Extension of the BLT Equation to Incorporate Electromagnetic Field Propagation

Fredrick M. TescheChalmers M. Butler

Holcombe Department of Electrical and Computer Engineering336 Fluor Daniel EIB

Clemson University, Clemson, SC 29634-0915 USA

This research was supported by the U.S. Department of Defense

under MURI grant F49620-01-1-0436 toUniversity of Illinois at Chicago

andClemson University

University of HoustonUniversity of Illinois at Urbana-Champaign

University of Michigan

June 8, 2002

Extension of the BLT Equation – Slide 2/17

Outline of Presentation

Introduction

Review of the Derivation of the BLT Equation

Extension of the BLT Equation

Summary

Extension of the BLT Equation – Slide 3/17

Overview

The BLT equation for analyzing transmission line networks permits a system-level analysis of the EM effects on large systems– This is the basis of the CRIPTE code, and its predecessor,

QV7TA

In this MURI effort, we wish to extend the formulation of the BLT equation to take into account the following:– EM field propagation and coupling to the network

– EM penetration through apertures

– EM scattering from nearby bodies (including cavities)

Extension of the BLT Equation – Slide 4/17

Illustration of BLT Equation Extension

We wish to include non-conducting paths in the interaction sequence diagram– To model aperture or diffusive penetrations

Conventional BLT conducting interaction path

New, non-conductive BLT interaction path

Extension of the BLT Equation – Slide 5/17

Outline of Presentation

Motivation

Review of the BLT Equation

Extension of the BLT Equation

Summary

Extension of the BLT Equation – Slide 6/17

The BLT Equation for a Single Line Network

Consider a single transmission line “network”

- +Vs

Is

ExcitationNode #1 Node #2

Node #1 Node #2Excitation

ZL1ZL2

+V1

-

+V2

-

I2I1 I(x)

+V(x)

-

0 xs x L

Observationpoint

x

Zc,

Forward travelingwave

Backward travelingwave

V_(x)

V+(x)

TransmissionLine

LinearGraph

V ref1

V inc2

V ref2

V inc1

The BLT Equation provides the voltage or current responses at the ends (junctions) of the line

This is done by manipulating the equations for the incident and reflected traveling waves at the loads.

And including the excitation of forward and backward traveling wave components on the line by the excitation sources.

Extension of the BLT Equation – Slide 7/17

The BLT Equation for the Load Voltages

The BLT equation for the load voltage responses is written in a simple matrix form as

s

s

xLscs

xscs

eIZV

eIZV

S

S

21

21

2

1

–where the excitation vector for the lumped sources is given as

2

1

1

2

1

2

1

2

1

10

01

S

S

e

e

V

VL

L

Load voltages at each end of the line

Matrix involving load reflection coefficients

Inverse matrix involving line propagation

Excitation vector

Extension of the BLT Equation – Slide 8/17

The BLT Equation for Incident Field Excitation

The BLT equation for a lumped voltage source can be viewed as a Green’s function

– The response is found byintegrating over the line to incorporate the tangential E-field excitation of the line.

The same functional form of the BLT equation is valid for incident field (plane-wave) excitation:

2

1

1

2

1

2

1

2

1

10

01

S

S

e

e

V

VL

L

Note the field coupling functions F1 and F2

)(

)(

1

1

2 2

1

)cos1(

)cos1(

2

1

F

FE

ee

edE

S

S inc

jkLjkL

jkLinc

Only a change in the source vector is necessary:

Extension of the BLT Equation – Slide 9/17

Outline of Presentation

Motivation

Review of the Derivation of the BLT Equation

Extension of the BLT Equation

Summary

Extension of the BLT Equation – Slide 10/17

Extension of the BLT Equation to Include EM Field Propagation

Consider the following simple problem– Involving transmission line responses (the “conventional”

BLT problem)

– And EM field propagation from the source to the linePrimary source

Incident E-fields

Induced line current

Scattered E-field

Load #1

Load #2

Transmission Line

EM FieldSource(s)

P & M rs

ro

s

o

FieldObservation

PointEinc

r1

Responses: E-fields and load voltages

Extension of the BLT Equation – Slide 11/17

Extension of the BLT Equation (con’t.)

We define a signal flow graph including both the transmission line and the field coupling paths

Node 1

Node 2

Node 3

Node 4

Tube 1Tube 2

Source

Transmission line tube

EM field propagation "tube"

E ref2,3

V ref1,2

E inc2,3 E inc

2,4

E ref2,4

V inc1,2

V ref1,1V inc

1,1Incident and reflected E-fields at the ends of the EM field propagation path

Transmission line “tube”

“Regular” nodes where the incident and reflected waves are related by a reflection coefficient.

