Transcript Analogue and digital techniques in closed loop regulation applications Digital systems • Sampling of analogue signals • Sample-and-hold • Parseval’s theorem.

Analogue and digital techniques in
closed loop regulation applications
Digital systems
• Sampling of analogue signals
• Sample-and-hold
• Parseval’s theorem
Sampling process
T
Continuous
Function
f(t)
Sampler
(t)
Reconstruction ?
Series of
samples
From time domain to frequency domain
Fourier transform
y(t) is a real function of time
We define the Fourier transform Y(f)
A complex function in frequency domain f

Y (f ) 
 2 jft
y
(
t
)
e
dt



y( t ) 
2 jft
Y
(
f
)
e
df


Y(f) is the spectral or harmonic representation of y(t)
Frequency spectrum
From time domain to frequency domain
Example of Fourier transform
Y(f)
y(t)
t
-T
T
Y ( f )  2T
Real even functions
sin( 2Tf )
2Tf
1.2
1
0.8
0.6
0.4
Series1
0.2
0
-6
-4
-2
-0.2
-0.4
0
2
4
6
From time domain to frequency domain
NB: Comments on unit with Fourier function
Use of =2f instead of f

Y()  a  y( t )e  jt dt


1
j t
y( t ) 
Y
(

)
e
df

2a  
a  1,
1
1
,
2 2
Fourier series
a sin(t )
Linear
System b()
 Fourier series
 Fourier transform
ab() sin(t  )
Periodic function
f (t )
t
T
f (t )  f (t  kT)
f (t ) 

2 jnt / T
c
e
n
n  
cn
T
t , k
1
cn 
T
T / 2

f (t )e
 2 jnt / T
T / 2
1
dt 
T
= Complex Fourier Coefficients
f(t) a series of frequencies multiple of 1/T
T/2

f (t )e nt dt
T / 2
Fourier coefficients for real functions
1
c n  (a n  jb n )
2
1
cn  (a n  jb n )
2


a0
f (t) 

[a n cos(nt )  b n sin(nt )]
2 n 1
2
an 
T
T
2
 f (t) cos(nt)dt

2
bn 
T
T
2
a0 
2
T
T
2
 f (t)dt

T
2
T
2
2

T
 f (t) sin(nt)dt

T
2
Principle


a0
f (t ) 

[a n cos(nt )  b n sin(nt )]
2 n 1

T

a0
  {f (t )   [a n cos(nt )  b n sin(nt )]}2 dt
2 n 1

0
  0,
To be minimal

0
a n

0
b n
Analysis in frequency domain
f (t )  f (t )(t )
'
n 
( t )  T  ( t  nT)
n  
• F ()= Fourier transform of f(t)
•  ()= Fourier transform of (t)
• F’ ()= Fourier transform of f’(t)
F' ()  F()  ()
Convolution in the frequency domain
Analysis of ()
n 
(T)  T  ( t  nT)
Periodic function
n  
(t ) 
Decomposition in Fourier series
n 
c e
n  
 t / 2
1
cn 
f ( t )e

T  t / 2
 2 jn
t
T
n=-1
 t / 2
1
dt 
T( t )e

T  t / 2
n=0
n=1
0
2
T
cn  1, n
 2
T
 2 jn
2 jn
t
T
n
t
T
2n
T
dt  1
Analysis of ()
n 
n
2n
()   (f 
)   ( 
)
T n  
T
n  
n 
5
t
4
t
3
t
n 
2
t
1
t
0
1
t
n 
2
t
3
t
4
t
n
( t  nT)   (f 
)

T
n  
n  
5
t

f
2
Transform of f’(t)
f ' ( t )  f ( t )( t )
F' ()  F()  ()
Convolution


2n
F' ()   F()(  )d   F()  ( 
 )d
T
n  





2n
2n
F' ()    F()( 
 )d   F( 
)
T
T
n    
n  

2n
F' ()   F( 
)
T
n  

Transform of f’(t)
F' () 
2
s 
 2f s
T
1
fs 
T

 F(  n )
s
n  
F' ()
F(  2s ) F(  s ) F() F(  s ) F(  2s )





 2s
 s
0
s
2s

Aliasing
F' ()

0
s
 s
s
s
2
2
Primary components
Fundamental components
Complementary
Complementary
components
components
The spectra are overlapping (Folding)
s
 Folding frequency
2
Requirements for sampling frequency
• The sampling frequency should be at least twice as large as
the highest frequency component contained in the continuous
signal being sampled
• In practice several times since physical signals found in the
real world contain components covering a wide frequency range
•NB:If the continuous signal and its n derivatives are sampled
at the same rate then the sampling time may be:
n 1
T 
2f h
f h  highest frequency component
Can we reconstruct f(t) ?
f’(t)
f(t)
f °(t)
Sampler
Filter
F' ()
1
2c
Window
Wc ()





