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Electronics - Designing Filter Circuits

Free essay example:

DEN 109 Electrical Technology                

DEN 109 Electrical Technology

Lab Experiment 1: Designing Filter Circuits

By Ali Asghari

DEN 109 Electrical Technology

Lab Experiment 1: Designing Filter Circuits

Introduction

The aim of this laboratory experiment was to devise a filter circuits using passive element which consisted of resistors and capacitors. In the experiment we had to design an active filter using operational amplifiers and operational amplifier elements. The filters had different acceptability range of frequencies. The filters are an electronic circuit which performs signal processing which intends to remove undesired signals to and from components but the filter will encourage the desired signals.

Filters are used in communication systems to pass those frequencies containing the desired information and to reject the remaining frequencies which are not within the range we are looking for. In stereo systems, for example, filters can be used to separate a particular band of frequencies for increased or decreased emphasis by the output acoustical system such as amplifier, speaker, etc. Filters are employed to filter out any unwanted frequencies, generally called noise, due to the nonlinear characteristics of some electronic devices or signals picked up from the surrounding medium.

The experiment is to design the filters and consequently look at the difference between the two types of filters using recorded data through practical experiments.

Background Theory

There are two types of filters in this experiment and each one consists of different components. In this segment I will be looking at each filter more directly to distinguish both the explicit differences as well as the implicit differences. Filters provide a way of taking in signals and allowing those frequencies to pass that are desired and prevent unwanted frequencies to go through.

A passive filter is a type of electronic filter that is made only from passive elements - in contrast to an active filter; it does not require an external power source (beyond the signal). Since most filters are linear, in most cases, passive filters are composed of just the four basic linear elements and they are:

  • Resistors - A device used to control current in an electric circuit by providing resistance
  • Capacitors - An electric circuit element used to store charge temporarily, consisting in general of two metallic plates separated and insulated from each other by a dielectric
  • Inductors - One that inducts, especially a device that functions by or introduces inductance into a circuit
  • Transformers - A device used to transfer electric energy from one circuit to another, especially a pair of multiply wound, inductively coupled wire coils that effect such a transfer with a change in voltage, current, phase, or other electric characteristic

The second part of the experiment consisted of designing an active filter. An active filter is one which consists of all similar elements which exists in a passive filter but it also has an operational amplifier which is used to provide gain. An active filter is also based on the principle of the resistor-capacitor [RC] design, but employs an op amp in the design.  The op amp is useful, as it can supply gain. This means the input can be amplified by a set amount to provide the output. Any component that can produce gain is classed as active. Therefore, in general, there are two classifications of filters:

  • Passive filter
  • Active filter

Resistors and capacitors were used in the buildup of the filter to manipulate the current flow. From ohms law, one of widely used equation in electronics, were V=IR, its voltage equal to current multiplied by resistance.  The current in the equation could either be direct or alternating current where both types can have any frequencies above 0 Hz.

The level of impedance (R) of a capacitor is reliant on the frequency of current (I) through it.

Impedance = 1/ jωC→ C is the capacitance which is expressed in Farads

But ω = 2πf→ represents frequency in radians and fis the frequency in cycles measured in Hz.

A voltage divider constructed using resistors and capacitor will explicitly show that output voltage is dependent on the frequency of the input.  The higher the frequency of the input signal, the lower the impedance of the capacitor, therefore the two are working vice-versa.

In the case of a voltage divider circuit on the left, the output       voltage from the potential divider will decrease for higher input frequencies. Here it’s concerned with low pass filter which will mean that output voltage is high for low frequencies and therefore reduce to near enough zero for higher frequencies.

Figure 1: Low pass filter voltage divider

RC Filter Circuit or Passive Filter:

The passive filter circuit is sometimes drawn in different ways but only after careful observation, it shows itself that it’s just another voltage divider. All frequencies perfectly is able to pass through a low pass filter until a desired cut-off point is given, here is would enable all the desired frequencies and reject all other frequencies that doesn’t fit in the correct range.

