Lab_2
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Dec 6, 2023
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ECE 2024
Lab 2: RLC filter design and analysis
NOTE: Your name is required!
By putting your name on this worksheet, you certify that what is presented is entirely your own
work.
The Virginia Tech Honor Code will be strictly enforced.
You are allowed to discuss steps and procedures with other classmates, but you are not
allowed to discuss specific answers.
If you get stuck, do not wait for help!
Jump on Piazza to get answers to your questions, and
learn more in a less-frustrating way while you are at it.
Frustration is counterproductive, and
we are here to help.
The purpose of this lab is to learn about impedance analyzers, AC frequency sweeps, and how
to use your knowledge of circuits to design an RLC filter.
Plenty of directions are given and you must read them carefully to save yourself time.
Follow the steps in order, and have fun with your first circuits lab!
There are four parts to this lab:
A.
Design calculations
B.
Simulate the circuit
C.
Using the AD2 impedance analyzer
D.
Build and test the circuit
Page 1 of 7
Name:
Required Materials:
●
Your breadboard
●
Your AD2 oscilloscope with the USB cable, and your laptop
●
10mH inductor
●
Film or ceramic capacitor(s) as needed
●
Resistors as needed
General tips to help you succeed:
a.
There are no expected “exact” answers.
i.
We don’t expect everyone to have lab-grade precision in these experiments.
Remember, the components you use have tolerances and are not perfect. If you
get a reading that you think is wrong, please talk to the GTA, ULAs, or your
professor.
b.
Record the most stable reading you see on your digital multimeter (DMM):
c.
Carefully wire up your breadboard circuit.
Ensuring it is correct as you go will save you
a lot of time and frustration.
d.
If you need help, post a question on Piazza or see a TA, ULA, or your professor at office
hours.
Office hours are posted on the Canvas homepage.
e.
Remember how to connect your power supply to a circuit and how to measure current
through a resistor. Observe the following diagram.
For measuring current, you need to
BREAK the circuit in order to measure in series (series elements have the same
current).
Page 2 of 7
A. Design calculations
See the circuit shown below for a band-pass filter (BPF). This is a parallel resonant RLC
circuit. Refer to section 6-7 in your text book for parallel resonance. We want to design the
center frequency
f
o
of the BPF to be around 5kHz. Also, we want to observe the change of
its gain when the input frequency changes. Finally, we want to use an AD2 impedance
analyzer to measure its impedance.
1.
We want to limit the peak current of the AD2 to around 10 or 11 mA, so as to not create
a strain on the unit for all frequencies, or our laptop’s USB supply/battery. What resistor
R1 value do you choose from your part kit to ensure this with a 5 V source? Show your
calculation.
_________ Ohms
2.
From the diagram above, derive the circuit transfer function
H
(
ω
)
=
V
o
V
i
to be in the
form of
N
(
jω
)
D
(
jω
)
=
H
(
jω
)
. The transfer function should look like
(
jω
)
N
0
(
jω
)
2
D
0
+
(
jω
)
D
1
+
D
2
.
N
0
,D
0
, D
1
and
D
2
are coefficients to be found. Show your derivation. (Hint: Let
Z
1
=
R
1
and
Z
2
=
L
1
∨
¿
C
1
. Use
Z
2
=
[
1
jωL
+
jωC
]
−
1
to start.
H
(
ω
)
=
V
o
V
i
=
N
(
jω
)
D
(
jω
)
=
Z
2
Z
1
+
Z
2
.
)
3.
Convert
H
(
jω
)
found in step 2 to its polar form. That is to find
|
H
(
ω
)
|
and
∠
H
(
ω
)
. Show your derivation.
4.
Obtain a
1mH
inductor from your kit and use your DMM to measure the resistance
across the inductor from your kit. Record its resistance. This is the winding resistance
inside your inductor as R2 shown below. Calculate the inductor Q
s
(Quality factor) at
5kHz and record it below.
Q
s
:_______
Page 3 of 7
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5.
If Q
s
is greater than 10, the resonant frequency will not change much if we still use X
L
=
X
C
to find the resonant frequency. Find the capacitor value C from the resonant
frequency
f
o
(5kHz).
C:________ Farads
6.
From step 2, we can find the Q of the circuit from
D
(
jω
)
.
