1. No category

EECE360
Lecture 3
Topics
Laplace Transform,
Transfer Functions
! 
Review
! 
! 
Today
! 
Dr. Oishi
Linearization through Taylor’s Series approximation
Physical systems:
! 
Electrical and Computer Engineering
! 
! 
University of British Columbia, BC
http://courses.ece.ubc.ca/360
! 
! 
Chapter 2.3-2.5
[email protected]
EECE 360
1
How to Linearize
! 
! 
"y =
#f (x)
"x
#x x = x
0
EECE 360
F = ma
"f (x)
"x x = x
f0
"  1st law or the junction rule (KCL): For a given junction or
node in a circuit, the sum of the currents entering equals the
sum of the currents leaving.
"  2nd law or the loop rule (KVL): Around any closed loop in a
circuit, the sum of the potential differences across all
elements is zero.
0
x
!x 0
3
!
Newton's second law of motion: the relationship between an
object's mass m, its acceleration a, and the applied force F
"  For electrical systems: Kirchhoff’s laws
!
!
For mechanical systems: Newton’s laws
! 
f (x) = x 2
!
2
Physical Laws of Process
! 
f (x)
!
Laplace transform
Transfer functions
EECE 360
Identify an operating point
!
Perform Taylor series expansion
and keep only
st
constant and 1 derivative terms
x
Spring-mass-damper system
RLC circuits
DC motor
!
EECE 360
4
Spring-Mass-Damper System**
! 
! 
Two-Mass System**
F(t) : equivalent force on
mass m
The signs of -kx and -cx are
negative because these forces
oppose the motion of f(t)
#m!x! = f (t ) $ Kx + K 1( x1 $ x) + C1( x!1 $ x! )
"
!m1!x!1 = $ K 1( x1 $ x) $ C1( x!1 $ x! )
Assume the motion directions: x>0; x1-x>0
EECE 360
5
EECE 360
Series RLC Circuit**
6
Parallel RLC Circuit
**
Applying Kirchhoff’s current law:
dv(t) 1
1
C
+ v(t) +
dt
R
L
Input: v(t);
EECE 360
Output: i(t)
Input: r(t);
7
EECE 360
!
t
" v(t) = r(t)
0
Output: v(t)
8
The Laplace Transform
! 
! 
The Laplace Transform
The method of Laplace transforms converts a
calculus problem (the linear differential
equation) into an algebra problem.
The solution of the algebra problem is then
fed backwards through a the Inverse Laplace
Transform and the solution to the differential
equation is obtained.
! 
! 
! 
! 
EECE 360
9
The Laplace Transform
! 
! 
EECE 360
10
Laplace Transform Properties
Like the Fourier transform, the Laplace
transform is an integral transform
Alternately the Laplace variable s can be
considered to be the differential operator:
! 
Linearity:
! 
Differentiation:
! 
! 
EECE 360
Pierre-Simon Laplace
(1749-1827)
Laplace proved the stability of
the solar system. He also put
the theory of mathematical
probability on a sound footing
“All the effects of Nature are
only the mathematical
consequences of a small
number of immutable laws.”
Studied, but did not fully
developed the Laplace
transform
11
Final value theorem:
Initial value theorem:
EECE 360
12
Important Laplace Tr. Pairs
Important Laplace Tr. Pairs
! 
! 
! 
! 
! 
Impulse function
Step function
Exponential decay
Sine and cosine
See Dorf and Bishop, Table 2.3, p. 47 and
Table D.1 (App. D) online.
EECE 360
See Dorf and Bishop, Table 2.3, p. 47 and
Table D.1 (App. D) online.
EECE 360
13
Laplace Transforms
! 
Region of convergence:
! 
! 
! 
f(t) F(s)
14
Example
R.o.C.
Where the transform integral
converges
The Laplace transform
exists only when the
integral converges!
! 
! 
If
for all positive t, then
Consider the spring-mass-damper system
Assuming the system is initially at rest (all initial
conditions at zero), then
output
will converge for s > a.
input
EECE 360
Damped oscillations
15
EECE 360
Transfer
function
16
Transfer Function
! 
