Series and Parallel Circuits Lab Report PDF

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Physics 2240

10/9/17

Experiment 5 Series and Parallel Circuits

Austin Ciervo

Abstract In the experiment, we performed resistor and ammeter calibration, as well as calculated the equivalent resistances of series and parallel circuits. It was found that through proper equipment and resistor calibration, we can calculate the measured equivalent resistances of series and parallel circuits with sufficient precision and accuracy. Although the measuring devices had some electrical noise and the resistors has some inconsistency, we were still able to calculate equivalent resistances using Ohm’s Law with very low percent differences between theoretical Req and measured Req.

Introduction The goal of this experiment is to understand series and parallel circuits, calculate their equivalent resistance, and construct them in the laboratory. There will be four circuits which the resistance will be theoretically and experimentally calculated. The voltage and currents of the circuits will be measured to calculate the experimental resistance.

Theory A device which obeys Ohm’s Law generally can be described as a resistor, which has a resistance R. Ohm’s Law: V = IR Equation 1 Resistors of two or more can be placed in series or parallel. An equivalent resistor is a single resistor that can replace complex components in a circuit and produce equal total current when equal total voltage is applied. For a series circuit, the equivalent resistance equals the sum of the resistances. Req = R1 + R2 Equation 2 For a parallel circuit, the resistances add as reciprocals. 1/Req = 1/R1 + 1/R2 Equation 3 1/Req = R1 + R2 / R1R2 Taking the reciprocal of both sides, a new expression for calculating equivalent resistances in parallel is obtained. Req = R1R2 / R1 + R2 Equation 4 When a circuit has two resistors in parallel and another resistor in series, this condition can be dealt by combining the resistors in parallel into an equivalent resistance using Equation 4. Then combine that equivalent resistance with the single resistor in series using Equation 2. This will result in finding the equivalent resistance for the whole circuit. The current through each resistor is the same in a series circuit. There is however a voltage drop across each resistor, which may vary depending on the resistor. In a parallel circuit, the voltage drop across each resistor is the same, but the current through each one may vary. Power Power is the rate at which work is done for a system. Electrical power is defined as: P = IV Equation 5 P is measured in watts, I is the current in amperes, and V is the voltage drop across the device measured in volts. P = IV =I(IR) = I2R Equation 6 R is the resistance measured in ohms. Power is directly proportional to resistance and current squared.

Experimental Procedure Resistor Calibration The purpose of resistor calibration is to increase the precision of each resistor. The gold band on each resistor indicates a precision of +/- 5%. This precision can be improved to +/- 1% after calibration. First, locate two 100 Ω resistors, two 330 Ω resistors, and two 560 Ω resistors. Then, label the 100 Ω resistors R3 and R4, the 330 Ω resistors R1 and R6 and the 560 Ω resistors R2 and R5. Next, insert a resistor in between the spring clips on the calibration circuit. Open the signal generator on the computer software, set the waveform for a triangle at 10 Hz and an Amplitude for 10 V, then select auto. Click record and run the program for 10 seconds before stopping. Select the data, scale to fit, then click linear fit. The slope of the linear fit is the measured resistance. Repeat these steps for each resistor and record each resistors’ measured resistance. Ammeter Calibration The internal ammeter measures voltage drops across a small resistor (~0.1Ω). The ammeter measures voltages of 0.01 mV since the sensitivity is about 0.1 Ma. Electrical noise can result in significant zero error. We can achieve a precision of 0.1-0.2 mA by averaging over several seconds. This calibration procedure can correct systematic errors. First, insert the resistor closest to 100 Ω into the calibration circuit. Then correct the first column of table 2 using Ohm’s Law: I = V/R. Next, open the signal generator, set Output 1 for DC waveform and a DC voltage of 0 V. Make sure the device is on, then click record. Wait a couple seconds, when you see the measured current stop varying, click stop. Record your data into the second column of table 2. Subtract value of column 1 from column 2 to get the value for column 3. Repeat these steps and increase the voltage by one until 7 V is reached. Setup: Series and Parallel Circuits There will be four circuits which need to be configured. A series circuit, a parallel circuit, a simple series parallel circuit, and a complex series parallel circuit. The series circuit will consist of R2 and R1 placed in series. The parallel circuit will consist of R2 and R1 placed in parallel. The simple series parallel circuit will consist of R2 and R1 placed in parallel and R3 is placed in series with them. The complex series parallel circuit will consist of R4 and R6 placed in series, R1 and R5 placed in series, and R3 and R2 placed in series. Each series group is them placed in parallel with each other. Series and Parallel Circuits First open the signal generator. Then, select output 1, set DC waveform at 15 V, click auto and record data until the values stop varying. Enter the values into i-Measured in Table 3. Next, identify the current that is closest to i-Measured in the second column of Table 3. Enter this value in the i-Corrected column. Next, apply Ohm’s Law: R = V/I to calculate the measured resistance using i-Corrected and the measured. Repeat these steps for each circuit. Finally, Calculate each theoretical resistance and the % difference between theoretical Req and the measured Req.

