Lab 7_ RC Circuit

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Electrical Engineering

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Dec 6, 2023

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Lab 7: RC Circuit Willem Brotha Deborah Harris 10/5/2023 Abstract: In this lab we studied the time behavior in charging and discharging a capacitor. The instruments used were similar to those of previous labs such as a power supply, oscilloscope, capacitor, RC circuit board, function generator, and Logger Pro software. Procedure: In part 1 of the lab we were tasked with measuring the long time constant on the oscilloscope. After plugging in our RC Circuit board to the power supply and oscilloscope, we turned our voltage up to 8.0V. We then charged and discharged the capacitor and observed that the line on the graph curved upward indefinitely. Beginning our experiment we charged our RC circuit board and observed that the graph created an upward line indicating charge. Noticing that our graph did not line up exactly on 0 seconds we started logging our data from -2 seconds and adjusted the remaining values accordingly. We simultaneously logged our voltage by increasing our horizontal knob on the oscilloscope 1.2 seconds and recorded the values. Furthermore, we discharged the capacitor and noticed that the line on the graph curved downward indefinitely indicating that the capacitor was now discharged. We then repeated the steps for charging by logging our results. After recording the values in the graphs below we utilized the Python program to perform a linear fit. Using the function np.log() we took the natural log of our y-axis data for discharge and used the equation ln VC = ln V0 - t/RC to find the value and error of time constant RC. This resulted in a slope of -0.36854078873715923, which is equal to the negative inverse of our time constant. We proceeded to create another plot with our values for charge and discharge on the same graph where they cross at time T ½. We found that the two graphs intersect at a time of 1.74s and Voltage of 4.42 giving us T ½ =1.74s. Using equation 2 from the lab manual we found the time constant of RC as 2.51 ± 0.0069s. Finally we compared our values of RC constant with the values of RC using the individual values of R & C. The actual values of R&C and R=220Ω and C=0.01F, giving us a theoretical value of 2.2 ± 0.0102s. Experimental Data: Our recorded voltages and their corresponding times are shown in the following two data tables. Table one shows the voltages over time when we charged the circuit, and table two shows the voltages over time when we discharged the circuit. Charge:

Time ± 0.001 (seconds) Voltage at Time T ± 0.001 (V) 0 0.160 1.200 3.360 2.400 5.600 3.600 6.560 4.800 7.200 6.000 7.600 7.200 7.760 8.400 7.840 9.600 7.920 10.800 8.000 Discharge: Time ± 0.001 (seconds) Voltage at Time T ± 0.001 (V) 0 8.080 1.200 5.600 2.400 3.040 3.600 1.840 4.800 1.120 6.000 0.800 7.200 0.480 8.400 0.320 9.600 0.240 10.800 0.160 Data Analysis: We can find the time constant value of a circuit using the equation Tau = RC where R is resistance and C is the capacitance of a capacitor. This Tau value, otherwise known as the RC value, can be found through three different methods. The first method involves calculating the

RC value by taking the Ln of the equation VC(t) = V0(1-e^-t/(RC)) to get Ln(VC(t)) = Ln(V0)+(-t/RC). We can then plot Ln(VC(t)) over time to get the following graph: The slope of the graph is equal to the negative inverse of our RC value. Therefore -1/slope = -1/(-0.36854078873715923) = 2.7134s. Error of the RC value can be calculated using the error propagation rules for power functions: δm = |n|*|m|*δp/|p| = |-1|*|2.7134|*(0.009841273449800129/| 2.7134|) = ±0.009841s. The second method involves finding the T1/2 value, which is the time at which the capacitor is charged to 50%. The equation T1/2 = Ln(2)*RC allows us to solve for RC. When the lines of the charge and discharge values overlap, that is when the capacitor is charged to 50%. This is shown in the following graph:

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The two graphs intersect at T = 1.74s and V = 4.42V, meaning T1/2 = 1.74s. Using the aforementioned (t1/2 = RC ln(2)) we can solve for RC: t1/2/ln(2) = RC. Plugging in our experimental values gives us: 1.74s/ln(2) = 2.51s. Error for the RC value can be calculated using the error propagation rules for multiplication and division: δm = |A|*δp = |ln(2)|*(0.01s) = ±0.00693s. The third method involves using the actual resistance and capacitance values to get RC. The actual values of R and C are R = 220 Ohms and C = 0.01F. These two values gives us the theoretical value of RC, which is equal to R*C = 220 Ohms * 0.01F = 2.2s. The error to this value is found through this equation: δm = |m|*(δp/|p|+δq/|q) = |2.2|*(1/|220|+1*10^-6/|0.01|) = ±0.0102s. This value is similar but not equal to the values that we got through method one and two. And since their error bars do not overlap, the difference between the values is significant. Method one had the greatest error between the two values, which was found through the percent error equation: percent difference = (|measured - expected|/expected) * 100%. When we plug in our found values we get this equation: (|2.7134-2.2|/2.2)*100% = 23%. The same equation can be applied to the RC value of method two, and the difference between the two values came out to be 14%. Conclusion: In this lab we studied how the voltage of a system changes as we charge or discharge the values. We successfully found three different values for the time constant of the system using three different methods. These values came out to be 2.7134 ± 0.009841s, 2.51 ± 0.00693s, and 2.2 ± 0.0102s respectively. The two experimental values had a significant difference to the theoretical value, and these were 23% and 14%. Both differences could possibly be a result of a very small sample size for our data points. For both table one and two we only plotted ten different values over time. To truly get accurate slopes and intercepts for our calculations, we would have had to plot hundreds of different values to get more accurate graphs. More accurate graphs would have led to more accurate calculations, which would probably have led to smaller percent errors.