Experiment To Theoretically And Experimentally Verify Circuit Analysis Methods Such As Nodal Analysis And Thévenin Equivalent Circuit Analysis

The objective of this experiment is to theoretically and experimentally verify circuit analysis methods such as Nodal analysis and Thévenin Equivalent circuit analysis. This experiment will be split into two sections: Section 1: Nodal Analysis and Section two: Thévenin’s Theorem.

Nodal Analysis:

Nodal Analysis is the method developed through Kirchhoff’s current law (KCL) to break down a circuit into its nodal voltages. This allows for the determination of potential differences around a circuit to be found. A node is the junction point of any part of a network to ground, where ground is your reference node with its voltage equal to zero. Applying KCL allows you to individually set up a string of equations for each node where the incident sum of the currents entering and exiting the node must equal to zero. Through a direct relationship of current and voltage using Ohms Law, V=IR, these strings of equations are then simultaneously solved for the voltage at each of the unknown nodes. Crammer’s Rule is also a very useful operation that can be conducted to solve more complex systems of equations.

This method of analysis becomes very useful when determining the voltage differences in a complex circuit that has many current sources involved. Nodal analysis can also be incorporated into other circuit analysis techniques such as Thévenin’s theorem.

Thévenin’s Theorem:

Thévenin’s Theorem is another circuit analysis method that allows for the simplification of any two-terminal dc network containing voltage, current and resistance sources to one of just a single equivalent voltage and resistance circuit. Applying a variation of superposition, all the current sources are converted to open circuits, which allows for the easy determination of RTh, the Thévenin resistance. The current sources are then reinstated before replacing the voltages sources with short circuits. From this, ETh, the Thévenin voltage, can then be determined using Ohm’s Law. By removing each internal resistive component (the voltage and current sources), an ideal voltage or current source can be obtained for the final Thévenin equivalent circuit.

Thévenin’s Theorem is a very effective tool when analysing circuits that are in neither parallel nor series configurations or for analysing changing circuits, without having to re-analyze the unaffected network parts. This method of analysis is considerably useful when dealing with complex circuits or circuits with power or battery systems. For more complex circuits Nodal Analysis or Mesh Analysis (using KVL to find current in any part of a planar circuit) can be used to determine the ETh voltage.

Method

Within this experiment, three means of recording circuit readings were set up in order to compare and assess the validity of the results. Three identical circuits were set up, a computer simulation in Multisim, a theoretical calculation using a hand-drawn circuit and the physical building on a breadboard. These three mediums compared many independent variables with the theoretical expectation of obtaining a single corresponding controlled variable, the output values of components.

Section 1: Nodal Analysis

Using a table for contrasting outputs, each source was measured in its own environment and recorded against each other. Within the theoretical circuit, Nodal Analysis was applied by first identifying all the nodes (including the reference node). Two nodes, including the reference node, was found and hence three equations in terms of voltage were constructed [Figure 3]. These were then simultaneously solved and the values for the potential difference at these nodes were recorded.

Section 2: Thévenin’s Theorem

Using a second table for contrasting outputs, a similar process was undertaken as previous with each source being measured and recorded against each other. Using the same theoretical circuit from Section 1, Thévenin’s Theorem was applied by first replacing the voltage source with a short circuit and determining RTh. The voltage source was then reinstated and the current source replaced by an open circuit in order to calculate ETh using Ohm’s Law. RTh and ETh were then used to construct a final Thévenin equivalent circuit.

Results and Discussion

The values of the resistors used were manually measured using a multimeter and then recorded (Table I). These new values were used to calculate the currents for both the measured and theoretical circuits in order draw conclusions as to whether or not Nodal Analysis and Thévenin’s Theorem is a valid circuit analysis method. The expected values were entered into the simulated circuit where it was found that each resistor used was approximately 0.03% under the expected values.

Table II: Resistive Values of the Nodal and Thévenin Circuit

The circuits were constructed, calculated and measured in order to determine the current resulting from each identical circuit. The Nodal Analysis Method was validated by the three tests (Table II), which showed the results being very close and producing an overall approximation of 1.72mA. This was an expected result as it was expected that all values would be within 0.01% of each other. Since all results proved the same outcome for different methods of working, Nodal Analysis was therefore proven a valid method for circuit analysis.