! 
Transfer Function
Defined as the ratio of the Laplace
transform of the output to that of the
input
Describes dynamics of a LTI system
! 
Time domain
y(t) = r(t) " g(t)
r(t)
g(t)
Laplace
transform
! Frequency domain
!
!
R(s)
G(s)
! 
Y (s)
!
EECE 360
17
Transfer Function
! 
! 
! 
! 
EECE 360
!
Y (s) = R(s)G(s)
!
EECE 360
!
18
Transfer Function
Differential equation replaced by algebraic
relation Y(s)=G(s)R(s)
Note that if R(s)=1 then Y(s)=G(s) is the
impulse response of the system
Note that if R(s)=1/s, the unit step function,
then Y(s)=G(s)/s is the step response
The magnitude and phase shift of the
response to a sinusoid at frequency ! is given
by the magnitude and phase of the complex
number G(j!) (see Chapter 8)
19
! 
! 
Using the Final Value Theorem, the static or
D.C. gain of a transfer function G(s) is given
by G(0):
Let G(s) = N(s)/D(s), then
! 
! 
EECE 360
Zeros of G(s) are the roots of N(s)=0
Poles of G(s) are the roots of D(s)=0
20
Second Order System Poles
**
Second Order System Poles
**
Damping ratio
Damping ratio
Natural frequency
Natural frequency
In the underdamped case
(complex roots due to the
quadratic):
In the underdamped case
(complex roots due to the
quadratic):
s1, 2 = !"#n ± j#n 1 ! " 2
s1, 2 = !"#n ± j#n 1 ! " 2
with " < 1
with " < 1
Critical damping
EECE 360
21
Natural Response (no input)
EECE 360
22
Transfer Functions
! 
A transfer function is
! 
! 
! 
! 
EECE 360
23
EECE 360
strictly proper when the degree of the
denominator is greater than that of numerator
proper if those degrees are equal
improper if the degree of the numerator is
greater than than of the denominator
Physical systems are proper or strictly proper
24
RLC Network
Example: Spring-mass-damper
! 
! 
Find transfer function G(s)
between the position x1(t)
and the forcing function r(t)
Key assumption: IC are zero
! 
! 
! 
Voltage divider
EECE 360
25
Example: DC Motor
! 
! 
! 
Note that x2(t) does not
appear explicitly. (Why?)
Dorf and Bishop,
Example 2.4, p. 56
EECE 360
26
Example: DC Motor
Power actuator
Robotic manipulators, disk
drives, machine tools, etc.
Armature-controlled
! 
! 
! 
EECE 360
x1(t)=x2(t)=0
dx1/dx=dx2/dt=0
Input: voltage applied to
armature Va (=Ea)
Output: angle of rotation "#
Control variable: current ia
27
EECE 360
28
Example: DC Motor
! 
! 
! 
Armature circuit:
Motor relations:
ea (t ) ! eb(t ) = Raia + La
Example: DC Motor
dia
dt
Armature circuit:
! 
Ea ( s ) ! Eb( s ) = ( Ra + sLa ) Ia ( s )
Tm(t ) = Kiia (t )
eb(t ) = Kb!m(t )
Motor relations:
! 
Mechanical
Response: Jm"˙ m (t) = Tm(t) # Td (t) # Bm"m (t)
Mechanical
Response:
! 
#Tm( s ) = KiIa ( s )
"
! Eb( s ) = Kb$m( s )
Jms"m (s) = Tm(s) # Td (s) # Bm"m (s)
!
EECE 360
EECE 360
29
30
!
Example: DC Motor
Summary
Using Laplace transform, we can represent this DC
motor by the block diagram below
Td(s)
! 
Today
! 
! 
! 
! 
Next
! 
Notice inherent feedback in the model
Transfer function : H(s) =
EECE 360
!
Laplace transform
Final Value Theorem
Transfer functions
! 
Operational Amplifiers
Block diagram models
" m (s)
E a (s)
31
EECE 360
32