Frequency Response in Passive Filters
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Passive Filters
Chapter 14
Frequency Response
As a frequency-selective device, a filter can be used to limit the
frequency spectrum of a signal to some specified band of
frequencies.
A filter is a
passive filter
if it consists of only passive elements
R
,
L
,
and
C
.
A filter is an
active filter
if it consists of active elements (such as
transistors and op amps) in addition to passive elements
R
,
L
,
C
.
Passive Filters
Chapter 14
Frequency Response
There are four types of filters based
on the frequencies they pass or
reject:
1.
Lowpass filter
, passes low
frequencies and rejects high
frequencies.
2.
Highpass filter
, passes high
frequencies and rejects low
frequencies.
3.
Bandpass filter
, passes
frequencies within a frequency
band and blocks frequencies
outside the band.
4.
Bandstop filter
, passes
frequencies outside a frequency
band and blocks frequencies
within the band.
2.
3.
4.
1.
Passive Filters
Chapter 14
Frequency Response
Passive Filters
Chapter 14
Frequency Response
A typical lowpass filter is formed when
the output of an
RC
circuit is taken off
the capacitor.
The transfer function is
Note that
The half-power frequency, or the cutoff
frequency
ω
c
, is obtained by setting the
magnitude of
H
(
ω
) equal to 1/√2.
Passive Filters
Chapter 14
Frequency Response
A lowpass filter can also be formed when the output of an
RL
circuit is taken off the resistor.
Of course, there are many other circuits for lowpass filter.
●
Find the transfer
function of lowpass
RL
filter.
The cutoff frequency
ω
c
can be found by
through
Passive Filters
Chapter 14
Frequency Response
A highpass filter is formed when the
output of an
RC
circuit is taken off the
resistor.
The transfer function is
Note that
The cutoff frequency is
A highpass filter can also be formed when
the output of an
RL
circuit is taken off the
inductor.
●
Find the transfer
function of highpass
RL
filter.
Passive Filters
Chapter 14
Frequency Response
A bandpass filter is formed when the
output of an
RLC
circuit is taken off the
resistor.
The transfer function is
Note that
The bandpass filter passes a
band frequencies
ω
1
<
ω
<
ω
2
.
ω
0
is the center frequency,
which is equal to the resonant
frequency.
●
What is resonant
frequency?
Passive Filters
Chapter 14
Frequency Response
Now we derive how to find the cutoff
frequencies
ω
1
and
ω
2
from the transfer
function of
At cutoff frequency, the magnitude of the
transfer function is 1/√2.
Passive Filters
Chapter 14
Frequency Response
●
We solve 2 quadratic equations, each
with 2 solutions.
●
There are in total 4 solutions of
ω
but
2 of them are negative and not
possible solution (not physical).
Passive Filters
●
Bandwidth
●
Resonant frequency
●
High cutoff frequency
●
Low cutoff frequency
Chapter 14
Frequency Response
Passive Filters
Chapter 14
Frequency Response
A bandpass filter can also be formed by cascading
a lowpass filter with
ω
2
=
ω
c
with a highpass filter
ω
1
=
ω
c
.
●
Write the transfer function
of this kind of bandpass
filter.
Passive Filters
Chapter 14
Frequency Response
A filter that prevents a band of frequencies
between two designated values from passing is
variably known as a
bandstop
,
bandreject
, or
notch
filter
.
A bandstop filter is formed when the output of an
RLC
circuit is taken off the LC series.
The transfer function is
Note that
Passive Filters
Chapter 14
Frequency Response
The bandstop filter passes a band frequencies
ω
<
ω
1
and
ω
>
ω
2
.
ω
0
is the center frequency, which is equal to the
resonant frequency.
The cutoff frequencies are
The resonant frequency and the cutoff frequencies for bandpass
filter and bandstop filter are identical.
Passive Filters
Chapter 14
Frequency Response
Several conclusions regarding passive filters:
1.
The maximum gain of a passive filter is unity (one). To
generate a gain greatre than unity, we should use an active
filter.
2.
There are other ways to get the types of filters discussed until
now, while only using passive elements of
R
,
L
, and
C
.
3.
The filters discussed until now are the simple types. Many
other filters have sharper and complex frequency response.
Example
1
Chapter 14
Frequency Response
Example
1
Chapter 14
Frequency Response
Example
1
Chapter 14
Frequency Response
Practice Problem
Chapter 14
Frequency Response
Example
2
Chapter 14
Frequency Response
●
What is
ω
1
and
ω
2
?
Practice Problem
Chapter 14
Frequency Response
1.
(
05.RBY+11.AMS−9
)
Homework 8
Chapter 1
4
F
requency Response
2.
(
Oct.05.20.23
)
Deadline: Wednesday, 10 April 2019.
Frequency response in passive filters plays a crucial role in signal processing by selectively allowing or blocking specific frequencies. This article explores the concept of passive filters, including lowpass, highpass, bandpass, and bandstop filters, their characteristics, and how they can be implemented using RL and RC circuits. Different transfer functions and cutoff frequencies are discussed, shedding light on designing and analyzing frequency-selective circuits.