D
(
jω
)
=
(
jω
)
2
+
(
jω
)
(
ω
o
Q
)
+
ω
o
2
. Use this expression and substitute component values to
calculate the qualify factor Q of the circuit. Alternately, we can calculate Q by converting
R1 to a resistor connected in parallel with the inductor and the capacitor, and use
R
ω
o
L
or
ω
o
RC
to calculate the Q of the circuit. Compare both calculations. Show
your calculations and results below. Calculate the bandwidth (BW) of the filter with
(
f
o
Q
)
. Refer to section 6-7 in your text book for bandwidth discussion. Show your
calculation.
Q (calculated from transfer function): ___
Q (calculated from
R
ω
o
L
or
ω
o
RC
): ___
Show your calculations.
BW :________ Hertz
Page 4 of 7
7.
Calculate the impedance of the circuit at resonance. To precisely calculate the
impedance, the winding resistor should be converted to a resistor in parallel. That is to
convert the series L1 and R2 in the diagram above to a parallel L2 and a R3 shown
below. (Formula:
L
p
=
L
s
[
1
+
1
Q
2
]
, R
p
=
R
s
[
Q
2
+
1
]
)
See figure 1 in this
reference
or (7-79)
in this
reference
. Since the L
s
becomes L
p
, their resonant frequency will change slightly,
calculate the new resonant frequency based on L
p
.
Maximum impedance: _______Ω at (frequency)______Hz
New resonant frequency based on L
p
______Hz
Show your calculations.
B. Simulate the circuit in LTspice
8.
Model your circuit as the diagram shown in step 4 with LTspice. Do a small-signal
analysis (an AC Analysis) with its input set at 1
∠
0
0
and measure the output with the
following settings. Select “Type of sweep” in Decade. Enter “Number of points per
decade” to 1000, Enter “Start frequency” to 500 and “Stop frequency” to 50k. Screenshot
your circuit and simulation result below. This plot is the frequency response of the circuit.
9.
On the same plot, attach a cursor to be placed at the peak voltage of the frequency
response and use “Label Curs. Pos.” under “Notes & Annotations” in the pull-down menu
of the “Plot Settings” to place a label for the present cursor position. Move the cursor to
the left side of the peak and find a Mag where it’s 3dB lower than the peak then place
another cursor label. Do the same thing by moving the cursor to the right side of the
peak then place the 3
rd
cursor label. Write their measurement below. Screenshot your
simulation with cursor labeled below.
Peak (
f
o
¿
: _______dB at (frequency) _______ Hertz
Page 5 of 7
Corner
freq
, <
f
o
(
f
L
¿
: _______dB at (frequency) _______
Hertz
Corner
freq
, >
f
o
(
f
H
¿
: _______dB at (frequency) _______
Hertz
10.Compare the frequency at peak in step 9 to the desired
f
o
(
5
kHz
)
frequency. Also
compare to the new resonant frequency calculated in step 7. Are they the same? Is the
simulated result closer to the desired
f
o
or closer to the new resonant frequency?
Comment on your observation. Also, find the bandwidth from step 9 and compare it to
the BW as found in step 6. (Note. The winding resistance calculated was neglected in
step 6.) Comment on if neglecting the winding resistance is appropriate. Show your
calculation.
11.Continue from the simulation in step 8. Change the voltage source to an AC input with
1V, 3000Hz and do a
transient analysis
. (Set the input to “SINE”, and “Amplitude” set to
1, “Frequency” set to 3k and “Ncycles” set to 5 then set simulate to 1ms.) Use cursors to
measure two maximum values between input and output. Find their angle difference
between input and output with the formula (
θ
=
Δt
T
×
360
o
¿
, and calculate the
amplitude gain in dB
. Screenshot your measurement by labeling the cursor positions
and paste it below. Show your calculation.
Gain at 3kHz: _______dB
Phase angle at 3kHz: ________ degrees
Show your calculations.
12.Compare your measurement in step 11 and calculation to the
|
H
(
ω
)
|
and
∠
H
(
ω
)
found in step 3 by letting
ω
=
2
π ×
3000
and substituting the actual component values
to step 3. Comment on your comparison and explain why if they are different.
20log
|
H
(
ω
)
|
from step 3 at ω=6000π : ________ dB
∠
H
(
ω
)
from step 3 at ω=6000π : ________ degrees
Show your calculations.
