Physics 122 Lab 7: Circuits II

Abstract: The main objective of this experiment was to test and verify Kirchhoff’s two rules: the loop rule and junction rule in a complex circuit. The voltage and current of each of the six resistors in the circuit were measured using a DMM. These values were then used to calculate the resistance of each resistor. The calculated resistor values were compared to the measured DMM values and the nominal values. By doing so, Ohm’s Law was proved because the values were consistent. The second part of the experiment involved using the current and voltage values obtained in part 1 to prove Kirchhoff’s loop and junction rule. The results, although were not zero, complied with Kirchhoff’s two rules within experimental error supporting Kirchoff’s rules regarding the voltage and circuit in a closed DC circuit.

Ruby Bahniwal Partner: Jesse Olson 2011-07-25 T.A. Ali Introduction:
The purpose of this experiment was to learn how to assemble a circuit properly, Figure 1 below, using the equipment provided and to experimentally verify Kirchhoff’s rules by measuring the voltage or potential drop across each resistor along with the current for each branch in the circuit constructed.
Electric circuits have the ability to direct and control the flow of electricity and the energy conveyed. It can be constructed using various combinations of batteries, resistors and capacitors. These circuits are found in common everyday devices including cars, washing machines, and electrical lights. An electrical circuit is a unique network that has a closed loop giving a return path for the current. It works to carefully direct the flow of electrons from a voltage source through an electric path, forming a loop.1
For example, a battery moves electrons that are formed at the negative terminal to the positive terminal through a closed circuit. This creates the current, which forces the electrons inside the circuit to move in a direction opposite of the current. Moreover, experiments have shown that electrons do not travel in a linear straight motion; rather they collide with atoms present in the conductor. This results in the production of internal friction, which is referred to as resistance. Resistance can be using Ohm’s Law.
Ohm’s Law is the relationship between the current flowing through resistance, R and the potential drop across it. Ohm’s Law states the voltage or electric potential in direction proportional to the product of the current and the resistance where current is in Amps (A), voltage in volts (v), and resistance in Ohms (Ω). Therefore, the relationship: expresses Ohm’s law (shown below).1
R = ∆V / I … (1)

