Download Resistor Inductor Capacitor Series Circuits

General Physics Experiment 1
Resistor Inductor Capacitor Series Circuits
Objectives:
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To observe the relationships between the voltage and current across resistors, inductors, and
capacitors in series combinations as the frequency of the source is varied.
To observe resonance in an R-L-C circuit.
Equipment:
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Cables, 3, with din connectors
Pasco Signal Generator PI-9587
Pasco Circuit Board with 100 and 330 microfarad Capacitors, 8 millihenry Inductor having about
5 S resistance, and a 10 S Resistor
Multimeter
Capacitance meter with 200 microfarad range
Pasco Signal Interface II with Science Workshop and Graphical Analysis software
Physical Principles:
The voltage, V, across a resistor varies in phase with a current, I, through it and is related to it by
V R ' VRo sin(Tt)
(1)
In this equation VR is the voltage across the resistor, VR0 is the maximum voltage or amplitude, and T = 2Bf
with f the frequency of the current.
The voltage across an ideal capacitor (one with negligible resistance) reaches a maximum one quarter cycle
after the current does while the voltage across an ideal inductor (one with negligible resistance) reaches a
maximum one quarter cycle before the current does. The voltage builds up on the capacitor as the current
deposits charge on it, and when it is charged the current is zero. Since the current is increasing most rapidly
when it is zero the voltage on the inductor is greatest then. Voltages across ideal capacitors and inductors
vary with time according to the equations below.
VC '
Q
' VCo sin(Tt & 90E)
C
(2)
VL ' L
)I
' VLo sin(Tt % 90E)
)t
(3)
Experiment 1 Page 1
where C is the capacitance in farads and L is the inductance in henries.
The amplitudes of the voltages across the three elements discussed are given by
VRo ' RIo
(4)
1
I
TC o
(5)
VLo ' T L Io
(6)
V Co '
where T = 2B f and f is the frequency. The coefficients of Io in equations (5) and (6) are called the
capacitive and inductive reactances, XC and XL.
XC '
1
TC
XL ' T L
and
(7)
For a series R-L-C circuit the current and voltage amplitudes are related by
V S ' Io Z
(8)
where Z is called the impedance and is determined by
Z ' R 2 %(X L & X C )2
(9)
The voltage leads the current if XL is larger than XC by an angle given by
N ' arctan
XL & XC
(10)
R
and lags the current by the angle N given by (9) (a negative angle) when XL is smaller than XC. When XC
= XL the resulting current amplitude is maximum and the circuit is in resonance. The resonance condition
is
TL '
1
TC
(11)
which gives
T'
1
LC
and f '
1
2B L C
(12)
The above equations are correct if ideal capacitors and inductors are used, where their resistances are
small compared to their inductances. For capacitors this is generally true but for inductors it is true only
for higher frequencies. To correct for the effect of the inductor resistance, the value R in equations (9) and
(10) must be replaced with R + RL.
Experiment 1 Page 2
For an R-L circuit, as shown in figure 1, the following phasor
diagram and equations can be written
tanNL '
tanNS '
L
T
RL
L
T
R%RL
X
(13)
Z
Z
N
NS
(14)
R
L
TL
L
R
R
L
For an R-C circuit, as shown in figure 2, the following equation
Figure 1 Phasor diagram for a R-L
and diagram can be written.
series circuit.
cotNS ' RC T
(15)
X
R
R
Procedure:
N
S
1
Z
T C
Series R-L Circuit
Caution: Do not turn on the Pasco Signal Generator
until the lab instructor checks your circuit.
Figure 2 Phasor diagram for a
series R-C circuit.
With an ohmmeter measure and record the values of
the resistance R (about 10 ohms) of the resistor and
RL (about 5 ohms) of the inductor.
Set up a series R-L circuit, with channel A of the
Pasco interface box connected across the resistor.
Be certain that the RED probe wire is on the positive
side of the resistor. Connect channel B of the
interface box across the inductor again with the RED
probe wire on the positive side of the inductor.
