Download H2- PHYS102 - Honors Lab-2H-LRC_resonance

PHYS-102
Honors Lab 2H
Series Resonance in LRC Circuits
1.
Objective
The objectives of this experiment are:
•
•
2.
To demonstrate the phenomenon of resonance in an LRC circuit
To measure the frequency response of an RLC circuit, measure its resonance
frequency and the associated quality factor Q.
Theory
Resistance, capacitance and inductance are electrical properties of circuit elements
that can influence the overall behavior of a circuit. One can express them in terms of the
potential difference or voltage V measured across the element and the current I flowing
through it. The following equations define the three properties, resistance R, capacitance
C and inductance L.
Vc = Q/C
Æ
I = C dV/dt
(1a)
VL = L dI/dt
(1b)
VR = IR
(1c)
where Q is the electrical charge so that current I = dQ/dt. Alternatively, capacitance C is
related to how much energy can be stored in the electric field between the charged plates
of a capacitor while inductance L is related to how much energy can be stored in the
magnetic field of a current carrying coil. Capacitance and inductance, which depend on
the geometry of the capacitor and inductor elements, have units of Farad and Henry,
repectively. In this experiment, we will investigate the behavior of a circuit (See Fig. 1)
consisting of a series connection of a resistor, capacitor and inductor and driven by a
time-varying or sinusoidal (AC) current with a frequency f = ω/2π Hz.
Fig. 1. The Series LRC circuit.
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The capacitor, labeled C in Fig. 1, consists of two overlapping plates of metal separated
by an insulating dielectric material while the inductor, labeled L is usually manufactured
by winding a wire into a coil.
Like the resistor R, the capacitor C and inductor L can pass AC currents through.
However, like the resistor that presents a resistance to current flow, the capacitor and
inductor individually affect the current passing through them through a capacitive
reactance Xc and inductive reactance XL, respectively. The effect of the reactance is to
change the time or phase relationships between the current I(t) passing through the
device and the voltage V(t) measured across it. We can thus compare the effect of these
three circuit elements on the voltage across and current flowing through them at time t:
Resistor:
The voltage V(t) across a resistor and current I(t) flowing through it are
said to be in phase with each other.
Capacitor:
The voltage V(t) and current I(t) are 90º out of phase. The magnitude of
the current I(t) is proportional to the rate of change dV/dt of voltage across
the capacitor. The voltage lags current by 90º. This means that if current
is maximum at one instant, voltage won’t be maximum until a quarter of a
period later.
Inductor:
The voltage and current are 90º out of phase, except that the voltage now
leads the current by 90º. The magnitude of the voltage V(t) across the
capacitor is proportional to the rate of change dI/dt of current through it.
The magnitudes of the reactances associated with the capacitor and inductor, driven at an
applied frequency f are:
Xc = 1/(2πfC) = 1/ωC
and
XL = 2πf/L
= ωL
(2)
where the units of reactances are ohms. The instantaneous voltages across the three
elements can then be written as:
Vc = IXc
VL = I XL
VR = IR
V = IZ
Vc,rms = IrmsXc
VL,rms = Irms XL
VR,rms = IrmsR Vrms = IrmsZ
(3a)
(3b)
where Z = the effective impedance (measured in ohms) of the whole circuit, representing
the relationship between the driving voltage and the resulting current for the whole
circuit. The root-mean-square (rms) voltages and currents are particularly important
because these are what multimeters detect when switched to AC mode. For sinusoidal
driving voltages and currents, root-mean-square values are smaller than the amplitude by
a factor of √2. For instance, Irms = I/√2 and Vrms = V/√2.
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One can conveniently display the instantaneous voltages across the three elements
of the LRC circuit on a voltage-phase diagram, where on the x-axis is plotted phase and
the y-axis is plotted the instantaneous magnitude of a voltage. A snapshot picture of such
a “phasor” diagram is shown in Fig. 2 where VR, VL and VC clearly do not lie on the
same axis because both VL leads VR and VC lags VR by 90º. Hence, the net voltage drop
across all three circuit elements is not the algebraic sum of the individual voltage drops
but the vector or phasor sum of these.
VL
VR
VC - VL
V
VC
Fig. 2. The Voltage-Phase Diagram for the Series RLC Circuit at an instant of time.