In second part of the experiment a circuit (Figure 1), was analyzed to determine the current in each component of the circuit and the potential difference across each component. This analysis was done using Kirchhoff’s rule described below. The Junction Rule We also know that charge is conserved. This leads to Kirchhoff’s first rule: the junction rule. The junction (or node) rule asserts that the net current into (or out of) a node is zero (i.e. ‘current in is equal to current out’). This is seen by:
ΣIin=ΣIOut … (2)
Where, if the resistor is in the same direction as the current the voltage is negative, and if the resistor is in the opposite direction as the current, the voltage is positive. The Loop Rule This rule asserts that the net change in electric potential around any closed loop in a circuit is zero. This is a statement of energy conservation, since a charge that moves around any closed path and returns to its original point must have no change in potential energy. If we apply this conservation law to a closed loop in a circuit, then ΣΔV=0 … (3) Where ΔV is the potential difference across each component in the loop as measured while ‘traversing’ the loop in an arbitrary but fixed direction (i.e. either clockwise or counter clockwise. 1. If you cross a cell (or battery) from the negative to the positive terminal, then this is a potential increase (or decrease if from positive to negative). 2. If you cross the resistor in the same direction as the one you assumed for the current then count the potential difference as a decrease (or increase if your direction of traversal is opposite the assumed direction of the current). To make sense of the second convention, remember: the direction of conventional current is from higher electric potential to lower electric potential.1 Procedure:
This experiment consisted of two parts. The first part of the experiment involved measuring the voltage drop across each resistor, the current in branch of the circuit and using Ohm’s law to calculate the resistance of each resistor for the circuit. The second part of the experiment was carried out to verify both of Kirchhoff’s rules using experimental setup. Part 1- Using Ohm’s Law to Calculate the Resistance of Each Resistor with measured voltage and current values
In the first part of the experiment two apparatus’ were used. The DC Power Supply Agilent E3610A was used to supply voltage and the Digital Multimeter, DMM, Fluke 8010A was used to measure voltage across each of the resistors and the current in the circuit. A breadboard was setup identical to the circuit diagram in figure 2 using the required resistors and pre-cut conducting wires. The resistors were R1= 220Ω with a coloured band code of red/red/brown/gold, R2= 130Ω with brown/orange/brown/gold, R3= 470Ω with yellow/purple/brown/ gold, R4= 130Ω, R5= 220Ω, and R6= 330Ω with orange/orange/brown /gold. Figure 2: Circuit setup used to verify Kirchhoff’s Rules.
Once the breadboard was assembled, the DC Power supply was connected to the circuit. This was done by connecting one end of a pre-cut and stripped wire to the positive bus strip on the breadboard, while connecting the other end to the beginning point on the circuit. This was repeated using the negative bus strip, which was connected to the end of the circuit instead. The circuit was then connected to a DMM to measure the voltage and current through each of the resistors. Once properly assembled, the DC power supply was turned on and set to 5V and 5 volts was run through the circuit using the DC supply, alligator clips and wires. This was to ensure that a consistent continuous voltage potential in the circuit and the voltages across each of the six resistors was measured and recorded in Table I (refer to results section).
Once the circuit had a power supply, the voltage across each resistor in the circuit was measured using wires and the Digital Multimeter (DMM). The DMM was turned on and then the second button for V was pressed in. The black wire was then connected into the “Common” outlet and the red wire into the red “V/kΩ/S” outlet. The opposite ends of the black and red wires were pressed against the metal wire on each resistor. This step was repeated several times and the voltage drop across each resistor was measured and recorded in Table I (refer to results). The DC Supply was continually checked to insure 5 volts were supplied.
Once the voltage drop across each resistor was determined, the DMM was set to measure current through each branch of the circuit. The DMM was turned on and the third button for mA was pressed in. When determining the current for a single resistor, one end of the resistor was removed from the breadboard and one end of a pre-cut and stripped wire was placed at that exact location. The opposite ends of the black and red wires were then connected. One of the ends was connected to the pre-cut and stripped wire and the other was connected to the resistor. The current across each of the six resistors was measured using the method above and recorded in table II (refer to results section). This process is illustrated in Figure 3 below. Part II- Verification of Kirchhoff’s Rules for Given Circuit
The circuit was then disassembled and resistance for each of the six resistors was measured using the DMM. In order to do this, the red wire connected to “V/kΩ/S” outlet and the second button for V was pressed in. The resistance for each resistor was measured by placing the red and black ends of the wire on the metal part of the resistor. Moreover, using the measured voltage and current values from part I, the resistance across each resistor was calculated under the assumptions of Ohm’s Law (Equation [1]). The nominal value for each resistor was also determined using the colour codes illustrated on the resistors. The measured resistance, calculated resistance and nominal resistance values were all recorded in table III for comparative purposes. Part II- Verification of Kirchhoff’s Rules for Given Circuit The second part of the experiment was carried out to test Kirchhoff’s two rules, by using the data collected in part 1. According to Kirchhoff’s first rule, the net change in electric potential around any closed loop in a circuit is zero. Seven loops were observed in this circuit as shown in Figure 4 below. Using equation 2, the voltage for each loop should equal zero in order to comply with the loop rule. Figure 2: The image above illustrates the seven loops in the circuit used to verify the loop rule. Furthermore, according to Kirchhoff’s second rule, the current at each junction should equal zero. This was again verified using the instruments mentioned above and applying the currents direction, into and out of the junction into equation 3. This is illustrated in figure 5 below. Figure 3: The image above illustrates the direction of the current at each junction on the circuit.
Results:
Part 1A- Determining the voltage drop across each resistor For part 1 of the experiment, a circuit was assembled (Figure 2) to measure the voltage drop across each of the six resistors and the current in each branch of the current. This part of the experimented was conducted to experimentally verify both of Kirchhoff’s rules. Using the assembled breadboard circuit, we measured the voltages across the resistors using the DMM while supplying the circuit with a 5V DC power supply. The measured voltage values for each resistor are shown below in Table I.

Table I: The voltage drop across each resistor was determined by using the DMM to measure the voltage at each resistor for the breadboard circuit. These values were then used to experimentally prove Kirchhoff’s loop rule, which asserts that the net change in electric potential energy around any closed loop in a circuit is zero. This was determined using equation (3). | Vsupply (V) | Vacross resistor (V) | VR1 | 5.00 | 2.49 ± 0.01 | VR2 | 5.00 | 0.62 ± 0.01 | VR3 | 5.00 | 1.88 ± 0.01 | VR4 | 5.00 | 0.88 ± 0.01 | VR5 | 5.00 | 1.62 ± 0.01 | VR6 | 5.00 | 0.26 ± 0.01 |

