RC & RL, Phase Measurements Fromal Lab Report PDF

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This the formal lab report for AC circuits, Part 1 which is RC & RL, Phase Measurements lab...

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Prof. Arya Akmal 12.10.2019

AC Circuits, Part 1 RC & RL, Phase Measurements Abstract: In alternating current (AC) circuits, the voltage oscillates in a sine wave pattern. In this experiment, we applied concepts of phasor diagrams to experimentally identify relationships between a change in frequency and its resulting phase in an AC circuit containing a resistor and capacitor in series. We used function generator, oscilloscope, 120 Ω(measured 120.8 Ω) resistor, and 0.1 μF (measured 0.099 μF) capacitor to construct an RC circuit in series. Then we played with frequency setting for the generator such that the amplitude of the V R signal would be approximately 1/2 of the generator amplitude which leads to the phase angle 56.14° which is almost close to 60° by 4.6%. Then we used 9 different frequencies (23.1852-127.062 kHz) and measured Δt(s) and period(T(s)) for each of them. Then we plotted the tanɸ vs. ω(Hz) graph. The slope of the graph corresponds to the 1/RC. Our measurements verified that the slope is perfectly close to the 1/RC value by 0.34%. We also used DMM in AC mode for voltage measurements for VR, VL, VC, and ε (RMS) to find

ɸ . Then we compared the phase(ɸ) results from arccos(ɸ) to

the direct measurements of the dual-trace oscilloscope. The phase(ɸ ) results were perfectly close by 1.6% on average to each other. We also verified the Pythagorean relation and V E(RMS) values were close by 4.9% on average to the

√V

2 R

+V C 2 (v) values. Then we constructed an LR circuit

in series (L22mH and R=1500Ω) . We used eight different frequencies (6.606-1260.933 kHz). Then we used DMM in AC mode to find V R and VL (RMS) in order to find tanɸ. The slope of the tanɸ vs. ω(Hz) graph which corresponds to L/R, verified our measurement by 8.14% difference. The relative phase between R&L is, as predicted, ~90º, and will remain constant independent of frequency. However, we cannot measure VR and VL with an oscilloscope using only one R and

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one L because they do not have mutual ground. For the last part of the experiment, we constructed the LR circuit containing two 22mH inductors and two resistors R=1500(1512) Ω. Due to slight differences between the values of the components, we would not get exactly 90° for the phase(ɸ ) and calculated ɸ =84.25°. Although we did obtain some experimental error, our results for phase values make sense and coincides with the theoretical values as we expected as most of the error is due to measurement precision.

Introduction: Direct current (DC) circuits involve current flowing in one direction. In alternating current (AC) circuits, instead of a constant voltage supplied by a battery, the voltage oscillates in a sine wave pattern. In this lab, all elements are in series and they have the same current i=Icos(⍵t). The voltage across each

component

can

be

expressed

in

terms

of

the

current

as

vR=Ri=RIcos(⍵t)=VRcos(⍵t) vL=Ldi/dt=-⍵LIsin(⍵t)=-X Lsin(⍵t)=-VLsinsin(⍵t) vc=

❑ q 1 1 Isin (⍵t)= XcIsin(⍵t )=¿ Vcsin(⍵t) = ∫ ❑idt= ⍵C C C ❑

In these formulas, upper-case symbols are for the amplitudes, and lower-case symbols are used for time-varying quantities. These three voltages have a relative phase. This diagram shows the voltage phasor for and RL and RC circuit. tan ɸ

=VL/VR

L

CosɸC=VR/ε

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The phase relationships can be expressed in terms of the impedances rather than the voltages. From Pythagorean theorem, we know that

.

If the amplitude of each phasor is divided by I, then we have The impedance of the entire circuit is Z=ε/I and XL is the inductive reactance and XC is the capacitive reactance. We know that XL=ωL and XC=1/ωC. Consequently, we have tanɸ=XL/R=ωL/R or tanɸ=XC/R=1/ωCR.

Equipment & Procedure: The equipment that we used for this experiment were: DMM, Signal Generator, Oscilloscope, 120 Ω(measured 120.8 Ω) resistor, two 1500 Ω (measured 1508 Ω) resistor, 0.1 μF (measured 0.099 μF) capacitor, and 22mH inductor. First, we set up an RC series circuit across the sine generator in such a way that the generator voltage, and the resistor voltage can be simultaneously displayed on the oscilloscope(Figure below). Then we calibrated the oscilloscope across channel 1 and channel 2 to ensure that data on the screen of the oscilloscope are accurate. We connected channel 1 of the oscilloscope across the resistor and channel 2 across the function generator. We had to find a frequency setting for the generator such that the amplitude of the V R signal is approximately 1/2 of the generator amplitude. Then we used 9 different frequencies from 23.1852 kHz to 127.062 kHz and measured Δt(s) and period(T(s)) for each of them(Figure2). We also used DMM in AC mode for voltage measurements for V R, VL, VC, and ε (RMS).

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Then we constructed an LR circuit in series using 22mH inductor and R equals to 1500 Ω. We used eight different frequencies from 6.606 kHz to 1260.933 kHz. Then we used DMM in AC mode to find VR and VL (RMS) in order to find tanɸ=V L/VR for all those eight different frequencies. Then we plotted tanɸ vs. ω(Hz) graph to analyze the slope in terms of L and R, and compared it with the actual components value. For the last part of the exam, we constructed and LR circuit using two 22mH inductors and two resistors R=1500(1512) Ω. Then we used the dual trace oscilloscope to measure V L and V R simultaneously. Then we had to find a frequency such that the two signals have approximately the same amplitude. Then we found a frequency such that the two signals have approximately the same amplitude, and we measured the relative phase between the two signals on the screen.

