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# Albany Medical College The RLC Circuit Lab Report Experiment 11: Physics Answers 2021

Albany Medical College The RLC Circuit Lab Report Experiment 11: Physics Answers 2021

## Albany Medical College The RLC Circuit Lab Report Experiment 11: Physics Answers 2021

**Question Title:**

Albany Medical College The RLC Circuit Lab Report Experiment 11

**Full Question:**

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Florida Institute of Technology

© 2020 by J. Gering

PHY 2092 IN – LAB WORKSHEET

Experiment 11 The RLC Circuit

Name: ________________________

Date: __________

Section # ________

Read the following and fill in the blank lines with the correct equation or response. This

worksheet will be graded out of 11 points and averaged in with all other worksheets.

First, we apply Kirchhoff’s voltage rule to a series RLC circuit driven by an oscillating voltage

given byV0 cos (wt). Kirchhoff’s rules states that the voltage drops (losses of energy) across the

resistor, inductor and capacitor equal the applied voltage (the energy source). In other words,

VL + VR + VC + V0 cos (wt) = 0

Next, 3 substitutions are made:

(1)

Faraday’s Law gives VL = – L (dI / dt)

Ohm’s Law gives

VR = ________.

The definition of capacitance gives VC = ________.

All three terms have minus signs to indicate the current loses voltage (energy) when it passes

through these elements. Also, the cosine driving voltage can be written as a complex

exponential using Euler’s relation:

eiwt = cos(wt) + i sin(wt)

where i = (-1)1/2

Since only the real part of complex exponentials are physically meaningful, we can legitimately

write V0 coswt = V0 eiwt. Substituting all this into Eqn. (1) yields the following.

L

d 2q

dq 1

+ q = V0 eiω t

2 + R

dt

dt C

(2)

Next, take the time derivative of Eqn. (2) and write the result below. The first term is given.

d3q

L 3 +

dt

____________________________

(3)

Now, Rewrite Eqn. (3) by using dq / dt = I. This returns the equation to one involving only

second-order time derivatives. Whew!

________________________________________________________

11 – 4

(4)

Florida Institute of Technology

© 2020 by J. Gering

Since the applied voltage oscillates, we’ll guess that another sinusoidal function would solve this

differential equation. So, consider a trial solution of the form I(t) = Io eiwt where Io may be a

complex number. To see if this guess is a correct, we substitute this into Eqn. (4) which gives

1 ⎤ iωt

⎡

2

iωt

= iωV0 e

⎢⎣− Lω + iRω + C ⎥⎦ I0 e

(5)

By dividing throughout by iw eiwt and using 1 / i = -i, we obtain a form of Ohm’s Law found in

Eqn. (5). Use the next two lines to perform this division and rearranging.

[

____________________________ I0 = V0

]

(6)

[

____________________________ I0 = V0

]

(7)

Then you should arrive at Eqn. (8):

⎡

⎛ ωL − 1 ⎞⎤ I = V

R

+

i

⎜

⎟

0

⎢⎣

⎝

ωC ⎠⎥⎦ 0

(8)

So, the trial solution is correct if Eqn. (8) holds. Interpreting Eqn. (8) as Ohm’s Law

means the quantity in brackets [ ] represents the total resistance of the circuit. It is called the

impedance and is given the symbol Z. So Ohm’s Law is then: Z I0 = V0. The impedance of this

circuit consists of real and imaginary parts. The real part is the resistance R and the imaginary

part is called the reactance, X. The reactance itself has two parts:

the inductive reactance

the capacitive reactance

XL = w L

XC = 1 / (w C)

(9)

(10)

You can think of inductive and capacitive reactance as A.C. resistances for the inductor and the

capacitor. Hence the impedance can be rewritten as

Z = R + i ( XL – XC )

(11)

Here the complex number Z has been written in the form of: a + ib, where both a and b are real

numbers. This is in direct analogy with 2-dimensional vectors where a is the x component and b

is the y component. Complex numbers can also be written in terms of a magnitude and a phase

factor. This is in direct analogy with 2-dimensional vectors expressed in plane-polar

coordinates.

Z = | Z | e if.

(12)

In this way, | Z | is the length (magnitude) of the vector and phi is the angle measured up from

the positive x axis. This angle is usually written as a theta in plane polar coordinates.

11 – 5

Florida Institute of Technology

© 2020 by J. Gering

Let us examine the real part of the impedance, which is the physically meaningful part.

However, we must develop a few mathematical tools for dealing with complex numbers.

