Boise State University
Department of Electrical and Computer Engineering
ECE 212L – Circuit Analysis and Design Lab
Experiment #1: Waveform Measurements
1
Objectives
The objectives of this laboratory experiment are:
• to review and apply the definitions of performance measures of periodic waveforms.
2
Theory
v(t)
Vmax
0
to
T
to +T
2T
t
Vmin
Figure 1: Example of a Periodic Waveform
Consider the above periodic waveform with period T in seconds. Recall that the period is equal
to the smallest time interval over which the waveform repeats itself. The period also includes any
dead time when the waveform is equal to zero. Mathematically,
f (t + T ) = f (t) for any time t
(1)
The frequency f of this waveform is equal to the number of cycles over one second. Mathematically,
the frequency of a periodic waveform is equal to the reciprocal of the period and it has units of
cycles per second (cps) or Hertz (Hz). Thus,
f
=
1
T
(2)
The angular frequency of the waveform is equal to
ω = 2πf
=
2π
T
(3)
and has units of radians per second (rad/s). The angle variable θ = ωt changes by
θ = ωT
= 2π = 360o
(4)
1
over a period T. The zero-to-peak or peak value Vp of this waveform is defined as the largest
absolute value (peak absolute value) of this waveform. Mathematically,
Vp = max {|Vmax |, |Vmin |}
≥ 0
(5)
The peak-to-peak value Vpp of this waveform is equal to the range of this waveform between its
maximum and minimum values. Mathematically,
Vpp = Vmax − Vmin ≥ 0
(6)
The average value of a periodic waveform is defined as
Vavg =
1
T
∫
T
∫
v(t) dt ⇐⇒ T Vavg
0
T
v(t) dt
=
(7)
0
Using the relationship on the right, this average value can be interpreted as the value of a constant
(DC) waveform that covers the same area between itself and the time axis over a period. Note
that the area under the original waveform is counted positively if the waveform is positive and
negatively if the waveform is negative. Thus, a sine waveform has zero average value over a period
as the positive area cancels the negative area under the sine wave.
The effective value Vef f of a periodic waveform is defined as the root of the mean (average) of the
squared waveform. For this reason, it is often referred to as the rms value of the waveform or Vrms .
Mathematically,
√
Vef f
1
T
= Vrms =
∫
T
0
1
T
2
v 2 (t) dt ≥ 0 ⇐⇒ Vrms
=
∫
T
v 2 (t) dt
(8)
0
Using the above relationships, the rms value squared can be interpreted as the average value of the
waveform squared over a period T. Note that the rms value of a DC (constant) waveform is equal
to the absolute value of the DC waveform.
A periodic waveform v(t) having a nonzero average voltage can be considered as the superposition
of a DC (constant) waveform and an AC waveform having zero average. Thus,
v(t) = Vdc + vac (t)
where
Vavg =
1
T
∫
(9)
T
v(t) dt =
0
1
T
∫
T
0
Vdc
1
dt +
T
∫
0
T
vac (t) dt = Vdc + 0 = Vdc
(10)
The rms value of this mixed waveform is then computed as
√
Vrms =
√
=
√
Vac =
1
T
∫
√
T
v 2 (t)
dt =
0
∫
0
T
1
T
∫
0
T
[Vdc + vac (t)]2 dt
]
[
2 + v 2 (t) + 2v (t)V
Vdc
ac
dc dt =
ac
2 +V2
Vdc
ac
=
where
1
T
√
1
T
∫
√
1
T
(11)
∫
0
T
[
]
2 + v 2 (t) dt
Vdc
ac
(12)
(13)
T
0
2 (t) dt
vac
(14)
2
is the rms value of the AC waveform. (Refer to the Appendix for the rms value of typical AC
waveforms encountered in the laboratory.)
