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Filters
 Filter is a frequency-selective circuit that passes a specified band of
Chapter 4: Active filters frequency and blocks or attenuates signals of frequency outside this band.
 Passive filter: employ only passive elements such as capacitors,
inductors and resistors.
 Active filter: in addition to resistors and capacitors, it make use of
active devices like transistor and op-amps.©
By: Behailu T.  The main disadvantage of passive filter:
a) the amplitude of the output signal is less than that of the input signal,
i.e., the gain is never greater than unity,
b) the load impedance affects the filters characteristics

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Review: Decibel (dB) Review: dBm

 The decibel is a logarithmic measurement of the ratio of one power to another or  The dBm is a unit for measuring power
one voltage to another. levels referenced to 1 mW.
 Power gain is expressed in decibels (dB) by:  Positive dBm values represent power
Ap(dB) = 10 log Ap where Ap is the actual power gain, Pout / Pin levels above 1 mW, and negative dBm
values represent power levels below 1 mW.
 Voltage gain is expressed in decibels by :
 Because the decibel (dB) can be used to
Av (dB) = 20 log Av where Av is the actual voltage gain, Vout / Vin represent only power ratios, not actual
If Av > 1, the dB gain is positive. power, the dBm provides a convenient way
If Av < 1, the dB gain is negative and is usually called attenuation. to express actual power output of an
amplifier or other device.

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Review: Frequency Response Review: Frequency Response

0 dB Reference
 In amplifiers, the coupling and bypass Mid-band  It is often convenient in amplifiers to assign a certain
capacitors appear to be shorts to ac at the value of gain as the 0 dB reference. This does not
mid-band frequencies. mean that the actual voltage gain is 1 (which is 0
Pass-band
dB); it means that the reference gain, no matter
 At low frequencies the capacitive reactance of stop-band
what its actual value, is used as a reference with
these capacitors affect the gain and phase Low-frequency magnitude stop-band stop-band
which to compare other values of gain and is
shift of signals, so they must be taken into Bode plots
therefore assigned a 0 dB value.
account.  Many amplifiers exhibit a maximum gain over a
 The frequency response of an amplifier is the Band-pass response curve certain range of frequencies and a reduced gain at
change in gain or phase shift over a specified frequencies below and above this range.
Mid-band
range of input signal frequencies.  The maximum gain occurs for the range of
stop-band  The term normalized means that the frequencies between the upper and lower critical
 Usually, expressed in logarithmic scale called midrange voltage gain is assigned a value frequencies and is called the midrange gain, which
Bode plot. of 1 or 0 dB. is assigned a 0 dB value.
High-frequency response curve  Any value of gain below midrange can be referenced
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Review: Frequency Response Review: Frequency Response: Critical Frequency

 For example, if the midrange voltage gain of a  A critical frequency (also known as cutoff frequency or corner frequency) is a
certain amplifier is 100 and the gain at a certain frequency at which the output power drops to one-half of its midrange value.
frequency below midrange is 50, then this reduced  This corresponds to a 3 dB reduction in the power gain, as expressed in dB by
voltage gain can be expressed as: the following formula:
= 20 50
100 = −6 = 10log 0.5 = −3dB
( )

 This indicates that it is 6dB below the 0 dB

reference.  Also, at the critical frequency the voltage gain is 70.7% of its midrange value and
is expressed in dB as:
 Halving the output voltage for a steady input
voltage is always a 6dB reduction in the gain. ( ) = 20log 0.707 = −3dB
 Correspondingly, a doubling of the output Decibel values corresponding to doubling
and halving of the voltage gain.
voltage is always a 6dB increase in the gain.

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Active Filter: Advantages Active Filter: Advantages

1) Flexibility of gain and frequency adjustment: since op-amps can provide a 4) Digital integration: analog filters and digital circuitry can be implemented
voltage gain, the input is not attenuated. Filter amplification be used to either on the same IC chip
shape or alter the frequency response of the filter circuit by producing a more 5) Filtering functions: active filters can be used to realize a wide range of
selective output response. The maximum frequency response of an active filtering functions than passive filters
filter is limited to the Gain-Bandwidth product (or open loop gain) of op-amp. 6) Provision of gain: active filter can provides gain, whereas a passive filters
2) No loading effect: because of the high input resistance of op-amp, active often exhibits a significant loss.
filters do not cause loading of the input source
3) Reduction of cost and size: active filters are less expensive than passive filters
because of the availability low-cost op-amp and the absence of inductors.

