Compact Equivalent Circuit Models for the Skin Effect

MicroelectromagneticMicroelectromagnetic DevicesDevices GroupGroup TheThe UniversityUniversity ofof TexasTexas atat AustinAustin 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: [email protected] www home page: http://weewave.mer.utexas.edu/

DarpaDarpa ElectronicElectronic PackagingPackaging andand InterconnectInterconnect DesignDesign andand TestTest ProgramProgram D.D. NeikirkNeikirk TexasTexas AdvancedAdvanced TechnoloTechnologygy ProProggramram MicroelectromagneticMicroelectromagnetic DevicesDevices GroupGroup TheThe UniversityUniversity ofof TexasTexas atat AustinAustin 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 Resistance: increases Resistance: increases Inductance: uniform Inductance: decreases Inductance: constant, current distribution 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 L6 technique R6 - e.g., H. A. Wheeler, L5 skin R “Formulas for the 5 effect L4 skin-effect,” Proceedings of model R the Institute of Radio 4 L Engineers, vol. 30, pp. 3 R3 412-424, 1942. L2 ¥ essentially an application of R2 transverse resonance L1

¥ lumping based on uniform step R size tends to generate large Lext 1 ladders Cext

δ z

3 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 M−1 = 1 ⋅ M− j−i R R + = RR Ri ∑()RR i i 1 σπ r2 for an M-deep ladder j=0 this leads to radii of rings: inductances: −1 M  M  µ ⋅ − M− j−n+1 ()ri−1 ri ri = r⋅ ∑  ∑RR()  L =   i 2π⋅r j= i+1 n=1 i

[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] Yen's results for a single circular wire ) π 0 1

/8 1x10 5x10 µ

internal inductance 1x101 1x10-1 blue: exact green: Yen, 4 deep red: Yen, 10 deep

resistance

1x10-2 1x100 Normalized Inductance (units of 1x10-2 1x10-1 1x100 1x101 1x102 Normalized Resistance (units of Rdc) π µ Normalized Angular Frequency (units of 8 Rdc/ 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

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"Compact" ladders

¥ problem: Yen's approach tends to 4 underestimate both resistance and 3 inductance 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) L 2 3 2 - R2 = RR R1 , R3 = RR R1 , R4 = RR R1 L ¥ L i / L i+1 = LL , a constant (< 1) 1 2 - L2 = LL L1 , L3 = LL L1

6 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: R ()RR 3 + ()RR 2 + RR + (1 − 1 ) = 0 Rdc

internal

- given the low frequency internal inductance Llf , then L1 and LL are related by: 2 2 2  2  2 Linternal   2   1  +  + 1  1 +  1  + 1 + − lf  + 1   1  + =   1    1 1    1 0 LL RR LL  RR  RR  L1  RR RR  

¥ 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 ω bandwidth (dc to max) ¥ scales in terms of radius compared to minimum skin depth (that occurs at highest frequency) 2 δ max = ω max µoσ

R1 (and hence RR): L1 (and hence LL): internal R wire radius L R 1 = 0.53 lf = ⋅ 1 δ 0.315 Rdc max L 1 R dc

8 Results for single circular wire )

π 0 1

/8 1x10 5x10

µ RR = 2.5, LL = 0.290

internal inductance

blue: exact 1x101 1x10-1 red: new 4-ladder

resistance

1x10-2 1x100 Normalized Inductance (units of Normalized Inductance (units of 1x10-2 1x10-1 1x100 1x101 1x102 Normalized Resistance (units of Rdc) π µ Normalized Angular Frequency (units of 8 Rdc/ o)

9 Errors for single circular wire

30% 80% resistance inductance 70% 25% 60% 20% Yen 4-ladder 50%

15% Yen 10-ladder 40% 30% 10% 20% 5%

Percent Resistance Error Percent Resistance 10% new 4-ladder

0% 0% Percent Internal Inductance Error 1x10-2 1x10-1 1x100 1x101 1x102 1x10-2 1x10-1 1x100 1x101 1x102 Normalized Angular Frequency 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

