Compact Equivalent Circuit Models for the Skin Effect

Transcription

1 Microelectromagnetic Devices Group The University of Texas at 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 7871 for further information, please contact: Professor Dean Neikirk, phone www home page: D. Neikirk Darpa Electronic Packaging and Interconnect Design and Test Program Texas Advanced Technology Program

2 Microelectromagnetic Devices Group The University of Texas at Austin 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: R dc 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?

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 , 194. L 6 L 5 L4 L 3 L R 6 R 5 R 4 R 3 skin effect model essentially an application of transverse resonance L1 R lumping based on uniform step size tends to generate large ladders Lext C ext R 1 δ z 3

4 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 R i R i+1 = RR R i = for an M-deep ladder this leads to 1 σπ r M 1 j=0 ( RR j i )M radii of rings: M M r i = r RR j= i+1 n=1 ( ) M j n+1 1 inductances: L i = µ ( r i 1 r i ) π r 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 , 198]

5 Yen's results for a single circular wire Normalized Inductance (units of µ/8π) 1x10 0 1x10-1 internal inductance blue: exact green: Yen, 4 deep red: Yen, 10 deep resistance 5x10 1 1x10 1 1x10-1x10 0 1x10-1x10-1 1x10 0 1x10 1 1x10 Normalized Angular Frequency (units of 8πR dc /µ o ) Normalized Resistance (units of Rdc) selection of ladder length and RR determines accuracy: - m = 4 (i.e., 4 resistors, 3 inductors), minimum error occurs for RR =.31 - m = 10, minimum error for RR = D. Neikirk Darpa Electronic Packaging and Interconnect Design and Test Program Texas Advanced Technology Program

6 Microelectromagnetic Devices Group The University of Texas at Austin "Compact" ladders problem: Yen's approach tends to underestimate both resistance and inductance 4 3 can a "short" ladder produce a good approximation? 1 - "de-couple" resistance and inductance in a 4-long ladder - each shell such that R i / R i+1 = RR, a constant (> 1) L 3 L - R = RR R 1, R 3 = RR R 1, R 4 = RR 3 R 1 L i / L i+1 = LL, a constant (< 1) L 1 - L = LL L 1, L 3 = LL L 1 6

7 Fitting parameters for 4-long ladder "unknowns" constrained by asymptotic behavior at low frequency - given the dc resistance R dc, then R 1 and RR are related by: ( RR) 3 + ( RR) + RR + (1 R 1 ) = 0 R dc 1 LL - given the low frequency internal inductance L lf internal, then L 1 and LL are related by: RR 1 LL + 1 RR + 1 RR + 1 L internal lf L 1 RR RR + 1 = 0 only "free" fitting parameters are R 1 and L 1 (or equivalently, RR and LL) - R 1 and L 1 tend to dominate the high frequency response

8 Best fit for single circular wire "universal" fit possible over specified bandwidth (dc to ω max ) scales in terms of radius compared to minimum skin depth (that occurs at highest frequency) δ max = ω max µ o σ R 1 (and hence RR): R 1 wire radius = 0.53 R dc δ max L 1 (and hence LL): internal L lf L 1 = R 1 R dc 8

9 Results for single circular wire Normalized Inductance (units of µ/8π) 1x10 0 1x10-1 internal inductance blue: exact red: new 4-ladder RR =.5, LL = 0.90 resistance 5x10 1 1x10 1 1x10-1x10 0 1x10-1x10-1 1x10 0 1x10 1 1x10 Normalized Angular Frequency (units of 8πR /µ ) dc o Normalized Resistance (units of Rdc) 9

10 Percent Resistance Error 30% 5% 0% 15% Errors for single circular wire resistance 10% 30% 0% 5% new 4-ladder 10% 0% 0% 1x10-1x10-1 1x10 0 1x10 1 1x10 1x10-1x10-1 1x10 0 1x10 1 1x10 Normalized Angular Frequency Yen 4-ladder Yen 10-ladder inductance 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 80% 70% 60% 50% 40% Percent Internal Inductance Error 10 D. Neikirk Darpa Electronic Packaging and Interconnect Design and Test Program Texas Advanced Technology Program

