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Active Filter
A. Marzuki
Table of Contents
1 Introduction
2 First- Order Filters
3 Second-Order Filters
4 Other type of Filters
5 Real Filters
6 Conclusion
Active filters Introduction
•Consist of only amplifiers, resistors, and capacitors.
•Use resistors and capacitors in their feedback loops, to
synthesize the desired filter characteristics.
•have high input impedance, low output impedance, and
virtually any arbitrary gain.
•They are also usually easier to design than passive filters.
•They lack inductors.
.
•Performance at high frequencies is limited by the gainbandwidth product of the amplifying.
•The op amp-based active filter can achieve very good
accuracy, provided that low-tolerance resistors and capacitors
are used.
•Active filters will generate noise due to the amplifying
circuitry, but this can be minimized by the use of low-noise
amplifiers and careful circuit design.
First-Order Transfer Functions
A standard form of the transfer function of a first-order low-pass filter is
(1)
Where TLP (j0) is the value of TLP (s) at DC and ωo pole frequency, It is common practice to
normalize both the magnitude and frequency. Normalizing (1) yields
(2)
First order has 6dB roll off and increase rate
(3)
(4)
(5)
First-Order Filter Realizations
A first-order filter has only one pole and zero. For stability, the pole must be on the
negative real axis but the zero may be on the negative or positive real axis. A single
amplifier low-gain realization of a first order low-pass filter is shown in below
The transfer function for this filter is
(6)
The low-frequency (ω << ωo) gain magnitude and polarity are set by K. First-order high-pass filters
can also be realized in a single-amplifier low-gain circuit.
The transfer function for this filter is given in (4), and in this case THP(j∞) = K. Note that the first-order
filters exhibit no peaking and the frequency-dependent part of the magnitude response approaches ±20
dB/decade.
More….
Transfer function for low pass filter shown below is
DC Gain is
And o is 1/CR2
More….
Alternative to low pass Filter with transfer function
Where the gain
More….
high pass Filter with transfer function
High Frequency Gain is
o is 1/CR1
More….
Alternative to high pass Filter with transfer function
Where the gain, K =
o
Second-Order Transfer Functions
The standard form of a second-order low-pass
filter is given as
(7)
(8)
The poles of the above transfer function (8) of are
(9)
Normalization of in both amplitude and frequency gives
(10)
(11)
(9)
(12)
(13)
Frequency Response (Magnitude and Phase)
If Q is greater than 0.707, then the normalized magnitude response has a peak value of
(14)
(15)
Butterworth
1.
The first, and probably best-known filter approximation is the Butterworth or maximally-flat
response.
2.
nearly flat passband with no ripple. The rolloff is smooth and monotonic, with a low-pass or highpass rolloff rate of 20 dB/decade (6 dB/octave) for every pole.
3.
Thus, a 5th-order Butterworth low-pass filter would have an attenuation rate of 100 dB for every
factor of ten increase in frequency beyond the cutoff frequency.
General Equation for amplitude response
(16)
Amplitude Response Curves of Low Pass Filter
For various order.
More Type Of Filters
Chebyshev
Another approximation to the ideal filter is the Chebyshev or equal ripple response. As the latter name implies, this
sort of filter will have ripple in the passband amplitude response. The amount of passband ripple is one of the
parameters used in specifying a Chebyshev filter. The Chebyschev characteristic has a steeper rolloff near the
cutoff frequency when compared to the Butterworth, but at the expense of monotonicity in the passband and
poorer transient response.
Bessel
exhibits approximately linear phase shift with frequency, so its action within the passband simulates a
delay line with a low-pass characteristic. The higher the filter order, the more linear the Bessel's
phase response.
Elliptic
The cutoff slope of an elliptic filter is steeper than that of a Butterworth, Chebyshev, or Bessel, but the
amplitude response has ripple in both the passband and the stopband, and the phase response is
very non-linear. However, if the primary concern is to pass frequencies falling within a certain
frequency band and reject frequencies outside that band, regardless of phase shifts or ringing, the
elliptic response will perform that function with the lowest-order filter.The elliptic function gives
a sharp cutoff by adding notches in the stopband. These cause the transfer function to drop to zero
at one or more frequencies in the stopband. Ripple is also introduced in the passband. An elliptic
filter function can be specified by three parameters (again excluding gain and cutoff frequency):
passband ripple, stopband attenuation, and filter order n. Because of the greater complexity of the
elliptic filter, determination of coefficients is normally done with the aid of a computer.
Second-Order Filters
These filters are easy to design and have been extensively used in many applications. The first
realization is a low-pass Sallen and Key filter.
Low Pass Filter
.
High Pass Filter
High Order Filters
The filter order is the number of poles. An N-pole active filters have rolloff rate of Nx6dB/octave and
increase rate of Nx6dB/octave. For Butterworth Filter, the solution is solved by taking first and
second derivatives of the Transfer function.
/1.082
/0.9241
/2.613
/0.3825
3dB =1/RC
4-pole Butterworth High Pass Filter
More….
Low pass Filter
1.082
0.9241
2.613
0.3825
3dB =1/RC
4-pole Butterworth Low Pass Filter
Other type of Filters
Switched Capacitor Filter,
1.
need no external resistors, capacitors or inductors,
2.
No internal resistor
3.
cutoff frequencies are set to a typical accuracy of 0.2% by an external clock frequency.
4.
filters whose cutoff frequencies are variable over a wide range simply by changing the clock
frequency.
Switched Current Filter,
1.
need no resistors, capacitors or inductors,
2.
filters whose cutoff frequencies are variable over a wide range simply by changing the clock
frequency.
Design Steps
1.
2.
3.
4.
Hand analysis for simple filter designs.
Employ synthesis program for complex designs.
Use analysis EDA tools such as SPICE and etc.
Prototyping and Characterization.
Real Active Filters
Mores
Conclusion
1.
2.
3.
4.
Cover First Order and its basic circuit.
Talked on 2nd Order and its basic circuit.
Brief description on Switched Capacitor Filter.
Active circuits are real components and there are many applications that use
them.
References
Sedra and Smith, Microelectronic Cicruits, 4th Edition, Oxford, 1998.
D. A. Neaman, Electronic Circuit Analysis and Design, 2th Edition, McGRAW-HILL,
2001.
The Circuits and Filters Handbook, Second Edition
Continuous-Time Active Filter Design
Kerry Lacanette, “A Basic Introduction to Filters–Active, Passive, and SwitchedCapacitor,” National Semiconductor, Application Note 779, April 1991.