PHYS 196L - Exp. 9 RL Circuits
Name
Lab Partner
Introduction:
In this lab, we will investigate the properties of an electrical circuit that contains an
inductor, or coil, like the ones you used in the Faraday’s Law experiment. Specifically, we will
look at how an inductor can affect the phase angle between output and input, as well as the
gain for a range of frequencies.
Theory:
Previously, we learned about resistors and how they can decrease the amount of current
flowing in a circuit. As we jump into the realm of AC circuits, we will see that resistance is
actually the real part of a complex variable, called Impedance:
𝑍 = 𝑅 + 𝑗𝑋.
Here, 𝑍 is the impedance, 𝑅 is resistance, and 𝑋 is called reactance. The 𝑗 is simply the
imaginary number,
𝑗 = √−1,
but to avoid confusion with a variable for current, electrical engineers typically do not use 𝑖 to
denote the imaginary number.
Now, the impedance of an inductor,
𝑍𝐿 = 𝑗𝜔𝐿 ,
is a function of frequency. As frequency increases, the impedance of the inductor increases. We
can analyze impedance graphically by plotting it in the complex plane, with the horizontal axis
being real numbers, and the vertical axis is imaginary:
Image from http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html
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Pictured above, the magnitude of the impedance vector forms an angle, 𝜙, with the real axis. As
frequency increases, so does the angle, heading towards +90� (a capacitor would head
towards -90�). Frequency, as you have seen it in previous experiments, is in Hertz, denoted
with 𝑓. Angular frequency, 𝜔, is simply equal to 2𝜋𝑓.
Now, in this lab we will be setting up a series RL circuit:
Here, V_out denotes the voltage being measured across the inductor. Measuring the voltage in
this way creates something called a Voltage Divider and these are an extremely important
concept in the world of circuit analysis.
Voltage Divider
According to Kirchhoff’s Voltage Law, the input voltage is equal to the sum of the voltage
drops across the resistor and the inductor:
𝑉𝑖𝑛 = 𝑉𝑅 + 𝑉𝐿 .
Following Kirchhoff’s Current Law, we know the current through each component in series is the
same:
𝐼𝑅 = 𝐼𝐿 ,
𝑉𝑅
𝑉
= 𝐿.
𝑅
𝑗𝜔𝐿
Substituting 𝑉𝑅 back into the sum of voltages, we get an expression containing only the input
voltage and the voltage across the inductor, or the output voltage we want to measure:
𝑅𝑉
𝑉𝑖𝑛 = 𝑗𝜔𝐿𝐿 + 𝑉𝐿 .
After a couple steps of algebra, we finally arrive at our very important Voltage Divider equation,
𝑗𝜔𝐿
),
𝑅+𝑗𝜔𝐿
𝑉𝐿 = 𝑉𝑖𝑛 (
or the general form,
𝑍
2
𝑉𝑜𝑢𝑡 = 𝑉𝑖𝑛 𝑍 +𝑍
,
1
2
where 𝑍2 is the impedance component we are measuring the voltage across. Now, recalling that
voltage gain, 𝐴𝑉 , is the ratio of output voltage to input voltage,
𝑉
𝐴𝑉 ≡ 𝑉𝑜𝑢𝑡,
𝑖𝑛
we have an expression for the gain of our series RL circuit. Note that the gain is a function of
frequency. As with any complex number, the gain can be separated into its real and imaginary
parts. This requires quite a few steps of algebra but once done, it can help us mathematically
determine the circuit’s phase between real and imaginary for any frequency.
𝑗𝜔𝐿
𝐴𝑉 = 𝑅+𝑗𝜔𝐿,
2
𝑗𝜔𝐿
𝑅−𝑗𝜔𝐿
𝐴𝑉 = 𝑅+𝑗𝜔𝐿 ⋅ 𝑅−𝑗𝜔𝐿,
𝐴𝑉 =
1
𝜔2 𝐿2 −𝑗𝜔𝑅𝐿
,
𝑅2 +𝜔2 𝐿2
2 2
𝐴𝑉 = 𝑅2 +𝜔2 𝐿2 (𝜔 𝐿 − 𝑗𝜔𝑅𝐿),
𝑅
𝜙 = 𝑡𝑎𝑛−1 (𝜔𝐿).
If you examine these expressions closely, you can conclude that as the impedance of
the inductor increases with increasing frequency, it will eventually equal the impedance of the
resistor. When 𝑅 = 𝜔𝐿, the phase goes to 45° and the point at which this occurs is called the
cut-off frequency. This occurs when
𝜔=
𝑅
= 𝜔𝑐 ,
𝐿
Procedure:
Connect the circuit pictured below:
You are using the large inductor and the 3.3 kΩ resistor (or another resistor close to 3k, just
make sure you take note of the resistance). Everything should be connected in series. The
black cord of the voltage sensor can connect to the ground, or black output port on the Pasco
Interface. Make sure your Pasco Interface is turned on and open Pasco Capstone. In Capstone,
under Hardware Setup, make sure the Voltage Sensor is present on Channel A. On the Output
1 port, add the Output Voltage/Current Sensor.