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Active Filter: Disadvantages Active Filters

Types:
1. Provides limited bandwidth: active components have a finite BW which
limits the application of active filters to the audio-frequency range, passive 1) Low Pass Filter (LPF)
filters do not have such an upper-frequency limitation and can be used up to 2) High Pass Filter (HPF)
approximately 500MHz. 3) Band Pass Filter (BPF)
2. Sensitive to component drift: due to manufacturing tolerances or/and 4) Band Stop Filter (BSF)
environmental changes; in contrast passive filters are less affected by such
factors Applications:
3. Power supply requirement: active filters require power supplies whereas  Televisions
passive filters do not.  Telephone
4. Distortion: active filters can handle only a limited range of signals  Biomedical Equipment
magnitudes; beyond this range they introduce unacceptable distortion Fig.4.0. General diagram of an active filter.
 Telecommunication systems
 Radar system
5. Noise: active filters use resistors and active elements which produce  Space satellite
electrical noise.
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1. Active Low Pass Filter 1. Active Low Pass Filter

 If a voltage gain greater than one is required we can use the following non-
 The most common and easily inverting op-amp config.
understood active filter is the Active
Low Pass Filter.
 Its principle of operation and frequency
response is exactly the same as those of
passive filter, the only difference is it Gain,
uses an op-amp for amplification and
Fig 4.2. First Order Low pass
gain control. Filter with higher gain
 The simplest form of a low pass active The transfer function of a first–order low pass filters is given as:
filter is shown in fig 4.1

Fig 4.1. First Order Low pass Filter &

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14

1. Active Low Pass Filter: Analysis 1. Active Low Pass Filter: Analysis
Using VDR at point x: 1
(s) Which gives the cutoff frequency fc at 3-dB is
( )= (s) = (s) =
+ +1 1+

The output voltage of the no-inverting amplifier is:

The magnitude and phase angle of the filter gain is
(s)
( )= 1+ ( )= 1+
1+
Which gives the voltage transfer function H(s) as
( )
H(s) = = , substituting s = jω
( )
( )
At very low frequencies, ƒ < ƒc K
( )
H(jω) = ( )
= where DC gain k = 1 + Fig 4.2. First Order Low pass Filter At the cut-off frequency, ƒ = ƒc √2
with higher gain
At very high frequencies, ƒ > ƒc
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Example-1: LPF Design Solution …

Design a non-inverting active low pass filter circuit that has a gain of ten at low Thus the final low pass filter circuit along with its frequency response is given
frequencies, a high frequency cut-off frequency of 159Hz and an input impedance of below as:
10kΩ.
20 log 10 = 20
Solution: Given parameters: k = 10, = 159 , = = 10kΩ
For LPF we know the DC-gain is given by:

=1+ = 10 , Lets assume =1 Ω→ =9 Ω

From formula, we can find a value for

Low Pass Filter Circuit Frequency Response Curve

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1. Active Low Pass Filter: 2nd order 2. Active High Pass Filter
 As with the passive filter, a first-order low-pass active filter can be converted  An Active High Pass Filter can be created by
into a second-order low pass filter simply by using an additional RC network in combining a passive RC filter network with
the input path. an operational amplifier to produce a high
pass filter with amplification
 The frequency response of the second-order low pass filter is identical to that of
the first-order type except that the stop band roll-off will be twice the first-order  Unlike Passive High Pass Filters which have
filters at 40dB/decade (12dB/octave). Therefore, the design steps required of the an “infinite” frequency response, the
second-order active low pass filter are the same. maximum pass band frequency response of
an active high pass filter is limited by the
open-loop characteristics or bandwidth of
the operational amplifier being used, making
them appear as if they are band pass filters
When cascading together filter circuits to form higher-order with a high frequency cut-off determined by
filters, the overall gain of the filter is equal to the product of
the selection of op-amp and gain.
each stage.
Fig.4.4 Active High Pass Filter circuit with
Chapter 4: Active Filters © ECE, AASTU, 2024 Fig 4.3. Second Order Low pass Filter 19 Chapter 4: Active Filters © ECE, AASTU, 2024
frequency response curve 20
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2. Active High Pass Filter 2. Active High Pass Filter

