Experiment 17 - Experimental Error Analysis

Transcription

1 Experiment 17 - Experimental Error Analysis Student ID: Group number: 44 Module code: ELEC273 Department of Electrical Engineering and Electronics October 20, 2015 Abstract This report presents an error analysis process of the measurements of several electronic parameters in an amplifier circuit. The concrete lab section includes the common emitter amplifier circuit construction, and corresponding DC and AC analysis of the circuit. The measurement data were all recorded in a scientific way with each correponding uncertainty. Also, the experimental errors caused by either the apparatus themselves or the experimental process were merticulously classified and analised. A reasonably comprehensive error analysis of experimental data is demonstrated. Declaration I confirm that I have read and understood the University s definitions plagiarism and collusion from the Code of Practice on Assessment. I confirm that I have neither commited plagarism in the completion of this work nor have I colluded with any other party in the preparation and production of this work. The work presented here is my own and in my own words except where I have clearly indicated and acknowledged that I have quoted or used figures from published or unpublished source (including the web). I understand the consequences of engaging in plagarism and collusion as described in the Code of Practice on Assessment (Appendix L) 1

2 Contents 1 Introduction Experimental Purpose Backgroud and Theory Materials and Methods Apparatus Experimental Procedure DC Bais of a Common Emitter amplifier Amplification of AC signals Result DC Anlysis Theoretical Result Experimental Result AC Anlysis Theoretical Result Experimental Result Discussion Error Analysis Uncertainties Systematic Error Propagated Error Experimental Process Limitation Conclusion Achievement Errors and Limitations Futural Improvement Question 15 7 Reference 15

3 1 Introduction 1.1 Experimental Purpose The objectives of this lab section are firstly to familiarise the experimenters with how to classify, report and reduce different types of errors, and secondly to practice building a common emmiter amplifier circuit and analysing it in terms of both the DC bias settings and the AC signals. Figure 1: Common Emitter Amplifier Circuit 1.2 Backgroud and Theory Measuring physical quantities properly in an experiment is an essential skill for electronic engineers since virtually every experiment requires data measurement to some extent. However, errors caused by either apparatus or accidental inaccurate conductances are not aviodable. Error includes systematic error, personal error and random error. Systematic error is caused by the apparatus itself. For exampe, the digital/anologe instruments are actually not up-to-state so that contain error to some degree. Personal error is usually due to imperfect experimental conductance such as misreading and wrong calculation. Random error refers to the error occurring in different times of conductance of the same mesurement under the same conditions. When representing the results, a number called uncertainty should follow the measured value. For digital instrument, the uncertainty is the measurement times 0.5 percentage while for analog instrument, the uncertainty is half of the minimum scale of the equipment. 1

4 2 Materials and Methods 2.1 Apparatus 1. EB2025T DC power supply 2. FG200 2MHz function generator 3. HAMEG 35MHz Analog Oscilloscope 4. TENMA Digital Multimeter 5. SK10 bread board 6. Transistor BC109, resistances and capacitances 7. Multisim 13.0, icircuit (simulation software) Figure 2: Instrument Picture 2

5 2.2 Experimental Procedure DC Bais of a Common Emitter amplifier Oscilloscope Calibration: Firstly, Channel One was connected to the 0.2VPP point through a probe. Then, after adjusting the voltage division and period, the displayed waveform on the screen of oscilloscope was a square wave with expected frequency, as shown in Figure 3. Figure 3: Calibration Waveform However, when conducting the same process of this calibration for Channel Two, a fault of the equipment was found Channel Two did not display a square wave but simply showed a flat line in the middle of screen. Fortunately, this lab did not require the use of channel two therefore we sticked to this oscilloscope anyway. Power Supply Connection: Firstly, two wires (one is red and one is black) were used to connect the DC power supply outputs and two parallel lines on the side of the bread board to serve as poles. Next, the power supply output was adjusted to be 15V and this output was tested by the oscilloscope using proper probe. Then, with the aid of the flat line and voltage division indication, the approximate value of the DC power was doubble checked to be 15V though with certain degree of uncertainty. Additionally, a good method of wring can be seen in Figure 4, which was learnt from the demonstrator. This wiring style could enable the connection to be more stable. Figure 4: Wiring Method 3