Data & Analysis: We were able to find the frequency setting for the generator that V gen amplitude=2.2*5=11V and VR

=1.2*5=6V which shows that VR

amplitude

amplitude

is approximately ½ of the Vgen

amplitude.

According to the trigonometric relations above, this should correspond to a phase of approximately 60° or � /3 radians. We measure T(period)=6.4E-6s and the

know that

ϕ 2π = . Then we calculated ∆t T

∆ t =1E-6s. We

ϕ=0.98 radians . The π/3 value is 1.047radians.

We could verify this by a direct measurement of the relative phase of the two signals by 4.6% difference.

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The data table below shows the measured values for the RC circuit.

Based on the measured data of the above table, we plotted the graph of tanɸ(rad) vs. 1/ω(1/Hz). From the formula, we know that tanɸ =1/ωRC. This means that the slope of the graph should be close to the value of 1/RC. Our measurements verified that the slope of the graph which is 83333 is perfectly close by 0.34% to the value of the 1/RC with R=120(120.8) Ω resistor and C=0.1(0.099) μF which is 83617. Then we used RMS voltage data and calculated cos( ɸ) and using arccos(ɸ ), we calculated phase(ɸ ) value. We also know that the AC voltmeters and DMM’s usually measure AC voltages in RMS units (VRMS=Vamplitude/ ɸ2 (rad)=2�Δt/ T

ɸ(rad)=acos(VR/ε)

0.60989633

0.672845817

0.743930931

0.786529566

0.881102490

0.862846594

0.97991857

0.958771726

1.072617497

1.090923165

1.192683148

1.156689171

1.153278110

1.169314763

1.292826949

1.248742206

1.386476981

1.342386043

√ 2 ).

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Then we compared the phase(ɸ ) results from arccos(ɸ ) to the direct measurements of the dualtrace oscilloscope. The phase(ɸ ) results were perfectly close by 1.6% on average to each other. In the next analysis, we used our RMS voltage data to show that the Pythagorean relation which is ε=

√V

2 R

+V C2 holds for each set of voltages.

Our measurements verified the Pythagorean relation and VE(RMS) values were close by 4.9% on average to the

√V

2 R

+V C2 (v) values.

The data table below shows the measured values for the RL circuit using L=22mH inductor and R=1500 Ω (1508Ω) resistor. Then we plotted the graph of tanɸ(rad)=V LRMS/VRRMS vs. ω(Hz).

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The slope of the graph above is 1.33645E-05 which also represents L/R value because tanɸ =ωL/R. The value of the L/R for the actual components is 1.45503E-05. It shows that the slope of the graph is almost close to the L/R value by 8.14%. For the last part of the experiment, we wanted to measure the V L and VR. We know that the relative phase between the resistor and the inductor is, as predicted, ~90º, and will remain constant independent of frequency. If we only were using one resistor and one inductor, we would predict ~90º for the relative phase between R & L. Since we cannot measure V R and VL on an oscilloscope using only one resistor and one inductor because they do not have mutual ground, we added the second resistor and inductor with the same values. We constructed the LR circuit containing two 22mH inductors and two resistors R=1500(1512) Ω. However, the angle that we measured was not exactly 90º. We found a frequency f=10.638 kHz such that V R and VL

had the same amplitude. We measured

∆ t =22 µs and using the formula

ϕ 2π = ∆t T

, we

calculated the ɸ =84.25° which is not exactly 90°, and it was off by -5.7° to 90°. Because the values of the components (two resistors and two inductors) are not exactly the same, we would not get exactly 90° for the phase(ɸ).

Summary and Conclusion 1. R=V /I ∧C=Q /V → RC =(V /I )×(Q /V )=Q / I ≡ Coulomb/(Coulomb / second )=second 2. 1/ ⍵ C=V / ⍵ q , ⍵=1/time →1/⍵ C=V /i≡ O h ms

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V L =− L di/dt → ⍵ L=⍵ ×V /( di/dt), ⍵=1 /time →⍵ L=V /di≡ Oh ms 3. V =V 0 cos (⍵t+ ɸ)→ dV /dt=−V 0 ⍵sin (⍵ t +ɸ).cos (¿ 2+ɸ )=− sin (ɸ )→ dV /dt =−V 0 ⍵cos (⍵t +ɸ+¿ 2 ) 4. If the internal resistance is not small, then inductor can be replaced by parallel(R&L). As noted, ⍵ L

behaves as a resistor and ⍵ is angular frequency. We know that V L=-L di/dt. As

we calculated in part 3, there will always be a phase difference of �/2 between the current and voltage across the inductor no matter what the frequency is. In this experiment, we applied concepts of phasor diagrams to experimentally identify relationships between a change in frequency and its resulting phase in two AC circuits containing a resistor and capacitor in series (RC) and inductor and capacitor in series (RL) . Our measurements for the phase values verified the theoretical values with slight errors. Part

Points

Pre-lab

10

Participation/raw data

10

Cover page

1

Abstract

10

Introduction

2

Equipment & Procedure

2

Score...