Post on 26-Mar-2020
The University of Texas at AustinThe University of Texas at Austin
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program
Microelectromagnetic Devices GroupMicroelectromagnetic Devices Group
D. Neikirk D. Neikirk
Compact Equivalent Circuit Models for the
Skin EffectSangwoo Kim, Beom-Taek Lee, and Dean P. NeikirkDepartment of Electrical and Computer Engineering
The University of Texas at AustinAustin, TX 78712
for further information, 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 AustinThe University of Texas at AustinMicroelectromagnetic Devices GroupMicroelectromagnetic 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: RdcInductance: uniform current distribution
Resistance: increasesInductance: decreases
Low frequencies Mid frequencies High frequencies
Resistance: increasesInductance: 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
δ 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 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
Li =µ ⋅ ri−1 − ri( )
2π ⋅ ri
ri = 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:
Ri Ri+1 = RR Ri = 1
σ π r2 ⋅ RR( )M− j−i
j=0
M−1
∑for an M-deep ladder
this leads to
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program D. Neikirk D. Neikirk
Yen's results for a single circular wire
1x100
1x101
5x101
1x10-2
1x10-1
1x100
1x10-2 1x10-1 1x100 1x101 1x102 No
rmal
ized
Res
ista
nce
(u
nit
s o
f R
dc)
No
rmal
ized
Ind
uct
ance
(u
nit
s o
f µ/
8π)
Normalized Angular Frequency (units of 8πRdc/µo)
• 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: exactgreen: Yen, 4 deepred: Yen, 10 deep
resistance
internal inductance
The University of Texas at AustinThe University of Texas at AustinMicroelectromagnetic Devices GroupMicroelectromagnetic 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)
- R2 = RR R
1 , R
3 = RR 2 R
1 , R
4 = RR 3 R
1
• L i / L
i+1 = LL , a constant (< 1)
- L2 = LL L
1 , L
3 = LL 2 L
1
6
1
2
34
L1
L2
L3
Fitting parameters for 4-long ladder
• "unknowns" constrained by asymptotic behavior at low frequency
- given the dc resistance Rdc
, then R1 and RR are related by:
- given the low frequency internal inductance Llf
internal, then L1
and LL are related by:
• only "free" fitting parameters are R1 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 − R1Rdc
) = 0
1LL
2+ 1 + 1
RR
2 1LL
+ 1RR
2+ 1
RR+ 1
2
−Llf
internal
L11 + 1
RR
1RR
2+ 1
2
= 0
• "universal" fit possible over specified bandwidth (dc to ωmax)
• scales in terms of radius compared to minimum skin depth (that occurs at highest frequency)
Best fit for single circular wire
8
R1 (and hence RR): L
1 (and hence LL):
δmax = 2ωmaxµoσ
R1Rdc
= 0.53wire radius
δmax
Llfinternal
L 1= 0.315 ⋅
R1
Rdc
Results for single circular wire
9
1x100
1x101
5x101
1x10-2
1x10-1
1x100
1x10-2 1x10-1 1x100 1x101 1x102 No
rmal
ized
Res
ista
nce
(u
nit
s o
f R
dc)
No
rmal
ized
Ind
uct
ance
(u
nit
s o
f µ/
8π)
Normalized Angular Frequency (units of 8πRdc/µo)
blue: exactred: new 4-ladder
resistance
internal inductance
RR = 2.5, LL = 0.290
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program D. Neikirk D. Neikirk
0%
10%
20%
30%
40%
50%
60%
70%
80%
1x10-2 1x10-1 1x100 1x101 1x102
Per
cen
t In
tern
al In
du
ctan
ce E
rro
r
Normalized Angular Frequency
Errors for single circular wire
10
0%
5%
10%
15%
20%
25%
30%
1x10-2 1x10-1 1x100 1x101 1x102
Per
cen
t R
esis
tan
ce E
rro
r
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 AustinThe University of Texas at AustinMicroelectromagnetic Devices GroupMicroelectromagnetic Devices Group
Results for coaxial cable
• can account for both inner (signal) and outer (shield) conductors
11
1x100
1x101
1x102
1.5x10-7
1.7x10-7
1.9x10-7
2.1x10-7
1x10
5
1x10
6
1x10
7
1x10
8
1x10
9
5x10
9
Res
ista
nce
(O
hm
/m)
Ind
uct
ance
(H
/m)
Frequency (Hz)
example:inner radius a = 0.1 mmshield radius b = 0.23 mmshield thickness 0.02 mmfmax = 5 GHz
