PHYS 3322 Modern Laboratory Methods I
AC R, RC, and RL Circuits
Purpose
For a given frequency, doubling of the applied voltage to resistors, capacitors, and inductors
doubles the current. Hence, each of these circuit elements obeys Ohm’s Law and are called
linear devices. Resistors, capacitors, and inductors are passive devices in the sense that they do
not have a built in power supply. The frequency response of these circuit elements renders them
useful for many applications, a few of which will be studied in this and a subsequent experiment.
This experiment focuses on increasing your understanding of the frequency response of resistive,
capacitive, and inductive impedance in multi-component circuits both theoretically and
experimentally by use of the transfer functions for each circuit element.
Background
The AC RC circuit, as well as the AC RL circuit, falls into the realm of circuits used to filter
unwanted signals. The properties of these passive filters are dependent on the values of R, C,
and L, as well as the frequency. Figure 1a shows how an RC filter might be inserted into a
circuit and Figure 1b shows the magnitude of the transfer function VR/VIN versus the input
frequency.
1.0
0.8
VIN
|VR/VIN|
VOUT=VR
0.6
0.4
0.2
0.0
101
102
103
104
105
frequency (Hz)
(a)
(b)
Figure 1. a) Circuit diagram for the series RC circuit as a high pass filter. b) The
magnitude of the transfer function VR/VIN for the series RC circuit with R=1 kΩ and
C=0.039 µF. Note that the magnitude of the transfer function is plotted versus a log
frequency scale.
With the setup shown in Figure 1a, low frequencies are filtered or attenuated while high
frequencies are passed with little attenuation. Hence, this circuit arrangement is called a high
pass filter.
The range of frequencies that are filtered is typically called the stopband and the range of
frequencies that are not filtered is typically called the passband. Even though the relative
smoothness of the transfer function of Figure 1 does not allow identification of a single
frequency that divides the stop band from the pass band, a meaningful “cutoff” frequency for this
filter can still be defined. The cutoff frequency for this filter, as well as for the low pass filter, is
the frequency for which the magnitude of the transfer function is decreased by the factor 1/ 2
from its maximum value. The factor 1/ 2 appears to an arbitrary choice, however, at the cutoff
frequency the average power delivered by the circuit is one-half the maximum average power.
Thus the cutoff frequency is also called the half power frequency. In the pass band, the average
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AC R, RC, and RL Circuits
power delivered by the circuit is at least 50% of the maximum average power. To be more
specific, the pass band is defined as the range of frequencies in which the amplitude of the output
voltage is at least 70.7% of the maximum possible amplitude.
For the filter shown in Figure 1, the cutoff frequency and the 70.7% amplitude point is marked
on the graph with straight lines. The cutoff frequency for the RC circuit is, in the ideal case,
ωc = 2πf c =
1
RC
(1)
Equipment
•
Multimeter
•
47 Ω resistor
•
Function generator
•
4.7 µF capacitor
•
Oscilloscope
•
10 mH inductor
Procedure
The lab consists of three parts: the R circuit, the RC circuit, and the RL circuit. The theoretical
expressions for each circuit, assuming ideal circuit elements, were obtained in “Transfer
Functions of R, RC, and RL Circuits”. For the circuits you will experimentally determine the
magnitude of the transfer functions and phase relationships as a function of frequency as a
comparison to the theoretical expressions as well as experimental values for the magnitude of the
current and impedance.
For your convenience, questions regarding each circuit are included within each section to assist
in making and entering your observations in the lab notebook. These questions are the bare
minimum to consider and answer and should by no means be the minimum you, as the student
should consider. The answers to the questions, in report form, must be included in the lab write
up. It is also advisable to complete the data acquisition, plotting, etc., for each section before
continuing on to the next.
Obtain the appropriate circuit elements listed above and use the appropriate multimeter to
determine the exact values of the resistor, capacitor, and inductor.
THE R CIRCUIT
Resistive impedance effects in an R circuit: Measurement of VR/VIN
function
generator VIN
ch 2 – VIN
oscilloscope
R
ch 1 – VR
Figure 2. Set up for measuring VR/VIN and the
phase of the current through the resistor
Set up the circuit shown in Figure 2. All ground connections are made to the ground connection
of the function generator. Set the function generator to output a sinusoidal voltage and set the
amplitude to the midpoint of its range. In this circuit the output of the function generator is read
from the oscilloscope via channel 2 and the voltage across the resistor is read from channel 1.
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AC R, RC, and RL Circuits
Using both inputs, the phase shift between the input voltage and the voltage across the resistor
can be determined. To view the frequency on the oscilloscope, press [Time] on the front panel
followed by [Freq] on the display menu. To view the phase shift, press [Time], then on the
display menu, press [Next Menu] twice, followed by [Phase].
•
Determine the magnitude of the transfer function VR/VIN as well as the phase difference
between the input signal and signal across the resistor as a function of frequency over the
range 20 Hz to 2500 Hz.
•
On the same graph, plot the measured transfer function as well as a theoretical curve for the
transfer function as a function of frequency.
•
On a separate graph, plot the phase difference between the input voltage and the voltage drop
across the resistor.
