Reluctance/Inductance Matrix under

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Gordon Chambers
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1 Generating Stable and Sparse Reluctance/Inductance Matrix under Insufficient Conditions Y. Tanji, Kagawa University, Japan T. Watanabe, The University it of Shizuoka, Japan H. Asai, Shizuoka University, Japan /22/28 ASP-DAC 8

2 Background Wire Packages Via VLSI s PCB s RLC Network ) ( ) ( ) ( ) ( ) ( t t t dt d t t T b i v L C i v A A G = + MNA Eqn. Fast Simulation Using Sparse Inductance/Reluctance Matrices /22/28 2 ASP-DAC 8

3 To speed up the transient simulation, we should make the inductance or reluctance matrix sparse. Positive definiteness of the inductance or reluctance matrix must be guaranteed for stable circuit simulation. Method Fault Shift & Truncate ICCAD995 Inaccurate Double Inverse DAC Inaccurate INDUCTWISE TCAD 3 No guaranteed Stable Wire Duplication TCAD 4 Zero initial Condition Only Block K ASP-DAC 4 No guaranteed Stable Band Matching ASP-DAC 5 Restricted Structure /22/28 ASP-DAC 8 3

4 To get a stable and sparse reluctance matrix, offdiagonal elements of reluctance matrix must be negative. Most previous works with guaranteed stability are based on a fine discretization of conductors. [ +9 ] [ ] [H Henry] 3 Inductance [ +9 ] [ Henry ] 4 2 Reluctance [Henry ] Positive OffDiagonal Elements Diagonal Element [Column Index] 2 4 [Column Index] [Column Index] /22/28 ASP-DAC 8 4

5 Even if the off-diagonal elements are all negative, this does not mean positive definite. Negative Eigen Value 2 2 = T 3 To get the stable and sparse reluctance matrix using the previous methods, the conductors must be more finely discretized to make the reluctance matrix diagonally dominant. we present how to obtain the stable and sparse reluctance/inductance matrix under insufficient discretization. /22/28 ASP-DAC

6 Generating Sparse Reluctance Matrix Inductance (INDUCTWISE) Eigen Value.23 reluctance [.4,.,.4, 3.5, 3.7] Eigen Value[ [.4,.,.4, 3.4, 3.7 ] Negative off-diagonal elements are found relatively small. Truncation affects a small change to the eigen values. /22/28 ASP-DAC 8 6

7 Three methods for generating g stable and sparse reluctance matrix are proposed. Direct Truncation is accurate but check of positive definiteness is necessary. Enforcing Positive Definiteness Enforcing Diagonally Dominance provide a provably positive definite reluctance matrix but are not accurate. /22/28 ASP-DAC 8 7

8 Direct Truncation Truncation Cholesky Decomposition An off-diagonal element is truncated, if the absolute value is less than a threshold. If the Cholesky decomposition is successfully done, the truncated matrix is positive definite. /22/28 ASP-DAC 8 8

9 Enforcing Positive Definiteness a a 2 a 3 - a 3 a 3 ã a 2 a 2 a 22 a 23 a 2 a 22 a 23 a 3 a 32 a 33 a 3 - a 3 a 32 ã 33 Positive Negative Positive Definite The off-diagonal element is shifted and truncated and the absolute value is added to the diagonal. Shift and Truncation /22/28 ASP-DAC 8 9

10 Enforcing Diagonally Dominance A diagonally dominant matrix can be extracted from the original matrix by removing the negative definite matrices. Positive Definite and Diagonally Dominant = diag (2,,) + + diag (, 2, 3) Negative Definite

11 [ +9 ] Sparse Reluctance Matrix [Henry ] Diagonal Element dt 5 ep exact edd exact: no truncation dt: direct truncation ep: enforcing positive definiteness edd: enforcing diagonally dominance [Column Index] /22/28

