ES 3: Introduction to Electrical Systems
Laboratory 2: Measurement of Time-Varying Signals
I. GOALS:
In this lab you will learn how to use two new instruments: the function generator and the oscilloscope. The function generator creates accurate, time-varying voltage signals with user-controlled frequency and amplitude. The oscilloscope is capable of measuring and displaying rapidly changing voltages.
You will also study capacitors and inductors in this lab. Using a function generator and an oscilloscope, you will verify the capacitor relationship
then you will use this expression to measure capacitance.
We will also continue our investigation of audio speakers. In Laboratory #1 you measured the resistance of a voice coil. The fact that the voice coil is a coil means that it also acts like an inductor. You will use an LCR meter to measure the inductance (L) of a speaker. We will introduce the concept ofelectrical impedance and you will investigate how the impedance of an audio speaker affects the fidelity of the speaker.
II. PRE-LAB QUESTIONS:
Read this entire document before coming to lab. In particular, review the use of the oscilloscope and the function generator in the Supplemental Information at the end of this lab and then answer the following Pre-Lab questions.
Figure 1 shows the display of an oscilloscope. Channel 1 of the oscilloscope is connected directly to a DC power supply. Channel 2 is connected to a function generator that is producing a sine wave output.
1. What is the DC voltage produced by the power supply?
2. What is the frequency (f) produced by the function generator?
(, where T is the period of a signal)
3. What is the amplitude of the sine wave produced by the signal generator (zero-peak)?
4. What is the DC offset of the sine wave?
5. What is the Trigger Voltage setting for this oscilloscope?
Figure 1. An Oscilloscope Display
III.EXPERIMENTS:
A. Transitioning from DC to AC Measurements
In this first experiment you will study the slow discharging of a rather large capacitor. Using a 100μF nonpolarized electrolytic capacitor, charge the capacitor to 5 volts. Do this by temporarily holding one lead of the capacitor to the COMMON terminal and the other lead to the +5V terminal of the DC power supply. The capacitor will charge to +5V almost instantly. Remove the capacitor from the power supply and connect it to the Tenma voltmeter (set to measure DC voltage, RANGE=20V). Observe the decay of the capacitor voltage. Note that the voltage slowly decreases as the charge stored in the capacitor “bleeds off” through the voltmeter.
How many seconds does it take for the capacitor to discharge from 5.0 volts down to 4.9 volts?
What is dv/dt Δv/Δtin units of volts per second?
What is the current flowing out of the capacitor and through the voltmeter?
Hint:and C = 100x10-6 F.
What is the resistance of the voltmeter (Rvm)?Hint: .
Repeat this experiment using a 1 μF capacitor. (You may need to increase the measured Δv.) Record your observations in your lab report. What can you conclude about the relationship between capacitance and the ability of a capacitor to store charge?
If the capacitor were 1 nF (nF = 10-9F) this would be a very difficult measurement witha voltmeter because the voltage would drop too quickly to observe. For measuring rapidly changing voltages, engineers use an oscilloscope.
B. Using the Function Generator and the Oscilloscope
Next, we will introduce two new lab instruments. The function generator allows us to create time-varying voltages with controlled waveshapes, amplitudes, and frequencies. The oscilloscope is essentially a voltmeter that allows us to measure and graph rapidly varying voltages. In preparation for these experiments, please review the use of the oscilloscope and the function generator in the supplemental materials at the end of this lab manual (Section V).
Set up the function generator to produce the following voltage signal:
v(t) = sin(2πft) volts …where f = 1000 Hz = 1 kHz.
Guidance: First choose the waveform to be a sinewave, and then select the frequency range of the function generator to be 1k. While viewing the green LED display, adjust the frequency knob until the frequency is 1.00 kHz (±0.02 kHz). Connect the oscilloscope to the function generator, paying attention that the ground leads of each instrument are connected together. Using the oscilloscope to measure the output voltage of the function generator, adjust the amplitude (AMPL) of the function generator to 1.0V(i.e., 2.0V peak-to-peak). You may want to use the MEASURE function of the oscilloscope.
