Lab 9 AC Filters and Resonance
Name
L9-1
Date
Partners
Lab 9 - AC Filters and Resonance
OBJECTIVES
• To understand the design of capacitive and inductive filters.
• To understand resonance in circuits driven by AC signals.
OVERVIEW
In a previous lab, you explored the relationship between impedance (the AC equivalent of resistance) and frequency for a resistor, capacitor, and inductor. These relationships are very important to people designing electronic equipment. You can predict many of the basic characteristics
of simple AC circuits based on what you have learned in previous labs.
Recall that we said that it can be shown that any periodic signal can be represented as a sum of
weighted sines and cosines (known as a Fourier series). It can also be shown that the response
of a circuit containing resistors, capacitors, and inductors (an “RLC” circuit) to such a signal
is simply the sum of the responses of the circuit to each sine and cosine term with the same
weights.
Recall further that if there is a current of the form
I(t) = Imax sin (ωt)
(9.1)
flowing through a circuit containing resistors, capacitors and/or inductors, then the voltage
across the circuit will be of the form
V (t) = Imax Z sin (ωt + ϕ) .
(9.2)
Z is called the impedance (and has units of resistance, Ohms) and ϕ is called the phase shift (and
has units of angle, radians). The peak voltage will be given by
Vmax = Imax Z.
University of Virginia Physics Department
PHYS 1429, Spring 2013
(9.3)
Modified from P. Laws, D. Sokoloff, R. Thornton
L9-2
Lab 9 AC Filters and Resonance
Figure 9.1 shows the relationship between V and I for an example phase shift of +20◦ . We
say that V leads I in the sense that the voltage rises through zero a time ∆t before the current.
When the voltage rises through zero after the current, we say that it lags the current.
Figure 9.1:
The relationship between ϕ and ∆t is given by
!
∆t
= 360◦ × f ∆t
T
where T is the period and f is the frequency.
ϕ = 2π
(9.4)
For a resistor, ZR = R and there is no phase shift (ϕR = 0). For a capacitor, ZC = XC = 1/ωC and
ϕC = −90◦ while for an inductor, ZL = XL = ωL and ϕL = +90◦ . In other words:
VR = Imax R sin(ωt)
(9.5)
VC = −Imax XC cos(ωt)
(9.6)
VL = Imax XL cos(ωt)
(9.7)
and
XC is called the capacitive reactance and XL is called the inductive reactance.
Let us now consider a series combination of a resistor, a capacitor and an inductor shown in
Figure 9.2. To find the impedance and phase shift for this combination we follow the procedure
we established before.
From Kirchhoff’s loop rule we get:
V = VR + VL + VC
(9.8)
Adding in Kirchhoff’s junction rule and Equations (9.5), (9.6), and (9.7) yields
Vmax sin (ωt + ϕRLC ) = Imax [R sin (ωt) + (XL − XC ) cos (ωt)]
Once again using a trigonometric identity1 and equating the coefficients of sin (ωt) and cos (ωt),
we get
1
sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 AC Filters and Resonance
L9-3
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Figure 9.2:
Vmax cos (ϕRLC ) = Imax R
and
Vmax sin (ϕRLC ) = Imax (XL − XC )
Hence the phase shift is given by
XL − XC
(9.9)
R
and the impedance of this series combination of a resistor, an inductor, and a capacitor is given
by:
tan (ϕRLC ) =
ZRLC = Vmax /Imax =
q
R2 + (XL − XC )2
(9.10)
The magnitudes of the voltages across the components are then
VR,max = Imax R =
VL,max = Imax XL =
R
Vmax
(9.11)
XL
Vmax
ZRLC
(9.12)
ZRLC
and
XC
Vmax
ZRLC
Explicitly considering the frequency dependence, we see that
VC,max = Imax XC =
(9.13)
R
VR,max Vmax = p
2
R + (ωL − 1/ωC)2
(9.14)
ωL
VL,max Vmax = p
2
R + (ωL − 1/ωC)2
(9.15)
and
VC,max Vmax = p
University of Virginia Physics Department
PHYS 1429, Spring 2013
1/ωC
R2 + (ωL − 1/ωC)2
(9.16)
Modified from P. Laws, D. Sokoloff, R. Thornton
L9-4
Lab 9 AC Filters and Resonance
This system has a lot in common with the forced mechanical oscillator that we studied in the
first semester. Recall that the equation of motion was
F = ma + bv + kx = m ẍ + b ẋ + kx
(9.17)
Similarly, Equation (9.8) can be written as
1
q
(9.18)
C
We see that charge separation plays the role of displacement, current the role of velocity, inductance the role of mass (inertia), capacitance (its inverse, actually) the role of the spring constant,
and resistance the role of friction. The driving voltage plays the role of the external force.
