L09-1
Name ________________________ 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
(1)
I (t )  I max sin t 
flowing through a circuit containing resistors, capacitors and/or
inductors, then the voltage across the circuit will be of the form
(2)
V  t   I max Z sin t    .
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  I max Z .
University of Virginia Physics Department
PHYS 2419, Fall 2011
(3)
Modified from P. Laws, D. Sokoloff, R. Thornton
L09-2
Lab 9 - AC Filters & Resonance
Figure 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 1
The relationship between  and t is given by
 t 
  2   or   360 f t
T 
where T is the period and f is the frequency.
(4)
For a resistor, Z R  R and there is no phase shift ( R  0 ). For
a capacitor, ZC  X C  1 C and C  90 while for an
inductor, Z L  X L   L and L  90 . In other words:
VR  I max R sin(t )
(5)
VC   I max X C cos(t )
(6)
and
VL  I max X L cos(t )
(7)
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 2. To find the
impedance and phase shift for this combination we follow the
procedure we established before.
R
V
L
C
Figure 2
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 - AC Filters & Resonance
L09-3
From Kirchhoff’s loop rule we get:
V  VR  VL  VC
(8)
Adding in Kirchhoff’s junction rule and Equations (2), (5), (6),
and (7) yields
Vmax sin t   RLC   I max  R sin t    X L  X C  cos t  
Once again using a trigonometric identity1 and equating the
coefficients of sin t  and cos t  , we get
Vmax cos RLC   I max R
and
Vmax sin RLC   I max  X L  X C 
Hence the phase shift is given by
X L  XC
(9)
R
and the impedance of this series combination of a resistor, an
inductor, and a capacitor is given by:
tan  RLC  
Z RLC  Vmax I max  R2   X L  X C 
2
(10)
The magnitudes of the voltages across the components are then
VR ,max  I max R 
R
Vmax
(11)
VL ,max  I max X L 
XL
Vmax
Z RLC
(12)
VC ,max  I max X C 
XC
Vmax
Z RLC
(13)
Z RLC
and
Explicitly considering the frequency dependence, we see that
VR ,max Vmax 
VL ,max Vmax 
and
VC ,max Vmax 
1
R
R 2    L  1 C 
2
L
R    L  1 C 
2
2
1 C
R 2    L  1 C 
2
(14)
(15)
(16)
sin(   )  sin( ) cos(  )  cos( )sin(  )
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
L09-4
Lab 9 - AC Filters & 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
(17)
F  ma  bv  kx  mx  bx  kx
Similarly, Equation (8) can be written as
1
(18)
V  Lq  Rq  q
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.
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 LC and the common
reactance of the capacitor and inductor at resonance as X LC ,
we see that, at resonance
X LC  X C LC   X L LC 
so
LC  1
LC
(19)
X LC  L C
(20)
and
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
X
(21)
VC ,max LC   VL ,max LC   LC Vmax
R
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 - AC Filters & Resonance
L09-5
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,
(22)
Q  VC ,max LC  Vmax
Hence,
Q  X LC R  L C R
(23)
Figure 3 (below) shows the voltage across a capacitor
(normalized to the driving voltage) as a function of frequency
for various values of Q .
Figure 3
In this lab you will continue your investigation of the behavior
of resistors, capacitors and inductors in the presence of AC
signals. In Investigation 1you will explore the relationship
between 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.
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
L09-6
Lab 9 - AC Filters & Resonance
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 4 (below).
[For clarity, we don’t explicitly show the voltage probes.]
R
V
L
C
Figure 4
Prediction 2-1: At very low signal frequencies (less than
10 Hz), will I R ,max and VR ,max be relatively large, intermediate
or small? Explain your reasoning.
Prediction 2-2: At very high signal frequencies (well above
3,000 Hz), will the values of I R ,max and VR ,max be relatively
large, intermediate or small? Explain your reasoning.
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 - AC Filters & Resonance
L09-7
1. On the axes below, draw qualitative graphs of X C vs.
frequency and X L vs. frequency. Clearly label each curve.
XC
and
XL
Frequency
2. On the axes above (after step 1) draw a curve that
qualitatively represents X L  X C vs. frequency. Be sure to
label it.
3. Recall that the frequency at which Z is a minimum is
called the resonant frequency, f LC and that the common
reactance of the inductor and the capacitor is X LC . On the
axes above, mark and label f LC and X LC .
Question 2-1 At f LC will the value of the peak current, I max ,
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 : __________
5. Use your measured values to calculate the resonant
frequency, the reactance of the capacitor (and the inductor)
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
L09-8
Lab 9 - AC Filters & Resonance
at resonance, and the resonant amplification factor. Show
your work. [Don’t forget the units!]
f LC : __________
X LC : __________
Q : __________
Activity 2-1: 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 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 2-1.
6. Repeat for the other frequencies in Table 2-1.
Table 2-1
f (Hz)
VR ,max (V)
VL ,max (V)
VC , max (V)
50
100
200
400
800
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 - AC Filters & Resonance
L09-9
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.]
f LC ,exp : __________
Question 2-2: 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 2-3: From these voltages, calculate Q and discuss
the agreement between this experimental value and your
calculated one.
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
L09-10
Lab 9 - AC Filters & Resonance
Question 2-4: Calculate your experimental value of X LC and
discuss the agreement between this value and your calculated
one.
Prediction 2-5: What will we get for Q if we short out the
resistor? Show your work.
9. Short out the resistor.
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 2-6:
Discuss the agreement
experimental value and your predicted one.
University of Virginia Physics Department
PHYS 2419, Fall 2011
between
this
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 - AC Filters & Resonance
L09-11
Activity 2-2: 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 4).
R
V
L
C
Figure 5
Question 2-7: 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 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
L09-12
Lab 9 - AC Filters & Resonance
Question 2-8: Which circuit element (the resistor, inductor, or
capacitor) dominates the circuit at frequencies well above the
resonant frequency? Explain.
Question 2-9a: In the circuit in Figure 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 2-9b: If your answer to Question 2-9a 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 2-10a: In the circuit in Figure 5, will the current
through the resistor always be in phase with applied voltage
from the signal generator? Explain your reasoning.
Question 2-10b: If your answer to Question 2-10a was no,
then which will lead for frequencies below the resonant
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
Lab 9 - AC Filters & Resonance
L09-13
frequency (current or voltage)?
Which will lead for
frequencies above the resonant frequency (current or voltage)?
1. Continue to use L09A2-1 RLC Filter.
2. Reconnect the circuit shown in Figure 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 2, and set the amplitude of the signal to 2 V.
Question 2-11: 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 2-12: Which leads – applied voltage, current or
neither – when the AC signal frequency is higher than the
resonant frequency? Discuss agreement with your prediction.
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton
L09-14
Lab 9 - AC Filters & Resonance
Question 2-13: At resonance, what is the phase relationship
between the current and the applied voltage?
5. Use this result to find the resonant frequency.
f LC , phase : __________
Question 2-14: Discuss how this experimental value compares
with your calculated one.
Question 2-15: 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.
University of Virginia Physics Department
PHYS 2419, Fall 2011
Modified from P. Laws, D. Sokoloff, R. Thornton