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Active Low Pass Filter

The most common and
easily understood active
filter is the Active Low
Pass Filter. Its principle of
operation and frequency
response is exactly the
same as those for the
previously seen passive
filter, the only difference
this time is that it uses an
op-amp for amplification
and gain control.

Thomas Kugelstadt . Op Amps for Everyone Chapter 16
Active Filter Design Techniques. Available online :
http://www.ti.com/lit/ml/sloa088/sloa088.pdf Enero 2012 2

Active Low Pass Filter

This first-order low pass
active filter, consists
simply of a passive RC
filter stage providing a low
frequency path to the
input of a non-inverting
operational amplifier. The
amplifier is configured as
a voltage-follower (Buffer)
giving it a DC gain of one,
Av = +1 or unity gain as opposed to the previous passive RC filter which
has a DC gain of less than unity. The advantage of this configuration is
that the op-amps high input impedance prevents excessive loading on
the filters output while its low output impedance prevents the filters
cut-off frequency point from being affected by changes in the
impedance of the load.

Active Low Pass Filter with Amplification

The frequency response of the circuit will be the same as that for the
passive RC filter, except that the amplitude of the output is increased by
the pass band gain, AF of the amplifier. For a non-inverting amplifier
circuit, the magnitude of the voltage gain for the filter is given as a
function of the feedback resistor (R2) divided by its corresponding input
resistor (R1) value and is given as:



Gain of a first-order low pass filter:

Where:
AF = the pass band gain of the filter, (1 + R2/R1)
ƒ = the frequency of the input signal in Hertz, (Hz)
ƒc = the cut-off frequency in Hertz, (Hz)

1. At very low frequencies, ƒ < ƒc,

2. At the cut-off frequency, ƒ = ƒc,

3. At very high frequencies, ƒ > ƒc,



Example No1

Design a non-inverting active low pass filter circuit that has a gain of ten
at low frequencies, a high frequency cut-off or corner frequency of
159Hz and an input impedance of 10KΩ. Assume a value for resistor R1 of
1kΩ.


Second-order Low Pass Active Filter

As with the passive filter, a first-order low pass active filter can be
converted into a second-order low pass filter simply by using an
additional RC network in the input path. The frequency response of
the second-order low pass filter is identical to that of the first-order
type except that the stop band roll-off will be twice the first-order
filters at 40dB/decade (12dB/octave). Therefore, the design steps
required of the second-order active low pass filter are the same.


Active High Pass Filters

The basic operation of an Active High Pass Filter (HPF) is exactly the
same as that for its equivalent RC passive high pass filter circuit, except
this time the circuit has an operational amplifier or op-amp included
within its filter design providing amplification and gain control. Like the
previous active low pass filter circuit, the simplest form of an active
high pass filter is to connect a standard inverting or non-inverting
operational amplifier to the basic RC high pass passive filter circuit as
shown.


Active High Pass Filters

This first-order high pass filter, consists simply of a passive filter
followed by a non-inverting amplifier. The frequency response of the
circuit is the same as that of the passive filter, except that the
amplitude of the signal is increased by the gain of the amplifier.
For a non-inverting amplifier circuit, the magnitude of the voltage gain
for the filter is given as a function of the feedback resistor (R2) divided
by its corresponding input resistor (R1) value and is given as:

Low frequencies

Cut-off

High frequencies


Second-order High Pass Active Filter

As with the passive filter, a first-order high pass active filter can be
converted into a second-order high pass filter simply by using an
additional RC network in the input path. The frequency response of
the second-order high pass filter is identical to that of the first-order
type except that the stop band roll-off will be twice the first-order
filters at 40dB/decade (12dB/octave). Therefore, the design steps
required of the second-order active high pass filter are the same.


Cascading Active High Pass Filters

Higher-order high pass filters, such as third, fourth, fifth, etc are
formed simply by cascading together first and second-order filters. For
example, a third order high pass filter is formed by cascading in series
first and second order filters, a fourth-order high pass filter by
cascading two second-order filters together and so on.
Then an Active High
Pass Filter with an
even order number
will consist of only
second-order filters,
while an odd order
number will start
with a first-order
filter at the
beginning as shown.


Active Band Pass Filter

The Active Band Pass Filter is a frequency selective filter circuit used in
electronic systems to separate a signal at one particular frequency, or a range of
signals that lie within a certain "band" of frequencies from signals at all other
frequencies. This band or range of frequencies is set between two cut-off or
corner frequency points labelled the "lower frequency" (ƒL) and the "higher
frequency" (ƒH) while attenuating any signals outside of these two points.