This graph represents a perfect result which in most cases the experimental result is not as fruitful therefore the resulting experimental graph looks more like a continual curve rather then this perfect conclusion.

Figure 2: Ideal low pass filter response

Cut-off frequency:

Cut off frequency is the boundary line where desired frequencies are passed but reject the undesired frequencies. Here, we are concerned with low pass filters which sometimes instead of cut-off frequency it’s called the corner frequency which defines the end of the pass band.

Which normally at a precise point where reaction drops -3 image00.png(70.7%) from the pass band response (Floyd, 2005)

The cut-off frequency occurs where:

image01.png

Or, image10.pngimage16.png

Thereforeimage17.png

But where active filters are concerned, two networks are formed by image18.pngand as well as image19.pngand image20.pngtherefore the cut-off frequency is given by the following formula:

image21.png

Equipments

The experiment requires the following equipments to compare the performance of an active and passive low filter to compare practical results with the theoretical values.

  • Resistors, capacitors, op amps
  • Bread board
  • Signal generator
  • Oscilloscope
  • Digital multi-meter
  • DC power supply

Experiment Part 1: RC Circuit as a Passive Low-Pass Filter

This is the method to set-up an RC circuit as a passive low pass filter:

Firstly the oscilloscope was turned on and waveform generator then the sinusoidal waveform was selected by pressing the right button on the front panel of the waveform generator. Then connect the probe from the oscilloscope to the probe from the waveform generator made sure that channel 1 was selected to obtain the right reading off the wave-generator. On the oscilloscope there was a curve which looked like the sinus graph which the reader had to get the peak to peak voltage of the waveform and measure the frequency of the waveform from the display. Ensured that the frequency obtained from the oscilloscope is similar to what the waveform generator showed. The frequency and amplitude was changed off the waveform generator through the adjustment of the knobs on the front panel then recorded the changes in the frequency and amplitude of the waveform shown on the generator. Then an RC low pass filter was constructed as show below.

image02.png

Figure 3: RC Low-pass filter

Then a series of measurement were performed where it started from 0 dc to 2 kHz. To be able to calculate gain Vout / Vin this is calculated from wave generator and then the gain is normalized. Normalisation is a process where any quantity is divided by itself to give a maximum value of 1 and to establish a dimensionless level of range. Gain can calculated in dB through image03.png where the maximum value should be 0.

Here are the results for part 1 of the experiment:

Fc

Vin

Vout

Gain:

Vout/Vin

Normalisation

Gain dB

0

0

0.0

0.0

0.000

0.000

2

20

5.2

0.3

0.347

-9.202

4

20

9.6

0.5

0.640

-3.876

6

20

12.5

0.6

0.833

-1.584

8

20

14.5

0.7

0.967

-0.294

10

20

15.0

0.8

1.000

0.000

12

20

14.0

0.7

0.933

-0.599

14

20

14.5

0.7

0.967

-0.294

16

20

14.0

0.7

0.933

-0.599

18

20

14.0

0.7

0.933

-0.599

20

20

13.5

0.7

0.900

-0.915

22

20

13.0

0.7

0.867

-1.243

24

20

12.5

0.6

0.833

-1.584

26

20

12.0

0.6

0.800

-1.938

28

20

12.0

0.6

0.800

-1.938

30

20

11.0

0.6

0.733

-2.694

35

20

10.0

0.5

0.667

-3.522

40

20

9.0

0.5

0.600

-4.437

45

20

8.0

0.4

0.533

-5.460

50

20

7.5

0.4

0.500

-6.021

60

20

6.6

0.3

0.440

-7.131

70

20

6.0

0.3

0.400

-7.959

80

20

5.5

0.3

0.367

-8.715

90

20

4.8

0.2

0.317

-9.988

100

20

4.0

0.2

0.267

-11.481

200

20

2.3

0.1

0.150

-16.478

500

20

1.0

0.1

0.067

-23.522

1,000

20

0.5

0.0

0.033

-29.542

2,000

20

0.3

0.0

0.017

-35.563

The graph below represents the dB gain obtained through the first part of the experiment.

