Electronics
ECE 211
Fall 2023
Lecture 13
Filters
Mohamed ELsheikh
Ain Shams University
ICL
References
• OPAMPS for Everyone, Ch:16-Active Filter Design Techniques,
Texas Instruments.
2 2
Outline
• What is a filter ?
• Types of filters
• Passive Filter Implementations
• Higher Order Filters
• Filters Transformations
3
What is a filter ?
• A filter is a circuit capable of passing (or amplifying) certain
frequencies while attenuating other frequencies.
• Thus, a filter can extract important frequencies from signals that also
contain undesirable or irrelevant frequencies
• Typical Applications:
o Radio communications
o DC power supplies
o Audio electronics
o Analog-to-digital conversion
4
Electronic Filters (discrete & integrated)
 Filters are special circuits which perform signal processing functions
like to remove (filter) unwanted frequency components from the input
signal, to enhance desired ones, or both.
 This can be done in analog or digital domain. Now, we are concerned
with analog filter as the signal is still analog amplified by preceding
amplifier.
Realization:
Function:
1. Passive:
1. Low-pass
- RC
2. High-pass
- LC
3. Bandpass
- Ceramic
4. Bandstop (notch)
- SAW
5. All-pass (equalizer)
2. Active:
- RC opamp
- Others……
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Simple RLC Passive Filter & Symbols
LPF
HPF
BPF
BSF
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Filters Parameters
• Response curves are used to describe how a filter behaves. A response
curve is simply a graph showing an attenuation ratio (VOUT / VIN) versus
frequency.
• Attenuation is commonly expressed in units of decibels (dB).
• Important parameters:
o 3dB Frequency (f3dB). or "minus 3dB frequency“ or cutoff frequency:
• It is the input frequency that causes the output signal to drop by -3dB
relative to the input signal.
• It is the frequency at which the output power is reduced by one-half
(which is why this frequency is also called the "half-power frequency"), or
at which the output voltage is the input voltage multiplied by 1/√2 .
• For low-pass and high-pass filters there is only one -3dB frequency.
However, there are two -3dB frequencies for band-pass and notch filters—
these are normally referred to as f1 and f2.
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Filters Parameters
• Center frequency (f0):
o The center frequency, a term used for band-pass and notch filters, is a
central frequency that lies between the upper and lower cutoff
frequencies.
o The center frequency is commonly defined as either the arithmetic
mean (see equation below) or the geometric mean of the lower cutoff
frequency and the upper cutoff frequency.
• Bandwidth (β or B.W.):
o The bandwidth is the width of the passband, and the passband is the
band of frequencies that do not experience significant attenuation
when moving from the input of the filter to the output of the filter.
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Filters Parameters
Important parameters:
• Stopband frequency (fs):
o This is a particular frequency at which the attenuation reaches a specified value.
o For low-pass and high-pass filters, frequencies beyond the stopband frequency are
referred to as the stopband.
o For band-pass and notch filters, two stopband frequencies exist. The frequencies
between these two stopband frequencies are referred to as the stopband.
•
Quality factor (Q):
o The quality factor of a filter conveys its damping characteristics. In the time domain,
damping corresponds to the amount of oscillation in the system’s step response.
o In the frequency domain, higher Q corresponds to more (positive or negative) peaking in
the system’s magnitude response.
o For a bandpass or notch filter, Q represents the ratio between the center frequency and
the -3dB bandwidth (i.e., the distance between f1 and f2).
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Filters Parameters
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Passive RC Filter Circuits
1 𝑗 𝜔𝐶
1
𝑣in =
𝑣in
𝑅 + 1 𝑗 𝜔𝐶
𝑗𝜔𝑅𝐶 + 1
𝑣out
1
𝐴=
=
𝑣in
1 + 𝜔 2 𝑅2 𝐶 2
𝑣out =
LPF
HPF
𝑅
𝑗𝜔𝑅𝐶
𝑣out =
𝑣in =
𝑣in
𝑅 + 1 𝑗 𝜔𝐶
𝑗𝜔𝑅𝐶 + 1
𝑣out
𝜔2 𝑅2 𝐶 2
𝐴=
=
𝑣in
1 + 𝜔 2 𝑅2 𝐶 2
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Passive RLC Filter Circuits
If the inductor and capacitor are in parallel there is a positive resonance.
𝑍𝐿𝐶 =
BPF
𝑣out
𝑗𝜔𝐿
=
=
𝑣in
𝑅 1 − 𝜔 2 𝐿𝐶 + 𝑗𝜔𝐿
𝑗𝜔 𝐿 𝑗 𝜔𝐶
𝑗𝜔𝐿
=
1 𝑗 𝜔𝐶 + 𝑗𝜔𝐿 1 − 𝜔 2 𝐿𝐶
𝑍𝐿𝐶
𝑣out =
𝑣
𝑅 + 𝑍𝐿𝐶 in
𝜔 2 𝐿2
𝑅 2 1 − 𝜔 2 𝐿𝐶 2 + 𝜔 2 𝐿2
BSF
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Cut-off Frequency Normalization
Another Way to look to it:
Substituting:

• For frequencies Ω >> 1, the rolloff is 20 dB/decade.
• For a steeper rolloff, n filter stages can be connected in series.
• To avoid loading effects, op amps, operating as impedance
converters, separate the individual filter stages
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Example of Filtration on two tones
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Cascading of filter stages
• In the case that all filters
have the same cut-off
frequency, fC, the
coefficients become:
• fC of each partial filter is 1/
times higher than fC of the
overall filter.
