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Compact Equivalent Circuit Models for the Skin Effect
The University of Texas at Austin
The University of Texas at Austin
Darpa Electronic Packaging and Interconnect Design and Test Program
Texas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test Program
Texas Advanced Technology Program
Microelectromagnetic Devices Group
Microelectromagnetic Devices Group
D. Neikirk
D. Neikirk
Compact Equivalent
Circuit Models for the
Skin Effect
Sangwoo Kim, Beom-Taek Lee, and Dean P. Neikirk
Department of Electrical and Computer Engineering
The University of Texas at Austin
Austin, TX 78712
for further information, Skin-Effect/' >please contact:
Professor Dean Neikirk, phone 512-471-4669
e-mail: neikirk@mail.utexas.edu
www home page:
http://weewave.mer.utexas.edu/ The University of Texas at Austin
The University of Texas at Austin
Microelectromagnetic Devices Group
Microelectromagnetic Devices Group
Origin of frequency dependencies in
transmission line series impedance
2
can frequency independent ladder circuits be
synthesized to accurately model frequency
dependent series impedance of line?
Uniform Current: dc
Non-Uniform: proximity
Non-Uniform: skin
depth & proximity
Resistance: R
dc
Inductance: uniform
current distribution
Resistance: increases
Inductance: decreases
Low frequencies
Mid frequencies
High frequencies
Resistance: increases
Inductance: constant,
infinite conductivity (high
frequency) limit 3
R-L ladder circuits for the skin effect
use of R-L ladders is classical
technique
- e.g., H. A. Wheeler,
Formulas for the
skin-effect, Proceedings of
the Institute of Radio
Engineers, vol. 30, pp.
412-424, 1942.
essentially an application of
transverse resonance
lumping based on uniform step
size tends to generate large
ladders

z
skin
effect
model
Lext
L1
R4
R3
R2
R1
L2
L3
L4
R6
R5
L5
L6
Cext Non-Uniform "step" size for compact
ladders
for lossy transmission lines and bandwidth limited Skin-Effect/' target='blank' class='doin' >signals, can
use increasingly long step size as propagate along line
- line acts like a low pass filter, so as you propagate along the line the
effective bandwidth decreases, allowing longer steps
for a skin effect equivalent circuit of a circular wire, Yen et al.
proposed use of steps such that the resistance ratio RR from
one step to the next is a constant
L
i
=
µ r
i 1 r
i
(
)
2

r
i
r
i
=
r RR
( )
M j n
+
1
n
=
1
M




1
j
=
i
+
1
M [C.-S. Yen, Z. Fazarinc, and R. L. Wheeler, Time-Domain Skin-Effect Model for Transient
Analysis of Lossy Transmission Lines, Proceedings of the IEEE, vol. 70, pp. 750-757, 1982]
radii of rings:
inductances:
R
i
R
i
+
1
=
RR
R
i
=
1

r
2 RR
( )
M j i
j
=
0
M 1 for an M-deep ladder
this leads to Darpa Electronic Packaging and Interconnect Design and Test Program
Texas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test Program
Texas Advanced Technology Program
D. Neikirk
D. Neikirk
Yen's results for a single circular wire
1x10
0
1x10
1
5x10
1
1x10
-2
1x10
-1
1x10
0
1x10
-2
1x10
-1
1x10
0
1x10
1
1x10
2
Normalized Resistance (units of Rdc)
Normalized Inductance (units of
µ
/8 )
Normalized Angular Frequency (units of 8 Rdc/
µ
o)
selection of ladder length and RR determines accuracy:
-
m = 4
(i.e., 4 resistors, 3 inductors), minimum error occurs for RR = 2.31
-
m = 10
, minimum error for RR = 1.37
5
blue: exact
green: Yen, 4 deep
red: Yen, 10 deep
resistance
internal inductance The University of Texas at Austin
The University of Texas at Austin
Microelectromagnetic Devices Group
Microelectromagnetic Devices Group
"Compact" ladders
problem: Yen's approach tends to
underestimate both resistance and
inductance
can a "short" ladder produce a good
approximation?
- "de-couple" resistance and inductance in a
4-long ladder
- each shell such that
R
i
/ R
i+1
=
RR , a constant (> 1)
- R
2
=
RR R
1
, R
3
=
RR
2
R
1
, R
4
=
RR
3
R
1
L
i
/ L
i+1
=
LL , a constant (< 1)
- L
2
=
LL L
1
, L
3
=
LL
2
L
1
6
1
2
3
4
L1
L2
L3 Fitting parameters for 4-long ladder
"unknowns" constrained by asymptotic behavior at low
frequency
- given the dc resistance R
dc
, then R
1
and RR are related by:
- given the low frequency internal inductance L
lf
internal
, then L
1
and LL are related by:
only "free" fitting parameters are R
1
and L
1
(or equivalently, RR
and LL)
- R
1
and L
1
tend to dominate the high frequency response
RR
( )
3
+
RR
( )
2
+
RR
+
(
1 R
1
R
dc
)
=
0
1
LL




