[SOLVED] Ecse597/ecse472 assignment 5 part a and b

Question I
A linear circuit has poles at S=-5, -3, -2, -7, and -8.
i. If the circuit is simulated using the Forward Euler method with step size h, for what
values of h is the simulation stable?
ii. If the circuit is simulated using the Backward Euler method with step size h, for what
values of h is the simulation stable?
iii. If the circuit is simulated using the Trapezoidal Rule with step size h, for what values
of h is the simulation stable?Question II
Consider the following Adams-Moulton integration method:
�” = �”$% + ‘(
%)
�”̇ + +(
%)
�̇
“$% + (
%)
�̇
“$)
Derive the difference equation corresponding the to the above method.Question III
Your circuit simulator needs to perform an AC analysis at 1000 frequency points, as well as
compute the sensitivity of 5 output nodes with respect to three different parameters. What is
the total number of Sparse orderings, L/U factorizations and forward backward substitutions
required for the full computation (including both the frequency response and its sensitivity) if:
1. We use the perturbation method.
2. We use the differentiation method.
3. We use the Adjoint method. Preliminaries
1. You can reuse and build upon your previous code for this assignment.
2. We provide you with the following additional functions that help handle time domain
sources.a. volsine.m and cursine.m are two functions that add time domain sinusoidal
voltage and current sources respectively. They are used in the netlist. The
netlists used in the questions below should provide examples. Also read the
comments in the functions for the required syntax.
b. BTime.m is a function that evaluates and returns the time domain value of the
right hand side vector b(t) as a function of time.3. In your submission please provide all code in a zip file in a way that allows us to run the
testbenches ourselves (include all code, not just the recent one).4. Also submit a pdf file containing the answers to the questions, the output plots and the
code for functions you have written for this assignment.Question I: Backward Euler
Write a function called transient_beuler.m with the following header:
function [tpoints,r] = transient_beuler(t1,t2,h,out)
% [tpoints,r] = beuler(t1,t2,h,out)
% Perform transient analysis for LINEAR Circuits using Backward Euler
% assume zero initial condition.
% Inputs: t1 = starting time point (typically 0)
% t2 = ending time point
% h = step size
% out = output node% Outputs tpoints = are the time points at which the output
% was evaluated
% r = value of the response at above time points
% plot(tpoints,r) should produce a plot of the transient responseYour function should use Backward Euler with a constant step size to compute the transient
response of a linear circuit.
1. Test your function by running the provided Testbench_Question1.m file. This file
simulates the netlists Circuit_chebychev_filter_TD.m (provided in the assignment) in the
time domain (transient) and Circuit_chebychev_filter (also provided – this is the same
circuit you used before to test your fsolve) in the frequency domain using your fsolve.m
function which you developed in past assignments. In your submission, include the code
for the new function transient_beuler.m as well as the output plot of the testbench
function.
2. Explain the relation between the two plots in the output figure.Question II Trapezoidal Rule
Write a function transient_trapez.m with the following header:
function [tpoints,r] = transient_trapez(t1,t2,h,out)
% [tpoints,r] = Transient_trapez(t1,t2,h,out)
% Perform Transient Analysis using the Trapezoidal Rule for LINEAR
% circuits.
% assume zero initial condition.
% Inputs: t1 = starting time point (typically 0)
% t2 = ending time point
% h = step size
% out = output node
% Outputs tpoints = are the time points at which the output
% was evaluated
% r = value of the response at above time points
% plot(tpoints,r) should produce a plot of the transient responseYour function should use Trapezoidal Rule with a constant step size to compute the transient
response of a linear circuit.
1. Test your function by running the provided Testbench_Question2.m file. This scripts also
runs your code from Question 1 and compares BE to TR. In your submission, provide the
code for the function you wrote as well as the plot from the testbench.2. By examining the plot, what can you deduce about the BE and TR methods?Question III Forward Euler
Write a function transient_feuler.m with the following header:
function [tpoints,r] = transient_feuler(t1,t2,h,out)
% [tpoints,r] = Transient_feuler(t1,t2,h,out)
% Perform Transient analysis for LINEAR circuit using Forward Euler
% This function assumes the C matrix is invertible and will not work
% for circuits where C is not invertible.
% assume zero initial condition.
% Inputs: t1 = starting time point (typically 0)
% t2 = ending time point
% h = step size
% out = output node
% Outputs tpoints = are the time points at which the output
% was evaluated
% r = value of the response at above time points
% plot(tpoints,r) should produce a plot of the transient responseYour function should use the Forward Euler method with a constant step size to compute the
transient response of a linear circuit. You may assume that the C matrix of your MNA is
invetible for the purposes of this assignment (this means that your function will not work for
most circuits).1. In order to test your code, run the provided test bench script Testbench_Question3.m
which simulates the circuit in the provided netlist Q3BEcircuit.m.
2. Examine and comment on what you learn from the output files.3. It can be shown that, when the C matrix is invertible, the poles of the circuit are the
eigenvalues of the matrix −�#$� (note the negative sign). Determine the stability
condition of the forward euler method for this circuit and experimentally verify your
results by running simulations (note the eig function in matlab computes the
eigenvalues of a matrix).Question IV Nonlinear Circuits
Write a function nl_transient_beuler.m with the following header:
function [tpoints,r] = nl_transient_beuler(t1,t2,h,out)
% [tpoints,r] = beuler(t1,t2,h,out)% Perform transient analysis for NONLINEAR Circuits using Backward Euler
% Assume zero initial condition.
% Inputs: t1 = starting time point (typically 0)
% t2 = ending time point
% h = step size
% out = output node
% Outputs tpoints = are the time points at which the output
% was evaluated
% r = value of the response at above time points
% plot(tpoints,r) should produce a plot of the transient responseYour function should use Backward Euler with a constant step size to compute the transient
response of a nonlinear circuit. For now, you only need to support diodes and you can reuse the
code you wrote before and that we have supplied to you.Test your circuit by running the provided script Testbench_Question4 which simulates the
rectifier circuit in netlist Circuit_Rectifier.m also provided.