Part 1B-Determining the current in each branch of the circuit Table II: The current through each resistor in the circuit supplied with a 5V DC power supply was determined using the DMM. The values obtained were than used in equation 1, to determine the calculated resistance for each resistor. | Vsupply (V) | I through resistor (A) | I0 | 5.00 | 11.46 ± 0.04 | I1 | 5.00 | 11.46 ± 0.04 | I2 | 5.00 | 4.90 ± 0.02 | I3 | 5.00 | 4.04 ± 0.02 | I4 | 5.00 | 6.58 ± 0.03 | I5 | 5.00 | 7.38 ± 0.03 | I6 | 5.00 | 0.80 ± 0.01 |

Using the lab manual and the coloured band code on each resistor, the nominal resistance value (Rnominal) for each resistor was determined. The resistance DMM value determined by placing the red and black ends of the wire on the metal part of the resistor. As indicated by the results, the measured DMM resistance and nominal resistance values are very similar. In order to calculate the resistance value, Rcalculated, equation 1 was used. When using Ohm’s Law (equation [1]), the voltage across each resistor (Table I), and the current through each resistor (Table II) were used.
Table III: Measured resistance, calculated resistance and nominal resistance values | Rnominal (Ω) | RDMM (Ω) | Rcalculated (Ω) | R1 | 220 | 214 | 216 ± | R2 | 130 | 128 | 127 | R3 | 470 | 465 | 465 | R4 | 130 | 129 | 134 | R5 | 220 | 216 | 219 | R6 | 330 | 326 | 325 |

Once the voltage across each resistor and current at each junction were determined in part 1, these values were then used to verify Kirchhoff’s two rules: the loop rule and the junction rule.
Part 2A-Verifying Kirchhoff’s First Rule: The Loop Rule
For testing the junction rule, the current values for each branch in the circuit were inserted into equation 2. In order to prove this rule, the net current into (or out of) a junction should be zero. Our results were fairly consistent with the junction rule that even though our values were not zero, they were very close. Differences may be attributed to experimental error associated with the experiment itself.

Table IV: Summary of the equations used to verify the junction rule at each junction along with net result. Junction #: | Equation: | Values Inserted into Equation: | Current (mA) | 1 | I1-I4-I2=0 | 11.46-6.58-4.90 | -0.02 | 2 | I3+I6-I2=0 | 4.04+0.80-4.90 | -0.06 | 3 | I4-I5+I6=0 | 6.58-7.38+0.80 | 0 | 4 | I3-I5-I0=0 | 4.04+7.38-11.46 | -0.04 |

Part 2B-Verifying Kirchhoff’s Second Rule: The Junction Rule
In order to prove Kirchhoff’s second rule, the loop rule, the net change in electric potential around any closed loop in a circuit should be zero. Our results were again fairly consistent with the loop rule even though our values were not zero; they were very close to the expected value of zero. Differences may be attributed to experimental error associated with the experiment itself. Our results justified Kirchhoff’s loop rule respectively.
Table V: Summary of the equations used to verify the loop rule using the voltage values across each of the six resistors in the circuit and the net result. Loop # | Equation | Values Inserted into Equation | Voltage Value (V) | 1 | V6 + V2 – V4 | 0.26V+0.62-0.88 | 0.00 V | 2 | V3 – V6 – V5 | 1.88V-0.26-1.62 | 0.00 V | 3 | Vsupply –V1 – V2 – V3 | 5.00V-2.49V-0.62V-1.88V | 0.01 V | 4 | Vsupply – V1 –V4 – V5 | 5.00V-2.49V-0.88V-1.62V | 0.01 V | 5 | V2 – V4 – V5 + V3 | 0.62V-0.88V-1.62V+1.88V | 0.00 V | 6 | Vsupply – V1 –V4 + V6 – V3 | 5.00V-2.49V-0.88V+0.26V-1.88V | 0.01 V | 7 | Vsupply – V1 –V2 - V6 – V5 | 5.00V-2.49V-0.62V-0.26V-1.62V | 0.01V |

Calculations:
Sample Calculation of the error for the measured voltage drop across the resistors values (Table I): 0.1 × (2.48V) + (0.01) = 0.0125
100

1 significant figure =0.01

Sample Calculation of the error for the measured current values for each branch of the circuit (Table II):
0.3 × (11.46 mA) + (0.01) = 0.0444
100

1 significant figure =0.04

Sample Calculation of the error for the resistance of each resistor determined using Ohm’s Law (Equation [2]). These values are illustrated in Table III above:
0.01 × 100 = 0.40% (ΔV)
2.48

0.04 × 100 = 0.35% (I)
11.46

When determining the error associated with a product, convert the error into percent values (see above) and then add the percentages them together. This is known as the relative error.

