Experiment 10
Inductors in AC Circuits
Preparation
Prepare for this week's experiment by looking up inductors, self inductance, Lenz's Law,
inductive reactance, and RL circuits.
Principles
An inductor is made of wire formed into a coil. The inductor has an inductance, L, which is a
property that depends on the geometry of the wire coils.
When a time varying current flows through the coils an emf is generated such that
ε = - dΦ
B
dt
,
where ΦB is the magnetic flux through the coil and is equal to
ΦB = IL .
An ideal inductor has no resistance, so the voltage across it is due only to the induced emf. If the
current through the inductor is varying sinusoidally so that
I = I 0 sin(ω t) ,
where ω = 2π f , then the voltage across the inductor will be
VL =
ε
=
dΦB
d
dI
=
( IL ) = ( I0 Lsin(ω t)) = I0 ω Lcos(ω t),
dt
dt
dt
π⎞
⎛
VL = I 0 X L cos(ω t) = I0 X L sin ⎜ ω t+ ⎟ .
2⎠
⎝
Comparing the equations above we see that the inductive reactance, XL, is a function of the
frequency such that
X L = ω L = 2π f L .
Consider an inductor in a single loop circuit with a resistor, R, and a sinusoidally varying voltage
source of amplitude VZ and frequency ω . Now a real inductor will have some resistance, RL ,
which must be included along with the resistor, R, in the circuit. Then the total impedance, Z, for
an RL circuit will be
Z = (R+RL )2 +X L2 .
This means that the voltage across the inductor is
always the sum of two voltages that are π/2 out of
phase as given by
VL = I 0 X L2 +RL2 .
Rearranging this equation to solve for the inductive
reactance gives
Figure 1. An RL circuit.
XL =
2
L
2
0
V
-RL2 .
I
We can note that the current amplitude, I 0 , is related to the amplitude of the voltage across the
resistor by the expression
V
I0 = R .
R
Substituting this back into the expression for the inductive reactance gives
XL =
VL2 R 2 2
-RL .
VR2
So we see that reactance can be found experimentally if the current and voltages across the
inductor and resistor are known.
The phase angle between the current and the signal
generator voltage is given by:
XL
.
R+RL
The voltage across the entire circuit is given by
tanφ =
VZ = I 0 Zsin ( ω t+φ ) .
The magnitude of the current will be
V
I0 = Z .
Z
Because the inductive reactance depends on frequency, this is a filter circuit. In this case the
inductive reactance increases with increasing frequency so the magnitude of the current and the
resistor voltage both decrease as the frequency is increased.
Equipment
1
1
1
1
2
3
2
1
1
3
2
2
1
protoboard
multimeter and leads
oscilloscope
signal generator with power cord
three prong adapters
BNC connectors
small wires
2 kΩ resistor
25 mH inductor
18" banana wires, one red, two black
24" banana wires, one red, one black
spade lugs
resistor substitution box
Procedure
Measure and record voltages to at least two
significant digits. Measure and record the resistor,
inductor resistance, and the inductance to three
significant figures.
1.
Use the impedance bridge to measure the
inductance of the inductor. Use the DMM
to measure the resistance of the resistor and
the inductor. If your inductor shows infinite
resistance consult your instructor. Use
ground isolators (three to two prong
adapters) on the oscilloscope and the signal
generator. Build your circuit as shown
in Figure 2.
Figure 2.
2.
Set the generator voltage to a convenient value like 4 Vpp. Make sure that the signal
generator voltage remains constant; it may drift. Be sure that the calibration knobs remain
off, otherwise the data will be worthless.
3.
Wire the resistor and inductor in series with the inductor closer to ground. Record the
frequency and the peak-to peak-voltages across the resistor and the inductor for 11
frequencies between 5 kHz and 25 kHz in increments of 2 kHz. Use the same steps as the
last experiment:
1.
2.
3.
4.
5.
6.
Set the frequency and record its value.
Check the calibration knobs on the 'scope.
Check the signal generator voltage and adjust it if necessary.
Measure and record VR.
Measure and record VL.
Go to step 1.
4.
If the inductor voltage does not change as you change the frequency, call your lab
instructor.
5.
Wire the inductor and the resistance
substitution box in series as you did in the
last experiment. Be sure that the resistor
box is not set to zero resistance. Set the
frequency to about 5000 Hz. Set the
'scope and adjust the frequency so that one
complete wave covers 4 divisions on the
screen. This means that, on the screen, 4
divisions equals a phase shift of 2π
radians. Two divisions equal π radians,
etc.
Figure 3
6.
Calculate the value of R that will make φ=π/4 and set the sub box to that value. Observe
the phase shift on the screen. The two traces should be separated by half a division. Set
the box to a smaller value and record the phase angle. Set the box to a larger value and
record the phase angle.
7.
Be sure your instructor approves your data before you put your equipment away.
Data
Data for the first part of the experiment should consist of the measured inductance and
resistances, the frequencies, and the voltages. For the second part it should consist of the
frequency, the substitution box settings, and the phase angles.
Analysis
1.
Calculate the theoretical and experimental reactances for each frequency and find the
percent error for each. Calculate the current for each frequency. Calculate the phase shift
for each frequency.
2.
For the part of the experiment where you kept frequency constant and changed the
resistance calculate the theoretical and experimental phase shifts. Calculate the current for
each resistance.
3.
Use semilog paper to graph Vout /Vin for both the resistor and the inductor.
4.
Calculate the frequency for which your Vout /Vin for the resistor and the inductor should be
equal. Does this agree with your data?
Questions
1.
Draw the schematics for this experiment and explain how you took the data.
2.
Write at least a paragraph in which you derive the equation for the voltage across an
inductor in an AC circuit. Discuss the frequency dependence of X L . Discuss what
happens to the current in an RL circuit as the frequency changes. How does this compare
to the way the current changes as a function of frequency in an RC circuit?
3.
Write out the units for f and L and show that the unit of X L is ohms.
4.
What happened to the current this time as the frequency increased when the resistance was
constant? What happened to the phase angle?
5.
Discuss the similarities and difference between RC and RL filters.
If it applies to you, write "I have not cheated on this lab report" and sign your name.
Grading
4 pts
Data and Analysis.
2 pts
Each for questions 1, 3, and 4.
5 pts
Each for questions 2 and 5.