Microelectromagnetic
Microelectromagnetic Devices
Devices Group
Group
The
The University
University of
of Texas
Texas at
at Austin
Austin
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, please contact:
Professor Dean Neikirk, phone 512-471-4669
e-mail: neikirk@mail.utexas.edu
www home page:
http://weewave.mer.utexas.edu/
D.
D. Neikirk
Neikirk
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Microelectromagnetic
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Origin of frequency dependencies in
transmission line series impedance
Low frequencies
Mid frequencies
High frequencies
Uniform Current: dc
Non-Uniform: proximity
Non-Uniform: skin
depth & proximity
Resistance: Rdc
Inductance: uniform
current distribution
Resistance: increases
Inductance: decreases
Resistance: increases
Inductance: constant,
infinite conductivity (high
frequency) limit
• can frequency independent ladder circuits be
synthesized to accurately model frequency
dependent series impedance of line?
2
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
L6
R6
L5
R5
L4
R4
L3
R3
L2
R2
L1
R1
L ext
C ext
δz
3
skin
effect
model
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
Ri Ri+1 = RR
M −1
1
M − j−i
Ri =
⋅
RR
(
)
∑
2
σ
π
r
j=0
for an M-deep ladder
this leads to
radii of rings:
ri = r ⋅
inductances:
 M

M − j −n+1
∑  ∑ ( RR)


j = i+1  n=1
M
−1
Li =
µ ⋅ (ri−1 − ri )
2π ⋅ ri
[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]
1x100
5x101
internal inductance
1x101
1x10-1
blue: exact
green: Yen, 4 deep
red: Yen, 10 deep
resistance
1x10-2
1x10-2
1x10-1
1x100
1x101
1x100
1x102
Normalized Resistance (units of Rdc)
Normalized Inductance (units of µ/8π)
Yen's results for a single circular wire
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
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D. Neikirk
Neikirk
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Microelectromagnetic
Microelectromagnetic Devices
Devices Group
Group
"Compact" ladders
• problem: Yen's approach tends to
underestimate both resistance and
inductance
4
3
2
1
• can a "short" ladder produce a good
approximation?
- "de-couple" resistance and inductance in a
4-long ladder
- each shell such that
L3
• R i / R i+1 = RR , a constant (> 1)
- R2 = RR R1 , R3 = RR R1 , R4 = RR R1
2
• L i / L i+1 = LL , a constant (< 1)
- L2 = LL L1 , L3 = LL 2 L1
6
3
L2
L1
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:
( RR)3 + ( RR)2 + RR + (1 −
R1
)=0
Rdc
- given the low frequency internal inductance Llfinternal, then L1
and LL are related by:
2
2
internal 
2
2

