Physics 241 Lab: RLC Circuit – AC Source
http://bohr.physics.arizona.edu/~leone/ua/ua_spring_2010/phys241lab.html
Name:____________________________
Section 1: The RLC Circuit
1.1.
In a circuit where an inductor, resistor and capacitor (RLC) are connected in series and driven
by a sinusoidal voltage source, the properties of the RC circuit and RL circuit that you studied
previously combine in a straightforward manner. Let’s summarize the results for the LRC circuit that
you should already suspect.
The voltage across each component oscillates at the same frequency as the driving frequency of
the source, drive. The properties of the inductor and capacitor are frequency dependent, which makes
the circuit respond differently to different driving frequencies.
The resistor voltage is chosen as the voltage reference because it is the only Ohmic device and
therefore can be used to determine the current in the circuit:
R
1
V (t) resistor  Vsource sin drivet  and therefore I(t)  Vsource sin drive t ,
Z amplitude
Z amplitude
where Z [Ohm] is the total circuit impedance, Z  R 2   L   C  . Also, C [Ohm] is the capacitive
1
reactance  C 
and L [Ohm] is the inductive reactance  L   driveL .
 driveC
Is the current in the circuit increased or decreased when the difference between the capacitive
reactance and inductive reactance is increased?
Explain your answer.
Note that
2
2
 L  C   C   L  . Your answer and explanation:
2
1.2.
The driving source voltage is given with a phase shift with respect to the resistor:
V ( t ) source  Vsource sin drivet   source  ,
amplitude
   C 
where the source phase shift is given by  source  tan1 L
. This shows that the source voltage
 R 
will be in phase with the resistor voltage when  L   C because tan1 0  0 . It also shows that if
 L   C , then  source  0 (the source voltage leads the resistor voltage), and if  L   C , then  source  0
(the source voltage lags the resistor voltage).
Is it possible for the source voltage to ever be 180o out of phase with the resistor voltage? Use
your graphing calculator to graph y=arctan(x) and find when y =  or –. Your answer and
reasoning:
The capacitor voltage is phase shifted to lag the resistor voltage by 90o:


 
V ( t ) capacitor  C Vsource sin drive t  .
Z amplitude 
2 
The inductor voltage is phase shifted to lead the resistor voltage by 90o:


 
V ( t ) inductor  L Vsource sin drive t  .
Z amplitude 
2 
Sometimes these four boxed (important) voltage equations are rewritten using the fact that
V
Iamplitude  source amplitude :
Z
V (t) resistor  R  Iamplitude sin drive t 
1.3.
V ( t ) source  Vsource sin drive t   source  (no change)
amplitude

 
V (t) capacitor   C  Iamplitude sin drive t  

2 

 
V (t) inductor   L  Iamplitude sin drive t  

2 
If R = 10 , L = 10 H, C = 10 F, Vsource amplitude = 10 volts and fdrive = 10 Hz, find the voltage
amplitudes of the inductor and capacitor. Your work and answers:
1.4. It is useful to examine an example graph showing the relationships between each of the
component’s voltages:
In this example, C is larger than L so that Vcapacitor amplitude is larger than Vinductor amplitude. Note that
Vresistor amplitude is larger than the other two voltage amplitudes, which indicates that R is larger than C
and L. This need not be the case since you could always choose to use a smaller resistor in your
circuit. Since the capacitor voltage lags the resistor voltage by /2 while the inductor voltage leads the
resistor voltage by /2, the capacitor and inductor voltages themselves are 180o out of phase
(/2 + /2 = ).
Conservation of energy indicates that the sum of the voltages of the three components at any
instant of time must equal the voltage of the source at that time. Therefore, if you add the three
graphed voltages, you obtain the source voltage: Vsource ( t )  VL ( t )  VR ( t )  VC ( t ) . The sum of the three
component voltage functions in the previous graph is shown below with a dotted line.
Notice that in this example the capacitor voltage amplitude is larger than the inductor voltage
amplitude. This causes the source voltage to reach it’s amplitude a little after the resistor. Recall that
   C 
 source  tan1 L
 and since C is larger than L, you obtain source < 0. This means that the
 R 
source voltage lags the resistor voltage.
At driving frequencies where L > C, you find that VL > VC, and this causes the source voltage
to lead the resistor voltage.
Finally, imagine that you are able to adjust the source frequency while leaving Vsource,amplitude
V
constant. Since Iamplitude  source amplitude , the current in the resistor can be maximized by minimizing Z.
Z
Remember that Z depends on drive since both C and L each depend on drive. Since
 Z  R 2   L  C  , Z is minimized when C is equal to L so that L – C, = 0. This gives
Zminimum = R.
The driving frequency at which L – C, = 0 occurs is called the resonant frequency. This can
1
  driveL . Some algebra (that should appear in your
be found by setting C equal to L so that
 driveC
1
lab report) gives  drive 
.
LC
resonance
1.5. Imagine that you have an RLC circuit being driven sinusoidally at resonance. Assume you
have placed the inductor voltage and capacitor voltage on the two channels of your oscilloscope
(correctly inverting one of the channels). Make a quick sketch of what you should see on the
oscilloscope screen. Think carefully about what the voltage amplitudes should be for these two
components at resonance. Your quick sketch below:
2
Section 2:
Let’s practice the math for an example of a sinusoidally driven series RLC circuit.
2.1. Calculate each of the basic RLC circuit parameters in SI units though not necessarily in the
order given.. Write the numerical value with correct SI units for each listed parameter:
C
L
Z
VR,amplitude


