and i.
©Copyright 2003 Peter O. Brackett
Understanding Field-Theoretic Devices
from a Circuit-Theoretic View
Brought to you by…
Ohm, Kirchoff and the Operational Amplifier rules!
Ideal Op Amp Analysis Rules
 Output impedance is zero.
 Input impedance is infinite
 Negative feedback forces
differential inputs to be equal.
©Copyright 2003 Peter O. Brackett
What exactly are
ElectroMagnetic (EM) waves?

Everyone has direct
experience with a
variety of physical
waves.
 But ElectroMagnetic
waves are invisible
and mysterious.
 The short answer is…
“EM Waves are solutions to wave equations.”
©Copyright 2003 Peter O. Brackett
What is a “wave equation”?
Start with a Transmission Line Model
©Copyright 2003 Peter O. Brackett
One Dimensional Transmission
Line Voltage Wave Equation
2
d v
2



v
2
dx
The Propagation Constant
  ( R  pL)(G  pC)    j 
The Characteristic Impedance
( R  pL)
Zo  (v / i ) 
(G  pC )
©Copyright 2003 Peter O. Brackett
Wave Equation Solutions
d 2v
2



v
2
dx
The possible solutions to such a wave equation are the “Waves” and are
sums of the form.
 x
x
v  ae
 be
Here we see that v is the sum of a forward and backward wave. Many such
waves may exist on a transmission line. However, since the wave equation
is second order, only two such waves, when they exist, can generally be
uniquely determined at any point. Forward and backward waves can be
resolved in terms of a reference impedance, usually, but not necessarily,
assumed to be the characteristic impedance Zo of the line because Zo
relates v and i on the line. The choice of any reference impedance then
allows a unique resolution of the sum of the forward and backward waves in
terms of that reference impedance.
©Copyright 2003 Peter O. Brackett
Resolving Forward and
Backward Waves
Forward and backward waves “a” and “b”
on a transmission media are “hidden” in the
line voltage “v” as a sum v = a + b.
 How can forward and backward waves
within the line voltage v be resolved and
physically separated for analysis or to
produce useful results?

Forward
Vin
= Reference
Vout ( Rx  R)

Vin ( Rx  R)
Vout
v=a+b
Backward
A simple Op Amp bridge circuit can
resolve forward from backward waves.
©Copyright 2003 Peter O. Brackett
Analysis of the Op Amp Bridge Circuit
©Copyright 2003 Peter O. Brackett
 = b/a = (Zin – R)/(Zin + R)
The Bridge is a Reflectometer
Op Amp bridge circuit connected to a transmission line meters the
“reflected voltage” and is thus a “Reflectometer”. The output of
the Reflectometer divided by its’ input is the Reflection Coefficient
“rho” at the front end of the transmission line.
Another view of waves…
Wave Variables (a, b)
Electrical Variable (v, i)
The wave variables (a, b) are computed by the Reflectometer and
are just simple linear combinations of the electrical variables (v, i):
Incident Voltage Wave: a = (v + Ri)
Reflected Voltage Wave: b = (v – Ri)
In vector-matrix format:
 a  1 R  v 
b   1  R   i 
  
 
The wave vector equals a transformation matrix times the electrical vector!
Waves = M * Electricals
Geometrically this transformation from electrical
to wave co-ordinates looks like:
 a  1 R  v 
b   1  R   i 
  
 
i
b
(1, 1)
Reflectometer
Mapping M
a
v
Electrical
Variables
Wave
Variables
(1+R, 1-R)
Simplified Reflectometer
Schematic
Wave
Variables
b

a
R
 a  1 R  v 
b   1  R   i 
  
 
Electrical
Variables
Reflectometer Computes the
Reflection Coefficient at the Port
Wave
Variables
R
b

a
 a  1 R  v 
b   1  R   i 
  
 
Electrical
Variables
R
R
Port 1
Reflectometer
Port 2
Reflectometer
Scatterometer
Two Back-to-Back Reflectometers
Scatterometer
The Scatterometer is a Vector Reflectometer
The Reflectometer Computes a Scalar Reflection Coefficient
While the Scatterometer Computes a Scattering Matrix
b1 = s11*a1 + s12*a2
b2 = s21*a1 + s22*a2
The Scattering Matrix
b1 = s11*a1 + s12*a2
b2 = s21*a1 + s22*a2
 b1   s11 s12   a1 
b    s



