Download 2.2 Electric circuit

Extended Essay in Physics
Investigation of influence of the frequency of the electric
current on the performance of transformer:
How does the frequency of current influence the input and output powers
of the transformer and what is the effect on the efficiency of the
transformer?
Word Count: 3025
1
Table of contents
Abstract......................................................................................................................3
1. Introduction..........................................................................................................4
1.1 Background.....................................................................................4
1.2 Objective..........................................................................................4
2. Equipment description and setup.................................................................4
2.1 General description.........................................................................4
2.2 Electric circuit.................................................................................5
3. Procedure..............................................................................................................5
3.1 Preparation......................................................................................6
3.1.1 Design of the electric circuit.......................................6
3.1.2 Gathering the data.......................................................6
3.1.3 The rate and amount of measurements.......................7
3.2 Conducting the experiment............................................................7
4. Data collection and processing...............................................................7
4.1 Graphs..............................................................................................7
4.2 Raw data..........................................................................................9
4.3 Processed data...............................................................................11
5. Data presentation and analysis............................................................13
5.1 Presentation....................................................................................13
5.2 Analysis...........................................................................................14
5.2.1 Uncertainties............................................................15
6. Conclusion and evaluation..........................................................................18
6.1 Conclusion.......................................................................................18
6.2 Evaluation.......................................................................................18
7. Bibliography....................................................................................................19
2
Abstract
This essay investigates how does the different frequency of electric current influence
the performance of transformer. Essay investigates how does the change of frequency influence the
input power, output power and efficiency of transformer. The research was done by applying the
current of 10 volts to primary coil of the transformer and then measuring both the voltage and
current in both primary and secondary coils of transformer. The product of voltage and current gave
the power in primary and secondary coils. The ratio of power in secondary coil and power in
primary coils gave the efficience of the transformer. The frequency was being controlled and being
changed from 5 Hz to 1 000 Hz with the help of power amplifier and the data from ampere-meters
and voltmeters were represented in a graphical way.
The investigation had not shown a significant change in the efficiency of transformer
when the frequency of the current was being changed in increasing intervals from 5 to 100 Hz. By
going to the frequencies higher than 100 Hz, the efficiency was decreasing in an increasing rates.
Powers in both primary and secondary coils have been almost constant in the interval from 5 to 100
Hz, but were decrasing in linear relation starting from 100 and continuing till the 1 000 Hz. Even
though the resistances of ammeters and voltmeters may have influenced the results of the
investigation, it can be clearly seen from the graphs that the change of frequency had the biggest
impact on the performance of transformer.
Word Count: 253
3
1. Introduction
1.1 Background
Transformers are not only being used in power lines in order to minizmize the heat losses when
distributing the electricity, but also in a daily-used appliances. Many devices, for example,
computers or mobile phones use transformers to change the voltage to the one that is needed to
properly work with that device. One important aspect of transformers is that they are usually
designed to be working in certain frequency of electric current. As there are different types of
transformers, including those which work in an official electric frequency of Europe – 50 Hz, those
that work in high frequencies (usually 440 Hz) and those that are designed to work in North
America and other regions with the supplied current of 60 Hz, it would be interesting to investigate
how do transformers perform in different frequencies of the current.
1.2 Objective
The objective of this study is to investigate how does the change of frequency influence the input
power, output power and efficiency of the transformer. Therefore there are three research questions
that are very closely related to each other and which require the same methods of investigation:

How does the change of frequency of the current influence the input power of the
transformer?

How does the change of frequency of the current influence the output power of the
transformer?

How does the change of frequency of the current influence the efficiency of the transformer?
2. Equipment description and setup
2.1 General description
The list of equipment:

Electric wires that connected the devices in electric circuit which is shown in Picture 1.

2 Nova computers which were gathering the data from both ammeters and voltmeters.

2 ammeters one of which was connected to the primary coild and the other – to the
secondary.
4

2 voltmeters one of which was connected to the primary coild and the other – to the
secondary.

Power amplifier „Xplore GLX“ that produced a current of 10 volts and was able to change
the frequency of it.


