Lab 4 - First Order Transient Response of Circuits
Lab Performed on October 22, 2008 by Nicole Kato, Ryan Carmichael, and Ti Wu
Report by Ryan Carmichael and Nicole Kato
E11 Laboratory Report – Submitted November 7, 2008
Department of Engineering, Swarthmore College
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Abstract:__________________________________________________________
Different circuits were assembled from a voltage source, resistor, and a single energystoring passive element, either a capacitor or an inductor. A circuit that contained a capacitor,
voltage source, timer, potentiometer, speaker and resistors, was also assembled using a
breadboard. Tests were then performed to examine the behaviors of Vin and Vout over time for
each configuration, and to determine how the time constant varied with changes in the magnitude
of the capacitance or inductance of the passive elements. The experimental results were, for the
most part, close to the predicted, theoretical values.
Introduction:_______________________________________________________
Capacitors and inductors are two of three passive elements that can be used in a circuit.
Ideally, these two passive elements are not able to dissipate or generate energy, but can store and
release energy in a circuit. Capacitors and inductors can be used in very simple circuit designs,
such as that for a blinking light, or in very advanced designs, such as those used in radio
reception. In addition to being unable to dissipate energy, ideal capacitors do not have inductance
or resistance, and the voltage across the element cannot change instantaneously. Similarly, ideal
inductors do not have capacitance or resistance, and the current across this element cannot
change instantaneously. However, in reality, both capacitors and inductors exhibit small amounts
of characteristics that they ideally would not have.
Theory:___________________________________________________________
A capacitor (C) is composed of two
closely and evenly spaced conducting plates. If
there is no charge placed on a capacitor, it will
remain neutral. Initially, when a charge is placed
on a capacitor, it will act as a short circuit. Over
time, one plate of the capacitor will become
Figure 1: Capacitor Symbol
more positively charged and the other more
negatively charged. This allows the capacitor to store energy. When the capacitor cannot store
any more energy, ideally, it will not allow anymore current to go through, as if it were an open
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circuit. If the energy source is removed, the capacitor will begin to release its stored energy to
keep the energy of the circuit constant until it runs out of energy.
An inductor (L) is composed of a conducting wire that is tightly wound or coiled. If there
is no charge placed on an inductor, it will remain neutral. Initially, when a charge is placed on an
inductor, it will act as an open circuit. Over time, one side
of the inductor will become positively charged and the
other negatively charged. This causes the inductor to store
Figure 2: Inductor Symbol
energy. When the inductor cannot store any more energy,
ideally it will allow all of the current to flow through, as if
it were a short circuit. If the energy source is removed, the inductor will also begin to release its
stored energy to keep the circuit constant until it runs out of energy.
A circuit with only one type of passive element that stores energy is called a first order
circuit– or more specifically, in our case, an RC or RL circuit. An RC circuit is composed of a
resistor and a capacitor, while an RL circuit is composed of a resistor and an inductor. If a circuit
is a first order circuit, we can find either the current or voltage for t greater than zero using the
following equation:
X(t) = X(∞) + ( X(0+) - X(∞) ) e^(-t / τ)
(Equation 1)
X(∞) is the value of either current or voltage when time is infinity, X(0+) is the value of
either the current or voltage right after t, or time becomes 0. τ, the time constant, is RC for an RC
circuit and L/R for an RL circuit.
-
+
For a capacitor, we know that the voltage from 0 to 0 must be equal because it cannot
-
+
change instantaneously. For an inductor, we know that the current from 0 to 0 must be equal
because it cannot change instantaneously. We know that when a capacitor cannot store any more
energy, it will act as an open circuit (so when t goes to ∞, the current through the capacitor must
be 0). We know that when an inductor cannot store any more energy, it will act as a short circuit
(so when t goes to ∞, the voltage through the inductor must be 0).