V = Lq̈ + Rq̇ +
As we saw in the mechanical case, this electrical system displays the property of resonance.
It is clear that when the capacitive and inductive reactances are equal, the impedance is at its
minimum value, R. Hence, the current is at a maximum and there is no phase shift between the
current and the driving voltage.
Denoting the resonant frequency as ωLC and the common reactance of the capacitor and inductor
at resonance as XLC , we see that, at resonance
XLC ≡ XC (ωLC ) = XL (ωLC )
so
.√
ωLC = 1 LC
(9.19)
and
XLC =
p
L/C
(9.20)
At resonance the magnitude of the voltage across the capacitor is the same as that across the
inductor (they are still 180◦ out of phase with each other and ±90◦ out of phase with the voltage
across the resistor) and is given by
XLC
Vmax
(9.21)
R
In analogy with the mechanical case, we call the ratio of the amplitude of the voltage across the
capacitor (which is proportional to q, our “displacement”) at resonance to the driving amplitude
the resonant amplification, which we denote as Q,
VC,max (ωLC ) = VL,max (ωLC ) =
Q ≡ VC,max (ωLC ) Vmax
(9.22)
.
L/C R
(9.23)
Hence,
Q = XLC /R =
p
Figure 9.3 shows the voltage across a capacitor (normalized to the driving voltage) as a function
of frequency for various values of Q.
In this lab you will continue your investigation of the behavior of resistors, capacitors and inductors in the presence of AC signals. In Investigation 1, you will explore the relationship between
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 AC Filters and Resonance
L9-5
Figure 9.3:
peak current and peak voltage for a series circuit composed of a resistor, inductor, and capacitor.
You will also explore the phase difference between the current and the voltage. This circuit is an
example of a “resonant circuit”. The phenomenon of resonance is a central concept underlying
the tuning of a radio or television to a particular frequency.
INVESTIGATION 1: The Series RLC Resonant (Tuner) Circuit
In this investigation, you will use your knowledge of the behavior of resistors, capacitors and
inductors in circuits driven by various AC signal frequencies to predict and then observe the
behavior of a circuit with a resistor, capacitor, and inductor connected in series.
The RLC series circuit you will study in this investigation exhibits a “resonance” behavior that
is useful for many familiar applications such a tuner in a radio receiver.
You will need the following materials:
• Voltage probes
• Multimeter
• 510 Ω resistor
• test leads
• 800 mH inductor
• 820 nF capacitor
Consider the series RLC circuit shown in Figure 9.4 (below). [For clarity, we don’t explicitly
show the voltage probes.]
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
L9-6
Lab 9 AC Filters and Resonance
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Figure 9.4:
Prediction 1-1: At very low signal frequencies (less than 10 Hz), will the maximum
values of current through the resistor and voltage across the resistor be relatively large,
intermediate or small? Explain your reasoning.
Prediction 1-2: At very high signal frequencies (well above 3,000 Hz), will the values
of IR,max and VR,max be relatively large, intermediate or small? Explain your reasoning.
1. On the axes below, draw qualitative graphs of XC vs. frequency and XL vs. frequency.
Clearly label each curve.
XC
an d
XL
F req u en c y
2. On the axes above (after step 1) draw a curve that qualitatively represents XL − XC vs.
frequency. Be sure to label it.
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 AC Filters and Resonance
L9-7
3. Recall that the frequency at which Z is a minimum is called the resonant frequency, fLC
and that the common reactance of the inductor and the capacitor is XLC . On the axes
above, mark and label fLC and XLC .
Question 1-1: At fLC will the value of the peak current, Imax , in the circuit be a maximum
or minimum? What about the value of the peak voltage, VR,max , across the resistor?
Explain.
4. Measure the 510 Ω resistor (you have already measured the inductor and the capacitor):
R:
L:
C:
5. Use your measured values to calculate the resonant frequency, the reactance of the capacitor (and the inductor) at resonance, and the resonant amplification factor. Show your
work. [Don’t forget the units!]
fLC :
XLC :
Q:
Activity 1-1: The Resonant Frequency of a Series RLC Circuit.
1. Open the experiment file L09A2-1 RLC Filter.
2. Connect the circuit with resistor, capacitor, inductor and signal generator shown in Figure 9.4. [Use the internal generator.]
3. Adjust the generator to make a 50 Hz signal with amplitude of 2 V.
4. Connect voltage probe VPA across the resistor, VPB across the inductor, and VPC across
the capacitor.
5. Use the Smart Tool to determine the peak voltages (VR,max ,VL,max , and VC,max ). Enter the
data in the first row of Table 9.1.
6. Repeat for the other frequencies in Table 9.1.
7. Measure the resonant frequency of the circuit to within a few Hz. To do this, slowly adjust
the frequency of the signal generator until the peak voltage across the resistor is maximal.
