Prelab 4 Law (1)

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Electrical Engineering

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Jan 9, 2024

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Pre-Lab 4 Assignment - EECS 230 1. Review magnetically coupled circuits from Lab Exercise 3. 2. Read through the entire lab exercise. 3. Summarize the experimental procedures in your pre-lab write-up (1 paragraph per experiment): a. Experiment 4.1 i. Using a vector network analyzer to measure the resonant frequency of shielded-loop resonators, and find the sweet spot of two special loops (5 and 9 cm). The setup includes attaching a coaxial cable to the network analyzer's reflection port and then we calibrate the device. Two shielded-loop resonators of different sizes are connected to the cable. The experiment aims to find the resonant frequency by observing the minimum magnitude of the input reflection coefficient on a Smith chart and then we use a log-magnitude plot. The procedure is repeated for both resonators so we can find both resonant frequencies. b. Experiment 4.2 i. This experiment aims to de-embed the effect of feedlines from shielded-loop resonators and determine their inductance and capacitance. We are doing this by observing changes in the reflection coefficient at different frequencies while removing the influence of the feedlines. Like in experiment 4.1, we use the machine to find the best frequency that the loop behaves at best. We do this for both 5 and 9 cm. We want to know about the loop's electronic characteristics. To do this, we need to measure the reflection coefficient at two other frequencies close to the best one. This lets us understand how the loops work without the cables affecting response/output. c. Experiment 4.3 i. Experiment 4.3 has a pretty simple purpose, as we are using the data from the previous experiments and calculating R, L and C. The lab doc gives us many formulas we can use to calculate each of the variables. There is no setup, procedure, or measuring for this experiment, just calculations. 4. Answer Questions 1 and 2 of Experiment 4.1. a. (Pre-Lab) From eq. (4.3) and eq. (4.4), which loop should have a higher capacitance, the 5 cm or the 9 cm one? (Hint: The transmission line characteristics are identical for the two loops.) Equation 4.3: Zin = -j*Z0/(β*l) = -j/(w*C'*l) so C = C'*l Equation 4.4: C' = sqrt(μ*ε)/Z0 = sqrt(εr)/(c*Z0) The transmission line characteristics should be the same for both loops so the capacitance relies only on the length of the loop. Obviously, 9 cm is longer than 5 cm so the 9 cm loop should have the higher capacitance.
b. (Pre-Lab) From eq. (4.7), which loop should have a higher inductance? 𝐿 = μ 𝑟 (ln 𝑙𝑛 (32 𝑟 /d) − 1. 75) Which means inductance relies on relative permeability and the loop dimensions (r and d, which are in the equation as r/d). The loops are made of the same material so the relative permeability will be the same for both loops. This leaves the r and d of each loop, so larger loop size will mean larger inductance.
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