University of California, Santa Cruz
Department of Electrical Engineering
EE 101L - Introduction to Electronic Circuits Laboratory
Winter 2016
EXPERIMENT 4: First and Second Order Circuits
In this experiment, you will work with a first-order circuit and a second-order circuit, and compare
their performance with theoretical expectations.
First-Order Circuits: In order to derive the mathematical behavior of the first-order circuit (RL or
RC circuit), a differential equation with an initial condition needs to be constructed based on the circuit in
question, which has the form
dy(t)
+ τ −1 y(t) = K
dt
(K is constant).
(1)
Once the differential equation is solved, it is noted that the solution has the form
f (t) = Ae−t/τ + B
(A and B are constants).
(2)
The time constant τ is called as such because the voltage (or current) on a circuit element has reached certain
values. For 1 time constnat, t = τ , the exponential term in the voltage (or current) will have e−1 ≈ 36.8% of the
its initial value. When t = 2τ , or 2 time constants, the exponential voltage (or current) will have e−2 ≈ 13.5%
of its initial value. We can see that by the time t = 5τ , the exponential term will be approximately 0, and
hence the voltage (or current will have reached its final value.
For the first part of the experiment, you will determine the time constant both experimentally and
theoretically, and compare your results. Then, you will be asked to adjust the time constant of the circuit by
making the appropriate modifications to the circuit, and verifying your circuit through experimentation.
Second-Order Circuits: RLC second-order circuits have more complex behavior than first-order
circuits, in that there is not a time constant. There is a damping behavior associated with the transient
response of the circuit (over-damped, under-damped, and critically-damped), which depends solely on the
circuit elements themselves. The behavior that will be tested in this experiment deals with the steady-state
response of the circuit, which depends on the circuit elements and the frequency of the input source. The
measured voltage response depends on which circuit element is being measured, and all circuit elements will
experience different but related responses. The current through an series RLC-circuit will reach a maximum
at the resonance frequency (which you will derive on your own time). Since the resistor voltage is directly
proportional to the current, it also reaches a maximum at this frequency.
For this portion of the experiment, you will build an RLC-circuit, and measure the amplitude and
phase of the voltage across each element for various frequencies, and plot versus the theoretical responses. To
obtain theoretical response, you will need to understand phasors, which we will cover briefly in lab. Using
this, the measurements can be compared to the resonance frequency of the circuit.
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Part 1: First-Order RC-Circuit
1. For the following circuit, set the function generator to square-wave, 1V peak-to-peak, and f = 500 Hz.
R1
2k
V1
square-wave
1V amplitude
1kHz
C1
0.1u
2. Connect channel 1 on your scopre to measure the input voltage on your circuit, and channel 2 to measure
the voltage across the capacitor.
3. Set the window to view 1 period of the input and output waves together on the scope. Make sure the
image is very clear.
4. Using the scope, shift the image so that the period begins with the input voltage transitioning from
minimum to maximum value. Use the measurement tools on the scope to place the vertical cursor (a)
at the time where the voltage begins to increase, and use curser (b) to find the time where the voltage
difference∆V is at 63.2% from its maximum value (note: the maximum value is the peak-to-peak value).
Record the difference in time ∆t from cursor ”a” to ”b” as the time constant τ . Take a screenshot of
this image.
5. Using the same approach from above, find the differences in voltage ∆V for each multiple of time
corresponding to 2τ, 3τ, 4τ, and 5τ .
6. Knowing what the theoretical time constant is for this circuit, determine the percent difference between
the theoretical value and experimental value. Plot these together and on the same axes using Excel,
and clearly label your plot.
7. Using the math function on the scope, take the difference between channel 1 and 2 in order to display the
voltage of the resistor. Does this waveform make sense? (take a screenshot) Use KVL as an argument to
find the theoretical plot of the resistor voltage, and plot 1 period of it to compare with your screenshot.
8. Using the same series RC-circuit, design a circuit that has a time constant of τ = 0.5 ms. Connect
it to the scope in the same exact manner as before, except that the frequency of the input voltage
must be adjusted. What is a good frequency to use, such that we can observe the time constant of the
circuit experimentally while viewing one period of the voltage waveforms? Take a screenshot of this to
demonstrate your result, including measuring the appropriate time difference and displaying it as was
done earlier.
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Part 2: Second-Order RLC-Circuit
1. Build the following circuit (exact value of inductor pending w.r.t. what BELS can provide us).
L1
10mH
C1
1uF
R1
1k Ohms
V1
2. For the source voltage, set the function generator to sinewave, with amplitude 2 V.
3. Connect channel 2 to measure the resistor voltage, and measure the peak-to-peak voltage across the
resistor for the following frequencies:
f = 80, 90, 100, 200, . . . , 1k, 2k, . . . , 10k, 20k, 30k, 40k Hz.
At what frequency does the voltage seem to reach a maximum?
4. From the circuit, calculate the resonance frequency. Theoretically, what should the peak-to-peak voltage
be across the resistor at this frequency? Compare it with your data and discuss.
5. Plot 20 log(VR /Vin ) v.s. frequency (VR =resistor voltage peak-to-peak, Vin =input voltage peak-to-peak),
and set frequency on a logarithmic scale.
6. Obtain the theoretical amplitude ratio for the voltages above, and plot on the same axes as the plot in
step 5.
7. Use theory and insight gained in this experiment to explain how the plots for the voltage across the
inductor and the capacitor look versus frequency will look.
8. How would you rearrange and connect the circuit and scope to verify your answer from step 7? Draw
detailed circuit diagrams, including how the scope would be connected, and what functions would be
used on the scope, if any.
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