source
VC,amplitude
VL,amplitude
Iamplitude 
2.2. Next write equations using the numerical results from 2.1 to describe the time-dependent
behavior of each component of the circuit.
Write a time-dependent equation for each listed item:
Vsource (t)
VR (t) 


VC ( t )
VL ( t ) 


I( t )
Qcap (t) 

2.3. Imagine that you are free to adjust the driving frequency of the source voltage. Use the
previous results to find the driving frequency that allows the largest current to flow through this RLC
circuit, fdrive .
resonance
Your answer:
Section 3:
In electronics, you may use the oscilloscope to simultaneously measure the voltage of
two adjacent components. Connecting your inductor, capacitor and resistor in series should be done
with some consideration of which components you will measure with your oscilloscope since you only
have two channels to measure with. The three circuit configuration examples below and the questions
that follow are used to illustrate this point
3.1. In circuit A, which pairs of components can be separately measured on the oscilloscope with a
middle ground technique? (You are free to choose the location of the oscilloscope ground.)
Your answer:
3.2. In circuit B, which pairs of components cannot be separately measured on the oscilloscope with
a middle ground technique? (You are free to choose the location of the oscilloscope ground.)
Your answer:
3.3. In circuit C, which pairs of components can be separately measured on the oscilloscope with a
middle ground technique? (You are free to choose the location of the oscilloscope ground.)
Your answer:
Therefore, as you make measurements on the RLC circuit, you often have to rearrange the components
depending on which two components you wish to measure.
3.4. Sketch a circuit configuration along with the proper placement of the three oscilloscope leads
(red 1, red 2 and ground) in order to be able to separately measure the time dependence of the solenoid
and resistor simultaneously.
Your complete sketch:
3.5. Sketch a circuit configuration along with the proper placement of the three oscilloscope leads
(red 1, red 2 and ground) in order to be able to separately measure the time dependence of the source
and resistor simultaneously.
Your complete sketch:
Section 4:
The explanation of the 90o phase shifts of the inductor and capacitor voltages from the
resistor voltage can be explained using calculus.
Conservation of energy for the sinusoidally driven RLC circuit gives:
Vsource ( t )  VL ( t )  VR ( t )  VC ( t ) .
dI(t)
Substituting Ohm’s Law VR ( t )  I ( t )  R , the definition of inductance VL (t)  L
and the definition
dt
Q(t)
gives:
of capacitance VC (t) 
C
Q(t)
dI(t)

.
 RI(t)
Vsource (t)  L

dt resistor term
C

capacitor term
inductor term

First notice that first two terms on the right hand side of the equation show that the time
dependence of the inductor is related to the resistor by a time derivative,
Q(t)
dI(t)
.
Vsource (t) 

L
 RI(t)
C
dt





the inductor voltage is proportional to the
time derivative of the resistor voltage

Since the resistor voltage is described by a sine function, V (t) resistor  Vresistor sin drivet , the inductor
amplitude
voltage must be described by a cosine function. But the cosine function leads the sine function by /2,
so the inductor voltage leads the resistor voltage by  /2.
dQ(t)
Next remember the definition of current I(t) 
to find that
dt
dQ(t)
Q(t)
dI(t)
Vsource (t)  L
R

