Sinusoidal Response of RC & RL Circuits Sachin Mehta Reno Nevada

1. Sinusoidal Response of RC & RL Circuits
Written By: Sachin Mehta
Reno, Nevada

2. 2
Objective:
When varying frequencies are applied to RC and RL circuits, analysis of the sinusoidal responses
of the respective circuits can be accomplished somewhat easily. By using a function generator,
an oscilloscope, and a few other circuit elements, we will create both an RC & RL circuit similar
to the previous lab. The major difference, however, is the implementation of a sine wave as
the response—instead of a step (square) wave.
Equipment used:
While doing the circuit analysis, we used several devices; one of which was a multimeter.
Specifics of the multimeter such as tolerance, power rating, and operation are discussed
below. Two other pieces of electronics that had to be employed were the power supply (or
voltage source) and the breadboard. Along with the electronic devices, resistors of
different values were used (also discussed more in depth below).
a) Breadboard: This device makes building circuits easy and practical for students
learning the curriculum. Instead of having to solder each joint, the student can build
and test a particular circuit, and then easily disassemble the components and be on
their way. The breadboard used in this experiment was a bit larger than usual, and
consisted many holes, which act as contacts where wires and other electrical
components, such as resistors and capacitors, can be inserted. Inside of the
breadboard, metal strips connect the main rows together (five in each row) and
connect the vertical columns on the side of the board together. This means that
each row acts as one node.
b) Multimeter: Made by BK Precision Instruments, the 2831c model that we used
during this lab allowed us to measure different currents, resistances, and voltages
from our circuits. Of the four nodes on the front of the device, we used only three:
the red, black, and bottom white. In theory, either of the white nodes could have
been used, but the 2 Amperes node was more than capable since we barely even hit
the 15 mA mark. The following ratings are manufacturer specifications for the
device:
DC Volts – 1200 Volts (ac + dc peak)
Ohms – 450 V dc or ac rms
200 mA – 2 A --- 2 A (fuse protected)
The use of the multimeter in this lab was mainly to ensure the correct values of
resistors, capacitors, and inductors were being used.
c) Wire Jumper Kit: This kit was not required to complete lab two, however the use of
the wires made building the circuits more manageable. The wires were 22 AWG
solid jumper wires that varied in length. The wire itself was copper, PVC insulated,
and pre-stripped at ¼ inch.

3. 3
d) Resistors: This lab involved the use of many different resistors; different values for
the various circuits that had to be constructed. A resistor, in the simplest definition,
is an object which opposes the electrical current that is passed through it. So: the
higher the Ohm value, the more a resistor will impede the current. Typically made
of carbon, each resistor is color coded so identification can be made easier. The first
three colored bands on the passive element are used to calculate the resistance
using the following equation: R = XY * 10Z. Where X and Y and are the first two
bands and Z is the third. The rightmost colored band however gives the tolerance
rating of the resistor itself.
e) Inductors: The inductor is a passive element designed to store energy in the
magnetic field it has. Inductors consist of a coil of wire wrapped around each other
and the ones that were used in lab were of pretty big size, compared to the resistors
and capacitors. An important aspect of the inductor is that it acts like a short circuit
to direct current (DC) and the current that passes through an inductor cannot
change instantaneously.
f) Oscilloscope: An oscilloscope is a test instrument which allows you to look at the
'shape' of electrical signals by displaying a graph of voltage against time on its
screen. It is like a voltmeter with the valuable extra function of showing how the
voltage varies with time. A graticule with a 1cm grid enables you to take
measurements of voltage and time from the screen.
g) Function generator: A function generator is a device that can produce various
patterns of voltage at a variety of frequencies and amplitudes. It is used to test the
response of circuits to common input signals. The electrical leads from the device
are attached to the ground and signal input terminals of the device under test.
h) In addition to the previously listed equipment a new utility was put to work in order
to provide a second opinion, as well as a solid foundation of the circuits. However,
the piece of equipment in question was not a machine, but software. Known as
Multisim, it is a well-known program used by technicians, engineers, and scientists
around the globe. Based on PSPICE, a UC Berkeley outcome, Multisim allows users to
put together circuitry without the need for soldering or a permanent outcome.
Mistakes can be corrected by the click of the mouse and there will be no loss in stock
prices in doing so. Even students who wish to practice on the breadboard can use
Multisim as there learning tool. Without it, electric circuit analysis would not be as
well known, and practical in today’s ever-changing and technologically advanced
society.

