ECE 546 Lecture 16 MNA and SPICE
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1 ECE 546 Lecture 16 MNA and SPICE Spring 2018 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois ECE 546 Jose Schutt Aine 1
2 Nodal Analysis The Node oltage method consists in determining potential differences between nodes and ground (reference) using KCL For Node 1: Z Z Z 1 m A B C ECE 546 Jose Schutt Aine 2
3 Nodal Analysis For Node 2: Z Z Z n 0 C D E Rearranging the terms gives: ZA ZB Z C Z C Z A Z C ZC ZD Z E Z E m n Defining: GA, GB, GC, GD, GE Z Z Z Z Z A B C D E ECE 546 Jose Schutt Aine 3
4 Nodal Analysis Rearranging the terms gives: GA GB GC GC 1 GAm G G G G G C C D E 2 E m The system can be solved to yield 1 and 2. ECE 546 Jose Schutt Aine 4
5 Nodal Analysis For Node 1: j j5 2 j2 0 2 j2 3 j4 5 For Node 2: ECE 546 Jose Schutt Aine 5
6 Nodal Analysis Rearranging the terms gives: j j 2 j2 10 j j 2 j2 3 j4 5 0 ECE 546 Jose Schutt Aine 6
7 Nodal Analysis j j2 2 5 [Y] [v]=[i] ECE 546 Jose Schutt Aine 7
8 Nodal Analysis - Solution j j j j j j j j0.5 ECE 546 Jose Schutt Aine 8
9 Nodal Formulation 1 2 Node 1: 3 0 Node 2: Node 3: ECE 546 Jose Schutt Aine 9
10 Nodal Solution Arrange in matrix form: [G][]=[I] Use Gaussian elimination to form an upper triangular matrix Solve for 1, 2 and 3 using backward substitution This can always be solved no matter how large the matrix is ECE 546 Jose Schutt Aine 10
11 Why SPICE? Established platform Powerful engine Source code available for free Extensive libraries of devices New device installation procedure easy ECE 546 Jose Schutt Aine 11
12 SPICE From Netlist Parser Device Stamp I=Y To Solver SPICE Directory Structure spice3f4 conf examples lib man notes patches src tmp util doc helpdir scripts man1 man3 man5 bin include lib unsuppo lib skeleton ckt cp dev fte hlp inp mfb mfbpc misc ni sparse mac asrc bjt bsim1 bsim2 cap cccs ccvs csw dio disto ind isrc jfet ltra mes mos1 mos2 mos3 mos6 res sw tra urc vccs vcvs vsrc ECE 546 Jose Schutt Aine 12
13 MOS SPICE Parameters Symbol Description alue Units L drawn Device length (drawn) 0.35 m L eff Device length (effective) 0.25 m t ox Gate oxide thickness 70 A N a Density of acceptor ions in NFET channel cm -3 N d Density of donor ions in PFET channel cm -3 Tn NFET threshold voltage 0.5 Tp PFET threshold voltage -0.5 Channel modulation parameter Body effect parameter 0.3 1/2 sat Saturation velocity m/s n Electron mobility 400 cm 2 /s p Hole mobility 100 cm 2 /s k n NFET process transconductance 200 A/ 2 k p PFET process transconductance 50 A/ 2 C ox Gate oxide capacitance per unit area 5 ff/m 2 C GSO,C GDO Gate source and drain overlap capacitance 0.1 ff/m C J Junction capacitance 0.5 ff/m 2 C JSW Junction sidewall capacitance 0.2 ff/m R poly Gate sheet resistance 4 /square R diff Source and drain sheet resistance 4 /square ECE 546 Jose Schutt Aine 13
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 ECE 546 Jose Schutt Aine 14
15 Motivations Z o Z o C - Loads are nonlinear - Need to model reactive elements in the time domain - Generalize to nonlinear reactive elements ECE 546 Jose Schutt Aine 15
16 Time-Domain Model for Linear Capacitor For linear capacitor C with voltage v and current i which must satisfy i dv C dt Using the backward Euler scheme, we discretize time and voltage variables and obtain at time t = nh v v hv ' n1 n n1 ECE 546 Jose Schutt Aine 16
17 Time-Domain Model for Linear Capacitor After substitution, we obtain v' n1 i C n1 so that n1 vn 1 vn h C i The solution for the current at t n+1 is, therefore, C C i 1 v 1- v h h n n n ECE 546 Jose Schutt Aine 17 17
18 Time-Domain Model for Linear Capacitor i v C Backward Euler companion model at t=nh Trapezoidal companion model at t=nh ECE 546 Jose Schutt Aine 18
19 Time-Domain Model for Linear Capacitor Step response comparisons o (volts) 0.2 Exact Backward Euler Trapezoidal Time (ns) ECE 546 Jose Schutt Aine 19