.
dt
dt 



 C
capacitor term
inductor term
resistor term

Notice that the second two terms on the right hand side show that the time dependence of the capacitor
voltage is related to the antiderivative of the resistor voltage,
dI(t)
dQ(t) Q(t)
Vsource (t)  L

R

dt
C
dt




the capacitor voltage is proportional to the
antiderivative of the resistor voltage

Since the resistor voltage is described by a sine function, V (t) resistor  Vresistor sin drive t , the capacitor
amplitude
voltage must be described by a negative cosine function. But the negative cosine function lags the
sine function by /2, so the capacitor voltage lags the resistor voltage by  /2.
4.1. Prove that you truly understand this argument by using the definition of current to rewrite the
inductor term with a second time derivative of charge, compare this to the capacitor term and explain
Your work and explanations:
why the inductor and capacitor are 180o out of phase.
Section 5:
There is a related circuit that should also be mentioned, the LC circuit (undriven but
with an initially charged capacitor).
Imagine that the capacitor was initially charged and placed into this circuit. At the instant
before charge began flowing from the capacitor, there would be an electric field inside the capacitor
and no fields or current inside the inductor.
1
The electric field in the capacitor contains energy, E energy of  CV 2 . As charge begins to flow
2
capacitor
E -field
from the capacitor, energy stored in the electric field inside the capacitor decreases. However, there is
no resistor in the circuit to remove energy by creating heat so the total energy of the circuit must be
conserved. Where does this energy go?
As current begins to flow through the inductor, the solenoid will have a magnetic field created
1
inside it. The energy stored in the inductor’s magnetic field is E energy of  LI 2 . The maximum current
2
inductor
B-field
will be reached at the moment the capacitor is completely discharged. This means that the moment
when all the charge has left the capacitor, all the energy stored in the electric field inside the capacitor
will have been transformed into energy stored in the magnetic field inside the inductor,
E energy of  E energy of .
capacitor
E -field
inductor
B-field
After some time, the charge will flow back onto the capacitor (in the reverse direction) and the
current will momentarily return to zero, E energy of  E energy of . And so it will go, energy swinging back
capacitor
E -field
inductor
B-field
and forth between the electric field of the capacitor and the magnetic field of the inductor, but at all
times the total energy is conserved E total  E energy of  E energy of .
energy
capacitor
E -field
inductor
B-field
d 2Q(t)
for the
dt 2
undriven LC circuit. (Simply plug in zero for the missing components.) Your work and answer:
5.1.
Use the same process part 4 of the lab to write an equation relating Q(t) and
Section 6:
Extra lab technique tips.
An RLC circuit has a resonant driving frequency fresonance that maximizes the circuit current
amplitude. Therefore, a plot of the current amplitude vs. driving frequency will show peak current
amplitude at the resonant driving frequency. However, this graph can appear different depending on
the resistance used in the circuit:
The sharpness of the resonance peak is measure by the quality factor Q. The larger the quality