Having trouble getting a sufficiently high output voltage? Make sure that the ATT(-20dB) button is off and the AMPL knob is pushed in. Both of these controls attenuate the output of the function generator.
Increase and decrease the amplitude (AMPL) of the function generator both above and below 1.0V. Describe the result that you observe on the oscilloscope.
Increase and decrease the frequency of the function generator both above and below 1 kHz. Describe the result that you observe on the oscilloscope.
Connect a 16Ω audio speaker to the function generator and repeat the two experiments above. In your lab report, comment on what you hear and correlate these sounds to your observations regarding frequency and amplitude.
C. Measuring a Time-Varying Current
Next, you will need to measure the current flowing through a 1 μF capacitor that is driven by a sinusoidal voltage source Vin(t) = 1.0sin(2π1000t) volts.
Theoretically,
i(t) = =
= 1.0Cωcos(ωt)
= (1x10-6)(2π1000)(1.0)cos(2π1000t)
= 0.0063cos(2π1000t) amperes.
(Eq. 1)
The multimeter can measure current, but not if the current is rapidly changing (i.e., changing at high frequency) so the circuit on the left side of Fig. 2 is not practical. The oscilloscope can measure high frequency signals, but it only measures voltage. Therefore, in order to measure the current through the capacitor, we need to measure the voltage that forms across a small sensing resistor (RS = 10Ω) in series with the capacitor as shown in Fig. 2. Once we know the voltage across RS, we can calculate the current through RSand the capacitor by applying Ohm’s Law and Kirchhoff’s current law.
Figure 2. Measurement of the current flowing through a capacitor (1 μF) is not practical using an ammeter (left). A practical measurement is to insert a small current-sensing resistor into the circuit and then measure the voltage across the resistor using an oscilloscope (right).
Caution: Both the function generator and the oscilloscope are grounded instruments. This means that the outer conductor of the BNC connectors on these instruments are all connected to the ground node of the building’s electrical system. Make certain that the ground of the signal is connected to the ground of the oscilloscope (usually, this means keeping the black alligator clips tied together.) Failure to follow this advice will short-circuit the function generator and the experiment will provide you with useless data.
Measure the zero-to-peak amplitude of the input signal from the function generator and the zero-to-peak amplitude of the voltage across the current-sensing resistor (RS) as shown in Fig. 2above. Convert the voltage across RSinto a current by dividing by the measured resistance of RS. Record the data and compare the measured current with the theoretical resultthat was derived above.
Frequency = 1kHz
Vin = 2.0V (pk-pk)= 1.0V(0-pk)
VRs = ___V (pk-pk)= ____V(0-pk)
Ic = _____ mA(0-pk)
Next, decrease the frequency from 1000 Hz to 500 Hz and repeat the current measurement. (Make sure that the function generator voltage remains 10 V(zero-pk).)
Frequency = 0.5kHz
Vin = 2.0V (pk-pk)= 1.0V(0-pk)
VRs = ___V (pk-pk) = ____V(0-pk)
Ic = _____ mA (0-pk)
Finally, double the frequency from 1000 Hz to 2000 Hz. (Again, making certain that the function generator output voltage remains 10 V(zero-pk).)
Frequency = 2kHz
Vin = 2.0V (pk-pk)= 1.0V(0-pk)
VRs = ___V (pk-pk) = ____V(0-pk)
Ic = _____ mA (0-pk)
Don’t forget to measure the actual DC resistance of the 10Ω resistor!
RS=____Ω – _____Ω (test leads) = _____ Ω
Note: If VRs is too small to measure accurately, simply increase Vin .
This is an example of a high pass filter. The amplitude of the measured signal is greater as the frequency increases. In fact, if you study Eq. (1) you can see that the amplitude of the current is proportional to the radian frequency, ω.
What would you expect the output signal (i.e., the current or the voltage across RS) to be if the frequency is reduced to zero (ω=0)?
D. Determining an Unknown Capacitance
In Section C you found the current flowing through aknown capacitor and compared that current to the theoretical current found from. It is also possible, however, to measure the AC capacitor voltage, the signal frequency, and the current flowing in an unknown capacitance, and then determine the value of that capacitance using. Capacitance meters use this technique.
