1. M2-3S Active Filter (Part II)
Biquadratic function filters
Positive feedback active filter: VCVS
Negative feedback filter: IGMF
Butterworth Response
Chebyshev Response
Active Filter (Part 2)

2. Biquadratic function filters
Realised by:
Positive feedback
(II) Negative feedback
Active Filter (Part 2)

3. Biquadratic functions
(III) Band Pass
(IV) Band Stop
(V) All Pass
Biquadratic functions
(I) Low Pass
(II) High Pass
Active Filter (Part 2)

4. Low-Pass Filter
Active Filter (Part 2)

5. High-Pass Filter
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6. Band-Pass Filter
Active Filter (Part 2)

7. Band-Stop Filter
Active Filter (Part 2)

8. Voltage Controlled Votage Source (VCVS) Positive Feedback Active Filter (Sallen-Key)
By KCL at Va:
Therefore, we get
where,
Re-arrange into voltage group gives:
(1)
Active Filter (Part 2)

9. Substitute (2) into (1) gives
But,
(2)
Substitute (2) into (1) gives or
(3)
In admittance form:
(4)
* This configuration is often used as a low-pass filter, so a specific example will be considered.
Active Filter (Part 2)

10. VCVS Low Pass Filter In order to obtain the above response, we let:
Then the transfer function (3) becomes:
(5)
Active Filter (Part 2)

11. we continue from equation (5),
Equating the coefficient from equations (6) and (5), it gives:
Now, K=1, equation (5) will then become,
Active Filter (Part 2)

12. Simplified Design (VCVS filter)
Comparing with the low-pass response:
It gives the following:
Active Filter (Part 2)

13. Example (VCVS low pass filter)
To design a low-pass filter with and
Let m = 1
 n = 2
Choose
Then
What happen if n = 1?
Active Filter (Part 2)

14. VCVS High Pass Filter
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15. VCVS Band Pass Filter
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16. Infinite-Gain Multiple-Feedback (IGMF) Negative Feedback Active Filter
Note: because no current flows into v+, v- terminals of op-amp. Therefore, from KCL at node v- :
Vo/Z5+ Vx/Z3 = 0
substitute (1) into (2) gives
(3)
Active Filter (Part 2)

17. rearranging equation (3), it gives,
Or in admittance form: Z1 Z2 Z3 Z4 Z5 LP R1 C2 R3 R4 C5 HP C1 R2 C3 C4 R5 BP
Filter
Value
Active Filter (Part 2)

18. IGMF Band-Pass Filter Band-pass:
To obtain the band-pass response, we let
*This filter prototype has a very low sensitivity to component tolerance when compared with other prototypes.
Active Filter (Part 2)

19. Simplified design (IGMF filter)
Comparing with the band-pass response
Its gives,
Active Filter (Part 2)

20. Example (IGMF band pass filter)
To design a band-pass filter with and
With similar analysis, we can choose the following values:
Active Filter (Part 2)

21. Butterworth Response (Maximally flat)
Butterworth polynomials
where n is the order
Normalize to o = 1rad/s
Butterworth polynomials:
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22. Butterworth Response
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23. Second order Butterworth response
Started from the low-pass biquadratic function
For
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24. Bode plot (n-th order Butterworth)
Butterworth response
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25. Second order Butterworth filter
Setting R1= R2 and C1 = C2
Now K = 1 + RB/ RA
Therefore, we have
For Butterworth response:
We define Damping Factor (DF) as: 
Active Filter (Part 2)

26. Damping Factor (DF)
The value of the damping factor required to produce desire response characteristic depends on the order of the filter.
The DF is determined by the negative feedback network of the filter circuit.
Because of its maximally flat response, the Butterworth characteristic is the most widely used.
We will limit our converge to the Butterworth response to illustrate basic filter concepts.
Active Filter (Part 2)

27. Values for the Butterworth response
Roll-off
dB/decade
1st stage
2nd stage
3rd stage
Order
poles DF 1
-20
optional 2
-40
1.414 3
-60
1.000 4
-80
1.848
0.765 5
-100
1.618
0.618 6
-120
1.932
0.518
Active Filter (Part 2)

28. Forth order Butterworth Filter+ -
R2 8.2 k
C F
Vout
+15 V
R1 8.2 k
RB 1.5 k
RA 10 k
R3 8.2 k
R4 8.2 k
-15 V
C F C2
0.01 F C4
RB 27 k
RA 22 k
741C
Active Filter (Part 2)

29. Chebyshev Response (Equal-ripple)
Where  determines the ripple and
is the Chebyshev cosine polynomial defined as
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30. Chebyshev Cosine Polynomials
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31. Second order Chebychev Response
Example: 0.969dB ripple gives  = 0.5,
Roots:
Note:
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32. Roots Roots of first bracketed term Roots of second bracketed term or
Note: or
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