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					IV. Active filters
1. Frequency response with RC filter and LC filter
2. Active filters: an overview
         Frequency domain
         Time domain
3. Filter types
         Butterworth and Chebyshev filters
         Bessel filter
         Filter comparison

1. Frequency response with RC filter and LC filter
Let us remind simple, ordinary RC filter:

    Vin                                       Vout


                                                         Fig.1. Simple RC filter.
The output depends on the input:
Vout  Vin                          .
             1   R C 
                  2       2   2 2

We assume R=const and C=const and frequency  changes its value from 0 to infinity .
Therefor from the above formula results:
     The filter is not an amplifier, it will rather weaken signal.
     The biggest value of output Vout=Vinput is obtainable for =0. For any >0, Vout<Vinput.
     The frequency for which the term RC  1 is called 3dB frequency.
What does the 3dB frequency mean? Let us check it up.
Vout       1      1
               
Vinput 1  11
              2    2

And the same in decibels:

2011-04-20                                           1                              Renata Kalicka
         Vout            1
20 log           20 log     10 * 0.3010dB  3.010dB  3dB
         Vinput           2

In one octave (octave means twice a frequency) the output drops to half. The simple RC
filter has a fall off=6dB per octave: 6dB/octave. Fig. 2 shows characteristic of RC filter.

                      1           0dB

                                                                   gentle slope
            1/       =0.707       -3dB

                       0.5         -6dB

                                                                                                       6dB/octave skirt
                     0.25         -12dB

                       0          -                                                            1 octave

                              0                                        3dB                '              ''=2'

                             Fig.2. Characteristic of simple RC filter.
Such filters are sufficient only if signals being rejected by the filter are situated far from the
desired pass-band:
                                        to the


                                                 input frequency

 Fig.3. Lower frequencies that are able to pass from input to output of the filter while higher
                                                        frequencies are rejected.
Very often however, frequency of undesired disturbances is very close to the useful signal. In
that case, an approximation of the ideal "brick-wall" frequency response is required

2011-04-20                                                                    2                                           Renata Kalicka
                                   Fig.4. "Brick-wall" frequency response.
Within f  0  f0 ,   2f the input voltage Vin appears on the output, and Vout=Vin. Beyond f0,
(f>f0) the output should be equal to 0.
Simple RC filter does not enable to get the desired shape from Fig.4. How to get closer to the
ideal? One possible way is to increase number of basic RC section from Fig.1. Frequency
response of multisection RC filters is shown in Fig. 5.

                                                       the original       the normalized

                                                        0    f3dB     f   0    1    f/f3dB

Fig.5. Frequency response of multisection RC filters. A and B are linear plots, while C is
logarithmic. B and C responses have been normalized (scaled) for 3dB attenuation at unit
frequency. Point 2 on horizontal axis for A means just f=2Hz, point 2 for B means f=2*f3dB.
Point 1.0 for C means log(f/f3dB)=1, therefore f/f3dB=10 and f=10*f3dB.

To compare characteristics closer they are adjusted in such a way that f3dB for every individual
filter is assumed to equal to 1.

2011-04-20                                       3                                     Renata Kalicka
    As it results from the figure, when more sections are added, the 3dB point moves down in
    frequency. For n=2 (see A) the filter is more "low-pass" then it is for n=1: f3dB,n=2<f3dB,n=1.
    Simple RC filter gives 6dB/octave, 2 such sections give 2*6=12dB/octave, 3 sections
    18dB/octave and so on. But this simple cascading causes that the input impedance of the next
    section loads seriously the output of previous section. Output of multisection filter is getting
    smaller, step by step, and finally it becomes to small to be useful. The final falloff is steeper
    (see C) but the knee is still soft (see B) and does not remind "brick-wall" characteristic. We
    might conclude: many soft knees do not a hard knee make.
    Let us set together the assets and the drawbacks of cascaded RC filter:
                         cons                                                pros
                                                
        Input resistance loads the previous section          Skirt is steeper. For n sections the final
    degrading the response.                               falloff=n*6dB/octave.
       Many soft knees do not a hard knee make.

    Fig.6. A good passive LC bandpass filter (inductances are in mH and capacitances are in pF).
    Bottom: measured response of the circuit.

    Including L element into design (see Fig.6) brings some magic, which can not be performed
    without L, but there are also serious drawbacks:
       The higher the requirements (remember desired wall-brick shape) the more complex the
        filter, therefore it becomes bulky and expensive.
       The response is slightly degraded.

