MOM-SO: A FAST AND FULLY-AUTOMATED METHOD HIGH-SPEED CABLES FOR RESISTANCE AND INDUCTANCE COMPUTATION IN

Transcription

1 MOM-SO: A FAST AND FULLY-AUTOMATED METHOD FOR RESISTANCE AND INDUCTANCE COMPUTATION IN HIGH-SPEED CABLES Utkarsh R. Patel 1, Bjørn Gustavsen 2, Piero Triverio 1 1 Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto 2 SINTEF Energy Research 17th IEEE Workshop on Signal and Power Integrity (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

2 MOTIVATION MOTIVATION Cables: backbone of many ICT system! Where Signal Integrity issues may arise. Broadband & accurate models needed! (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

3 MOTIVATION MOTIVATION First step: compute the per-unit-length parameters. Can be time consuming! (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

4 MOTIVATION MOTIVATION 1.5 First step: compute the per-unit-length parameters. Can be time consuming! Examples: USB 2.0 Firewire, Ethernet, HDMI, cable bundles,... mm Power Drain Signal 1 Ground Signal 2 Shield mm 115 round strands Finite elements analysis: 20 minutes!...just for characterizing a cable! (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

5 MOTIVATION MOTIVATION Similar issue in power engineering Underground & submarine cables Power umbilical by Aker Solutions (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

6 MOTIVATION MOTIVATION Similar issue in power engineering Underground & submarine cables Predict transients caused by faults, lightning, switching converters,... Bandwidth: 1 Hz to 1 MHz Power umbilical by Aker Solutions (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

7 MOTIVATION STATE OF THE ART ANALYTIC FORMULAS! Fast! Skin Effect % No Proximity Effect NUMERICAL METHODS: VOLUMETRIC Finite elements (FEM), PEEC,...! Proximity and skin effect % Slow: must mesh the whole volume of the conductors % Very fine mesh needed at high frequency (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

8 MOTIVATION STATE OF THE ART SURFACE-BASED METHODS Unknowns only on the conductors surface Wires modelled with impedance boundary conditions Coperich, Ruehli, Cangellaris 2000 Morsey, Okhmatovski, Cangellaris 2004 Chakraboty, Jandhyala 2005 De Zütter, Knockaert 2005 Qian, Chew, Suaya 2007 and many other... Less unknowns -> much faster, less memory Focus on rectangular or arbitrarily-shaped conductors (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

9 MOTIVATION PROBLEM OUR CONTRIBUTION Broadband R(ω), L(ω) computation for cables with round conductors V z = [R(ω) + jωl(ω)] I (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

10 MOTIVATION PROBLEM OUR CONTRIBUTION Broadband R(ω), L(ω) computation for cables with round conductors V z = [R(ω) + jωl(ω)] I Goal: handle hundreds of conductors in less than a minute (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

11 MOTIVATION PROBLEM OUR CONTRIBUTION Broadband R(ω), L(ω) computation for cables with round conductors V z = [R(ω) + jωl(ω)] I Goal: handle hundreds of conductors in less than a minute We start from De Zütter & Knockaert (2005) 1 Surface admittance operator (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

12 MOTIVATION PROBLEM OUR CONTRIBUTION Broadband R(ω), L(ω) computation for cables with round conductors V z = [R(ω) + jωl(ω)] I Goal: handle hundreds of conductors in less than a minute We start from De Zütter & Knockaert (2005) 1 Surface admittance operator 2 Electric field integral equation (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

13 MOTIVATION PROBLEM OUR CONTRIBUTION Broadband R(ω), L(ω) computation for cables with round conductors V z = [R(ω) + jωl(ω)] I Goal: handle hundreds of conductors in less than a minute We start from De Zütter & Knockaert (2005) 1 Surface admittance operator 2 Electric field integral equation 3 Method of moments (new: no numerical integration!) (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

14 MOTIVATION PROBLEM OUR CONTRIBUTION Broadband R(ω), L(ω) computation for cables with round conductors V z = [R(ω) + jωl(ω)] I Goal: handle hundreds of conductors in less than a minute We start from De Zütter & Knockaert (2005) 1 Surface admittance operator 2 Electric field integral equation 3 Method of moments (new: no numerical integration!) 4 New: adaptive discretization 5 New: fully-automated (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

