Compact Equivalent Circuit Models for the Skin Effect

The University of Texas at AustinThe University of Texas at Austin

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: [email protected]

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 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


"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



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


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)



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


• 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



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


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)



4 µm4 µm

20 µm



• 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)



10 µm

5 µm

10 µm


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



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

Documents

Compact Equivalent Circuit Models for the Skin Effect