6 Circuit Construction and testing: Firstly, the resistor values were all measured using the digital multimeter and thier tolerances were recorded based on thier color bars. Next, the two resistor R1 and R2 were plumped onto the bread board according to the circuit configuration shown in Figure 5. Figure 5: Bias Setting [1] Then, the voltage between the negative pole and the middle point of the two resistor R1 and R2 was measured using the digital multimeter and oscilloscope respectively. Following that, the transistor was inserted, and the other two resistor R3 and R4 were added to the circuit as shown in Figure 5. Then, the voltage of emiiter, colltector and base were all measured in the same way as we did for the middle point volatge. Effect of changing transistor: Right after the previous step, all those voltages were measured and recorded again for another transistor plugged into the circuit board after finishing the previous steps of the measurement of the first transistor. 4

7 2.2.2 Amplification of AC signals Circuit Constrcution Firstly, capacitors and a resistor serving the load were added to the circuit as shown in Figure 3. Then, the variation resistor was first adjusted to its biggest value which is approximately 10k ohm using a screw before being inserted onto the board. Figure 6: Amplifier Configuration [1] Testing the circuit First, the output pin of the generator was connected to the input point of the circuit as shown in Figure 3. Then, the frequency of the signal from the generator was set to be 1000Hz. After that, the displayed sine wave of the C1 was adjusted to have amplitude of 50mV by rotating the voltage control on generator and the variation resistor. Next, the output voltage measured across the R L was obtained, and the corresponding gain was calulated. After that, the amplitude of the output was increased to the state where the signal observed on the screen of oscilloscope became distorted so that the maximum input signal the circuit could handle was found. Finally, the output smplitude was decreased back to 50mV and repeat this testing process for another transistor. 5

8 3 Result 3.1 DC Anlysis Theoretical Result Analysing the biasing setting circuit: For the circuit given in Figure 3, it is already known that R 1 = 22kΩ, R 2 = 4.7kΩ, R C = 1.3kΩ, R E = 330Ω, V CC = 15V, V BE(on) = 0.7V, and β = 290. Figure 7: Equivalent Circuit of Bias Setting Using the Thevenin equivalent circuit in Figure 7, we have R T H = R1 R2 = (22)(4.7) = kΩ (1) V T H = R 2 V CC = 4.7 (15) = V (2) R 1 + R Writing the Kirchhoff voltage law equation around the BE loop, we obtain I BQ = V T H V BE(on) R T H + (1 + beta)r E = The collector current is ( )(330) = A (3) I CQ = βi BQ = (290)( ) = A (4) And the emitter current is I EQ = (1 + β)i BQ = ( )( ) = A (5) 6

9 The quiescent CE voltage is then VCEQ = VCC ICQ RC IEQ RE = V (6) The emitter voltage is then VEQ = RE IEQ = ( )(0.33) = V (7) The collector voltage is then VCQ = VCEQ + VEQ = V (8) VBQ = VT H RT H IBQ = = V (9) The base voltage is then Experimental Result The experimentally measured values of the base voltage(vb ), emitter voltage(ve ), and collector voltage(vc ) can be seen in Figure 8. Figure 8: Voltages of the First Transistor The absolute error is Errorabs = xexp x [1] (10) The error percentage is Error% = Errorabs 100% xexp [1] (11) The uncertainty for digital instrument like multimeter is U ncertainty = Actual 0.5% 7 [1] (12)

10 Based on the experimental data as shown in Figure 8 and the three formulas above, the data table below as can be seen in Table 1 can be obtained (actual values are measured using multimeter). VB VE VC Expected Actual ± Uncertainty Abs.error %error % % % Table 1: Voltages of the First Transistor (Multimeter) VB VE VC Expected Actual ± Uncertainty Abs.error %error % % % Table 2: Voltages of the First Transistor (Oscilloscope) Similarly, for the second transistor, the measured data can be seen in figure 9. Figure 9: Voltages of the Second Transistor Substituting the data into the same three formulas, the obtained data set is Table 3 (actual values are measured using multimeter). VB VE VC Expected Actual ± Uncertainty Abs.error %error % % % Table 3: Voltages of the Second Transistor (Multimeter) VB VE VC Expected Actual ± Uncertainty Abs.error %error % % % Table 4: Voltages of the Second Transistor (Oscilloscope) 8

