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Recap Filters
ABE425 Engineering
Tony Grift, PhDDept. of Agricultural & Biological EngineeringUniversity of Illinois
Agenda
Recap complex numbersRelationship Laplace, frequency (Fourier) domainRelationship time, s and frequency domainsdecibel notation (dB)RC circuit as a Low-Pass and High-Pass filterBode plotsCombination filters
Complex number in complex plane
jyxess j
s
Argument of sAbsolute value of s (aka Modulus or Magnitude)
Operations on complex numbers cont.
1
2
1 1 1 1
2 2 2 2
j
j
s x jy s e
s x jy s e
2121212121 ** jjj essesesss
2121
2
12121 // jjj e
s
sesesss
Multiplication/divisionusing Euler’s notation
Operations on complex numbers cont.
111111
jj essess
211111
11 ** sesesss jj
Complex conjugate
Multiplying a complex number by its conjugate gives a real number
Relation Laplace and Fourier Transform
sFdtetftfL st
0
s-domain (Laplace Domain)
Time domain
jFdtetftfF tj
-domain (Frequency Domain)j
Time domain
js
Transient response (step, impulse)
Frequency response (filters)
Relation time, s and frequency domain
tUtUtU OOIN
OC UU INU
i
ssG
sU
sU
sUsUssU
IN
O
OOIN
1
1
Time domain
Laplace (s)-domain
jjG
jU
jU
IN
O
1
1
-domainjjs
Concept of impedance (Capacitor)
cCC
CC idt
dUC
dt
dQCUQ
dtiC
U cC 1
CC CUQ
Current
ImpedanceVolt
2 11
jiCj
jUsisC
sU cCFL
cC
Cj
jZ
1
Concept of impedance (Inductor (coil))
dt
diLU L
L
CurrentImpedanceVolt
2 jiLjjUssLisU LLFL
LL
dt
diLU L
L
LjjZ
Low-Pass filter using RC network
Derivation transfer function with impedance
OC UU INU
iiO U
RCjU
CjR
CjU
1
11
1
jjG
jU
jU
IN
O
1
1
Decibel notation
Addition is much simpler than multiplication
Notation in Bel (after Alexander Graham Bell)
For Power
For Voltages (Power ~ Voltage2)
In deciBel (0.1 Bel)
Belin log10 P
log*2 log 10210 UU
(dB) deciBelin log*20Belin log*2 1010 UU
Transfer function of RC circuit is complex number
jjG
jU
jU
IN
O
1
1
OC UU INU
i
RC circuit as a Low-Pass filter
Filter response has a Absolute value (Magnitude of complex number) andPhase (argument of complex number)
Analyze three points:Very low frequencies
‘Corner’ frequency
Very high frequencies
1
jjG
jU
jU
IN
O
1
1
1
1
RC Filter response at very low frequencies
Magnitude
Magnitude in dB
Phase (argument)
dB01log*20 10 dB
jG
j
jG
1
1
deg0 jG
1
11 jG
RC Filter response at corner frequency
Magnitude
Magnitude in dB
Phase (argument)
dB32
1log*20 10
dBjG
j
jG
1
1
deg45 jG
j
jG
1
11
RC Filter response at very high frequencies
Magnitude
Magnitude in dB
Phase (argument)
j
jG
1
1
deg90 jG
j
jG1
1
jjjG
jjjG
jjG
dB
dB
dB
1log20dB20
10
1log2010
1log20dB6
2
1log202
1log20
1010
1010
10
Summary 1st order low pass filter characteristics
G j dB
G j Phase
1 1
1 1020* log 1 0dB 1 0 0degj
1 1
1 j 10 1
20* log 32
dB
1
45deg1 j
1 1
j
-6 dB / octave or -20 dB / decade
190deg
j
OC UU INU
RC circuit as a Low-Pass filter: Bode plot
-40
-30
-20
-10
0
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
-90
-45
0
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
bode([0 1],[1 1])
High-pass filter using RC network
High-Pass filter characteristics
OC UU INU
1 1O i i
j RCR
U U Uj RCR
j C
1O
IN
U j jG j
U j j
RC circuit as a High-Pass filter
Filter response has a Absolute value (Magnitude of complex number) andPhase (argument of complex number)
1
1G j j
j
1
1O
IN
U jG j j
U j j
1
1dBdBdB
G j jj
Summary 1st order High Pass filter characteristics
G j dB
G j Phase
1
1
j
+6 dB / octave or +20 dB / decade 1
90 0 90deg1
j
1
1
j
j 10 1
20* log 32
dB
190 45 45deg
1j
j
1 j
j
1020* log 1 0dB 1
90 90 0degjj
OC UU INU
RC circuit as a High-Pass filter: Bode plot
-50
-40
-30
-20
-10
0
Mag
nitu
de (
dB)
10-2
10-1
100
101
102
0
45
90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
bode([1 0],[1 1])
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
INU OC UU
Buffer
LOW HIGHdB dBdB
LOW HIGH
G j G G
G j G G
The End
