Determination and Validation of Effective Resistance and Kirchhoff’s Rules

Creo Baylon, Rhei Juan, Joseph MabajenNational Institute of Physics, University of the Philippines, Diliman, Quezon City

AbstractThis experiment aims to confirm the validity of the two Kirchhoff's laws as well as the equational consequence of different orientations for resistors in a circuit. This is achieved by obtaining the current and voltage reading at crucial points in the given circuit containing three resistors with differing resistance values and are oriented in different ways: series, parallel and the combination of the two. The obtained values will then be compared to the theoretical values calculated using the given the equational quantities of resistance as well as the two laws themselves.

I. IntroductionCircuit is formally defined as “a path between two or more points along which an electrical current can

be carried.”[1] Circuits can vary based on their general structure; whether if the circuit components are connected in series, parallel, or a combination of the two. The experiment the researchers performed concerned each of the three said kinds of circuit in order to study the relationship(s) existing between the different parameters (e.g. voltage, current) in that circuit.

The total amount of resistance that a circuit applies to a source is dependent on the orientation of the resistors in it. The effective resistance that a circuit with resistors connected in series is:

Rtotal=R1+R2+R3 …+Rn (1)

Where Rtotal is the total amount of resistance of the circuit while the individual terms in the right hand

equation represent the resistances of the nindividual resistors. The resistance equation for a parallel circuit on the other hand is given by:

Rtotal=1

1R1

+1R2

+1R3

+…1Rn

(2)

This application can be seen in the reflected amount of voltage or current in the given source. In a series, the voltage is reduced each time it encounters a resistor component and always drops to zero when finishing a loop, while the current is split in any junction it encounters. These are the two Kirchhoff’s Laws. Kirchhoff’s Voltage Law may also be applied to formulate the following equation:V=V 1+V 2+…V n (3)

V=I 1 R1+ I 2 R2+… In Rn (4)

which applies to any closed loop within the circuit. Kirchhoff’s Current Law, on the other hand, has the following application.I initial=I 1+ I 2+… I n (5)

Where I initial is the current of the first line behind the junction and I n terms are the individual current values of the lines after the junction.

The objective of the study is to verify the equations correlating the resistances of given resistors connected in varying ways: series, parallel and a combination of both, as well as verifying and compare the overall effect on the Voltage and Current of each component in the given circuit to the theoretical Voltage and Current calculated using the two Kirchhoff’s Laws.

II. MethodologyThree different resistors were chosen and their resistances were determined basing on the color bands they

had. Using an ohmmeter, the resistance was also measured and compared with the determined resistance based on the color bands. The resistors, named R1, R2 and R3 were connected in series, as seen in Figure 1.

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Figure 1: Setup for the determination of the effective resistance of specific simple circuits

The effective resistance was measured using the ohmeter and compared again with the theoretical value through percent deviation. A 5-V power supply ws connected to the resistor network to form a circuit. The voltage and current passing through each resistor was recorded to calculate the resistance of each resistor. The same steps were then repeated for resistors in parallel and R3 and R2 parallel with each other while R1 is in series with it.

Figure 2: Circuit used in the validation of Kirchoff’s rules

In order to test and verify the Kirchoff’s rules, the circuit components were constructed as seen in Figure 2. A 4.5V power supply was added by connecting it to R2 and the joint where R3 and the 5-V power supply is. The output voltages of the 5-V and 4.5V power supplies were measured, and the currents and voltages within each resistor and was compared to the theoretical currents and voltages determined using the equations that apply to Kirchoff’s rules.

III. Results and DiscussionIn order to determine the effective resistances of the circuit diagrams to be used in the experiment, it is vital

for the researchers to first measure the resistance of each circuit component needed in the experiment. Due that reason, the researchers first determined the amount of resistance of each resistor through the use of the color banding scheme, and the values were compared with the values acquired with the use of an ohmmeter. The summary of data recorded, together with their percent deviation is summarized in table 1 below.

Table 1. Theoretical and experimental resistance values.Resistor Color Band Color Reading (Ω) Ohmmeter Reading (Ω) Average

Deviation (%)R1 Brown-Black-Red 1000 975 2.5R2 Brown-Red-Red 1200 1193 0.583333333R3 Brown-Green-Red 1500 1472 1.866666667

To further the information that can be interpreted from the experiment, the researchers then constructed three different circuit diagrams using the breadboard and the resistors, in which their respective effective resistances were calculated. Their theoretical resistances were then compared with the experimental effective resistances computed from the resistors’ ohmmeter readings. The data obtained by the researchers are summarized below in table 2.