Questions: resistive impedance
Compare the results of your measurements with your theoretical predictions. Discuss any
discrepancies and consider possible sources of error.
Is a resistor a reactive (i.e. frequency dependent) component in an AC circuit?
Looking at the phase differences, how much does the current lead or lag the applied voltage?
Does this circuit possess any filtering capabilities? If so, how would the circuit be set up? What
is the cutoff frequency? How does the cutoff frequency compare to the ideal cutoff frequency?
Discuss the impedance effects that cause the circuit to behave as a filter.
THE RC CIRCUIT:
Reactive impedance effects in an RC circuit: Measurement of VR/ VIN
Set up the circuit shown in Figure 3. With this setup the input voltage VIN is read via channel 2
and VR is read via channel 1.
VIN
ch 2 – VIN
C
R
oscilloscope
ch 1 – VR
Figure 3. Setup for measuring VR/VIN, and the
phase difference between VR and VIN.
•
Over the range 20 Hz to 2500 Hz obtain the magnitude of VR/VIN and the phase difference
between VR and VIN.
•
Plot the measured impedance of the circuit along with the calculated impedance on linear
scales as a function of frequency.
•
Plot the measured current and the calculated current on linear scales as function of
frequency.
•
For both the measured and theoretical values plot |VR/VIN| versus log f.
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AC R, RC, and RL Circuits
•
Plot the measured phase difference data along with the theoretical values for the phase
difference on linear scales.
Reactive impedance effects in an RC circuit: Measurement of VC/ VIN
Set up the circuit shown in Figure 4. With this setup the input voltage VIN is read via channel 2
and VC is read via channel 1.
VIN
ch 2 – VIN
R
oscilloscope
C
ch 1 – VC
Figure 4. Setup for measuring VC/VIN, and
the phase difference between VC and VIN.
•
Over the range 20 Hz to 2500 Hz obtain the magnitude of VC/VIN and the phase difference
between VC and VIN.
•
For both the measured and theoretical values plot |VC/VIN| versus log f.
•
Plot the measured phase difference data along with the theoretical values for the phase
difference on linear scales as function of frequency.
Questions: The RC circuit
Compare the results of your measurements with your theoretical predictions. Discuss any
discrepancies and consider possible sources of error.
Is a capacitor a reactive (i.e. frequency dependent) component in an AC circuit?
Looking at the phase differences, does the current lead or lag the applied voltage?
Does the voltage across the capacitor lead or lag the applied voltage?
Does the voltage across the capacitor lead or lad the voltage across the resistor?
Does this circuit possess any filtering capabilities? If so, how would the circuit be set up? What
is the cutoff frequency? How does the cutoff frequency compare to the ideal cutoff frequency?
Discuss the impedance effects that cause the circuit to behave as a filter.
THE RL CIRCUIT:
Reactive impedance effects in an RL circuit: Measurement of VR/ VIN
Set up the circuit shown in Figure 5. With this setup the input voltage VIN is read via channel 2
and VR is read via channel 1.
VIN
ch 2 – VIN
L
oscilloscope
R
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ch 1 – VR
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AC R, RC, and RL Circuits
Figure 5. Setup for measuring VR/VIN, and the phase difference between VR and VIN.
•
Measure VR and VIN and the phase difference between VR and VIN over the frequency range
20 Hz to 2500 Hz.
•
Plot the measured impedance along with the calculated impedance on linear scales as
function of frequency.
•
Plot the measured current and the calculated current on linear scales as function of
frequency.
•
For both the measured and theoretical values plot |VR/VIN| versus log f.
•
Plot the measured phase difference along with the theoretical values for the phase difference
on linear scales as function of frequency.
Reactive impedance effects in an RL circuit - Measurement of VL/ VIN
Set up the circuit shown in Figure 6. With this setup the input voltage VIN is read via channel 2
and VL is read via channel 1.
VIN
ch 2 – VIN
R
oscilloscope
L
ch 1 – VL
Figure 6. Setup for measuring VL/VIN, and
the phase difference between VL and VIN.
•
Measure VL and VIN and the phase difference between VL and VIN over the frequency range
20 Hz to 2500 Hz.
•
For both the measured and theoretical values plot |VL/VIN| versus log f.
•
Plot the measured phase difference data along with the theoretical values for the phase
difference on linear scales as function of frequency.
Questions: The RL circuit
Compare the results of your measurements with your theoretical predictions. Discuss any
discrepancies and consider possible sources of error.
Is an inductor a reactive (i.e. frequency dependent) component in an AC circuit?
Does the current lead or lag the applied voltage?
Does the voltage across the capacitor lead or lag the applied voltage?
Does the voltage across the capacitor lead or lad the voltage across the resistor?
Does this circuit possess any filtering capabilities? If so, how would the circuit be set up? What
is the cutoff frequency? How does the cutoff frequency compare to the ideal cutoff frequency?
Discuss the impedance effects that cause the circuit to behave as a filter.
Other Questions
Three sinusoidal signals at100 Hz, 500 Hz, and 1000 Hz are superimposed. You wish to
measure the 1000 Hz signal. Explicitly, how would you do this?
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