Lab 9 – Introduction to RC Circuits in the Time and Frequency-Domains

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Dec 6, 2023

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Lab 9 – Introduction to RC Circuits in the Time and Frequency-Domains Student: Celine Habr Instructor: Dr. Iman Salama Lab Partner: Mateo Wu Chen November 6, 2023

1. Introduction The goal of this experiment was to examine how alternating current (AC) voltage signals impact a resistor-capacitor (RC) circuit. Initially, a function generator was linked to an RC circuit to create a square wave. The charge and discharge of the capacitor, which led to a decay in the signal, was monitored using an oscilloscope. Subsequently, the experiment continued with the identical setup, but with a sine wave input, to investigate how the frequency of the signal influenced the voltage across the capacitor and to note the variances between the capacitor's voltage and that of the source. 2. Results 2.1 Part 1: Transient Signals With An Rc Circuit With Square Waves Firstly, we connected the function generator to a 20 kΩ resistor in series with a 0.1 μF capacitor – the circuit diagram for this is seen in figure 1 , while the actual built circuit is figure 2 . The function generator was then set to produce a 1V amplitude square wave with a DC offset of 1V. Figure 1: Simple RC circuit used for this lab where R is 20 kΩ and C is 0.1 μF Source: Dr. Iman Salama. “Lab 9 - Getting started with Analog to Digital Conversion and Sampling.” Northeastern University. 23 October 2023.

Figure 2: Completed RC circuit on protoboard with oscilloscope connected across the capacitor and to the signal generator. We then determined the frequency that should be used to observe the exponential decay behavior of the voltage across the capacitor by first finding the time constant τ. In this circuit, , where R is the equivalent resistance in the circuit, being 20kΩ, and C is the capacitance τ = 𝑅𝐶 of the capacitor, 0.1 μF. Hence, the time constant τ for this circuit was In order to see the capacitor fully charge and discharge, the 2000 × 0. 1 × 10 −6 = 0. 002 ?. period of the wave should be at least 10 times the value of τ, so T = 0.02s. Therefore, the frequency was set to ? = 1 0.2 = 50 𝐻𝑧.

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Next, we connected the oscilloscope across the capacitor in order to measure the voltage across it, seen in figure 2 . This produced a waveform that was shaped as expected, seen in figure 3 , with the charging/discharging of the capacitor resembling exponential curves. We then used the waveform displayed on the oscilloscope to find the experimental value of the time constant τ. Figure 3: Square wave input (yellow) and voltage output across capacitor (green) shown on oscilloscope. We then calculated the time constant τ from the output waveform using the equation . (1) 𝑉 𝑐 (?) = 𝑉 0 ? − ∆? τ

By using the cursor on the oscilloscope, we found the peak value of voltage across the capacitor at t = 0, which was found as V 0 = 1.452V. Then, to choose the value of V c (t) we multiplied V 0 by e –1 obtaining V c (t) = 0.534V. We then found the corresponding times (on the x-axis) of the voltages and found the difference, Δt, which was 2.5 ms. These values were substituted in Eq. 1 , 0. 534 = 1. 452 ? − 2.50×10 −3 τ and we then solved for τ, obtaining a value of 2.5 ms. This value agrees with the theoretical value found earlier, being 2 ms. 2.2 Part 2: A Simple Rc Circuit With Sine Waves In this part of the lab, the same RC circuit was used, seen in figure 1 and 2 , but the function generator settings were changed to compare the waves at different frequencies. The function generator was set to produce a 1V amplitude sine wave with a frequency of 80Hz. We then measured the voltage across the capacitor in the circuit on the oscilloscope. The amplitude of the function generator input waveform was compared to that of the voltage across the capacitor by using the cursor. It was found that the amplitude of the input signal (1.09 V) is larger than the output voltage amplitude (750 mV). This occurs because the voltage drops across the resistor. In addition, the phase shift between the two waveforms was found to be 0.75 or This was calculated using the equation 6 25 π. (2) ∆ϕ = ∆? × ? 0 × 2π where f 0 is the fundamental frequency, which in this case is 80. These measurements were then done for frequencies of 0.1, 1, 10,100, 1000, and 10000 Hz. However, the waveforms for the 0.1 Hz trial and the phase shift of the 1 Hz waves were unreadable on the oscilloscope. This was because the input and output waveforms were too close together. The results are seen in table 1 . As a result, it can be concluded that as the frequency increases, the phase shift between the input and output signal increases, and the difference between their amplitudes gets larger with the output decreasing.

Frequency (Hz) Amplitude (mV) Phase shift (rad) Input Voltage Output Voltage 0.1 n/a n/a n/a 1.0 237.5 230.0 n/a 10 998.5 950.8 0.38 80 1090 750.0 0.75 100 988.5 387.0 0.89 1000 988.5 97.5 1.46 10000 988.5 20.5 1.62 Table 1: Frequency and its effect on the phase angle and amplitude of the signal output 3. Results Part 1 of the experiment demonstrated that the voltage over a capacitor displays an exponential reaction when subjected to a square wave input in an RC circuit. The voltage-time graph for the capacitor confirmed this behavior, showing exponential increases and decreases that corresponded with the abrupt shifts in the square wave signal's input voltage. The exponential nature was further justified by Eq. 1 using two points on the graph, in order to find the experimental time constant τ. The experimental value was determined to be 2.4 ms while the theoretical value, which was found by multiplying the resistance and capacitance values, was found to be 2 ms. These two values are relatively close to each other indicating that they agree. This value was found to concur with the theoretical expectation of τ, which was calculated using the known values of resistance and capacitance. Part 2 concluded that at different frequencies the capacitor has different behaviors. The trend seen in table 1 is that as the frequency increases, the phase shift between the input and output signal increases, and the difference between their amplitudes gets larger with the output decreasing. At higher frequencies, the reactance of the capacitor drops, meaning that it will eventually get shorted since it cannot charge or discharge fast enough, justifying the phase shift between the waves. The capacitor only had time to partially charge before the source voltage signal changed direction. This explains why the amplitude of the voltage through the capacitor

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decreased as the frequency increased. At low frequencies, there was enough time for the capacitor to fully charge, which is why the amplitude of the voltage source and the capacitor voltage were closer in value. 4. References Dr. Iman Salama. “Lab 9 - Getting started with Analog to Digital Conversion and Sampling.” Northeastern University. 23 October 2023.