L




1 
1
lf
 1  + 1 +
1 + 1   1  + 1
+
+
+ 1 −


 = 0
 LL 

RR  LL   RR 
RR 
RR    RR 
L1  

2
1 2 1
• only "free" fitting parameters are R1 and L1 (or equivalently, RR
and LL)
- R1 and L1 tend to dominate the high frequency response
Best fit for single circular wire
• "universal" fit possible over specified
bandwidth (dc to ωmax)
• scales in terms of radius compared to minimum
skin depth (that occurs at highest frequency)
δ max =
R1 (and hence RR):
R1
wire radius
= 0. 53
Rdc
δ max
8
2
ω max µ o σ
L1 (and hence LL):
L internal
lf
L1
= 0.315 ⋅
R1
R dc
9
1x100
RR = 2.5, LL = 0.290
5x101
internal inductance
1x10-1
blue: exact
red: new 4-ladder
1x101
resistance
1x10-2
1x100
1x10-2
1x10-1
1x100
1x101
1x102
Normalized Angular Frequency (units of 8πRdc/µo)
Normalized Resistance (units of Rdc)
Normalized Inductance (units of µ/8π)
Results for single circular wire
Percent Resistance Error
30%
80%
resistance
inductance
70%
25%
20%
Yen 4-ladder
15%
Yen 10-ladder
60%
50%
40%
30%
10%
20%
5%
10%
new 4-ladder
0%
1x10-2 1x10-1
1x100
1x101
1x102
Normalized Angular Frequency
1x10-2 1x10-1
1x100
1x101
0%
1x102
Percent Internal Inductance Error
Errors for single circular wire
Normalized Angular Frequency
• 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
10
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Microelectromagnetic
Microelectromagnetic Devices
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Results for coaxial cable
1x102
R2 in
L1 in
L2 out
R2 out
blue: exact
red: circuit
1x100
L 1 out
1.5x10-7
Frequency (Hz)
Lext
R1 in
R1 out
• can account for both inner (signal) and
outer (shield) conductors
11
example:
inner radius a = 0.1 mm
shield radius b = 0.23 mm
shield thickness 0.02 mm
fmax = 5 GHz
Inductance (H/m)
1.7x10-7
5x109
L2 in
R3 out
resistance
1x109
R3 in
L3 out
1x108
L 3 in
R4 out
1x107
R4 in
2.1x10-7
1.9x10-7
1x101
1x106
c
total inductance
1x105
a
Resistance (Ohm/m)
b
Inclusion of proximity effects
• 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 (Lext) to force current
redistribution
12
Twin lead with proximity effect
φ
inner face
R4 / z
L3/ z
R3 / z
L2/ z
2h
R2 / z
L1/ z
• more flux coupling at inner
faces
- quarter from angle φ
sin(φ ) =
R1 / z
2Lext
outer face
1 − ( r h )2
R4 /(1- z)
L3/(1- z)
R3 /(1- z)
• two branches required
• weight skin effect by ζ
L2/(1- z)
L1/(1- z)
R2 /(1- z)
ζ = φ/π
2Lext
13
R1 /(1- z)
7.5x10-9
4x101
conformal mapping
approximation
3x101
conformal mapping
approximation
2x101
circuit model
1x101
6.5x10-9
circuit model
Lexternal
0x100
1x107
7.0x10-9
6.0x10-9
5.5x10-9
Inductaance per length (H/cm)
Resistance per length (Ohm/cm)
Results for closely coupled twin lead
5.0x10-9
1x108 1x109 1x1010 1x1011
Frequency (Hz)
1x106 1x107 1x108 1x109 1x1010 1x1011
Frequency (Hz)
• example for 1 mil diameter Al wires on 2 mil centers
- φ = 60 ˚
14
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• observation:
- regardless of
geometry of
transmission line, for
frequencies greater
than about 3Rdc/Llf,
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
15
Normalized Resistance (units of R/Rdc)
Generalized circuit generation
1x102
ωmax
1x101
Rmax
R ≈ Rmax ⋅
1x100
5x10-1
1x10-1
3 Rdc
1x100
ω ω max
L total
lf
1x101
1x102
1x103
Normalized angular frequency (units of Rdc/Llfinternal )
General fitting procedure
• Objective: force high frequency circuit response to pass
through Rmax at ωmax
- high frequency asymptotic behavior of 4-ladder is
circuit
Z hf
≈
(
)
R1 ⋅ ( 1 + RR−1 ) + j ω L1
R1 R1 ⋅ RR−1 + j ω L1
(eq. 1)
• for a given choice of RR, from dc requirements find R1:
(
)
R1 = Rdc RR3 + RR2 + RR + 1
• require that Rcircuit = Rmax at ωmax:
Rmax = R1
(
)
ω
L 
RR−1 ⋅ 1 + RR−1 +  max 1 
 R1 
(
1 + RR
)
−1 2
ω
L 
+  max 1 
 R1 
2
(eq. 2)
2
(eq. 3)
Generalized fitting procedure
•so L1 is given by:
L1 =
(
)
Rdc RR3 + RR2 + RR + 1 (1 + 1 RR)
ω max
(
(
Rmax − Rdc 1 + RR2
)
)
Rdc RR3 + RR2 + RR + 1 − Rmax
