Exploration of Passive and Active Filters Circuits

By,
David.R.Gilson

 

 

 

 

Abstract

In this experiment, the properties of two types of passive filter circuit were investigated (Low Pass and Band Pass). Then the behaviour of an active filter circuit was analysed.

The cut off frequency, roll off, and frequency response of both filter types were measured and compared with theory.

Data from an active band pass filter was used to reconstruct a wave form from a Thandor signal generator. The active filter was interfaced with a BBC micro computer, so that it's analysis could be displayed.

 

1) Introduction

In many communication systems (radio, telecommunication) there is a constant need to separate different frequency components of an incoming signal. For example, in Amplitude Modulation (AM) radio communication. In AM systems a message signal is superimposed onto a carrier wave. The resulting wave form is one of constant frequency but and varying amplitude. A simple circuit, such as the one shown in figure one, will separate a high frequency carrier signal from a lower frequency band message signal.

 

Figure One, a simple filter circuit

The circuit in figure one, is what is known as a passive filter circuit. There are many types of passive filter circuit, all of which can be classified as variations of the following, high pass, low pass, and band pass. In this experiment, high pass filters are not considered.

 

Figure Two, a simple low pass filter circuit

Combinations of filter circuits can be formed, which are known as ladder networks. In figure two, a simple low pass filter, the input is applied across AB and the filtered output is across CD. Only the fraction of low frequency components of the voltage AB is seen at CD. High frequency signals are rejected. The rejection effect is increased as the ladder of filter is built up. [1]

The Band Pass filter

The band pass filter, as it's name suggests, only filter out a band of frequencies. Hence, for higher and lower frequencies the impedance of the circuit is high. The terminology for this is that, the high and low frequencies form the stop band of the filter. A simple band pass filter is shown in figure three.

 

Figure Three, a band pass filter circuit

It can be shown that the lowest frequency that a band pass filter will let through is [1],

w1=(LC1)^- (1)

and the highest frequency that it will let through is [1],

w2=((C2+4C1)/(LC1C2))^ (2)

It can be seen that the stop band exists below w1 and above w2. Conversely, the pass band is between w1 and w2.

The Low Pass filter

A low pass filter, only allows low frequency signals through to it's output. Following the same principal as for the band pass filter, the pass band of the low pass filter is below the cut off frequency and the stop band is above the cut off frequency. It is obvious that the cut off frequency is the limiting frequency value between stop and pass bands. Considering the circuit in figure 2 it can be shown that the cut off frequency is,

wo=2.(LC)^- (3)

Comparing the similarity between equations one and three, it is easy to see that analogy that in the lower stop band, the band pass filter is behaving like a low pass filter.

 

Active Filters

These are introduced here because they are later used in the final part of the experiment. The active filter used in this experiment is simple an array of narrow band pass filters. Each band pass filter allows a specific frequency through. This means that an incoming wave form can be frequency analysed and reconstructed (if required) based on it's readings.

Attributes of filters

There are certain attributes that can be measured for a filter that can used to describe it's performance.

Frequency Response
This is usually measured in Decibels (dB) against log(f). The decibel is a relative measure of output, often used in acoustic systems. For this purpose, it is defined in terms of the input amplitude. We therefore say,

xdB = 10.log10(VI/xvolts) (4)

Cut Off frequency
This is defined as the frequency corresponding to an output of -3 dB. This also equivalent to the output amplitude falling to half of the input amplitude.

"Roll Off"
This determines the "sharpness" of the filter and is measured in dB per Octave. This represents how quickly the output falls once the frequency is around the stop band threshold. An octave represents a frequency separation by a factor of two from some reference value [2]. In general we would write,

n octaves = fo.2^n (5)
(n=1,2,3...)


If we extend this to non-integer octaves, we can write,

x octaves = fm = fo.2^x (6)


Rearranging this to calculate the octave value, x, in terms of fm yields,

x=Ln(fm)/(Ln(2).Ln(fo)) (7)


However, by definition, we would like x to be zero when fm=fo. So we make a final adjustment to equation 7.

x= Ln(fm/fo)/(Ln(2).Ln(fo)) (8)

 

2) Experimental Procedure

Passive Filters

The first half of the experiment was two test the various properties of the Band and Low pass filters. Below are schematics of the respective Band and Low pass filter circuits used in the experiment.

Figure Four, band pass filter used in experiement.

Figure Five, the low pass circuit used in the experiment.

The resistor in figure five, was put there to match the Low Pass filter to the oscilloscope (which was connected to the output). If the resistor had not been used, there would have been reflection effects which would have adversely affected the readings. The matching of impedances was not effected different frequencies, since the matching condition was frequency independent.

The measurements were taken by incrementally changing the input frequency (from a standard laboratory signal generator) and noting the new output on the cathode ray oscilloscope (CRO) screen, until the output reached -3 dB.

