Resistor Networks: Practical Experiments Report

Section 1: Abstract

The experiments conducted during the two-week term focused on resistor networks and their theoretical principles, which were tested through practical experiments. Various equipment was used to measure important variables, such as voltage and resistance. After conducting the experiments, a comparison was made between the measured values and the values calculated using relevant formulae, which will be discussed in detail later in the report. The primary equipment used in these experiments was the P-IE-R board, featuring printed circuits with soldered resistors.

Section 2: Introduction

Section 2.1: First Experiment Theory

The main theoretical concept in the first experiment is Thévenin's theorem, which states that complex linear circuits with multiple voltage sources and resistors can be simplified into a single resistor and a single ideal voltage source [1]. To apply this theorem, we need to understand the relationship between voltage, current, and resistance, as described by Ohm's law [2]:

^V/_I = ^R/_t

This equation shows that voltage (V) is directly proportional to current (I), with resistance (R) as the constant of proportionality.

It results in a linear relationship between voltage and current, represented graphically by V-I curves.

To determine the ideal voltage source using Kirchhoff's law, we apply the principle that the sum of voltage differences in a closed loop is zero [3]. By applying this law, we can calculate the voltage across the loop of the circuit.

For calculating the total resistance in a circuit, we need to identify whether the resistors are in series or parallel. For resistors in parallel, we use the formula:

¹/_Rt = ¹/_R1 + ¹/_R2 [4]

In the case of series resistors, we simply add them together to find the total resistance, and then we can calculate the current using Ohm's law:

V_R = I * R_t

Section 2.2: Second Experiment Theory

In the second experiment, we explore the concept of a potential divider circuit. A potential divider involves voltage distributed across two components, typically resistors, between an input voltage and a zero voltage state [5]. It results in a fraction of the input voltage being present between the connected resistors.

In addition to understanding potential dividers, knowledge of short circuits and open circuits is crucial. A short circuit occurs when a conductor connects across a voltage source's terminals, potentially damaging the source [6]. Open circuits, on the other hand, are circuits where no current can flow, but voltage remains present.

Section 2.3: Third Experiment Theory

In the third experiment, the same circuit as in experiment 2 is used, but with additional components, including a 68kΩ resistor, which is later replaced by a 22kΩ resistor. The objective of this experiment is to determine whether a 30V voltage source is safe for an 18W resistor. The 18W rating of the resistor indicates the power it can dissipate before experiencing damage due to high temperatures, as explained in more detail in [7].

To calculate the maximum voltage that can be applied to the resistor without exceeding its power rating, we need to solve two equations for certain values. The first equation relates power dissipation by the resistor (P) to current (I) and resistance (R) using the formula:

P = I² * R

Given the values of power dissipated (0.125W) and resistance (68kΩ), we can calculate the current in the circuit, which is approximately 0.04287mA. Using Ohm's law [2], we can then calculate the maximum voltage:

V = I * R

By applying Ohm's law, we find that the maximum safe voltage for the 18W resistor is approximately 2.91547V. Therefore, a 30V voltage source is not safe to use in this context.

Section 2.4: Wheatstone Bridge

The Wheatstone bridge circuit, invented by Sir Charles Wheatstone, consists of four resistors arranged in a diamond shape. Among these resistors, three have known values, while one has an unknown value [8]. This circuit operates in a condition known as the balanced bridge, where the voltage between two specific terminals becomes zero [8]. In this balanced state, the voltages across the resistors positioned opposite each other are equal, resulting in the equation:

Vx * V2 = V3 * V4 [8]

By applying Ohm's law [2] to this equation, we can simplify it to:

Ix * Rx = I2 * R2 = I3 * R3 = I4 * R4 [8]

Here, all the current values cancel each other out, leaving us with a simplified formula for calculating the unknown resistor:

Rx = (R2 * R3) / R4

In this experiment, if we input the known resistor values, such as R2 (2.7kΩ), R3 (1.61kΩ), and R4 (1.61kΩ), we can calculate the value of the unknown resistor (Rx) as 2.7kΩ.