First, the complex conjugate of an imaginary number is called z-star and is written as:

Z* = a – ib

(13)

Then the magnitude of this number (the length of the complex vector) is given by

|Z|

= ( Z Z* )1/2

(14)

Substituting a’s and b’s into Eqn. (14) gives

|Z|

= (a2 + b2 )1/2

Say, Eqn. (15) is just a form of whose Theorem?

And just as with vectors:

(15)

________________________

f = tan-1 (b / a)

(16)

Using Eqns. (13) and (14) on Eqn. (11), we can express the square of the magnitude of the

impedance as

|Z|2

= R2 + ________________________

(17)

Using Eqn. (16) allows us to write:

⎛ Lω − 1

⎜

ωC

φ = tan-1 ⎜

R

⎜

⎝

⎞

⎟

⎟

⎟

⎠

(18)

Here, phi is interpreted as a phase angle. It determines how the phase of the current differs from

the phase of the input voltage. Referring to Ohm’s Law, the real part of the current can be

written as

I(t) =

V0

cos(ω t − φ )

Z

(19)

In this form, I(t) reaches a maximum when the impedance |Z| is a minimum. When this

happens, the circuit is said to resonate. Eqn. (17), is a minimum when |Z| = R, or when

________________________ = 0

11 – 6

(20)

Florida Institute of Technology

© 2020 by J. Gering

Substitute Eqns. (9) and (10) into Eqn. (20) and rewrite it below.

________________________ = 0

Now solve for the angular frequency w =

(21)

(22)

Resonance occurs only when the driving frequency, w, reaches this value. Resonance in

an RLC circuit can be described in other terms. Notice that at resonance Eqn. (6) indicates that

the impedance equals the total, normal resistance R. In other words at resonance, XC and XL are

equal and cancel one another out. In still other terms, Z = |Z| eif implies that at resonance the

phase factor f = 0. Using tan-1 (0) = 0 in Eqn. (18) gives another way of determining Eqn. (22).

During the experiment, you will notice the input voltage and VR are only in phase at resonance.

During the experiment, you measure VR and compare it with the input voltage V0. By

virtue of Ohm’s Law and R being a constant, VR is always directly proportional to the current.

Hence, resistors are called linear circuit elements. So, the voltage across the resistor is always in

phase with the current flowing in the circuit. By looking at VR on the oscilloscope, you are

equivalently looking at the behavior of the current. The only difference is a scaling factor (the

value of the resistance).

As you perform the experiment, you will find that the current (akin to VR) and the input

voltage are not always in phase with one another. When f > 0, the current reaches a maximum