Connect channel C across the signal generator
terminals with the RED probe wire on the positive
Figure 3 Pasco Signal Generator.
side of the inductor. Set the multimeter to 20 volt
AC operation. Ask the lab instructor to check your
circuit. Turn the voltage on the signal generator all the way down, and then turn it on. Now, raise the
voltage on the signal generator until the voltage displayed on the multimeter is approximately 3.5 Volts.
Experiment 1 Page 3
Never exceed 4 Volts on the multimeter. For
frequencies of about 30 Hz, 60 Hz, 100 Hz, 150 Hz,
and 200 Hz complete the table 1 by following the
instructions listed below.
Science Workshop Setup
Double click on the Science Workshop icon. Click
on the din plug icon (to the right of the graph icon),
drag it onto analog channel A and double click on
Voltage Sensor. In the same way drag the din plug
icon onto analogue channels B and C, selecting
Voltage Sensor in each case. Double Click on
Sampling Options and click on Time under Stop
Conditions. Enter a stop time of 0.1s and click on
OK. Set the Periodic Samples rate to 10,000 and
click on OK. Click on REC to collect a set of data.
Click on the graph icon and drag it onto the Voltage
Figure 4 AC circuit board for R - L
connections
tR
tL
tS
Cursor icon
Rescale icon
Input selection icon
Figure 5 Graphs of the voltages in an R - L circuit, channel A shows the resistor voltage, channel B
shows the inductor voltage and channel C shows the signal generator voltage.
Experiment 1 Page 4
icon below channel A. Click on the rescale icon and a graph of the resistor voltage versus time is shown.
Click on the Input selection box at the lower left of the graph window (second row, second icon), click on
Analog B and click on Voltage to display the inductor voltage graph. Repeat this to select and display the
signal generator voltage from Analog C.
Click on the cursor icon and place the vertical cursor line through the zero crossing point for the resistor
voltage then read the value below the time axis. Repeat this for the zero crossing just before that of the
resistor voltage for the inductor and signal generator voltages as shown in figure 5 and record these values
in table 1 as tR , tL and tS .
Compute the phase shifts in radians from NL = T(tR-tL) and NS = T(tR-tS).
Plot a graph of tan NL on the y axis versus T on the x axis. From equation (13) the slope = L/RL.
Determine the value of the inductance from this relation.
Plot a graph of tan N on the y axis versus T on the x axis. From equation (14) the slope = L/(R+RL).
Determine the value of the inductance from this relation.
The two inductances calculated should be very close to each other. Calculate the % error between them.
Table 1 Series R-L circuit
f
T
tR
tS
tL
NL
NS
tan NL
tan NS
Series R-C Circuit
Turn the amplitude on the signal generator down and replace the inductor with a 330 µF capacitor. Then
increase the gain until the voltage is about 3.5 volts. Repeat the measurements performed in the R-L
experiment and complete table 2. Do this for frequencies of 20 Hz, 30 Hz, 50 Hz, 80 Hz, and 150 Hz.
You need not include phase shifts for the capacitor in table 2 since they are all very nearly 90E. In your
journal calculate at least one capacitor phase shift. Click on the cursor icon and place the horizontal cursor
at the top of each voltage, then read the value at the left of the voltage axis.
Experiment 1 Page 5
Plot a graph of VRo/VCo on the y axis vs T on the x
axis and compute the value of the capacitance from
t
h
e
relation slope = RC. Dividing equation (4) by
equations (5) gives Vro/VCo = RC T showing that the
slope is RC Plot a graph of 1/tanN on the y axis vs
T on the x axis. From equation (15) the slope =
L/(R+RL). Determine the value of the inductance
from this relation. These values should be close to
each other. Find the % error between them.
Figure 6 AC circuit board for R - L
connections
Table 2 Series R-C circuit
f
T
tR
tS
NS
VR0
VC0
VR0'VC0
tan NS
Resonance in a Series R-L-C Circuit
Turn the source voltage amplitude to zero. Add the inductor to the circuit so everything is connected in
series. Increase the amplitude to about 3.5 volts. For a frequency of about one fifth of the resonant
frequency describe and explain the phase and voltage relationships. Repeat this for a frequency of about
5 times the resonant frequency. Find resonance by adjusting the frequency until the voltage across the
resistor is maximum. Compare this to the theoretical resonant frequency.
Experiment 1 Page 6