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

-Fields versus Circuits- Understanding 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 The Propagation Constant The Characteristic Impedance ©Copyright 2003 Peter O. Brackett

Wave Equation Solutions 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. The possible solutions to such a wave equation are the “Waves” and are sums of the form. ©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?

A simple Op Amp bridge circuit can resolve forward from backward waves. = Reference Vin Vout Forward Backward ©Copyright 2003 Peter O. Brackett v = a + b

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

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. The Bridge is a Reflectometer  = b/a = (Zin – R)/(Zin + R)

Another view of waves… Wave Variables (a, b) Electrical Variable (v, i) Incident Voltage Wave: a = (v + Ri) Reflected Voltage Wave: b = (v – Ri) In vector-matrix format: 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 Geometrically this transformation from electrical to wave co-ordinates looks like: v i a b (1, 1) (1+R, 1-R) Reflectometer Mapping M Electrical Variables Wave Variables

Simplified Reflectometer Schematic R Wave Variables Electrical Variables

R Reflectometer Computes the Reflection Coefficient at the Port Wave Variables Electrical Variables

Scatterometer Two Back-to-Back Reflectometers R R Port 1 Reflectometer Port 2 Reflectometer

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 Scatterometer

The Scattering Matrix b1 = s11*a1 + s12*a2 b2 = s21*a1 + s22*a2 B = S A

Simplified Schematic of a Scatterometer Scatterometer R

A Few Scattering Matrices Is the complex frequency variable

Examining Waves in an L-C Filter by Cascading Scatterometers R RRR Low Pass L-C Filter a1 b1 a2 b2 Waves Electricals (v, i)

More Scattering Matrices Clockwise Circulator v1 v2 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 v2 v3 RR R

Electronic Circulator Full Schematic of the Three Port Circulator Three Reflectometers in a Ring! v1 v2 v3

Four Port Circulator – Four Reflectometers in a Ring R RR R

Full Schematic of Four Port Circulator

More Scattering Matrices R Isolator v1 v2 v3 Isolator 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

More Scattering Matrices R R Gyrator v1 v2 V3 = 0 Gyrator i2 = -v1/R v2 = i1*R i2 i1 Grounding Port 3, makes v3 = 0 and so causes the bottom amplifier to invert thus b3 = - a3, b1 = - a2, b2 = a1 Waves Electricals

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. FIN For a copy of this copyright Power Point presentation, in Adobe PDF format, send an to Peter Brackett at the above address requesting a copy of the Adobe file of the IEEE Melbourne “Waves” presentation. ©Copyright 2003 Peter O. Brackett