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

ECE 546 – Jose Schutt-Aine 2 The Node Voltage method consists in determining potential differences between nodes and ground (reference) using KCL For Node 1: Nodal Analysis

ECE 546 – Jose Schutt-Aine 3 For Node 2: Rearranging the terms gives: Defining: Nodal Analysis

ECE 546 – Jose Schutt-Aine 4 Rearranging the terms gives: The system can be solved to yield V 1 and V 2. Nodal Analysis

ECE 546 – Jose Schutt-Aine 5 For Node 1: For Node 2: Nodal Analysis

ECE 546 – Jose Schutt-Aine 6 Rearranging the terms gives: Nodal Analysis

ECE 546 – Jose Schutt-Aine 7 Nodal Analysis [Y] [v]=[i]

ECE 546 – Jose Schutt-Aine 8 Nodal Analysis - Solution

ECE 546 – Jose Schutt-Aine 9 MNA Formulation Node 1:Node 2: Node 3:

ECE 546 – Jose Schutt-Aine 10 Solve for V 1, V 2 and V 3 using backward substitution Arrange in matrix form: [G][V]=[I] Use Gaussian elimination to form an upper triangular matrix This can always be solved no matter how large the matrix is MNA Solution

ECE 546 – Jose Schutt-Aine 11  Established platform  Powerful engine  Source code available for free  Extensive libraries of devices  New device installation procedure easy Why SPICE ?

ECE 546 – Jose Schutt-Aine 12 SPICE Directory Structure SPICE

ECE 546 – Jose Schutt-Aine 13 SymbolDescriptionValueUnits L drawn Device length (drawn)0.35 mm L eff Device length (effective)0.25 mm t ox Gate oxide thickness70A NaNa Density of acceptor ions in NFET channel 1.0  cm -3 NdNd Density of donor ions in PFET channel 2.5  cm -3 V Tn NFET threshold voltage0.5V V Tp PFET threshold voltage-0.5V Channel modulation parameter0.1V -1  Body effect parameter0.3V 1/2 V sat Saturation velocity 1.7  10 5 m/s nn Electron mobility400cm 2 /Vs pp Hole mobility100cm 2 /Vs knkn NFET process transconductance200  A/V 2 kpkp PFET process transconductance50  A/V 2 C ox Gate oxide capacitance per unit area5 fF/  m 2 C GSO,C GDO Gate source and drain overlap capacitance0.1 fF/  m CJCJ Junction capacitance0.5 fF/  m 2 C JSW Junction sidewall capacitance0.2 fF/  m R poly Gate sheet resistance4  /square R diff Source and drain sheet resistance4  /square MOS SPICE Parameters

ECE 546 – Jose Schutt-Aine 14 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 Problems

ECE 546 – Jose Schutt-Aine 15 - Loads are nonlinear - Need to model reactive elements in the time domain - Generalize to nonlinear reactive elements Motivations

ECE 546 – Jose Schutt-Aine 16 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 Time-Domain Model for Linear Capacitor

ECE 546 – Jose Schutt-Aine 17 After substitution, we obtain so that The solution for the current at t n+1 is, therefore, Time-Domain Model for Linear Capacitor

ECE 546 – Jose Schutt-Aine 18 Backward Euler companion model at t=nh Trapezoidal companion model at t=nh Time-Domain Model for Linear Capacitor

ECE 546 – Jose Schutt-Aine 19 Time-Domain Model for Linear Capacitor

ECE 546 – Jose Schutt-Aine 20 Time-Domain Model for Linear Inductor

ECE 546 – Jose Schutt-Aine 21 If trapezoidal method is applied Time-Domain Model for Linear Inductor

ECE 546 – Jose Schutt-Aine 22 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. The Diode

ECE 546 – Jose Schutt-Aine 23 Nonlinear transcendental system  Use graphical method Solution is found at itersection of load line characteristics and diode characteristics Diode Circuits

ECE 546 – Jose Schutt-Aine 24 NPN Transistor Describes BJT operation in all of its possible modes 24 BJT Ebers-Moll Model

ECE 546 – Jose Schutt-Aine 25 Problem: Wish to solve for f(x)=0 Use fixed point iteration method: With Newton Raphson: therefore, Newton Raphson Method

ECE 546 – Jose Schutt-Aine 26 Newton Raphson Method (Graphical Interpretation)

ECE 546 – Jose Schutt-Aine 27 x k+1 is the solution of a linear system of equations. Forward and backward substitution. A k is the nodal matrix for N k S k is the rhs source vector for N k. Newton Raphson Algorithm

ECE 546 – Jose Schutt-Aine 28 Newton Raphson Algorithm

ECE 546 – Jose Schutt-Aine 29 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 Newton Raphson - Diode

ECE 546 – Jose Schutt-Aine 30 After manipulation we obtain Newton Raphson

ECE 546 – Jose Schutt-Aine 31 Newton-Raphson Method Procedure is repeated until convergence to final (true) value of V D which is the solution. Rate of convergence is quadratic. Where is the value of V D at the kth iteration Use : Diode Circuit – Iterative Method

ECE 546 – Jose Schutt-Aine 32 Newton-Raphson representation of diode circuit at kth iteration Newton Raphson for Diode

ECE 546 – Jose Schutt-Aine 33 Companion Current Controlled

ECE 546 – Jose Schutt-Aine 34 Let x = vector variables in the network to be solved for. Let f(x) = 0 be the network equations. Let x k be the present iterate, and define Let N k be the linear network where each non-linear resistor is replaced by its companion model computed from x k. General Network

ECE 546 – Jose Schutt-Aine 35 Companion model General Network

ECE 546 – Jose Schutt-Aine 36 Nonlinear Reactive Elements

ECE 546 – Jose Schutt-Aine 37 General Element