15 SURFACE ADMITTANCE OPERATOR SURFACE ADMITTANCE OPERATOR ε, µ, σ ε out, µ 0 APPROACH Sample configuration: 2 conductors (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

16 SURFACE ADMITTANCE OPERATOR SURFACE ADMITTANCE OPERATOR ε out, µ 0 ε out, µ 0 APPROACH Sample configuration: 2 conductors Replace conductors with surrounding medium (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

17 SURFACE ADMITTANCE OPERATOR SURFACE ADMITTANCE OPERATOR ε out, µ 0 J (p) s J (p) s (θ) = 1 2πa p N p ε out, µ 0 J n (p) n= N p e jnθ APPROACH Sample configuration: 2 conductors Replace conductors with surrounding medium On conductors surface: introduce equivalent current density (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

18 SURFACE ADMITTANCE OPERATOR SURFACE ADMITTANCE OPERATOR ε out, µ 0 J (p) s J (p) s (θ) = 1 2πa p N p ε out, µ 0 J n (p) n= N p e jnθ APPROACH Sample configuration: 2 conductors Replace conductors with surrounding medium On conductors surface: introduce equivalent current density Apply equivalence theorem to maintain the E field outside unchanged (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

19 SURFACE ADMITTANCE OPERATOR SURFACE ADMITTANCE OPERATOR ε out, µ 0 J (p) s J (p) s (θ) = 1 2πa p E (p) z (θ) = N p ε out, µ 0 J n (p) n= N p N p E n (p) n= N p e jnθ e jnθ APPROACH Sample configuration: 2 conductors Replace conductors with surrounding medium On conductors surface: introduce equivalent current density Apply equivalence theorem to maintain the E field outside unchanged Current value can be related to the field E z on the conductors surface (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

20 SURFACE ADMITTANCE OPERATOR SURFACE ADMITTANCE OPERATOR From equivalence theorem [De Zütter & Knockaert, 2005]: J (p) n [ kap J n (ka p) k outa p J n (k outa p ) ] µj n (ka p ) µ o J n (k out a p ) }{{} Y n (p) = 2π jω E (p) n (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

21 SURFACE ADMITTANCE OPERATOR SURFACE ADMITTANCE OPERATOR From equivalence theorem [De Zütter & Knockaert, 2005]: J (p) n J (1) N 1. J (1) N 1 J (2) N 2. J (1) N 1. = 2π jω [ kap J n (ka p) µj n (ka p ) Y (1) N 1... = Y (1) N 1 Y (2) k outa p J n (k outa p ) µ o J n (k out a p ) N 2... N 2... Y (2) ] E (p) n E (1) N 1. E (1) N 1 E (2) N 2. E (1) N 1. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

22 SURFACE ADMITTANCE OPERATOR SURFACE ADMITTANCE OPERATOR From equivalence theorem [De Zütter & Knockaert, 2005]: J (p) n J (1) N 1. J (1) N 1 J (2) N 2. J (1) N 1. = 2π jω [ kap J n (ka p) µj n (ka p ) Y (1) N 1... = Y (1) N 1 Y (2) k outa p J n (k outa p ) µ o J n (k out a p ) N 2... N 2... Y (2) ] E (p) n E (1) N 1. E (1) N 1 E (2) N 2. E (1) N 1. J = Y s E Surface admittance operator (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

23 ELECTRIC FIELD INTEGRAL EQUATION ELECTRIC FIELD INTEGRAL EQUATION E z ( r) = jωµ 0 J( r )G ( r, r ) d r V z (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

24 ELECTRIC FIELD INTEGRAL EQUATION ELECTRIC FIELD INTEGRAL EQUATION E z ( r) = jωµ 0 J( r )G ( r, r ) d r V z Field on conductor #3: superimpose effects of current density on #1, #2 #1 #3 (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

25 ELECTRIC FIELD INTEGRAL EQUATION ELECTRIC FIELD INTEGRAL EQUATION E z ( r) = jωµ 0 J( r )G ( r, r ) d r V z Field on conductor #3: superimpose effects of current density on #1, #2, #2 #1 #3 (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