0,5 ohm resistor which served as a load in order not to have all the current flowing through
the voltmeter so the voltage and current could me measured properly.
Step-down transformer that can step-down the voltage over 19 times.
2.2 Electric circuit
Figure 1.1
Figure 1.2
5
As it can be seen in Figure 1.2 the primary coil of transformer is connected to the polygon
ACEBDF and the secondary coild is connected to the polygon GHJLKI.
Alternating current (later: AC) source in AB represents current of 10 volts that is being
produced by power amplifier.
Ampermeters in BD and HJ parts are connected in series and they measure the current
flowing through respectively primary and secondary coils.
Voltmeter in CD measures the voltage across the primary coil of tranformer.
Voltmeter in KL is connected in parallel with a resistor, because in other case voltmeter
would be connected in series with ammeter and which would result in near to infinity
equivalent resistance in secondary coil.
3. Procedure
3.1 Preparation
In the stage of preparation it was important to design how will the circuit look, how
the data will be measured, analyzed and interpreted.
3.1.1 Design of the electric circuit
The most important was to decide how will the ammeters and voltmeters be arranged
in a circuit so that the data could be measured accurately. I have put ammeter in part BD so that it
would be in series with the AC source. Therefore, the voltmeter which is connected in parallel with
the primary coil and the series of AC source will measure the coltage which is of course the same
for parts CABD and EF as they are parallel.
Also, one change was made to the design of a circuit during the process. At first, the
resistor was not put in part IJ. As the data was being gathered, it did not look realistic (the current
value was near to zero) and thus I have made an assuption that the voltmeter is connected wrong in
the circuit. As the voltmeter has to be connected in parrallel, it needed some kind of load in parallel
with it, so that the current would almost not flow through the voltmeter. For that purpuse a resistor
of value of 0,5 ohms was chosen..
3.1.2 Gathering the data
As my school did not have four different multimeters for measuring the voltages and
current in both primary and secondary coild, I have decided to use Nova computers and its ammeter
and voltmetrs for this procedure. Two voltmeters and two ammeters that were used in this
investigation required to be connected to Nova computers so the data from them could be gathered
6
and analyzed. As Nova computer has 4 ports, two voltmeters and two ammeters were connected to
it. It was strange to see that the voltage from the secondary coil was zero. As it did not sound
logical, assumption was made that there is a short connection in the Nova computer might have
occured. Therefore another Nova computer was turned on. This time the ammeter and voltmeter
from primary coil were connected to one computer and the ones from secondary coil to another
computer. This time the data gathered seemed much more logical – the voltage was decreasing and
the current was increasing, just like the step-down transformer should perform.
3.1.3 The rate and amount of measurements
It was also important to decide how much measurements should be taken with the
Nova computers and what the rate of theirs should be. I have set the maximum 10 000
measurements per second, so that the graph which had to be produced would be as accurate as
possible as that the peaks of sinusoids would be detected.
3.2 Conducting the experiment
The experiment was being conducted with in total 46 different frequencies (I have measured even
more, but the accuracy after 1000 Hz has drastically decreased as the maximum rate of
measurement was not as big). I first started with 5 Hz and was increasing the frequency by 5 Hz,
but later the intervals between two frequencies were being increased as the changes were not as big.
4. Data collection and processing
4.1 Graphs
The Nova computers represented the gathered data in graphs – the sinusoids of voltage and current
by time. After the experiment there were four sets of graphs:
1. Voltage in primary coil
2. Current in primary coil
3. Voltage in secondary coil
4. Current in secondary coil
To show how did it look, in Figures 2.1 and 2.2 the voltage and current in primary coil can be seen
represented in the graph, when frequency is equal to 10 Hz.
7
Figure 2.1
Figure 2.2
8
One of the problems faced which that can also be seen in the graph was that there was
an offset for both voltage and current (which has been changing every time and thus could not be
automatically eliminated). Therefore, it was not enough just to look where the peak of sinusoid is.
In order to the most accurate data for both voltage and current I decided to interpolate the data using
Graphical Analysis program which when set to search for a sine function, finds the most accurate
