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Kirchhoff’s Law 5/9/2018

UNIVERSITY OF

Southampton "I am aware of the requirements of good academic practice and the potential penalties for any breaches".

Karen Vanessa Perez Lopez ID29666619

Electricity and Electronics Assistant: Claire McGregor Wednesday 9th May 2018

UNIVERSITY OF

Kirchhoff’s Laws

Southampton ABSTRACT The purpose of this experiment was to check Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL); which establish that the currents entering a node is equal to the sum of the currents leaving from the law of conservation of energy and that the sum of the voltages in a closed loop is equal to the resultant emf, respectively. Theoretical values for the current and voltage across each resistor were obtained by solving the equations of each loop and experimental values were obtained by measuring the voltage and the current of each resistor with the DMM.

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Kirchhoff’s Laws

Southampton TABLE OF CONTENTS

1. Introduction…………………………………………………………...4 2. Objective………………………………………………………………4 - 5 3. Theory…………………………………………………………………5 - 6 4. Apparatus…………...…………………………………………………6 – 7 5. Experimental Method………………………………………………….8 6. Data Analysis………………………………………………………….8 7. Results…………………………………………………………………8 – 12 8. Discussion…………………………………………………………….13 9. Conclusion……………………………………………………………14 10. References……………………………………………………………14

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UNIVERSITY OF

Kirchhoff’s Laws

Southampton 1. INTRODUCTION It is well known how important are the electrical circuits for innovation and research, for which it is important to study the properties that govern these electrical systems, such as ohm’s law, or Kirchhoff’s laws, which will be discussed in this report, it is important to know how the electric potential varies in some segment of the electrical configuration, which is of vital importance to make the "arrangements" of elements of a circuit, understanding arrangement, as the way in which the elements of an electric circuit are organized, for this case resistors. Kirchhoff's laws establish a postulate of great importance for the study of physical electronic or therefore for the study of circuits, where it is stated that the sum of the currents entering a node is equal to those that come out, starting from the theory of a conservation of energy, this report first tries to confirm this, but beyond that pretends to understand how this really happens and the great utility it has when calculating the currents that circulate through a segment of a configuration, for which an arrangement will be made in an electrical circuit, with resistors organized in series and parallel and some aspects will be analysed as the relationship of the currents in different points of the system. 2. OBJECTIVE The objective of this experiment is to check out the theory of multi-loop circuits by building the circuit shown in Figure 1 and comparing the experimental values for currents and potential differences with theoretical predictions according to Kirchhoff’s current and voltage laws.

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UNIVERSITY OF

Kirchhoff’s Laws

Southampton

Figure 1. A simple two-loop network.

We wish to know the currents through R1, R2 and R3 and the relative potential of the nodes A, B and C, with respect to D (earth,0V). 3. THEORY 3.1 Law of conservation of energy:The law of conservation of energy is considered one of the fundamental laws of physics and constitutes the first principle of thermodynamics. It states that the total energy of an isolated system remains constant or that the energy is not created or destroyed, it only transforms, which implies that in certain conditions the mass can be considered as a form of energy.i 3.2 iiKirchhoff’s Current Law: In an electrical circuit, it is common for current nodes to be generated. A node is the point of the circuit where more than one terminal

of an electrical component is joined.

Therefore, the sum of the entering currents is equal to the currents coming out of the node.iii Figure 2.

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UNIVERSITY OF

Kirchhoff’s Laws

Southampton

3.3 Kirchhoff’s Voltage Law: States that in all closed loop the sum off all the voltage drops is equal to the resultant emf, and the algebraic sum is equal to zero. 

An arbitrary direction of current circulation is assigned to each loop.



The sign that is given to each voltage drop depends on the sense of circulation chosen for

Figure 3

the current. (When the direction of the loop goes against the current the sign is positive).iv 3.4 Percent Error: The percent error is a useful tool that allows us to compare how accurate an estimated value is. The percent error gives us the difference between the experimental value and the theoretical value and helps us to determine how close is our estimated value to the theoretical value.v The formula is given by : −¿ experimental |¿ theoretical |∗100 ¿ theoretical

%error=

4. APPARATUS 4.1 Electronics Breadboard vi

(0)

UNIVERSITY OF

Kirchhoff’s Laws

Southampton

Figure 4.

4.2 DMM vii

Figure 5. Digital Multimeter

4.3 Resistors viii

Figure 6. ix

Figure 7.

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Kirchhoff’s Laws

Southampton

5. E XPERIMENTAL METHOD Firstly, the resistors were tested, and their tolerance levels were recorded. Secondly, using the electronic breadboard, resistors and cables the circuit was constructed. Thirdly, the P.S.U1 and the P.S.U2

voltage output was set to 15V and to 5V,

respectively. Then, the current limit for both supplies was set to 0.1A and the dc output for both power supplies were switched on. After that, using the same DMM, the three potential differences V AD, VBD and VCD were measured. Finally, from the values for the voltages across R1,R2 and R3 the three unknown currents I 1, I 2 and I 3 were determined. 6. DATA ANALYSIS The table below shows that all the resistors used to build the circuit are within 1% tolerance level. Table 1. Tolerance levels of the resistors.

