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TA’s Name:
ABSTRACT
This laboratory report examines a two-part experiment carried. The first experiment was carried out to test both the Kirchhoff’s loop and junction rule. The results of the experiment have been presented in the report. Using the results collected the loop and junction. The second part of the experiment involved determining the Kirchhoff’s loop and junction rule. The maximum power transfer equation has been derived in the report.
Objectives
The object of the experiment is to test both the Kirchhoff’s loop and junction rule. The experiment was also aimed at determining how maximum power can be drawn from a non-ideal battery.
Procedure
Part 1:
- The circuit was set up as shown in figure 1 below. The circuit consisted of a power supply, 3 resistors, and 2 decade resistance boxes.
- A digital multimeter was used to measure the resistances of the resistors in the circuit
- The output voltage of the power supply was set to 4.0 v
- Using the digital multimeter the potential difference across of each of the resistors was read and the side with the higher potential was noted
- Kirchhoff’s loop rule was tested for loops abcda, cdefc, and abcfeda shown in figure 1 below
- The digital multimeter was also used to measure the current flowing through each of the resistors and the direction of current was noted
- Kirchhoff’s Junction rule was tested for junctions c and d
- The results of the tests were recorded
Part 2:
- The circuit was set up as shown in figure 2 below. The circuit consisted of a power supply and two decade resistance boxes.
- The power supply was set to 10.0 V and R0 was set to 100 Ω
- The load resistance RL was varied from 10 Ω to 1280 Ω. For each value of RL the resulting values of I (from DMM) and VRL (from the DVM) were recorded
- The results obtained above were checked by varying RL from 1280 Ω to 10 Ω
Experimental Data [15 Points]
Voltage Reading across Resistors Part 1
Current Measurements Part 1
Voltage Measurements Part 2
Current Measurements Part 2
Graph of Power vs. Resistance
Results [20 Points]
Statement of Necessary Equations:
The required equations are the Kirchhoff’s rules, which are based on the principle of conservation of charge. The first rule is the junction rule, which states that the algebraic sum of all currents flowing into a node is equal to the currents flowing out of the node.
The second rule to be used is the Kirchhoff’s second law or the loop rule. This rule states that the sum of the voltage around a closed loop is zero.
The maximum power transfer equation also needs to be derived.
The Kirchhoff equations are as stated below for both the junction and loop rule.
Junction rule:
Loop rule:
Maximum Power transfer equation
PL= (ERS+RL)2RL
Opening the brackets
= (E2RS2RL+2RS+RL)
In order to find the maximum RL we differentiate the denominator, with respect to RL and equating it to zero.
ddRL=(RS2RL+2RS+RL)
0 =- RS2RL2+1
Therefore, for maximum power transfer
RL = RS
Define the parameters:
Explain derivation (where do equations come from?)
The equations are based on the principle of conservation of charge.
For the maximum power transfer the equation comes from the differentiation of the PL expression provided from the lab manual with respect to RL.
Calculations proving Node Rule
For junction C
Currents entering the junction = 2.93 A
Currents leaving the junction = 2.29 A and 0.2 A
Sum of currents entering and leaving the junction
2.93 – (2.29 + 0.2)
= 0.44 A
For junction D
Currents entering the junction = 8.7 A
Currents leaving the junction = 8.6 A and 0.2 A
Sum of currents entering and leaving the junction
8.7 – (8.6 + 0.2)
= -0.1 A
Calculations proving Loop Rule
For loop abcda
Voltages within the loop = 0.3 V, 3.006 V and - 3.039 V
Sum of voltages around the loop
(0.3 + 3.006) - 3.039
= 0.267 V
For loop cdefc
Voltages within the loop = 0.3 V, 0.903V and - 0.906 V
Sum of voltages around the loop
(0.3 + 0.903) - 0.906
= 0.297
For loop abcfeda
Voltages within the loop = 2.93 V, 8.6 V, - 2.29 V and – 8.7 V
Sum of voltages around the loop
2.93 + 8.6 – (2.29 + 8.7)
= 0.54 V
Calculation of Power
When RL = 10, V = 0.901 and I = 0.0848
P=IV
P = 0.901 * 0.0848
= 0.07 W
P=I2R
P = 0.08482 * 10
= 0.07 W
Calculations of Error
Comparing ε to measured voltage with percent error
ε = 10 V
Measured voltage = 11.5 V
Error = (11.5 – 10) / 10
= 0.15
= 15%
Comparing R0 to measured resistance with percent error
R0 = 100
Measured Resistance = 102
Error = (102 – 100) / 100
= 0.02
= 2%
Comparing P=IV to P=I2R with numbers and equations
The value of power obtained from both P=IV and P=I2R is the same. Therefore, there is no difference in using both formulas.
Discussion and Analysis:
Was the Node Rule confirmed?
Yes, the node rule was confirmed. However, there were errors in the experiment that affected the calculated value.
Was the Loop Rule confirmed?
Yes, the loop rule was confirmed. However, there were errors in the experiment that affected the calculated value.
Do measured ε and calculated ε agree? Why or why not?
The measured ε and calculated ε do not agree. This is mainly because there were errors in the experiment.
Do measured and calculated R0 agree? Why or why not?
The measured and calculated R0 do not agree. This is mainly because there were errors in the experiment.
Are the two different ways of calculating Power Consistent? Why or why not?
Yes, the different ways of calculating power is consistent.
Maximization of Power in terms of RL
PL= (ERS+RL)2RL
Opening the brackets
= (E2RS2RL+2RS+RL)
In order to find the maximum RL we differentiate the denominator, with respect to RL and equating it to zero.
ddRL=(RS2RL+2RS+RL)
0 =- RS2RL2+1
Therefore, for maximum power transfer
RL = RS
Conclusion
Was Objective Completed
Yes
Application to Real World
In industrial systems to determine the load resistance required in order to determine the maximum power transfer through the system.
