Lecture 5 Active Filter (Part II)

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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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Biquadratic function filters

• Realised by
• Positive feedback
• (II) Negative feedback

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• (III) Band Pass
• (IV) Band Stop
• (V) All Pass

Biquadratic functions

• (I) Low Pass
• (II) High Pass

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Low-Pass Filter
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High-Pass Filter
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Band-Pass Filter
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Band-Stop Filter
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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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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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VCVS Low Pass Filter
In order to obtain the above response, we let
Then the transfer function (3) becomes
(5)
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we continue from equation (5),
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Simplified Design (VCVS filter)
Comparing with the low-pass response
It gives the following
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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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VCVS High Pass Filter
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VCVS Band Pass Filter
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Infinite-Gain Multiple-Feedback (IGMF) Negative
Feedback Active Filter
substitute (1) into (2) gives
(3)
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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 R1 R2 C3 C4 R5
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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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Simplified design (IGMF filter)
Comparing with the band-pass response
Its gives,
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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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Butterworth Response (Maximally flat)
Butterworth polynomials
where n is the order
Normalize to ?o 1rad/s
Butterworth polynomials
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Butterworth Response
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Second order Butterworth response
Started from the low-pass biquadratic function
For
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Bode plot (n-th order Butterworth)
Butterworth response
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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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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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Values for the Butterworth response
Roll-off dB/decade 1st stage 1st stage 2nd stage 2nd stage 3rd stage 3rd stage
Order Roll-off dB/decade poles DF poles DF poles DF
1 -20 1 optional
2 -40 2 1.414
3 -60 2 1.000 1 1.000
4 -80 2 1.848 2 0.765
5 -100 2 1.000 2 1.618 1 0.618
6 -120 2 1.932 2 1.414 2 0.518
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Forth order Butterworth Filter
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Chebyshev Response (Equal-ripple)
Where ? determines the ripple and is the
Chebyshev cosine polynomial defined as
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Chebyshev Cosine Polynomials
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Second order Chebychev Response
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Roots
or