ENME 482L Lab 3: Second Order Systems
By
Aaron Caprarola, Justin Mitchell,
& Anthony Snukis
Conducted: October 3, 2006
Instructor: Dr. Uri Tasch
T.A: Priya Narayanan
INTRODUCTION
The objective of this lab was to study the effects of input voltage and frequency
on the outputs of second order RLC (Resistor-Inductor-Capacitor) circuits and find the
time domain and frequency domain response for the circuits. The first part of the
experiment was to use a D.C. power supply and oscilloscope to capture the response of
the capacitor voltage from the DC source with various sized resistors. Emphasis was
placed on the transient response for different sized resistors. The second part of the
experiment utilized the same procedures as the first only the source was replaced with a
function generator. The magnitude and phase of the output voltage in relation to the
input was the main relevance of this part.
METHODS & MATERIALS
A National Instruments ELVIS system was used to conduct this experiment. The
ELVIS system is an integrated signal generator, oscilloscope, and breadboard.
In
addition, an HP oscilloscope was used. For the first part of this experiment, the time
domain of an RLC circuit was analyzed. The circuit in Figure 1 [I] was constructed.
Channel one of the HP oscilloscope was connected to measure the input voltage produced
by the DC source. Channel two was connected to measure the capacitor’s voltage.
R1, R2, R3
5V /
FuncOut
INPUT
L=58mH
CH A+
C=1F
Groun
d
R1 = 10 
ACH
0+
OUTPUT
ACH
0-
CH B+
R2 =
330 
R3 = 1000

(RL = 88  )
CH B-
Figure 1: RLC Circuit with varying resistors and inputs.
In order to determine the capacitor’s response to a step input, a 5V DC input signal
was suddenly applied, and the results captured on the oscilloscope. This step was
repeated for each of the three different resistors whose values were 10  , 330  , and
1000  respectively. It is important to note that these were the provided values not the
actual values of the resistors.
The second part to the experiment was to analyze the frequency response of the
circuit to an AC input signal.
To accomplish this only the source of the input was
changed from 5V DC to the FuncOut on the NI ELVIS board, the rest of the circuit
remained the same as shown in Figure 1. The input signal was set to a 2 volts peak-topeak sine wave, at a frequency of 5 Hz. The amplitude of the capacitor voltage was
measured using the oscilloscope, as well as the phase difference compared to the input
signal. To observe the behavior of the RLC circuit at different input frequencies, this
procedure was repeated at frequencies of 10 Hz, 100 Hz, 150 Hz, 200, Hz, 500 Hz, 1
kHz, 10 kHz and 100 kHz for each of the three resistors used in the first part of the
experiment.. The resulting output signals observed on the oscilloscope were compared to
the theoretical values.
Results
In the first part of this lab, the time response of a second order RLC circuit was
captured, using three different resistors.
Using three different resistors gave three
different damping ratios, ξ. Table 1, given below, tabulates the resistors and their
respective damping ratios, calculated from the transfer function, equation 1 [II].
Table 1
R (ohms)
ξ
10
330
1000
0.021
0.685
2.076
1
Vout ( s)
 n2
LC
G(S ) 

 2
R
1
Vin ( s)
s  2 n   n2
2
s  s
L
LC
(eq.1)
The 10 ohm and 330 ohm resistor both gave values of ξ less than 1, therefore making the
circuit underdamped. The 1000 ohm resistor gave a value of ξ greater than 1, making the
system overdamped. Using this theory, the transfer function predicts that the 10 ohm and
330 ohm resistors should oscillate when an input is applied, while the 330 ohm resistor
will ease into its steady state value. Although the three different resistors provide three
different damping ratios, the natural frequency remains the same for all three. The
natural frequency was calculated from the transfer function to be 4,152.3 rad/sec.
The first circuit was constructed using the 10 ohm resistor. A 5 volt input voltage
was applied. The acquired data was plotted using Matlab, as shown in Figure 2.
Voltage vs. time; 10 ohm resistor
9
8
7
Output Voltage (Volts)
6
5
4
3
2
1
0
0.005
0.01
0.015
time (seconds)
0.02
0.025
0.03
Figure 2: Voltage vs. time, 10 ohm resistor
The plot in figure 2 demonstrates the behavior expected of an underdamped second order
circuit to a step response. There is initially a transient response, which is the oscillation
present immediately after the input is applied. As time goes on, the transient response
disappears, and the only response present is due to the forced response, or the step input.