26 ELECTRIC FIELD INTEGRAL EQUATION ELECTRIC FIELD INTEGRAL EQUATION E z ( r) = jωµ 0 J( r )G ( r, r ) d r V z Field on conductor #3: superimpose effects of current density on #1, #2, and #3. #2 #1 #3 (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

27 ELECTRIC FIELD INTEGRAL EQUATION ELECTRIC FIELD INTEGRAL EQUATION E z ( r) = jωµ 0 J( r )G ( r, r ) d r V z Field on conductor #3: superimpose effects of current density on #1, #2, and #3. #2 FIELD ON CONDUCTOR # p #1 #3 E (p) z = jωµ 0 q J s (q) ( r )G ( r, r ) d r (p) V z G ( r, r ): Green s function of homogeneous medium. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

28 ELECTRIC FIELD INTEGRAL EQUATION METHOD OF MOMENTS E (p) z = jωµ 0 q J s (q) ( r )G ( r, r ) d r V (p) z (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

29 ELECTRIC FIELD INTEGRAL EQUATION METHOD OF MOMENTS N p E n (p) n= N p e jnθ = jωµ 0 q J s (q) ( r )G ( r, r ) d r V (p) z (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

30 ELECTRIC FIELD INTEGRAL EQUATION METHOD OF MOMENTS N p E n (p) n= N p e jnθ = jωµ 0 q N q n= N q (q) Jn e jnθ G ( r, r ) d r V (p) 2πa q z (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

31 ELECTRIC FIELD INTEGRAL EQUATION METHOD OF MOMENTS N p E n (p) n= N p e jnθ = jωµ 0 q N q n= N q (q) Jn e jnθ G ( r, r ) d r + 2πa q P [R pq (ω) + jωl pq (ω)]i q q=1 (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

32 ELECTRIC FIELD INTEGRAL EQUATION METHOD OF MOMENTS N p E n (p) n= N p e jnθ = jωµ 0 q N q n= N q (q) Jn e jnθ G ( r, r ) d r + 2πa q P [R pq (ω) + jωl pq (ω)]i q q=1 Apply Method of Moments with Galerkin projection E = jωµ 0 GJ + U[R(ω) + jωl(ω)]i (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

33 ELECTRIC FIELD INTEGRAL EQUATION COMPUTATION OF THE GREEN S MATRIX To obtain G: must compute THOUSANDS double integrals of the form 2π 2π [ (x p x q + a p cos θ a q cos θ ) 2 + G (p,q) n,n = 1 (2π) π ln (y p y q + a p sin θ a q sin θ ) 2 ] e j(nθ n θ) dθdθ Numerical integration: quite time-consuming! (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

34 ELECTRIC FIELD INTEGRAL EQUATION COMPUTATION OF THE GREEN S MATRIX To obtain G: must compute THOUSANDS double integrals of the form 2π 2π [ (x p x q + a p cos θ a q cos θ ) 2 + G (p,q) n,n = 1 (2π) π ln (y p y q + a p sin θ a q sin θ ) 2 ] e j(nθ n θ) dθdθ Numerical integration: quite time-consuming! We derived analytical expressions for it! Makes algorithm very fast and more accurate. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

35 ELECTRIC FIELD INTEGRAL EQUATION COMPUTATION OF THE P.U.L. RESISTANCE AND INDUCTANCE Surface admittance operator J = Y se Electric field integral equation E = jωµ 0GJ + U[R(ω) + jωl(ω)]u T J (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

36 ELECTRIC FIELD INTEGRAL EQUATION COMPUTATION OF THE P.U.L. RESISTANCE AND INDUCTANCE Surface admittance operator J = Y se Electric field integral equation E = jωµ 0GJ + U[R(ω) + jωl(ω)]u T J Solve for [R(ω) + jωl(ω)] to get : R(ω) + jωl(ω) = [ U T (1 jωµ 0 Y s G) 1 Y s U ] 1 FOR DETAILS SEE: U. R. Patel, B. Gustavsen, and P. Triverio, An Equivalent Surface Current Approach for the Computation of the Series Impedance of Power Cables with Inclusion of Skin and Proximity Effects, Submitted to IEEE Trans. on Power Delivery, 2013 (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

37 ELECTRIC FIELD INTEGRAL EQUATION EXAMPLE: USB 2.0 CABLE Power Shield mm Drain Signal 1 Signal Round strands Aggregated in 6 lines Geometry from West and Jandhyala, Ground mm (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