 2   21 s22  a2 
B=SA
 s11 s12 
S

 s21 s22 
R
Scatterometer
 b1   s11 s12   a1 
b    s



 2   21 s22  a2 
Simplified Schematic of a Scatterometer
A Few Scattering Matrices
 pL
 pL  2 R

 2R
 pL  2 R

2R 
pL  2 R 

pL 
pL  2 R 
1

1  2 pRC

 2 pRC
1  2 pRC

2 pRC 
1  2 pRC 

1

1  2 pRC 
1

1  2 pL / R

 2 pL / R
1  2 pL / R

2 pL / R 
1  2 pL / R 

1

1  2 pL / R 
  pC
 pC  2 / R

 2/ R
 pC  2 / R

p    j Is the complex frequency variable
2/ R 
pC  2 / R 

 pc 
pC  2 / R 
Low Pass L-C Filter
b2
a1
Waves
b1
R
R
R
R
a2
Electricals
(v, i)
Examining Waves in an L-C Filter by Cascading
Scatterometers
More Scattering Matrices
v1
v2
 b1  0 0 1  a1 
b    1 0 0   a 
 2 
 2
b3  0 1 0  a3 
Clockwise Circulator
v3
Clockwise Circulator
Provided all ports are terminated
properly, power going in to Port 1
comes out Port 2, power going into
Port 2 comes out Port 3, etc…
Electronic Circulator
Three Port Circulator
Three Reflectometers in a Ring
v1
R
 b1  0 0 1  a1 
b    1 0 0   a 
 2 
 2
b3  0 1 0  a3 
R
R
v3
v2
Electronic Circulator
Full Schematic of the Three Port Circulator
Three Reflectometers in a Ring!
v2
v1
v3
R
R
R
R
Four Port Circulator – Four Reflectometers in a Ring
Full Schematic of Four Port Circulator
More Scattering Matrices
v1
Isolator
v2
 b1  0 0  a1 
b    1 0   a 
 2
 2 
Isolator
R
v3
R
Provided all ports are terminated by
resistance R; power going in to Port 1
comes out Port 2, power going into
Port 2 is isolated and goes nowhere.
(Dissipates in the grounded resistor R)
More Scattering Matrices
Gyrator
v1
v2
i2
i1
R
R
 b1  0 1  a1 
b    1 0   a 
 2
 2 
Waves
Gyrator
i2 = -v1/R
v2 = i1*R
Electricals
V3 = 0
Grounding Port 3, makes v3 = 0 and so
causes the bottom amplifier to invert
thus b3 = - a3, b1 = - a2, b2 = a1
Full Gyrator Schematic
Three Reflectometers in a ring: Port 3 is grounded
Understanding Field-Theoretic
Device Electrodynamics
-Fields versus Circuits
Circulators
 Isolators
 Gyrators
 Filters
The operation of “wave devices”
is often explained only in terms of
field theory and material properties
using the“wave variables” a and b.
In this presentation we examined their
operations in terms of circuit theory
and the “electrical variables” v and i.
©Copyright 2003 Peter O. Brackett
Waves
The Scatterometer and Other Curious Circuits
Peter O. Brackett, Ph.D.
[email protected]
FIN
For a copy of this copyright Power Point presentation, in Adobe PDF format, send an
email to Peter Brackett at the above email address requesting a copy of the Adobe file of
the 11-2003 IEEE Melbourne “Waves” presentation.
©Copyright 2003 Peter O. Brackett