Using the Matlab generated plot in figure 2, values can be found for the rise time,
settling time, percentage overshoot, and peak time. Also, using the values of the damping
ratio and natural frequency previously found, the same values can be calculated
theoretically.
Table 2 presents the values found by measurement, as well as the
calculated values.
Rise time (s)
Peak time (s)
Settling time (s)
% Overshoot
Table 2
Measured
2.46E-04
6.14E-04
5.04E-03
65.13%
Theoretical
≈2.65E-4
7.568E-04
4.587E-02
93.61%
The measured rise time and peak time are both similar to their respective theoretical
values. However, the measured settling time is much quicker than the calculated value.
One possible source of this error is that only the impedances of the inductor, capacitor,
and resistor were taken into account. There are also impedances due to the breadboard
and input/output connections. For the same reason, the percentage overshoot is less than
the theoretical value.
During the next part of the experiment, the ten ohm resistor was replaced with a
330 ohm, and the previous procedure was repeated. Although the circuit is theoretically
underdamped, it was found that it behaved more like a critically damped one. The
voltage rose to its steady state value very quickly, with no overshoot or oscillation. The
time response is shown in Figure 3.
Voltage vs. time, 330 ohm resistor
6
5
Output Voltage (Volts)
4
3
2
1
0
0.005
0.01
0.015
time (seconds)
0.02
0.025
0.03
Figure 3: Voltage vs. time, 330 ohm resistor
The measured values for the settling time and rise time are presented with the theoretical
underdamped values in Table 3. Although the circuit is theoretically underdamped, table
two indicates that the percentage overshoot is theoretically only 5.21%, which is much
closer to 0 than the circuit with the 10 ohm resistor, whose value was 93.61%. This
circuit does not behave ideally for the same reason as before. The impedances not
accounted for would further damp this system, possibly to the extent where the damping
ratio is greater than one.
Rise time (s)
Peak time (s)
Settling time (s)
% Overshoot
Table 3
Measured
5.42E-04
N/A
7.89E-04
N/A
Theoretical
≈4.82E-4
1.038E-03
1.406E-03
5.21%
For the final portion of the first part of the experiment, a 1000 ohm resistor was
used in the circuit. This gives a damping ration of 2.076, or an overdamped system. The
5 volt input was applied as before, and the data was acquired. The resulting time
response is shown in Figure 4.
Output Voltage vs. time, 1000 ohm resistor
6
5
Output Voltage (volts)
4
3
2
1
0
0.005
0.01
0.015
time (seconds)
0.02
Figure 4: Voltage vs. time, 1000 ohm resistor
0.025
0.03
The response in figure 4 indicates that the circuit behaves like an overdamped one, as
expected. The input is applied at time t = 0, and the output voltage rises to its steady state
value. The measured values of the rise time and settling time are presented in Table 4.
The values confirm the analysis. The overdamped system in figure 3 has a larger rise
time than the critically damped circuit in figure 2. A critically damped circuit has the
fastest possible rise time without oscillation. Although the overdamped system is second
order, it behaves in a similar matter as a first order circuit.
Table 4
Rise time (s)
Settling time (s)
Measured
5.42E-04
7.89E-04
Conclusion
In the first part of this lab, the objective was to study the response of the voltage
across a capacitor in an RLC circuit as a function of input. It was shown that the
response consisted of a transient and steady state part. The type of transient response was
dependent on the resistor used. It was shown that for resistors with small values, the
system was underdamped, and the transient response of the output was a voltage
oscillating around its steady state value. As the value of the resistance was increased to
its critically damped value and beyond, the transient response involved no oscillation, and
the rise time increased.
The transfer function was used to derive the damping ratio, which enabled us to
predict the behavior of the system. The 10 ohm circuit produced an underdamped
response, as expected. The 1000 ohm resistor produced an overdamped response, also as
expected. However, the damping ratio of the 330 ohm circuit was found to be 0.685,
implying that the circuit should be underdamped, but it behaved as a critically damped
one instead. One way to correct this discrepancy, and improve the other results, would be
to measure the impedance of the other components in the circuit. Examples of some of
these components include the breadboard and connectors.
Reference
[I]
Tasch, Uri
Second Order Systems
03 October 2006
[II]
Nise, N.S.
Control Systems Engineering
Fourth Edition, 2004
Contributions
Snukis, Anthony
Title page, Introduction, Methods
Caprarola, Aaron
Time Domain Analysis; Results, Conclusion, Editing
Mitchell, Justin
Frequency Domain Analysis; Results, Conclusion