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ECE 546 Lecture - 12 MNA and SPICE

ECE 546 Lecture - 12 MNA and SPICE. Spring 2014. Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu. Nodal Analysis. The Node Voltage method consists in determining potential differences between nodes and ground (reference) using KCL.

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ECE 546 Lecture - 12 MNA and SPICE

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  1. ECE 546 Lecture -12 MNA and SPICE Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jesa@illinois.edu

  2. Nodal Analysis The Node Voltage method consists in determining potential differences between nodes and ground (reference) using KCL For Node 1:

  3. Nodal Analysis For Node 2: Rearranging the terms gives: Defining:

  4. Nodal Analysis Rearranging the terms gives: The system can be solved to yield V1 and V2.

  5. Nodal Analysis For Node 1: For Node 2:

  6. Nodal Analysis Rearranging the terms gives:

  7. Nodal Analysis [Y] [v]=[i]

  8. Nodal Analysis - Solution

  9. MNA Formulation Node 1: Node 2: Node 3:

  10. MNA Solution Arrange in matrix form: [G][V]=[I] Use Gaussian elimination to form an upper triangular matrix Solve for V1, V2and V3 using backward substitution This can always be solved no matter how large the matrix is

  11. Why SPICE ?  Established platform  Powerful engine  Source code available for free  Extensive libraries of devices  New device installation procedure easy

  12. SPICE SPICE Directory Structure

  13. MOS SPICE Parameters

  14. Problems • Nonlinear Devices • Diodes, transistors cannot be simulated in the frequency domain • Capacitors and inductors are best described in the frequency domain • Use time-domain representation for reactive elements (capacitors and inductors) • Circuit Size • Matrix size becomes prohibitively large

  15. Motivations • Loads are nonlinear • Need to model reactive elements in the time domain • - Generalize to nonlinear reactive elements

  16. Time-Domain Model for Linear Capacitor For linear capacitor C with voltage v and current i which must satisfy Using the backward Euler scheme, we discretize time and voltage variables and obtain at time t = nh

  17. Time-Domain Model for Linear Capacitor After substitution, we obtain so that The solution for the current at tn+1 is, therefore,

  18. Time-Domain Model for Linear Capacitor Backward Euler companion model at t=nh Trapezoidal companion model at t=nh

  19. Time-Domain Model for Linear Capacitor

  20. Time-Domain Model for Linear Inductor

  21. Time-Domain Model for Linear Inductor If trapezoidal method is applied

  22. The Diode • Diode Properties • Two-terminal device that conducts current freely in one direction but blocks current flow in the opposite direction. • The two electrodes are the anode which must be connected to a positive voltage with respect to the other terminal, the cathode in order for current to flow.

  23. Diode Circuits Nonlinear transcendental system  Use graphical method Solution is found at itersection of load line characteristics and diode characteristics

  24. BJT Ebers-Moll Model NPN Transistor Describes BJT operation in all of its possible modes

  25. Newton Raphson Method Problem: Wish to solve for f(x)=0 Use fixed point iteration method: With Newton Raphson: therefore,

  26. Newton Raphson Method (Graphical Interpretation)

  27. Newton Raphson Algorithm xk+1 is the solution of a linear system of equations. Forward and backward substitution. Ak is the nodal matrix for Nk Sk is the rhs source vector for Nk.

  28. Newton Raphson Algorithm

  29. Newton Raphson - Diode It is obvious from the circuit that the solution must satisfy f(V) = 0 We also have The Newton method relates the solution at the (k+1)th step to the solution at the kth step by

  30. Newton Raphson After manipulation we obtain

  31. Diode Circuit – Iterative Method Newton-Raphson Method Use: Where is the value of VD at the kth iteration Procedure is repeated until convergence to final (true) value of VD which is the solution. Rate of convergence is quadratic.

  32. Newton Raphson for Diode Newton-Raphson representation of diode circuit at kth iteration

  33. Current Controlled Companion

  34. General Network • Let x = vector variables in the network to be solved for. Let f(x) = 0 be the network equations. Let xk be the present iterate, and define • Let Nk be the linear network where each non-linear resistor is replaced by its companion model computed from xk.

  35. General Network Companion model

  36. Nonlinear Reactive Elements

  37. General Element

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