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

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Albany Medical College The RLC Circuit Lab Report Experiment 11

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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 Purchase answer to see full attachment [/vc_column_text][vc_message message_box_color="success" icon_fontawesome="fas fa-check-circle"]This question was handled by a Studyhelp247 Physics tutor and the student left a positive review. 19+ custom answers have been given to students for this question in [month] [year] alone. 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