Create a table in Capstone with User-Entered Data called ‘Frequency’ (units of Hz) and the
second column a User-Entered Data Set called ‘Gain.’ The rows in the Frequency column
should start at 100 Hz and increase in 100 Hz increments until 1 kHz, then begin increasing by
1 kHz until 4 kHz. Switch the recording mode (button on bottom) to Fast Monitor so that
Capstone doesn’t mess up on the sample rate. Set the Voltage Sensor to 20 kHz. In the
Calculator, create the following two calculations:
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V_out = sqrt(avg([Voltage, Ch A (V)]^2))
V_in = sqrt(avg([Output Voltage, Ch 01 (V)]^2))
A_v = [V_out] / [V_in]
Make sure your ‘^2’ is just outside the bracket for the voltage in both the input and output
voltage. Next, open the Signal Generator and set a Sine wave with Voltage Amplitude of 5 Volts
and a Frequency of 100 Hz and click ‘On.’ Drag down a Digits display and for the measurement,
select your A_v calculation. Click ‘Monitor’ (like Record from before) for the data to begin
displaying. After a couple seconds for the averages to settle, click ‘Stop’ and record the value
for A_v under Gain in your table at the 100 Hz row. Then, go to the signal generator and change
the Frequency to your next step in the table, click ‘Enter’ to set that frequency, then click
‘Monitor’ and wait for the averages. Repeat this until you have filled up your table. Once you
have finished your table, drag down a graph and for the y-axis, select Gain. For the x-axis,
select Frequency.
Post a picture of your Gain vs Frequency graph here:
Now, we need to find the cut-off frequency. Look over your table with the different Gain values
and determine which frequency is closest to 0.707. Keep adjusting the frequency in the Signal
Generator until you achieve a Gain as close to 0.707 as possible. Don’t forget to turn on and off
the Monitor mode when changing the frequency otherwise the averages won’t settle and you will
get incorrect results.
What frequency did you find the Gain to be as close to 0.707 as possible and what was that
measured Gain?
Now that you have a cut-off frequency and you know the value of your resistor, solve for the
inductance, L, of the large coil. (Don’t forget 𝜔 = 2𝜋𝑓)
Next, make sure your Signal Generator is turned off, and open a new page in Capstone. Drag
down an Oscilloscope widget. It’s simply called ‘Scope.’
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Add another y-axis to this scope display:
Now that you have 2 y-axes, make one of them the Voltage from Channel A and the other y-axis
the Output Voltage Ch 01.
Click the button to make the scope trace, which makes the voltage wave stop moving so fast:
Now, check that your Signal Generator is still on your cut-off frequency. Turn on the Signal
Generator and click Monitor. You should now see the waveforms on the oscilloscope. Click
stop. Next, click the delta tool:
This will allow you to measure certain points on the waveform. If you right-click on the crosshair
that appears, a menu will appear. Click on ‘show delta tool’ so that a second crosshair appears.
With this delta tool, you can place one on each of the peaks of the 2 different waveforms, the
peaks closest to each other. This will tell you a delta-t, or time between peaks. Determine the
period of your waveform by inverting the frequency.
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Divide your delta tool’s time by the period of your waveform. This is the fraction of a full period
that your two waveforms differ by, A.K.A. the phase shift. Multiply that ratio by 360� (a full
period) and you will have the phase shift between input and output.
What is your measured phase shift?
At cut-off, the frequency should be 45�. What is the percent error between your measured
phase and the theoretical phase?
FUN EXERCISE!
Turn off the Signal Generator. Unplug all your wires except for the voltage sensor in Channel A.
Have one lab partner hold onto the metal part of the red banana cord of the voltage sensor. As
long as you have your data for the previous parts, delete the scope plot and drag down a new
one (this is the easiest way to reset the image). Click on the trace again, and click Voltage
Channel A for the measurement. Click monitor on and while it is running, try zooming on the
horizontal and vertical axes separately to get a good view of a sinusoidal waveform. Once you
have a clean(ish) sine wave in sight, click stop. Use the delta tool to find the period of the
waveform and use that to calculate its frequency.
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What is the frequency of this mystery waveform?
Where does this sine wave come from if nothing is connected? The walls! This is called line
noise and it is from all the power in the building alternating at 60 Hz. When you plug into the wall
outlet, that is about 115/120 Volts RMS at 60 Hz. With every wire in the building alternating at
this frequency, it is easy for electronics to pick up this signal.
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