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Name_____________________________ Date_________________ Day/Time of Lab______________
Partner(s)___________________________ _____________________________ Lab TA_____________
LAB 5:
AC CIRCUITS
Objectives
 To understand the behavior of resistors, capacitors, and inductors in AC Circuits
 To understand the physical basis of frequency-dependent impedance
 To observe and understand the phase differences between current and voltage for resistors,
capacitors, and inductors
 To observe oscillations in an RCL circuit
Introduction
In this lab you will explore the behavior of standard circuit elements in cases where the input voltage
varies and the time structure of the voltage becomes important.
In AC circuits, the behavior of circuit elements like inductors and capacitors is in some ways similar
to the behavior of resistors. With a resistor in a DC circuit, the resistance determines how much
current will flow through it when a voltage is applied to it, according to Ohm's Law.
In AC circuits there is a quantity called impedance associated with each circuit element that acts like
the resistance in Ohm's Law. In fact, the peak voltage Vmax is related to the peak current, Imax by the
relationship Imax = Vmax/Z, where Z is the impedance.
For a resistor, the impedance is just the same as the resistance. As you will see, the capacitor and
inductor have more complicated behavior which is related to their ability to store and release
electrical energy.
In this lab, some of the results are based on ratios of measured voltages. In this experiment, all
voltage measurements with the oscilloscope are peak-to-peak values, and can be left in units of cm
of vertical deflection on the oscilloscope. Since amplitudes, rms values and peak-to-peak values are
all proportional to one another, the same vector addition rules apply to all. However, if you want to
compare voltages on two oscilloscope channels in terms of vertical deflection, their vertical
deflection scales must be the same. Note that phase difference measurements can also be made
by looking at the difference in the times where the two signals cross the axis. The time difference δt
= t2-t1 can be related to the phase difference between the two signals (δφ) by dividing by the period
(T) (which gives the difference in terms of a fraction of one oscillation) and multiplying by 2π:
δt/T = δφ/2π.
A schematic diagram of the components used in the lab and the connections between them is
shown below. The series circuit consists of the function generator and two or three of the elements
R, L, C, labeled in the diagram as A, B, or C. Either channel of the oscilloscope can be used to
measure the voltage across C, C + B, or C + B + A (= Vo, where Vo is the output voltage of the
function generator). The ground connections of the function generator and the oscilloscope must
always be connected together since this connection already exists through the grounding
connections in the power cord (round prong of plug) of each unit. In order to measure the voltage
across either B or A alone, that element must be placed in the position of element C, so that one of
its terminals is connected to ground. These permutations in the positions of the circuit elements are
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
2
required throughout the experiment in order to make the appropriate measurements and to
maintain the ground connections as mentioned above.
Overloading of the function generator may cause distortion of the sinusoidal voltage waveform. If
this occurs, turn down the amplitude control on the function generator until the proper waveform is
restored while increasing the sensitivity of the oscilloscope to obtain a reasonable height on the
scope.
INVESTIGATION 1: IMPEDANCE OF CIRCUIT ELEMENTS
You will need the following materials for this investigation:




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Oscilloscope
Function Generator
Lucite boards with resistors, capacitors, and inductors
scope probe
plenty of leads to build circuits
The values of the L, C, and R components are listed on the frames on which the components are
mounted; as well, each resistor is color coded with its resistance. The internal resistance, RL, of
each inductor is labeled with its value; the nominal internal resistance of the function generator, RG,
is 600 ohms.
Activity 1-1: Impedance of a Resistor
1. Assemble the simple circuit shown below. In addition to the appropriate set-up for the first
activity, it is also a good circuit to build just to get a decent signal on the oscilloscope and to
check that everything is working properly.
Function Generator
Vin
Ground
a
R=2.8kΩ
600 Ω
Vout
c
red
a
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
black
b
c
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
3
2. Set the frequency generator to create a 50 kHz sine wave, and display both the input signal (Vin,
measured between “a” and “c”) and the output signal (Vout measured between “b” and “c”) on the
scope. Set the oscilloscope trigger controls to "Internal", "Auto", and "+ slope". Adjust the time
and voltage per division so you can see two or three full periods of the oscillation. Make sure
both scope channels are set to the same voltage setting.
3. On the axes below, sketch the input and output voltages as measured by the scope. Mark the
appropriate units for time and voltage per division.
Volts
Volts
time
Input Voltage
time
Output Voltage
Question 1-1: Is there any observable difference in phase (time offset) between the two signals? Is
this what you would have expected? Why or why not?
4. Now, vary the signal frequency over a wide range using the function generator. Observe the ratio
of the amplitudes Vin and Vout as you do this. Note any differences as the frequency varies.