38 ELECTRIC FIELD INTEGRAL EQUATION USB CABLE: RESISTANCE Resistance p.u.l. [ Ω /m] R(p,p) R(s1,s1) R(p,s1) 1 khz - 10 GHz 50 points Excellent agreement between MoM-SO and FEM! 10 1 R(s1, s2) MoM SO FEM Frequency [Hz] (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

39 ELECTRIC FIELD INTEGRAL EQUATION USB CABLE: INDUCTANCE Inductance p.u.l [H/m] 8 x 10 7 L(s2,s2) 7 L(s1,s1) L(p,p) 3 MoM SO FEM 1 khz - 10 GHz 50 points FEM: 19.5 min MoM-SO: 34.4 s Speed up: 34X 2 L(p,s1) & L(p,s1) L(s1,s2) Frequency [Hz] (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

40 ELECTRIC FIELD INTEGRAL EQUATION Current density. Differential excitation. f = 1 MHz (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

41 AUTOMATIC ORDER ESTIMATION AUTOMATIC ORDER ESTIMATION HOW TO CHOOSE SMARTLY THE NUMBER OF BASIS FUNCTIONS N p N p J n (p) n= N p e jnθ (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

42 AUTOMATIC ORDER ESTIMATION AUTOMATIC ORDER ESTIMATION HOW TO CHOOSE SMARTLY THE NUMBER OF BASIS FUNCTIONS N p TRIAL & ERROR Same N p for all conductors Increase N p until results stop changing % Slow, overestimation! N p J n (p) n= N p e jnθ (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

43 AUTOMATIC ORDER ESTIMATION AUTOMATIC ORDER ESTIMATION HOW TO CHOOSE SMARTLY THE NUMBER OF BASIS FUNCTIONS N p TRIAL & ERROR Same N p for all conductors Increase N p until results stop changing % Slow, overestimation! ADAPTIVE ESTIMATION Customized N p for each conductor Convergence detected with energy-based criterion J (p) N p 2 + J (p) N p 2 < β Np 1 n= N p+1 J n (p) 2 J (p) n N p J n (p) n= N p e jnθ Frequency = 0.1 MHz Basis Function (n) (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

44 AUTOMATIC ORDER ESTIMATION AUTOMATIC ORDER ESTIMATION HOW TO CHOOSE SMARTLY THE NUMBER OF BASIS FUNCTIONS N p TRIAL & ERROR Same N p for all conductors Increase N p until results stop changing % Slow, overestimation! ADAPTIVE ESTIMATION Customized N p for each conductor Convergence detected with energy-based criterion J (p) N p 2 + J (p) N p 2 < β Np 1 n= N p+1 J n (p) 2 J (p) n N p J n (p) n= N p e jnθ Frequency = 0.9 MHz Basis Function (n) (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

45 AUTOMATIC ORDER ESTIMATION AUTOMATIC ORDER ESTIMATION HOW TO CHOOSE SMARTLY THE NUMBER OF BASIS FUNCTIONS N p TRIAL & ERROR Same N p for all conductors Increase N p until results stop changing % Slow, overestimation! ADAPTIVE ESTIMATION Customized N p for each conductor Convergence detected with energy-based criterion J (p) N p 2 + J (p) N p 2 < β Np 1 n= N p+1 J n (p) 2 J (p) n N p J n (p) n= N p e jnθ Frequency = 4281 MHz Basis Function (n) (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

46 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

47 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

48 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

49 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

50 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

51 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

52 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e+03 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

53 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e+03 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

54 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e+02 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

55 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e+01 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

56 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e+00 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

57 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e 01 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

58 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e 02 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

59 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e 03 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

60 AUTOMATIC ORDER ESTIMATION ADAPTIVE ESTIMATION Proposed procedure: 1 Initialization: N p = 1, ω = ω max. 2 Compute G, R, and L. 1.5 x Freq = e 03 MHz If conductor p does not satisfy convergence criterion: increase N p and go to step 2. (m) Move to previous frequency point Update G. Compute R, and L 6 Convergence criterion satisfied for N p 1? If so, decrease N p! Go to step (m) x (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

61 AUTOMATIC ORDER ESTIMATION Current density Order N p for each conductor (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