11 Microelectromagnetic Devices Group The University of Texas at Austin Results for coaxial cable b 1x10 Lext L 3 in L in L 1 in R 1 in c R 4 in R 3 in R in can account for both inner (signal) and outer (shield) conductors a L 3 out L out L 1 out R 1 out R 4 out R 3 out R out Resistance (Ohm/m) 1x10 1 1x10 0 1x10 5 resistance 1x10 6 total inductance 1x10 7 blue: exact red: circuit 1x10 8 Frequency (Hz) 1x10 9 example: inner radius a = 0.1 mm shield radius b = 0.3 mm shield thickness 0.0 mm f max = 5 GHz 5x10 9.1x x x x10-7 Inductance (H/m) 11

12 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 (L ext ) to force current redistribution 1

13 Twin lead with proximity effect φ inner face L 3 /z R 4 /z R 3 /z h L /z L 1 /z R /z more flux coupling at inner faces - quarter from angle φ sin( φ) = 1 ( rh) two branches required weight skin effect by ζ L ext outer face L3/(1- z) L /(1- z) L 1 /(1- z) R 1 /z R 4 /(1- z) R 3 /(1- z) R /(1- z) ζ = φ / π Lext R 1 /(1- z) 13

14 Results for closely coupled twin lead Resistance per length (Ohm/cm) 4x10 1 3x10 1 x10 1 1x10 1 0x10 0 conformal mapping approximation circuit model 1x10 7 1x10 8 1x10 9 1x x10 11 Frequency (Hz) circuit model L external conformal mapping approximation 1x10 6 1x10 7 1x10 8 1x10 9 1x x10 11 Frequency (Hz) 7.5x x x x x x10-9 Inductaance per length (H/cm) example for 1 mil diameter Al wires on mil centers - φ = D. Neikirk Darpa Electronic Packaging and Interconnect Design and Test Program Texas Advanced Technology Program

15 Microelectromagnetic Devices Group The University of Texas at Austin Generalized circuit generation 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 Normalized Resistance (units of R/Rdc) 1x10 1x10 1 1x10 0 5x10-1 R max total 3 R dc L lf ω max R R max ω ω max 1x10-1 1x10 0 1x10 1 1x10 1x10 3 Normalized angular frequency (units of R dc /L lf internal ) 15

16 General fitting procedure Objective: force high frequency circuit response to pass through R max at ω max - high frequency asymptotic behavior of 4-ladder is ( ) R 1 ( 1 + RR 1 ) + j ω L 1 R circuit 1 R 1 RR 1 + j ω L 1 Z hf for a given choice of RR, from dc requirements find R 1 : R 1 = R dc ( RR 3 + RR + RR + 1) (eq. 1) (eq. ) require that R circuit = R max at ω max : RR RR 1 ( )+ ω max L 1 R R max = R 1 1 ( 1 + RR 1 ) + ω max L 1 R 1 (eq. 3)

17 Generalized fitting procedure so L 1 is given by: ( ) RR L 1 = R dc RR 3 + RR + RR + 1 ω max ( ) and finally by LL is found using the dc requirement: ( ) R max R dc 1 + RR ( ) R max R dc RR 3 + RR + RR + 1 internal (eq. 4) L LL lf + LL 1 RR RR RR 1 1 RR 3 RR RR ( ) + ( + + ) ( ) = L 1 (eq. 5) where L lf internal total = L lf external Lhf (eq. 6)

18 Summary of procedure find low and high frequency behavior - R dc, L lf total, L hf external, R max at single high frequency ω max - could be determined by either calculation or measurement iterate to find optimum RR - since R 1 > R max, RR is bounded below such that: R max ( RR) 3 + ( RR) + RR + 1 R dc - constraint on real value for L 1 produces an upper bound (eq. 7) RR +1 < R max R dc - hence RR must satisfy the inequality 1 + RR < R max R dc < RR 3 + RR + RR

