1. Lecture 5 Active Filter (Part II)
Biquadratic function filters
Positive feedback active filter: VCVS
Negative feedback filter: IGMF
Butterworth Response
Chebyshev Response
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EE3110 Active Filter (Part 2)

2. Biquadratic function filters
Realised by:
Positive feedback
(II) Negative feedback
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EE3110 Active Filter (Part 2)

3. EE3110 Active Filter (Part 2)
(III) Band Pass
(IV) Band Stop
(V) All Pass
Biquadratic functions
(I) Low Pass
(II) High Pass
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EE3110 Active Filter (Part 2)

4. EE3110 Active Filter (Part 2)
Low-Pass Filter
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EE3110 Active Filter (Part 2)

5. EE3110 Active Filter (Part 2)
High-Pass Filter
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EE3110 Active Filter (Part 2)

6. EE3110 Active Filter (Part 2)
Band-Pass Filter
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EE3110 Active Filter (Part 2)

7. EE3110 Active Filter (Part 2)
Band-Stop Filter
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EE3110 Active Filter (Part 2)

8. EE3110 Active Filter (Part 2)
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)
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EE3110 Active Filter (Part 2)

9. EE3110 Active Filter (Part 2)
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.
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EE3110 Active Filter (Part 2)

10. EE3110 Active Filter (Part 2)
VCVS Low Pass Filter
In order to obtain the above response, we let:
Then the transfer function (3) becomes:
(5)
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EE3110 Active Filter (Part 2)

11. EE3110 Active Filter (Part 2)
we continue from equation (5),
Equating the coefficient from equations (6) and (5), it gives:
Now, K=1, equation (5) will then become,
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EE3110 Active Filter (Part 2)

12. Simplified Design (VCVS filter)
Comparing with the low-pass response:
It gives the following:
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EE3110 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?
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EE3110 Active Filter (Part 2)

14. EE3110 Active Filter (Part 2)
VCVS High Pass Filter
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EE3110 Active Filter (Part 2)

15. EE3110 Active Filter (Part 2)
VCVS Band Pass Filter
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EE3110 Active Filter (Part 2)

16. Infinite-Gain Multiple-Feedback (IGMF) Negative Feedback Active Filter
substitute (1) into (2) gives
(3)
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EE3110 Active Filter (Part 2)

17. EE3110 Active Filter (Part 2)
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
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EE3110 Active Filter (Part 2)

18. EE3110 Active Filter (Part 2)
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.
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EE3110 Active Filter (Part 2)

19. Simplified design (IGMF filter)
Comparing with the band-pass response
Its gives,
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EE3110 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:
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EE3110 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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EE3110 Active Filter (Part 2)

22. EE3110 Active Filter (Part 2)
Butterworth Response
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EE3110 Active Filter (Part 2)

23. Second order Butterworth response
Started from the low-pass biquadratic function
For
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EE3110 Active Filter (Part 2)

24. Bode plot (n-th order Butterworth)
Butterworth response
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EE3110 Active Filter (Part 2)

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: 
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EE3110 Active Filter (Part 2)

26. EE3110 Active Filter (Part 2)
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.
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EE3110 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
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EE3110 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
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EE3110 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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EE3110 Active Filter (Part 2)

30. Chebyshev Cosine Polynomials
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EE3110 Active Filter (Part 2)

31. Second order Chebychev Response
Example: 0.969dB ripple gives  = 0.5,
Roots:
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EE3110 Active Filter (Part 2)

32. EE3110 Active Filter (Part 2)
Roots
Roots of first bracketed term
Roots of second bracketed term or
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EE3110 Active Filter (Part 2)