4. 4
Theory:
The RC and RL circuits examined in this lab will consist of either a capacitor or an inductor in
series with a resistor. This will make an oscillatory motion in regards to the voltage running
through the circuit. By using the function generator as an input—and setting it to various
frequencies—we will be able to simultaneously view the output & input waveform on the
oscilloscope. We will also be able to calculate the output voltages, phase shift and resistor
currents at specified frequencies, and compare the data to Multisim simulations. Since we are
investigating circuits of the first-order, the mathematical equations which have to be used are
not as simple as previous equations such as Ohm’s law, for example. The following are the
equations needed to complete this lab experiment on the analysis of sinusoidal responses.
𝑉𝑜𝑢𝑡 =
𝑅
√ 𝑅2 + 𝑋 𝑐
2
𝑉𝑖𝑛
or
𝑉𝑜𝑢𝑡 =
𝑅
√ 𝑅2 + 𝑋 𝐿
2
𝑉𝑖𝑛
𝑋 𝑐 =
1
2𝜋𝑓𝐶
𝑋 𝐿 = 2𝜋𝑓𝐿
𝑉𝑅𝑀𝑆 =
𝑉𝑜𝑢𝑡
√2
𝑃ℎ𝑎𝑠𝑒 𝑆ℎ𝑖𝑓𝑡 =
∆𝑡
𝑡
∗ 360 𝑜
V= I*R

5. 5
Procedure:
The first portion of this lab involved building a circuit similar to the one from the previous lab.
It is an RC circuit with a 100Ω resistor in series with the capacitor equal to 0.1μF. The input is
the function generator with a signal at 320 Hz at a voltage level of 10 V. It is important to
remember that this voltage is the peak to peak voltage—not the peak voltage alone. Making
sure to select the sine wave button on the function generator—we constructed the circuit
shown in Fig. 1 on the breadboard, except we used a different resistor value.
Fig. 1: RC Circuit constructed for Part I
Table 1: Experimental Measurements for RC Circuit in Part I
Frequency Input Voltage Output
Voltage (Vp-p)
Output Voltage
(VRMS)
V-I Time
Difference
Phase Shift
(Degrees)
Resistor
Current
320 Hz 10 V 352 mV 82.4 mV 800 µs 0 .5490 mA
3.2 kHz 10.2 V 2.12 V 688 mV 64 µs 40 .0046 mA
32 kHz 10.2 V 8.6 V 2.75 V 2.4 µs 75.5 .0183 mA
320 kHz 9.92 V 10.1 V 3.35 V .5µs 90 .0223 mA
The data in table 1 was collected by measuring the waveforms on the oscilloscope with the
voltage cursor and time cursor. By comparing input and output waveforms, we were able to
detect the time difference between the two waves—which in turn helped us gain a better idea
about sinusoidal steady-state analysis and other circuit analysis methods. The right most-
farthest column in table 1 is the current that is passing through the resistor. Usually, in the
past, we have used the multimeter to measure current in the circuits we have built. But for this
lab, we used a different method. Using the following equation, we were able to determine the
RMS current (IRMS): IRMS =
𝑉 𝑅𝑀𝑆
𝑅
Since we were able to measure the RMS voltage with the oscilloscope, and we knew the value
of the resistor we used in our circuit—we could easily solve for this current. This RMS voltage is
the average voltage across R and essentially, by manipulating Ohm’s Law, we could solve for
current.

6. 6
Section 2.b. of Part I asked us to calculate the output voltage, time difference, phase shift, and
resistor current for the ideal values. Using 0.1μF and 150 Ω as the resistance, I was able to
calculate the output voltage:
𝑉𝑜𝑢𝑡 =
𝑅
√ 𝑅2 +𝑋 𝑐
2
𝑉𝑖𝑛 =
150
√1502 +49732 × 10 𝑉 = .301 𝑚𝑉
𝑉𝑅𝑀𝑆 =
𝑉𝑜𝑢𝑡
√2
=
301 𝑉
√2
= .213 𝑚𝑉
Since the voltage and current are in phase—there is no time difference—meaning that there is
a phase shift of 0 degrees, as well.
IRMS =
𝑉 𝑅𝑀𝑆
𝑅
=
.213 𝑚𝑉
150𝛺
= .00142 A
𝐹𝑜𝑟 3.2 𝑘𝐻𝑧: 𝑉𝑜𝑢𝑡 =
𝑅
√ 𝑅2 +𝑋 𝑐
2
𝑉𝑖𝑛 =
150
√1502 +497 2 × 10.2 𝑉 = 2.94 𝑉
𝐹𝑜𝑟 32 𝑘𝐻𝑧: 𝑉𝑜𝑢𝑡 =
𝑅
√ 𝑅2 +𝑋 𝑐
2
𝑉𝑖𝑛 =
150
√1502 +49.72 × 10.2 𝑉 = 9.68𝑉
𝐹𝑜𝑟 320 𝑘𝐻𝑧: 𝑉𝑜𝑢𝑡 =
𝑅
√ 𝑅2 +𝑋 𝑐
2
𝑉𝑖𝑛 =
150
√1502 +4.972 × 9.92 𝑉 = 9.91𝑉
For the four different frequencies asked for, the ideal output voltages came out to be quite
close to that of the experimental measurements. For example, the percent difference between
the measured 320 kHz output voltage and the calculated output voltage is:
|9.91 − 10.1|
9.91
× 100 = 1.92%
With such a miniscule percent discrepancy, we can conclude that the experimental we
performed, and the circuit we constructed was very close to the real thing. It was precise,
accurate, and definitely intriguing. Notice how the output voltage increased steadily as the
frequency was increased by a factor of 10.
On the other hand though, the RMS output voltage which was calculated for each respective
frequency did not compare well to the measured values of the RMS voltages. For example, the
VRMS measured for 320 Hz was 82.4 mV, but the calculated was .213 mV. This large discrepancy
could have been due to error in the experiment, or error in the calculations done by hand.
Since the peak to peak output voltages came out to be consistent, it is not likely that the
equipment malfunctioned in any way. The reasonable assumption would be that the equation

