Waves The Scatterometer and Other Curious Circuits Peter O. Brackett, Ph.D. [email protected] ©Copyright 2003 Peter O. Brackett

WavesWavesThe Scatterometer and Other Curious Circuits

Peter O. Brackett, Ph.D.

[email protected]

©Copyright 2003 Peter O. Brackett

-Fields versus Circuits--Fields versus Circuits- Understanding Field-Theoretic Device Understanding Field-Theoretic Device

ElectrodynamicsElectrodynamics

Circulators Isolators Gyrators Filters

The operation of “wave devices”is often explained in terms of field theory and material propertiesusing 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.


Understanding Field-Theoretic Devices Understanding Field-Theoretic Devices from a Circuit-Theoretic Viewfrom 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.


What exactly are What exactly are ElectroMagnetic (EM) waves?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.”EM Waves are solutions to wave equations.”©Copyright 2003 Peter O. Brackett

What is a “wave equation”?What is a “wave equation”?Start with a Transmission Line ModelStart with a Transmission Line Model


One Dimensional Transmission One Dimensional Transmission Line Voltage Wave EquationLine Voltage Wave Equation

22

2

d vv

dx

( )( )R pL G pC j

The Propagation ConstantThe Propagation Constant

The Characteristic ImpedanceThe Characteristic Impedance

( )( / )

( )

R pLZo v i

G pC


Wave Equation SolutionsWave Equation Solutions2

22

d vv

dx

Here we see that v is the sum of a Here we see that v is the sum of a forwardforward and and backwardbackward wave. Many such wave. Many such waves may exist on a transmission line. However, since the wave equation 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 is second order, only two such waves, when they exist, can generally be uniquely determined at any point. Forward and backward waves can be uniquely determined at any point. Forward and backward waves can be resolved in terms of a reference impedance, usually, but not necessarily, resolved in terms of a reference impedance, usually, but not necessarily, assumed to be the characteristic impedance Zo of the line because Zo 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 relates v and i on the line. The choice of any reference impedance then allows a uniqueallows a unique resolution of the sum of the forward and backward waves resolution of the sum of the forward and backward waves in in terms of that reference impedance.terms of that reference impedance.

x xv ae be

The possible solutions to such a wave equation are the “Waves” and are The possible solutions to such a wave equation are the “Waves” and are sumssums of the form. of the form.


Resolving Forward and Resolving Forward and Backward WavesBackward 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?

A simple Op Amp bridge circuit can resolve forward from backward waves.

= ReferenceVin

Vout

( )

( )

Vout Rx R

Vin Rx R

Forward

Backward


v = a + b

Analysis of the Op Amp Bridge CircuitAnalysis of the Op Amp Bridge Circuit©Copyright 2003 Peter O. Brackett

Op Amp bridge circuit connected to a transmission line meters theOp Amp bridge circuit connected to a transmission line meters the““reflected voltage” and is thus a reflected voltage” and is thus a “Reflectometer”.“Reflectometer”. The output of The output of the Reflectometer divided by its’ input is the Reflection Coefficientthe Reflectometer divided by its’ input is the Reflection Coefficient““rho” at the front end of the transmission line. rho” at the front end of the transmission line.

The Bridge is a ReflectometerThe Bridge is a Reflectometer

= b/a = (Zin – R)/(Zin + R)

Another view of waves…Another view of waves… Wave Variables (a, b) Wave Variables (a, b) Electrical Variable (v, i)Electrical Variable (v, i)

Incident Voltage Wave: a = (v + Ri)Reflected Voltage Wave: b = (v – Ri)

In vector-matrix format:

1

1

a R v

b R i

The wave variables (a, b) are computed by the Reflectometer and are just simple linear combinations of the electrical variables (v, i):

The wave vector equals a transformation matrix times the electrical vector!