[Use the results from Table 2-1 to help you locate the resonant frequency.]
fLC,exp :
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
L9-8
Lab 9 AC Filters and Resonance
f (Hz)
50
VR,max (V)
VL,max (V)
VC,max (V)
100
200
400
800
Table 9.1:
Question 1-2: Discuss the agreement between this experimental value for the resonant
frequency and your calculated one.
8. Use the Smart Tool to determine the peak voltages at resonance.
Vmax :
VR,max :
VL,max :
VC,max :
Question 1-3: From these voltages, calculate Q and discuss the agreement between this
experimental value and your calculated one.
Question 1-4: Calculate your experimental value of XLC and discuss the agreement
between this value and your calculated one.
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 AC Filters and Resonance
L9-9
Prediction 1-3: What will we get for Q if we short out the resistor? Show your work.
9. Short out the resistor. Make sure that Data Studio oscilloscope does not trigger from the
VR signal.
10. Measure Q. [You may have to lower the signal voltage to 0.5 V.] Show your work. Explicitly indicate what you had to measure.
Q:
Question 1-5: Discuss the agreement between this experimental value and your predicted
one.To what remaining resistance in the circuit does the measured value of Q correspond?
Calculate the value of the Rremaining in the circuit and discuss where it is coming from.
Activity 1-2: Phase in an RLC Circuit
In previous labs, you investigated the phase relationship between the current and voltage in an
AC circuit composed of a signal generator connected to one of the following circuit elements:
a resistor, capacitor, or an inductor. You found that the current and voltage are in phase when
the element connected to the signal generator is a resistor, the current leads the voltage with a
capacitor, and the current lags the voltage with an inductor.
You also discovered that the reactances of capacitors and inductors change in predictable ways
as the frequency of the signal changes, while the resistance of a resistor is constant – independent of the signal frequency. When considering relatively high or low signal frequencies in a
simple RLC circuit, the circuit element (either capacitor or inductor) with the highest reactance
is said to “dominate” because this element determines whether the current lags or leads the voltage. At resonance, the reactances of capacitor and inductor cancel, and do not contribute to the
impedance of the circuit. The resistor then is said to dominate the circuit.
In this activity, you will explore the phase relationship between the applied voltage (signal generator voltage) and current in an RLC circuit.
Consider again our RLC circuit (it is the same as Figure 9.4).
Question 1-6: Which circuit element (the resistor, inductor, or capacitor) dominates the
circuit at frequencies well below the resonant frequency? Explain.
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
L9-10
Lab 9 AC Filters and Resonance
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Figure 9.5:
Question 1-7: Which circuit element (the resistor, inductor, or capacitor) dominates the
circuit at frequencies well above the resonant frequency? Explain.
Question 1-8: In the circuit in Figure 9.5, will the current through the resistor always be
in phase with the voltage across the resistor, regardless of the frequency? Explain your
reasoning.
Question 1-9: If your answer to Question 1-8 was no, then which will lead for frequencies
below the resonant frequency (current or voltage)? Which will lead for frequencies above
the resonant frequency (current or voltage)?
Question 1-10: In the circuit in Figure 9.5, will the current through the resistor always
be in phase with applied voltage from the signal generator? Explain your reasoning.
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 AC Filters and Resonance
L9-11
Question 1-11: If your answer to Question 1-10 was no, then which will lead for
frequencies below the resonant frequency (current or voltage)?
frequencies above the resonant frequency (current or voltage)?
Which will lead for
1. Continue to use L09A2-1 RLC Filter.
2. Reconnect the circuit shown in Figure 9.5. Connect voltage probe VPA across the resistor,
VPB across the inductor, and VPC across the capacitor.
3. Start the scope and set the signal generator to a frequency 20 Hz below the resonant
frequency you measured in Investigation 1, and set the amplitude of the signal to 2 V.
Question 1-12: Which leads – applied voltage, current or neither – when the AC signal
frequency is lower than the resonant frequency? Discuss agreement with your prediction.
4. Set the signal generator to a frequency 20 Hz above the resonant frequency.
Question 1-13: Which leads – applied voltage, current or neither – when the AC signal
frequency is higher than the resonant frequency? Discuss agreement with your prediction.
Question 1-14: At resonance, what is the phase relationship between the current and the
applied voltage?
5. Use this result to find the resonant frequency.
fLC,phase :
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton
L9-12
Lab 9 AC Filters and Resonance
Question 1-15: Discuss how this experimental value compares with your calculated one.
Question 1-16: How does this experimental value for the resonant frequency compare
with the one you determined by looking at the amplitude? Comment on the relative
“sensitivities” of the two techniques.
Please clean up your lab area
University of Virginia Physics Department
PHYS 1429, Spring 2013
Modified from P. Laws, D. Sokoloff, R. Thornton