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Compact Equivalent Circuit Models for the Skin Effect

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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: − M M 1 M− j−n+1 µ ⋅()r − − r r = r⋅ ∑ ∑()RR L = i 1 i i i π⋅ j= i+1 n=1 2 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] 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) Normalized Angular Frequency (units of 8πR /µ ) dc 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 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" 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 δmax = ωmaxµoσ R1 (and hence RR): L1 (and hence LL): R wire radius Linternal R 1 = 0.53 lf = ⋅ 1 δ 0.315 Rdc max L 1 Rdc 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 1x10-2 1x10-1 1x100 1x101 1x102 Normalized Resistance (units of Rdc) Normalized Angular Frequency (units of 8πR /µ ) dc 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 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 10 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 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) 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 ˚ 14 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 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 Normalized angular frequency (units of R /L ) noncircular dc lf 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 R1 ()R1⋅RR + j ω L1 Z circuit ≈ (eq. 1) hf −1 R1⋅()1 + RR + j ω 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 + RR2 < max < RR3 + RR2 + RR +1 Rdc 18 Summary of procedure • start with RR at lower bound (eq.
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