13.View the circuit as a one-port network and find its
impedance
by sweeping an ac small
signal from 1k to 50kHz. The method is to go back to step 8 and plot the source voltage.
Then, right click on the title of the plot of the source voltage and enter an expression to
divide the source voltage by the circuit current to obtain impedance. The circuit current
Page 6 of 7
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can use the same current through the series resistor R1 which is I(R1). Compare the
maximum impedance to the impedance found in step 7 to see if they are the same.
Comment on your result. Also, record the impedance at 2kHz for later use, including its
phase angle.
Maximum impedance: _______Ω at (frequency)______Hz
Impedance: _______Ω at 2kHz
14.
Measure the current
I
c
1
through C1 at resonance by right clicking on the vertical scale
on the left of the plot and change it to “linear” representation. Also, measure the circuit
current through R1 at a frequency that is
(
1
5
)
of the resonant frequency. Check to see
if
I
c
1
is Q times larger than
I
R
1
. Refer to page 21 in this
reference
. Comment on
your measurement and paste your measurement below.
C. Build and test your circuit with AD2
15.Build the circuit as shown in step 4 on your breadboard. The R2 in step 4 is the winding
resistance inherent to the 1mH inductor. Combine two or three capacitors to make the
actual capacitor value closer to the desired value. Use your DMM to verify the capacitor
value. Take a picture with your DMM to show your verification and paste it below. Set up
the network analyzer in your AD2 by connecting Wavegen (yellow) and Scope ch1
(Orange) to the input and connecting Scope ch2 (Blue) to the output. Connect ground
(Black), ch1 negative (Orange/White) and ch2 negative (Blue/White) together. Sweep a
sinusoidal voltage 1V from 500Hz to 50kHz and record its frequency response. Find the
maximum output in dB and record the center frequency
f
o
.
Screenshot your
measurement below. Compare it to the peak voltage in simulation found in step 9. Are
they the same? If not, how much difference? Why? Comment on your measurement.
Peak voltage: _______V,
At
f
o
:________ Hertz
16.Continue from step 15. Measure the output at 3kHz and its phase angle. Record their
values below. Change the AD2 setup to generate a 3kHz, 1V sinusoidal signal with
Wavegen and measure it with the Scope ch1 and ch2 in time domain. Find the phase
angle between peaks or between zero-crossings by using formula (
θ
=
Δt
T
×
360
o
¿
.
Calculate the output gain in dB. Record your measurement below. Screenshot your
measurement. Compare it to step 12 and comment on your result.
From frequency response: Voltage gain: ______dB at 3kHz
Phase angle: ________ degrees
Page 7 of 7
From time-domain output:
Voltage gain: ____ at 3kHz, or ________dB
Phase angle: ________ degrees
Comment on your result.
17.Use the AD2 as an impedance analyzer to measure the impedance of the BPF from
1KHz to 50kHz. See the “help” in the “Waveform” program to learn how to use the
impedance analyzer. Obtain a 1kΩ resistor from your kit and place it on the left side of
the R1. (R1, L1 and C1 in the first diagram of the lab are the DUT below). DUT means
Device Under Test. The “Resistor” below in the diagram is the abovementioned 1kΩ.
Connect the Wavegen (yellow) and Scope ch1 (Orange) to the left side of the 1kΩ and
Scope ch2 (Blue) to the right side of the 1kΩ resistor. Connect ground (Black), ch1
negative (Orange/White) and ch2 negative (Blue/White) together. Run the analyzer and
find the impedance at 2kHz including its phase angle. Also, find the maximum
impedance. At what frequency the maximum impedance occurs? Is it the resonant
frequency
f
o
? Is the measured resonant frequency closer to the new resonant
frequency found in step 7? What is the phase angle at resonant frequency? Compare to
the simulation in step 13 and step 7. Screenshot your measurement and comment on
your result.
Impedance: ______Ω at 2kHz
Phase angle: ________ degrees
maximum impedance:_________ Ω
At frequency________ Hertz
Phase angle at resonant frequency___ degrees
Comment on your observation.
Screenshot your measurement
18.
Take a picture of you circuit and AD2 connection. Paste it below.
Page 8 of 7
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9. Replace the filter capacitor with another one of value 10µF and repeat step 8.
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