7. 7
used to calculate the VRMS is incorrect, or a mistake was made with the voltage and time cursors
when handling the oscilloscope during the lab. In order to calculate the phase for each of the
trials, the following equation was utilized:
Phase = arctan (
1
𝑅𝐶𝑤
)
 For 320 Hz---- phase = arctan (33.16) = 1.54
 For 3.2 kHz----- phase = arctan(3.316) = 1.28
 For 32 kHz------ phase = arctan(.3316) = .3202
 For 320 kHz---------phase = arctan(.03316) = .0331
Using the output voltages that were measured on the oscilloscope, we were able to determine
the peak voltage—and from there the RMS voltage that should have resulted from the
measured output voltage.
 For example, for 3.2 kHz, the peak to peak output voltage was 2.12 V. Dividing this by 2
gives us 1.06 Volts as the peak voltage. And finally multiplying by .707 gives a
theoretical RMS voltage of .749 V. Instead the cursors we used on the oscilloscope
measured the RMS voltage at .688 V.
- The percent difference between these two values is:
|. 749 − .688|
. 749
× 100 = 8.14%
The time delay could be deduced using the fact that the current leads voltage by anywhere
from 0 to 90 degrees in an RC circuit. Simply put, V0 leads Vin. This time difference can be
analyzed by studying the input and output waveforms on the oscilloscope reading. During the
lab, we did our best to estimate the correct time difference, but it all depended on how close
you looked at the reading.
4) For this section, we used Multisim to perform a simple AC sweep of the circuit—and then
created a Bode plot. A Bode plot is a plot of both the voltage and phase and can help a great
deal in circuit analysis—especially when sinusoidal responses are being considered. Making
sure to set the frequency limits from 1 Hz to 500 kHz, and specifying the measurement probe to
read voltage across the two probes, a simulation was put together. It can be seen in Fig. 3.
Fig. 2: Multisimsimulationof RCcircuit

8. 8
Fig. 3: AC Sweep of RC circuit using MultiSim
 When determining if this would be a high pass or a low-pass filter, it was necessary for
me to look up these terms in the textbook. I found that a high pass filter ends up being
a high and steady voltage, whereas a low-pass filter ends up at a lower voltage and
approaches zero in fact. Looking at the AC sweep, I determined this RC circuit to have a
high-pass filter.
 When determining the half-power point, we had to find the spot at which the resistance
value equals the reactance. In addition, the output would be equal to 2-.5 times the
voltage input. This means that the half power point would occur at the following
voltage:
10 * 2-.5 = 7.07 V
Meaning that the frequency that the half-power point would occur at would be
approximately 20 kHz.
Part II of the lab involved the implementation of an RL circuit, similar to the one made for the
last lab (step response). The inductor and resistor were placed in series—and constructed on a
breadboard. The beginning frequency we used was 800 Hz at a 10 V peak to peak sine wave.
From there, we again kept increasing the frequency, but kept the input voltage the same. This
would ensure that we could compare the figures, and see how a change in the frequency can
drastically manipulate the circuit we are working with. The circuit constructed for this section is
shown in Fig. 4, but instead of the 100 Ω resistor that is shown—we had to use 150 Ω for all the
calculations, since the function generator had a resistance in and of itself.
Fig.4: RL circuitconstructed for partII