Data R1 = 330 Ω

R2 = 560 Ω

328 +/- 0.062

577 +/- 0.20

Table 1: Measured Resistance Values R3 = 100 Ω R4 = 100 Ω R5 = 560 Ω 98.3 +/- 0.014

96.3 +/- 0.0091

578 +/- 0.18

Theory Current (mA) =0V/R: 0 mA =1V/R: 10.2 mA =2V/R: 20.3 mA =3V/R: 30.5 mA =4V/R: 40.7 mA =5V/R: 50.9 mA =6V/R: 61.0 mA =7V/R: 71.2 mA

Table 2: Ammeter Calibration Data A Current (mA) A Corrected (mA) 4.5 -4.5 14.6 -4.4 24.7 -4.4 34.8 -4.3 45.1 -4.4 55.3 -4.4 65.8 -4.8 76.4 -5.2

Circuit Diagram 1 2 3 4

Table 3: Resistance Summary i Measured i Corrected Measured (mA) (mA) Req Ω 20.2 15.7 954 78.1 72.9 205 53.3 48.9 306 43.2 38.8 385

Analysis

Theoretical Req Ω 905 209 307 386

R6 = 330 Ω 328 +/- 0.061

% Difference 5.51 1.91 0.326 0.259

Figure 1. Output Current (A) VS Output Voltage (V) As you increase the output voltage, the current through the resistor increases. The slope (m) of the linear fit is the measured resistance of the resistor.

Discussion of Results From the results given in Table 1, you can see that the measured resistance of each resistor isn’t exactly the value that it is rated to be. This is because each resistor has a precision of +/- 5%. The results from Table 2 show that there is electrical noise within the ammeter which can be calibrated to achieve a more precise output. The first column shows what the current output of the resistor should be in a perfect scenario. But since the is electrical noise within the ammeter, the actual current output is displayed in the second column. If you subtract the values from column 1 from column 2, you will get the corrected value needed to offset the actual current to achieve the theory current. This value is used to account for the electrical noise within the device. The results from Table 3 show the resistance summary of the series and parallel circuits. The second columns values were attained by analyzing each circuit diagram and calculating the equivalent resistances of the resistors using Equation 2 and Equation 4. The third column values were attained by running the signal generator for each circuit and recording the current flowing through each circuit. Each value from Table 3 column 3 was compared with the values we got in Table 2, column 2. Whichever values were close to each other, we then used its corresponding A corrected values from Table 2 column 3. If you subtract the A

corrected value from i-Measured, you will then get i-Corrected. The measured Req was attained my using Ohm’s Law R = V/I, where V equals 14.97 V and I equals the values calculated in i- Corrected. Lastly, the percent difference between theoretical Req and measured Req was calculated. This percent difference of Req may be caused by electrical within our measuring devices and the

Conclusions The results from this lab experiment support the theory because each of the measured results received and very low percent difference from the theoretical calculations. In table 1 the output values of resistance for each resistor closely matched the resistance rated value. In table 2 we accounted for electoral noise within the measuring device which minimized our zero error. In table 3, Using Ohm’s Law, we were able to calculate the measured Req of a circuit with very low percent difference compared to the theoretical Req. Errors can be reduced by using higher precision resistors and measuring devices. Given that we had electrical noise within our measuring devices and a slight inconsistency in each resistor, we can conclude that the methods used in this experiment were sufficiently precise and accurate....