AC-CH2
-
Upload
shiraz-husain -
Category
Documents
-
view
217 -
download
0
Transcript of AC-CH2
-
7/30/2019 AC-CH2
1/40
Chapter 2. Mathematical Foundation1
Chapter 2. Mathematical Foundation
1. Complex-Variable Concept
2. Frequency-Domain Plots
3.Differential Equations
4. Laplace transform
5. Inverse Laplace transform by partial-fraction expansion
6. Summary
-
7/30/2019 AC-CH2
2/40
Chapter 2. Mathematical Foundation2
2-1 Complex Variable
z x j y= + ejz R R = =
cos
sin
x R
y R
=
=
2 2
1tan
R x y
yx
= +
=
cos sinje j = + * jz x j y Re = =
rectangular form polar form
2* 2 2 2z zz x y R= = + =
2 3 4, , , , nj j j j z
-
7/30/2019 AC-CH2
3/40
Chapter 2. Mathematical Foundation3
2-1 Complex Variable
( ) : ~ ratioal functionG s C C
2
10( 2)( )
( 1)( 3)
sG s
s s s
+=
+ +zeropole
order of pole simple pole
2 2
10( 2)( )
( 2 2)( 3)
sG s
s s s
+=
+ + +s C
-
7/30/2019 AC-CH2
4/40
Chapter 2. Mathematical Foundation4
2-2 Frequency-Domain Plots
Polar Plot, Bode Plot, Magnitude-phase Plot
magnitude, phase( ) ( ) ( )G j G j G j =
Fig 2-8
-
7/30/2019 AC-CH2
5/40
Chapter 2. Mathematical Foundation5
2-3 Differential Equations
linear ODE1
1 1 01( ) ( ) ( ) ( ) ( )
n n
nn nd y t d y t dy t a a a y t f t
dt dt dt
+ + + + =L
2
2
( )sin ( ) 0
d tm mg t
dt
+ =l nonlinear ODE
2
1 0 02
( ) ( )( ) ( )
d y t dy t a a y t b f t
dt dt + + =
state equation & output equation
1 2
( )( ) ( ); ( ) ( )
dy tx t y t x t y t
dt= = =&
1 2
2 0 1 0 0 1 1 2 0
( ) ( ) ( );( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
x t y t x tdy t
x t y t a y t a b f t a x t a x t b f tdt
= =
= = + = +
& &
& &&
-
7/30/2019 AC-CH2
6/40
Chapter 2. Mathematical Foundation6
2-3 Differential Equations
x Ax Bu= +&
state equation
; ;n q px R y R u R
1 1
0 1 02 2
( ) ( )0 1 0 ( )( ) ( )
x t x t f ta a bx t x t
= +
&
&
[ ] [ ]1
2
( )( ) 1 0 0 ( )
( )
x ty t f t
x t
= +
output equation
y Cx Du= +
~ ; ~ ; ~ ; ~A n n B n p C q n D q p
-
7/30/2019 AC-CH2
7/40
Chapter 2. Mathematical Foundation7
2-4 Laplace transform
0( ) ( )stF s f t e dt
= ( ) ( )f t F s
( ) ( )kf t kF s
1 2 1 2( ) ( ) ( ) ( )f t f t F s F s+ +
( ) ( ) (0)d
f t sF s fdt
1 ( 1)( ) ( ) (0) (0)n
n n n
n
df t s F s s f f
dt
L
-
7/30/2019 AC-CH2
8/40
Chapter 2. Mathematical Foundation8
2-4 Laplace transform
0
1
( ) ( )
t
f d F ss 1 1
1 10 0 0
1( ) ( )
nt t t
n nf d dt dt F s
s
L L
( ) ( ) ( )Tssf t T u t T e F s
( ) ( )te f t F s
1 2 1 2( ) ( ) ( ) ( )f t f t F s F s
1 2 1 2( ) ( ) ( ) ( )f t f t F s F s
convolution
-
7/30/2019 AC-CH2
9/40
Chapter 2. Mathematical Foundation9
2-4 Laplace transform
0lim ( ) lim ( )t s
f t sF s
=
0lim ( ) lim ( )t s
f t sF s
=
Initial-value theorem
Final-value theorem
If ( ) does not have poles on or to the right of the imaginary axis,sF s
-
7/30/2019 AC-CH2
10/40
Chapter 2. Mathematical Foundation10
2-5 Inverse Laplace transform
partial-fraction expansion
-
7/30/2019 AC-CH2
11/40
Chapter 2. Mathematical Foundation11
2-6 Application of the Laplace Transform
Ex2-6-11
( ) ( ) ( )y t y t f t
+ =&(0) 0; ( ) ( )sy f t u t= =
1 1 1( ) ( ) ( )1
( )
sY s Y s Y ss
s s
+ = =
+
( ) ( )1
t
Y s y t ess
= =
+
-
7/30/2019 AC-CH2
12/40
Chapter 2. Mathematical Foundation12
2-6 Application of the Laplace Transform
Ex2-6-1
2 22 ( )n n n sy y y u t + + =&& & (0) (0) 0y y= =&
2
2 2 2 2
21
( ) ( 2 ) 2
n n
n n n n
s
Y s s s s s s s
+= =
+ + + +
0 1