Complex Eigenvalues
Complex Eigenvalues
In this lecture we consider again a system of n linear homogeneous
equations with constant coefficients
x0 = Ax,
where the coefficient matrix A is real-valued, and we explore the
eect of complex eigenvalues.
Complex
Dierential Equations with Discontinuous Forcing
Functions
Dierential Equations with Discontinuous Forcing
Functions
Discontinuous forcing functions are routinely used by engineers to
analyze systems.
Dierential Equations with Discontinuous Forcing
Functio
Definition of the Laplace Transform
The Laplace Transform
The Laplace transform oers another technique for solving linear
dierential equations with constant coefficients. It is a particularly
useful technique when the right side of a dierential equation i
Step Functions
Step Functions
Some of the most interesting elementary applications of the
transform method occur in the solution of linear dierential
equations with discontinuous or impulsive forcing functions.
Equations of this type frequently arise in t
Solution of Initial Value Problems
Solution of Initial Value Problems
In this lecture we show how the Laplace transform can be used to
solve initial value problems for linear dierential equations with
constant coefficients.
Properties of the Laplace Trans
The Convolution Integral
The Convolution Integral
The product of Laplace transforms arises naturally in the solution
of initial value problems.
The Convolution Integral
Theorem
If F (s) = Lcfw_f (t) and G (s) = Lcfw_g (t) both exist for s > a
then
H(s) =
Homogeneous Linear Systems with Constant
Coefficients
Homogeneous Linear Systems with Constant Coefficients
Now that we have discussed properties of linear systems, we are
ready to turn to the task of finding the general solution to linear,
homogeneous eq
Impulse Functions
Impulse Functions
In this lecture, we discuss the delta function, which is a model for
a force that concentrates a large amount of energy over a short
time interval.
Impulse Functions
Such problems often lead to dierential equations of t
Basic Theory of Systems of First Order Linear
Equations
Basic Theory of Systems of First Order Linear Equations
In this lecture we look at some of the special properties of linear
systems.
Basic Theory of Systems of First Order Linear Equations
We can wri
Introduction to Systems of First Order Linear
Equations
Introduction to Systems of First Order Linear Equations
Systems of simultaneous ordinary dierential equations arise
naturally in problems involving several dependent variables, each of
which is a fun
Homogeneous Equations with Constant
Coefficients
Finding a Fundamental Set of Solutions
Consider the nth order linear homogeneous equation
a0 y (n) + a1 y (n
1)
+ + an
1y
0
+ an y = 0,
where a0 , a1 , . . . , an are real constants and a0 6= 0.
It is natur
The Method of Undetermined Coefficients
The Method of Undetermined Coefficients
A particular solution Y of the nonhomogeneous nth order linear
equation with constant coefficients
a0 y (n) + a1 y (n
1)
+ + an
1y
0
+ an y = g (t)
can be obtained by the meth
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
In this lecture we review some results from linear algebra that are
important for t
Finite State Machines
L14
Shared Inputs and Outputs:
A
B
N
C
N
N
N
N
W
W
D
En
N
N
X
X
D
En
shared
Bus
N
D
N
N
Y
Y
D
En
N
N
Z
Z
D
En
4
4
A Bus Controller:
page 1 of 9
Ensc 252-1157
Fall 2015
L14
Finite State Machines
L14
Swapping Register Contents:
D
WrA
N
Ensc 252-1157
Fall 2015
Registers with Operations
L13
L13
Signal Integrity:
voltage
0.0
0.5
1.0
1.5
2.0
2.5
Timing Parameters for Combinational Circuits:
combo
X
old value
new value
tcd
Y
1
old value
Y
1
Y
X
1
Y
M
tpd
new value
combo
N
X
tpd =
combo
N
tcd
L10
Hierarchical Design
Ensc 252-1157
Fall 2015
L10
VHDL-Array Declarations:
An array-type must first be declared before the array-type can be used in the declaration of a signal. The
array-type declaration must specify 3 pieces of information.
The name o
ENSC 220 (2015-3)
HW 3 (Due Oct 17, 2015, 4:00pm)
For all these problems, you may consider the opamp is powered by V power supply
1. Determine the gain Vo/Vs for the circuit shown below. Consider the opamp to be ideal.
2. Determine the gain. Opamp is idea
Problem Set
06
First Name:
Cheng
Last Name:
Chen
Student Number:
3
1
0
3
1
0
2
8
4
Ensc 252-1157
Fall 2015
Ensc 252-1157
Fall 2015
Show ALL your work IN DETAIL. Provide explanations when necessary.
4.30
f x1x 2 x3 x1x 2x3 x1x 2 x3
x1 x 2x 3
x3
x1 x 2
x1 x
ENSC 220 (2015-3)
HW2 Due 5th October 2015 4:00pm
1. A student acquires a 50mV, 1mA (Meter-A) and a 50mV, 2mA (Meter B) DArsonval
movements and converts both the meters to measure 100V full scale individually. A potential
divider consisting of 70k and 30k
Problem Set
03
First Name:
Angus
Last Name:
Chen
Student Number:
3
2
0
6
1
0
2
7
3
Ensc 252-1157
Fall 2015
Ensc 252-1157
Fall 2015
5.9)
Type a complete explanation here
Create the truth table for adding two bit signed integers.
Where C2 exord with C1 is t
Boyce/DiPrima 10th ed, Ch 1.1: Basic
Mathematical Models; Direction Fields
Dieren'al equa'ons are equa'ons containing deriva'ves.
The following are examples of physical phenomena involving
rates of change:
Mo'on of uids
Mo'on of mechanical systems
Boyce/DiPrima 10th ed, Ch1.3: Classication
of Differential Equations
The main purpose of this course is to discuss proper2es of
solu2ons of dieren2al equa2ons, and to present methods
of nding solu2ons or approxima2ng them.
To provide a framework for
Boyce/DiPrima 10th ed, Ch 1.2: Solutions of
Some Differential Equations
Recall the free fall and owl/mice dieren3al equa3ons:
v = 9.8 0.2v,
p = 0.5 p 450
These equa3ons have the general form y' = ay - b
We can use methods of calculus to solve diere
Boyce/10ed/Ch7
Chapter Review Sheets for
Elementary Differential Equations and Boundary Value Problems, 10e
Chapter 7: Systems of First Order Linear Equations
Definitions:
Systems of ODE's
Linear vs. Nonlinear Systems
Solution
Homogenous and Nonhomoge
Table of useful Laplace transforms
Some useful Laplace transform facts (where F(s) : cfw_f(t) and 0(5) : cfw_g(t), a and
b are constants, and c is a nonnegative constant):
Function Laplace transform
af(t) +bg(t) aF(s)+bG(s)
Ht) 5F(
Boyce/10ed/Ch1
Chapter Review Sheets for
Elementary Differential Equations and Boundary Value Problems, 10e
Chapter 1: Introduction
Definitions:
Differential Equation Mathematical Model
Direction (Slope) Field Equilibrium Solution
Rate (growth) constan
Journal Problems, Set 9
ODE: Introduction to Differential Equations
Math 310 - D100 (Summer 2015)
Quiz on Friday, July 24, 2015
Complete this assignment by Wednesday in your homework journal. This will give you plenty of
time to make sure you understand t
Journal Problems, Set 10
ODE: Introduction to Differential Equations
Math 310 - D100 (Summer 2015)
Quiz on Friday, July 31, 2015
Complete this assignment by Wednesday in your homework journal. This will give you plenty of
time to make sure you understand