(eq. 4)
•and finally by LL is found using the dc requirement:
(
) (
LL−2 + LL−1 RR−1 + 1
2
)
+ RR−2 + RR−1 + 1
2
−
Linternal
lf
L1
(
)
RR−3 + RR−2 + RR−1 + 1
2
(eq. 5)
where
external
Llfinternal = Llftotal − Lhf
(eq. 6)
= 0
Summary of procedure
• find low and high frequency behavior
- Rdc, Llftotal, Lhfexternal, Rmax at single high frequency ωmax
- could be determined by either calculation or measurement
• iterate to find optimum RR
- since R1 > Rmax, RR is bounded below such that:
Rmax
≤ ( RR)3 + ( RR)2 + RR + 1
Rdc
- constraint on real value for L1 produces an upper bound
RR2 +1 <
Rmax
Rdc
- hence RR must satisfy the inequality
1 + RR2 <
18
Rmax
< RR3 + RR2 + RR + 1
Rdc
(eq. 7)
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 R ≈ Rmax ⋅
response
ω ω max
• increment RR, find new error
- continue until error is minimized
19
D.
D. Neikirk
Neikirk
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at Austin
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Microelectromagnetic
Microelectromagnetic Devices
Devices Group
Group
Examples for generalized fitting
• series equivalent per unit length circuit for transmission line is
R4
L3
R3
L2
R2
L1
R1
Lhfexternal
• verification of circuit model using:
- experimental results for closely coupled twin lead
• experimentally measured resistance and inductance data
• fit to experimental resistance, calculation for Llftotal, Lhfexternal
- full volume filament calculations for wide range of rectangular
geometries
• parallel thick plates
• coplanar lines
• parallel square bars
20
Closely coupled twin lead
2 mm
5x10-9
1x106
1x105
1x104
1x103
0
1x10-9
0x100
Frequency (Hz)
Frequency (Hz)
• Rdc = 0.01 Ω/m , Llftotal = 4.1 x 10-7 H/m , Lhfexternal = 1.77 x 10-7 H/m
• fmax = 9.33 x 105 Hz , Rmax = 0.193 Ω/m
→
21
RR = 2.34 , LL = 0.782
3x106
1x10-4
8x10-5
2x10-9
1x106
5
3x10-9
1x105
10
4x10-9
1x104
15
Error(%)
1x10-3
20
blue: experimental
red: circuit
1x103
blue: experimental
red: circuit
green: error
Inductance (H/cm)
25
3x106
Resistance (Ohm/cm)
3x10-3
0.2 mm
4 µm
4 µm
Parallel thick plates
20 µm
2.8
blue: volume filament
red: circuit
green: error
4
3
10
2
1
2
1x10-2 1x10-1 1x100 1x101
Frequency (GHz)
0
1x102
Total Inductance (nH/cm)
5
Error(%)
Resistance (Ohm/cm)
100
2.6
2.4
2.2
blue: volume filament
red: circuit
2
1x10-2 1x10-1 1x100 1x101
Frequency(GHz)
• Rdc = 431 Ω/m , Llftotal = 2.7 x 10-7 H/m , Lhfexternal = 2 x 10-7 H/m
• fmax = 1 x 1010 Hz , Rmax = 1650 Ω/m
→
22
RR = 1.54 , LL = 0.523
1x102
4 µm
Coplanar lines
6
5
4
3
1x101
2
1
3x100
1x10-2 1x10-1 1x100 1x101
Frequency (GHz)
0
1x102
Total Inductance (nH/cm)
blue: volume filament
red: circuit
green: error
20 µm
6
Error(%)
Resistance (Ohm/cm)
1x102
4 µm
5.5
5
4.5
4
blue: volume filament
red: circuit
3.5
1x10-2 1x10-1 1x100 1x101
Frequency (GHz)
• Rdc = 431 Ω/m , Llftotal = 5.7 x 10-7 H/m , Lhfexternal = 4 x 10-7 H/m
• fmax = 1 x 1010 Hz , Rmax = 2460 Ω/m
→
23
RR = 2.07 , LL = 0.351
1x102
5 µm
10 µm
Parallel square bars
10 µm
10
8
1x103
6
4
2
1x102
1x107
0
1x108 1x109 1x1010 1x1011
Frequency (Hz)
Inductance (H/m)
blue: volume filament
red: circuit
green: error
5x10-7
12
Error (%)
Resistance (Ohm/m)
1x104
5x10-7
4x10-7
4x10-7
blue: volume filament
red: circuit
3x10-7
1x107
1x108 1x109 1x1010 1x1011
Frequency (Hz)
• Rdc = 350 Ω/m , Llftotal = 4.8 x 10-7 H/m , Lhfexternal = 3.22 x 10-7 H/m
• fmax = 5 x 1010 Hz , Rmax = 5160 Ω/m
→
24
D.
D. Neikirk
Neikirk
RR = 2.36 , LL = 0.448
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Microelectromagnetic
Microelectromagnetic Devices
Devices Group
Group
The
The University
University of
of Texas
Texas at
at Austin
Austin
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
25
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 Rdc, Llftotal, Lhfexternal, and Rmax 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
D.
D. Neikirk
Neikirk
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