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Results for coaxial cable

2 b 1x10 a total inductance 2.1x10-7 c

1.9x10-7 1x101 in out R4 R4 in L out L3 3 resistance 1.7x10-7 Inductance (H/m)

in out Resistance (Ohm/m) R3 R3 blue: exact in out red: circuit L2 L2 in out 0 -7 R2 R2 1x10 1.5x10 in L out 5 6 7 8 9 9 L1 1 1x10 1x10 1x10 1x10 1x10 5x10 Frequency (Hz) out L R in R ext 1 1 example: ¥ can account for both inner (signal) and inner radius a = 0.1 mm outer (shield) conductors shield radius b = 0.23 mm shield thickness 0.02 mm

fmax = 5 GHz

11 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 2Lext R1/z - quarter from angle φ outer face R4/(1- z) 2 sin(φ) = 1 − (rh) L3/(1- z) R3/(1- z) ¥ two branches required L2/(1- z) ¥ weight skin effect by ζ R2/(1- z) L1/(1- z) ζ = φ / π

2Lext R1/(1- z)

13 Results for closely coupled twin lead

4x101 7.5x10-9 conformal mapping approximation 7.0x10-9 3x101 conformal mapping 6.5x10-9 approximation 2x101 6.0x10-9 circuit model circuit model 1x101 5.5x10-9 Lexternal Inductaance per length (H/cm) Resistance per length (Ohm/cm) Resistance per length 0x100 5.0x10-9 1x107 1x108 1x109 1x1010 1x1011 1x106 1x107 1x108 1x109 1x1010 1x1011 Frequency (Hz) Frequency (Hz)

¥ example for 1 mil diameter Al wires on 2 mil centers - φ = 60 û

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Generalized circuit generation

¥ observation: - regardless of 2 geometry of 1x10 ω transmission line, for max frequencies greater than about 3R /L , dc lf R resistance increases 1x101 max as √ω ¥ can force single 4-long R ≈ R ⋅ ωω ladder circuit response to max max

pass through a given high 0 1x10 total frequency point with √ω 3Rdc Llf Normalized Resistance (units of R/Rdc) -1 dependence 5x10 1x10-1 1x100 1x101 1x102 1x103 - should work for internal noncircular Normalized angular frequency (units of Rdc/Llf ) geometries, even with strong proximity effects

15 General fitting procedure

¥ Objective: force high frequency circuit response to pass ω through Rmax at max - high frequency asymptotic behavior of 4-ladder is ⋅ −1 + ω R1 ()R1 RR j L1 Z circuit ≈ (eq. 1) hf ⋅ + −1 + ω R1 ()1 RR j L1

¥ for a given choice of RR, from dc requirements find R1:

= 3 + 2 + + R1 Rdc ()RR RR RR 1 (eq. 2)

ω ¥ require that Rcircuit = Rmax at max: 2 − −  ω L  RR 1 ⋅()1 + RR 1 +  max 1   = R1 Rmax R1 2 (eq. 3) − 2  ω L  ()1 + RR 1 +  max 1  R1  Generalized fitting procedure

•so L1 is given by:

3 + 2 + + + − + 2 Rdc ()RR RR RR 1 ()1 1 RR Rmax Rdc ()1 RR L = 1 ω 3 + 2 + + − max Rdc ()RR RR RR 1 Rmax (eq. 4) •and finally by LL is found using the dc requirement: internal −−−2 − −2 L − − −2 LL211++ LL() RR11+++() RR21 RR −+++lf ()RR321 RR RR 10= L1 (eq. 5) where internal = total − external Llf Llf Lhf (eq. 6) Summary of procedure

¥ find low and high frequency behavior total external ω - Rdc, Llf , Lhf , Rmax at single high frequency max - could be determined by either calculation or measurement ¥ iterate to find optimum RR

- since R1 > Rmax, RR is bounded below such that: R max ≤ ()RR 3 + ()RR 2 + RR + 1 (eq. 7) Rdc

- constraint on real value for L1 produces an upper bound R RR2 +1 < max Rdc - hence RR must satisfy the inequality R 1 + RR 2 < max < RR 3 + RR 2 + RR +1 Rdc