Table III: Currents running through three identical circuits – Nodal Analysis

When testing Thévenin’s Theorem the same process was applied to the circuits used for the Nodal Analysis method. After finding ETh and RTh, the current was calculated using Ohms Law, V=IR and producing a current value of 1.72mA. This was a compatible value to the ones produced using Nodal Analysis and hence validates the Thévenin’s Theorem. The circuits were also rebuilt to the expected Thévenin’s equal circuit and the definition of Thévenin’s Theorem was proven valid, ‘the simplification of any bilateral dc network to one of just a single equivalent voltage and resistance circuit’. The calculation, the simulated and measured circuits all produced very similar values to what was expected and the results were recorded and compared (Table III).

Table III: ETh and RTh Calculated Across Three Circuits – Thévenin’s Theorem

From the gathered results across Nodal Analysis and Thévenin’s Theorem the proven results where as expected, with the measured results being the greatest since there were external factors such as the wires and digital millimeter resistances that were not taken into account for the simulation and calculations. This was also seen at the start of the experiment from the measured values of the resistors when compared to the theoretically expected ones.

The behaviour of voltage and current in an electrical system plays an integral part to be able to theoretically prove the validity of Nodal Analysis. When voltage passes a load, a potential difference will occur across the load and the voltage exiting the load will decrease, however current across a load will remain constant. Moreover, at a node (where a circuit splits into 3 or more ways), the current will divide itself across the branches and the voltage will remain constant through each branch. This property of electrical circuits leads to the development of Kirchhoff’s current law where current going into a node will equal the current exiting the node.

This concept is applied to the Nodal Analysis method used in the experiment where, combined with Ohms Law, allows for the voltage to be determined by calculating the current flowing in each loop [Figure 3]. Once the current was determined through each loop, a simple Ohm’s Law calculation could be performed to find the value of any unknown within the circuit. The same principle was applied to Thévenin’s circuit using the voltage divider rule.

When undertaking this experiment, sources of marginal error occurred in the calculated values due to early rounding however the differences remained within a 0.01% uncertainty of the other two values obtained from the other systems. As a result, a high level of accuracy was still maintained throughout. The experiment proved valid due to the methods being compared across three separate circuit systems. The two methods were not only compared with the simulated and measured circuits but also compared between the two circuit analysis methods, where both results produced the same value of 1.72 mA.

Not only was circuit analysis techniques observed and verified, but the newly learnt concepts of Nodal Analysis and Thévenin’s theorem were re-enforced and practiced as a way of understanding the properties of electrical systems. A gained understanding in the behavior of circuits was achieved, as through the physical manipulation of the built circuits, recalculating values and testing these against measured values, a better knowledge of how voltage and current changes within a circuit was achieved.

Improvements for future experiments in Nodal analysis and Thévenin’s theorem would be to repeat the two sections of the experiment two more times with different levels of complexity in the circuit arrangement. This would provide a more reliable set of results that can be collated and graphed in order to determine more accurate averages and uncertainties when moving between the physically measured and theoretically calculated circuits. A further improvement would be to test the circuit for not only the initial current but for multiple other points around the circuits.

Conclusion

Two circuit analysis techniques, Nodal Analysis and Thévenin’s Theorem were implemented on three identical circuits set up in three different environments, a digital simulation, a physical measurement, and theoretical calculation, in order to prove the validity of these methods. Within Nodal Analysis, the current values recorded in each of the tests were ≈ 1.72 mA and proved consistent for all the tested circuits, validating that the potential voltages across each node can be determined through this method of analysis.

Moreover, Thévenin’s Theorem was verified by comparing the Thévenin’s equivalent circuit with that of the initial circuits. The current value of 1.72 mA was also compatible with the nodal circuits that further strengthened the consistency in the results. Expected results of ETh and RTh were achieved and as a result, proved that any bilateral DC circuit could in fact be simplified to one with an equivalent voltage and resistance source.

References

1. Raad, R. and Jarrett, L. (2019). ENGG104 – Lecture 4/13.
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