Elements of AC Circuits - The Series RLC circuit - Physics

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5 Οκτ 2013 (πριν από 4 χρόνια και 1 μήνα)

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Experiment 12: Elements of AC Circuits
-

The Series RLC circuit


Purpose:

To study the behavior of a resistor, capacitor and inductor in a series ac circuit.


Prior to Lab:

Sketch a phasor diagram for an ideal LRC circuit consisting of an
33


F
capacitor,
a 70 mH inductor, and a
47

resistor connected to a
5

v,
150

Hz source. Be sure
you calculate X
L

and X
C
. Show these calculations.


Theory:


For a series RLC circuit, Kirchhoff’s rules demand that the current be a constant throughout.
Furthermore, the pote
ntial differences across each element must combine to equal the source
voltage. However, each element also has a unique phase relation between the potential
difference and the current, so the potential differences must add according to the rules for
combin
ing phasors for the circuit. If we look at each element in turn we can investigate the
individual behavior and then the collective behavior of the elements in the circuit.


For example, the capacitor’s potential difference is determined by the charge on i
t, while the
current is the rate of change of charge. This fact leads to a phase difference between current
and voltage of 90
o

with the potential difference lagging the current by that phase angle. The
effective potential difference across the capacitor V
C

equals the effective current I times the
capacitive reactance X
C
.



Figure 1. The apparatus wired and ready to read data. Note that the meter on the
right

is
connected to the
DMM Output of the Function Generator

and is indicating the frequency output
of
the source. The meter on the
left

is set to read ac volts, and is reading the source voltage at
this time.



Procedure:

In order to completely analyze the behavior of the circuit, we will measure the resistance R and
five different potential differences

a
t two different frequencies. We will also determine the
resonant frequency of the circuit
.


1.
BEFORE

connecting wires to the circuit, use a DMM set on the 2
00



range to measure the
resistance between connection C and D in Fig.
2

or
3
. The instrumental uncertainty for
measuring resistance is 0.5% of the reading plus 1 digit.


2.
Connect one DMM to the D
V
M Output of the function generator
. Set this DMM to measur
e
20VDC.
Note that the frequency measured

is the product of the reading on this meter and the
multiplier setting on the function generator
.

Connect the other DMM between A and D initially
and set it to measure 20 VAC
.


3.
Connect the ac source to the circ
uit board
using

the 10


resistor
and the 100

F capacitor.

Set the
function generator

for
5.0 V

at a frequency of
120 Hz
. Note: Be certain the output is
set for the sine wa
ve function
.






Figure
2
. Above is a
diagram

of the apparatus used in
the experiment.


Figure
3
. Above is the schematic wiring
diagram representing the apparatus and its
source. Note that the inductor is not ideal, and
we have represented any (
power) loss in the
circuit by a

resistance, r, in series with the
inductor.


4.
Compl
ete the measurements listed below for the frequency set in step 3

by connecting the
probe wires of the DMM set to measure ACV to the connectors listed
. Record this data in the
first line of a table in your data notebook which looks like the sample below.
W
hen finished
,
change the frequency output of the function generator to 250 Hz and repeat the measurements
to fill in the second line of the table in your data notebook.


a. V
source

is measured by connecting the probes at A and D.

b. V
C

is measured by conne
cting the probes at A and B.

c. V
Lr

is measured by connecting the probes at B and C.

d. V
LR

is measured by connecting the probes at B and D.

e. V
R

is measured by connecting the probes at C and D.


f

R

V
source

V
C

V
Lr

V
LR

V
R
















5.

To measure t
he resonant frequency of the
circuit

c
onnect

the
DMM
used to

measure ACV
across points A and C to measure the combined voltage across the inductor and capacitor.
Adjust the frequency output of the function generator until this measured
voltage is a
minimum
. For
this frequency setting

the combined reactance of the inductor and capacitor
is
theoretically zero
. This
minimum
voltage occurs at the resonan
t

frequency of the circuit.
Record
the measured resonant

frequency
, the voltage across R,

and the voltage acr
oss the
capacitor and inductor
.


Analysis:

1. Determine the current in the circuit by
calculating

the current in the resistor.
Note:

We will
use I (and V
R

since it has the same phase) for reference throughout our analysis.




2. Dete
rmine the capacitance of the circuit from its reactance, the current calculated above,
and the frequency of the source.



and


3. Determine the inductance of the circuit from three different voltage measure
ments, the
current calculated in step 1
,

and the frequency of the source. Follow the instructions
carefully; this can be complicated.


a. Determine the phase of V
LR

with respect to V
R

using the law of cosines.













Figure 5. A phasor diagram showing the relationship
between the measured voltages across the inductor,
resistor and combination.

b. Determine the fraction of V
LR

due V
L

alone.




c. Determine t
he inductance of the circuit from its reactance using the current determined
in step 1 and the V
L

determined above.



and


d. Determine the loss in the inductor and write it as a resistance using the curre
nt found in
step 1.



and


4. Determine the phase between the source voltage and the current




5. Calculate the resonant frequency of the circuit from the values you determined for L andC.




Report:

1.
For the first frequency

that

you collected data, and u
sing R, r, X
L

and X
C
, draw to scale a
phasor diagram to determine Z and


.
Make a second drawing for the second frequency for
which you collected data.
Assume the phase angle for the capacitor is 90
o
.
Answer these
questions for each drawing:
Does V
source

= IZ? Is the phase angle in the drawing close to
phase angle you calculat
ed in

step 4 of the analysis.

2. Are the values for C and L close to the manufacturer’s values? Explain.

3. What is the per cent difference between the measured resonant frequency and that
calculated in step 5 of the analysis?

4
.
Using the voltage across t
he resistor recorded in step 5of the procedure, calculate the
current in the circuit at resonance.

If you now divide the current you just determined into

the voltage across the inductor and capacitor
you
also
measured in step 5 of the procedure,
do you get the same value for the resistance of the inductor that you got in step 3d of the
analysis?
Calculate a
per cent difference

if it exists
? Try to explain any difference.

5
. Discuss your results

paying particula
r attention to instrumental error and other sources of
experimental error
.