btec hnc - electrical and electronic principles - investigate complex waves

Investigate Complex WavesElectrical and Electronic Principles

By Brendan Burr

Brendan Burr BTEC Higher National Certificate in ElectronicsInvestigate Complex Waves

Table of Contents

TABLE OF CONTENTS - 2 -

TASK 1 - 5 -

1.1 Calculate the r.m.s value of a voltage waveform given by :- - 5 -

volts - 5 -

Solution:- - 5 -

1.2 A complex voltage waveform has an r.m.s value of 220 V and it contains 25% third harmonic and 15% fifth harmonic. Determine - 6 -

(a) The r.m.s value of the fundamental and each harmonic. - 6 -Solution:- - 6 -

(b) The maximum value of the fundamental and each harmonic. - 7 -Solution:- - 7 -

(c) The frequency of the harmonics if the frequency of the fundamental is 60 Hz.- 7 -

Solution:- - 7 -

(d) The equation of the voltage waveform. - 7 -Solution:- - 7 -

1.3 Determine the average power in a 50 Ω resistor if the current i flowing through it is represented by :- - 8 -

mA - 8 -Solution:- - 8 -

2


1.4 A voltage waveform v, represented by :- - 9 -

volts - 9 -

is applied to a circuit and the resulting current i is given by :- - 9 -

amps - 9 -

Determine :- - 9 -

(a) The total active power in Watts supplied to the circuit - 9 -Solution:- - 9 -

(b) The overall power factor. - 11 -Solution:- - 11 -

TASK 2 - 12 -

2.1 Using an Excel Spreadsheet show that the following Fourier Series represents a rectangular voltage waveform of period 2π and amplitude of 2V by plotting a graph of at least 2 cycles of the waveform. - 12 -

- 12 -

Use values of x from 0 to 360 degs in steps of 10 degs for each cycle. - 12 -

Note that the value of x in above Fourier Series is represented in radians. - 12 -Solution:- - 12 -

2.2 The values of a voltage waveform v volts at different times in a periodic cycle are given by the following tabulated data :- - 13 -

(a) Using Excel Spreadsheet copy the above tabulated data into the appropriate cells. - 13 -

Solution:- - 13 -

(b) Draw a graph of voltage v against angle x degrees over 1 cycle only - 13 -Solution:- - 13 -

(c) Using Excel Spreadsheet S/W complete Table 1 of data - 14 -Solution:- - 14 -

(d) Using the Tabulated data find all of the Fourier coefficients correct to 2 decimal places using the appropriate formulae. - 15 -

Solution:- - 15 -

(e) Write the equation for v. - 16 -

3


Solution:- - 16 -

(f) Using Excel Spreadsheet draw a graph of voltage v against angle x degrees by using the equation for v. Use the same values for x as in above table: - 16 -

Solution:- - 16 -

(g) Compare the graph obtained in 2.2 (b) with the graph obtained in 2.2 (f): - 17 -

Solution:- - 17 -

(h) Explain the differences in these results. - 18 -Solution:- - 18 -

(i) Justify the results are valid. - 18 -Solution:- - 18 -

(j) Explain why a numerical method of harmonic analysis is required. - 18 -Solution:- - 18 -

EVALUATION - 19 -

CONCLUSION - 19 -

BIBLIOGRAPHY - 20 -

Books - 20 -

Catalogues - 20 -

Websites - 20 -

4


TASK 1

1.1 Calculate the r.m.s value of a voltage waveform given by :-

volts

Solution:-

5


1.2 A complex voltage waveform has an r.m.s value of 220 V and it contains 25% third harmonic and 15% fifth harmonic. Determine

(a) The r.m.s value of the fundamental and each harmonic.

Solution:-

Volts

6


(b) The maximum value of the fundamental and each harmonic.

Solution:-

Volts

(c) The frequency of the harmonics if the frequency of the fundamental is 60 Hz.

Solution:-

(d) The equation of the voltage waveform.

Solution:-

7


1.3 Determine the average power in a 50 Ω resistor if the current i flowing through it is represented by :-

mA

Solution:-

8


1.4 A voltage waveform v, represented by :-

volts

is applied to a circuit and the resulting current i is given by :-

amps

Determine :-

(a) The total active power in Watts supplied to the circuit

Solution:-

9


10


(b) The overall power factor.

Solution:-

11


TASK 2

2.1 Using an Excel Spreadsheet show that the following Fourier Series represents a rectangular voltage waveform of period 2π and amplitude of 2V by plotting a graph of at least 2 cycles of the waveform.

Use values of x from 0 to 360 degs in steps of 10 degs for each cycle.

Note that the value of x in above Fourier Series is represented in radians.

Solution:-

Task 2.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 100 200 300 400 500 600 700

Degrees

Vo

lts

12


2.2 The values of a voltage waveform v volts at different times in a periodic cycle are given by the following tabulated data :-

x degrees 30 60 90 120 150 180 210 240 270 300 330 360v (volts) 62 35 -38 -64 -63 -52 -28 24 80 96 90 70

(a) Using Excel Spreadsheet copy the above tabulated data into the appropriate cells.

Solution:-

See Task 2.2b.