The Cut off frequency was sought first (readings being took along the way). Equation four was rearranged to estimate the minimum output top look for. A point to note however, an input amplitude of 5 Volts was used for the Low Pass filter, at this level the entire frequency range of the signal generator could not get an output from the Band Pass filter above -3 decibels. To get around this apparent difficulty the input amplitude for the Band Pass filter was decreased to 0.5 Volts. This allowed a set of data well describing the output attenuation to be collected. All of the sought quantities could then be derived from the data collected on both filters as described above.

Active Filter

A schematic of the apparatus used in this part of the experiment is shown in figure six.

Figure Six, active filter set up.

For this part of the experiment the Thandor signal generator was connected to active filter and set to produce a sine wave. The active filter was connected to a BBC micro computer, with a combination of software and hardware interfacing. The output from the filter was printed out (this print out is supplied in the appendix). The active filter was (arbitrarily) set to it's eleventh filter and it's second frequency band. The data from this could be used with the settings of the signal generator to reconstruct the incoming wave form. Measurements were taken three times of the same filter/band/signal because the readings were floating with every scan. So the final results are actually means of the three repetitions.

3) Results

Passive Filters : Frequency Response

Figure seven shows the frequency response of both passive filters. (Note, dB's for each filter are calculated relative to their own input levels.)

Figure Seven, graph showing comparison of frequency response between band and low pass filter circuits.

Below is a table showing the slopes and intercepts for the least square fitted lines.

 

Slope

Error (slp)

Intercept

Error (itc)

Band Pass Filter

-1.872E+01

2.326E-11

7.463E+01

9.458E-11

Low Pass Filter

-2.534E+01

6.273E-10

9.275E+01

2.346E-09

 

Table One, least square fit analysis of frequency response.

Passive Filters : Cut Off Frequency

From theory there were two cut off frequencies for the band pass filter, however with the bandwidth of the supplied signal generator there was only cut off frequency found. Below is a table of the theoretical values calculated and the apparent cut off frequencies from the measured data.

 

 

 

 

Band Pass (Hz)

 

Low Pass (Hz)

 

Lower

Upper

 

Theoretical

2.132E+04

6.674E+04

4.472E+04

Experimental

1.45E+04

 

6.000E+03

 

Table Two, Theoretical and Experimental cut off frequencies.

Passive Filters : Roll Off

The "Roll Off" for both filters are plotted on the same graph for a direct comparison. (Note, dB's for each filter are calculated relative to their own input levels.)

Figure Eight, comparison of roll off between low pass and band pass filters.

 

 

Again, below is a table of the slope and intercept for the least square fitted lines.

 

Slope

Error(slp)

Intercept

Error(itc)

Band Pass

-5.131E+01

3.100E-14

6.134E-01

1.454E-15

Low Pass

-6.497E+01

1.799E-12

-9.788E-01

3.351E-14

 

Table Three, least squares fit analysis of roll off (dB/Oct) of both filters.

Active Filters

The distribution of amplitudes in each channel are shown in figure nine.

Figure Nine, bar chart of the channel amplitude from the acitve filter.

There were 11 frequency components counted in the wave form analysis. Below is a table of mean amplitudes and respective frequency and channel number.

 

 

Channel

Frequency (Hz)

Mean Amplitude (V)

1

78

0.18

2

156

0.25

3

200

0.71

4

250

2.57

5

315

2.33

6

400

0.22

7

500

0.00

8

625

0.02

9

800

0.05

10

1000

0.07

11

1250

0.06

12

1600

0.04

13

2000

0.00

14

2500

0.00

15

3200

0.00

16

4000

0.00

 

Table Four, Amplitudes of corresponding frequency components.

From these values the original wave form could be reconstructed and plotted. This wave done by adding successive sine waves together.

 

Figure Ten, Graph of reconstructed wave form using data from the active filter.

4) Conclusion

Passive Filters : Frequency Response

It is clear from figure seven, that the band pass filter has the best frequency response. The slope is much greater than that for the low pass filter.

Passive Filters : Cut Off Frequency

It is clear that the experimental values far under estimate the theoretical values. Since, even for the Band Pass filter there was only one apparent cut off frequency this indicates likely errors in the execution of the experiment.

Passive Filters : Roll Off

It is clear from figure eight, that the band pass filter has the sharpest roll off.

Final Note for passive filters

Given all the above statements, the reader should also bear in mind that for the band pass filter the input signal amplitude had to the lowered from the value used for the low pass filter, before any useful data could be yielded. If the response of the filter depends on signal amplitude aswell as frequency, the above is not a far comparison.

 

Bibliography

[1] "Electricity and Magnetism", Bleany, B.I. and Bleany, B. OUP, Vol. 1, Chapter 9, pp 269-279 (1989).

[2] "Physics, 2nd Ed.", Ohanian, Hans, C. W.W.Norton & Company, p.441 (1989). ISBN 0-393-95750-0

 

Appendix

Included here is the original print out from the BBC micro of the readings. Given that the signal generator was supposed to be set to 3.5 Volts, the values are curious. Given that the effect of varying amplitude are linear, for purpose of the experiment the values were normalised to the 3.5 Volt limit.