0.40% + 0.35% = 0.75%

Next, convert the relative error into absolute error by dividing the percent value by 100 and multiply by the determined resistance value obtained by using Ohm’s Law (equation [2]):

0.75% × 216 Ω = 1.62
100%
2 significant figure =1.6

Sample calculation for the calculated resistance value for each resistor using Ohm’s Law.

R= ΔV I
R= 2.48 V 0.01146 A
R= 216.4 Ω
R= 216 Ω

Discussion: For part 1 of this experiment, the voltage across each resistor and the current through each resistor in the complex circuit (Figure 1) were measured using a DMM. Using these values, the resistance value for each resistor was determined by Ohm’s Law (equation 1). The value for the calculated resistance was compared to both the nominal and measured resistance values, which is illustrated in Table III above. The resultant resistance values were quite even though slight discrepancies existed. These discrepancies can be attributed with error associated with the experiment and when taken into account the values overlap and are consistent with each other. Error may be attributed to the measuring range on the DMM because if the DMM is at a lower range, it is more inaccurate in measurement depending on the magnitude of the voltage or current being measured. Overall, the calculated, nominal and DMM values for the resistance of each resistors was consistent, supporting Ohm’s Law. The second part of the experiment was conducted to test the junction rule and loop rule using the measured voltage and current values obtained in part 1.
From Table IV, the values were very close to zero; therefore, they were consistent with the loop rule that states: the net change in electric potential around any closed loop in a circuit is zero. Some values were not exactly zero; however, were close enough to support Kirchhoff’s loop rule regarding the conservation of energy. Referring to Table IV, the voltage across R1 was slightly different. This resulted in obtaining values not exactly zero for loops 3, 4, 6 and 7. Using the error calculated for the voltage value for R1 in Table I, a maximum value of 2.50 V is possible taking the error into account. When using the maximum value instead of the calculated R1 value, for the verification of the loop rule, the net value would be exactly zero. Therefore, the results obtained in this part of the experiment, satisfy both Kirchhoff’s loop rule and the principle of conservation of energy. Considering the possible sources of error in this lab, the DMM as well as the DC supply can held accountable. The instruments have error associated with them depending on how properly they are used in the lab. For examples, even though DC supply may read 5V, the actual voltage value may be not as accurate.
From Table V, the values obtained using the junction rule were very close to zero. Only junction 3 resulted in an absolute value of 0 mA using Kirchhoff’s junction rule. However, the other junction values were very close to the accepted value of zero because the values were in mA; therefore, the values were only off by values in the hundredths. The difference between the values obtained experimentally and the actual value of 0 mA can be attributed once again to error associated with the experiment itself. The current values through each resistor in the circuit were measured using the DMM (Table II) in part 1 to verify Ohm’s Law. When the resistance for each resistor was calculated using equation 1, the values were not identical to the nominal and measured values. Therefore, the current values obtained in part 1 were slightly off impacting the results obtained in part 2 to verify the junction rule. When taking the error for each of the current values obtained in part 1 (Table II) into account, the junction calculations would all equal zero; therefore, these values are fairly accurate in proving Kirchhoff’s junction rule, and the principle of conservation of charge respectively.
Overall, this experiment was successful in verifying Kirchhoff’s rules in a complex DC circuit. Moreover, obtaining consistent values for the calculated, nominal and DMM resistance values demonstrated Ohm’s Law across each resistor.
Conclusion:
At the end of this experiment, we verified Kirchhoff’s two rules: the junction rule and the loop rule by setting up a closed circuit as shown in Figure 2. The unique property of this circuit was that it could not be simplified into one equivalent resistor. In part 1 of the experiment, the DC power supply of 5V was supplied and the voltage across each resistor was measured using a DMM. Moreover, the current in each branch, or junction of the circuit was measured. Using the two measured values and equation 1, the resistance across each of the resistors was calculated. The calculated resistance values were then compared to the measured and nominal values. The results were fairly consist with each other and were in agreement with Ohm’s Law. For the second part of the experiment, the voltage and current values obtained in part 1 were used to verify Kirchhoff’s rules by inserting them into equation 2 and 3. These results are present in Table 4 and 5 above. As observed, some of the calculations did not yield a net result of zero, but were fairly close. This difference can be attributed to experimental error associated with the experimental setup of the lab such as: error associated with the instruments (DMM and breadboard circuit), other possible sources of error could include human error. When taking such errors into consideration, the assumption can be made that these results verify and support Kirchhoff’s rules.

Works Cited:
[1] Faughn, Jerry S., Serway, Raymond A., and Vuille, Chris. College Physics, 8thed; Brooks/Cole: Belmont, 2008, Chp 17.