Extension of the BLT Equation to Incorporate
Electromagnetic Field Propagation
Fredrick M. Tesche
Chalmers M. Butler
Holcombe Department of Electrical and Computer Engineering
336 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 to
University of Illinois at Chicago
and
Clemson University
University of Houston
University of Illinois at Urbana-Champaign
University of Michigan
June 8, 2002
Outline of Presentation
 Introduction
 Review of the Derivation of the BLT Equation
 Extension of the BLT Equation
 Summary
Extension of the BLT Equation – Slide 2/17
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 3/17
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 4/17
Outline of Presentation
 Motivation
 Review of the BLT Equation
 Extension of the BLT Equation
 Summary
Extension of the BLT Equation – Slide 5/17
The BLT Equation for a Single Line Network
 Consider a single transmission line “network”
The BLT Equation provides the voltage or current
responses at the ends (junctions) of the line
1
I
T
ran
sm
ssi
i
o
n
Z
L
n
ie
L
1
+
V
1
0
s
+
-V
Z
c
,
Is
xs
x
(
I)
+
V
x)
(
x
This is done N
by
manipulating
od
e#1includingthe
E
xcit
at
ion for
O
bser
va
ion
t
And
theequations
excitation
of the
forward
and
it
incident and reflected
traveling
waves
at the
loads. pon
backward
traveling
wave
components
on the line by
2
I
+
V
2
-
Z
L
2
x
L
N
ode#2
the excitation sources.
inc
V
1
L
n
i ear
G
rap
h
ef
r
V
1
N
ode#1
E
xcit
at
ion
F
ow
rad
ra
r
t ven
il g
w
ave+
V
x)
(
_
x)
V(
B
ackw
ar
dt
aveling
r
w
ave
Extension of the BLT Equation – Slide 6/17
inc
V
2
ef
r
V
2
N
ode#2
The BLT Equation for the Load Voltages
 The BLT equation for the load voltage responses is written in
a simple matrix form as
1
0   1 e   S1 
V1  1  1
  L
 
  

1  2   e
  2  S2 
V2   0
L
Load voltages
at each end of
the line
Matrix involving
load reflection
coefficients
Inverse matrix
involving line
propagation
Excitation
vector
–where the excitation vector for the lumped sources is given as
 S1   12 Vs  Z c I s exs 
  1
  L  xs  



V

Z
I
e
S
c s
 2  2 s

Extension of the BLT Equation – Slide 7/17
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 by
integrating 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:
L 1
0   1 e   S1 
V1  1  1
 
  
   L
0
1


  2  S2 
2  e
V2  
 Only a change in the source vector is necessary:
 S1  E d
S   2
 2
inc


F ( ) 
 e
1 
inc  1

E
 jkL  jkL(1cos )

 F ( )
e
e

1
 2



jkL (1cos )


Extension of the BLT Equation – Slide 8/17
Note the field
coupling functions
F1 and F2
Outline of Presentation
 Motivation
 Review of the Derivation of the BLT Equation
 Extension of the BLT Equation
 Summary
Extension of the BLT Equation – Slide 9/17
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 line
Primary source
EM Field
Source(s)
Load #1
Incident E-fields
P&M
Field
Observation
Point
Responses: Efields and load
voltages
rs
r1
o
s
Einc
Induced line current
ro
Scattered E-field
Load #2
Transmission Line
Extension of the BLT Equation – Slide 10/17
Extension of the BLT Equation (con’t.)
 We define a signal flow graph including both the
“Regular” nodes where the
transmission
line and the field coupling paths
incident and reflected waves are
related by a reflection coefficient.
Incident and reflected
voltage waves at the
A “field
coupling”
transmission lineField
ends
propagation
node
“tube”
Einc
2,
3
N
o
d
e2
V inc
1,
2
Transmission
line “tube”
N
o
d
e4
e
rf
V
1,
2
inc
S
o
u
rce
E
2,
4
N
o
d
e3
rf
Ee
T
u
b
e2
2,
3
Incident and reflected Efields at theTra
ends
ofsio
the
nsm
is
nlineEM
ube
t
field propagation path
e
rf
E
2,
4
Vinc
1,
1
E
M
ieldpropagat
f
ion"t
ube"
Extension of the BLT Equation – Slide 11/17
T
u
b
e1
e
rf
V
1,
1
N
o
d
e1
The Extended BLT Propagation Equation
 As in the case of the transmission line BLT equation, we
Transmission line
terms
can define a propagation
relationship between Field
the coupling
incident
propagation subbetween the incident Ematrix
and reflected voltages
and normalized E-fieldsfield
ononthe
the line and the
EM field
traveling voltage waves
Extended BLT Equation
tubes:
propagation sub-
F ( ) 