(b) Draw a graph of voltage v against angle x degrees over 1 cycle only

Solution:-

Task 2.2b

-80-60-40-20

020

406080

100120

30 80 130 180 230 280 330

Degrees

Vo

lts

13


(c) Using Excel Spreadsheet S/W complete Table 1 of data

Solution:-

14


(d) Using the Tabulated data find all of the Fourier coefficients correct to 2 decimal places using the appropriate formulae.

Solution:-

15


(e) Write the equation for v.

Solution:-

(f) Using Excel Spreadsheet draw a graph of voltage v against angle x degrees by using the equation for v. Use the same values for x as in above table:

Solution:-

Task 2.2f

-80

-60

-40

-20

0

20

40

60

80

100

120

30 80 130 180 230 280 330

Degrees

Vo

lts

16


(g) Compare the graph obtained in 2.2 (b) with the graph obtained in 2.2 (f):

Solution:-

I will begin the comparison with the different entries of “v”. There is clearly a difference in the values of “v” when the derived equation is used for calculating the value of Voltage.One thing I noticed was how the differences in values have a tendency of cancelling one another out, for example at 60 and 90 degrees where the values are out by 7 volts in either phase. This tendency has resulted in there being very little effect on the summation of all of the values, which in turn has kept the coefficients at a very similar number.

With the two graphs there are clear similarities with regards to form, but there are also a few anomalies. These anomalies come straight from the two above charts, at 60 and 330 degrees. The effect this has on the waveform is obvious, there are three clear areas where the curve flattens out, rather than continuing in an obvious path. These are between 60 - 90 degrees, 120 - 210 degrees, and 300 – 360 degrees.

Without Equationx v30 6260 3590 -38

120 -64150 -63180 -52210 -28240 24270 80300 96330 90360 70

With Equationx v30 6560 2890 -31

120 -67150 -64180 -50210 -28240 22270 80300 100330 85360 73

17


(h) Explain the differences in these results.

Solution:-

The differences in the results have partly derived from the fact that there was a limitation on the number of decimal places allowed. All of the coefficients that I calculated had many more numbers that had to be cut off and rounded, which begins to make other calculations inaccurate. Another reason is that the formula for calculating the Fourier Coefficients is only an approximate method. Because of this the values are very unlikely to be exactly correct, causing deviances in the answers for the coefficients and then anomalies in the form of the graph.

(i) Justify the results are valid.

Solution:-

The Voltage values that have been calculated from the equation follow a very similar trend as to what is expected. This suggests to me that the values are approximately correct, but slightly out.

(j) Explain why a numerical method of harmonic analysis is required.

Solution:-

In the real world waveforms are produced by an electronic circuit and through measuring equipment such as oscilloscopes you can read them, but the equation is often not provided. Harmonic analysis can be used to provide an equation through careful derivation of points on the waveform, which is often shown as volts/time. Using this information can enable you to calculate coefficients and then the overall equation of the voltage, so at any point in time the voltage can be determined.This can be useful for predicting what the voltage might be at a certain time in a circuit, when an irregular waveform is present, which may help when fault finding.

18


EvaluationFor Task 1.1 I had to calculate the RMS voltage from a complex waveform equation. This was relatively easy once the formula was given to us. It was simply a case of knowing what information had to be extracted from the equation to be able to successfully calculate the RMS value.Task 1.2 presented a problem which was the reverse of being supplied with an equation, instead we had to make the equation from the information given. Again I found this straight forward after receiving the formulas and an example to understand what information needed to be placed where.For Task 1.3, I had to work out the average power when the current (given in the equation) flowed through a 50 ohm resistor. The general calculation for the power when I know the current and resistance isn’t a problem when the current is static, i.e DC, however the thought of the current equation made the question seem difficult. After going through an example everything became clear and it was very obvious what the process was to successfully calculate the power. So after thinking it was going to be difficult, I actually managed to work out the answer fairly easily.1.4 seemed at first to be difficult, because it is difficult to comprehend the waveform from the equation without plotting it on Graphmatica or MS Excel. I soon realised that it was just another number crunching Task, which involved knowing where to enter the relevant data.Task 2 moved onto the subject of Harmonic Analysis, which required the information to be plotted using MS Excel. We usually take a short cut and enter the equations into Graphmatica as this saves a lot of time and can be written like the equation, rather than in MS Excel where the equation has to be entered in a form that the software recognises.The bulk of the time was taken up by entering formula in the correct way into multiple cells to determine the coefficients and the values for the voltage. My results came out as expected, and there was only a slight deviation to the exact voltage values, reasons of which I have explained above.

ConclusionTo conclude, I enjoyed completing this assignment. It has broadened my understanding of the methods of splitting a waveform into segments to create a formula which can enable you to make an equation for the waveform. This new knowledge can also fit into the work at I do onsite as AgustaWestlands. I am please, again, with my presentation of the assignment and the accuracy of my work, these two attributes example the large amounts of time I put into these assignments and the results I am fortunate to get back from them.

19


Bibliography

Through guidance from my lecturer, the following text books, catalogues and websites I was able to complete this assignment:

Books

N/A

Catalogues

N/A

Websites

N/A

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btec hnc - electrical and electronic principles - investigate complex waves

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