20 Time-Domain Model for Linear Inductor v di L dt Backward Euler : in in hi n i n1 v L n1 L L v i i h h n1 n1 n 1 1 ECE 546 Jose Schutt Aine 20
21 Time-Domain Model for Linear Inductor If trapezoidal method is applied h i i i i 2 n1 n n1 n 2L 2L v i v h h n1 n1 n ECE 546 Jose Schutt Aine 21
22 The Diode + I - 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. ECE 546 Jose Schutt Aine 22
23 Diode Circuits out I I e D S D D / T 1 RI RI ( ) S D D D D D Nonlinear transcendental system Use graphical method I D Diode characteristics s /R Load line (external characteristics) out Solution is found at itersection of load line characteristics and diode characteristics S D ECE 546 Jose Schutt Aine 23
24 BJT Ebers-Moll Model NPN Transistor C B E I vbe / T vbc / T 1 S 1 S ie e I e F I ic IS e e R vbe / T S vbc / T 1 1 I I vbe / T S vbc / T 1 1 S ib e e F R F F R 1 R 1 F R Describes BJT operation in all of its possible modes ECE 546 Jose Schutt Aine 24
25 Newton Raphson Method Problem: Wish to solve for f(x)=0 Use fixed point iteration method: Define F( x) x K( x) f ( x) : x F( x ) x K( x ) f( x ) With Newton Raphson: k1 k k k k 1 1 df K( x) [ f( x)] dx 1 therefore, : x x [ f( x )] f( x ) k1 k k k ECE 546 Jose Schutt Aine 25
26 Newton Raphson Method (Graphical Interpretation) ECE 546 Jose Schutt Aine 26
27 Newton Raphson Algorithm 1 : k1 k k k N R x x A f x Akxk 1 Akxk f( xk) Sk. x k+1 is the solution of a linear system of equations. A x k S k LU fact Forward and backward substitution. A k is the nodal matrix for N k S k is the rhs source vector for N k. ECE 546 Jose Schutt Aine 27
28 Newton Raphson Algorithm voltage controlled current controlled 0. k 0, gives 0, i0 1. Find k, ik compute companion mod els. G,,, k Ik Rk Ek 2. Obtain A, S. 3. Solve A x S. c k k k k k C c 4. xk 1 Solution 5. Check for convergence xk 1 xk. If they converge, then stop. 6. k 1 k, and go to step 1. ECE 546 Jose Schutt Aine 28
29 Newton Raphson - Diode It is obvious from the circuit that the solution must satisfy f() = 0 We also have 1 Is / t f '( ) e R t The Newton method relates the solution at the (k+1)th step to the solution at the kth step by f( k ) k 1 - k f '( ) k1 k - k k E I s R 1 Is e R t e k k / t / t 1 ECE 546 Jose Schutt Aine 29
30 ECE 546 Jose Schutt Aine 30 After manipulation we obtain k k k E g J R R / k t s k t I g e / ( -1) - k t k s k k J I e g Newton Raphson
31 Diode Circuit Iterative Method Newton-Raphson Method Use: 1 xk1 xk f '( xk) f ( xk) ( k1) ( k) ( k) 1 ( k) x x f '( x ) f( x ) out D D S D / T f( D) ISe 10 R 1 IS D / T f '( D ) e R ( k ) D ( k1) ( k) D D ( k ) D S IS e R 1 IS e R T Procedure is repeated until convergence to final (true) value of D which is the solution. Rate of convergence is quadratic. ( k ) D ( k ) D / / T T T 1 Where is the value of D at the kth iteration ECE 546 Jose Schutt Aine 31
32 Newton Raphson for Diode Newton-Raphson representation of diode circuit at kth iteration g k I s t e k / t J I e g / ( k t k s -1) - k k ECE 546 Jose Schutt Aine 32
33 Current Controlled Companion R k dh() i di ii k E h( i ) R i k k k k ECE 546 Jose Schutt Aine 33
34 General Network 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 A f ( x ) Jacobian of f at x x k k k Let N k be the linear network where each non-linear resistor is replaced by its companion model computed from x k. I g ( ) j j j ECE 546 Jose Schutt Aine 34
35 General Network P P jk jk jk Companion model G k dg( ) d k I g G k k k k ECE 546 Jose Schutt Aine 35
36 Nonlinear Reactive Elements i n+1 i n C(v) q() n+1 h J n n+1 g k J k J n - dq q f() v, i dt qn 1 qn h dt ttn - dq 1 - qn 1 qn f ( vn 1) or, in 1 in 1( vn 1) h h h ECE 546 Jose Schutt Aine 36
37 General Element ECE 546 Jose Schutt Aine 37
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