Experiment 5 Report

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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. 5: Resonant Circuits, LC Filters and Mutual Inductance Om Patel, 605518179 Experiment Introduction and Theory The labs in this experiment aim to understand resonance circuits and their frequency response, lowpass and bandpass LC Filters, and mutual inductance. A series resonance circuit consists of a resistor, capacitor, and an inductor in series as shown below in Figure 1. To represent the function this circuit’s frequency response can be analyzed through, you can use a simple voltage divider as shown in Equation 1. Figure 1: Series RLC resonance circuit 𝐻(?ω) = 𝑉 ??? 𝑉 ? = ? ?+?ω?+ 1 ?ω𝐶 = ??𝐶ω 1+??𝐶ω−?𝐶ω 2 = 1 1+??( ω ω 0 − ω 0 ω ) where , ω 0 = 1 ?𝐶 ? = 1 ? ? 𝐶 Equation 1: transfer function of circuit By analyzing this transfer function, you can see when omega is extremely small or large, the function returns a very small value since the denominator becomes huge. This shows why this circuit is a bandpass filter that is surrounded around omega_0. The frequency response peaks when omega is equal to omega_0 and the phase is zero. Mutual Inductance is the phenomenon when the flux in one coil is caused by current through another coil. It is a method of energy transmission without contact. The voltage induced across the second coil in terms of the first coil is shown in Equation 2. As you can see when the distance between the coils, L, increases the Mutual Inductance, M, is decreased. ? 2 (?) ? 1 (?) = ? 21 ? 1 Equation 2: Mutual Inductance

Lab 1: Bandpass filter based on series resonance circuit Introduction: In this part of the experiment, we will explore the bandpass filter based on the series resonance circuit. We will use an inductor that is 150mH, a capacitor that is 10nF, and a resistor that is 2200 ohms. After constructing an RLC circuit, we will analyze the frequency response plots that the network analyzer in the AD2 outputs. Figure 2: Serial LCR circuit (output across resistor) Figure 3: Serial CRL circuit (output across inductor) Figure 4: Serial LRC circuit (output across the capacitor)

Measured Data: Component Measured Value Theoretical Value Resistor (Ω) 2151 2200 Inductor 1 (mH) 148.3 150 Inductor 2 (mH) 149.4 150 R L1 (Ω) 163.5 250 R L2 (Ω) 248.2 250 Capacitor (nF) 10.29 10 Table 1: Measured and theoretical values for circuit elements Circuit Measurement Measured Value Theoretical Value LCR f r (Hz) 4107.8 4074.2 (degrees) θ ? 0.2535 0 Bandwidth 2701.2 2300 Q (quality factor) 1.521 1.588 CRL f r,peak (Hz) 4693.5 4700 (degrees) θ ?,?𝑒𝑎? 63.94 65 f r,+90 (Hz) 3999.8 4100 (degrees) θ ?,+90 90.01 90 LRC f r,peak (Hz) 3704.7 3700 (degrees) θ ?,?𝑒𝑎? -72.21 -65 f r,-90 (Hz) 4121.5 4100 (degrees) θ ?,−90 -90.20 -90 Table 2: Measured and Theoretical Data from Lab 1

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Figure 5: LCR bode plots (peak) Figure 6: LCR bode plots (bandwidth) Figure 7: CRL bode plots (peak) Figure 8: CRL bode plots (+90) Figure 9: LRC bode plots (peak) Figure 10: LRC bode plots (-90) Discussion: 1. The circuit is a bandpass filter because the capacitor acts as an open circuit at low frequencies while the inductor acts as an open circuit at high frequencies. This causes the magnitude bode plot to start at 0 and approach 0. In the middle range of frequencies, the magnitude peaks at the frequency that the elements are chosen to “activate”. 2. The measured circuit resonant frequency was 4107.8 Hz, compared to the theoretical 4074.2. 3. The theoretical quality factor for the LCR circuit is 1.771. 4. The filter 3dB bandwidth is 2701.2 Hz. 5. The measured quality factor for the LCR circuit is 1.521.

6. The measured and theoretical quality factors are very similar, the measured is lower than the theoretical value. 7. The theoretical quality factor, adding the inductor resistance to the total resistance was 1.588. This theoretical value is much closer to the measured value. 8. Bode plots and crucial points of interest are recorded above. Lab 2: Mutual Inductance Introduction: In this part of the experiment, we will explore the mutual inductance of a system. We will vary the location (1-6cm) of one inductor relative to another and record its mutual inductance. After constructing the circuit, we will analyze the output voltage from the AD2 AC voltage input. Figure 11: Constructed Circuit for Mutual Inductance lab Figure 12: Wire diagram for Mutual Inductance lab Measured Data: Distance (cm) V out (V) V s (V) M (mH) k Log (M) Log(1/D) 1 3.458 0.488 21.006015 0.14112 -1.678 0.0000 2 3.458 0.084 3.615789 0.02429 -2.442 -0.3010 3 3.458 0.030 1.291353 0.00868 -2.889 -0.4771 4 3.458 0.012 0.516541 0.00347 -3.287 -0.6021 5 3.458 0.006 0.258271 0.00174 -3.588 -0.6990 Table 3: Data from the Mutual Inductance lab

Figure 13: Plot to visualize the relationship between distance and mutual inductance, n = 2.71 Discussion: 1. As the proximity of the second coil increases from the primary coil, the voltage across the primary decreases. 2. The equation is as follows: . ? 2 (?) = ? 21 ? 1 ? 1 (?) 3. Mutual Inductance is always smaller than the self-inductance of the coils. All values are recorded in the tables above. 4. The mutual inductance seems to scale linearly with the reciprocal of the distance between the two coils as shown in Figure 13. The slope that relates both these variables is 2.71. 5. General Motors things using mutual inductance is safer than charging with a power cord because you eliminate the energy transfer through current which can be safer since there is a lower possibility of short circuits or failure due to the voltage differential.

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