Obtain an ‘unknown’ capacitor from the TA and determine its capacitance by substituting it for the 1 μF capacitor used in the test circuit of Section C. Describe the measurement technique, your calculations, and the value of the ‘unknown’ capacitance in your lab report.
E. Audio Speaker Inductance and Impedance
As discussed in Laboratory#1, many audio speakers create sound by driving a current through a voice coil. The voice coil is positioned within a region of a permanent magnetic field. The magnetic field generated by the current in the voice coil forces the voice coil to move and this movement generates sound. In Lab 1, we found that the voice coil has a DC resistance of 4-16Ω due to the long, thin wire used in its fabrication. Because it is a coil of wire, the voice coil also has an inductance.
Use the lab’s LCR meter to measure the resistance and inductance of the speaker’s voice coil. A ‘dissected’ speaker, with its voice coil and cone partially removed, will be provided.
Using f=1kHz and a 0.25 volt DRIVE, find R (pressR+Q) and L (pressL+Q). Please ask the TA for assistance if needed.
Next connect a working speaker to the LCR meter. Listen to the sound generated by the speaker as you make the measurement! The LCR meter is applying a sinusoidal voltage to the speaker and measuring the resulting current flowing through the speaker. From the inductor relationship, , the value of L can be deduced in a manner similar to the capacitance experiment performed earlier in this lab. Using your own words, describe this concept in your lab report.
We can model the voice coil of an audio speaker as a resistance (Rcoil) in series with an inductance (Lcoil) as shown in Fig. 3.
Figure 3. The cross section of an audio speaker connected to an AC voltage source (left) and the equivalent circuit model of the speaker (right).
Any inductor resists the flow of current, particularly at high frequencies: to see this consider an inductor driven by a current source of i(t) = Iosin(ωt) amperes. The voltage that develops across the inductor is v(t) = = = (ωL) Iocosωt. The ratio of voltage to current is typically considered to be a measure of electrical resistance. For example, Ohm’s law tells us that R = v/i. In the case of the inductor we find that
(Ω)
Because both sine and cosine have the same shape and are restricted to values between -1 and +1, we will simplify this analysis and only consider the amplitude of the voltage across the inductor and amplitude of the current through theinductor. Sometimes the signal amplitudes are referred to as magnitudes, |v| =(ωL Io)and |i| = Io:
= ωL (in units of Ω)
In other words, the ratio tells us how much an inductor resists or impedes the flow of current. Technically, this is called the magnitude of the impedance.
|ZL| ωL (ohms)
At the upper end of the human hearing range (f=20kHz) and atthe lower end of the human hearing range (f=20 Hz), find the impedance of your speaker’s inductance. Compare these two impedances with the measured resistance of the voice coil andcomment on the effect that impedance has on the current flowing through a speaker, and, therefore, on the speaker’s performance. Remember,ω is the radian frequency and ω = 2πf!
One final note: A capacitor also exhibits an electrical impedance. The magnitude of the impedance of a capacitor is determined by the relationship .
IV.LAB REPORT:
All required data, questions, and written comments can be found within the framed text blocks. In your report, tabulate or plot all of the requested data and answer the questions in a manner that flows naturally.
Lab Report Guidelines are listed on the website.
V.SUPPLEMENTARY INFORMATION:
The Oscilloscope
Along with the voltmeter, ammeter, and ohmmeter, the oscilloscope(also called a scope) is one of the most important electrical measurement tools. A voltmeter allows us to measure a DC or average voltage, but an oscilloscope lets us “see” a time-varying voltage signal, v(t). In this lab you will use a 2-channel digital oscilloscope similar to Figure A.
Figure A. The Tektronix TDS220 oscilloscope
Most people think that it is best to learn how to use a scope by simply playing around with it. Here are a few basics to get you started.