Table 2. Experimental and Theoretical effective resistance values.Circuit Theoretical Reff (Ω) Experimental Reff (Ω) Deviation (%)

1 3700 3630 1.891891892

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2 400 396 13 1666.67 1637 1.78019644

After subsequently adding a 5 V power supply to each of the three circuit diagrams previously used, the researchers then proceed in obtaining the experiment voltage (V) and current (I) values for each of the resistor component. Using the said values, they then computed the respective resistance values and the process was repeated until all the resistors for each circuit were examined. The tabulation of data values gathered and computed is presented in the table below.

Table 3. Calculated resistance values from experimental V and I measurements.

Resistor 1 Resistor 2 Resistor 3Circuit V (V) I (mA) R (Ω) V (V) I (mA) R (Ω) V (V) I (mA) R (Ω)

1 1.46 1.366 1068.81 1.76 1.367 1287.49 2.2 1.368 1608.192 4.65 5.1 911.765 4.39 4.3 1020.93 4.92 3.5 1405.713 3.26 3.1 1051.61 2.22 1.71 1298.25 2.19 2.08 1052.88

Analyzing the above table, it can be said that in a series circuit, the amount of current that passes through each resistor is more or less equal, regardless the amount of resistance it has. Additionally, the voltage readings on the resistors in a parallel circuit is also equal to one another. Both are important observations in analyzing the two kind of circuits.

Using the calculated resistance values from table 3, the researchers then computed for the experimental effective resistances for each of the circuit. This was done in order to compute for average percent deviations from the theoretical effective resistances presented earlier in table 2. The values computed in this part of the experiment is then presented in table 4.

Table 4. Calculated experimental Reff from V and I measurements.Circuit Itotal (mA) Vtotal (V) Experimental Reff (Ω) Theoretical Reff (Ω) Deviation (%)

1 1.367 5.42 3964.886613 3700 7.1590976492 12.9 4.6533 360.7209302 400 9.8197674423 3.445 5.465 1586.357039 1666.67 4.818768011

As observed in table 4, the percent deviation between of experimental and theoretical effective resistances only ranges from 4% to 10%, which implies that the formulas used for computing the theoretical R eff is consistent with the results of experimentation, varying only due to the systematic errors made by inaccurate equipments.

After constructing the two-loop circuit required to test Kirchhoff’s loop and junction rule, the researchers then measured the voltage and current passing through each resistor in the said circuit using the multimeter. This was done in order to verify the consistency of the gathered data values to the theoretical values computed and acquired from the computations. The summary of data recorded, together with their percent deviation is summarized in table 5 below.

Table 5. Data values for the two-loop circuit.Resistor Vexp (V) Vtheo (V) Deviation (%) Iexp (mA) Itheo (mA) Deviation (%)

1 1.66 1.646341463

0.829629655 1.612 1.688555 4.53375815

2 1.67 1.14 46.49122807 1.12 0.955574 17.207043 3.39 3.36 0.892857143 2.74 2.282608 20.0381318

In testing Kirchhoff’s rules, the researchers assigned the clockwise loop as the direction of current used in computing for the theoretical currents for each resistors. Looking at the average % deviation in table 5, the percent deviations ranges from less than 1% up to almost 50% error, nonetheless the experiment yielded positive results and was consistent with the theoretical data computed using the concepts of Kirchhoff’s rules.

Deviations on the experimental data gathered in the experiment were the result of systematic errors present in the measuring devices used. Such factors are the dust inside the breadboards that may affect its performance, the miniscule rust on the surfaces of alligator clips, and in the multimeter itself, and other objects unseen to the naked eye that may have affected the circuit used.

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IV. Conclusion Based from the results presented in tables 1, 2, and 4, the deviations for the experimental data gathered can be

considered acceptable. If the materials used the two-loop circuit had better conductivity and accuracy, the percent deviation would have been zero. In general, the small difference between the current and voltage measured in a series and parallel circuit, respectively, proves that current is consistent through all the resistors in a series circuit, the same being said on the voltage in a parallel circuit.

Furthermore, with the deviations in table 5 being at most, less than 50%, it can be said that Kirchhoff’s junction and loop rule holds true in any circuit analysis, as it was consistent with the theoretical data of the experiment.

References1."Circuit.” Techtarget, n.d. Web. 08 September 2015. <http://whatis.techtarget.com/definition/circuit>

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