values for it. The way program interpolates the data can be seen in Figure 3.
Figure 3
4.2 Raw data
Using the interpolation method that is explained above, data about voltage and current in both
primary and secondary coils was gathered and represented in a table which is shown in Figure 4
below.
9
Figure 4
Frequency Voltage
f (Hz)
V 1 (V)
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
110
120
130
140
150
160
170
180
190
200
220
240
260
290
320
360
400
450
500
550
600
670
740
820
900
1000
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
10,00
9,99
9,97
9,96
9,97
9,98
9,99
9,98
9,96
9,98
9,96
9,96
9,94
9,89
9,95
9,98
9,98
9,98
9,91
9,89
9,90
9,87
9,86
9,86
9,86
9,80
9,81
Current
I1 (A)
0,0166
0,0166
0,0165
0,0165
0,0165
0,0165
0,0165
0,0165
0,0165
0,0165
0,0164
0,0165
0,0165
0,0165
0,0165
0,0165
0,0164
0,0164
0,0164
0,0163
0,0162
0,0161
0,0161
0,0161
0,0161
0,0159
0,0158
0,0157
0,0157
0,0156
0,0155
0,0153
0,0151
0,0149
0,0146
0,0142
0,0138
0,0134
0,0128
0,0124
0,0119
0,0113
0,0106
0,0098
0,0092
0,0086
10
Voltage
V 2 (V)
0,528
0,528
0,528
0,528
0,528
0,528
0,528
0,528
0,528
0,528
0,527
0,527
0,527
0,527
0,527
0,527
0,526
0,526
0,527
0,527
0,526
0,525
0,523
0,522
0,521
0,520
0,521
0,520
0,520
0,520
0,520
0,516
0,516
0,514
0,512
0,508
0,508
0,500
0,498
0,504
0,489
0,479
0,472
0,456
0,445
0,432
Current
I2(A)
0,252
0,252
0,252
0,252
0,252
0,252
0,252
0,252
0,251
0,251
0,251
0,251
0,251
0,250
0,250
0,250
0,249
0,249
0,249
0,248
0,247
0,246
0,245
0,244
0,244
0,243
0,242
0,240
0,240
0,238
0,235
0,232
0,229
0,225
0,220
0,213
0,207
0,198
0,187
0,175
0,169
0,160
0,148
0,136
0,125
0,113
4.3 Processed data
In order to answer all three research questions, three formulas had to be used. In order
to calculate the power in primary coil, it is suggested to use a formula:
𝑃1 = 𝑉1 ∗ 𝐼1
In order to calculate the power in the secondary coil, the same formula has to be used, only with the
other quantities:
𝑃2 = 𝑉2 ∗ 𝐼2
The efficiency of transformer is equal to the ratio of power in secondary coil and power in primary
coil:
𝑒=
𝑃2
𝑉2 ∗ 𝐼2
∗ 100 % =
∗ 100 %
𝑃1
𝑉1 ∗ 𝐼1
Data about powers in primary and secondary coils and efficiency are represented in a table which is
shown in Figure 5 below.
11
Figure 5
Frequency
f (Hz)
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
110
120
130
140
150
160
170
180
190
200
220
240
260
290
320
360
400
450
500
550
600
670
740
820
900
1000
Power
P1 (W)
Power
P2 (W)
0,166
0,166
0,165
0,165
0,165
0,165
0,165
0,165
0,165
0,165
0,164
0,165
0,165
0,165
0,165
0,165
0,164
0,164
0,164
0,163
0,162
0,161
0,160
0,161
0,161
0,159
0,158
0,156
0,157
0,155
0,154
0,152
0,149
0,148
0,146
0,142
0,138
0,133
0,127
0,123
0,117
0,111
0,105
0,096
0,090
0,084
12
0,133
0,133
0,133
0,133
0,133
0,133
0,133
0,133
0,133
0,133
0,132
0,132
0,132
0,132
0,132
0,132
0,131
0,131
0,131
0,131
0,130
0,129
0,128
0,127
0,127
0,126
0,126
0,125
0,125
0,124
0,122
0,120
0,118
0,116
0,113
0,108
0,105
0,099
0,093
0,088
0,083
0,077
0,070
0,062
0,056
0,049
Efficiency
e (%)
80,2
80,2
80,6
80,6
80,6
80,6
80,6
80,6
80,3
80,3
80,7
80,2
80,2
79,8
79,8
79,8
79,9
79,9
80,0
80,2
80,3
80,5
79,9
79,3
79,1
79,6
80,0
79,8
79,6
79,7
79,2
78,7
79,1
78,0
77,3
76,4
76,4
74,6
73,6
71,8
70,4
68,8
66,8
64,3
61,5
57,9
5. Data presentation and analysis
5.1 Presentation
Figure 6
Figure 7
13
Figure 8
5.2 Analysis
By looking to the graphs in Figures 6 and 7 which respectively represent powers in primary and
secondary coild, it was hard to detect a single equation which would describe how does it behave. It
was neither a single quadratic nor a single linear equation. In my opinion, the power in both graphs
was decreasing in a very small rate from 5 to 100 Hz and that decrease could be described by a
linear equations. As it can be seen in the graphs, equations of power from 5 to 100 Hz in form of
y=mx + b are represented.
The second thing that I have observed is that powers from 100 to 1000 Hz are decreasing in a much
bigger rate, but still linearly. In my opinion, there are several ways how to describe such behaviour
of the transformers:
Impedance. The impedance is influenced by both resistance and the reactance. I assume that the
resistance of the system does not change while we are changing the frequency. On the other hand
the reactance is directly proportional to the frequency in AC circuit. The reactance formula is given
by:
14
𝑋𝐿 = 2𝜋𝑓𝐿 , where XL – reactance, f – frequency, L - inductance. Therefore:
𝑉
𝐼 = 2𝜋𝑓𝐿 , where I – current, V – voltage.
Even though, we can see from the formula that higher frequency influences a smaller current, it
does not explain why does the current decrease linearly, but not by inverse function as the above
formula suggest. Firstly, it happens because, as we can see from the Figure 4, voltage does not stay
the same throughout the process. Secondly, another phenomenon, in my opinion, has an influence in
this experiment:
Skin effect. Skin effect is basically the tendency of alternating current to become distributed densest