L
factor, the sharper the peak. Q is found to be given as Q  resonance
for an RLC circuit in series
R
driven sinusoidally.
In today’s lab you usually want to use as small of a resistance as possible (while still being
safe) in order to obtain a nice sharp peak in the current amplitude. This will make it easier to find the
resonance frequency. However, it is not always desirable to have a large quality factor. In some
mechanical systems (bridges, buildings, etc.) a pronounced response to a driving frequency can cause
destruction.
The most accurate way to find fresonance is to utilize the fact that at resonance, VR(t) and Vsource(t)
are exactly in phase with each other with equal amplitudes. You should place each of these voltages
on your oscilloscope channels and examine an XY formatted display. The resonance frequency is
easily found because you will see an ellipse when VR(t) and Vsource(t) are out of phase and a diagonal
line when they are in phase. You see a straight line when they are in phase because both voltages must
reach zero simultaneously.
Note that if R is chosen small enough, the resistance of the very long wire of the solenoid may
become an appreciable resistance of the circuit. When this occurs, the amplitude of the resistor voltage
will be smaller than the source voltage amplitude so that you will see a different angle of tilt (not 45o)
of the in phase resonance line shown above. This is because Vsource  VR  Vresistance of solenoid  VL  VC .
Section 7:
Construct an RLC circuit driven by a sinusoidal driving voltage of Vsource
with
fdrive = 3,500 Hz, R  1 k , C  0.1 F , and L  50 mH .
frequency the same until 7.8 of this section of the lab.
 5 Volts
amplitude
You must keep this driving
7.1. Quickly measure the voltage amplitudes of each component separately using a single channel
of your oscilloscope. Double check that your source voltage amplitude is 5 Volts first. Record your
observations:
7.2. Now use two-channel observations to simultaneously measure the resistor voltage and the
inductor voltage to determine their respective phase shift. Your observed inductor voltage phase
shift:
7.3. Now use two-channel observations to simultaneously measure the resistor voltage and the
capacitor voltage to determine their respective phase shift. Your observed capacitor voltage phase
shift:
7.4. Now use two-channel observations to simultaneously measure the inductor voltage and the
capacitor voltage to determine their respective phase shift. Your observed capacitor voltage phase
shift:
7.5. Now use two-channel observations to simultaneously measure the resistor voltage and the
source voltage to determine their respective phase shift. Note that a phase shift of /2 corresponds to a
shift along the time axis of ¼ of oscillation cycle T. Therefore, if tshift is the amount of time lag or time
lead that you measure between the resistor amplitude and the source amplitude, you can set up the
following equivalent ratios to find the phase shift of the source source,
t shift

 source .
1   
 T   
4  2 
2  t shift
This gives  source 
  drive  t shift .
T
Another way to think about how to measure the source voltage phase shift is to examine the
extra amount of time it would take for the resistor voltage to come into phase with the source voltage if
the source voltage was frozen in time:
sin drive t  t shift   sin drive t   source .
For these sine functions to be equal, their arguments must be equal so that you find  source   drive  t shift .
Use this equation to experimentally determine the source voltage phase shift source from the
resistor voltage. Your observed source voltage phase shift:
7.6. Theoretical calculations: Use the labeled values for inductance, capacitance and resistance that
you have in your circuit as well as your actual driving frequency to calculate the following RLC
circuit parameters in SI units. Your work and answers:

C
L
Z

fresonance
resonance
7.7. Comparing theory and observation: Use your results from 7.6 along with your observed source
voltage amplitude to theoretically calculate the voltage amplitudes of the resistor, inductor and
capacitor in SI units. Then compare each of these theoretically computed amplitudes to your
observations in 7.1. Your calculations, theoretical results and comparisons:
V 
 resistor
2
7.8.
Since Presistor  I  Vresistor
, the power converted to heat by the resistor oscillates in
R
time as a squared sine function. The average value of a squared sine function over a complete
2


Vresistor 
1
1  amplitude 
2
. Use this formula to
oscillation cycle is average sin( drive t   . This means Presistor 
2
2
R
average
find the average power loss of your circuit in SI units. Your answer:


