### Capacitor Self-Resonance

```Experiment No. 2.
Voltage Dividers, DC & AC Signals, and Batteries
By:
Prof. Gabriel M. Rebeiz
The University of Michigan
EECS Dept.
Ann Arbor, Michigan
Read Chapters 1-3 in the additional course notes (Audio Electronics & Batteries).
Read this experiment and answer the pre-lab questions before you come to the lab.
1.0 Voltage Divider; KVL and KCL Rules:
Equipment:
Agilent 34401A Multimeter
Agilent E3631A Triple Power Supply
Experiment Set-up:
1. Measure the resistance of the 1 KΩ, 2.4 KΩ and 5.1 KΩ resistors using the
Agilent 34401A multimeter.
2. Connect the following circuit on your proto-board. You should have learned the
layout of the protoboard and how to put components in it during the Lab Lecture. If
in doubt, ask you lab instructor for help.
A
I
1.0 Kž
B
2.4 Kž
C
+
10 V
–
5.1 Kž
3. Connect the Agilent E3631A power supply to your circuit and set it at +10V.
4.
Using the Agilent 34401A multimeter, measure the node voltages VA, VB, VC
(with respect to ground).
Also, measure the voltage drop, VAB and VBC. Check that the KVL rule is
satisfied around the loop.
Compare the measured voltages with your pre-lab calculations.
5.
Break the circuit at node A (or node B or node C), and measure the current I
in the circuit using the Agilent 34401A multimeter.
Compare this value with your pre-lab calculation: Write down a brief statement
summarizing this comparison, but you may calculate the percentage difference in
6. Connect now the 3.3 KΩ and 4.7 KΩ resistors from node C to ground as shown
below:
A
1 Kž
B
2.4 Kž
C I3 3.3Kž
I1
+
10 V
–
1
D
I2
5.1 Kž
4.7 Kž
Measure the 3.3 KΩ and 4.7 KΩ resistors using the Agilent 34401A
multimeter.
Measure VA, VB, VC, VD with respect to ground. (Be sure that the Agilent
34401A multimeter is set for voltage measurement).
Measure the currents I1, I2, I3. Check that the KCL rule is satisfied. Write a
brief statement on the validity of KCL, based on your measurements.
2.0 Linear Circuits in Time Domain:
Turn off the power supply.
Disconnect your circuit board from the power supply (remove both cables with banana
plugs).
1. Set the Agilent 33120A to give a 10 kHz sinewave with Vppk = 2 V and connect it
to circuit of page 30. Use a cable with a BNC connector on one end and two
alligator clips (for the signal and ground connections) on the other end. Use probe
wires, connect them to node A and the ground of your circuit.
A
1 KΩ
B
2.4 KΩ
C I3 3.3 KΩ
I1
+
I2
5.1 KΩ
~
Vs = 10 KHz
D
4.7 KΩ
–
2.
Measure VBppk, VCppk and VDppk and plot the waveforms on your notebook
(voltage vs. time).
3. Set the source to give a 10 kHz triangular wave with Vppk = 2 V.
Measure VCppk and VDppk and plot the input (Vs) and output waveforms
(VC, VD) on your lab notebook.
(You will note that in linear circuits, the output signals have exactly the same
shape as the input signal but with different amplitudes.)
3.0 Batteries and Internal Resistance:
Equipment: Agilent 34401A Multimeter
As seen in Chapter 2 of Additional Course Notes, the battery voltage may be around
1.4-1.5V when open circuited (no current) and drops to 1.3–1.4V when in operation.
Actually, the battery voltage depends on the discharge current and time. A nice way to
model this voltage drop is to include an internal resistance (Rb) in the battery model.
Figure on p. 32 shows an ideal battery and a real battery. Notice that we cannot
access the inside nodes of Rb and therefore, the internal resistance is calculated from
measured external V/I values:
Rb
2
2
Vb
Ideal
Model
1
1
2
2
Vb
Real
Model
1
For a new C-type battery with a full rated capacity of 7100 mAH, the internal
resistance can be as low as 0.1-0.2 Ω at a 500 mA discharge current (50 mV–100 mV
drop). When the battery is 1/3 full (66% of the capacity is removed), the internal
resistance will increase to around 0.6 Ω (0.3 V drop). If the battery is nearly empty, the
internal resistance will be as high as 1–2 Ω resulting in a voltage drop of 0.5–1 V (from
a 1.5V source) and making the battery useless.
Experiment Set-Up:
1. Take two C-type batteries and connect them in series.
a. Measure the open-circuit voltage, Vb = Voc (no current from the batteries). It
should be around 2.5–2.9V, depending on the “age” of the batteries.
2. Connect a 10 Ω load resistor across the battery terminals.
3. Connect an ammeter in the circuit loop.
Remember:
loop.
Current is measured in series in a circuit
Voltage is measured in parallel across an
element.
Refer to p. 12-13 of this Manual.
The experiment should look like this:
For a load resistor of 10 Ω:
1.
Measure the current in the loop (I).
2.
Measure the voltage across the battery terminals (Vb).
3.
Repeat steps 1 and 2 every minute or so for a total of 3 measurements.
Calculate the internal resistance of the “battery source” for all three
measurements. Notice, this is the resistance of two C-cell batteries in
series. What is the internal resistance of one C-cell battery for each case?
4.0 Sawtooth Waveform in Time and Frequency Domain:
3
1. Set the Agilent 33120A waveform generator to give a 10 KHz, 3 Vppk sawtooth
waveform. Connect the function generator to the scope.
2.
Measure the waveform period and peak-to-peak amplitude using the cursors
function. Sketch the waveform (several periods) in your lab notebook. Pay
attention to the slope of the sawtooth since you will need it in your lab report.
3.
Go to the frequency domain (review p. 21-22 for the FFT set-up). Measure
the frequencies in kHz and amplitudes in dBV of the fundamental (fo) and the
harmonics (2fo, 3fo, 4fo, 5fo, 6fo). Note that unlike the square-wave
spectrum, the sawtooth spectrum does contain both even and odd
harmonics. Record your data in a table form.
4
Experiment No. 2.
Voltage Dividers, DC & AC Signals, and
Batteries
Pre-Lab Assignment
1. Calculate VA, VB, VC and I in the circuit of experiment #1 (on p. 30) for a 10V DC input
voltage.
2. a. Calculate the time it takes for the C-type battery with a full rated capacity of 7100 mAH
to reach its 1/3 capacity value, at a discharge current of 200 mA.
b. The measured open-circuit voltage of a battery (Vb = Voc) is 1.5 V. The measured
output voltage is 1.24 V for a 500 mA discharge current. Calculate the internal
resistance value (Rb).
c. A unit requires three AA 1.5 V batteries connected in this fashion. What is the
operating voltage (Vb) of the unit?
+
–
–
+
2
+
Vb
–
+
5
–
1
Experiment No. 2
Voltage Dividers, DC & AC Signals, and
Batteries
Lab Report Assignment
1. Using the measured values of the resistors, calculate VA, VB, VC and I for the circuit on p.
30. Compare with measurements (down to mV levels), and show difference in percent.
2. For the circuit with the additional 3.3 kΩ and 4.7 kΩ resistors (on p. 31), calculate VA, VB,
VC, VD, I1, I2, I3 and compare with measurements and show difference in percent.
Why did VC change? Explain using KVL and KCL (as clearly as you would do it on a midterm).
3. Summarize your measurements of the battery circuit. State Voc and Rs of each battery.
From the measured battery voltage (Vb) under a 10 Ω load, calculate the power delivered
by the battery to the load. Also, using the internal resistance of the battery, calculate the
power dissipated in the battery. Estimate the percent of capacity removed using Fig. 3.7 in
4. a) Take the measured amplitudes (Vrms) of the 10 kHz sawtooth signal at fo, 2fo, 3fo,
4fo, 5fo, 6fo, and plot the following waveform from t=0 to t=0.5 msec. The y-axis
should be between -2 and +2 V.
V1 ( t ) = 2 V ( fo ) sin (2 π ( fo ) t) − 2 V (2 fo ) sin (2 π (2 fo ) t)
+
2 V (3 fo ) sin (2 π (3 fo ) t ) − 2 V ( 4 fo ) sin (2 π ( 4 fo ) t )
+
2 V (5 fo ) sin (2 π (5 fo ) t) −
V in rms!
2 V (6 fo )sin ( 2 π (6 fo ) t )
(The derivation of the Fourier series decomposition of a sawtooth signal can be found on pp.
699-701 of “Electric Circuit Analysis” by D.E. Johnson et al. (3rd Edition) as well as in
many other textbooks.
b) Using the same amplitudes of all 6 harmonics, plot:
V2 ( t) =
2 V ( fo )sin (2 π ( fo ) t) +
2 V (2 fo ) sin ( 2 π ( 2 fo ) t )
+
2 V (3 fo ) sin (2 π (3 fo ) t ) + 2 V ( 4 fo ) sin (2 π ( 4 fo ) t )
+
2 V (5 fo ) sin (2 π (5 fo ) t) +
V in rms!
2 V (6 fo )sin ( 2 π ( 6 fo ) t )
Compare V1(t) and V2(t) with your lab measurements. Write a brief comment.
Recall that sin(ωt) = -sin(ωt + 180˚), and therefore V1(t) and V2(t) differ only by
phases of their even harmonic components. Notice that it is important to know
phase (or sign) of the harmonics to get the time domain representation from
frequency. The FFT spectra measured with Agilent 54645A oscilloscopes do
contain phase information.
the
the
the
not
These experiments have been submitted by third parties and Agilent has not tested any of the experiments. You will undertake any of
the experiments solely at your own risk. Agilent is providing these experiments solely as an informational facility and without review.
6
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WITH THE USE OF ANY OF THE EXPERIMENTS.
```