Recap Filters
Tony Grift, PhD
Dept. of Agricultural & Biological Engineering
University of Illinois
ABE425 Engineering
Agenda
Recap complex numbers
Relationship Laplace, frequency (Fourier) domain
Relationship time, s and frequency domains
decibel notation (dB)
RC circuit as a Low-Pass and High-Pass filter
Bode plots
Combination filters
Complex number in complex plane
s  s e j  x  jy
 Argument of s
s Absolute value of s (aka Modulus or Magnitude)
Operations on complex numbers cont.
Multiplication/division
using Euler’s notation
s1  x1  jy1  s1 e j1
s2  x2  jy2  s2 e j2
s1 * s2  s1 e j1 * s2 e j2  s1 s2 e j 1 2 
s1 / s2  s1 e
j1
/ s2 e
j 2

s1
s2
e j 1 2 
Operations on complex numbers cont.
Complex conjugate
s1  s1 e j1  s1  s1 e j1
Multiplying a complex
number by its conjugate
gives a real number
s1 * s1  s1 e j1 * s1 e  j1  s1
2
Relation Laplace and Fourier Transform
Time domain
Time domain

F  f t  
L f t    f t e dt  F s 
 st

f t e jt dt  F  j 

0
s-domain (Laplace Domain)

j -domain (Frequency Domain)
s  j
Transient response
(step, impulse)
Frequency response
(filters)
Relation time, s and frequency domain
Time domain
i
U IN t   U O t   UO t 
U IN
UC  UO
Laplace (s)-domain
U IN s   sU O s   U O s 
U O s 
1
 Gs  
U IN s 
1  s
s  j
j -domain
U O  j 
1
 G j  
U IN  j 
1  j
Concept of impedance (Capacitor)
QC  CU C 
dQC
dU C
C
 ic
dt
dt
QC  CU C
1
U C   ic dt
C
1
1
L2F
U C s  
ic s  
U C  j  
ic  j 






sC
j

C
 Current
Volt
Impedance
1
Z  j  
j C
Concept of impedance (Inductor (coil))
di L
UL  L
dt
di L
UL  L
dt
2F
U L s   sLiL s  L

U L  j   
jL iL  j 




Volt
Z  j   jL
Impedance Current
Low-Pass filter using RC network
Derivation transfer function with impedance
U IN
UC  UO
1
1
jC
UO 
Ui 
Ui
1
1  j 
RC
R

j C
U O  j 
1
 G j  
U IN  j 
1  j
Decibel notation
Addition is much simpler than multiplication
Notation in Bel (after Alexander Graham Bell)
For Power
10
logP in Bel
For Voltages (Power ~ Voltage2)
10
 
log U 2  2*10 logU 
In deciBel (0.1 Bel)
2*10 logU  in Bel  20*10 logU  in deciBel (dB)
Transfer function of RC circuit is complex number
i
U IN
UC  UO
U O  j 
1
 G j  
U IN  j 
1  j
RC circuit as a Low-Pass filter
Filter response has a
Absolute value (Magnitude of complex number) and
Phase (argument of complex number)
U O  j 
1
 G j  
U IN  j 
1  j
Analyze three points:
Very low frequencies
  1
‘Corner’ frequency
  1
Very high frequencies
  1
RC Filter response at very low frequencies
Magnitude
  1  G  j  
Magnitude in dB
1
1
G j  dB  20*10 log1  0dB
Phase (argument)
G j   0 deg
1
G  j  
1  j
RC Filter response at corner frequency
Magnitude
1
  1  G j  
1 j
1
G  j  
1  j
Magnitude in dB
G j  dB
 1 
 20* log
  3dB
 2
10
Phase (argument)
G j   45deg
RC Filter response at very high frequencies
Magnitude
  1  G j  
1
j
1
G  j  
1  j
Magnitude in dB
 1 

  G  j  dB  20 log
 j 
 1 
 1 
10
10
  6dB  20 log

2  G  j  dB  20 log
 2 j 
 j 
 1 
 1 
10
10
  20dB  20 log

10  G  j  dB  20 log
 10 j 
 j 
10
Phase (argument)
G j   90deg
Summary 1st order low pass filter
characteristics
UC  UO
U IN
G  j 
  1
  1
  1
1
1
1
1 j
1
j
G  j  dB
Phase
20*10 log 1  0dB
 1  0 j   0deg
 1 
20*10 log 
  3dB
2


-6 dB / octave or
-20 dB / decade
 1 

  45deg
1

j


 1 

  90deg
 j 
RC circuit as a Low-Pass filter: Bode plot
bode([0 1],[1 1])
Bode Diagram
Magnitude (dB)
0
-10
-20
-30
Phase (deg)
-40
0
-45
-90
-2
10
-1
10
0
10
Frequency (rad/sec)
1
10
2
10
High-pass filter using RC network
High-Pass filter characteristics
UC  UO
U IN
UO 
j RC
R
1
R
jC
UO  j 
U IN  j 
Ui 

1  j RC
 G  j  

j
1  j
Ui
RC circuit as a High-Pass filter
Filter response has a
Absolute value (Magnitude of
complex number) and
Phase (argument of complex
number)
UO  j 
 1 
 G  j    j  

U IN  j 
1

j



G  j  dB
1
 j dB 
1  j
dB
 1 
G  j     j    

1

j



Summary 1st order High Pass filter
characteristics
U IN
  1
  1
  1
UC  UO
G  j 
G  j  dB
Phase
j
1
+6 dB / octave or
+20 dB / decade
j
1 j
 1 
20*10 log 
  3dB
2


20*10 log 1  0dB
1
  j       90  0  90deg
1
 1 
  j    
  90  45  45deg
1

j


j
j
 1 
  j    
  90  90  0deg
j



RC circuit as a High-Pass filter: Bode plot
bode([1 0],[1 1])
Bode Diagram
0
Magnitude (dB)
-10
-20
-30
-40
Phase (deg)
-50
90
45
0
-2
10
-1
10
0
10
Frequency (rad/sec)
1
10
2
10
Band-Pass filter through cascading
Cascade of High-Pass and Low-Pass filters to
obtain a Band-Pass filter
Since the sections are separated
by a buffer: Add absolute values
in dB;s. Add phase angles
G  j  dB  GLOW
dB
 GHIGH
dB
G  j   GLOW  GHIGH
Buffer
U IN
UC  UO
The End