In circuit 1.a, we know that the resistor is 1kΩ and the capacitor 1µF; therefore, we can
find the time constant as RC or 0.001s. If we are looking at the rising edge of the circuit, Vin will
initially be zero at 0- and will become 1 at 0+. At Vc(0-), the voltage is zero since there is no
voltage going through the capacitor. Since we also know that the voltage across a capacitor
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cannot change instantaneously because the capacitor
initially acts as a short circuit, when Vin initially
equals 1V the Vc(0+) is also zero. At time infinity, the
capacitor will act as an open circuit and Vc(∞) will
become 1V at steady state. Using equation 1 we can
see that Vc(t) is equal to V(∞) + ( V(0+) - V(∞) ) e^(-t
/ τ) or 1 + (0-1)e^(-t/.001) V.
In circuit 1.b, we know that the resistor is 2kΩ
Figure 3: Circuit 1.a (Vin = 1V,
R = 1kΩ, C = 1µF)
and the capacitor 1µF therefore we can we can find
the time constant as RC or 0.002s. At Vc(0-) the
voltage is zero since there is no voltage going though
the capacitor. Since we also know that the voltage
across a capacitor cannot change instantaneously
because the capacitor initially acts as a short circuit,
when Vin initially equals 1V the Vc(0+) will remain at
zero. At time infinity, the capacitor will act as an open
circuit and Vc(∞) will become 1V at steady state.
Figure 4: Circuit 1.b (Vin = 1V,
R = 2kΩ, C = 1µF)
Using equation one we can see that Vc(t) is equal to 1 + (0-1)e^(-t/.002) V.
In circuit 1.d, we know that the resistor is 1kΩ
and the capacitor 1µF therefore we can we can find
the time constant as RC or 0.001s. If we are looking at
the falling edge of the circuit, Vin will initially be 1V
at 0- and will become 0V at 0+. At Vc(0-) the voltage
is 1 since Vin has been going through the capacitor for
a long time and the capacitor has reached a steady
Figure 5: Circuit 1.d (Vin = 1V,
R = 1kΩ, C = 1µF)
state. This also means that Vc(0+) will remain 1V
when Vin becomes 0V. At time infinity the capacitor will discharge and Vc(∞) will go to 0V.
Using equation one we can see that Vc(t) is equal to 0 + (1-0)e^(-t/.001) V.
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In circuit 2.a, we know that the resistor is 1kΩ
and the capacitor 1µF therefore we can we can find the
time constant as RC or 0.001s. If we are looking at the
rising edge of the circuit, Vin will initially be zero at 0and will become 1 at 0+. At Vc(0-), the voltage is zero
as there is no voltage going through the capacitor. Since
we also know that the voltage across a capacitor cannot
change instantaneously because the capacitor initially
Figure 6: Circuit 2.a (Vin = 1V,
R = 1kΩ, C = 1µF)
acts as a short circuit, when Vin initially equals 1V, the Vc(0+) is zero. At time infinity, the
capacitor will act as an open circuit and Vc(∞) will become 1V at steady state. Using equation
one, we can see that Vc(t) is equal to 1 + (0-1)e^(-t/.001) V.
In circuit 3.a, we know that the resistor is 1kΩ
and the inductor is 112mH therefore we can we can find
the time constant as L/R or 0.000112s. If we are looking
at the rising edge of the circuit, Vin will initially be zero
at 0- and will become 1V at 0+. At t=0- we know that if
there is no voltage going though the circuit, the current
will be zero, so VL(0-) will be 0V. At 0+ the Vin will
Figure 7: Circuit 3.a (Vin = 1V,
R = 1kΩ, L = 112mH)
become 1, but the inductor will act as an open circuit so
there will be no current going through the inductor and therefore the voltage will remain 0V. At
VL(∞), the inductor will act as a short circuit, current will become Vin /R which means that
VL(∞) will be Vin or 1V. Using equation one, we can see that Vc(t) is equal to 1 + (0-1)e^(t/.000112) V.
Circuit 4.a can be redrawn as circuits 4.b and
circuit 4.c. We are given the information that when Vc
is less that V1, the switch opens, but when Vc is greater
than V2, the switch closes. When the switch is open, we
can see that the Vc is charging the capacitor to V2 and
when the capacitor has reached V2, the switch opens to
allow the capacitor to dissipate back to V1. The
capacitor dissipating energy causes a sound from the
Figure 8: Circuit 4.a (Ra = 33kΩ,
Rb = 20kΩ, C = 0.1µF, Vcc = 5V)
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speaker. With this information we can use circuit 4.c to find that the time constant, τ1, before Vc
is greater than V2 is (RA + RB)C or 5.30 ms, and that after the switch closes the time constant, τ2,
after Vc is greater than V2 is RBC or 2.00 ms.
