Experiment 6 Report

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University of California, Los Angeles *

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110L

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Electrical Engineering

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Apr 3, 2024

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ECE 110L Winter 24’ Instructor: Mesghali, Farid Exp. 6: Transient Response of the 1st-Order Circuits Om Patel, 605518179 Experiment Introduction and Theory In this set of labs, we will be exploring the natural and step response of first-order capacitive and inductive circuits along with designing and measuring an advanced RC Filter. As we have learned throughout the last couple of labs a capacitor stores energy in the form of an electrostatic field between two plates as current flows through them and an inductor is a passive element that stores energy within magnetic fields generated by a change in current. To analyze the circuits we must derive the linear differential equations for them. Equation 1: RC linear differential equation ? 0 + ?𝐶 ?? 0 ?? = ? ? Equation 2: RC step response ? 0 (?) = 𝑉(1 − ? ( −? ?𝐶 ) ) Equation 3: RL linear differential equation ? 0 + 𝐿 ? ?? 0 ?? = ? ? Equation 4: RL step response ? 0 (?) = 𝑉(1 − ? ( −?? 𝐿 ) ) The time constant is a concept that states the time the circuit needs to stake the step response to gain a value of (~0.63) of its final value. These constants are in the above 1 − ? −1 equations as well. For the RC circuit, the time circuit is R*C while for the RL circuit, the time constant is L/R. The Twin-T Notch filter is a type of RC filter that is used to reject signals at a certain frequency. It serves the opposite purpose from that of a bandwidth filter; instead of accepting signals from a certain frequency, it rejects them. Another reason why this type of filter is preferred is because it only consists of resistors and capacitors which are smaller in size compared to an inductor. The following equations help to explain the filter's characteristics. Equation 5: Twin-T Notch Filter Gain Eq. 𝑉 ??? 𝑉 ? = ( ? 0 ? − ? ? 0 )+?(?+ ? ? −1) ( ? 0 ? − ? ? 0 )+?(?+ ? ? + 1 ? + 1 ? ) , ?ℎ?𝑟? ? 0 = 1 ?𝐶 Equation 6: Gain Eq. at max frequency 𝑉 ??? 𝑉 ? = ? 2 (?+1)−?? ? 2 (?+1)+?+? , ?ℎ?? ? = ? 0 Equation 7: B value for Gain to be zero ? = ? ?+1 , ??𝑟 ???? = 0 Lab 1: RC Circuit Analysis Introduction: For this lab, we will construct an RC circuit that is shown in Figure 1 below using a 4.7 nF capacitor and 33kΩ resistor. Using the waveform generator, output a square wave of 1V peak-to-peak. Ensuring that the appropriate frequency with a period of 5-10 times the time constant to allow the output voltage to reach a steady state. Measure the voltage across the capacitor using the 2+ and 2- pins from the AD-2.

Figure 1: RC Circuit diagram for Lab 1 Figure 2: RC Circuit constructed for Lab 1 Figure 3: CR Circuit constructed for Lab 1 Measured Data: Component Measured Value Theoretical Value Resistor (Ω) 32890 33000 Capacitor (nF) 4.59 4.70 Time constant RC (μs) 143 151 Time constant CR (μs) 143 151 Table 1: Measured and theoretical values for circuit elements/analyzed variables

Figure 4: RC data showing frequency band Figure 5: RC data showing tau Figure 6: CR data showing frequency band Figure 7: CR data showing tau Discussion: 1. The experimental time constants is very similar to the theoretical values. a. 143 compared to the intended 150 for both the RC and CR circuits. 2. Zooming in on the time resolution led to an increase in sampling rate which made the measurements more smooth and readable. 3. With an increase in the input frequency, the signal did not reach a steady state because the distance between the peaks shrank.

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Lab 2: RL Circuit Analysis Introduction: For this lab, we will construct an RL circuit that is shown in Figure 1 below using a 150 mH inductor and 1kΩ resistor. Ensuring that the internal resistance of the inductor is taken into consideration. Using the waveform generator, output a square wave of 1V peak-to-peak. Ensuring that the appropriate frequency with a period of 5-10 times the time constant to allow the output voltage to reach a steady state. Measure the voltage across the inductor using the 2+ and 2- pins from the AD-2. Figure 8: RL Circuit diagram for Lab 2 Figure 9: RL Circuit constructed for Lab 2 Figure 10: LR Circuit constructed for Lab 2 Measured Data: Component Measured Value Theoretical Value Resistor (Ω) 985 1000 Inductor (mH) 150 150 R L (Ω) 249.6 250 Time constant RL (μs) 123 122 Time constant LR (μs) 119 122 Table 2: Measured and theoretical values for circuit elements/analyzed variables

Figure 11: RL data showing frequency band Figure 12: RL data showing tau Figure 13: LR data showing frequency band Figure 14: LR data showing tau Discussion: 1. The experimental time constants is very similar to the theoretical values. a. 123 compared to the intended 122 for the RL circuit. b. 119 compared to the intended 122 for the LR circuit. 2. Yes, when calculating the theoretical time constant of the RL circuit, the resistance of the inductor should be included because of significant resistance to the circuit in relation to the 1k ohm resistor. 3. The response of the RL circuit is similar other than the offset presented from the internal resistance of the inductor to the RC circuit. a. RC response looks very similar to that of the LR response. b. CR response looks very similar to that of the RL response.

Lab 3: Twin-T Notch Filter Introduction: For this lab, we will construct a 60Hz notch filter as shown in figure 15 below. In this circuit, k=1 and b=½ with respect to the original twin-t notch filter circuit setup. This circuit will be constructed using a capacitor value of 2C and a resistor value of R/2 which we constructed by putting two capacitors and two resistors in parallel. We will measure the frequency response and the null frequency. Measure the output voltages using the 2+ and 2- pins from the AD-2. Figure 15: Twin-T Filter circuit diagram Figure 16: Constructed Twin-T Filter Circuit Measured Data: Component Measured Value Theoretical Value Resistor 1 (Ω) 27250 26500 Resistor 2 (Ω) 27050 26500 Resistor 3 (Ω) 27200 26500 Resistor 4 (Ω) 27120 26500 Capacitor 1 (nF) 96 100 Capacitor 2 (nF) 101 100 Capacitor 3 (nF) 98.8 100 Capacitor 4 (nF) 97 100 Notch frequency (Hz) 59.65 61.57 Vo/Vi @fn (dB) -53.64 -40 Vo/Vi @fn' (dB) -35.23 -30 Vo_min/V0_max (%) 0.112 NA Table 3: Measured and theoretical values for circuit elements/analyzed variables

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Figure 17: Twin-T Filter data showing frequency band Figure 18: Twin-T Filter data showing tau Discussion: skipped as per instruction on board Signature of Professor: Prelab: Lab 1: - Find the time constant (t) for R-C and C-R circuits with R = 33 kΩ and C = 4.7 nF Lab 2: - Find the time constant (t) for R-L and L-R circuits with R = 1 kΩW and L = 150 mH. Lab 3: - Find the notch frequency of a Twin-T filter, where R = 26.5 kΩ and C = 0.1 μF