11 As required by the lab script, the actual values were measured using oscilloscope, and the corresponding data table is shown in Table 2 and Table 4. Since the voltage selected division is 5V, which is big compared to the tiny readable measurement differences between two transistor, the actual results of two transistor are virtually the same. 3.2 AC Anlysis Theoretical Result The small-signal hybrid- parameters are Figure 10: Equivalent Circuit of Amplifier r π = βv T I C = (290)(0.026) = Ω (13) The input signal is The output signal is The gain is g m = I C = V T V π = here we regard r o is infinitely big, therefore = A/V (14) R 1 R 2 r pi R 1 R 2 r π + R S V S (15) V O = gmv π (r o R C ) (16) A V = V O V π = g m (r o R C R L ) (17) A V = g m (R C R L ) = ( )( (1200)(1300) ) = (18)

12 3.2.2 Experimental Result The amplitude graph of the 50mV input signal, the corresponding output siganl, and the amplitude of the output siganl when it strated to distort were all obtained using the oscilloscope. For the first transistor, these three amplitudes are demonstrated in figure 11. Figure 11: Amplitudes of the First Transistor For the second transistor, those three amplitude graphs are presented in Figure 12. Figure 12: Amplitudes of the Second Transistor Reading from the graphs, both two transistors shared virtually the same gain since they have got quite similar amplitude graphs. V IN = 50 ± 5mV, V O = 4.9 ± 0.1V, V distort = 9.5 ± 0.2V; And for the second transistor, V O = 5.1 ± 0.1V, V distort = 9.8 ± 0.2V It can be clearly seen that the experimental result of the gain is approximately 98, which means that the amplifier actual amplifying function is to amplify the signal for around 100 times. And based on the calculated gain and the amplitude when waveform strated to distort, the maximum input can be calculated out to be around 97 ± 5mV for the first transistor and 96 ± 5mV for the second transistor. The similation software were used to double check the expected theoretical results. It was found that the BC109BP transistor component in Multisim 13.0 does not share the same parameters with the experimental transistor since the simulation result of gain is -106, while the icircuit result verified the calculated theoretical result by giveing a gain of -145, which is closer to the calculated expected value. 10

13 The simulation result of gain using software Multisim 13.0 is presented below in Figure 13. Figure 13: Simulation Result using Multisim 13.0 The simulation result of the gain using software icircuit is presented below in Figure 14. Figure 14: Simulation Result using icircuit 11

14 4 Discussion Error Analysis Uncertainties The measured value are presented with uncertainties in Table 5. Since oscilloscope is analog instrument, the uncertainty is the half of the minimum scale which is determined by the voltage division selection for each volatge measurement (the voltage division of each measurement can be seen in Figure 8 and Figure 9). VB VE VC VIN VOU T Vdistort gain VM AXIN First Transistor Second Transistor 2.5 ± 0.5V 2.0 ± 0.5V 8.0 ± 0.5V 50 ± 5mV 4.9 ± 0.1V 9.5 ± 0.2V 98 ± 4 97 ± 5mV 2.5 ± 0.5V 1.9 ± 0.5V 8.0 ± 0.5V 50 ± 5mV 5.1 ± 0.1V 9.8 ± 0.2V 102 ± 4 96 ± 5mV Table 5: Resistor Data Set Systematic Error The resistor actual values were not equal to the expected values as can be seen in Figure 15. Figure 15: Resistor Measurement Result As can be clearly seen in Figure 15, the actual resistance of each resistor is not the same with the expected value of it. This error is systematic error witch exists in equipment itself. The resistors used in this lab had tolerance of 5%, wihch contributed to the error as well. Apart from this typical error in resistance, all the other measurement eqiupment contains systematic error to different extent. To reduce this type of error, the most direct method is to use devices and components with accuracy as high as possible. Also, conducting more times of measurement to obtain average value should reduce the error as well. 12

15 The data table of these resistors are presented be low in table 6. Expected Actual ± Uncertainty Abs.error %error R 1 22k % R 2 4.7k % R 3 1.3k % R % R L 1.2k % Table 6: Presentation of results with uncertainties Propagated Error This type of error typically exists in the calculation of gain. This is because: the gain is obtained by dividing the measured V IN by the measured V OUT, which means that both the numerator and the demoniator are read from the instrument. Two measured experimental values will conduct a calculation reasult with error including both two measurement error. Similarly, the maximum input signal calculation which is obtained using measured experimental values contains this type of error. The concrete calculation process is gain gain = V IN V IN + V OUT V OUT [1] (19) Substituting the concrete values of experimental result, we obtain the uncertainty of gain gain = ( )(98) = 4 (20) 4.9 gain = ( )(102) = 4 (21) 5.1 These two gains are the same bacause the uncertainty is usually one significant number. 4.2 Experimental Process Limitation Inadequate times of measuremen For each transistor, the measurements of volatages should have been done for more times. Which means that the same transistor should be plugged into the circuit for many times and the each time, the same measurement process should be conducted. In this way, more groups of data could have been recorded and the average measured data x can be worked out using this group of number rather than simply using one data as what was done in this report. The random error might increase because of this limitation. The scale selection of oscilloscope 13