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Presentation Transcript
Linear Circuit Analysis 2 Lecture 8 Dr.-Ing. Erwin Sitompul President University http://zitompul.wordpress.com 2 0 1 9 President University Erwin Sitompul LCA2 7/1
Chapter 14 Frequency Response Passive Filters As a frequency-selective device, a filter can be used to limit the frequency spectrum of a signal to some specified band of frequencies. A filter is a passive filter if it consists of only passive elements R, L, and C. A filter is an active filter if it consists of active elements (such as transistors and op amps) in addition to passive elements R, L, C. President University Erwin Sitompul LCA2 8/2
Chapter 14 Frequency Response Passive Filters There are four types of filters based on the frequencies they pass or reject: 1. Lowpass filter, passes low frequencies and rejects high frequencies. 2. Highpass filter, passes high frequencies and rejects low frequencies. 3. Bandpass filter, passes frequencies within a frequency band and blocks frequencies outside the band. 4. Bandstop filter, passes frequencies outside a frequency band and blocks frequencies within the band. 1. 2. 3. 4. President University Erwin Sitompul LCA2 8/3
Chapter 14 Frequency Response Passive Filters President University Erwin Sitompul LCA2 8/4
Chapter 14 Frequency Response Passive Filters A typical lowpass filter is formed when the output of an RC circuit is taken off the capacitor. The transfer function is Note that The half-power frequency, or the cutoff frequency c, is obtained by setting the magnitude of H( ) equal to 1/ 2. President University Erwin Sitompul LCA2 8/5
Chapter 14 Frequency Response Passive Filters The cutoff frequency ccan be found by through A lowpass filter can also be formed when the output of an RL circuit is taken off the resistor. Of course, there are many other circuits for lowpass filter. Find the transfer function of lowpass RL filter. President University Erwin Sitompul LCA2 8/6
Chapter 14 Frequency Response Passive Filters A highpass filter is formed when the output of an RC circuit is taken off the resistor. The transfer function is Note that The cutoff frequency is A highpass filter can also be formed when the output of an RL circuit is taken off the inductor. Find the transfer function of highpass RL filter. President University Erwin Sitompul LCA2 8/7
Chapter 14 Frequency Response Passive Filters A bandpass filter is formed when the output of an RLC circuit is taken off the resistor. The transfer function is Note that The bandpass filter passes a band frequencies 1< < 2. 0is the center frequency, which is equal to the resonant frequency. What is resonant frequency? President University Erwin Sitompul LCA2 8/8
Chapter 14 Frequency Response Passive Filters Now we derive how to find the cutoff frequencies 1 and 2from the transfer function of R L ( ) = H + ( 1 ) R j C At cutoff frequency, the magnitude of the transfer function is 1/ 2. 1 R L = + ( 1 ) R j C 2 + = ( 1 ) 2 R j L C R 2 ( 1 R ) L C = 1 + = 2 2 ( 1 ) 2 R L C R 2 = 2 2 ( 1 ) L C R 2 ( 1 R ) L C + = 2 2 2 R R R 2 = ( 1 ) R L C 2 ( 1 R ) L C + = 1 2 R R 2 President University Erwin Sitompul LCA2 8/9
Chapter 14 Frequency Response Passive Filters = = + ( 1 ) R L C ( 1 ) R L C + = = 2 2 1 0 1 0 LC RC LC RC + + 2 2 2 2 4 4 RC R C LC R LC RC R C LC R L LC = = 2 2 2 2 1 R L 1 R L = + = + 2 2 LC 2 2 L LC 2 2 1 R L R L 1 R L R L = + + = + + 2 1 2 2 LC 2 2 LC We solve 2 quadratic equations, each with 2 solutions. There are in total 4 solutions of but 2 of them are negative and not possible solution (not physical). President University Erwin Sitompul LCA2 8/10
Chapter 14 Frequency Response Passive Filters 2 1 R L R L = + + 1 2 2 LC Low cutoff frequency 2 1 R L R L = + + 2 2 2 LC High cutoff frequency R L = 2 1 Bandwidth Resonant frequency President University Erwin Sitompul LCA2 8/11
Chapter 14 Frequency Response Passive Filters A bandpass filter can also be formed by cascading a lowpass filter with 2= cwith a highpass filter 1= c. Write the transfer function of this kind of bandpass filter. President University Erwin Sitompul LCA2 8/12
Chapter 14 Frequency Response Passive Filters A filter that prevents a band of frequencies between two designated values from passing is variably known as a bandstop, bandreject, or notch filter. A bandstop filter is formed when the output of an RLC circuit is taken off the LC series. The transfer function is Note that President University Erwin Sitompul LCA2 8/13
Chapter 14 Frequency Response Passive Filters The bandstop filter passes a band frequencies < 1and > 2. 0is the center frequency, which is equal to the resonant frequency. The cutoff frequencies are The resonant frequency and the cutoff frequencies for bandpass filter and bandstop filter are identical. President University Erwin Sitompul LCA2 8/14
Chapter 14 Frequency Response Passive Filters Several conclusions regarding passive filters: 1. The maximum gain of a passive filter is unity (one). To generate a gain greatre than unity, we should use an active filter. 2. There are other ways to get the types of filters discussed until now, while only using passive elements of R, L, and C. 3. The filters discussed until now are the simple types. Many other filters have sharper and complex frequency response. President University Erwin Sitompul LCA2 8/15
Chapter 14 Frequency Response Example 1 President University Erwin Sitompul LCA2 8/16
Chapter 14 Frequency Response Example 1 President University Erwin Sitompul LCA2 8/17
Chapter 14 Frequency Response Example 1 President University Erwin Sitompul LCA2 8/18
Chapter 14 Frequency Response Practice Problem President University Erwin Sitompul LCA2 8/19
Chapter 14 Frequency Response Example 2 R L 150 L = = Bandwidth 2 1 150 0.2387 H = = 2 100 L 2 100 = = = Resonant frequency 1 LC 0 1 = = 2 200 1 0.2387C C 2 (0.2387)(2 2.653 F 200) = What is 1and 2? President University Erwin Sitompul LCA2 8/20
Chapter 14 Frequency Response Practice Problem President University Erwin Sitompul LCA2 8/21
Chapter 14 Frequency Response Homework 8 1. (05.RBY+11.AMS 9) 2. (Oct.05.20.23) Deadline: Wednesday, 10 April 2019. President University Erwin Sitompul LCA2 8/22