a
c
b
blue: exactred: circuit
resistance
total inductance
R1in
L1in
R2in
R3in
R4in
L2in
L3in
R1out
L1out
R2out
R3out
R4out
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 φ
• two branches required• weight skin effect by ζ
ζ = φ / π
Twin lead with proximity effect
13
φ
2h
sin φ( ) = 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 ProgramTexas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program D. Neikirk D. Neikirk
0x100
1x101
2x101
3x101
4x101
1x107 1x108 1x109 1x1010 1x1011
Res
ista
nce
per
len
gth
(O
hm
/cm
)
Frequency (Hz)
5.0x10-9
5.5x10-9
6.0x10-9
6.5x10-9
7.0x10-9
7.5x10-9
1x106 1x107 1x108 1x109 1x1010 1x1011
Ind
uct
aan
ce p
er le
ng
th (
H/c
m)
Frequency (Hz)
14
Results for closely coupled twin lead
• example for 1 mil diameter Al wires on 2 mil centers- φ = 60 ˚
Lexternal
conformal mapping approximation
conformal mapping approximation
circuit modelcircuit model
The University of Texas at AustinThe University of Texas at AustinMicroelectromagnetic Devices GroupMicroelectromagnetic Devices Group
• observation:- regardless of
geometry of transmission line, for frequencies greater than about 3R
dc/L
lf,
resistance increases as √ω
• can force single 4-long ladder circuit response to pass through a given high frequency point with √ω dependence
- should work for noncircular geometries, even with strong proximity effects
Generalized circuit generation
15
5x10-1
1x100
1x101
1x102
1x10-1 1x100 1x101 1x102 1x103
Nor
mal
ized
Res
ista
nce
(uni
ts o
f R/R
dc)
Normalized angular frequency (units of Rdc/Llfinternal )
R ≈ Rmax ⋅ ω ωmax
ωmax
Rmax
3 Rdc Llftotal
• Objective: force high frequency circuit response to pass through R
max at ω
max
- high frequency asymptotic behavior of 4-ladder is
• for a given choice of RR, from dc requirements find R1:
• require that Rcircuit
= Rmax
at ωmax
:
General fitting procedure
(eq. 1)
(eq. 2)
(eq. 3)
Z hfcircuit ≈
R1 R1 ⋅RR−1 + j ω L1( )R1 ⋅ 1 + RR−1( ) + j ω L1
R1 = Rdc RR3 + RR2 + RR + 1( )
Rmax = R1
RR−1 ⋅ 1 + RR−1( ) + ωmax L1R1
2
1 + RR−1( ) 2+ ωmax L1
R1
2
Generalized fitting procedure
•so L 1 is given by:
•and finally by LL is found using the dc requirement:
where
(eq. 4)
(eq. 5)
L1 =Rdc RR3 + RR2 + RR + 1( ) 1 + 1 RR( )
ωmax
Rmax − Rdc 1 + RR2( )Rdc RR3 + RR2 + RR + 1( ) − Rmax
Llfinternal = Llf
total − Lhfexternal (eq. 6)
LL LL RR RR RRL
RR RR RRlf− − − − − − − −+ +( ) + + +( ) − + + +( ) =2 1 1 2 2 1 2 3 2 1 21 1 1 0
internal
1L
Summary of procedure
• find low and high frequency behavior- R
dc, L
lftotal, L
hfexternal, R
max at single high frequency ω
max
- could be determined by either calculation or measurement• iterate to find optimum RR
- since R1 > R
max, RR is bounded below such that:
- constraint on real value for L1 produces an upper bound
- hence RR must satisfy the inequality
18
(eq. 7)
1 + RR2 < RmaxRdc
< RR3 + RR2 + RR + 1
RmaxRdc
≤ RR( )3 + RR( )2 + RR + 1
RR2 +1 < RmaxRdc
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program D. Neikirk D. Neikirk
Summary of procedure
• start with RR at lower bound (eq. 7)
• calculate R1 from eq. 2
• calculate L1 from eq. 4
• calculate LL from eq. 5
• use resulting 4-ladder to calculate circuit response over interval from 3Rdc/Llf to ωmax
(interval over which √ω behavior holds)- find error between circuit and assumed
response
• increment RR, find new error- continue until error is minimized
19
R ≈ Rmax ⋅ ω ωmax
The University of Texas at AustinThe University of Texas at AustinMicroelectromagnetic Devices GroupMicroelectromagnetic Devices Group
20
Examples for generalized fitting
• series equivalent per unit length circuit for transmission line is
• verification of circuit model using:- experimental results for closely coupled twin lead
• experimentally measured resistance and inductance data• fit to experimental resistance, calculation for L
