Experiment 4 Report

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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. 4: Circuits in Sinusoidal Steady State Om Patel, 605518179 Experiment Introduction and Theory The set of labs in this experiment aims to verify the principle of Ohm’s law for resistive and non-resistive circuits along with finding amplitude and phase of current and voltage in RC and RL circuits. When dealing with sinusoidal voltage and alternating current sources, the impedance is not equal to the resistance as it was with DC circuits. Electrical impedance allows us to read relative amplitudes and phases of the voltage and current. The equation below describes Ohm’s law of adhering to AC circuits. 𝑉 = 𝑍𝐼 Equation 1: Ohm’s law for AC circuits where Z is the electrical impedance, V is the voltage supplied, and I is the driven current Looking at resistors, the impedance is simply just its measured resistance. Looking at inductors, the impedance increases as frequency increases which is shown by . Looking at capacitors, the 𝑍 = 𝑗𝐿ω impedance decreases as frequency increases which is shown by . With an applied sinusoidal 𝑍 = 1 𝑗𝐶ω voltage input, the current with an inductor and capacitor is also sinusoidal but just 90 degrees out of phase. In an inductor, the current is lagging(-90) while in the capacitor, the current is leading(+90). Bode plots are a tool to see the variation in the transfer function V out /V s with changing frequency. It shows the magnitude and phase of the frequency response. By analyzing a bode plot, one can pull out the measured 3dB frequency or break frequency which shows the variation of the dB Gain. At higher frequencies, the slope or roll-off rate can be interpreted. Lab 1: Frequency domain analysis of 1st-order circuits Introduction: In this part of the experiment, we built four circuits corresponding to the shown diagrams below while applying a sinusoidal input of 100 mV (70.7 mVrms) and frequency ranging from 100hz to 100kHz. From this, we used the AD2 software to create bode plots. With the C-R circuit, an additional plot at 1 kHz was created. Figure 1: Wire diagrams for Lab 1 (clockwise starting at top left: R-C, C-R, R-L, L-R)

Figure 2: Constructed Circuit for Lab 1 (R-C) Figure 3: Constructed Circuit for Lab 1 (C-R) Figure 4: Constructed Circuit for Lab 1 (R-L) Figure 5: Constructed Circuit for Lab 1 (L-R) Measured Data: Component Measured Value Theoretical Value Resistor 1 (Ω) 5120/5050(part 3) 5100 Resistor 2 (Ω) 98.8 100 Capacitor (nF) 10.04/9.88(part 3) 10 Inductor (mH) 3.1 3.3 R Inductor (Ω) 9.2 10 Table 1: Measured and theoretical values for circuit elements

Type Component Measured Value Theoretical Value R-C (Hz) 𝑓 𝑜 3201 3096 (degrees) θ 𝑐 -44.13 -45.95 C-R (Hz) 𝑓 𝑜 3383 3096 (degrees) θ 𝑐 44.03 44.05 V 0 /V i (dB) @1kHz -11.006 -10.300 (degrees) @1kHz θ 𝑐 73.028 72.23 Z t @1kHz 16881.8 16108.8 V 0 (mV) @1kHz 28.16 28.80 I (µA) @1kHz 5.577 5.924 R-L (Hz) 𝑓 𝑜 5601 5544.75 (degrees) θ 𝑐 39.85 42.45 (adjusted 39) L-R (Hz) 𝑓 𝑜 5435 5544.75 (degrees) θ 𝑐 -43.99 -47.55 Table 2: Measured and theoretical values for magnitude and phase

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Figure 6: Bode Plot for Lab 1 (R-C) Figure 7: Bode Plot for Lab 1 (C-R) Figure 8: Bode Plot for Lab 1 (R-L) Figure 9: Bode Plot for Lab 1 (L-R) Figure 10: Bode Plot for Lab 1 (C-R @1kHz)

Discussion: 1. The measured magnitude and phase values/plots were very similar to the theoretical values. All values meet theoretical values with appropriate margin. This is discussed below for each circuit type. a. R-C: i. Magnitude starts at 0 then drops off significantly ii. Phase goes from 0 to -90 as frequency increases iii. 3dB frequency → measured 3201Hz, compared to 3096Hz iv. 3dB phase → measured -44.13, compared to -45 b. C-R: i. Magnitude increases significantly steadying out to 0 ii. Phase goes from 90 to 0 as frequency increases iii. 3dB frequency → measured 3383Hz, compared to 3096Hz iv. 3dB phase → measured 44.03, compared to 45 v. @1kHz, measured -11.006dB, compared to -10.3dB vi. @1kHz, measured 73.03 degrees, compared to 72.23 degrees vii. @1kHz, measured 5.58µA, compared to 5.924µA c. R-L: i. Magnitude starts at 0 then drops off significantly ii. Phase goes from 0 to -90 as frequency increases iii. 3dB frequency → measured 5601Hz, compared to 5545Hz iv. 3dB phase → measured 39.85, compared to 39 d. L-R: i. Magnitude increases significantly steadying out to 0 ii. Phase goes from 0, rises, and then goes back down to 0 as frequency increases (due to inductors internal resistance) iii. 3dB frequency → measured 5435Hz, compared to 5545Hz iv. 3dB phase → measured -43.99, compared to -47.55

Signature of Professor: Prelab: Lab 1: - Calculate the 3dB frequency (cut-off frequency) of all the circuits. - Plot the magnitude and phase bode-diagrams of V 0 /V s (for all the circuits)

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