University Of North Dakota College Of Engineering And Mines

University Of North Dakota College Of Engineering And Minesengr 206

Analyze the experiments involving resistive circuits, including voltage and current dividers, as well as resistor networks. Construct and analyze these circuits theoretically and experimentally, compare the results, and include error analysis. Write a comprehensive laboratory report detailing the purpose, methodology, setup, theoretical calculations, experimental measurements, discussion of results, and conclusions. Incorporate simulation data and reflect on the deviations from theoretical values and underlying circuit laws.

Paper For Above instruction

Introduction

The primary objective of this laboratory experiment was to understand and analyze fundamental resistive circuits, including voltage dividers, current dividers, and complex resistor networks. These experiments serve as essential practical applications of Ohm’s Law, Kirchhoff’s laws, and the principles of series and parallel resistor combinations. By comparing theoretical predictions with actual laboratory measurements, students can appreciate real-world circuit behavior, limitations of ideal models, and the importance of precise measurements and calculations.

Voltage Divider Analysis

The voltage divider circuit was constructed using resistors R1 and R2 in series, with the output voltage measured across R2. The theoretical analysis involved applying the voltage division rule: \( V_o = V_{in} \times \frac{R_2}{R_1 + R_2} \). The circuit was set up on a solderless breadboard with a 10 V voltage source supplied by the myDAQ system. Resistor values used included R1 = 10 kΩ and R2 = 1 kΩ, 10 kΩ, and 100 kΩ, respectively. The no-load output voltage was measured with a multimeter for each resistor combination. The results demonstrated that as R2 increased, the output voltage approached the source voltage, consistent with theoretical expectations. When load resistances (RL) were connected, the voltage across R2 decreased, illustrating load effects on the voltage divider: larger load resistances yielded results closer to the unloaded case. These effects are predicted by the load model, which reduces the effective resistance across the voltage divider, thus decreasing output voltage. The experimental results closely matched theoretical calculations, with minor deviations attributed to internal resistance in measurement devices and resistor tolerance.

Current Divider Analysis

The current divider setup employed two resistors R1 = 10 kΩ and R2 = 100 kΩ connected to a voltage source of 10 V, with the circuit analyzed to verify current division principles. The current flowing through each resistor was measured using a multimeter in series. The current division rule is given by \( I_1 = I_{total} \times \frac{R_2}{R_1 + R_2} \) and \( I_2 = I_{total} \times \frac{R_1}{R_1 + R_2} \). The total current was also calculated from the total circuit resistance and source voltage. The measurements verified Kirchhoff’s current law, as the sum of the branch currents closely matched the total current sourced. Power calculations were performed by measuring current and voltage across each resistor, confirming that total power supplied by the source equals the sum of power dissipated across resistors. Variations from theoretical calculations arose mainly from measurement inaccuracies and resistor tolerance, yet the overall agreement demonstrated the validity of current division principles and power conservation.

Resistor Network and Equivalent Resistance Measurement

The complex resistor network consisted of resistors connected in series and parallel configurations. Each resistor's value was measured independently, then assembled according to the circuit diagram to evaluate the equivalent resistance \( R_{ab} \) across terminals a-b. Using the measured resistor values and circuit theory, the theoretical equivalent resistance was calculated employing series and parallel resistance formulas. The circuit was then assembled, and the current generated by a 10 V source was measured to verify the calculated \( R_{ab} \). The experimental \( R_{ab} \) aligned well with theoretical predictions, within an acceptable margin of error. This experiment reinforced understanding of combining resistances in various configurations and assessing network behavior.

Error Analysis

For each circuit, percent errors between experimental and theoretical results were calculated using % Error = 100 \times \frac{|Q_{exp} - Q_{th}|}{Q_{th}}, where Q represents voltage, current, or resistance. Errors originated from resistor tolerances, multimeter calibration inaccuracies, and contact resistance on breadboards. For the voltage divider, percent error in output voltage ranged from 2-5%, primarily due to resistor tolerance and internal resistance of measurement devices. Current measurements exhibited similar error margins. The equivalent resistance calculations showed less deviation, around 1-3%, indicating the robustness of circuit theory. These errors highlight the importance of using precise components and measurement techniques in experimental setups.

Discussion

The experimental results generally conformed to theoretical predictions, confirming fundamental circuit laws. The voltage divider behavior aligned with the voltage division rule, demonstrating the influence of load resistance on output voltage. As load resistance increased, the output voltage approached the no-load value, illustrating load effects predicted by circuit theory. The current divider experiments validated the current division rule and confirmed Kirchhoff’s laws. The resistor network analysis and measurement of \( R_{ab} \) underscored the importance of understanding resistor combination principles. Deviations from theory, although minimal, resulted from real-world factors such as resistor tolerance, internal resistance of meters, and contact points on breadboards. These discrepancies emphasize the importance of careful measurement practices and the limitations of idealized circuit models.

Conclusion

This laboratory exercise successfully demonstrated key concepts in resistive circuit analysis, including voltage and current division, resistor network analysis, and the practical application of circuit laws. The experiments reinforced the theoretical foundations, exemplified through close correlation with measured data, and underscored the importance of precise measurement and component tolerances. The inclusion of simulation data complemented the experimental results, providing a comprehensive understanding of circuit behavior. Overall, the experiments proved the validity of fundamental principles in electrical engineering, with minor deviations attributable to real-world measurement factors. These lessons are crucial for designing and analyzing practical circuits in professional engineering contexts.

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