Simple Active Band Pass Filter can be easily made by cascading together a single
Low Pass Filter with a single High Pass Filter as shown.


Active Band Pass Filter

This cascading together of the individual low and high pass passive filters
produces a low "Q-factor" type filter circuit which has a wide pass band.

The higher corner point (ƒH) as well as the lower corner frequency cut-off point
(ƒL) are calculated the same as before in the standard first-order low and high
pass filter circuits. Obviously, a reasonable separation is required between the
two cut-off points to prevent any interaction between the low pass and high pass
stages. The amplifier provides isolation between the two stages and defines the
overall voltage gain of the circuit.


Filter response


Fundamentals of Low-Pass Filters

The most simple low-pass filter is the passive RC low-pass network
shown:

For a steeper rolloff, n filter stages can be connected in series as
shown in Figure 16–3. To avoid loading effects, op amps, operating as
impedance converters, separate the individual filter stages:


In comparison to the ideal low-pass, the RC low-pass lacks in the following
characteristics:

•The passband gain varies long before the corner frequency, fC, thus
mplifying the upper passband frequencies less than the lower passband.
•The transition from the passband into the stopband is not sharp, but happens
gradually, moving the actual 80-dB roll off by 1.5 octaves above fC.
•The phase response is not linear, thus increasing the amount of signal
distortion significantly.

The gain and phase response of a low-pass filter can be optimized to satisfy
one of the following three criteria:

1) A maximum passband flatness,
2) An immediate passband-to-stopband transition,
3) A linear phase response.

For that purpose, the transfer function must allow for complex poles and
needs to be of the following type:


The transfer function of a passive RC filter does not allow further
optimization, due to the lack of complex poles.

•The Butterworth coefficients, optimizing the passband for maximum
flatness
•The Tschebyscheff coefficients, sharpening the transition from
passband into the Stopband
•The Bessel coefficients, linearizing the phase response up to fC


Quality Factor Q

The quality factor Q is an equivalent design parameter to the filter
order n. Instead of designing an nth order Tschebyscheff low-pass, the
problem can be expressed as designing a Tschebyscheff low-pass filter
with a certain Q. For band-pass filters, Q is defined as the ratio of the
mid frequency, fm, to the bandwidth at the two –3 dB points:

For low-pass and high-pass filters, Q represents the pole quality and is
defined as:


Quality Factor Q

High Qs can be graphically presented as the distance between the 0-dB
line and the peak point of the filter’s gain response.

In addition, the ratio

is defined as the pole
quality. The higher the Q
value, the more a filter
inclines to instability.


In applications that use filters, the amplitude response is generally
of greater interest than the phase response. But in some applications,
the phase response of the filter is important.
It might be useful to visualize the active filter as two cascaded
filters. One is the ideal filter, embodying the transfer equation; the
other is the amplifier used to build the filter.

Filter design is a two-step process. First, the filter response is
chosen; then, a circuit topology is selected to implement it. The
filter response refers to the shape of the attenuation curve. Often,
this is one of the classical responses such as Butterworth, Bessel, or
some form of Chebyshev. Although these response curves are usually
chosen to affect the amplitude response, they will also affect the
shape of the phase response


Filter complexity is typically defined by the filter ―order,‖ which is
related to the number of energy storage elements (inductors and
capacitors). The order of the filter transfer function’s denominator
defines the attenuation rate as frequency increases. The asymptotic
filter rolloff rate is – 6n dB/octave or –20n dB/decade, where n is the
number of poles. An octave is a doubling or halving of t he frequency;
a decade is a tenfold increase or decrease of frequency.


Phase response


The center frequency can also be referred to as the cutoff frequency
(the frequency at which the amplitude response of the single-pole, low-
pass filter is down by 3 dB—about 30%). In terms of phase, the center
frequency will be at the point at which the phase shift is 50% of its
ultimate value of –90° (in this case).


For the second-order, low-pass case, the transfer function has a
phase shift that can be approximated by:

The phase response of a 2-pole, high-pass filter can be approximated
by:

Where α is the damping ratio of the filter. It will determine the
peaking in the amplitude response and the sharpness of the phase
transition. It is the inverse of the Q of the circuit, which also
determines the steepness of the amplitude rolloff or phase shift.