image04.png

image05.png

Calculations:

Calculating the theoretical value for cut off frequency, it is demonstrated below, the cut off frequency for experiment 1 (RC filter) can be calculated as follows:

image06.png

We know the resistance and the capacitance values:

C = 0.1 * 10-6 F

R = 82000 Ω

image07.png

The actual cut of voltage is the 70.7% of the maximum value, therefore:

For experiment 1:

Vcut = 20 x 0.707 = 14.4 V

For experiment 1 we can linearly interpolate by using bottom and top end frequency values of the theoretical cut off frequency (19.41) therefore its between 10 Hz & 12 Hz. We also have the voltage output for these values, we can therefore calculate as follow:

Voltage Out (V)

15

14.4

14

Frequency (Hz)

10

y2

12

Solving for y2:


image08.png

y2 = [(14.4 – 15) (12 – 10)/ (14 – 10)] + 10

y2 = 9.7 Hz

Experiment Part 2:

Operational Amplifier Circuit as an Active Low-PassFilter

In this part of the experiment Butterworth filter was used which demonstrates very flat amplitude in its pass. The diagram below illustrates a two-pole Butterworth low-pass filter because the two pole filter uses two RC networks to produce a roll-off rate of –40 dB.

The active filter below is wired to give a unity gain (output voltage = input voltage) within the passband as it is connected as a voltage follower.

Method for Experiment Part 2:

Firstly the power supply was turned on then connected and adjusted the power supply to

+/- 15 V. Following that oscilloscope was used to verify the voltage displayed is close +/- 15 V. The dc power supply was switch off and the waveform generator was turned on following that probe 1 was connected to the waveform generator and selected channel 1. The peak to peak voltage of the waveform was viewed on the oscilloscope. Then the frequency was measured of the waveform display and ensured that frequency shown on the oscilloscope was the same as the one shown waveform generator.

Then constructed an active low pass filter;

image09.png

Figure 4: Active Low-pass filters using an op-amp.

Then a series of measurement were performed where it started from 0 dc to 2 kHz. To be able to calculate gain Vout / Vin this is calculated from wave generator and then the gain is normalized. Normalisation is a process where any quantity is divided by itself to give a maximum value of 1 and to establish a dimensionless level of range. Gain can calculated in dB through image03.png where the maximum value should be 0.

Here are the results for part 2 of the experiment:

Fc

Vin

Vout

Vout/Vin

Normalised

Gain

Gain dB

image03.png

0

0

0.0

0.000

0.000

0.000

2

5

4.8

0.923

0.997

-0.027

4

10

9.0

0.900

0.972

-0.247

6

14

12.5

0.926

1.000

0.000

8

15

13.0

0.867

0.936

-0.574

10

16

14.0

0.875

0.945

-0.491

12

18

13.5

0.750

0.810

-1.830

14

19

13.0

0.684

0.739

-2.628

16

19

12.0

0.649

0.701

-3.091

18

20

10.5

0.538

0.582

-4.708

20

20

10.0

0.500

0.540

-5.352

22

20

9.0

0.450

0.486

-6.267

24

20

8.5

0.425

0.459

-6.764

26

20

8.0

0.400

0.432

-7.290

28

20

7.0

0.350

0.378

-8.450

30

20

6.2

0.310

0.335

-9.504

35

20

5.0

0.250

0.270

-11.373

40

20

4.0

0.200

0.216

-13.311

45

20

3.0

0.150

0.162

-15.810

50

20

2.6

0.130

0.140

-17.053

60

20

2.0

0.100

0.108

-19.332

70

20

1.4

0.070

0.076

-22.430

80

20

1.2

0.060

0.065

-23.769

90

20

1.0

0.050

0.054

-25.352

100

20

0.8

0.040

0.043

-27.290

200

20

0.2

0.009

0.010

-40.247

500

20

0.1

0.004

0.004

-48.450

1,000

20

0.0

0.000

0.000

-67.290

2,000

20

0.0

0.000

0.000

-79.332

The graph below represents the dB gain obtained through the second part of the experiment.

image11.png

image12.png

Now to calculate the cut off frequency of experiment 2 (op-Amp filter) we can use the following formula:

image13.png

We know the resistance and the capacitance values:

C1 = 0.1 * 10-6 F

C2 = 0.1 * 10-6 F

R1 = 82000 Ω

R2 = 82000 Ω

image14.png

It is now important to compare the values of the theoretical cut off frequency against the actual cut off frequency.