7.9. When the driving frequency is at resonance, the voltages of the source and resistor are equal
and in phase. Use this fact to search the range of driving frequencies for the resonant driving
frequency using the technique described in part 6 of this lab. Compare this observed value to your
predicted value in 7.6. Your observed resonant frequency and comparison:
7.10. Collect data in order to make the graph of
Iamplitude vs. fdrive. A poor example is shown with a large
resistance is shown. Collect your data below and
construct your graph on graph paper now:
Section 8: (Open-ended question / creative lab design)
Pretend that you are an old-timey inventor. You have already
discovered the coiled wire device (now called inductor) and the metallic
parallel plates device (now called capacitor). Resistors were discovered
by a competitor of yours. It is still a sore subject around the workshop.
You have begun to combine these new components together in
order to observe their compound behavior. You already found that when
the inductor and capacitor are combined in series and driven sinusoidally,
the circuit produces a special resonant frequency where the current in the
circuit is a maximum. This means that civilization can now make
electronics that “select” specific driving frequencies and “suppress”
frequencies far from resonance.
You should try to find the behavior of the current through the circuit for various driving
frequencies. Try to deduce why this circuit may be said to have an “antiresonant frequency” and see
if you can find it.
Now you try combining the inductor and capacitor in parallel and drive them sinusoidally (see
figure). What electrical possibilities might this new compound device hold for the future of mankind?
Well, use your anachronistic oscilloscope to find out.
At the following prompts, design an experiment to find the behavior of a parallel RLC circuit
that is sinusoidally driven. Then implement your experiment and record your observations. You may
“cheat” by talking to other groups for ideas, but not “cheat” by already knowing the answer or looking
it up.
Your planned experiment, sketch of actual implementation and any theoretical calculations:
Your observations:
Your explanations & conclusion:
I, the physics 241 laboratory TA, have examined this worksheet and found it to be thoroughly
completed excepting any sections that I have marked herein. TA signature: ____________________
Report Guidelines: Write a separate section using the labels and instructions provided below. You
may add diagrams and equations by hand to your final printout. However, images, text or equations
plagiarized from the internet are not allowed!
 Title – A catchy title worth zero points so make it fun.
 Goals – Write a 3-4 sentence paragraph stating the experimental goals of the lab (the big
picture). Do NOT state the learning goals (keep it scientific). [~1-point]
 Concepts & Equations – [~9-points] Be sure to write a separate paragraph to explain each of
the following concepts.
 Explain the meaning and use of the four time dependent equations describing the
component voltages.
 Explain the meaning and use of the resonant frequency.
 Explain how energy is stored in a capacitor and how energy is stored in an inductor.
 Explain how to experimentally find an unknown inductance using an RLC circuit.
 Derive the equation for the resonant frequency.
 Derive the phase shifts of the inductor and capacitor using calculus.
2  t shift
.
 Derive why  source 
T
 Procedure & Results – Do not write this section. [~0-points]
 Conclusion – Write at least three paragraphs where you analyze and interpret the results you
observed or measured based upon your previous discussion of concepts and equations. It is all
right to sound repetitive since it is important to get your scientific points across to your reader.
Write a separate paragraph analyzing and interpreting your results from your open-ended
experiment. Do NOT write personal statements or feeling about the learning process (keep it
scientific). [~5-points]
 Graphs – All graphs must be neatly hand-drawn during class, fill an entire sheet of graph
paper, include a title, labeled axes, units on the axes, and the calculated line of best fit if
applicable. [5-points]
o The graphs from section 7.10.
 Worksheet – thoroughly completed in class and signed by your TA. [~5-points]
Introductory Electricity & Magnetism Laboratory Learning Outcomes:
Introductory electricity & magnetism laboratory students should be prepared for leadership roles in
an increasingly diverse, technological and highly competitive world. To this end, these students
should…
 Understand the role of science in our society – The resonance behavior of an RLC circuit.
 Have a firm grasp of the theories that form the basis of electricity and magnetism –
trigonometric functions modeling a complex circuit.
 Be able to apply the principles of physics to solve real world problems – Design an experiment
to determine the anti-resonant behavior of an multiple-component circuit.
 Be familiar with basic laboratory equipment, and should be able to design and carry out
experiments to answer questions or to demonstrate principles. – Skills developed throughout
lab especially the open-ended component. Use of modern digital oscilloscope.
 Be able to communicate their results through written reports – See the general report writing
guidelines in the syllabus and the specific report writing guidelines at the end of the handout.
Graduates of the introductory electricity & magnetism laboratory program should…
 Have a broad education that will allow them to succeed in diverse fields such as business, law,
medicine, science writing, etc. – Writing skills, critical thinking skills, creative problem-solving
skills, communicating ideas & information, teamwork skills, leadership skills, working with
time constraints.
 Have mastered the introductory theoretical techniques and electricity and magnetism
experimental techniques that are commonly expected for students at this level. – Across the
world, all introductory electricity and magnetism students must be able to
o Use a digital oscilloscope to measure rapidly changing voltages.
o Build circuits and measure their properties.
 Be familiar with the principles and practice of engineering and should be able to apply their
knowledge to solve state of the art problems, both individually and as part of a team. –
Individuals must be able to build and test simple circuits. Teams must devise an experiment to
discover the anti-resonance of a parallel RLC circuit.
Teaching Tips:
o V vs. t graph of components for LRC circuit.
o How voltage amplitude of components can exceed the source amplitude via subtraction.
o How resonance is achieved by minimizing the impedance.
o How R affects the “width” of the resonance.
Demonstrate together:
 The authentic assessment to explain hazards.
Instructional Equipment Needed:
