ECE 2101L Lab 12 (2)

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California Polytechnic State University, Pomona *

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2101L

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Electrical Engineering

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Feb 20, 2024

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ECE 2101L Section 8 Electrical Circuit Analysis II Lab LAB 12 Transfer function of AC Circuits 11/14/2023 ABSTRACT

In this lab, we examine the transfer function of AC circuits and how it plays a part in producing an output from a given input under different frequencies. We do this by constructing an op amp circuit and measuring the output voltage V out indicated in our schematic, given a certain frequency in our input. We also adjust circuit frequency to find various properties of the circuit, like maximum gain, minimum gain, maximum phase shift, minimum phase shift, and the frequencies where the optima in the aforementioned properties occur. Table of Contents 2

Page Abstract…………………………………………………………………………2 List of Figures…………………………………………………………………..5 List of Tables……………………………………………………………………4 I. Objectives………………………………………………………………...6 II. Pre-Lab…………………………………………………………………...6 III. Lab………………………………………………………………………..9 IV. Conclusion……………………………………………………………….13 LIST OF TABLES 3

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Table 1……………………………………………...9 Table 2……………………………………………...10 LIST OF FIGURES Figure 1………………………………………….…..7 4

Figure 2………………………………………….…..9 Figure 3……………………………………………..10 Figure 4………………………………………….….10 Figure 5……………………………………………..11 Objective: The objective of this lab is to examine the role of transfer function in determining a desired output for a given input in AC circuits under different frequencies 5

Pre-Lab Theory of transfer function 1) What is a transfer function?A transfer function represents the behavior of the output in relation with every possible input value. It is defined as the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable assuming all conditions equal zero. 2) Which of the following is correct? a) Transfer function is a real number. b) Transfer function is a complex number. c) Transfer function is either a real or complex number. d) All of the above. e) None of the above. 3) In an AC circuit, the transfer function H(w) and the input voltage is Vin (w). Calculate the output voltage Vout(w). ?(?) = ? ??? (?) ? ?? (?) ? ??? (?) = ?(?) · ? ?? (?) Task A2 Transfer function of AC Circuit with Op AmP 6

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1) Assuming the op amp is ideal calculate Vout(w) knowing Vin(w) ? ? =− 𝑅 𝐹 𝑅 1 × ? ? ?(?) = 1 ?? ||𝑅 2 = 𝑅 2 1+𝑅 2 ??𝐶 ? ? (?) =− ?(?) 𝑅 1 ? ?? (?) ? ? (?) =− 𝑅 2 1+𝑅 2 ??𝐶 𝑅 1 ? ?? (?) ? ? (?) =− 𝑅 2 𝑅 1 1 1+??𝐶𝑅 2 ? ?? (?) 2) Determine the circuit transfer function H(w)=Vout(w)/Vin(w) ?(?) = ? ? (?) ? ? (?) =− 𝑅 2 𝑅 1 1 1+??𝐶𝑅 2 3) Calculate the magnitude of H(w) |?(?)| = 𝑅 2 𝑅 1 1 1+(?𝐶𝑅 2 ) 2 4) Calculate the phase of H(w) 7

∠ =0.0659 ∠ -2.77 degrees (?(?)) =− ??? −1 (?𝐶𝑅 2 ) Schematic + PCB 8

Lab Table 1 Frequency (Hz) Calc H( ⍵ ) Calc V out Meas V out Error V out % Magnitude (gain) Phase deg. Amplitude , V Phase deg. Amplitude, V Phase, deg. Amplitude, V Phase 20 5.343 -89.99 5.60 -46.75 6.955V -48.98 24.19% 4.77% 80 5.299 -90 2.87 -76.77 4.315V -69.86 50.34% 9.00% 160 4.607 -83.29 1.95 -83.29 3.17V -75.60 56.42% 9.23% 320 3.269 -86.63 0.58 -86.63 2.02V -66.8 22.89% 22.89% 640 2.002 -88.31 0.24 -88.31 1.125V -77.7 12.01% 12.01% 9

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Plots from Table 1 Calculated magnitude of H( ⍵ ) vs. frequency Figure 3 Calculated phase of H( ⍵ ) vs. frequency Figure 4 10

Measured amplitude of V out vs. frequency Figure 5 Analysis: From the graphs, we can observe that they all behave inversely. Magnitude decreases as frequency increases. For the second graph, the calculated phase degree stayed nearly consistent for all different frequencies. Lastly, in the last graph we can observe how the amplitude decreases as the frequency increases. For frequency of 80 Hz, express the input voltage and output voltages as cosines in time domain. What do you observe from the expressions vin(t) and vout(t) with regard to the role of H(w)? ???(?) = 1. 5𝑐??(2π??) ????(?) = |?(?)| * 1. 5𝑐??(2π?? + ∠?(?)) At f=80Hz, and ???(?) = 1. 5𝑐??(160π?) ???? = |?(?)| * 1. 5𝑐??(2π?? + ∠?(?)) ????(?) = 1. 87𝑐??(160π? − 76. 77°) 11

We can observe that both equations are similar to one another however, H(w) does not play a role in finding Vin but is needed to calculate Vout. We are taking the equation of Vin and adding H(w) to find our final expression. Table 2 Frequency, Hz Gain Phase Shift Max Min Max Min 3 Hz 5.177 1.9 kHz 0.820 2 Hz 175.79 700 71.12 Method Proposal for Table 2 verification To verify results for Table 2, it would be useful to use the formula , to calculate and verify. Maximum phase shift occurs ??????? ???? |?(??)| ??? = 𝑅2/𝑅1 when , so ∠ Minimum phase shift ? = ∞ ?(??)|??? =− π − π/2 =− 3π/2 ?? ? = ∞. occurs at w=0 so ∠?(??)|??? =− π − 0 =− π ?? ? = 0. 12

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Conclusion For this lab, we studied the behavior of transfer functions by examining the output of the op amp at different frequencies. Based on our results, we can observe that as the frequency increases, the output for amplitude decreases. It is evident from our results that frequency is inversely proportional to the time interval of phase. We can also observe that the phase angle measurements are in close proximity from one another. 13