Looking at circuit 4.b, we can find the current to be 5V/15KΩ or 1/3mA. We can use this
information to find that V2 is equal to 5V-5kΩ/3mA or 3.33V, and V1 is equal to 5V –
2(5kΩ/3mA) or 1.67V. Using the
equation X(t) = X(∞) + ( X(0+) X(∞) ) e^(-t / τ) we can solve for
the voltage before the switch
closes as V(t) = 5 + (1.67 – 5)e^(188.7t1), and when we solve for t
when V(t) is 3.33, we get t1 to be
0.003674s. We can use the same
formula to get the equation V(t) =
3.33e(-500t2) and solve for t2
when V(t) is equal to 1.67 to get
Circuit 4.b
Circuit 4.c (Switch closes when Vc > V2)
Figure 9: Circuits 4.b and 4.c (Ra = 33kΩ, Rb = 20kΩ,
C = 0.1µF, Vcc = 5V)
t2 to be 0.001386. We can now solve for the frequency, fosc to be 1/(t1 + t2) or 197.6 s-1.
Procedure:_________________________________________________________
1. First we connected the Wavetek signal generator, set to produce a square wave function
that varied approximately between 0 and 1 volts to the oscilloscope.
2. We created Circuit 1a, as shown in the theory, and connected Vin from the function
generator. We also set the resistor to 1kΩ, and the capacitor to 1µF. *Note: our 1a is the
lab handout procedure’s 1d. In addition, our 1b is based off of our 1a not the lab handout
procedure’s 1a. We apologize for the confusion.
3. For more precise measurements, we made the scale on the oscilloscope as large as
possible, and set the triggering so that the rising edge of the Vin was displayed at the
center of the screen.
4. We took and recorded the following measurements:
a. The initial and final measurements of Vin and Vout
b. The time constant where the Vout has progressed through 63% of its period. .
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5. We downloaded and saved the following items:
a. The screenshot of Vin and Vout with the measured time constant
b. The data from Vout
6. We then doubled the resistance to 2kΩ and repeated step 4.
7. After, we predicted and observed what happens if the capacitance is doubled to 2 µF and
the resistance returned to its original value.
8. We returned the capacitance and resistance to their original values and saw what would
happen if we set the triggering to the minimum edge of the input at the center of the
screen, and then repeated steps 4 and 5.
9. We assembled Circuit 2 from the theory and set the resistance to 1kΩ and the capacitance
to 1 µF, with the function generator providing Vin.
10. We then repeated steps 3 through 8.
11. We created Circuit 3 from the theory and set the resistance to 1kΩ and the inductance to
112 mH, with the function generator providing Vin. Also, we set the function generator’s
frequency so that Vout would reach steady-state before Vin changed.
12. We then repeated steps 3 through 6.
13. We created Circuit 4 using a breadbox, and connected it to an oscilloscope.
14. We watched pin 6 and 3 in the oscilloscope for one period.
15. We then downloaded and saved a screenshot of the oscilloscope, and the data from pin 6.
16. We also measured and recorded the frequency of pin 6.
17. We determined why the circuit oscillated, and saw what happened when we put our
fingers across the leads of the resistors and capacitor.
18. We connected a 555 oscillator to the oscilloscope and attached the potentiometers at the
midpoints.
