Download ET 304b

ET 304b
Laboratory 5
Superposition Theorem With Ac and Dc Sources
Objective: Use the Superposition theorem to find the total response to sinusoidal ac and dc
inputs. Use an oscilloscope to measure ac waves with a dc offset. Measure composite signals
with digital multimeters and determine the rms values of the ac and dc components.
Theoretical Background
The superposition theorem is a technique for solving electric circuits that have more than
one source. Nodal and mesh analysis also can find solutions for complex networks that include
multiple sources, but require the solution of an entire set of voltages or currents. Another
limitation of these methods is that the sources must all be of the same current type: ac or dc. The
superposition theorem does not have this limitation, which makes it a valuable technique in
electronic circuit design.
The superposition theorem only applies to linear circuits and linear circuit responses. A
circuit must contain linear elements to be linear. A linear circuit element has a proportional
output for a given input. Components that follow Ohm's law are all linear elements. All
components covered in the lecture qualify as linear elements, including dependent sources that
have a linear relationship between their output and the controlling parameter. Linear responses
are the current through any branch and the voltage drop across any component. Power can not
be found directly by superposition because it is a non-linear function of current or voltage and
the component value.
Superposition is implemented by first identifying the circuit response of interest. This is
the voltage or current for which we will solve the circuit. Unlike the other network techniques it
will not be necessary to solve for other variables to find this response. The procedure for using
superposition is:
1.) de-active all independent sources except one. Replace de-activated voltage sources
by short circuits and de-activated current sources by open circuits.
2.) Calculate the desired response of the circuit for the remaining source.
3.) Repeat steps 1 and 2 until responses are found for all sources in the circuit.
4.) Add the individual response from each sources to get the total response to all the
circuit sources.
Superposition is especially useful in finding the response of a circuit to combined ac and dc
inputs. This is typical in electronic circuits.
Electronic circuits use active components such as transistors and integrated circuits to
amplify ac signals. A dc voltage is necessary for the active components to operate. The output
is generally a composite signal that has both a dc and ac component. In practice it is not possible
to de-activate the dc source and measure the ac response because the dc bias is necessary for
circuit operation. It is possible to measure the dc output only with the ac input de-activated.
The oscilloscope can effectively measure composite ac and dc signals. It is also possible
to measure the ac component separately using the oscilloscope. To isolate the ac signal from the
dc level, the vertical coupling switch must be in the proper position. The vertical coupling switch
is located below the vertical inputs and has three positions: ac, gnd, dc. The gnd position
grounds the input. The scope channel is referenced to zero with the input in this position. The
Spring 2002
exp504b.doc
other two positions determine how the input signal couples to the scopes vertical input amplifier.
Figure 1 shows that with the switch in the ac position the input connects to the vertical amplifier
through a capacitor, Cb.
Vert. Amp.
input
Cb
ac
To vertial
deflection circuits
dc
Figure 1. Vertical Coupling Switch Circuit.
The capacitor blocks the dc component of the input signal while passing the ac signal. Ac
coupling makes measurement of low level ac voltages possible even if a significant dc
component is present.
Moving the switch to the ac position will block the dc component and center the ac wave
about the ground position on the display. The ac portion can now be amplified if necessary by
decreasing the V/div scaling of the scope channel.Placing the coupling switch in the dc position
displays both the dc levels and the superimposed ac level.
Ac coupling will center any waveform that has a dc component, even pulses. The pulse
in Figure 2-a goes from 0 to +V when dc coupled to the scope. Ac coupling centers this wave
about the ground reference, as shown in Figure 2-b. This is not the correct display of the wave
since it only goes positive.
v(t)
v(t)
+V
+V/2
t
t
-V/2
b.)
a.)
Figure 2. Effects of Ac Scope Coupling on Positive Voltage Pulses.
Digital multimeters can also measure composite signals, but meter construction and the
type of waveform are a consideration. True rms digital multimeters display the rms values, or
effective value, of the total ac and dc components of the signal. The rms value of time varying
waves depends on the shape of the signal. True rms meters accurately respond to any waveform
and display the correct rms value. Average responding multimeter designs assume sinusoidal ac
inputs and calibrate the output to read correctly. Measuring a non-sinusoidal ac wave with an
average responding multimeter produces erroneous results.
Before measuring ac waves with a DMM, determine whether the instrument is a true rms
or average responding device. Also determine the frequency response limit of the instrument.
Instrument instruction manuals usually contain this information.
Spring 2002
2
exp504b.doc
Procedure
Before starting the lab, install 10x probes to both channels of the scope and
compensate them. Obtain frequency response information about the DMM used for
making the following measurements. Also determine if the DMM is true rms or average
responding.
1.)