later than the input voltage (the current lags the voltage), and when f < 0, the current reaches a
maximum earlier than the input voltage (the current leads the voltage).
This can be remembered by using the mnemonic: ELI the ICEman. To interpret this
mnemonic: E stands for Emf (i.e. the voltage), I stands for current, L represents the inductor, C
represents the capacitor. So for an inductor, L the Emf sits before (i.e. leads) the current I in the
word ELI. Thus voltage leads current in an inductor. Similarly, the current leads the voltage in a
capacitor (the letter I leads the letter E in the word ICE).
11 - 7
Florida Institute of Technology
© 2020 by J. Gering
Experiment 11
The R-L-C Circuit
Introduction
During this experiment, you will drive an R-L-C series circuit with a signal generator and use an
oscilloscope to measure the voltage across elements of the circuit. This circuit is unique because
the amount of A. C. current that flows through an R-L-C circuit depends upon the frequency of
the current itself. If the frequency is just right (called the resonant frequency), a very large
current will flow. This makes the R-L-C circuit tremendously important as a detector and
amplifier of weak, frequency dependent currents (like those in a radio or TV antenna). This is
the circuit that allows radios and televisions to “tune in” one signal from all the others.
Concepts
The R-L-C series circuit is the electrical analog to the mechanical, damped, driven, harmonic
oscillator. In other words, the motion of a mass forced to oscillate on a spring in the presence of
friction is exactly analogous to the variation in the current in an R-L-C circuit.
Mechanical System
(forced, damped oscillator)
Electrical System
(R-L-C circuit)
friction
velocity of mass
energy stored in spring
driving force: Fo cos wt
inertial mass: m
resistance to current flow (R)
current (I)
energy stored in the field between capacitor plates
impressed voltage: Vo cos wt of signal generator
self-inductance of inductor: L
This analogy is exact because the differential equations describing these two systems (Newton’s
Law of motion and Kirchhoff's voltage rule) are identical in form. Therefore, after solving one
equation, you need only substitute the corresponding variables of the other system to obtain the
solution for that system.
To prepare for this experiment, watch 9 minutes of Episode 38 in the Mechanical Universe
television series. This video segment contains excellent animations that provide a way of
thinking about what is going on in an RLC circuit connected to source of alternating voltage. Fast
forward to time index 9:00 and watch the next 9 minutes. A Canvas exercise for this experiment
develops the theory behind the R-L-C in wonderful detail. It is also recommended that you
review Lissajous figures in Appendix F.
https://www.youtube.com/watch?v=P-uyKrPlH9Q&list=PL8_xPU5epJddRABXqJ5h5G0dkXGtA5cZ&index=38
An R-L-C circuit resonates. Mechanical resonance occurs when the frequency of a driving force
matches the natural frequency of oscillation of the system. For example, when you push a child
on a swing at just the right frequency, the amplitude of oscillations reaches a large, maximum
11 - 1
Florida Institute of Technology
© 2020 by J. Gering
value. By the same token, when the signal generator’s driving frequency reaches the natural
resonant frequency of the R-L-C circuit, the current in the circuit reaches a maximum. This
current is measured by monitoring the voltage across the resistor. Resistors are linear circuit
elements (i.e. the current flowing through a resistor is directly proportional to the voltage across
the resistor). So, resonance occurs when the amplitude of the sinusoidal oscillations of VR reach
a maximum.
Procedure
1)
Watch an eight-minute video segment from episode 38 in the Mechanical Universe series,
then complete the worksheet. Give it to your instructor before you leave the laboratory
today. The mathematics introduced on the worksheet may be new but it is not difficult to
complete the blanks. You may work with your partner.
2)
Record the manufacturer’s value for, C (and its uncertainty). Record the manufacturer’s
values for the inductance and capacitance: L = 45 milli-Henries and C = 0.01 microFarads. Both values are ±10%. Measure and record the D.C. resistance of the resistor and
the resistance of the wire making up the inductor. Add these two resistances to obtain the
total D.C. resistance, R, in the circuit.
Scope
Figure 1. A signal generator driving a series R-L-C circuit.
3)
Set the signal generator to a sine wave, and build the circuit shown in Figure 1. Connect
the scope’s Channel 1 to the signal generator (not shown in Fig. 1). Display the voltage
across the resistor on Channel 2 (Fig. 2). Press the DUAL button to display both channels.
4)
Adjust the signal generator to 5-6 volts peak to peak amplitude and a few thousand Hertz.
Measure and record the peak-to-peak voltage of the input sine wave on Channel 1.
Remember: Do not change the signal generator's voltage knob during the experiment.
5)
Get an overall idea of how the RLC circuit responds to the alternating applied voltage.
Vary the signal’s frequency from 400 to 40,000 Hz and observe VR.
Question 1: How does the amplitude of VR change as the frequency increases?
Question 2: How does the phase of VR (relative to the input V0 ) change with frequency?
Question 3: When resonance occurs, how does VR, relate (visually) to V0 ?
6)
Use Eqn. 22 on the worksheet to calculate the angular resonant frequency of your circuit
using the best values for L and C. Convert this angular frequency to a linear frequency