19. We measured the frequency of the squealy oscillator when:
a. The button was not pushed
b. The button was pushed and
i. LED of blinky stage was on
ii. LED of blinky stage was off
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Results:___________________________________________________________
Circuit
Initial-Final Input Value
Initial-Final Output Value
Time Constant
(Cursor Measurement)
Time Constant
(Curve Fit)
Time Constant
(Theoretical)
1A
1B
1D
2A
3A
4
1.02 V 1.04 V 1.02 V 1.02 V 1.02 V
3.88 V
1.03 V 0.920 V 1.04 V 0.960 V 0.960 V 2.24 V
1.22 ms 2.00 ms 1.00 ms 1.09 ms 0.128 ms
-----1.15 ms
------
------
1.07 ms 0.112 ms 1.98 ms,
4.62 ms
1.00 ms 2.00 ms 1.00 ms 1.00 ms 0.112 ms 1.39 ms,
3.67 ms
Table 1: Results Table
In addition, the frequency of oscillation for circuit 4 is 196.4 Hz vs. a theoretical 197.7 Hz
Figure 10: Circuit 1a Vin, Vout vs. Time
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Figure 11: Circuit 1a Vout vs. Time Curve Fit
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Figure 12: Circuit 1b Vin, Vout vs. Time
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Figure 13: Circuit 1d Vin, Vout vs. Time
*We erred with the cursor method. The vertical cursor line should have been placed around 1 ms
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Circuit 2a: Oscilloscope Output with Cursor Determination of τ
Voltage
CH1 = Input
CH2 = Output
Time
Figure 14: Circuit 2a Vin, Vout vs. Time
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RC Circuit 2a Curve Fit
Voltage
(Volts)
Circuit 2a curve fit
General model Exp1:
f(x) = a*exp(b*x)
Coefficients (with 95% confidence bounds):
a=
0.913 (0.9121, 0.9138)
b=
-933.8 (-935.2, -932.4)
τ = -b -1 = 0.001071 (0.001069, 0.001073)
Goodness of fit:
SSE: 0.4912
R-square: 0.9979
Adjusted R-square: 0.9979
RMSE: 0.009954
Time (seconds)
Figure 15: Circuit 2a Vout vs. Time Curve Fit
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Circuit 3a: Oscilloscope Output with Cursor Determination of τ
Voltage
CH1 = Input
CH2 = Output
Time
Figure 16: Circuit 3a Vin, Vout vs. Time
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LC Circuit 3a Curve Fit
Voltage
(Volts)
Circuit 3a curve fit
General model:
f(x) = a*(1-exp(b*x))
Coefficients (with 95% confidence bounds):
a=
0.8528 (0.8524, 0.8532)
b=
-8947 (-8977, -8917)
τ = -b -1 = 0.0001117 (0.0001114, 0.0001121)
Goodness of fit:
SSE: 0.6301
R-square: 0.9934
Adjusted R-square: 0.9934
RMSE: 0.01127
Time (seconds)
Figure 17: Circuit 3a Vout vs. Time Curve Fit
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Circuit 4: Oscilloscope Output for the 555 Oscillator
Voltage
CH1 = Output
CH2 = Input
Time
Figure 18: Circuit 4 Vin, Vout vs. Time
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Circuit 4 Dissipating Energy Curve Fit
General model:
f(x) = a+(b-a)*exp(-x/τ2)
Coefficients (with 95% confidence
bounds):
a = 0.1849 (0.08446, 0.2853)
b=
3.528 (3.524, 3.533)
τ2 = 0.00198 (0.0019, 0.002061)
Goodness of fit:
SSE: 0.7867
R-square: 0.9967
Adjusted R-square: 0.9967
RMSE: 0.02537
Voltage
(Volts)
Time (seconds)
Figure 19: Circuit 4 (Dissipating) Vout vs. Time Curve Fit
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Circuit 4 Storing Energy Curve Fit
Voltage
(Volts)
General model:
f(x) = a + (b-a)*exp(-x/τ1)
Coefficients (with 95% confidence bounds):
a=
4.757 (4.724, 4.791)
b=
1.878 (1.875, 1.881)
τ1 = 0.004614 (0.004535, 0.004692)
Goodness of fit:
SSE: 2.425
R-square: 0.997
Adjusted R-square: 0.997
RMSE: 0.02542
Time (seconds)
Figure 20: Circuit 4 (Storing) Vout vs. Time Curve Fit
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Discussion:_________________________________________________________
In general, our experimental results matched up well with our theoretical results. In fact,
for circuits 1b, 1d, and 3a we were able to match our theoretical values to three or more
significant figures. For circuits 1b and 1d, this accuracy was somewhat due to luck as the cursor
method used to obtain τ isn’t nearly accurate enough to expect such precise results. For the other
circuits this method erred by an average of about 11%. For circuit 3a the cursor method gave
about a five percent error. The theory-matching result was obtained from a very accurate curve
fit that can be seen in Figure 17. A curve fit of such precision was unexpected as the errors that
effected circuit 2a (which also had a good curve fit) somehow canceled.