0
e L
0
 1 s 
matrix

0


a4


ref
inc


 V1,1 
 V1,1 
BLT
propagation equation for a single
line 0
F2 ( transmission
L
s )


e
0
0

 ref 
 inc 
jk ( ro  rs ) 
r
e


V
V
a
o
1
,
2
1
,
2



   S1  s 
4

xs
ref
inc
L


ro jkr
1 o   a 3 E 2inc
 a 3 E 2ref,3  

r
s
,3

0
0
0
e 2  s

jk ( ro  r1 ) 



c
s
1
1
ref
ro e
a3

 a 4 E 2inc

 S 2L
,4 
a 4 E 2, 4  


1

x

ref
inc

L

1
s
ro jkro
jk Z o
jk Z o


r
1
F1 ( o ) 
F2 ( o )2
e

c s
2
20 s
2 a 4 Z c
a4
 2 a 4 Z c

V

V
 0

 e
e  V

0  V
   V  Z I e

  V  Z I e



Note that the E-fields are
normalized by suitable
4-vector of
4-vector of
lengths a3 and a4 , which are
4x4that
propagation
and
reflected
incident4-vector
voltagewith the source
Note
both
the
coupling
and
typical dimensions of the
excitation
functions
Radiation
the terms
coupling
matrix
voltage
waves terms from
waves on
the
radiation
contain
the
same
nodes
traveling
on the
tubes voltage waves
tubes
functions F1 and F2 --- a consequence
of reciprocity
Extension of the BLT Equation – Slide 12/17
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:
 V1,ref
  1
1
 ref  
 V1, 2    0
 a3 E2ref,3   0


ref 
a
E
 4 2, 4   0
0
0
2
0
0
3
0
0

0   V1inc
,1
 inc 

0   V1, 2 



0  a3 E2inc
,3
 
inc 
 4  a4 E2, 4 
Ground plane
(cavity wall)
Node 2
3corresponds
2 correspond
= to
0is in
E-field
thisto
example)
1and
4 (which
It
through
this term that
reflection
voltage
corresponds
reflections
at node
3,E-field
which
onof
thea is
Node 3
theto
effect
local
usually
transmission
reflection
zero. groundplane
atline
node 4, in the
or cavity wall
absence ofcan
thebe
transmission
incorporated in the
line. The line
has
analysisalready been
taken into account in the
propagation and coupling
Extension of the BLT Equation – Slide 13/17
matrix.
Einc
Node 4
Source
Tube 2
Tube 1
Eref
Node 1
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:
 V1,1  1  1
0
0
0 

 
BLT
voltage
equation
lineBLT
 for a single transmission
The
extended
0
1


0
0
V
2
1
,
2



voltage equation
 a3 E 2 , 3   0
0
1  3
0 

 

0
0
1 4 
a4 E 2, 4   0
1
0   1 e1L  S11 
V1  1  1
eL
 1   
   0L  a F1 ( )S 
0
1


 4 2   2 
2  e
V2  
1




eL
 2
0


0
0
 3



ro jkro
jk Z o
jk Z o
 2 a Z F1  o   2 a Z F2  o  a e
4
c
4
c
4


F2 ( )

a4
ro jkro 
e

a3


 4 

Extension of the BLT Equation – Slide 14/17


0


ro e jk ( ro rs ) 

 S1  s 


rs

jl ( ro  r1 ) 
 S2  1  ro e

r1


0
Outline of Presentation
 Motivation
 Review of the Derivation of the BLT Equation
 Extension of the BLT Equation
 Summary
Extension of the BLT Equation – Slide 15/17
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 16/17
Work Currently Underway …
Excitation Region
"outside"
Shielded Region
"inside"
Load #1
EM Field
Source(s)
Transmission line within a
simple shielded region with
an aperture
Induced Load
Responses
P&M
Load #2
EM Field
Source
Shield and
Aperture
Transmission Line
Outside
Inside
Node 2
Interaction diagram, showing
both field propagation and
transmission line propagation
paths
Tube 1
Node 3
Source
Tube 2
Node 5
Node 4
Tube 4
Tube 3
Node 1
Tube 5
Extension of the BLT Equation – Slide 17/17
Node 6