 A first-order (single-pole) Active High Pass  The transfer function of a first–order high pass filter is found by:
Filter as its name implies, attenuates low
frequencies and passes high frequency
signals. It consists simply of a passive filter
section followed by a non-inverting
operational amplifier.  The magnitude and phase angle of the high pass filter is
 The frequency response of the circuit is the
same as that of the passive filter, except and
that the amplitude of the signal is increased
by the gain of the amplifier and for a non-  At very low frequencies, ƒ < ƒc
inverting amplifier the value of the pass
band voltage gain is given as 1 + R2/R1, the  At the cut off -frequency fc = f
same as for the low pass filter circuit
 At Very high Frequency ƒ > ƒc
Fig.4.4 Active High Pass Filter circuit with
Chapter 4: Active Filters © ECE, AASTU, 2024 frequency response curve 21 Chapter 4: Active Filters © ECE, AASTU, 2024 22

2. Active High Pass Filter 2. Active High Pass Filter

 For a first-order filter the frequency response curve of the filter increases by
20dB/decade or 6dB/octave up to the determined cut-off frequency point which is  First order active High pass filter using inverting operational amplifier circuit is
always at -3dB below the maximum gain value. shown in fig below.
 As with the previous filter circuits, the lower cut-off or corner frequency (ƒc) can
be found by using the same formula:
where = =
on the above circuit layout

 The corresponding phase angle or phase shift of the output signal is the same as
that given for the passive RC filter and leads that of the input signal. It is equal
to +45o at the cut-off frequency ƒc value and is given as:
First order active high pass filter
Frequency Response Curve

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2. Active High Pass Filter: Application Example-2: HPF Design

A first order active high pass filter has a pass band gain of two and a cut-off corner
 Applications of Active High Pass Filters are in frequency of 1kHz. If the input capacitor has a value of 10nF, calculate the value of
 audio amplifiers,
the cut-off frequency determining resistor and the gain resistors in the feedback
network. Also, plot the expected frequency response of the filter.
 equalizers or speaker systems to direct the high frequency
Solution:
signals to the smaller tweeter speakers or to reduce any
With a cut-off corner frequency given as 1kHz and a capacitor of 10nF, the value
low frequency noise or “rumble” type distortion. of R will therefore be:

=1+ = 2, ↔ = choose appropriate value of resistor

The data for the frequency response bode plot can be obtained by substituting the
values obtained above over a frequency range from 100Hz to 100kHz into the
equation for voltage gain:

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Solution: 3. Active Band Pass Filter

Frequency, ƒ Voltage Gain
( Hz )
Gain, (dB)
( Vo / Vin ) 20log( Vo / Vin )
 The principal characteristic of a Band Pass Filter or any filter for that matter, is its
 For different value of f, the corresponding ability to pass frequencies relatively unattenuated over a specified band or
100 0.20 -14.02
voltage gain is tabulated in table on the spread of frequencies called the “Pass Band”.
right. 200 0.39 -8.13
 Active Band Pass Filter is a frequency selective filter circuit used in electronic
 Bode plot can be drawn as follows: 500 0.89 -0.97
systems to separate a signal at one particular frequency, or a range of signals
800 1.25 1.93
that lie within a certain “band” of frequencies from signals at all other
1,000 1.41 3.01
frequencies.
3,000 1.90 5.56
 This band or range of frequencies is set between two cut-off or corner frequency
5,000 1.96 5.85
points labeled the “lower frequency” (ƒL) and the “higher frequency” (ƒH) while
10,000 1.99 5.98
attenuating any signals outside of these two points.
50,000 2.00 6.02