9. 9
Setting the function generator to 800 Hz and finding the 10 V peak to peak input gave us our
first experimental measurement for part II. Refer to table 2 for a complete list of data.
Table 2: Experimental Measurements for RL circuit in Part II
Frequency Input Voltage Output
Voltage
Output Voltage
(RMS)
V-I Time
Difference
Phase Shift
(Degrees)
Resistor
Current
800 Hz 10 V 9.8 V 3.31 V 0 0 .022 A
8 kHz 10 V 8.4 V 2.68 V 10µs 40 .018 A
80 kHz 10 V 1.68 V 55 mV 2.6µs 75.5 .366 mA
500 kHz 10 V 260 mV 56.1 mV 440 ns 90 .374 mA
Part 2.b. again involved calculating data by hand. Using the following equation allowed a quick
and easy way to determine theoretical values:
𝑉𝑜𝑢𝑡 =
𝑅
√ 𝑅2 + 𝑋 𝐿
2
𝑉𝑖𝑛 =
150
√1502 + 5.032
10 = 9.99 𝑉
- For 8 kHz:
150
√1502 +50.32 10 = 9.48 𝑉
- For 80 kHz :
150
√1502 +5032 10 = 2.86 𝑉
- For 500 kHz:
150
√1502 +31412 10 = .477 𝑉
 The values calculated above vary quite greatly from the measured values in table 2. This
could have been a result of inaccurate inductors, resistors, or equipment. Calibration of
equipment had probably not properly been done for a long time. Such simple things as
such could affect data deeply.
Calculations of other properties of the RL network, such as RMS voltage, time difference, and
phase are done in the following section. However, with the output voltage already being so
different from the calculated, it is likely that any other thing we calculate for comparison
purposes is not going to be a good reference. Regardless, the RMS voltage is calculated as
follows:
Vp =
𝑉𝑝−𝑝
2
where VRMS = Vp *.707
Trial 1) VRMS = 3.53 V
Trial 2) VRMS = 3.35 V
Trial 3) VRMS = 1.01 V

10. 10
Trial 4) VRMS = .169 V
Utilizing the percent discrepancy equation for trial 1:
|3.53 − 3.31|
3.53
× 100 = 6.23%
We can see that with a 6.23%, the measured values from the lab experiment don’t quite
compare to the theoretical values, which were calculated by hand. There is an assortment
of reasons for such behavior, such as wrong inductor values being used on the breadboard,
incorrect use of the oscilloscope, or even just plain human error. It is not uncommon to see
mistakes made with new equipment like the function generator and oscilloscope.
For the fourth trial (500 kHz)—Fig. 5 shows a screenshot I took of the oscilloscope output.
As the picture shows, there is a time difference of 440 ns. Using the cursors, we were able
to determine this time on the oscilloscope.
Fig. 5: Oscilloscope output of trial 4—with a frequency of 500 kHz
4) Using Multisim to create a Bode plot was necessary for this section, just as it was for the
previous section. The AC sweep is depicted in Fig. 5 below. You can see that the curve for
both the phase and magnitude (voltage) decrease as the frequency increases. Just as the
experimental data suggests—we have a low-pass filter in this bode plot.

11. 11
Fig. 5: AC sweep of RL circuit in Part II
 When determining if this was a low or high pass filter, I needed to refer back to the
text to examine the definitions more closely. After studying, I found that this plot
shows a low-pass filter since the voltage and phase start out at a higher degree and
fade & approach zero.
 To find the half-power point for this plot, we find where the input voltage is
multiplied by .707, and compare this voltage to the data we have in table 2. In doing
so, the half-power point would occur at approximately 15 – 20 kHz.
Conclusion:
The majority of this lab was spent studying the theory behind sinusoidal waves and the
responses of RC and RL circuits. With new material being introduced, such as complex
numbers, frequencies of all different ranges, and reactance; it was one of the more difficult labs
of the semester. In addition to this new material, we had to work with Multisim in a new
fashion and create never before seen or heard of “Bode Plots” to match with our data. Overall,
with circuit analysis becoming more and more important as time passes—and the use of
technology expanding ever so quickly—learning how to manipulate circuits in Multisim will
make future assignments much easier.
Overall, the data that was collected matched up well with the theory of what should happen as
frequency of a sine wave is increased. We saw the correct response of the RC and RL circuits
that we built on the breadboards. Viewing these sine waves and the integration of these
waveforms into the study of RC & RL circuits was a major learning outcome of this lab
experiment. Determining how various frequencies affect the output voltage, RMS voltage,
phase shift, and even the current through the different elements was interesting.

12. 12
However, when doing calculations by hand near the end of the lab experiment—we noticed
that some figures were ambiguous and did not correlate well with the data that we had
collected. There could be plenty of reasons for something like this to happen, but the most
common and most likely is human error: error on our part. Either when collecting the data and
working with the equipment, or when manipulating these new and fresh equations—mistakes
could have been made in either place or even both. If we had more time to cover this material,
things would have gone a lot smoother and a thorough understanding of the material would be
much greater.