Final Exam

EE382M

Spring 1997

You may use your class notes and text book while working this exam.

 

 

THIS EXAM IS DUE AT 12 NOON, MONDAY, MAY 12, AT MY ENS 635 OFFICE. TURNING IN YOUR EXAM LATE MAY RESULT IN REDUCED OR NO CREDIT!

 

 

 

 

The exam consists of 5 problems (7 pages). You have 3 hours to complete all of these problems. Show all of your work!!! Try to keep your answers organized and concise. The total number of points is 100. Please put your name on all pages, and if you attach sheets, make sure to staple them to the rest of the exam!

 

Problem 1: (20 pts)

For the circuit shown below, write the out the net list using the node numbering given in the figure.

branch name from node to node value

Now stamp this net list into the MNA matrix; fill out the INDEFINITE form first, using the variables ordered as v0, v1, v2, v3, v4, v5, and i1. Please put your final answer into the matrix given below; I have labeled the rows and columns for your convenience. Show your work (as necessary). BE NEAT!!!

 

 

Now write out the definite form:

 

 

If all the conductances were of comparable value, and were much larger than 1, what row/column swaps should be made to pick out the "best" first pivot value for LU factorization? Show the re-ordered matrix system below; make sure you show the new variable and source term ordering!

 

 

Problem 2 (25 pts.)

 

For the circuit below, the diode equation in the forward active region is approximated by

 

 

where vt = 0.026 volts.

 

 

(a) Using KVL/KCL, find the function that we would use in a fixed point iteration scheme, such as Newton-Raphson; i.e., find , where will be in terms of I1, G1, G2, and v.

(b) In an N-R iteration . For the circuit above, what is ?

(c) Graphically illustrate using the I-V curve of the nonlinear element how the iterations proceed.

(d) Draw and label the Newton-Raphson linearized circuit model for the above circuit when the diode is in the forward active region of operation. Use a linearized diode companion model such that no additional nodes are added to the problem. Show the expressions for the diode companion model elements at the n+1 iteration.

(e) Write the nodal equations (in matrix form; i.e., write out the MNA matrix) for this linearized circuit at iteration n+1. (You may write the equations by inspection.)

(f) Solve the linearized circuit equations above by Gaussian Elimination or LU factorization with G1=0, G2=1.0, and I1=1e-3. Perform the first Newton Raphson iteration (n=1). Start with an initial guess (n=0) of 0.55 volts for the diode voltage and 1.54e-5 amps for the diode current. Show your work.

 

 

Problem 3 (15 pts.)

When there are TR integration models in the circuit, there are some conductors which are a function of [Delta]t. This creates a matrix such as:

Solve for the values of [Delta]t that will make it unnecessary to re-order any rows or columns if full pivoting is being used during LU factorization. SHOW ALL OF YOUR WORK! There is a range of values that will work; give me the interval of [Delta]t's that will work. (Recall full pivoting means you re-order (swap rows and/or columns) the submatrix being factor to use the largest (magnitude) element as the pivot.) THIS IS NOT HARD!!!!!

 

Problem 4 (20 pts.)

 

Note Vin is a unit step input. All the R's are the same, and all the C's are the same, in this problem. Notice that node 4 has a "load" of 2C, while all the others are just C.

 

 

Follow this procedure (show your work!!):

 

a) Draw the appropriate dc circuit you use to find m0. What is m0 at nodes 3 and 4?

b) Now draw the appropriate dc circuit you use to find m1. What is m1 at nodes 3 and 4? Find these by setting up the MNA matrix you need to solve, then LU factor the matrix (don't do any re-ordering when you do the LU; just pivot off the elements as they naturally occurred for the node numbering I gave above).

 

 

Problem 5 (20 pts.):

Since the last problem could be reduced to dc circuits, we can also find the moments for the new problem shown below using Kron's formula:

(a) Find the change in m0 at nodes 3 and 4 due to the resistor that has been added between nodes 3 and 4. HINT: THIS IS VERY EASY!!!!

(b) Find the change in m1 at nodes 3 and 4 due to the resistor that has been added between nodes 3 and 4. Recall you can get this using the LU factorization you got in the last problem, and find the "Thevenin equivalent" for the network "before" the new resistor is added:

Get Rthev from: Get Voc from:

 

Now you have the new "voltages":