Filtre Tasarımı,
Butterworth ve Chebyshev
Yaklaşıklığı
DOÇ. DR. REVNA ACAR VURAL
H5
Filter Designs
Low Pass
High Pass
Band Stop
Band Pass
Differential Filtering
Filter Applications
Power Filtering transfer function zeros or transmission zeros
Audio Application
Band Stop
aM s z1 s z2 s z3 s zM
s p1 s p2 s p3 s pN
T( s )
ECG Application
transfer function poles or the natural poles
Order of filters
First order
Second order
Third and higher order
Cutoff frequency
Magnitude response is
maximally flat for Qp = 1/sqrt(2).
1
H s
s s2
1
0Qp 02
Filter Designs – Low Pass
Passive
Transfer function
Filter Designs – Low Pass
Active
Filter Designs – High Pass
Passive
Active
𝑉𝑜𝑢𝑡 𝑅2 𝑗𝜔𝐶1 𝑅2 𝑅2 𝑗𝜔𝐶1 𝑅1
=− =− =−
𝑉𝐺1 1 1 + 𝑗𝜔𝐶1 𝑅1 𝑅1 1 + 𝑗𝜔𝐶1 𝑅1
𝑅1 + 𝑗𝜔𝐶
1
Filter Designs – Bandwidth
Bandwidth
Cutoff frequency and center frequency
Q factor
Filter Designs – Band Pass
Passive
Active
Filter Designs – Band Stop
Background
Analog design
Digital design
Filter Applications – Band Stop
Picture 1 is the ECG signal; When we analyze these spectrum,
especially when we zoom in, we can see the 60 Hz power noise;
To erase the 60 Hz power noise, we need to use the Notch filter
to eliminate the signal at 60 Hz.
Filter Applications – Band Stop
Plot of the Notch Filter
1.05
0.95
| H( ejw ) |
0.9
0.85
0.8
0.75
0.7
0 20 40 60
Hz
80 100 120 140
After Notch filter, we can see that the
ECG signal is cleaner and 60 Hz power
noise is erased
Filter Designs – Differential
Concepts：
Differential Filter: Any filter with a differential input and a differential
output.
Single-Ended Signal：
-One of the signal terminals are grounded.
Differential Signal：
-Neither of the signal terminals are grounded.
Why do we need it ?
Couple with differential amplifiers.
Increase common mode rejection ratio, reduce noise and interference.
Filter Designs – Differential
Passive High-pass Low-pass
Active High-pass Low-pass
Filter Designs – Differential
How to design it？
Single-Ended to Differential-Ended Filter Translation.
Single-Ended LP Filter Differential LP Filter
*Figures from Texas Instruments Application Report “Design of Differential Filters for High-Speed Signal
Chains” by Ken Chan
Filter Designs – Differential
Approximation Problem
• Magnitude response of an ideal filter is defined as "'brickwall" characteristic.
• Ideal response can not be expressed as a rational function of angular frequency.
• Since the ideal low-pass filter is not physically realizable, it is possible to design a
physical circuit which approximates to the ideal characteristics
• Approximation problem can be defined as a curve fitting optimization.
Classical Approaches
b0
H (s)
s n bn1 s n b1 s b0
Butterworth Approximation • Chebyshev Approximation
n Chebyshev Polynomials
employs no ripples in the passband • meets hard specifications 1 s 1.965
2 s 2 1.097 s 1.102
• ripples exist in passband
long transition band n
1
Butterworth Polynomials
s 1
3 s 3 0.7378s 2 1.0222s 0.3269
4 s 4 0.952s 3 1.453s 2 0.742s 0.275
2 s 2 1.414s 1
5 s 5 0.7064s 4 1.4995s 3 0.6935s 2 0.4594s 0.0817
3 s 3 2s 2 2s 1
4 s 4 2.613s 3 3.414s 2 2.613s 1
5 s 5 3.236s 4 5.236s 3 5.236s 2 3.236s 1
36
Bode Plot
Steeper Roll-off, but what about stopband rejection?
s 2
1
z
H s
s s
2
p Qp p
1
p 2
H j
z
transfer function zeros or transmission zeros
aM s z1 s z2 s z3 s zM
s p1 s p2 s p3 s pN
T( s )
transfer function poles or the natural poles
1.Dereceden Filtre Fonksiyonları
43
Conventional Design Method
Components are assumed to be ideal and have infinite value during analog design
process. However, discrete components such as resistors and capacitors are
produced in approximate logarithmic multiples of a defined number of constant
values.
Typically produced 12 preferred values are known as E12 series which are 1.0, 1.2,
1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2, 10,… .
Component values of an analog filter are calculated due to the transfer function
and design criteria (cutoff frequency and quality factor).
Due to the preference of equal values for some discrete components, the obtained
values of the other components may not correspond to the preferred values.
Overall performance of the analog filter may decrease due to the nearest preferred
values of discrete components and error criteria value of the analog filter circuit
may increase under these circumstances.
44
VCVS Butterworth Active Filter
Op-Amp Design Criteria
Classical Design
Ideal Values E12 Values
ωc1, ωc2 = 10k rad/sec R1 1kΩ 1kΩ
Q1= 1/0.7654 R2 1kΩ 1kΩ
Q2 = 1/1.8478 C1 38.27 nF 39 nF
C2 26.13 nF 0.27 µF
R3 1kΩ 1kΩ
R4 1kΩ 1kΩ
C3 92.39 nF 0.1 µF
C4 0.2613 µF 0.1 µF
Δw 0 0.02549
ΔQ 0 0.05026
Total Error = 0.5Δw + 0.5ΔQ Total Error 0 0.03788
45
Design Parameters: The amplitude response of Butterworth
Resistors&Capacitors 4th order VCVS LP filter
Resistor (Ω) Capacitor (pF)
X 1000 10000 100000 1000 10000 100000
1 1k 10k 100k 1nF 10nF 100nF
1.2 1.2k 12k 120k 1.2nF 12nF 120nF
1.5 1.5k 15k 150k 1.5nF 15nF 150nF
1.8 1.8k 18k 180k 1.8nF 18nF 180nF
2.2 2.2k 22k 220k 2.2nF 22nF 220nF
2.7 2.7k 27k 270k 2.7nF 27nF 270nF
3.3 3.3k 33k 330k 3.3nF 33nF 330nF
3.9 3.9k 39k 390k 3.9nF 39nF 390nF
4.7 4.7k 47k 470k 4.7nF 47nF 470nF
5.6 5.6k 56k 560k 5.6nF 56nF 560nF
6.8 6.8k 68k 680k 6.8nF 68nF 680nF
8.2 8.2k 82k 820k 8.2nF 82nF 820nF
