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TOPIC 4: FREQUENCY
SELECTIVE CIRCUITS
1
INTRODUCTION
•Transfer Function
•Frequency Selective Circuits
2
TRANSFER FUNCTION
• The s-domain ratio of the Laplace
transform of the output (response) to the
Laplace transform of the input (source)
when all initial conditions are zero.
• The transfer function depends on what is
defined as the output signal.
3
DEFINITION
Y (s)
H (s) 
X ( s)
all ICs  0
 H ( j )  H ( j ) 
4
POLES AND ZEROS
• The roots of the denominator polynomial
are called the poles of H(s): the values of
s at which H(s) becomes infinitely large.
• The roots of the numerator polynomial
are called the zeros of H(s): the values of
s at which H(s) becomes zero.
5
FREQUENCY RESPONSE
• The transfer function is a useful tool to
compute the frequency response of a
circuit (i.e. the steady state response to a
varying-frequency sinusoidal source).
• The magnitude and phase of the output
signal depend only on the magnitude and
phase of the transfer function, H(j).
6
FREQUENCY RESPONSE
• Frequency response analysis is used to
analyze the effect of varying source
frequency on circuit voltages and
currents.
• The circuit’s response depends on:
– the types of elements
– the way the elements are connected
– the impedance of the elements
7
FREQUENCY SELECTIVE
CIRCUITS
• Frequency selective circuits is a circuits
that pass to the output only those input
signals that reside in a desired range of
frequencies.
• Can be constructed with the careful
choice of circuit elements, their values,
and their connections.
8
PASSIVE FILTERS
• Passband & Stopband
• Cutoff Frequency
• Bode plot
9
PASSIVE FILTERS
• Frequency selective circuits are also
called filters.
• Filters attenuate, that is weaken or lessen
the effect of any input signals with
frequencies outside a particular
frequency.
• Called passive filters because their
filtering capabilities depend only on the
passive elements (i.e. R,L,C).
10
PASSBAND & STOPBAND
• The signal passed from the input to the
output fall within a band of frequencies
called Passband.
• Frequencies not in a circuit’s passband
are in its Stopband.
• Filters are categorized by the location of
the passband.
11
FREQUENCY RESPONSE
PLOT
• One way of identifying the type of filter
circuit is to examine a frequency
response plot.
• Two parts: one is a graph of H(j)  vs
frequency. Called magnitude plot.
• The other part is a graph of (j) vs
frequency. Called phase plot.
12
TYPE OF FILTERS
•Low Pass Filter
•High Pass Filter
•Band Pass Filter
Band Reject Filter
13
FILTER’S FREQUENCY
RESPONSE
lowpass
highpass
c
c
bandpass
c1
c2
bandreject
c1
c2
14
MAGNITUDE
TYPE
H(0)
H(∞) H(ωC)@H(ωo)
LOWPASS
1
0
1/√2
HIGHPASS
0
1
1/√2
BANDPASS
0
0
1
BANDREJECT
1
1
0
15
CUT OFF FREQUENCY
• LPF and HPF have one passband and
one stopband, which are defined by the
cut off frequency that separates them.
• BPF passes a input signal to the output when
the input frequency is within the band defined
by the two cut off frequencies.
• BRF passes a input signal to the output when
the input frequency is outside the band defined
by the two cut off frequencies.
16
CUT OFF FREQUENCY
• The cutoff frequency (fc) is the frequency
either above which or below which the
power output of a circuit, such as a line,
amplifier, or filter, is reduced to 1/2 of the
passband power; the half-power point.
• This is equivalent to a voltage (or
amplitude) reduction to 70.7% of the
passband, because voltage, V2 is
proportional to power, P.
17
CUT OFF FREQUENCY
• This happens to be close to −3 decibels,
and the cutoff frequency is frequently
referred to as the −3 dB point.
• Also called the knee frequency, due to a
frequency response curve's physical
appearance.
18
CUT OFF FREQUENCY
VO ( jc )  H ( jc ) Vi
1

H max
2
6
1

H max
2
1

Vo max
2
19
BODE PLOT
• The most common way to describe the
frequency response is by so called Bode
plot.
• Bode lot is a log-log plot for amplitude vs
frequency and a linear-log plot for phase
vs frequency.
• Many circuits (e.g. amplifiers, filters,
resonators, etc.) uses Bode plot to
specify their performance and
characteristics.
20
FILTER’S RESPONSE BODE PLOT
21
LOW PASS FILTER
22
LOW PASS FILTER (LPF)
•
The filter preserves low frequencies
while attenuating the frequencies above
the cut off frequencies.
•
There are two basic kinds of circuits that
behave as LPFs:
a) Series RL
b) Series RC.
23
(a) LPF RL CIRCUIT
sL