Question 1-2: Did you see any variation in the amplitude Vout as you changed the frequency? Is there
any evidence that the impedance of a resistor is frequency-dependent? Was this what you would
have expected? Why?
Comment: What you should have seen is that the voltage across the resistor is exactly equal and
equal in phase with the input voltage. You also should not have seen any variation of this
relationship with input frequency. This implies that VR = IR ZR = IR R is valid at any frequency. What
this means for the rest of this lab is that by observing the voltage across the resistor in the circuit,
we can measure the phase of the current flowing in the circuit, since they are just related by a real,
multiplicative constant.
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
4
Activity 1-2: Impedance of a Capacitor
5. Assemble the RC circuit shown below.
Function
Generator
a
Function Generator
Ground
600 Ω
C=2nF
red
b
black
R=2.8kΩ
c
a
b
c
Prediction 1-1: Suppose that you replaced the signal generator with a battery and a switch. The
capacitor is initially uncharged, and therefore the voltage across the capacitor is zero. If you close
the switch, which quantity reaches its maximum value first: current in the circuit or voltage across
the capacitor? As charge builds up on the capacitor, and the voltage across the capacitor increases,
what happens to the current in the circuit? Explain.
Prediction 1-2: The actual AC voltage applied to the circuit by the signal generator is shown on the
axes that follow. Use your answers from the above questions to sketch with dashed lines your
prediction for the current as a function of time.
Input Voltage
Volts
time
Current
Voltage
time
Circuit Current, Capacitor Voltage
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
5
6. To observe VRC (= Vin) connect the oscilloscope probe at point "a" (across R and C) and the probe
ground connected at point “c”, where the black ground lead from the function generator is
connected.
7. Adjust the sec/div (Time Base) and the horizontal position knob, until one complete cycle covers
the scope display horizontally (as much as possible; see figure below). To measure α, the phase
difference between VR (which, remember, is proportional to the current in the circuit) and VRC,
let probe #1 remain connected as before across R and C, and connect probe #2 to point "b".
Why is it important that both traces are centered vertically as shown in the figure below on the
left (they need not have the same amplitude)? The phase difference α is then given by the
distance, x in cm, between the x-intercepts of the two traces according to:
α = ( x / D ) 360°
where D is the distance in cm of one cycle. (The relative positions in the diagram are for
illustration only.)
VR
VRC
x
D
Question 1-3: What phase difference do you measure? Go back to Prediction 1-2 and draw the
observed VR with a solid line on your “Current” graph. Does VR “lead” VRC, or does VRC reach a
maximum before VR? Do these answers match your predictions? Explain any differences you see.
8. Now, switch the positions of the capacitor and resistor in the circuit so that you are measuring
the voltage across the capacitor. Draw in and clearly label the voltage across the capacitor VC
with respect to the input voltage (VRC) on the axes under Prediction 1-2, above. Measure the
phase difference between VC and VRC as you did before with VR and VRC.
Question 1-4: What phase difference do you measure? What is the total phase difference between
VC and VR (= the current in the circuit)? Is this what you would have expected? Why?
9. While you are measuring the voltage on the capacitor, you can investigate the impedance of the
capacitor, ZC. Remember that VC = ICZC. Change the frequency of the generator and observe
what happens to VC as a function of frequency.
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
6
Question 1-5: Does the capacitor’s impedance increase or decrease with frequency? Can you
explain this using what you know about capacitors?
Comment: The impedance of a capacitor, XC, is called the capacitive reactance.
Activity 1-3: Impedance of an Inductor
1. Assemble the RL circuit shown below.
Function
Generator
a
Function Generator
Ground
600 Ω
L=25mH
red
b
black
R=500Ω
c
a
b
c
Prediction 1-3: Suppose that you replaced the signal generator with a battery and a switch. The
inductor initially has no current through it. If you close the switch, which quantity reaches its
maximum value first: current in the circuit or voltage across the inductor? (Hint: recall that when the
current through an inductor is changing, the induced voltage across the inductor opposes the
change.) As the current builds up in the circuit, what happens to the induced voltage across the
inductor? Explain.
Prediction 1-4: At the instant the current reaches its maximum value for this circuit, what do you
predict the magnitude of the induced voltage will be--maximum, minimum or zero? Why?
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
7
Prediction 1-5: The actual AC voltage applied to the circuit by the signal generator is shown on the
axes that follow. Use your answers from the above questions to sketch with dashed lines your
prediction for the current as a function of time on the following graph.
Input Voltage
Volts
time
Current
Voltage
time
Circuit Current, Inductor Voltage
2. To observe VRL (= Vin) connect the oscilloscope probe at point "a" (across L and C) and the probe
ground connected at point “c”, where the black ground lead from the function generator is
connected.