Section 3: Methods and Results

3.1 Experiment One: Thévenin's Equivalent Circuit

In the first experiment, we conducted measurements using the Thévenin's Equivalent Circuit on the P-IE-R board. The equipment used included an AVO meter (with a small internal resistance considered negligible), a P-IE-R board, and a digital multimeter (DMM).

The steps involved in this experiment were as follows:

1. Set up the power supply to 10V.
2. Connect the negative terminal via a black lead to the lower track and the positive terminal via a red wire to the upper track of the circuit.
3. Insert a resistor with a value of 2.2kΩ, determined using the resistor value color chart.
4. Use the DMM to measure the voltage across the terminals with the voltage output.
5. Set the AVO meter to measure the current across the terminals labeled V2 at the 10 mA range.

Measurements were taken with and without a short circuit [6]. Without a short circuit, the voltage was 10V, and the current was 4.5mA with the 2.2kΩ resistor. With a short circuit, the current dropped to 0.16276mA, and the AVO meter's resistance was measured at 0.127Ω.

Although a short circuit theoretically has zero voltage and resistance, in practice, there is a small internal resistance in the AVO meter. Plotting voltage and resistance against each other on a graph results in an upward-sloping relationship, consistent with Ohm's law [2]. When current and voltage are plotted, the pattern demonstrates the direct proportionality of voltage and current, with resistance as the constant gradient.

3.2 Experiment Two: Potential Divider

In the second experiment, we examined the potential divider circuit labeled as such on the P-IE-R board [5]. The circuit required the connection of a black lead to the 0V terminal and a red lead to the 10V power supply. The circuit comprised three resistors: r1, r2, and r3, with two terminal outputs (one on the bottom track and one on the middle track) and a jumper connecting the top track. R2 and r3 were in parallel, while r1 and r2 were in series. A DMM was connected to the middle and bottom track terminals to measure the circuit's output.

Measurements were taken for two scenarios: open circuit and short circuit [5]. In the open circuit, the input voltage was 0, resulting in an output voltage of 0.001V, a current of 0 (as it's an open circuit), and an immeasurable resistance due to resistance approaching infinity in the absence of current. In the short circuit, the input and output voltages were 10V and 0.02V, respectively. The resistance was 0.1287Ω due to the AVO meter's internal resistance, and the current was 0.0023mA.

To find the resistance that halves the output voltage, we calculated the current for a 10V input voltage, which was 2.27mA, and then used Ohm's law [2] to find the resistance value (2.2kΩ) that halves the voltage (5V). This configuration is similar to a Thévenin's Equivalent Circuit, where the voltage remains consistent at 5V across the circuit, and all the resistors can be simplified into one equivalent resistor.

3.3 Experiment Three: Maximum Voltage

The objective of this experiment was to calculate and measure the maximum voltage that an 18 Watt resistor can handle without causing component damage and to determine if 30V is a safe voltage. The calculation is discussed in more detail in section 2.3, with references to sources [7] and [9].

To calculate the maximum current, we used the formula P=IV, knowing the power rating (18W) and the maximum voltage (30V) [9]. This resulted in a maximum current of approximately 0.441176mA.

The experiment utilized the exp 2 part of the P-IE-R board. The method involved:

1. Turning off the power supply unit to ensure no voltage or current.
2. Connecting the negative terminal to the power supply via a black lead and double-ended crocodile clips.
3. Connecting one end of a 68kΩ resistor in series with another 2.2kΩ resistor to the positive terminal of the power supply, resulting in a total resistance of 70.2kΩ [4].
4. Applying 30V to the circuit to find the maximum current.
5. Measuring the current using the DMM, set to measure current in the 2.5mA range, which was measured at 0.4254mA.

Although the measured and calculated results are close, they differ due to the practical limitations of wires, which introduce small resistance. In conclusion, the measured results are acceptable in real-life situations where current flows through wires, and there is minimal resistance.