100,000 2.00 6.02

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3. Active Band Pass Filter: Frequency Response 3. Active Band Pass Filter
 The normalized frequency  Simple Active Band Pass
response and phase shift Filter can be easily made by
for an active band pass filter cascading together a single Low
is shown as follows. Pass Filter with a single High
Pass Filter as shown.
 The bandwidth of the filter is
therefore the difference
between these upper and lower -
3dB points

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Infinite Gain Multiple Feedback Active Filter Example-3: BPF Design

An active band pass filter that has a voltage gain Av of one and a resonant
 This active band pass filter circuit uses the full gain of the operational amplifier,
frequency, ƒr of 1kHz is constructed using an infinite gain multiple feedback filter
with multiple negative feedback applied via resistor, R2 and capacitor C2.
circuit. Calculate the values of the components required to implement the circuit.
Solution:
Firstly, we can determine the values of the two resistors, R1 and R2 required for the
active filter using the gain of the circuit to find Q as follows.
= 1 = −2 ∴ = - = 0.7071 = 0.7071 = ∴ =2
Let resistor R1 = 10kΩ then R2 = 20kΩ.
The center or resonant frequency is given as 1kHz, assuming that C = C1 = C2. we have
= 10000 = ∴ = =11.2nF

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4. Active Band Stop Filter 4. Active Band Stop Filter: Frequency Response
 A band Stop Filter known also as a Notch Filter, blocks/rejects frequencies that lie
between its two cut-off frequency points passes all those frequencies either side  The frequency response
of this range. curve of an ideal band
stop filter is therefore
 Combine low and high pass filter sections to produce another kind of RC filter given as:
network called a band stop filter that can block or at least severely attenuate a
band of frequencies within these two cut-off frequency points.
 The band stop filter, also known as a band reject filter, passes all frequencies with
the exception of those within a specified stop band which are greatly attenuated.
If this stop band is very narrow and highly attenuated over a few hertz, then the
band stop filter is more commonly referred to as a notch filter

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4. Active Band Stop Filter: Configuration 4. Active Band Stop Filter: Circuit configuration
The summing of the high  The transformation of this filter
pass and low pass filters characteristic can be easily
means that their frequency implemented using a single low pass
responses do not overlap, and high pass filter circuits isolated
unlike the band-pass filter. from each other by non-inverting
This is due to the fact that voltage follower, (Av = 1).
their start and ending  The output from these two filter
frequencies are at different circuits is then summed using a third
frequency points operational amplifier connected as a
voltage adder as shown.
Fig. Band Stop Filter Circuit
 The two non-inverting voltage followers can easily be converted into a basic non-inverting
amplifier with a gain of Av = 1 + Rƒ/Rin by the addition of input and feedback resistors.

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Example-4: BSF Design Notch Filter

Design a basic wide-band RC band stop filter with a lower cut-off frequency of  Notch filters are a highly selective, high-Q, form of the band stop filter which can
200Hz and a higher cut-off frequency of 800Hz. Find the geometric center be used to reject a single or very small band of frequencies rather than a whole
frequency, -3dB bandwidth and Q of the circuit. bandwidth of different frequencies. For example, it may be necessary to reject or
Solution: attenuate a specific frequency generating electrical noise .
Using both the low and high pass filters cut-off frequency formula and assuming a  Notch filters by design have a very narrow and very deep stop band around their
capacitor, C value for both filter sections of 0.1uF, we have center frequency with the width of the notch being described by its
1 selectivity Q in exactly the same way as resonance frequency peaks in RLC circuits.
= = 200 = 0.1 , = = 7958Ω 8 Ω
2 × 200 × 0.1 × 10  The most common notch filter design is the twin-T notch filter network
1
= = 800 = 0.1 = = 1990 2 Ω
2 × 800 × 0.1 × 10

= =400Hz and = − = 600 = = = 0.67

×
Fig .Basic Twin-T Notch Filter Design

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Notch Filter: Single Op-amp Twin-T

 The output from the twin-T notch
filter section is isolated from the
voltage divider by a single non-
inverting op-amp buffer.
 The output from the voltage divider is
fed back to “ground” point
of R and 2C.
 The amount of signal feedback,
known as the feedback fraction k, is
set by the resistor ratio and is given
as:

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