19 Summary of procedure start with RR at lower bound (eq. 7) calculate R 1 from eq. calculate L 1 from eq. 4 calculate LL from eq. 5 use resulting 4-ladder to calculate circuit response over interval from 3R dc /L lf to ω max (interval over which ω behavior holds) - find error between circuit and assumed response increment RR, find new error - continue until error is minimized R R max ω ω max 19 D. Neikirk Darpa Electronic Packaging and Interconnect Design and Test Program Texas Advanced Technology Program

20 Microelectromagnetic Devices Group The University of Texas at Austin Examples for generalized fitting series equivalent per unit length circuit for transmission line is R 4 L3 R 3 L L 1 R L hf external 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 lf total, L hf external - full volume filament calculations for wide range of rectangular geometries parallel thick plates coplanar lines parallel square bars R 1 0

21 Closely coupled twin lead mm 0. mm Resistance (Ohm/cm) 3x10-3 1x10-3 blue: experimental red: circuit green: error Error(%) Inductance (H/cm) 5x10-9 4x10-9 3x10-9 x10-9 1x10-9 blue: experimental red: circuit 1x10-4 8x10-5 1x10 3 1x10 4 1x10 5 Frequency (Hz) 1x10 6 3x x10 0 R dc = 0.01 Ω/m, L lf total = 4.1 x 10-7 H/m, L hf external = 1.77 x 10-7 H/m 1x10 3 f max = 9.33 x 10 5 Hz, R max = Ω/m RR =.34, LL = x10 4 1x10 5 Frequency (Hz) 1x10 6 3x10 6 1

22 Parallel thick plates 4 µm 4 µm Resistance (Ohm/cm) blue: volume filament red: circuit green: error 0 1x10-1x10-1 1x10 0 1x10 1 1x10 Frequency (GHz) Error(%) Total Inductance (nh/cm) µm blue: volume filament red: circuit 1x10-1x10-1 1x10 0 1x10 1 1x10 Frequency(GHz) R dc = 431 Ω/m, L lf total =.7 x 10-7 H/m, L hf external = x 10-7 H/m f max = 1 x Hz, R max = 1650 Ω/m RR = 1.54, LL = 0.53

23 Coplanar lines 4 µm 0 µm 4 µm Resistance (Ohm/cm) 1x10 1x10 1 blue: volume filament red: circuit green: error 3x x10-1x10-1 1x10 0 1x10 1 1x10 Frequency (GHz) Error(%) Total Inductance (nh/cm) blue: volume filament red: circuit 1x10-1x10-1 1x10 0 1x10 1 1x10 Frequency (GHz) R dc = 431 Ω/m, L lf total = 5.7 x 10-7 H/m, L hf external = 4 x 10-7 H/m f max = 1 x Hz, R max = 460 Ω/m RR =.07, LL =

24 Parallel square bars 10 µm 10 µm 5 µm Resistance (Ohm/m) 1x10 4 1x10 3 blue: volume filament red: circuit green: error x10 0 1x10 7 1x10 8 1x10 9 1x x10 11 Frequency (Hz) Error (%) Inductance (H/m) 5x10-7 5x10-7 4x10-7 4x10-7 3x10-7 blue: volume filament red: circuit 1x10 7 1x10 8 1x10 9 1x x10 11 Frequency (Hz) R dc = 350 Ω/m, L lf total = 4.8 x 10-7 H/m, L hf external = 3. x 10-7 H/m 4 D. Neikirk f max = 5 x Hz, R max = 5160 Ω/m RR =.36, LL = Darpa Electronic Packaging and Interconnect Design and Test Program Texas Advanced Technology Program

25 Microelectromagnetic Devices Group The University of Texas at 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 5

26 6 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 total external - requires knowledge of only R dc, L lf, L hf, and R max at single high frequency ω 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 Darpa Electronic Packaging and Interconnect Design and Test Program Texas Advanced Technology Program