please fill out table $6\mathrm{.}1$ through table $6\mathrm{.}6,$ then answer questions $1-6,$ thank you so much

Objective

This exercise examines the voltage and current relationships in series R$,$ L$,$ C networks. Of particular importance is the phase of the various components and how Kirchhoff's voltage law is extended for AC circuits. Both time domain and phasor plots of the voltages are generated.

Theory Overview

Each element has a unique phase response: for resistors, the voltage is always in phase with the current, for capacitors the voltage always lags the current by $90$ degrees, and for inductors the voltage always leads the current by $90$ degrees. Consequently, a series combination of R$,$ L $,$ and C components will yield a complex impedance with a phase angle between $+90$ and $-90$ degrees. Due to the phase response, Kirchhoff's voltage law must be computed using vector $($phasor$)$ sums rather than simply relying on the magnitudes. Indeed, all computations of this nature, such as a voltage divider, must be computed using vectors.

Equipment

$(1)$ AC function generator

model:

srn:

$(1)$ Oscilloscope

model:

srn:

Components

$(1)10$ nF

actual

$(1)10$ mH

actual

$(1)\backslash (1\backslash$mathrm$\{$k$\}\backslash$Omega $\backslash )$

actual:

Schematics

Figure $6\mathrm{.}2$

Procedure

RC Circuit

Using Figure $6\mathrm{.}1$ with Vin $=2$Vp$-$p sine at $10$kHz$,$R$=1$kOmega, and C$=10$nF$,$ determine the theoretical

capacitive reactance and circuit impedance, and record the results in Table $6\mathrm{.}1($the experimental

portion of this table will be filled out in step $5).$ Using the voltage divider rule, compute the resistor

and capacitor voltages and record them in Table $6\mathrm{.}2.$

Build the circuit of Figure $6\mathrm{.}1$ using R$=1$kOmega, and C$=10$nF$.$ Place one probe across the generator and

another across the capacitor. Set the generator to a $10$ kHz sine wave and $2$ V p$-$p$.$ Make sure that the

Bandwidth Limit of the oscilloscope is engaged for both channels. This will reduce the signal noise

and make for more accurate readings. Also, consider using Averaging for the acquisition mode,

particularly to clean up signals derived using the Math function.

Measure the peak$-$to$-$peak voltage across the capacitor and record in Table $6\mathrm{.}2.$ Along with the

magnitude, be sure to record the time deviation between V$\_($C$)$ and the input signal $($from which the

phase may be determined$).$ Using the Math function, measure and record the voltage and time delay

for the resistor $($ V$\_("$in $")-$V$\_($C$)).$ Compute the phase angle and record these values in Table $6\mathrm{.}2.$

Take a snapshot of the oscilloscope displaying V$\_($m$),$V$\_($C$),$ and V$\_($R$).$

Compute the deviations between the theoretical and experimental values of Table $6\mathrm{.}2$ and record the

results in the final columns of Table $6\mathrm{.}2.$ Based on the experimental values, determine the

experimental Z and X$\_($C$)$ values via Ohm's law $(\{$:i$=$V$\_($R$)//$R$,$X$\_($C$)=$V$\_($C$)//$i$,$Z$=$V$\_($in$)//$i$)$ and record back in Table $6\mathrm{.}1$

along with the deviations.

Create a phasor plot showing V$\_($in$),$V$\_($C$),$ and V$\_($R$).$ Include both the time domain display from step $4$ and

the phasor plot with the technical report.

RL Circuit

Replace the capacitor with the $10$ mH inductor $($i$.$e$.$ Figure $6\mathrm{.}2),$ and repeat steps $1$ through $6$ in like

manner, using Tables $6\mathrm{.}3$ and $6\mathrm{.}4.$

RLC Circuit

Using Figure $6\mathrm{.}3$ with both the $10$ nF capacitor and $10$ mH inductor, repeat steps $1$ through $6$ in

similar manner, using Tables $6\mathrm{.}5$ and $6\mathrm{.}6.$ Using a four channel oscilloscope: To obtain proper

readings, place the first probe at the input, the second probe between the resistor and inductor, and the

third probe between the inductor and capacitor. Probe three yields V$\_($C$).$ Using the Math function, probe

two minus probe three yields V$\_($L$),$ and finally, probe one minus probe two yields V$\_($R$).$ Assigning

Reference waveforms can be useful to see all of the signals together. Using a two channel

oscilloscope: Unfortunately, it will be impossible to see the voltage of all three components

simultaneously with the source voltage using a two channel oscilloscope. To obtain proper readings,

place the first probe at the input and the second probe across the capacitor in order to see the phase

and magnitude of V$\_($C$).$ Then, swap C and L $($placing the second probe across the inductor$)$ to see V$\_($L$),$

and finally, swap L and R $($with the second probe across R$)$ in order see V$\_($R$).$

Data Tables

RC Circuit

Table $6\mathrm{.}1$

Table $6\mathrm{.}2$

RL Circuit

Table $6\mathrm{.}3$

Table $6\mathrm{.}4$

RLC Circuit

Table $6\mathrm{.}5$

Table $6\mathrm{.}6$

Questions

What is the phase relationship between R$,$L$,$ and C components in a series AC circuit?

Based on measurements, does Kirchhoff's voltage law apply to the three tested circuits $($show work$)?$

In general, how would the phasor diagram of Figure $6\mathrm{.}1$ change if the frequency was raised?

In general, how would the phasor diagram of Figure $6\mathrm{.}2$ change if the frequency was lowered?