3. Adjust the sec/div (Time Base) and the horizontal position knob, until one complete cycle covers
the scope display horizontally (as much as possible; see figure below). As before, measure the
phase difference between VR (which, remember, is proportional to the current in the circuit) and
VRL, let probe #1 remain connected as before across R and L, and connect probe #2 to point "b".
Question 1-6: What phase difference do you measure? Go back to Prediction 1-5 and draw the
observed VR with a solid line on your “Current” graph. Does VR “lead” VRL, or does VRL reach a
maximum before VR? Do these answers match your predictions? Explain any differences you see.
4. Now, switch the positions of the inductor and resistor in the circuit so that you are measuring the
voltage across the inductor. Draw in and clearly label the voltage across the inductor VC with
respect to the input voltage (VRC) on the axes under Prediction 1-2, above. Measure the phase
difference between VC and VRC as you did before with VR and VRL.
Question 1-4: What phase difference do you measure? What is the total phase difference between
VL and VR (= the current in the circuit)? Is this what you would have expected? Why?
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
8
5. While you are measuring the voltage on the inductor, you can investigate the impedance of the
inductor, ZL. Remember that VL = ILZL. Change the frequency of the generator and observe what
happens to VL as a function of frequency.
Question 1-5: Does the inductor’s impedance increase or decrease with frequency? Can you explain
this using what you know about inductors?
Comment: The impedance of an inductor, XL, is called the inductive reactance. The actual functional
forms of the capacitive and inductive reactances are:
XC = 1/ωC = 1/(2πf C)
XL = ωL = 2πfL
When these elements are combined in circuit, the overall impedence is Z = R 2 + ( X L − X C ) 2 .
Phase differences result in this form instead of a straight sum. Remember that Vtot=IZ.
INVESTIGATION 2: RESONANCE IN RCL CIRCUITS
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. One of the most familiar uses of such a circuit is as a tuner in
a radio receiver.
Activity 2-1: Properties of RCL Circuits
6. Using the 500 Ω resistor, build the RCL circuit shown below.
Function Generator
Ground
600 Ω
red
black
a
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
b
c
d
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
9
Prediction 2-1: At very low signal frequencies (1kHz), will the maximum values of I through and V
across the resistor be relatively large, intermediate or small? Explain your reasoning.
Prediction 2-2: At very high signal frequencies (well above 300 kHz), will the maximum values of I
and V be relatively large, intermediate or small? Explain your reasoning.
Prediction 2-3: Based on your Predictions 2-1 and 2-2, is there some intermediate frequency where I
and V will reach maximum or minimum values? Do you think they will be maximum or minimum?
7. On the axes below, draw qualitative graphs for XC and XL as a function of frequency. Clearly
label each curve.
XC
or
XL
Frequency (Hz)
8. Remembering that Z = R 2 + ( X L − X C ) 2 , at what relative values of XC and XL will the total
impedance Z be a minimum? On the graphs above, mark this point. Explain your reasoning
here.
Question 2-1: At the frequency you labeled, will the value of the peak current, Imax, in the circuit be a
maximum or minimum? What about the value of the peak voltage, Vmax, across the resistor? Explain.
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia
10
9. The frequency you identified above is the resonant frequency of the circuit, which we will call f0.
Using the fact that this occurs when XC = XL and the expressions for XC and XL given in
Investigation 1, solve (algebraically) for f0 in terms of L and C and other constants. Then, put in
the numerical values of L and C to predict the frequency for your circuit.
f0 =
=
Hz for our values of L and C.
Activity 2-2: Resonance in RCL Circuits
1. Connect the ground leads of probe #1 to "d" and probe #2 to "c" and observe VR on channel 1 (or
A). Connect probe #2 to "a" and observe Vo = Vad = VRLC on channel 2 (B)
2. Start with a function generator at the frequency calculated above, f0. Vary the frequency ± 5 kHz
about this value as needed to observe the behavior of the voltage VR near resonance.
3. On the grid below, sketch the qualitative behavior of your circuit around your calculated
resonance frequency. Note here that you should graph the amplitude (peak-to-peak size) of the
oscillation so you can see how the impedance is changing. What is the measured value of f0 for
your circuit? How close was it to what you calculated?
Amplitude of
oscillation
frequency
End-of-lab Checklist:
Turn off the scope and the function generator.
Make sure you turn in:
 Your lab manual sheets with all predictions and questions answered and data filled in.
©University of Notre Dame
Physics Department
PHYS 11320, Fall 2005
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000,
and the University of Virginia