Waves = M * Electricals Waves = M * Electricals Geometrically this transformation from electrical Geometrically this transformation from electrical

to wave co-ordinates looks like:to wave co-ordinates looks like:

1

1

a R v

b R i

v

i

a

b(1, 1)

(1+R, 1-R)

ReflectometerMapping M

Electrical Variables

Wave Variables

Simplified Reflectometer Simplified Reflectometer SchematicSchematic

R

1

1

a R v

b R i

WaveVariables

ElectricalVariables

b

a

R

Reflectometer Computes the Reflectometer Computes the Reflection Coefficient at the PortReflection Coefficient at the Port

1

1

a R v

b R i

WaveVariables

ElectricalVariables

b

a

ScatterometerScatterometerTwo Back-to-Back Reflectometers

R R

Port 1Reflectometer

Port 2Reflectometer

The Scatterometer is a Vector ReflectometerThe Reflectometer Computes a Scalar Reflection Coefficient

While the Scatterometer Computes a Scattering Matrix

b1 = s11*a1 + s12*a2b2 = s21*a1 + s22*a2

ScatterometerScatterometer

The Scattering MatrixThe Scattering Matrix

b1 = s11*a1 + s12*a2b2 = s21*a1 + s22*a2

1 11 12 1

2 21 22 2

b s s a

b s s a

B = S A

11 12

21 22

s sS

s s

Simplified Schematic of a Scatterometer

1 11 12 1

2 21 22 2

b s s a

b s s a

ScatterometerR

A Few Scattering MatricesA Few Scattering Matrices

2

2 2

2

2 2

pL R

pL R pL R

R pL

pL R pL R

1 2

1 2 1 2

2 1

1 2 1 2

pRC

pRC pRC

pRC

pRC pRC

1 2 /

1 2 / 1 2 /

2 / 1

1 2 / 1 2 /

pL R

pL R pL R

pL R

pL R pL R

2 /

2 / 2 /

2 /

2 / 2 /

pC R

pC R pC R

R pc

pC R pC R

p j Is the complex frequency variable

Examining Waves in an L-C Filter by Cascading Examining Waves in an L-C Filter by Cascading ScatterometersScatterometers

R R R R

Low Pass L-C FilterLow Pass L-C Filter

a1

b1 a2

b2WavesWaves

Electricals Electricals (v, i)(v, i)

More Scattering MatricesMore Scattering Matrices

Clockwise Circulator

v1 v2

v3

1 1

2 2

3 3

0 0 1

1 0 0

0 1 0

b a

b a

b a

Clockwise CirculatorClockwise Circulator

Provided all ports are terminatedproperly, power going in to Port 1 comes out Port 2, power going into Port 2 comes out Port 3, etc…

Electronic CirculatorElectronic Circulator

Three Port CirculatorThree Reflectometers in a Ring

v1 v2

v3

R R

R1 1

2 2

3 3

0 0 1

1 0 0

0 1 0

b a

b a

b a

Electronic CirculatorElectronic Circulator

Full Schematic of the Three Port CirculatorThree Reflectometers in a Ring!

v1 v2

v3

Four Port Circulator – Four Reflectometers in a Ring

R

RR

R

Full Schematic of Four Port Circulator


R

Isolatorv1 v2

v3

1 1

2 2

0 0

1 0

b a

b a

IsolatorIsolator

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)

R


RR

Gyratorv1 v2

V3 = 0

1 1

2 2

0 1

1 0

b a

b a

GyratorGyrator

i2 = -v1/Rv2 = i1*R

i2i1

Grounding Port 3, makes v3 = 0 and socauses the bottom amplifier to invertthus b3 = - a3, b1 = - a2, b2 = a1

Waves

Electricals

Full Gyrator SchematicFull Gyrator Schematic

Three Reflectometers in a ring: Port 3 is groundedThree Reflectometers in a ring: Port 3 is grounded

Understanding Field-Theoretic Understanding Field-Theoretic Device ElectrodynamicsDevice Electrodynamics-Fields versus Circuits--Fields versus Circuits-

Circulators Isolators Gyrators Filters

The operation of “wave devices”is often explained only in terms of field theory and material propertiesusing 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.


WavesWaves

The Scatterometer and Other Curious Circuits

Peter O. Brackett, Ph.D.

[email protected]

FINFor 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.


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Waves The Scatterometer and Other Curious Circuits Peter O. Brackett, Ph.D. [email protected] ©Copyright 2003 Peter O. Brackett