11 - 2
Florida Institute of Technology
© 2020 by J. Gering
using w = 2pf. Adjust the signal generator to a frequency just below the resonant (linear)
frequency. With the signal generator’s voltage on CH 1 and VR on CH 2, press the Scope’s
X-Y button to produce a Lissajous figure. Fine tune the frequency so the Lissajous figure
is a straight line 45° to the horizontal. Measure the resonant frequency with the
oscilloscope. Estimate an error in your measurement.
7)
Using the manufacturer’s values or values given by your instructor, calculate the
propagated percent error in the theoretical resonant frequency. Use this percent error to
perform a d vs. sd comparison between theory and your measurement from the
oscilloscope. (Note: This procedure can be completed after the experiment has ended.)
8)
In an Excel spreadsheet, prepare a data table to record both f and w, as well as VR and either
VL or VC From the Scope, measure VR at the following (linear) frequencies: 400, 600, 800,
1000, 2000, 4000, 6000, 8000, 10000, 20000, 40000, 60000.
For a shorter (COVID-19) lab period, reduce the number of frequencies to
700, 1000, 3000, 5000, 8000, 10000, 15000, 30000
9)
Rewire the circuit with either the inductor or the capacitor last in the series before ground
(the choice is yours). Connect the scope across the inductor or capacitor and measure
either VL or VC at the same frequencies you measured VR.
Note: The capacitor and inductor present an additional resistance to the flow of alternating
current. These two A.C. resistances are given the symbols XC and XL. They are called the
capacitive and inductive reactance and they are measured in Ohms. You can compute
either reactance by applying Ohms Law: VL = I XL or VC = I XC, where the current, I, is
first obtained from I = VR / R.
10) Extend the spreadsheet and compute the current and either XL or XC for each frequency in
the data table.
11) Plot a graph of current versus w using a logarithmic axis for w.
12) Arrange the combined data / data analysis table and the graph so they will fit on one page.
Print this page, label the maximum on the fitted curve and record the corresponding
frequency. Question 4: According to theory, what does this frequency represent?
13) Depending on your earlier choice, plot a graph of either XL versus w or XC versus 1/w
(place the reactance on the y-axis). For each graph, fit a straight line through the points,
and display the equation of the fit on the graph. Also determine the error in the slope using
Excel. Arrange the graph and regression analysis on a separate page. Include the
appropriate units on all numbers of interest and then print the page. Question 5: What is
the physical significance of the slope?
14) Compare (using a d vs. sd test) the experimental value of either L or C with that given by
the manufacturer. Question 6: Do they agree? Question 7: What type of error does the
±10% represent? Question 8: What category does the error in the slope fall into? In your
report, fully discuss the success of the experiment in the context of the error analysis.
11 - 3
Procedures 4-7 (voltages are peak-to-peak)
Vin (V)
Frequency (Hz) V out (V)
3
400
3
4000
3
7500
0.5
2.6
3
3
40000
1.6
Theoretical Resonance Frequency
R total (Ohms)
R total Error (Ohms)
Error in L (H)
R (Ohms)
7390
Error in R (Ohms) R inductor (Ohms) Error in R L (Ohms)
112.7
0.9016
L (H)
0.05
Experimental Resonance Frequency
RF (Hz)
Res. Freq. Error (Hz)
Discrepancy (Hz)
Error in Discrep. (Hz
Success?
7140
Procedures 7-14, except #8
Frequency (Hz)
400
Omega (rad/s)
VL(V)
1(A)
X_L (Ohms)
X_C (Ohms)
V R (V)
0.70
0.80
V_C(V)
3.000
0.018
0.030
600
1.20
800
1000
0.053
0.064
2.900
2.830
2.650
2.200
1.30
2000
2.20
0.190
1.440
1.040
4000
6000
8000
10000
2.80
3.20
3.00
2.80
0.480
0.740
0.990
1.240
0.800
0.610
20000
2.40
2.100
2.700
1.60
40000
60000
0.250
0.080
0.039
1.00
2.850
Students whose last digit of their student number is ODD should analyze the voltages across the inductor.
Students whose last digit of their student number is EVEN should analyze the voltages across the capacitor.
Score
Points Possible
5
10
3
3
4
10
10
Experiment 11 Rubric
See appendix B for more details on each section of the report
5 Introduction What & why. 2-3 sentences
20 Data
Part 1
Theoretical resonance w = sqrt(1/LC) vs Experimental resonance (results from oscilloscope) & % difference
Part 2
Table of values from data on canvas
Plot 1 current vs log(w)
Plot 2 X L vs W OR X_C vs 1/w
Part 1
Sample theoretical calculations
30 Data Analysis % difference and error propagation calculations
Part 2
Sample calculations (reactance, current, etc) hint: V_L = 1* X_L and V_C = 1*X_C and I = V_RIR
Part 1 Use questions as a guide for physics of part 1 and 2
TABLE OF RESULTS
experiment theory 1 % difference
Setup
Physics
Error
40 Discussion
Results
Part 2
Setup
Physics
Error
Results
5 Conclusion
2-3 sentences summarizing experiments & results
10
6
4
4
40 Discussion
4
5
4
4
4
5
5
Totall
100
משם
CRT
COUP SOUPS
BK PRECISION
TRIG
LEVEL
INTENSITY
FOCUS
AUTO
HOLD OFF
TY
TRACE ROTATION
POWER
VERTICAL
POSITION POSITION
VAR SWEEP
ON
CH1 7
VAR —— VOLT/DIV
CH2Y
VOLT/DIV
VERT MODE
CHE
EZ
IMO 25PF
GND
CAL
הם של
2120B
DUAL TRACE OSCILLOSCOPE 30MHz
777
CRT
BK PRECISION
COUPS SOAP
AUTO
TRIG
LEVEL
INTENSITY
FOCUS
HOLD OFF
POSITION
NOAN
туу
LINE
MN
PULL CHOP
FULL
SLOPE
TV
EXT
POLL DAG
TRACE ROTATION
TRIGGER
HORIZONTAL
POWER
VERTICAL
POSITION
POSITION
VAR SWEEP
ON
TIME BASE
PULL TIG
TIME/DIV
CH1 X
VOLT/DIV
CHệ
VOLT/DIV
VAR
VERT MODE
VAR
UND
DUAL
IMPE
IMO 25
EXT TRIO
GND
CAL
CE
2120B DUAL TRACE OSCILLOSCOPE 30MHz
11
400000
POWER
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