12 Generating Sparse Inductance Matrix After getting g the reluctance matrix, the matrix is again inverted. Then, we can obtain the sparse inductance matrix. Direct Truncation Neumann Series Inversion & Direct Cholesky Truncation Decomposition Dense K K - L L - Sparse Neumann Enforcing oc K Series K - Diagonally Dominant /22/28 2

13 [ ] Sparse Inductance Matrix [H Henry] Diagonal Element inv exact exact: no truncation inv: inverse of truncated reluctance matrix -, -2 : threshold of direct truncation 2 [Column Index] /22/28 3

14 Neumann Series (/2) Dense Inversion & Direct Truncation K K - L L - Sparse Neumann Enforcing K Series K - Diagonally Dominant Cholesky Decomposition /22/28 ASP-DAC 8 4

15 Neumann Series (2/2) Inversion of the reluctance matrix is done using the Neumann series. Reluctance Matrix: Neumann Series: K K = D N D : Diagonal Matrix p = i L p D N D i = ( ) K D D ND L /22/28 ASP-DAC 8 5

16 Example x W W T Division into Unit Cells y Inductance and Capacitance Extraction PCB C L R K C2 Reluctance: 5 5: 7.8% positive : 7.% positive Capacitance: : diagonally dominant C2 /22/28 ASP-DAC 8 6

17 Package Model (R, L Matrices) PCB Model (L or K, C, R Matrices) Sparse Approximation SPICE Transient Analysis /22/28 ASP-DAC 8 7

18 Transient Response (Reluctance) /2.58 Thereshold: 9 Thereshold: 8.58 exact exact dt dt ep ep.56 edd.56 edd [V Volt] [V Volt] [ 9 ] [Time] 2 [ 9 ] [Time] dt: direct truncation, ep: enforcing positive definiteness, edd: enforcing diagonally dominance /22/28 8

19 Transient Response (Reluctance) 2/2 Thereshold: 9 Thereshold: 8 [Volt] dt exact ep edd [Vo lt] exact dt ep edd [ 9 ] [Time] [ 9 ] [Time] dt: direct truncation, ep: enforcing positive definiteness, edd: enforcing diagonally dominance /22/28 9

20 Transient Response (Inductance) Direct Truncation Neumann Series [Volt] exact dt (reluctance) 2 [Volt] exact edd (reluctance) Neumann [ 9 ] [ 9 ] [Time] [Time] dt: direct truncation, edd: enforcing diagonally dominance /22/28 2

21 CPU Time Comparison Reluctance Method Sparsity [%] CPU times [sec.] Direct Truncation.7 6 Direct Truncation 2 2.,9 Enforcing Positive Definiteness.7 5 Enforcing Positive Definiteness 2.,85 Enforcing Diagonally Dominance Enforcing Diagonally Dominance Inductance Method Sparsity [%] CPU times [sec.] Exact 2,8 Direct Truncation.38 Direct Truncation Neumann Series 2. 75

22 Conclusions Generating the sparse and stable reluctance/inductance t matrices with insufficient conditions is presented. When the discretiztion of conductors is not fine, we find positive-off diagonal elements. Even if the positive-off off diagonal elements were found, the positive definiteness was guaranteed after truncation. The key techniques are as follows: )check of positive definiteness by Cholesky decomposition 2)enforcing positive definiteness 3)enforcing diagonally dominance 4)Neumann series /22/28 ASP-DAC 8 22

23 SPICE Model of Reluctance (Block K Method) A row of reluctance matrix is represented with a inductance and voltage control voltage sources. Relation between current and voltage V = s LI s I = KV si i = N j= K ij V j V i = s K ii I i inductance N j=, j i K K ij ii V j voltage control voltage sources /22/28 ASP-DAC 8 23

Efficient Reluctance Extraction for Large-Scale Power Grid with High- Frequency Consideration

Efficient Reluctance Extraction for Large-Scale Power Grid with High- Frequency Consideration Shan Zeng, Wenjian Yu, Jin Shi, Xianlong Hong Dept. Computer Science & Technology, Tsinghua University, Beijing