Construct the circuit in Figure 3. Use the source transformation theorem to find an
equivalent ac voltage source for the given current source.
Vs1 12Vdc
R1
7.2k
R3
3.3k
Is1
2.5mA peak
f=5.0kHz
R2
2.2k
+
Vc
C
0.02uF
-
Figure 3. Lab Circuit 1.
2.)
3.)
4.)
5.)
6.)
De-activate the 12 Vdc source and replace it with a short circuit according to the
superposition theorem. Measure the resulting voltage across the capacitor, Vc using the
scope with dc coupling and record the result in Table 1. Make the same measurement
using a DMM and record the result in Table 1. Find the rms value of the scope
measurement by using the formula: Vrms =0.707Vpeak.
De-activate the equivalent ac voltage source and replace it with a short circuit. Activate
the dc source and measure the dc voltage appearing across the capacitor with a
oscilloscope and DMM. Keep the scope in the dc coupling mode. Record these
measurements in Table 1.
Activate both voltage sources in the circuit. With the scope in the dc coupling mode,
measure the voltage across the capacitor. Sketch the trace on the grids provided.
Determine the dc level by measuring the vertical offset from ground.
Switch the scope input to ac coupling and observe the waveform of Vc. Sketch the trace
on the grids provided. Measure the peak value of the wave and convert it to rms by using
the formula in step 2.
Solve the circuit in Figure 1 using the superposition theorem. Find the rms value of the ac
response and the dc level. Enter these values in Table 1. Calculate the percentage error
between the theoretical values and the measured values of the scope and the DMM
measurements using the following formula:
Spring 2002
3
exp504b.doc
7.)
Vtheoretical  Vmeasured
 100%  %error
Vtheoretical
Record the results in Table 2.
Construct the circuit in Figure 4. The source Vs1 is a 5.0 kHz, 3 V peak sine wave.
L
R1
1k
3 Vp
f=5.0 kHz
Vs2
Rc
100mH
R2
7.2k
+
VR
Vs1
+
5Vdc
C
0.01uF
R4
4.7k
-
-
.
Figure 4. Circuit 2.
8.)
9.)
10.)
11.)
12.)
The resistor, Rc, is the dc resistance of the inductor. Measure this value with the
multimeter and record it for later use.
De-activate the dc source and replace it with a short. Measure the ac voltage that appears
across the resistor R4,VR, with the scope. Make sure that the scope is dc coupled for this
measurement. Also measure the ac voltage at VR with the DMM. Record both of these
values in Table 3.
De-activate the ac source and replace it with a short. Activate the dc source. Measure
the dc voltage that appears across resistor R4 with the scope and the DMM. Record these
values in Table 3.
Activate both voltage sources in the circuit. With the scope in the dc coupling mode,
measure the voltage across the resistor, R4. Sketch the trace on the grids provided.
Determine the dc level by measuring the vertical offset from ground.
Switch the scope input to ac coupling and observe the waveform of VR. Sketch the trace
on the grids provided. Measure the peak value of the wave and convert it to rms by using
the formula in step 2.
Solve the circuit in Figure 2 using the superposition theorem. Find the rms value of the
ac response and the dc level. Enter these values in Table 3. Calculate the percentage
error between the theoretical values and the measured values of the scope and the DMM
using the formula in step 6. Enter the values in Table 4.
Spring 2002
4
exp504b.doc
Discussion Points
What is the procedure for finding voltages and currents using superposition? How does the
vertical coupling effect the display of the measured voltage? What type of coupling should be
used to measure dc signals with the scope? Low level ac with high level dc? Ac voltages only?
Do the theoretical values match the measured values within component tolerances? What type of
DMM is used for the experiment: true rms or average responding?
Spring 2002
5
exp504b.doc
Table 1. Circuit 1 Measurements.
Vc (ac rms)
Vc (dc)
Vc total response
Scope
DMM
Calculated
Table 2. Error Percentages for Circuit 1.
% Error DMM
% Error Scope
Vc (ac rms)
Vc (dc)
Circuit 2 Rc = __________
Table 3. Circuit 2 Measurements.
VR (ac rms)
VR (dc)
VR total response
Scope
DMM
Calculated
Table 4. Percentage Error Circuit 2.
% Error DMM
% Error Scope
VR (ac rms)
VR (dc)
Spring 2002
6
exp504b.doc
Channel 1 Volts/div ______ Channel 2 Volts/div _____ Time/div __________
Channel 1 Volts/div ______ Channel 2 Volts/div _______ Time/div __________
Spring 2002
7
exp504b.doc
Channel 1 Volts/div ______ Channel 2 Volts/div _____ Time/div __________
Channel 1 Volts/div ______ Channel 2 Volts/div _____ Time/div __________
Spring 2002
8
exp504b.doc