The amplitude of the net voltage V across the circuit is thus
V = [VR2 + (VL – VC)2]1/2
(4)
One can express the voltages in (4) in terms of the current I, reactances XL and XC, and
impedance Z in (3) so that:
V = IZ = [I2R2 + (IXL – IXC)2]1/2
(5)
We obtain the total impedance of the RLC circuit to be: Z = V/I = [R2 + (XL – XC)2]1/2
Solving for current, which can be thought of as the response of the circuit to a driving
voltage, one obtains
I = V/Z = V/[R2 + (XL – XC)2]1/2 = V/[R2 + (ωL – 1/(ωC))2]1/2
(6)
One can observe that the amplitude of the current has a resonance at the frequency where
(ωL – 1/(ωC))2 = 0. This happens when the capacitive and inductive reactances exactly
cancel each other out. This resonance occurs at the so-called resonance frequency:
ωo = 1/√(LC)
or equivalently
fo = 1/[2π√(LC)]
(7)
During resonance, energy is maximally being pumped into the system by a driving signal
(applied voltage) oscillating at the natural frequency of the system (LRC series circuit).
Some of the energy is being dissipated through Joule heating in the resistor R while the
rest is oscillating between the electric field E confined between the capacitor plates and
the magnetic field B confined in the coils of the inductor.
One can calculate the average power dissipated in the resistor R using Equation (6):
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Pave = Irms2R = (Vrms/Z)2R = Vrms2R/[R2 + (ωL – 1/(ωC))2]
(8a)
where Vrms is the rms voltage output of the signal generator. In this experiment, since
the coil is not a perfect inductor and has a DC resistance Ro, then the “R” in the
denominator of Equation (8) must be adjusted to a value of (R + Ro):
Pave = Irms2R = (Vrms/Z)2R = Vrms2R/[(R + Ro)2 + (ωL – 1/(ωC))2]
(8b)
In terms of the resonance frequency ωo = 1/√(LC) :
Pave = Vrms2Rω2/[(R + Ro)2ω2 + L2(ω2 – ωo2)2]
(8c)
When Pave is plotted as a function of angular frequency ω, this results in a peak
centered at the resonance frequency (ω = ωo). This makes sense since at resonance, all the
energy is dissipated in the resistive component of the circuit since the reactances cancel.
In general, one can also define a property of resonances called “quality factor” or
“Q” of the resonance as the ratio of energy dissipated to the energy stored in the system.
The larger the Q, the sharper the resonance peak. Alternatively, Q is the ratio of the
average power’s full-width-at-half-maximum Δω to the resonance frequency ωo.
Theoretically, the quality factor can be found to be :
Q = ωo/Δω = ωoL/(R+Ro)
(9)
3. Experimental Details
Overview of Experiment and Apparatus
In this experiment, the amplitude of the current through a series RLC circuit is
measured as a function of frequency. The inductance L = 840 mH is provided by a large
solenoidal coil of copper wire with an intrinsic electrical resistance Ro = 60-65 Ω.
Connected in series with the coil is another resistor R and a 1μF capacitor. The whole
RLC circuit is driven by a sinusoidal voltage with a fixed output voltage of about 10 v
peak-to-peak and whose frequency is varied. The voltage across the resistor is measured
and plotted versus driving frequency to experimentally determine the resonance
frequency. The dissipated power is calculated to determine the quality factor Q. To
determine the effect of increasing resistance, the experiment is repeated for a larger
resistance.
Experimental Procedure
1.
Using the digital multimeter (DMM) provided, measure and record the DC
resistances of the resistors and coil provided. The two leads must be plugged into the
DMM’s terminals labeled Ω and COM. If the DMM provided is equipped with
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capacitance meter, measure the capacitance of the capacitor provided. Record these in the
Data Table.
2.
Connect the inductor, resistor, capacitor and function generator provided
according to Fig. 1. Connect the voltmeter (in AC voltmeter mode) to measure the RMS
voltage VRMS across the resistor. For a sinusoidal driving voltage with amplitude VR,
VRMS = VR/√2.
If an oscilloscope is available, use it instead of the voltmeter to measure the
amplitude of the sinusoidal voltage across the resistor. The oscilloscope’s display shows
the voltage on the vertical axis and time on the horizontal axis. The scale on both vertical
and horizontal axes can be adjusted using the VOLTS/DIV and SEC/DIV knobs. Each
DIV refers to a large square on the display. Use channel 1 in your measurements. Using
the CH1 Menu button, select the following settings:
Coupling = AC
Volts/Div = Coarse
Probe = 1X
3.
Before driving the LRC circuit with a sinusoidal voltage, turn down the voltage
and frequency of the function generator provided. Set the waveform to produce a sine
wave at very low frequency, e.g. 100 Hz. Set the voltage output to be 10 V, peak-to-peak.
Before connecting the LRC circuit to the function generator, ask your TA to check
your circuit.
4.