Field propagation “tube”

A “field coupling” node

Incident and reflected voltage waves at the transmission line ends

Extension of the BLT Equation – Slide 12/17

1

)(

12

)(

1

4,24

3,23

2,1

1,1

42

41

4

3

4

2

4

1

4,24

3,23

2,1

1,1

1

0

0

0)(2

)(2

000

)(00

)(00

r

err

er

Ea

Ea

V

V

ea

rF

Z

Z

a

jkF

Z

Z

a

jk

ea

ra

Fe

a

Fe

Ea

Ea

V

V

rrjko

s

rrjko

s

inc

inc

inc

inc

jkroo

c

oo

c

o

jkro

sL

sL

ref

ref

ref

ref

o

so

o

o

S

S

The Extended BLT Propagation Equation

As in the case of the transmission line BLT equation, we can define a propagation relationship between the incident and reflected voltages and normalized E-fields on the tubes:

4x4 propagation and coupling matrix

4-vector with the source excitation functions

Transmission line propagation sub-matrix

EM field propagation sub-matrix

Field coupling terms between the incident E-field on the line and the traveling voltage waves

Radiation terms from the traveling voltage waves

Note that both the coupling and radiation terms contain the same functions F1 and F2 --- a consequence of reciprocity

s

s

xLscs

xscs

inc

inc

L

L

ref

ref

eIZV

eIZV

V

V

e

e

V

V

21

21

2

1

2

1

0

0

BLT propagation equation for a single transmission line

Extended BLT Equation

4-vector of reflected voltage waves on the tubes

4-vector of incident voltage waves on the tubes

Note that the E-fields are normalized by suitable lengths a3 and a4 , which are typical dimensions of the nodes

Extension of the BLT Equation – Slide 13/17

inc

inc

inc

inc

ref

ref

ref

ref

Ea

Ea

V

V

Ea

Ea

V

V

4,24

3,23

2,1

1,1

4

3

2

1

4,24

3,23

2,1

1,1

000

000

000

000

The Extended BLT Reflection Coefficient Matrix

Similarly, we define an extended reflection coefficient matrix, which is similar to the 2x2 matrix for the simple transmission line:

Node 1

Node 2

Node 3

Node 4

Tube 1Tube 2

Source1 and 2 correspond to voltage reflections on the transmission line

3 corresponds to E-field reflection at node 3, which is usually zero.

Ground plane (cavity wall)

Eref

Einc

It is through this term that the effect of a local groundplane or cavity wall can be incorporated in the analysis

4 (which = 0 in this example) corresponds to E-field reflection at node 4, in the absence of the transmission line. The line has already been taken into account in the propagation and coupling matrix.

Extension of the BLT Equation – Slide 14/17

1

)(

12

)(

1

44

24

14

33

24

2

14

1

4

3

2

1

4,24

3,23

2,1

1,1

1

0

0

1

22

00

)(1

0

)(1

0

1000

0100

0010

0001

r

err

er

ea

rF

Z

Z

a

jkF

Z

Z

a

jk

ea

r

Fa

e

Fa

e

Ea

Ea

V

V

rrjlo

s

rrjko

s

jkroo

c

oo

c

o

jkro

L

L

o

so

o

o

S

S

The extended BLT voltage equation

The Extended Voltage BLT Equation

The BLT reflection and propagation matrix equations can be combined just like the single transmission line case to yield the extended BLT equation for the load voltages and normalized E-fields:

2

1

1

2

1

2

1

2

1

10

01

S

S

e

e

V

VL

L

BLT voltage equation for a single transmission line

Extension of the BLT Equation – Slide 15/17

Outline of Presentation

Motivation

Review of the Derivation of the BLT Equation

Extension of the BLT Equation

Summary

Extension of the BLT Equation – Slide 16/17

Summary

A modified BLT equation, taking into account EM field propagation paths in addition to the usual transmission line propagation mechanisms, has been developed.– This required modifying the BLT propagation matrix to

include the field coupling to the transmission line and the EM scattering from the line.

– These features have been illustrated with a very simple example.

The next step in this development will be to include the more general case of EM shields with apertures, and multiple field paths, as shown in the next slide …….

Extension of the BLT Equation – Slide 17/17

Work Currently Underway …

Load #1

Load #2

Transmission Line

EM FieldSource(s)

P & M

Induced LoadResponses

Excitation Region"outside"

Shielded Region"inside"

Shield andAperture

EM FieldSource

Transmission line within a simple shielded region with an aperture

Node 1

Node 2

Node 3 Node 4Node 5 Node 6

Tube 1

Tube 2Tube 3 Tube 4

Tube 5

Source

Outside Inside

Interaction diagram, showing both field propagation and transmission line propagation paths