3.4 Experiment Four: Wheatstone Bridge

For this experiment, we utilized the part of the P-IE-R board labeled "exp 4 Wheatstone bridge." The setup was similar to previous experiments, with the power supply unit connected by attaching the positive terminal to "V" via a red wire and the negative terminal to "0V." The P-IE-R board already had three resistors soldered onto it: two with known values, labeled r3 and r4, with resistances of 1.62kΩ each, and one labeled Rx with an unknown value. The last resistor was replaced with a resistance decade box, which allowed us to increment resistance in certain ranges by a factor of 10, as indicated by the dial values, along with the corresponding Ohm values labeled above [8].

We connected a DMM meter across the terminals in the middle of the bridge. After connecting all the components, we turned on the power supply unit to 10V. Initially, all the dials on the decade box were set to 9 (approximately below 10kΩ), resulting in a current reading of 1.5262mA on the DMM.

To find the value of the unknown resistor (Rx) that would balance the bridge, we needed to lower the resistance on the decade box until the current reading on the DMM approached zero without changing signs from positive to negative. This demonstrated the inverse relationship between the dials and the current reading, with lower dial values leading to decreased current readings. We set the dials to 2 for the 1kΩ dial, 7 for the 100Ω dial, and 0 for the other dials. This adjustment reduced the current to 0.0161mA, indicating that the value of Rx required to balance the bridge was 2.7kΩ (for more information on balancing the bridge, see [8] and section 2.4).

Using the formula Rx * R2 = R3 * r4 from section 2.4, we found the value of R2 by substituting the known resistor values, resulting in 2.7 * R2 = 1.62 * 1.62, which gave us R2 = 2.7kΩ. To verify these results, we connected a DMM set up to measure resistance to the terminals of R2, which displayed a resistance value of 2.6671kΩ.

In conclusion, while there is a degree of difference between the measured and calculated values, this variation is expected due to the non-ideal nature of components, including slight deviations from specified resistor values.

Section 4: Conclusion

4.1 Thévenin's Equivalent Circuit

Throughout this experiment, I gained a comprehensive understanding of Thévenin's Equivalent Circuit and how it can be employed to simplify complex circuits into a single component using resistor series and parallel laws [1]. Additionally, I learned that an ideal short circuit is unattainable due to the internal resistance of the AVO meter [6]. Familiarizing myself with the equipment used in the experiments was crucial, as it deepened my understanding of their capabilities and how to set them up.

4.2 Potential Divider

Experiment two provided insights into the purpose and composition of a potential divider circuit. I learned how open circuits and short circuits affect resistance, current, and voltage, gaining practical knowledge of their effects.

4.3 Maximum Voltage

In experiment three, I discovered that resistors have power ratings, indicating the maximum safe operating conditions without damage. I learned how to calculate and measure the maximum voltage that a resistor can tolerate.

4.4 Wheatstone Bridge

The Wheatstone Bridge experiment introduced me to a circuit used to find the values of resistors with unknown values, utilizing a decade box. It required some time to understand how to operate the device [8]. The Wheatstone Bridge can be likened to a balanced weight scale, as it balances two voltages on opposite sides of the circuit, resulting in zero voltage across the bridge. This balance depends on the ratio between the two resistors placed across the bridge, which must be identical [8].

References

1. Tony R. Kuphaldt. "Lessons in Electric Circuits - Lesson 1, Chapter 2." Available at: http://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_2.html [Accessed on 18/11/2021].
2. Digilent. "Kirchhoff's Voltage Law." Available at: https://digilent.com/reference/learn/fundamentals/circuits/kirchhoffs-voltage-law.
3. Khan Academy. "Calculating Resistors in Parallel." Available at: https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-resistor-circuits/a/ee-parallel-resistors.
4. Electronics Notes. "What is a Potential Divider?" Available at: https://www.electronics-notes.com/articles/basic_concepts/voltage/voltage-potential-divider.php [Accessed on 22/11/2021].
5. Understanding DC Circuits. "Short Circuits" by Dale Patrick and Stephen Fardo, 1999, Page 17.
6. Electronics-Tutorials. "Resistor Power Rating." Available at: https://www.electronics-tutorials.ws/resistor/res_7.html [Accessed on 23/11/2021].
7. Thomas Floyd and David Buchla. "Electronics Fundamentals: Pearson New International Edition" (Published on 29/08/2013). Pages 245-300.