Physics 241 Lab – Matt Leone
Week 11: The Sinusoidally Driven Series RLC Circuit
Leone@physics.arizona.edu (email preferred), PAS 376, o. 520-621-6819
Office Hours: M & W 11:00-11:50, or by appointment. Consultation Room (PAS 372): F 12:00-12:50
http://bohr.physics.arizona.edu/~leone/phy241/phys241lab.html
General Comments:
• The driven series RLC circuit (often just called the RLC circuit) is usually the most
complicated circuit examined in introductory courses. We will build up to it today in pieces.
• You should practice for your practical.
Lab 11 – Discussion.
Series RLC Sinusoidally Driven:
The differential equation for this circuit (adding voltages around loop) is:
Vdrive (t) = VL (t) + VR (t) + VC (t) ,
which reduces to (using Ohm’s Law and the definitions of inductance and capacitance):
dI(t)
Q(t)
.
Vdrive (t) = L
+ RI(t) +
dt
C
dQ(t) !
But note that since
= I(t) we can write Q(t) = " I(t)dt . (You will use this idea to answer
dt
questions later in this lab). Thus we can write:
!
" I(t)dt .
dI(t)
Vdrive (t) = L
+ RI(t) +
!
dt
C
!
This is a very useful equation for us because if we know what the current in the resistor is doing as a
function of time, we will know what is going on in each component.
Here is the part that confuses students sometimes. You would think we would solve everything
from the perspective of !
the sinusoidally oscillating voltage supply, VDRIVE. It turns out to be easier to
solve everything from the perspective of the current in the resistor!
First, you need to know that if the oscillating voltage supply has angular frequency ωDRIVE, then
the voltage in each component of the circuit oscillates with that frequency. So working from the
perspective of the current in the resistor, write I(t) = Iamplitude cos(" drive t) . All this is saying is that the
current in the resistor oscillates sinusoidally (a cosine function oscillates sinusoidally). Now plug this
into the voltage differential equation.
cos(" drive t)dt
d cos(
!" drive t) + RIamp cos(" drive t) + Iamp #
,
Vdrive (t) = LIamp
dt
C
and work out the derivative and integral. (Note that “amp” means “amplitude”).
sin(# drive t)
Vdrive (t) = "L# drive Iamp sin(# drive t) + RIamp cos(# drive t) + Iamp
.
C# drive
!
Now, if you graph a sine and cosine function, you will see that the cosine leads the sine by a 90o phase
shift. And if you examine a negative sine and cosine you will see that the negative sine leads the
cosine by a 90o phase shift. Therefore, examining the previous equation, you should be able to
! the relative phases of VL, VR and VC. (Let’s do this together now, and compare to the
determine
following graph).
1
This graph shows the relative 90 o or 180o phase shifts of VL, VR and VC. It also shows that at
any given moment in time, VL, VR and VC must add to the source voltage. Notice that there is no
“nice” phase shift for the source voltage. We won’t worry about that in today’s lab, though if we had
more time we could test the equation for it (given later).
If you remember the lab from week 8, we discussed the following:
" R%
VR (t) = $ 'Vsource,amp cos((t) ,
#Z&
where Z is the impedance of the whole circuit and is given by
2
Z = R 2 + (" L # "C ) .
Now XL and XC are like the !
resistances of the inductor and capacitor in the circuit. They are given by
1
and " L = #L .
"C =
#C
! appearing in the last equation of the previous page).
(You can even see these factors
!
!
2
#" &
We also discussed VC (t) = % C (Vsource,amp sin()t ) . Notice how VC here lags behind VR by 90o
$ Z '
(compare sine and cosine). See how the capacitor voltage was related to the source voltage by the ratio
of the capacitor reactance to the total impedance.
$# '
Now we !
add a new one. VL (t) = "& L )Vsource,amp sin(*t ) . Pretty much the same story as the
% Z (
capacitor except that the negative sine function makes it 90o ahead of the resistor. Note that this also
means that the inductor and capacitor are 180o out of phase.
Finally, though we won’t be too concerned with it in this lab, the alternating voltage from the
! a mathematical description: V (t) = V
power source still needs
drive
source,amp cos("t + # ) where the phase
% $ # $C (
difference between the source and the resistor is given by " = tan#1' L
*.