Firstly we are required to calculate the actual values of the cut off voltage, in order to use linear interpolation to calculate the cut off frequencies. The actual cut of voltage is the 70.7% of the maximum value, therefore:

For experiment 2:

Since the value of input voltage is not the same throughout the experiment, I have decided that I will use an average value to calculate the cut-off frequency to obtain a more accurate value.

Therefore

Vcut = 17.748 x 0.707 = 12.55 V

For experiment 2 we will need to use the frequency values of 10 & 12Hz:

Voltage Out (V)

13

12.55

12

Frequency (Hz)

14

y2

16

Solving for y2

image08.png

y2 = 14.9 Hz

Therefore y2 for the first experiment and y2 for the second experiment are 9.7 Hz and 14.9 respectively.

Discussion

The findings of the experiment are concisely presented in the table below to make easier to analyse, therefore the conclusion is;

Experiments

Theoretical Calculation (Hz)

Experimental values (Hz)

1

14.4

9.7

2

12.55

14.9

As the table above show, the theoretical cut-off frequencies are 14.4 Hz and 12.55 Hz respectively, the value for the theoretical cut-off frequency was calculated using the equation provided at the beginning of the experiment. The theoretical results vary from the actual experimental cut-off frequency obtained from experiments 1 and 2. The experimental  results for the cut-off frequency from experiment 1 gave a value of 9.2 Hz, which depict a huge difference between the theoretical  and the experimental values which could have be caused by numerous errors when the actual circuit was being built through multiple inconsistencies. We should have had one person reading from oscilloscope instead at least five different people read it causing inaccuracies in the experiment.

Moreover for the second experiment the theoretical value for cut off frequency is 12.55 Hz and the experimental value is 14.9 Hz which show a difference off 2.35 Hz. This would suggest that the results obtained aren’t as reliable and accurate because the two experiments show completely different patterns. This could be due to the readings from the oscilloscope not being taken correctly and noise interference should also be considered, from these probable inaccurate interpretations the cut off frequency was calculated using linear interpolation which itself is just an estimate, which leads to more inaccuracy of the results. Interpolated result is calculated through combination of data which consists of multiple errors.

Calculated percentage error for experiments 1 and 2 are 36.2% and 18.7% respectively highlighting the fact there were some major error when conducting the experiments.

When calculating the percentage error for experiment 1 and 2 we can see that experiment 1 gave a percentage error of 36.3%, and experiment 2 gave a percentage error of 18.7%. As the percentage errors show neither of the results is anywhere accurate to have any significant importance. Again this could be due to the readings from the oscilloscope not being taken correctly.

image15.png

Comparing the two experiments it’s explicit that the gap between the two experiments are widening as you approach the 2000 Hz point but there as slight exception between 1 to 9 Hz where passive filter dB gain dropped further the Active filter dB gain which is against the overall pattern of how the two circuits work.

Conclusion

Electronic filters are electronic circuits which perform signal processing functions, specifically intended to remove unwanted signals from components and enhance wanted ones therefore such tool is very important in variety of systems. The experiments have given us a first hand experience and see how signal can be diverted and manipulated to serve a specific task. The elements which generates or produces electrical energy is called active elements where as passive elements are components which consume rather than produce energy. The experiment has overall very successful which even under crowded circumstances the results are roughly in the right path.

References

  • Lecture notes chapter 4 ‘operational amplifier’, ‘filter circuits’ in chapter 3
  • http://www.ajdesigner.com/phpinterpolation/linear_interpolation_equation.php
  • http://en.wikipedia.org/wiki/Electronic_filter

Name: Ali Asghari - (ex07028) - 070693030

        

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