62 AUTOMATIC ORDER ESTIMATION RESULTS: ADAPTIVE VS. TRIAL & ERROR ESTIMATION CPU Time (s) TE AD Two-wires Coaxial Bundle of micro-coax (232 strands) USB (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

63 AUTOMATIC ORDER ESTIMATION RESULTS: ADAPTIVE VS. TRIAL & ERROR ESTIMATION CPU Time (s) Max Error on R and L (%) TE AD TE AD Two-wires Coaxial Bundle of micro-coax (232 strands) USB (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

64 AUTOMATIC ORDER ESTIMATION RESULTS: ADAPTIVE VS. TRIAL & ERROR ESTIMATION CPU Time (s) Max Error on R and L (%) Problem size N (min-max) TE AD TE AD TE AD Two-wires Coaxial Bundle of micro-coax (232 strands) USB (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

65 AUTOMATIC ORDER ESTIMATION RESULTS: ADAPTIVE VS. TRIAL & ERROR ESTIMATION CPU Time (s) Max Error on R and L (%) Problem size N (min-max) TE AD TE AD TE AD Two-wires Coaxial Bundle of micro-coax (232 strands) USB Average number of unknowns per conductor: 3 to 7. MoM-SO can easily handle hundreds of conductors! (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

66 CONCLUSION CONCLUSION MoM-SO: fast technique to compute the resistance and inductance of cables with round conductors. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

67 CONCLUSION CONCLUSION MoM-SO: fast technique to compute the resistance and inductance of cables with round conductors. Accurate & broadband (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

68 CONCLUSION CONCLUSION MoM-SO: fast technique to compute the resistance and inductance of cables with round conductors. Accurate & broadband Fast: surface formulation, no meshing, no mesh refinement due to skin effects. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

69 CONCLUSION CONCLUSION MoM-SO: fast technique to compute the resistance and inductance of cables with round conductors. Accurate & broadband Fast: surface formulation, no meshing, no mesh refinement due to skin effects. Fast: developed analytic formulas for Green s function discretization. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

70 CONCLUSION CONCLUSION MoM-SO: fast technique to compute the resistance and inductance of cables with round conductors. Accurate & broadband Fast: surface formulation, no meshing, no mesh refinement due to skin effects. Fast: developed analytic formulas for Green s function discretization. Adaptive discretization (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

71 CONCLUSION CONCLUSION MoM-SO: fast technique to compute the resistance and inductance of cables with round conductors. Accurate & broadband Fast: surface formulation, no meshing, no mesh refinement due to skin effects. Fast: developed analytic formulas for Green s function discretization. Adaptive discretization Fully-automated: one-button-click technique (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

72 CONCLUSION CONCLUSION MoM-SO: fast technique to compute the resistance and inductance of cables with round conductors. Accurate & broadband Fast: surface formulation, no meshing, no mesh refinement due to skin effects. Fast: developed analytic formulas for Green s function discretization. Adaptive discretization Fully-automated: one-button-click technique Applications: high-speed cables (USB, HDMI,...), submarine and underground power cables. Acknowledgement: Work partially supported by Norwegian Research Council (RENERGI programme) and by a consortium of industry partners led by SINTEF Energy Research. (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24

73 Thank you! Photo author: Mkooiman, Source: Wikimedia

74 APPENDIX FOR FURTHER READING FOR FURTHER READING I D. De Zutter, and L. Knockaert, Skin Effect Modeling Based on a Differential Surface Admittance Operator, IEEE Trans. on Microwave Th. and Tech., vol. 53, no. 8, pp , Aug U. R. Patel, B. Gustavsen, and P. Triverio, An Equivalent Surface Current Approach for the Computation of the Series Impedance of Power Cables with Inclusion of Skin and Proximity Effects, Submitted to IEEE Trans. on Power Delivery, B. Gustavsen, A. Bruaset, J. Bremnes, and A. Hassel, A finite element approach for calculating electrical parameters of umbilical cables, IEEE Trans. Power Delivery, vol. 24, no. 4, pp , Oct T. West and V. Jandhyala, Universal serial bus signal integrity analysis for high-speed signaling, Technical Report, University of Washington, Dec (PATEL, GUSTAVSEN, AND TRIVERIO) MOM-SO SPI / 24