Vi (s)
Vo (s )
R
OUTPUT

24
Transfer Function
Vo
R
H (s) 

Vi R  sL
25
The voltage transfer function
R
L
H ( s) 
sR
L
• To study the frequency response,
substitute s=j:
R
Vo
L
H ( j )  
Vi
j  ( R )
L
26
Magnitude and Phase
R
L
 H ( j ) 
2
2
R
  ( L)
1 L
  ( j )   tan
R


27
When =0 and =
Vi
R
Vi
R
28
Qualitative Analysis
• At low frequencies (L<< R):
– jL is very small compared to R, and
inductor functions as a short circuit.
V0  Vi
V0  Vi
29
Qualitative Analysis
• At high frequencies (L>> R):
– jL is very large compared to R, and inductor
functions as a open circuit.
V0  0
V0  Vi  90
30
LPF Frequency Response
H(j)
1.0
0 
c

-90 
31
32
Cutoff Frequency
• At the cutoff frequency, voltage
magnitude is equal to (1/2)Hmax :
R
1
L
H (C ) 

2
2
R
2
  ( L)
R
C 
L
33
Ex.
• Electrocardiograph is an instrument that is used
to measure the heart’s rhythmic beat. This
instrument must be capable of detecting
periodic signals whose frequency is about 1 Hz
(the normal heart rate is 72 beats per minute).
• The instrument must operate in the presence of
sinusoidal noise consisting of signals from the
surrounding electrical environment, whose
fundamental frequency is 50 Hz- the frequency
at which electric power is supplied.
34
Ex.
i.
Choose values for R and L in the series
RL circuit such that the resulting circuit
could be used in an electrocardiograph
to filter out any noise above 10 Hz and
pass the electric signals from the heart
at or near 1 Hz. (choose L=100 mH)
ii. Then compute the magnitude of Vo at 1
Hz, 10 Hz, and 50 Hz to see how well
the filter performs.
35
Known quantities
• Inductor, L = 100 mH
• Cut off frequency, fc = 10 Hz
– therefore, c = 2fc = 20 rad/s
36
Find R
R  c L
 (20 )(100 10 )
3
 6.28 
37
Find the magnitude of Vo
• Using the transfer function, the output
voltage can be computed:
Vo  H ( j )  Vi
 Vo ( ) 
R
L
Vi
2
2
R
  ( L)
20

Vi
2
2
  400
38
Vo ( ) 
20
  400
2
2
Vi
f(Hz)
Vi 
Vo 
1
1.0
0.995
10
1.0
0.707
50
1.0
0.196
39
(b) LPF RC CIRCUIT
R

Vi
vo

C OUTPUT
40
Transfer Function
1
Vo
jC
H ( j ) 

Vi R  1
jC
1
RC

j  1
RC
41
Magnitude and Phase of
H(j)
Vo
 H ( j ) 

Vi
1
RC
2
  1 RC
1
  ( j )   tan RC


2
42
• Zero frequency (=0):
– the impedance of the
capacitor is infinite, and
the capacitor acts as an
open circuit.
– Vo and Vi are the same.
• Infinite frequency (=):
– the impedance of the
capacitor is infinite, and the
capacitor acts as an open
circuit.
– Vo is zero.
• Frequency increasing
from zero:
– the impedance of the
capacitor decreases
relative to the
impedance of the
resistor
– the source voltage
divides between the
resistive impedance
and the capacitive
impedance.
– Vo is smaller than Vi.
43
Cutoff Frequency
• The voltage magnitude is equal to (1/2)
Hmax at the cutoff frequency:
1
RC
H (C ) 
2
C  ( 1
RC
1
C 
RC
1

2
2
)
44
GENERAL LPF CIRCUITS
sL
OUTPUT

Vi (s)
Vo (s )
R

c
H (s) 
s  c
R

Vi
vo

C
OUTPUT
45
Ex.
•
For the series RC circuit of LPF:
a) Find the transfer function between the
source voltage and the output voltage
b) Choose values for R and C that will yield a
LPF with cutoff frequency of 3 kHz.
46
a) Find the transfer function
• The magnitude of H(j):
1
Vo
RC
H ( j ) 

Vi
j  1
RC
1
Vo
RC
H ( j ) 

2
Vi
( )  ( 1
RC
)
2
47
b) Find R & C
• R and C cannot be computed
independently, so let’s choose C=1F.
• Convert the specified cutoff frequency
from 3 kHz to c=2(3x10-3) rad/s.
48
Calculate R
1
R
c C
1