Butterworth Low-Pass Filters

The Butterworth low-pass filter provides maximum passband flatness.
Therefore, a Butterworth low-pass is often used as anti-aliasing filter
in data converter applications where precise signal levels are required
across the entire passband.


Tschebyscheff Low-Pass Filters

The Tschebyscheff low-pass filters provide an even higher gain rolloff
above fC. However, as the Figure shows, the passband gain is not
monotone, but contains ripples of constant magnitude instead. For a
given filter order, the higher the passband ripples, the higher the
filter’s rolloff.


Bessel Low-Pass Filters

The Bessel low-pass filters
have a linear phase response
over a wide frequency range,
which results in a constant
group delay in that frequency
range. Bessel low-pass
filters, therefore, provide an
optimum square-wave
transmission behavior.
However, the passband gain
of a Bessel low-pass filter is
not as flat as that of the
Butterworth low-pass, and
the transition from passband
to stopband is by far not as
sharp as that of a
Tschebyscheff low-pass filter

Second-Order Sections

A variety of circuit topologies exists for building second-order
sections. To be discussed here are the Sallen-Key and the
multiplefeedback. They are the most common and are relevant
topologies.

The general transfer function of a low-pass filter is

The filter coefficients ai and bi distinguish between Butterworth,
Tschebyscheff, and Bessel filters. The coefficients for all three types
of filters are tabulated for second order filters:


Sallen-Key Topology

The general Sallen-Key topology in Figure 16–15 allows for separate
gain setting via A0 = 1+R4/R3. However, the unity-gain topology in
the Figure is usually applied in filter designs with high gain accuracy,
unity gain, and low Qs (Q < 3).


Sallen-Key Topology

The transfer function of the circuit is:

For the unity-gain circuit (A0=1), the transfer function simplifies to:

The general transfer function of a low-pass filter is:

The coefficient comparison between this transfer
function and the general transfer function is:

Sallen-Key Topology

The coefficient comparison between this transfer
function and the general transfer function is:

Given C1 and C2, the resistor values for R1 and R2 are calculated
through:

In order to obtain real values under the square root, C2 must satisfy
the following condition:


Problem: Design a second-order Sallen-Key unity-gain Tschebyscheff
low-pass filter with a corner frequency of fC = 3 kHz and a 3-dB
passband ripple. Supposse C1=22 nF.

From the Coefficients Table obtain a1 and b1 for a second-order
filter.

a1 = 1.0650 and b1 = 1.9305.


Multiple Feedback Topology

The MFB topology is commonly used in filters that have high Qs and
require a high gain.

The transfer function of the circuit is:


Higher-Order Low-Pass Filters

Higher-order low-pass filters are required to sharpen a desired filter
characteristic. For that purpose, first-order and second-order filter stages are
connected in series, so that the product of the individual frequency responses
results in the optimized frequency response of the overall filter. In order to
simplify the design of the partial filters, the coefficients ai and bi for each filter
type are listed in the coefficient tables.


Fifth-Order Filter

The task is to design a fifth-order unity-gain Butterworth low-pass
filter with the corner frequency fC = 50 kHz.

First the coefficients for a fifth-order Butterworth filter are obtained:

Then dimension each partial filter by specifying the capacitor values
and calculating the required resistor values


First Filter

First-Order Unity-Gain Low-Pass With C1 = 1nF,


Second Filter

Second-Order Unity-Gain Sallen-Key Low-Pass Filter With C1 = 820 pF,


Third Filter

The calculation of the third filter is identical to the calculation of the
second filter, except that a2 and b2 are replaced by a3 and b3, thus
resulting in different capacitor and resistor values.
Specify C1 as 330 pF:

The closest 10% value is 4.7 nF.

With C1 = 330 pF and C2 = 4.7 nF, the values for R1 and R2 are:
R1 = 1.45 kΩ, with the closest 1% value being 1.47 kΩ
R2 = 4.51 kΩ, with the closest 1% value being 4.53 kΩ


Higher-Order High-Pass Filter

Likewise, as with the low-pass filters, higher-order high-pass filters are designed
by cascading first-order and second-order filter stages. The filter coefficients are
the same ones used for the low-pass filter design, and are listed in the
coefficient tables.

Third-Order High-Pass Filter with fC = 1 kHz

The task is to design a third-order unity-gain Bessel high-pass filter with the
corner frequency fC = 1 kHz. Consider C1 = 100nF


Bessel Coeficients


Butterworth Coeficients


Tschebyscheff Coefficients
for 3-dB Passband Ripple


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