HW #3
For the circuit shown below, the differential equation relating () to () is given by
()
+ () = ()
Dividing by and solving for the derivative,
1
()
[ () + ()] = ( (), )
=
(a) Beginning with the basic iterative equation
( + 1) = () +
where is an a
Review for Exam 1
No formula sheet
No calculator
Be able to solve a first order differential equation using time domain or s-domain
techniques
Be able to solve the state equation for a second order system
Simulation Diagrams and State Space Representation
Application of Numerical Integration Techniques to State Equations
The numerical integration techniques that we have considered for solving a first
order differential equation also apply to the solution of a set of first order
linear differential equation
AM
a)
b)
d)
j = u (an integrator)
sY(s) = [1(5) :> 17(5) = HS) = l :> pole at s = 0 (marginally stable)
(1(5) 3
yum (t) = C! I
j? = u (a double integrator)
3217(5) = cfw_1(3) :> H (s) = KS) = ~15 :> double pole at s = 0 (unstable)
U (s) s
ynar(t
HW #2 Solutions
(1) Given the state and output equations
()
= () + ()
() = () + ()
(a) Draw the corresponding simulation diagram
(b) Assuming () is a unit step function and (0) = 1, use
() = 0 + () ()
0
to solve the state equation, that is find (), 0.
=
General Solution of the State Equation
Given the time-invariant, continuous-time state space representation
() = () + ()
(1)
() = () + ()
is called the state vector and is composed of state variables
is an by matrix of constant elements ,
is a column
Application of Numerical Integration Techniques to State Equations
The numerical integration techniques that we have considered for solving a first
order differential equation also apply to the solution of a set of first order
linear differential equation
Masons Rule for Finding the System Transfer Function
Definition of Components of a Block Diagram and Signal Flow Graph (SFG)
Component
Signal or Variable:
Transfer Function or
Multiplying Factor:
Summing Junction:
Block Diagram
Directed Line Segment
Block
Differential Equation to Transfer Function
Using Laplace Transforms
The Laplace transform of a time-domain function () is denoted by () and is
defined by
() = cfw_() ()
0
(), the inverse Laplace transform, is written as
() = 1cfw_()
() and () form a un