 2s
 s
0
s
2s
 c
c
In the frequency domain
F()  F' ()Wc ()

Back to time domain
Convolution
F()  F' ()Wc ()
f (t )  f ' (t )  w c (t )
f ( t )   f ' () w c ( t  )d
Window in the time domain
1
w c (t) 
2c
c
e
2 j t
 c
sin( c t ) sin( 2f c t )
d 

c t
2f c t
f’(t) in the time domain
f ' ( t )  f ( t )( t ) 
n 
 f (nT)(t  nT)
n  
Back to time domain
f ( t )   f ' () w c ( t  )d
sin c ( t  )
f (t )    f (nT)(  nT)
d
c ( t  )
n  
  nT

sin c ( t  nT)
f ( t )   f (nT)
c ( t  nT)
n  

sin 2f c ( t  nT)
f ( t )   f (nT)
2f c ( t  nT)
n  

Reconstruction
f(t)
sin 2f c ( t  nT)
i( t ) 
2f c ( t  nT)
i( t  nT)  1
i( t  kT)  0
t
nT
(n+1)T
Interpolation functions
(n+2)T
(n+3)T
Delayed pulse train
t
T(1  )
T
T(1  )
T(2  )
2

T
0   1
T
2
1
1  jnT
1  j2 n
 jnt
Cn 
( t  T)e

e

e

T T
T
T
2
(t ) 

 jnT jnt
e
e 

n  

jn( t T )
e

n  
Analogue and digital techniques in
closed loop regulation applications
Zero-order-hold
Reconstruction of sampled data
To reconstruct the data we have a series of data
f (kT), k  1,2,3,4,....................n
1
f ( t )  f [kT]  f '[kT]( t  kT)  f ' '[kT]( t  kT) 2  ..
2!
Approximation
1
f '[kT]  {f [kT]  f [( k  1)T]}
T
1
f ' '[kT]  {f '[kT]  f '[( k  1)T]}
T
A device which uses only the first term f[kT] is called a
Zero-order extrapolator or zero-order-hold
Sample-and-Hold devices
kT
z  0 Source x e (t)
c
x s (t)
z 
x s (kT )
x e (t)
x s (kT )
k-2
k -1
k
k 1 k  2 k  3
t/t
x s (kT)  x e (kT).[u(kT)- u((k  1)T]
Droop
Sample-and-hold circuit
Input signal
Ts
Ta
Tp
t
Output signal
Hold mode
Ta = Acquisition time
Sample mode
Tp = Aperture time
Hold mode
Ts = Settling time
Laplace transform of output
x o (t)   x s (kT )
k

X o (s)    x e (kT ).[u(kT )- u((k  1)T ].e-st dt
0 k
Xo (s)   x e (kT)
( k 1) T
k
Xo (s)   x e (kT)[
k
st
[
u
(
k

T
)

u
((
k

1
)

T
)]
e
dt

kT
 ksT
e
s
e( k 1) T

]
s
1  esT
Xo (s) 
[ x e (kT).eskT ]
s
k
Transfer function
(t )
h(t )
F(s)
Impulse response
F(s)=L[h(t)]
h(t)
(t)
t
t
k=0
 x e (kT)e
k
sk T
1
1  e sT
X o (s) 
s
s  j
Transfer function
 jT
1 e
F()  X o () 
j
 2 j
1 e

j

s
2 e
 .
s
 j

s
2 sin(u)  ju
F()  .
.e
s u
2
s  2f s 
T

u
s
j

s
e

2j
s
(e
 j

s
)

)
s


s
sin(
1.2
1
0.8
0.6
0.4
Series1
0.2
0
-0.2
-0.4
0
1
2
3
4
5
6

s

)  j 
s
e s


s
sin(
F() 
Phase of F()
0

 2
 3
 4
 5
s
2s
3s
4s

Parseval’s theorem
x(t) and y(t) have Fourier transform X(f) and Y(f) respectively
x ( t )  y( t )   x () y( t  )d
Convolution
x ( t )  y( t )  F1[X(f )Y(f )]   X(f )Y(f )e 2 jft df
2 jf ' t
x
(
t
)
y
(
t
)
e
dt   X(f )Y(f 'f )df

 x (t ) y(t )dt   X(f )Y(f )df
 x (t ) y * (t )dt   X(f )Y * (f )df
f’=0
y(t)y*(t)
Parseval’s theorem
 x (t ) y * (t )dt   X(f )Y * (f )df
x(t)y(t)
 x (t )x * (t )dt   X(f )X * (f )df
 x(t)
2
2
dt   X(f ) df
This expression suggests that the energy of a signal
Is distributed in time with a density
x(t)
Or is distributed in frequency with density
2
X (f )
2