& R )
Also, an important equation to remember
is that Vsource = Iresistor Z . This is like V=IR.
!
amplitude
amplitude
There is a related concept that should also be discussed (and it is in figure 32.11 of your text).
!
Examine the following LC circuit.
!
Imagine that the capacitor was initially charged and placed into this circuit. At the instant before
charge began flowing off the conductor, there would be an electric field inside the capacitor and
nothing going on with the inductor. This electric field in the capacitor contains energy,
1
E energy of = CV 2 . Now as charge flows from the conductor decreasing the capacitor energy. But there
2
capacitor
E -field
is no resistor in the circuit to remove energy so the total energy must be conserved. Where does this
energy go?
Ampere’s law tells us that the solenoid will have a magnetic field inside it. This is given by
1
E energy of = LI 2 . The maximum current will be reached at the moment the capacitor is
2
inductor
!
B-field
completely discharged. This means that all the electric field energy in the capacitor will have
been transformed into magnetic field energy in the inductor, E energy of " E energy of . After some
!
capacitor
E -field
inductor
B-field
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 swinging back and forth, but
capacitor
E -field
at all times E total = E energy of + E energy of .
energy
capacitor
E -field
inductor
B-field
!
inductor
B-field
!
!
3
Let’s practice the sinusoidally driven series RLC circuit with an example.
First, calculate some basic parameters using these values. Use correct units.
χC
χL
Z
VR,amplitude
VC,amplitude
!
φsource
!
VL,amplitude
!
Next write equations to describe the time-dependent behavior of each element of the circuit:
Vsource (t)
!
!
VC (t)
VR (t)
!
VL (t)
Finally, imagine that you were able to adjust the source frequency while leaving Vsource,amp
!
Vsource
amplitude
constant. Since Iresistor =
, the current in the resistor can be maximized by minimizing Z
Z
amplitude
2
(remember that Z depends on ω since both χC and χL depend on ω , and Z = R 2 + ( " L # "C ) ). Find
the resonant frequency (both linear and angular) that maximizes the current in the resistor.
!
!
ωRESONANCE
fRESONANCE
4
Lab 11 – Procedure – write on this sheet and turn it in with your write-up.
1. Here you will double check previous results about the sinusoidally driven series RC circuit.
a. Sketch the circuit diagram for the driven series RC circuit. YOUR SKETCH:
b. Implement this circuit using a sinusoidal driving voltage of VMAX " 5 Volts and
frequency of about 1,000 HZ, R = 1,000 " , and C = 1x10 -7 Farads .
c. Use your oscilloscope to simultaneously measure the alternating voltages across the
!
resistor and capacitor using a middle ground.
!
!
d. Describe the relationship between phase of VR and VC with both words and a labeled
sketch. For this sketch you do not need to label the scale of your axes. Be sure to invert
the signal being measured “backwards” otherwise you will get the wrong answer.
YOUR ANSWER AND LABELED SKETCH:
e. Assume that the current passing through the resistor is described by
I(t) = Iamplitude cos(" drive t) and then use this and the concepts of “derivative” or
“indefinite integral” to explain why the alternating voltage on the resistor leads the
alternating voltage on the capacitor by 90o. Hint: Q(t) = " I(t)dt . YOUR ANSWER:
!
!
5
2.
Here you will examine the driven series RL circuit (alternating driving voltage).
a. Sketch the circuit diagram for the driven series RL circuit. YOUR SKETCH:
b. Implement this circuit using a sinusoidal driving voltage of VMAX " 5 Volts and
frequency of about 1,000 HZ, R = 1,000 " , and L " 50 mH .
c. Use your oscilloscope to simultaneously measure the alternating voltages across the
!
resistor and inductor using a middle ground.
!
!
d. Describe the relationship between phase of VR and VL with both words and a labeled
sketch. For this sketch you do not need to label the scale of your axes. Be sure to invert
the signal being measured “backwards” otherwise you will get the wrong answer.
YOUR ANSWER AND LABELED SKETCH:
!
e. Assume that the current passing through the resistor is described by
I(t) = Iamplitude cos(" drive t) and then use this and the concepts of “derivative” or
“indefinite integral” to explain why the alternating voltage on the inductor leads the
dI(t)
alternating voltage on the resistor by 90o. Hint: VL (t) = L
. YOUR ANSWER:
dt
!