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LPF Response Optimization
The gain and phase response of a low-pass filter can be optimized to satisfy one of the following
three criteria:
1) A maximum passband flatness
2) An immediate passband-to-stopband transition
3) A linear phase response
Therefore
 The Butterworth coefficients, optimizing the passband for maximum flatness.
 The Tschebyscheff coefficients, sharpening the transition from passband into the stopband.
 The Bessel coefficients, linearizing the phase response up to fC.
Note: the transfer function of a passive RC filter does not allow further optimization, due to the
lack of complex poles. The only possibility to produce conjugate complex poles using passive RLC
filters (used for high frequencies RF filters).
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Filter Response Comparison
Butteroworth
Techebyscheff
Phase
Group Delay
1717
Fourth-Order Passive RC Low-Pass with Decoupling Amplifiers
To avoid loading effects, op amps,
operating as impedance converters,
separate the individual filter stages.
Disadvantage of this technique:
Need n op-amps to realize nth order
filter!!
Remedy:
Use Active Filter to save up to 50% of
op-amps.
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2nd Order LPF Active Filter
𝐴(𝑠) =
𝐴0
1 + 𝜔𝑐 𝐶1 𝑅1 + 𝑅2 + 1 − 𝐴0 𝑅1 𝐶2 𝑠 + 𝜔𝑐2 𝑅1 𝑅2 𝐶1 𝐶2 𝑠 2
For the unity-gain circuit 𝐴0 = 1 , the transfer function simplifies to:
𝐴(𝑠) =
1
1 + 𝜔𝑐 𝐶1 𝑅1 + 𝑅2 𝑠 + 𝜔𝑐2 𝑅1 𝑅2 𝐶1 𝐶2 𝑠 2
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Butterworth Low-Pass Filters
The Butterworth coefficients, optimizing the passband for maximum flatness
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Low-Pass Active Filter Design
 The first and second-order filter stages are the building blocks for
higher-order filters.
 First-order filter:
 Given C1, the resistor value for R1 is determined.
 Often the filters operate at unity gain (A0=1) to lessen the stringent
demands on the opamp’s open-loop gain.
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Low-Pass Active Filter Design
 Second-order filter:
 Given C1 and C2, the resistor values for R1 and R2
are calculated through:
 In order to obtain real values under the square root, C2 must satisfy the
following condition:
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Cascading Filter Stages for Higher-Order Filters
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5th order Active LPF (Sallen-Key)
Design a fifth-order unity-gain Butterworth low-pass filter with the
corner frequency fC = 50 kHz.
Steps:
1. First the coefficients for a fifth-order Butterworth filter are
obtained from Table:
2. Then dimension each partial filter by specifying the capacitor
values and calculating the required resistor values.
o
First Filter Stage:
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5th order Active LPF (Sallen-Key)
Design a fifth-order unity-gain Butterworth low-pass filter with the
corner frequency fC = 50 kHz.
Steps:
3. Second Stage:
R2
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5th order Active LPF (Sallen-Key)
Design a fifth-order unity-gain Butterworth low-pass filter with the corner
frequency fC = 50 kHz.
Steps:
4. Third Stage:
• The calculation of the third filter is identical to the calculation of the
second filter, except that a2 and b2 are replaced by a3 and b3, thus
resulting in different capacitor and resistor values. Specify C1 as 330 pF,
and obtain C2 with:
 The closest 10% value is 4.7 nF.
• With C1 = 330 pF and C2 = 4.7 nF, the values for R1 and R2 are:
•
•
R1 = 1.45 kΩ, with the closest 1% value being 1.47 kΩ
R2 = 4.51 kΩ, with the closest 1% value being 4.53 kΩ
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5th order Active LPF (Sallen-Key)
Design a fifth-order unity-gain Butterworth low-pass filter with the
corner frequency fC = 50 kHz.
Steps:
• The Final Design of the Fifth-Order Unity-Gain Butterworth LowPass Filter.
• You can use LTSpice to check the AC simulations for maximum flat
passband!!
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High-Pass Active Filter Design
By replacing the resistors of a 2nd order low-pass filter with capacitors,
and its capacitors with resistors, a 2nd order high-pass filter is created.
where A∞ is the passband gain (=1)
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Unity-Gain 2nd order Sallen-Key High-Pass Filter
Given C,
Example: Design a 4th order HPF fC = 10kHz
The filter is composed of two cascaded 2nd order stages:
First Stage:
With C = 10nF, the values for R1 and R2 are:
R1 = 1.7 kΩ
R2 = 1.46 kΩ
Second Stage:
With C = 10nF, the values for R1 and R2 are:
R1 = 4.18 kΩ
R2 = 602 kΩ
2929
Cascading of LPF and HPF Stages
• Which one should work ? And Why ?
• R1 < R2
• R1 > R2
3030
LPF and BPF in Parallel
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Transformation from LPF to BPF
• This can happen by substituting each S by 𝟏 ∆𝛀 𝒔 + 𝟏 𝑺
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Sallen-Key 2nd order Band-Pass
OR
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Band-Reject Filter Design
• To generate the transfer function of a second-order bandrejection filter, replace the S term of a first-order low-pass
response with the transformation:
• Which result the gain to be:
• Similarly:
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Band-Reject Filter – Another Configuration
• It can be built using what is called
Twin-T configuration where the Q
for passive filter is 0.25.
• The quality factor can be increased
by using active filter:
3535
Band-Reject Filter – Better Configuration
• If we want to decouple between
the gain and Q, we can use the
Wien-Robinson configuration:
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Band-Reject Filter – Better Configuration
• To design using this configuration:
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