3
6
(2 )(3 10 )(110 )
 53.05 
49
HIGH PASS FILTER
50
HIGH PASS FILTER
• HPF offer easy passage of a high
frequency signal and difficult passage to
a low frequency signal.
• Two types of HPF:
– RC circuit
– RL circuit
51
a) CAPACITIVE HPF
C
Vi

vo

R
OUTPUT
52
s-Domain Circuit
1
Vi (s)
sC

Vo (s )
R

53
When =0 and =

Vi
vo
R

Vi
R
54
• Zero frequency (=0):
– the capacitor acts as an
open circuit, so there is
no current flowing in R.
– Vo is zero.
• Infinite frequency (=):
– the capacitor acts as an
short circuit and thus there
is no voltage across the
capacitor.
– Vo is equal to Vi.
• Frequency increasing
from zero:
– the impedance of the
capacitor decreases
relative to the
impedance of the
resistor
– the source voltage
divides between the
resistive impedance
and the capacitive
impedance.
– Vo begins to increase.
55
HPF Frequency Response
56
Transfer Function
s
H (s) 
s 1
RC
s  j ;
j
H ( j ) 
j  1
RC
57
Magnitude & Phase
 H ( s) 

 (1
2
RC
)
2
  ( j )  90  tan RC
1
58
Ex: INDUCTIVE HPF
• Show that the series RL circuit below also
acts as a HPF.
R
Vi

vo
L

59
Ex.
a) Derive an expression for the circuit’s
transfer function
b) Use the result from (a) to determine an
equation for the cutoff frequency
c) Choose values for R and L that will yield
HPF with fc = 15 kHz.
60
s-Domain circuit
R
Vi (s)

Vo (s )
sL

61
Transfer Function
s
H (s) 
sR
s  j ;
L
j
H ( j ) 
j  R
L
62
Magnitude & Phase
 H ( j ) 

2
R
  ( L)
2
• H()=1 andH(0)=0  HPF
63
Cutoff Frequency
H ( j ) 

2
R
  ( L)
2
c
1
 H (c ) 

2
2
R
2
c  ( L )
 c  R
L
64
R and L
• Choose R=500 , and convert fc to c:
LR
c
500


5
.
31
mH
3
(2 )(15 10 )
65
GENERAL HPF CIRCUITS
1
sC
OUTPUT

Vi (s)
Vo (s )
R

s
H (s) 
s  c
R
Vi (s)

Vo (s )
sL
OUTPUT

66
REMARKS
• The components and connections for LPF
and HPF are identical but, the choice of
output is different.
• The filtering characteristics of a circuit
depend on the definition of the output as
well as circuit components, values, and
connections.
• The cutoff frequency is similar whether
the circuit is configured as LPF or HPF.
67
BANDPASS FILTER
68
Bandpass Filter
• BPF is essential for applications where a
particular band or frequencies need to be
filtered from a wider range of mixed
signals.
• There are 3 important parameters that
characterize a BPF, only two of them can
be specified independently:
– Center frequency (and two cutoff frequencies)
– Bandwidth
– Quality factor
69
Cutoff freq. & Center freq.
• Ideal bandpass filters have two cutoff
frequencies, c1 and c2, which identify
the passband.
• c1 and c2 are the frequencies for which
the magnitude of H(j) equal (1/2).
• The center frequency, o is defined as the
frequency for which a circuit’s transfer
function is purely real.
• Also called as the resonant frequency.
70
Bandwidth,  and Quality
Factor, Q
• The bandwidth,  tells the width of the
passband.
• The quality factor, Q is the ratio of the
center frequency to the bandwidth.
• The quality factor describes the shape of
the magnitude plot, independent of
frequency.
71
a) BPF: Series RLC
L
vi
C

vo

R
72
At =0 and =
L
C

vi
vo
R

L
vi
C

vo

R
73
At =0 and =
• Zero frequency
(=0):
– the capacitor acts as
an open circuit and
the inductor behaves
like a short circuit, so
there is no current
flowing in R.
– Vo is zero.
• Infinite frequency
(=):
– the capacitor acts as
an short circuit and
the inductor behaves
like an open circuit, so
again there is no
current flowing in R.
– Vo is zero
74
Between =0 and =
• Both capacitor and inductor have finite
impedances.
• Voltage supplied by the source will drop
across both L and C, but some voltage
will reach R.
• Note that the impedance of C is negative,
whereas the impedance of L is positive.
75
• At some frequency, the impedance of C
and the impedance of L have equal
magnitudes and opposite sign cancel
out!
• Causing Vo to equal Vi
• This is happen at a special frequency,
called the center frequency, o.
76
BPF Frequency Response
77
Center Frequency
H max  H ( jo )
78
s-Domain Circuit
sL
Vi (s)
1
sC

Vo (s )