Connect the circuit to the function generator. Using the DMM, measure the rms
voltage across the resistor R as a function of frequency. Vary the frequency until the
VR,rms is maximum. Alternatively, you can also use the oscilloscope to measure either the
voltage amplitude or the VRMS. The frequency at which this occurs is the resonance
frequency fo of your LRC circuit. By varying the frequency between ¼ fo and 4fo, make
at least 15 measurements. Record these data in the provided Data Table. Use the back of
the sheet if you want to add more measurements.
5. Now, replace the 47-Ω resistor with a larger resistor provided. Repeat the same
measurements in Steps 3 and 4 and record the results.
KEEP EVERYTHING ABOVE THIS LINE. SUBMIT EVERYTHING BELOW THIS LINE.
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Physics 102 - Honors Lab 2H
Resonance in RLC Circuit
Name:_______________________ Sec./Group__________ Date:_____________
4. Experimental Data
Coil resistance Ro:
Capacitance:
A. R =
Ω
Frequency (Hz)
_______ Ω
_______ F
FWHM = _______________
B. R =
Ω
Frequency (Hz)
FWHM = _______________
Coil Inductance :
Quality Factor Q:
VR,rms (measured, volts)
_________ H
_________
PR,ave (mW)
Measured Quality Factor Q = _________________
VR,rms (measured, volts)
PR,ave (mW)
Measured Quality Factor Q = _________________
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5. Analysis of Data
5.1 If you were able to measure the capacitance using a DMM, calculate the percentage
difference between the measured capacitance and the capacitance value specified on the
capacitor’s packaging. Usually, the value printed on the packaging have commercial
tolerances of up to 20%.
5.2 Using Excel or any other spreadsheet software, calculate and tabulate on the Data
Sheet provided the average power dissipated in the resistor R. The measured average
power is obtained using Pave = VR,rms2/R. This R is the value of the resistor provided.
Plot Pave vs. ω (or alternatively, Pave vs. f) with properly-labeled vertical and horizontal
axes. Print and submit graphs for the case of large and small resistor R.
5.3 For both resistances, determine the measured value of the resonance frequency fo. Are
these different or the same ? Calculate the percentage difference between the measured
resonance frequencies for the two resistors relative to the theoretical values given by ωo =
1/√(LC).
5.4 Based on these graphs, determine the full-width-at-half-maximum (FWHM). Use this
to determine the measured quality factor Q for both LRC circuits. Compare the two
values and account for the difference.
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5.5 Use Equation (9) to calculate the theoretical Q’s for both LRC circuits. Based on
these, determine the percentage difference between measured and theoretical values of
Q.
5.4 What is the uncertainty in your measurement of the resonance frequency ? Make
reasonable estimates of various uncertainties in your measurements of experimental
quantities.
5.5 Estimate the maximum uncertainty introduced by these values and compare this with
your results.
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Physics 102 - Honors Lab 2H - Prelab
Resonance in RLC Series Circuit
Name:_______________________ Sec./Group__________
Date:_____________
1. Imagine that you are riding a swing in a playground, oscillating between two kinds of
energies: potential and kinetic and possessing an amplitude, frequency and a phase. In
terms of frequency and phase, describe how a friend should push on your swing so as to
maximize transfer of energy from him to you.
A circuit is made up of a capacitor and a dense coil of wire called an inductor, connected in series with a
time-varying voltage source. Such a system is analogous to the swing system, except that energy can be
stored in either the electric field of the charged capacitor or the magnetic field of the current-carrying
coil. If a resistance is included, energy can be dissipated through resistive heating in the resistor. This
circuit is called a series RLC circuit.
2. A series RLC circuit is driven by a sinusoidal current I(t) = I cos ωt. The amplitude of
the voltage oscillations across the resistor is IR The same idea can be applied to the
capacitance and inductance if we define, a capacitive reactance XC = (ωC)-1 and inductive
reactance XL = ωL so that the amplitude of the oscillating voltage drops across the
capacitor and inductor are IXC and IXL, respectively.
a. Calculate the amplitude (in units of volts) of the oscillating voltage drop across a
capacitor with C = 1μF and an inductor with L = 0.84 Henry when the frequency f = ω/2π
= 60 Hz. Assume that the amplitude I of the current is 0.1Ampere.
b. For the given capacitor and inductor above, at what frequency f = ω/2π are the
capacitive reactance XC and inductive reactance XL identical ?
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JUST FOR THE FUN OF IT: Effect of a Ferromagnet on LRC
resonance
Predict what happens to the value of
(a) the resonance frequency fo and
(b) the Q-factor
if you insert a ferromagnet such as an iron rod though the inductor coil of an LRC circuit.
Verify your predictions by inserting the provided iron rod through the inductor coil. How
does a permanent magnet change the inductance of the coil: Did it increase it or decrease
it ?
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