lftotal, L
hfexternal
- full volume filament calculations for wide range of rectangular geometries
• parallel thick plates• coplanar lines• parallel square bars
Lhf
external
L1
R4
R3
R2
R1
L2
L3
• Rdc
= 0.01 Ω/m , Llf
total = 4.1 x 10-7 H/m , Lhf
external = 1.77 x 10-7 H/m
• fmax
= 9.33 x 105 Hz , Rmax
= 0.193 Ω/m
→ RR = 2.34 , LL = 0.782
21
Closely coupled twin lead
0x100
1x10-9
2x10-9
3x10-9
4x10-9
5x10-9
1x10
3
1x10
4
1x10
5
1x10
6
3x10
6
Ind
uct
ance
(H
/cm
)Frequency (Hz)
8x10-51x10-4
1x10-3
3x10-3
0
5
10
15
20
25
1x10
3
1x10
4
1x10
5
1x10
6
3x10
6
Res
ista
nce
(O
hm
/cm
)
Err
or(
%)
Frequency (Hz)
blue: experimentalred: circuitgreen: error
blue: experimentalred: circuit
0.2 mm2 mm
• Rdc
= 431 Ω/m , Llf
total = 2.7 x 10-7 H/m , Lhf
external = 2 x 10-7 H/m
• fmax
= 1 x 1010 Hz , Rmax
= 1650 Ω/m
→ RR = 1.54 , LL = 0.523
Parallel thick plates
22
2
2.2
2.4
2.6
2.8
1x10-2 1x10-1 1x100 1x101 1x102
To
tal I
nd
uct
ance
(n
H/c
m)
Frequency(GHz)
2
10
100
0
1
2
3
4
5
1x10-2 1x10-1 1x100 1x101 1x102
Res
ista
nce
(O
hm
/cm
)
Err
or(
%)
Frequency (GHz)
blue: volume filamentred: circuitgreen: error
blue: volume filamentred: circuit
4 µm4 µm
20 µm
• Rdc
= 431 Ω/m , Llf
total = 5.7 x 10-7 H/m , Lhf
external = 4 x 10-7 H/m
• fmax
= 1 x 1010 Hz , Rmax
= 2460 Ω/m
→ RR = 2.07 , LL = 0.351
Coplanar lines
23
3.5
4
4.5
5
5.5
6
1x10-2 1x10-1 1x100 1x101 1x102
To
tal I
nd
uct
ance
(n
H/c
m)
Frequency (GHz)
3x100
1x101
1x102
0
1
2
3
4
5
6
1x10-2 1x10-1 1x100 1x101 1x102
Res
ista
nce
(O
hm
/cm
)
Err
or(
%)
Frequency (GHz)
blue: volume filamentred: circuitgreen: error
blue: volume filamentred: circuit
4 µm4 µm
20 µm
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program D. Neikirk D. Neikirk
• Rdc
= 350 Ω/m , Llf
total = 4.8 x 10-7 H/m , Lhf
external = 3.22 x 10-7 H/m
• fmax
= 5 x 1010 Hz , Rmax
= 5160 Ω/m
→ RR = 2.36 , LL = 0.448
Parallel square bars
24
3x10-7
4x10-7
4x10-7
5x10-7
5x10-7
1x107 1x108 1x109 1x1010 1x1011
Ind
uct
ance
(H
/m)
Frequency (Hz)
1x102
1x103
1x104
0
2
4
6
8
10
12
1x107 1x108 1x109 1x1010 1x1011
Res
ista
nce
(O
hm
/m)
Err
or
(%)
Frequency (Hz)
blue: volume filamentred: circuitgreen: error
blue: volume filamentred: circuit
10 µm
5 µm
10 µm
The University of Texas at AustinThe University of Texas at AustinMicroelectromagnetic Devices GroupMicroelectromagnetic Devices Group
25
Compact Equivalent Circuit Models for the Skin Effect
• small R-L ladders (four resistors, three inductors) can provide excellent equivalent circuit for circular conductors- good fit from dc to high frequency
- simple, analytic equations have been established that allow fast calculation of circuit element values for a specified maximum frequency, wire radius, and wire conductivity
• can be used directly to model transmission lines using coupled circular conductors with "weak" proximity effects- excellent fit for coaxial cable
- analytic result for twin lead as a function of wire separation
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program
Darpa Electronic Packaging and Interconnect Design and Test ProgramTexas Advanced Technology Program D. Neikirk D. Neikirk
Compact Equivalent Circuit Models for Skin and Proximity Effects in General Transmission Lines
• for arbitrary cross-section conductors or in the presence of strong proximity effects generalized procedure has been established- only one fitting parameter, easily determined via simple error
minimization- requires knowledge of only R
dc, L
lftotal, L
hfexternal, and R
max at single high
frequency ωmax
• can be determined by calculation or measurement
• excellent fit to detailed calculations for wide range of geometries- closely coupled twin lead- square to thick, narrow to wide plates- also tested for microstrip and strip line, similar excellent agreement
• should provide efficient technique for circuit simulation of lossy transmission lines
26