2
+
1
+
1
RR




2
1
LL
+
1
RR






2
+
1
RR
+
1






2 L
lf
internal
L
1
1
+
1
RR






1
RR






2
+
1












2
=
0 "universal" fit possible over specified
bandwidth (dc to max
)
scales in terms of radius compared to minimum
skin depth (that occurs at highest frequency)
Best fit for single circular wire
8
R
1
(and hence RR):
L
1
(and hence LL): max
=
2 max
µ
o R
1
R
dc
=
0. 53 wire radius max
L
lf
internal
L
1
=
0.315 R
1
R
dc Results for single circular wire
9
1x10
0
1x10
1
5x10
1
1x10
-2
1x10
-1
1x10
0
1x10
-2
1x10
-1
1x10
0
1x10
1
1x10
2
Normalized Resistance (units of Rdc)
Normalized Inductance (units of
µ
/8 )
Normalized Angular Frequency (units of 8 Rdc/
µ
o)
blue: exact
red: new 4-ladder
resistance
internal inductance
RR = 2.5, LL = 0.290 Darpa Electronic Packaging and Interconnect Design and Test Program
Texas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test Program
Texas Advanced Technology Program
D. Neikirk
D. Neikirk
0%
10%
20%
30%
40%
50%
60%
70%
80%
1x10
-2
1x10
-1
1x10
0
1x10
1
1x10
2
Percent Internal Inductance Error
Normalized Angular Frequency
Errors for single circular wire
10
0%
5%
10%
15%
20%
25%
30%
1x10
-2
1x10
-1
1x10
0
1x10
1
1x10
2
Percent Resistance Error
Normalized Angular Frequency
Yen 4-ladder
Yen 10-ladder
new 4-ladder
excellent fit possible over wide range of frequencies, from
low to high frequency
shorter ladders (three of less) give much larger errors
longer ladders improve accuracy very slowly
resistance
inductance The University of Texas at Austin
The University of Texas at Austin
Microelectromagnetic Devices Group
Microelectromagnetic Devices Group
Results for coaxial cable
can account for both inner (signal) and
outer (shield) conductors
11
1x10
0
1x10
1
1x10
2
1.5x10
-7
1.7x10
-7
1.9x10
-7
2.1x10
-7
1x10
5
1x10
6
1x10
7
1x10
8
1x10
9
5x10
9
Resistance (Ohm/m)
Inductance (H/m)
Frequency (Hz)
example:
inner radius a = 0.1 mm
shield radius b = 0.23 mm
shield thickness 0.02 mm
f
max
= 5 GHz
a
c
b
blue: exact
red: circuit
resistance
total inductance
R1in
L1in
R2 in
R3 in
R4 in
L2in
L3in
R1out
L1out
R2 out
R3 out
R4 out
L2out
L3out
Lext Inclusion of proximity effects
12
for transmission lines with "non-circular"
geometry must also account for proximity
effects
use high frequency behavior to estimate
current division over surfaces of conductors
- subdivide external inductance (L
ext
) to force current
redistribution more flux coupling at inner
faces
- quarter from angle

two branches required
weight skin effect by = / Twin lead with proximity effect
13 2h
sin ( )
=
1 r h
( )
2
inner face
outer face
L1/ z
R4/ z
R3/ z
R2/ z
R1/ z
L2/ z
L3/ z
2Lext
2Lext
L1/(1- z)
R4/(1- z)
R3/(1- z)
R2/(1- z)
R1/(1- z)
L2/(1- z)
L3/(1- z) Darpa Electronic Packaging and Interconnect Design and Test Program
Texas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test Program
Texas Advanced Technology Program
D. Neikirk
D. Neikirk
0x10
0
1x10
1
2x10
1
3x10
1
4x10
1
1x10
7
1x10
8
1x10
9
1x10
10
1x10
11
Resistance per length (Ohm/cm)
Frequency (Hz)
5.0x10
-9
5.5x10
-9
6.0x10
-9
6.5x10
-9
7.0x10
-9
7.5x10
-9
1x10
6
1x10
7
1x10
8
1x10
9
1x10
10
1x10
11
Inductaance per length (H/cm)
Frequency (Hz)
14
Results for closely coupled twin lead
example for 1 mil diameter Al wires on 2 mil centers
-

= 60
L
external
conformal mapping
approximation
conformal mapping
approximation
circuit model
circuit model The University of Texas at Austin
The University of Texas at Austin
Microelectromagnetic Devices Group
Microelectromagnetic Devices Group
observation:
- regardless of
geometry of
transmission line, for
frequencies greater
than about 3R
dc
/L
lf
,
resistance increases
as