near the surface of conductor as the frequency of a current increases. As the current flows denser
near the surface of a conductor, the resistance of the material increases, because the cross-section
area decreases:
𝑅 =
𝜌∗𝐿
𝐴
, where ρ-resistivity of a material, L – length of a material, A – cross-section area.
Even though these two phenomena (impedance and skin effect) with the decrease of voltage explain
why does power decrease, it does not explain why does the rate of decrease differs in intervals from
5 to 100 Hz and from 100 to 1000 Hz. I can make an assumption that the designers of this
transformers have solved how to minimize the losses due to frequency change from 5 to 100 Hz
which are the nearest to 50 Hz (which is the usual current being used in Europe). It is a very logical
design decision, as people do not usually use frequencies of current higher than 100 Hz.
Figure 8 shows a graph of efficiency from the frequency. As we can see, it is a cubic function of
form bx3 + cx2 +dx + a. From this graph we can see that the efficiency from 5 to 100 Hz almost
does not change. As it could already be predicted from the power functions, the efficiency almost
does not change in the interval from 5 to 100 Hz and is equal to about 80 %. From 100 Hz the slope
of a function becomes negative and starts to decrease even more as x approaches infinity.
5.2.1 Uncertainties
The uncertainties of power, efficiency and frequency can be seen in a figure 9 below.
15
Figure 9
Uncertainty of Uncertainty of Uncertainty of Uncertainty of
power P1 (W) power P2 (W) efficiency (%) frequency (Hz)
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000507
0,000506
0,000505
0,000504
0,000505
0,000505
0,000506
0,000505
0,000504
0,000505
0,000504
0,000504
0,000503
0,000500
0,000503
0,000504
0,000504
0,000504
0,000500
0,000499
0,000499
0,000497
0,000496
0,000496
0,000495
0,000492
0,000492
0,000293
0,000293
0,000293
0,000293
0,000293
0,000293
0,000293
0,000293
0,000292
0,000292
0,000292
0,000292
0,000292
0,000292
0,000292
0,000292
0,000291
0,000291
0,000291
0,000291
0,000291
0,000290
0,000289
0,000288
0,000288
0,000287
0,000287
0,000286
0,000286
0,000286
0,000285
0,000283
0,000282
0,000281
0,000279
0,000275
0,000274
0,000269
0,000266
0,000267
0,000259
0,000253
0,000247
0,000238
0,000231
0,000223
16
0,302
0,302
0,305
0,305
0,305
0,305
0,305
0,305
0,304
0,304
0,306
0,303
0,303
0,302
0,302
0,302
0,304
0,304
0,304
0,307
0,309
0,311
0,309
0,307
0,307
0,311
0,314
0,316
0,315
0,317
0,318
0,320
0,326
0,325
0,329
0,334
0,343
0,346
0,358
0,364
0,370
0,381
0,396
0,412
0,421
0,429
0,0025
0,01
0,0225
0,04
0,0625
0,09
0,1225
0,16
0,2025
0,25
0,3025
0,36
0,4225
0,49
0,5625
0,64
0,7225
0,81
0,9025
1
1,21
1,44
1,69
1,96
2,25
2,56
2,89
3,24
3,61
4
4,84
5,76
6,76
8,41
10,24
12,96
16
20,25
25
30,25
36
44,89
54,76
67,24
81
100
To calculate the uncertainties of power and efficiency I have used formulas:
𝛥𝑃
= 𝑃 ∗ √(
∆𝑉
𝑉
2
) +(
∆𝐼
𝐼
2
)
Where ΔP – uncertainty of Power, P – Power, ΔV = systematic error of voltage, V – voltage, ΔI –
systematic error of current, I – current.
2
2
𝑃
𝑃
√( ∆ 1 ) + ( ∆ 2 )
𝛥𝑒 = 𝑒 ∗
𝑃1
𝑃2
Where Δe – uncertainty of efficiency, e – efficiency, ΔP1 = uncertainty of input power, P1 – input
power, ΔP2 – uncertainty of output power, P2 – output power.
One of the most important things to take into account was the rate at which measurements were
taken. As it is mentioned before, both Nova computers could take 10 000 measurements per second
which is enough for measuring the current of low frequencies. On the other hand, 10 000
measurements per second might not be sufficient for higher frequencies, such as 1 000 Hz, because
then the percentage uncertainty is as high as 10 %. Therefore, by considering the data I have
gathered, I have decided to take frequency uncertainties into account by using this formula:
𝛥𝑓
2
𝑓
= 𝑓 ∗ √(10 000) which is equivalent to 𝛥𝑓 =
𝑓2
10 000
As we can see in the figure 6 and 7 above which represent power functions from frequency the
uncertainties in y axis (power uncertainties) are relatively small and many of them do not fit in the
straight line graphs. Only by considering these uncertainties the measurements could be said to very
inaccurate but the frequency uncertainties are very important while considering this. As we can see,
the uncertainties in x axis are relatively big and the straight line graph fits in the most of the error
bars of it. Latter uncertainties make the interpretation of a data more justifiable.
By analysing figure 8 which represents efficiency from frequency function we can see that the
uncertainties in y axis have more meaning than the ones in power functions as the formers are
relatively bigger. With addition to the uncertainties in x axis we can see that the polynomial
efficiency function fits in the most of the error bars.
17
6. Conclusion and evaluation
6.1 Conclusion
During the investigation several things were found out:

The power in primary and secondary coils is highest in range from 5 to 100 Hz which are
the nearest points to the frequency of 50 Hz for which this transformer was designed to
work.

The power in both primary and secondary coils drops linearly in range from 100 to 1 000
Hz. Assumption can be made that such dependency continues after 1 000 Hz also.

The efficiency of transformer does not highly depend on the frequency in range from 5 to
100 Hz as it stays almost constant – about 80 %.

The efficiency drops in increasing rates in range from 100 to 1 000 Hz. Assumption can be
made that such dependency continues after 1 000 Hz also.

Conclusion can be made that transformers similar to the one which was tested in
experiments can be easily used in frequencies that are close to the ones that they were built
for. For example, European transformer built for the currents of 50 Hz can be easily used in
North America where the frequency of the current is 60 Hz.

It is not advisable to use such transformers in frequencies bigger than 100 Hz as both the
power and the efficiency of a transformer starts to drop.
6.2 Evaluation
Even though the investigation can be said to be successful, there are several points on how could
this research could be improved:

The data might have been much more accurate if I had used bigger power loads for the
transformer. It was originally made to work in voltages of 230 V while the voltage I have
used in the primary coil was 10 V. On the other hand, the data was quite logical and the
voltage ratio is19 times – the same as it is written on the label of the transformer.

The measuring the devices could have been much more accurate. It is difficult to say
whether the voltmeters and ampere-meters gathered accurate data but it is obvious that the
uncertainty of frequency is too big. As the frequencies increased to as high as 1 000 Hz the
uncertainty of frequency increased to 10 per cent which does not give very accurate data.
18
That is because as we have a sinusoid of either voltage of current, the devices might not
detect the peak which are the most important values for the research but rather choose the
values that are smaller than the values of the peaks.

During the research more transformers could be used. One device might not show universal
results, because other transformers might perform a little bit differently in the conditions that
were created during the investigation.

Some external factors, such as the resistance of wires or the temperature of devices might
have slightly influenced the results of this investigation. If the all data could be taken into
account, the measurements could be much more accurate.
7. Bibliography

Inductive
Reactance
Formulae
&
Calculations.
http://www.radio-
electronics.com/info/formulae/inductance/inductor-inductive-reactance-formulaecalculations.php Last accessed 30 September 2014.

Formulas for calculating uncertainties. http://webpages.ursinus.edu/lriley/ref/unc/unc.html
Last accessed 20 December 2014.
19