The remaining circuits ranged from a reasonable 7% error for circuit 2a to a mediocre
15% error for circuit 1a to rather large 26% and 42% errors for τ in circuit 4. For circuit 1a, the
major error was a curve fit that did not accurately represent the oscilloscope data. As seen in
Figure 11, the curve fit doesn’t intersect any of the data points until about 0.2 V and after this
point the fit ranges between the upper and lower regions of the data points, failing to provide an
accurate fit.
Unlike in circuit 1a, fitting accuracy does not appear to be a major source of error for
circuit 2a. Instead, the major error for this circuit appears to come from the discrepancy between
real and ideal circuit elements. This can particularly be seen in the ideally flat square waves that
take a rounded shape due to the inability of the voltage source to make a perfect jump between
voltages. In addition, the voltage source has some impedance, which is a major contributor to the
error due to a difference between real and ideal circuit elements. Creating a curve fit for the data
helps to limit this error, but does not eliminate it. This error also appears in circuit 1a, but is
overshadowed by the somewhat more evident curve fit error.
Circuit 4 similarly has error due to nonideal circuit elements, but the time constant errors
are much larger than for circuit 2a because the circuit contains two time constants. However, the
frequency obtained from these time constants is very similar to the theoretical frequency (within
0.7% error). This occurs because, when the voltage source changes voltages, it overshoots briefly
before settling at its assigned voltage. As there is an overshoot in both the positive and negative
directions, the time constant will be greater than the theoretical value for one direction of the
jump, and smaller for the other direction. Additionally, each overshoot duration will be
approximately equal and opposite, so the sum of the time constants (or the period) will not be
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greatly altered from the theoretical value. As the period will remain the same, its inverse, the
frequency, will also remain near the theoretical value.
Conclusion:________________________________________________________
In conclusion, for the six major circuits evaluated (1a, 1b, 1d, 2a, 3a, and 4), we were
able to obtain our theoretical time constant values experimentally for three of the circuits, while
the other three circuits had varying levels of accuracy for the time constant value. All of these
errors, however, can be reasonably accounted by three major sources of error: poor curve fitting,
impedance in the voltage source, and the inability of a voltage source to make an ideal jump
between two voltages.
Acknowledgments:__________________________________________________
Cheever, Erik. "Curve Fitting with Matlab." Swarthmore College :: Home. 29 Oct. 2008
<http://www.swarthmore.edu/NatSci/echeeve1/Ref/MatlabCurveFit/MatlabCftool.html>.
Cheever, Erik. "E11 Lab (1st order time domain response) - Procedure." Swarthmore College
:: Home. 29 Oct. 2008
<http://www.swarthmore.edu/NatSci/echeeve1/Class/e11/E11L4/Lab4(Procedure).html>.
Cheever, Erik. "E11 Lab 3 (1st order time domain response) - PreLab." Swarthmore College ::
Home. 29 Oct. 2008
<http://www.swarthmore.edu/NatSci/echeeve1/Class/e11/E11L4/Lab4(Prelab).html>.
Cheever, Erik. "Measuring Time Constants on the TDS4013B Using Cursors." Swarthmore
College :: Home. 29 Oct. 2008
<http://www.swarthmore.edu/NatSci/echeeve1/Ref/TDS4013/MeasureTau/MeasureTau.h
tml>.
"Circuit Schematic Symbols." Oracle ThinkQuest Library . 29 Oct. 2008
<http://library.thinkquest.org/10784/circuit_symbols.html>.
"Electronics/Inductors - Wikibooks, collection of open-content textbooks." Main Page Wikibooks, collection of open-content textbooks. 29 Oct. 2008
<http://en.wikibooks.org/wiki/Electronics/Inductors>.
We would also like to thank Anne Krikorian for teaching us the way of the comma.
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Appendices:________________________________________________________
Figure 21: Button Not Pushed
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Figure 22: Button Pushed, Blinky On
*Note that the frequency is roughly half of the frequency with the button not pushed.
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Figure 23: Button Pushed, Blinky Off
*Note that the frequency returns to about that of when the button is not pushed.
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