Resistor Value (Ω)

Experimental Value (1000 Ω)

Tolerance Levels (1000 Ω)

10K

9.98 ± 0.01

0.02

20K

19.96 ± 0.01

0.02

20K

20.0 ± 0.01

0.00

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Kirchhoff’s Laws

Southampton 7. RESULTS 7.1 Theoretical values assuming experimental values for E1 and E2. DATA:

1

2

Loop 1 E1−I 1 R1 −I 3 R3=0

(2)

− E2 + I 3 R3− I 2 R 2=0

(3)

Loop 2

Node B I 1 =I 2 + I 3

(4)

SOLUTION: Loop 1 E1=I 1 R1 + I 3 R 3

(5)

15=I 1 × ( 20000) +I 3 × (20000 ) 15= ( I 2 + I 3 ) × ( 20000 ) +20000 I 3 9

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Kirchhoff’s Laws

15=20000 I 2+ 20000 I 3 +20000 I 3

Southampton

15=20000 I 2+ 40000 I 3 (÷ 10) 1.5=2000 I 2+ 4000 I 3

Loop 2

E2=I 3 R 3− I 2 R2

(6)

5=I 3 × ( 20000) −I 2 × (10000) 5=20000 I 3−10000 I 2 (÷ 5) 1=4000 I 3−2000 I 2

From loop 1 – loop 2 , hence I2

1.5=2000 I 2+ 4000 I 3 —

1=4000 I 3−2000 I 2

0.5=4000 I 2

I2 =

0.5 =125 μA 4000

Hence I3 from loop 2 1=4000 I 3−2000 I 2

I3 =

1+2000 I 2 4000

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UNIVERSITY OF

Kirchhoff’s Laws

Southampton

I3 =

−6 1+2000 × ( 125 × 10 ) 4000

I3 =

1.25 =312.5 μA 4000

Node B I 1 =I 2 + I 3 I 1 =125 μ+312.5 μ

I 1 =437.5 μA Therefore, the voltage across node B and D is: V BD =I 3 R3 −6

V BD =437.5 ×10 × 20 ×10

(7) 3

V BD =8.75 V

7.2 Experimental values

VAD = 15v, VBD = 6.31v and VCD = 5.06v

I1 =

I1 =

V AB V AD−V BD = R1 R1

(8)

15−6.31 =434.5 μA 20000 11

UNIVERSITY OF

Kirchhoff’s Laws

Southampton I2 = I2 =

(9)

6.31−5.06 =125 μA 10000

I3 =

I3 =

V BC V BD−V CD = R2 R2

V BD R3

( 10 )

6.31 =315.5 μA 20000

7.3 Table 2. Comparison of experimental and theoretical values.

Experimental Value

Theoretical Values

15V

15V

0%

6.31V

8.75V

38.6%

5V

5V

0%

I1

434.5µA

437.5µA

0.68%

I2

125µA

125µA

0%

I3

315.5µA

312.5µA

0.9%

Quantity

VAD (=E1)

VBD

VCD (=E2)

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%Error = ¿ theoretical−¿ experimen ¿ theoretical

|

UNIVERSITY OF

Kirchhoff’s Laws

Southampton

8. DISCUSSION The values of the currents and voltages were determined by solving the equations obtained after applying Kirchhoff’s laws to the circuit simultaneously; in addition, it can be seen that the values are almost the same. According to the data in Table 2, the similarity between the experimental and theoretical current and voltage values is clear. The error % oscillates between 0% and 40%, and between 0% and 1%; for the voltage and the current respectively. This congruence between the experimental and theoretical values suggests that Kirchhoff's laws, in effect, allow us to determine the current and voltage of elements in an electrical circuit. The relative difference between the theoretical and experimental values was due to energy losses in the system and errors in the measurement. In theory, only the resistors dissipate the electrical energy, but in fact the other elements of the circuit (cables, source, etc.) also do it but in a much smaller amount, since they have a small resistance to the flow of electric current that does not it is considered when applying Kirchhoff's laws. Even so, the determination of the current and voltage of resistors by this means is an excellent approximation to the real values, due to the low error they

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UNIVERSITY OF

Kirchhoff’s Laws

Southampton

present. As for the measurements, the correct reading of them and the imprecision of the instruments used to record them are very influential. 9. CONCLUSION By observing the values obtained both theoretically and experimentally, it demonstrates that the difference between them is minimal, therefore it is concluded that Kirchhoff’s laws are met. . 10. REFERENCES

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i ThoughtCo. (2018). What Is the Law of Conservation of Energy? [online] Available https://www.thoughtco.com/law-of-conservation-of-energy-605849. ii http://upload.wikimedia.org/wikipedia/commons/2/26/Kirchhoff_first_law.png iii Physics.uoguelph.ca. (2018). Kirchhoff's Current Law. [online] Available https://www.physics.uoguelph.ca/tutorials/ohm/Q.ohm.KCL.html. iv https://www.miniphysics.com/kirchhoffs-second-law.html v Astro.physics.uiowa.edu. (2017). Percent Error Formula | Imaging the Universe. [online] Available at: http://astro.physics.uiowa.edu/ITU/glossary/percent-error-formula/. vi http://www.nexuscyber.com/content/images/thumbs/0000028_full-size-solderless-breadboard.jpeg vii http://cdn.wonderfulengineering.com/wp-content/uploads/2015/08/Best-digital-DMMs-9.jpg viii https://cdn.shopify.com/s/files/1/1097/0232/products/IMG_0007-001_grande.JPG?v=1482867991 ix http://rayshobby.net/wordpress/wp-content/uploads/2011/08/10k-30k-annotate.jpg

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