R
79
Transfer Function
( R )s
L
H (s)  2
s  ( R )s  ( 1 )
L
LC
s  j;
j ( R )
L
H ( j ) 
2
  ( R ) j  ( 1 )
L
LC
80
Magnitude & Phase
H ( j ) 
 ( R L)
 2  ( 1
)  ( R ) j
LC
L
j ( R )
L
 H ( j ) 
2 2
2
1
R
[(
)   ]  [( ) ]
LC
L
 (R ) 
1
L 
  ( j )  90  tan 
( 1 ) 2 
 LC

81
Center Frequency
• For circuit’s transfer function is purely
real:
j o L  1
 o 
j o C
0
1
LC
82
Cutoff Frequencies
c1
R
 

2L
c 2
R


2L
 R 


 2L 
 R 


 2L 
2

2

1
LC
1
LC
83
Relationship Between
Center Frequency and
Cutoff Frequencies
o  c1  c 2
84
Bandwidth, 
  c1  c 2
R

L
85
Quality Factor, Q
o
Q

o L
1
Q


R
oCR
L
2
RC
86
Cutoff Frequencies in terms
of 

 
2
c1       o
2
2

2
 
2
c 2      o
2
2
2
87
b) BPF: Parallel RLC
R

vi
C
vo
L

88
GENERAL BPF CIRCUITS
• BPF series RLC:
( R )s
L
H ( s)  2
s  ( R )s  ( 1 )
L
LC
 o  1 ,   R
LC
L
• BPF parallel RLC:
(s
)
RC
H (s)  2
s s
(1 )
RC
LC
 o  1 ,   1
LC
RC
89
Remarks
• The general circuit transfer functions for
both series and parallel BPF:
s
H ( s)  2
2
s   s  o
90
BANDREJECT FILTER
91
Bandreject Filter
• A bandreject attenuates voltages at
frequencies within the stopband, which is
between c1 and c2. It passes
frequencies outside the stopband.
• BRF are characterized by the same
parameters as BPF:
– Center freq. (and two cutoff frequencies)
– Bandwidth
– Quality Factor
92
BRF: Series RLC
R

vi
L
vo
OUTPUT
C

93
At =0 and =
R

vi
L
vo
C

R

vi
L
vo
C

94
At =0 and =
• Zero frequency
(=0):
– the capacitor acts as
an open circuit and
the inductor behaves
like a short circuit.
– the output voltage is
defined over an
effective open circuit.
– Magnitude of Vo and
Vi are similar
• Infinite frequency
(=):
– the capacitor acts as
an short circuit and
the inductor behaves
like an open circuit,.
– the output voltage is
defined over an
effective open circuit.
– Magnitude of Vo and
Vi are similar
95
Between =0 and =
• Both capacitor and inductor have finite
impedances of opposite signs.
• As the frequency is increased from zero, the
impedance of the inductor increases and that
for the capacitor decreases.
• At some frequency between the two
passbands, the impedance of C and L are
equal but opposite sign.
• The series combination of L and C is that short
circuit so the magnitude of Vo must be zero.
This is happen at a special frequency, called
the center frequency, o.
96
BRF Frequency Response
97
Center Frequency
• The center freq. is still defined as the
frequency for which the sum of the
impedances of L and C is zero.
• Only, the magnitude at the center freq. is
minimum.
H min  H ( jo )
98
s-Domain Circuit
R

Vi (s)
sL
Vo (s )
1

sC
99
Transfer Function
sL  1
s 1
2
sC
LC
H ( s) 

R
2
R  sL  1
1
s

s

sC
LC
L
s  j ;
H ( j ) 
1
LC

2
R
1
   j
LC
L
2
100
Magnitude & Phase
H ( j ) 
1

1
LC
LC

2


2
2
 R 


 L 
2
 R

1
L


  ( j )   tan
2
1
(
)  
 LC

101
Center Frequency
• For circuit’s transfer function is purely
real:
j o L  1
 o 
j o C
0
1
LC
102
Cutoff Frequencies
2
R
1
 R 
c1  
   
2L
 2 L  LC
2
R
1
 R 
c 2 
   
2L
 2 L  LC
103
Relationship Between
Center Frequency and
Cutoff Frequencies
o  c1  c 2
104
Bandwidth, 
  c1  c 2
R

L
105
Quality Factor, Q
o
Q

o L
1
Q


R
oCR
L
2
RC
106
Cutoff Frequencies in terms
of 

 
2
c1       o
2
2

2
 
2
c 2      o
2
2
2
107
Remarks
H (s) 
R

sL
Vi (s)
Vo (s )
1

sC
Vi (s)
1
 o 
R
Vo (s )

1
LC
L
s2  1
LC
s
2
s 
1
LC
RC
 o  1
LC
  1
RC
H ( s) 

sC
LC
R
2
s  s 1
LC
L
 R
sL
s2  1
108
GENERAL BRF CIRCUIT
• The general circuit transfer functions for
both series and parallel BRF:
s 
H ( s)  2
2
s   s  o
2
2
o
109