16 When measuring the voltages, the scale selected were slightly too big. For example, in the DC bias setting section, the experimental data were measured using the biggest sacle 5V rather than smaller but more suitable scale such as 1V or even 50mV. This limitation might increase the reading error, or saying, personal error. Apparatus were not up-to-date It was found that the result measured using oscilloscope were smaller compared to that measured using multimeter. The oscilloscope used here has a non-functioning Channeal Two which indicated that this oscilloscope is not quite advanced and in perfect condition. This limitation might increase the systematic error. 5 Conclusion 5.1 Achievement The main achievements of this lab section are that the measurements of base voltage, emitter voltage, and collector volatge in the DC bias setting circuit generally agree with their expected theoretical values and the gain of the AC amplifier circuit was tested practically. The method of analysing errors was comprehensively learnt with the aid of the required lab exercises and the knowledge on the pre-lab introduction script. Also, the DC analysis and AC analysis of npn transistor amplifier were reviewed by calculating the theoratical results and their circuit construction were practiced successfully. 5.2 Errors and Limitations However, there are some experimental errors existing in the results obtained in the experiments. These errors include systematic error, personal error and random error. The general error percentage is around 5%-10%. Also, the measurement was not conducted for many times for each parameter. This made it impossible to calculate the standard error to represent the uncertainty, which increase the random error. 5.3 Futural Improvement To improve the accuracy of the experiment, both the instrument with higher quality, and more careful operations are required. For example, proper scales should be selected when measuring the voltages using oscilloscope and more groups of measurements should be conducted to obtain valid average value of the experimental result to reduce random error. 1. Firstly, in terms of the device perspective, it was found that the equipment in the lab were not all up-to-state and worked perfectly. For example, the oscilloscope used in this section has a faulty Channel Two which could not even generate square wave when being calibrated. 2. Also, the accuracy of experiment conductance process should be improved as well. First, suitable scale should be chosen to measure the voltages when using the oscilloscope. This particularly means that the scale should not be selected too big comared to the actual experimental value; Second, the measurement of one parameter should be carried out repeatatively for more than one time to obtain a group of more accurate results. 14

17 6 Question 1. What effect does changing the transistor have effect on the DC bias and hence the AC gain? Answer: For different transistor β, the DC and AC analysis result will be different. Different transistors have different β, and even if they are the same type, they should have slight different which result in slight different amplifying result and volatges as examined in this report. 2. Is there any affect of the input frequency (AC situation) on the response of the amplifier? Explain your answer. However, in this particular lab section, the two transistor in the same type were actually working quite similarly with virtually the same measurement results, which demonstrated that thwy have quite similiar parameters including β Answer: Different frequencyies affect the AC circuit function situation slightly. For example, in this circuit, the frequency value 1kHz is classified into so-called midband range frequency which means that the capacitor can be regarded as short circuit. The amplifier gain remains a constant when the frequency value is within the midband range [2]. Figure 16: Resistor Measurement Result 3. Which part of the experiment was most successful? Why? Answer: All the required parts of the lab section were conducted within the limited time. The experimental data were basically valid since the error persentages are generally around 5% which is not excessively big. 4. What changes could be made to improve the experiment? Answer: First, the scale selection of the voltage divison of the oscilloscope should be more reasonale. For example, if the measure value is just around 2 volts, then the voltage division should not be 5V. In this way, the personal error occurring when reading the value should be decresed. Second, a group of repeatative measurements should be carried out of each trnaistor so that a more valid average value of the measured data can be obtained. This can decrease the random error effectively, and in this case, the uncertainty can be expressed by standard error which is more accurate. Third, the apparatus used should be more up-to-date so that the systematic error can be reduced. 7 Reference 1 University of Liverpool, Experimental Error Analysis script,ver 3.3, Aug, D.A.Neamen Microelectronics: Circuit Analysis and Design, Fourth Edition,