# paraphrase

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__Introduction __

The purpose of this experiment was to gain experience finding the impedances, phasor voltages and currents within a circuit. In the procedure below, these values were measured and plotted, which helped to verify that the preliminary calculations were correct. The class was able to see how changing the frequency and capacitor values effects impedance, phasor voltage and current.

__Procedure__

**Preliminary Calculations:**

- For the circuit shown in Figure 9.5, the impedances, Z
_{1}, Z_{2}and Z_{s}were calculated. Z_{1}includes the inductor in series with the1 kΩ resistor, Z_{2}includes the capacitor in series with the 680Ω resistor and Z_{s}is the impedance of the entire circuit.

a.) f=1kHz and C=0.1μF:

Z_{1}= 1084.3 + j427.3

Z_{2}= 680 – j1591.5

Z_{s}= 1764.3 – j1164.2

b.) f=1kHz and C=1μF:

Z_{1}= 1084.3 + j427.3

Z_{2}= 680 – j159.2

Z_{s}= 1764.3 + j268.1

c.) f=500Hz and C=0.1μF:

Z_{1}= 1084.3 + j213.6

Z_{2}= 680 – j3183.1

Z_{s}= 1764.3 – j2969.5

**2.)** For each of the three cases in part 1, the phasors V_{1}, V_{2} and I were calculated.

a.) V_{1} = 0.55V_{S}cos(6283t + 59.4°)

V_{2} = 0.82V_{S}cos(6283t – 33.5°)

I = 4.7×10^{-4}V_{S}cos(6283t + 37.9°)

b.) V_{1} = 0.65V_{S}cos(6283t + 12.9°)

V_{2} = 0.39V_{S}cos(6283t – 21.8°)

I = 5.6×10^{-4}V_{S}cos(6283t – 8.6°)

c.) V_{1} = 0.32V_{S}cos(3142t + 70.4°)

V_{2} = 0.20V_{S}cos(3142t + 46.1°)

I = 2.9×10^{-4}V_{S}cos(3142t + 59.3°)

**Main Procedure:**

- The circuit shown in Figure 9.5 was constructed and the function generator was set to generate a 1V peak sinusoid with a frequency of 1kHz. A 0.1μF capacitor was used in this circuit. The oscilloscope was used to measure the magnitude of the phasors V
_{1}and V_{2}.

V_{1}=1.12±0.04 V with an angle of 43.2º.

V_{2}=1.40±0.04V with an angle of 37.4º.

V_{s}=1.88±0.04V with an angle of 0º.

- Next, the magnitude and phase angle of the current phasor I were found by observing the voltage across the 680Ω resistor.

V_{680Ω}=0.68V

I(t)=Icos(ωt+θ)

I(t)=(1mA)cos(6283t+17.4º)

- Phasor diagrams of V
_{1}, V_{2,}V_{s}and I were drawn. It is shown graphically that V_{s}=V_{1}+V_{2}.

V_{1}=1.12e^{j43.2º}=0.82+j0.76

V_{2}=1.40e^{-j37.4º}=1.11-j0.85

V_{s}=1.88e^{j0º}=1.88+j0

I=.001e^{j17.4º}=0.00095+j0.00030

- Using the measured values of V
_{1}, V_{2}, V_{s}and I, the impedances Z_{1}, Z_{2}and Z_{s}were calculated. The plot shows that Z_{1}+Z_{2}=Zs

Z_{1}=1120e^{j25.8º}=1008+j487

Z_{2}=1400e^{-j54.8º}=806-j1145

Z_{s}=1930e^{-j19.9º}=1815-j657

- Steps 1-4 were repeated for f=1kHz and C=1μF.

V_{1}=1.28±0.04 V with an angle of 15.8º.

V_{2}=0.76±0.04V with an angle of -20.2º.

V_{s}=1.88±0.04V with an angle of 0º.

I(t)=(1mA)cos(6283t+10.1º)

V_{1}=1.28e^{j15.8º}=1.23+j0.35

V_{2}=0.76e^{-j20.2º}=0.71-j0.26

V_{s}=1.88e^{j0º}=1.88+j0

I(t)=0.001e^{j10.1º}=0.00098+j0.00018

It can be seen in the graph below that Z_{1}+Z_{2}=Z_{s}.

Z_{1}=1280e^{j5.7º}=1273+j127.1

Z_{2}=760e-^{j30.2º}=656.8-j382.3

Z_{s}=1948e-^{j7.5º}=1931-j255

__Discussion __

When doing the calculations, it was necessary to be able to convert back and forth between polar and rectangular form. When adding or subtracting phasors, they first needed to be converted to rectangular form. When dividing or multiplying phasors, the polar form must be used. The impedances, phasor voltages, and phasor currents were graphed in rectangular form; however, it was helpful to use the polar form angle to confirm that the graph was plotted correctly. The graphs were also an effective way to visually verify that V_{1}+V_{2}=V_{s}.

While doing the pre-lab calculations, some of the students forgot that inductors have internal resistance that must be taken into account. If this additional resistance is not included in the equation, the calculated impedance values will not match the experimental impedance values. Therefore, when doing the pre-lab calculations, it was essential to add an 84.3W resistor after the inductor to account for this internal resistance.

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__Conclusion__

In this lab, the students gained experience calculating the impedances of inductors and capacitors in RLC series circuits. They also calculated phasor voltages and currents. All these values were verified experimentally and then plotted to visually show the relationships between impedances, phasor voltages, and phasor currents in different circuits. This helped to further the students’ knowledge and experience, which will help them in their future classes and careers.

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