    2011-04-20                                        4                                Renata Kalicka
   L elements are "lossy" that means they are not pure inductance, but in fact can be
    represent as R, L and C elements in series.
   L element is non-linear, which means it is not constant for different value of current and
Inductors as filter elements leave much to be desired. What we really need is a way to obtain
inductorless filter with characteristic of LC filter. The solution is, so called, active filter. It
uses R, C and op-amp.
2. Active filters: an overview
There are different kinds of active filters:
   When band is criterion, the following are being distinguished: high-pass, low-pass,
    band-pass and band-reject.

                      pass    low pass

                                      pass           band pass

                       pass        reject            pass
                                                  pass             f

                                             high pass      pass

   When shape is criterion, the following are being distinguished: with maximal flat
    characteristic, with maximal steepness of characteristic and with minimal distortion in
    output signal.
As we want to design inductorless filter we need an element which is able to mimic (=to
simulate) inductor. There are 2 interesting elements:
1. Negative Impedance Converter NIC.
2. Gyrator.
Negative Impedance Converter NIC
It converts impedance to its negative (opposition). This element converts C to its negative that
                  1        1                          j      1            1
is to L: z C         j    is converted to z NIC      j 2  jL, L  2 .
                 jC      C                         C     C           C

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                imaginary axis                                                   imaginary axis
                 impedance                real axis                               impedance
                                                                    j 2 L

    j                                                                                                 real axis

                       For type C element:                                                For type L element:
        current goes before voltage                                        voltage goes before current
        the bigger frequency the smaller impedance                         the bigger frequency the bigger impedance
                            j        1
To       sum up: z NIC         j 2 causes that       voltage goes before current, like it is for L. But the
                          C        C

bigger frequency gives the smaller impedance, not like it is for L.
This element converts C into a true L:
                       1        1
               zC         j    is converted to z GYR  jCR2  jL, L  CR2
                      jC      C

To sum up: z GYR  jCR2  jL causes that voltage goes before current, like it is for L. The
bigger frequency the bigger impedance again like it is for L.
This element, which consists of R, C and op-amp and is called an active filter, is able to
mimic any inductance in a filter. The field of active filters is currently very successful. It is
possible to obtain almost any desired form (shape) of filter characteristic.
         Frequency domain
Let us concentrate on frequency characteristics of filters. There are 3 of them shown in Fig. 7.

                                      Fig.7. Filter characteristics versus frequency.
(Pass band, stop band, ripple band is sometime written as passband, stopband, rippleband.
                                    Which is correct? No one knows. It is up to you.)

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Pass band – it is range of frequencies, which are not seriously dumped by the filter. The end
of the passband is usually (but not necessarily, another definitions are allowed, see ripple
band) defined as the f3dB.
Cut-off frequency – allocates the right edge of the passband.
Ripple band – (ripple = very small wave) it is the range of gain within which those very small
waves contain. The pass band may be defined by such a frequency fc for which the
characteristic leaves ripple band.
Transition region (skirt) – it starts from fc and ends at the beginning of stop band.
Stop band – the band of strong, serious attenuation. The definition says: stop band is the band
in which attenuation is –40dB (or –50dB, –60dB etc. and it depends on particular situation,
filter application).
Phase shift versus frequency. It should be linear for low-pas filter. If not – the output will
contain distortions. This means bad quality of noisy output signal.
Time delay is closely related with phase shift. It should be constant and it is constant as long
as phase shift is linear.
Imagine you have a tape recorder which time delay characteristic is not constant. Imagine that
the higher frequency the bigger time delay.
                        input signal is a group of notes                                               input signal is a group of notes
                        a)time delay is constant                                                       b)time delay is not constant, is bigger for bigger frequencies

        f1         f2                  f3                  f4   input signal           f1         f2                  f3                   f4         input signal

    delayed output signal,                                                         delayed output signal,
        f1         f2              f3                   f4                             f1            f2                        f3                           f4
    The input and the output sequence of notes are the same.                       The output sequence of notes is different from the input sequence of notes.

       Time domain
               Filters are also characterized in time domain (Fig.8). It is important when input has
form of step or pulse:

2011-04-20                                                                     7                                                                Renata Kalicka
     STEP            PULSE

                  Fig.8. Filter characteristic versus time. Right: step response.

Rise time t r – it is the time when response reaches 90% of its final value.
Settling time t s - it is time required to set output within specified levels (for instance 5% of
the final value) and stay there.
Overshoot – it is undesired feature. It shows how big is the first ripple. The figure shows 15%
overshoot. Its height is 15% of the final value.
Ringing – all the small and big ripples, waves, undesired disturbances.
All the above parameters seam to be self explanatory (with a little help of the figure).