6
3. Here you will examine the driven series RLC circuit (alternating driving voltage).
a. Sketch the circuit diagram for the driven series RLC circuit. YOUR SKETCH:
b. Implement this circuit using a sinusoidal driving voltage of VMAX " 5 Volts and
frequency of about 1,000 HZ, R = 1,000 " , C = 1x10 -7 Farads , and L " 50 mH .
Record the actual values used. YOUR ACTUAL VALUES:
!
!
!
!
c. Use your oscilloscope to simultaneously measure the alternating voltages across the
capacitor and inductor using a middle ground. (Therefore, the capacitor and inductor
must be adjacent in your circuit).
d. Describe the relationship between phase of VC and VL with both words and a labeled
sketch. For this sketch you do not need to label the scale of your axes. Be sure to invert
the signal being measured “backwards” otherwise you will get the wrong answer.
YOUR ANSWER AND LABELED SKETCH:
e. Assume that the current passing through the resistor is described by
I(t) = Iamplitude cos(" drive t) and then use this and the concepts of “derivative” and
“indefinite integral” to explain why the alternating voltage on the inductor and the
alternating voltage on the capacitor are 180o out of phase. YOUR ANSWER:
!
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f. Theoretical calculations: Using the given values for inductance, capacitance and
resistance that you have in your circuit, calculate the following and show your work.
YOUR ANSWERS WITH WORK:
1. χC
2. χL
3. Z
4. ωRESONANCE
5. fRESONANCE
g. Now examine the alternating voltage across the resistor on your oscilloscope.
h. Adjust the driving frequency until VR,AMP reaches a maximum. Compare this value to
your theoretical predictions. If this frequency is not close to your predictions, come get
me. Record the measured value of the resonant frequency of your circuit. YOUR
MEASUREMENT:
i. Calculate Iamplitude using Ohms law (valid only in the resistor) when driving at resonance
showing your work. YOUR CALCULATION WITH WORK:
!
!
j. Collect frequency dependent data of the
amplitude of the current in the resistor,
Iamp ( f ) vs. f. Use this data to make a
nice graph (large, on graph paper). It
should look something like the quick
sketch (right). Points will be awarded
for accuracy (hint: it should be a
different than my sketch in a way only I
know ). ATTACH YOUR GRAPH.
8
4. Here you will examine the driven RLC circuit with the inductor and capacitor in parallel with
each other and both in series with the resistor and alternating driving voltage. Note: you can
only make theoretical predictions for this circuit using the procedure of complex impedances.
a. Sketch the circuit diagram for this circuit. YOUR SKETCH:
b. Implement this circuit using a sinusoidal driving voltage of VMAX " 5 Volts and
frequency of about 1,000 HZ, R = 1,000 " , C = 1x10 -7 Farads , and L " 50 Henry .
Record the actual values used. YOUR ACTUAL VALUES:
!
!
!
!
c. Collect frequency dependent data of the amplitude of the current in the resistor, Iamp ( f )
vs. f. Use this data to make a nice graph (large, on graph paper). Points will be
awarded for accuracy. Hint: I made up the word “antiresonance” to describe what
happens here. ATTACH YOUR GRAPH.
!
5. Here you will play with the demo equipment.
a. Examine what happens when you insert a soft iron core into the inductor of the demo
equipment. YOUR OBSERVATIONS:
b. Explain what is happening in terms of the following concepts: “total impedance”,
“inductive reactance”, “capacitive reactance”, and “driving frequency”. YOUR
EXPLANATION:
6. CLEAN UP LAB STATION.
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Lab 11 – Report Guidelines
This week you will write a “standard” lab report. (Ellipsis means you should not need help
here on what to write about).
1. Title - …
2. Goals – …
3. Theory – Address the following concepts in your theory section
• Capacitive reactance
• Inductive reactance
• VL, VR, VC, and VSOURCE phase shifts
• Resonance in an RLC circuit.
4. Problems – Type and draw these up and attach to your report (for serious points)
1. Examine quick quizzes 33.2-33.4 then explain your answer to problem 57 in chapter 33.
2. Type up (and add to if need be) your answer to part b of procedure #5 of this lab.
5. Procedure – …
6. Results – Report the results of each section.
7. Conclusion – Discuss the main points of each section of this lab.
8. Attachments including 2 graphs.
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