Inputs: CH1 and CH2
Each “channel” of an oscilloscope accepts a separate voltage input. This scope has two channels (CH1, CH2), although some scopes have four or more channels. The input jack is a shielded BNC connector. The outside conductor of the BNC connector is ground! A common mistake is to connect this grounded lead to a node in your circuit that is not ground. This almost always causes your circuit to fail. Remember, always connect the grounded lead of the scope to the ground of your circuit. Then you may probe your circuit using the other lead –i.e., the lead that is connected to the inner pin of the BNC input of the scope. Each channel may be turned on or off by pressing the CH1 Menu or CH2 Menu button.
VOLTS/DIV:
The display screen of an oscilloscope displays the input voltage vs. time. The screen is divided by a grid, and each major grid line is called a “division” or DIV. The voltage axis is controlled by the knob labeled VOLTS/DIV. In Figure A, a sine wave is applied to channel 1. The VOLTS/DIV knob has been adjusted to 5.00V (as shown in the lower left corner of the scope’s screen). This means each division represents 5.00 volts. Where iszero volts? On the left side of the screen a small arrow and the number 1 are displayed (1). This is the zero volt marker for channel 1. You should notice that the peak of the sine wave is approximately 1.5 divisions above the zero marker. This means the amplitude of the sine wave is 1.5 divisions * 5 volts/division = 7.5 volts. Likewise, the minimum value of the sine wave is -7.5 volts.
CH1 MENU, CH2 MENU:
Pressing these buttons will display a large number of options on the right-hand side of the display screen. Most important is the COUPLING type. The options are AC or DC. DC COUPLING shows you the entire signal including any DC voltages. For example, if the input to channel 1 were v(t) = 10.0 + 0.2*sin(1000t) volts, the display will show a 0.2 volt sine wave located 10 volts above the zero marker level(1). It can be hard to see such a small signal (0.2 v) when a large DC voltage is present (10v). The AC COUPLING let’s you eliminate the DC part of the signal and examine just the AC part. In this particular example, AC COUPLING would cause the scope to display v(t) = 0.2*sin(1000t) volts even though the actual signal is v(t) = 10.0 + 0.2*sin(1000t) volts. Many circuits produce very small signals that are superimposed on DC voltages, so the AC COUPLING feature can be quite useful.
/POSITION:
The zero voltage marker(1)that is displayed on the screen can be adjusted up or down using this knob. This is useful if you are looking at two channels with overlapping voltages. Simply move channel 1 up and move channel 2 down to get a clearer view of each.
SEC/DIV:
As previously mentioned, the scope provides a view of voltage vs. time. The time axis is controlled by the SEC/DIV knob. This tells us how many seconds each horizontal division on the screen represents. Use this control to spread or compress the horizontal axis so that you can see the signal clearly. The SEC/DIV setting is indicated in the middle of the screen at the very bottom. In Figure A, the scope is set to 500μs per division. The period of the sine wave is approximately 2 divisions * 500μs per division = T= 1000μs = 1 ms. Therefore, the frequency of the sine wave is = 1 kHz.
POSITION:
This knob allows you to shift the signal horizontally on the screen, and functions just like the vertical position knob.
TRIGGER LEVEL:
On the far right side of the scope are the trigger controls. The scope can be thought of as a camera that takes a picture of the signal. The TRIGGER tells the scope when to take the picture. More precisely, the trigger tells the scope to begin taking and displaying the input signal once a certain voltage level is reached. Notice that there is a small triangle marker () on the right side of the screen. This marker shows the input voltage level that will trigger the scope. This arrow will move up or down as you rotate the TRIGGER LEVEL knob. It is critical that this marker() be positioned between the maximum and minimum voltage displayed on the screen, otherwise the “pictures” will be taken at random times, and the voltage trace will appear to jump around on the screen.
TRIGGER MENU:
This button gives you several options on the screen. The most important is the channel used to trigger the scope! If you are looking at channel 2, but triggering from channel 1, the display will be quite jittery. Also important is the trigger slope. The scope can be triggered when the input voltage increases above the trigger level (positive slope), or it can trigger as the input voltage drops below the trigger level (negative slope). The trigger data is displayed in the lower right corner of the screen. In Figure A, the trigger is set to CH1, _/ (i.e., positive slope), 80.0 mV trigger level. This setting causes the scope to begin taking and displaying voltage data the instant that the voltage applied to CH1 exceeds 80 mV.