3. Filter types
There are 3(4) main criterions when we talk of filter quality:
   The ultimate rate of falloff. It is usually n*6dB/octave.
   The flatness of passband (it talks of the ripples).
   Distortion caused by nonlinear phase shift (that is by not constant time delay).
   One may also care about time domain parameters: rise time, overshoot, settling time, etc.
Let us assume that we have chosen low-pass filter with falloff (steepness)=36dB/octave. That
means we have quite complicated filter, which consists of 6 elementary sections. We need 1
capacitor, or 1 inductor, for each section, so the required rate of falloff of filter response
determines the complexity of the filter.
Having the filter, which consists of n sections we call it "n-pole filter". The name "pole"
comes from a method of analysis, which is beyond the area of our interest.
Once we have decided the falloff, we can optimize the flatness but we can do that at the
expense of the falloff steepness. By allowing some ripples we can get steeper skirt.
There are available classic filters, being standards. The most important are 3 of them:

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The Butterworth filter – maximally flat passband characteristic.
The Chebyshev filter – the steepest transition from passband to stopband.
The Bessel – maximally flat (constant) time delay.
All the filters are available in low pass, high pass or band pass version.

      Butterworth and Chebyshev filters

Fig.9. Normalized low-pass Butterworth filter characteristics. Note the improved steepness for
                                                         the higher order filters.

The Butterworth filter produces the flattest passband response, at the expense of steepness
in the transition region from passband to stopband. It has poor phase characteristic (see
Fig.13). The output depends on the input as follows:
        Vout            1
                                        1
                   f      
                                        2
                  1              
                    fc            
                                  

where n is the order of filter (number of poles, individual sections), fc it is 3dB frequency.
As it results from Fig.9, increasing n flattens the passband and steepens the falloff.
In most applications we allow some small ripples (let us say 1dB ripples) throughout the
The Chebyshev filter is an alternative: by allowing some passband ripples it gives sharper
knee. The output depends on the input as follows:
        Vout                1
                                                1
                       2 2 f 
                  1   Cn  
                            f 
                            c 

where Cn is the Chebyshev polynomial of the first kind of degree n,  is a constant that sets
the passband ripples. Like the Butterworth, the Chebyshev has rather poor phase

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 Logarithmic scale of amplitude response and       Linear scale of amplitude response and
           normalized frequency.                           normalized frequency.
 Fig.10. Comparison of some common 6-pole low-pass filters. The same filters are plotted on
                             both linear and logarithmic scales.

As you can see both, the Butterworth and the Chebyshev, are tremendous improvements over
an ordinary 6-pole RC filter.
It should be notice, that active filters constructed with R and C of finite tolerance (accuracy is
usually 5%, 10%, sometimes is less) will deviate from the theoretically calculated response.
This means that in fact real Butterworth will produce some pass band ripples anyway. Fig.11
shows the worst case variation in resistor and capacitor value on filter response.

             Fig.11. The effect of R and C tolerance on active filter characteristic.

From this point of view, the Chebyshev is very rational design. It is called an equiripple
Suppose that we need a filter with flatness to 0.1dB within the passband and 20dB attenuation
at a frequency 25% beyond the top of the passband. It will require a 19-pole Butterworth and
only 8-pole Chebyshev. Fig. 12 shows how we specify filter frequency response graphically.

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                 Fig.12. Specifying filter frequency response parameters.

    Allowable range of filter gain – ripples - G passMIN  G pass  G passMAX .

    The minimum frequency at which the response leaves the passband - fcutoff .

    The maximum frequency at which the response enters the stopband - fstop .

    The minimum attenuation in the stop band - G stop .

    Bessel filter
In situation when the shape of output wave is paramount, a linear phase filter (constant time
delay filter) is desired. A filter whose phase shift varies linearly with frequency is equivalent
to a constant time delay for signals within the pass band. That means the waveform is not
distorted. The Bessel (called also the Thomson) has maximally flat time delay.

    Fig.13. Comparison of time delays for 6-pole Bessel and Butterworth low pass filters. The
        excellent time-domain performance of the Bessel minimizes waveform distortion.

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The poor time delay performance of the Butterworth gives nondesired effects such as
overshoot when driven with pulse signals. The price we pay for the Bessel's constancy of time
delay is less steep an amplitude response (see Fig.10).

   Filter comparison
Fig.14 gives more information about time-domain performance for the Chebyshev, the
Butterworth and the Bessel.

Fig.14. A step response comparison for 6-pole low-pass filters normalized for 3dB attenuation
                                            at 1Hz.

The Bessel is a very desirable filter where performance in time domain is important. The
Chebyshev, with its highly desirable amplitude versus frequency characteristic has the poorest
time domain performance of the 3. The Butterworth is in between in time domain and
frequency properties.

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