18 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 ω response over interval from 3Rdc/Llf to max (interval over which √ω behavior holds) ≈ ⋅ ωω - find error between circuit and assumed R R max max response ¥ increment RR, find new error - continue until error is minimized

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Examples for generalized fitting

¥ series equivalent per unit length circuit for transmission line is

R4 L3 R3 L2 R2 L1

external R Lhf 1 ¥ verification of circuit model using: - experimental results for closely coupled twin lead ¥ experimentally measured resistance and inductance data total external ¥ fit to experimental resistance, calculation for Llf , Lhf - 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 0.2 mm

3x10-3 25 5x10-9 blue: experimental blue: experimental red: circuit red: circuit 20 4x10-9 green: error 1x10-3 15 3x10-9

-9 10 Error(%) 2x10

5 Inductance (H/cm) -9 Resistance (Ohm/cm) 1x10

1x10-4 8x10-5 0 0x100 3 4 5 6 6 3 4 5 6 6 1x10 1x10 1x10 1x10 3x10 1x10 1x10 1x10 1x10 3x10 Frequency (Hz) Frequency (Hz) Ω total -7 external -7 ¥ Rdc = 0.01 /m , Llf = 4.1 x 10 H/m , Lhf = 1.77 x 10 H/m 5 Ω ¥ fmax = 9.33 x 10 Hz , Rmax = 0.193 /m → RR = 2.34 , LL = 0.782

21 4 µm 4 µm Parallel thick plates

20 µm 100 5 2.8 blue: volume filament red: circuit 4 green: error 2.6

3 2.4 10

2 Error(%)

2.2

Resistance (Ohm/cm) 1 blue: volume filament Total Inductance (nH/cm) red: circuit 2 0 2 1x10-2 1x10-1 1x100 1x101 1x102 1x10-2 1x10-1 1x100 1x101 1x102 Frequency (GHz) Frequency(GHz)

Ω total -7 external -7 ¥ Rdc = 431 /m , Llf = 2.7 x 10 H/m , Lhf = 2 x 10 H/m 10 Ω ¥ fmax = 1 x 10 Hz , Rmax = 1650 /m → RR = 1.54 , LL = 0.523

22 4 µm 4 µm Coplanar lines 20 µm

1x102 6 6 blue: volume filament red: circuit 5 green: error 5.5

4 5 3 4.5 Error(%) 1x101 2 4 Resistance (Ohm/cm) 1 blue: volume filament Total Inductance (nH/cm) red: circuit 3x100 0 3.5 1x10-2 1x10-1 1x100 1x101 1x102 1x10-2 1x10-1 1x100 1x101 1x102 Frequency (GHz) Frequency (GHz)

Ω total -7 external -7 ¥ Rdc = 431 /m , Llf = 5.7 x 10 H/m , Lhf = 4 x 10 H/m 10 Ω ¥ fmax = 1 x 10 Hz , Rmax = 2460 /m → RR = 2.07 , LL = 0.351

23 5 µm Parallel square bars 10 µm

10 µm

1x104 12 5x10-7 blue: volume filament red: circuit green: error 10 5x10-7 8

1x103 6 4x10-7

4 Error (%)

Inductance (H/m) 4x10-7 Resistance (Ohm/m) 2 blue: volume filament red: circuit 1x102 0 3x10-7 1x107 1x108 1x109 1x1010 1x1011 1x107 1x108 1x109 1x1010 1x1011 Frequency (Hz) Frequency (Hz)

Ω total -7 external -7 ¥ Rdc = 350 /m , Llf = 4.8 x 10 H/m , Lhf = 3.22 x 10 H/m 10 Ω ¥ fmax = 5 x 10 Hz , Rmax = 5160 /m → RR = 2.36 , LL = 0.448

24 DarpaDarpa ElectronicElectronic PackagingPackaging andand InterconnectInterconnect DesignDesign andand TestTest ProgramProgram D.D. NeikirkNeikirk TexasTexas AdvancedAdvanced TechnoloTechnologygy ProProggramram MicroelectromagneticMicroelectromagnetic DevicesDevices GroupGroup TheThe UniversityUniversity ofof TexasTexas atat AustinAustin 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 R , L total, L external, and R at single high ω dc lf hf max frequency max ¥ 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

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