Finally, there are two additional performance measures of periodic waveforms. The crest factor is
defined as the ratio of the peak (zero-to-peak) value to the rms value of a periodic waveform:
CF =
Vp
Vrms
(15)
Another factor called the form factor relates the rms value of a periodic waveform to the average
value of the rectified waveform, that is,
FF =
Vrms
|V |avg
(16)
where
|V |avg =
3
1
T
∫
T
0
|v(t)| dt
(17)
Equipment
• Agilent DSO5014A Digital Storage Oscilloscope
• Agilent 33220A Function/Arbitrary Waveform Generator
• HP/Agilent 34401A Benchtop Multimeter
• Fluke 115 True RMS Multimeter
3
4
4.1
Procedure
Waveform Measurements with Built-In Waveforms
Step 1: Set up a 1-kHz sine voltage with a 10 V peak-to-peak and a DC offset voltage of 2 V. Hook up
the output of the waveform generator directly to Channel 1 of your scope. Observe and record
the following performance measures for the sine waveform using the measurement capability of
the scope. Also, record the rms value of the periodic waveform using the benchtop multimeter
and the handheld multimeter.
Step 2: Repeat the above step with a 1-kHz square voltage, triangular voltage, and sawtooth voltage
having 10 V peak-to-peak and +2-V offset.
Step 3: Reset the DC offset voltage to zero and program the arbitrary waveform below (see Sec. 4.2).
Observe and record the same performance measures as for the built-in waveforms using the
scope, and the benchtop and handheld multimeters.
Oscilloscope
Waveform
Sine
Square
Triangular
Sawtooth
Arbitrary
4.2
Vavg
Benchtop
Multimeter
(DC+AC)Vrms
(AC)Vrms
DC V
Handheld
Multimeter
AC V
Programming an Arbitrary Waveform
v(t) (V)
4
3
2
1
0
0
T
2T
3T
4T
5
5
5
5
T
t (s)
Figure 2: Example of an Arbitrary Periodic Waveform
4
V̄
Ṽ
Step 4: Set your arbitrary function generator in the HIGH Z mode.
Step 5: Program the periodic waveform shown in Figure 2. (Programming instructions can be found
in the user manual, pp. 120-125.) Use the following instructions:
Press the Arb key.
Press the Edit Wform softkey.
Press the Create New softkey. Use the knob or the numeric pad to set the Cycle Period
to 1 ms, the High V Limit to 4 V, the Low V Limit to 0 V, the Int (interpolation) method
to Off, set the Init # Points to 5 and press Enter. Finally, press the Edit Points softkey.
Press the Voltage softkey and enter the voltage value for Point 1 as 4 V.
Press the Point # softkey and turn the knob to move to Point 2. Press the Time softkey
to set the time for the current point to 200 µs. (This softkey is not available for Point 1.)
Press the Voltage softkey and enter the voltage value for Point 2 as 3 V.
Repeat the above procedure for Points 3, 4, and 5. (See Table 1 for time and voltage
values.)
Press the End/Store softkey when done.
Press the Store in Non-Vol (non-volatile memory) softkey.
Select the Arb Mem 1 name and press the STORE ARB softkey.
Press the Output key and observe your waveform with an oscilloscope.
Table 1: Time and Voltage Values for Arbitrary Waveform
Point
1
2
3
4
5
Time Value
0s
200 µs
400 µs
600 µs
800 µs
5
Voltage Value
4V
3V
2V
1V
0V
5
Data Analysis and Interpretation
1. For the arbitrary waveform in Figure 2, show that
1
T
Vdc =
∫
T
v(t) dt = 2.00 V
(18)
0
2. For the arbitrary waveform in Figure 2, plot its AC waveform vac (t) and show that
√
Vac =
1
T
∫
√
T
2 (t) dt =
vac
0
1
T
∫
T
0
(v(t) − Vdc )2 dt =
√
2 ∼
= 1.41 V
(19)
3. For the arbitrary waveform in Figure 2,compute Vrms analytically using the definition:
√
1
T
Vrms =
∫
T
v 2 (t) dt =
√
∼ 2.45 V
6 =
(20)
0
and verify that
√
Vrms =
2 +V2
Vdc
ac =
√
6 ∼
= 2.45 V
(21)
from the earlier computations of Vdc and Vac .
4. Use three significant digits and fill out the theoretical (predicted) values for the ideal waveforms in steps 1 through 4 of the experiment:
Waveform
Sine
Square
Triangular
Sawtooth
Arbitrary
Vpp (V)
Vdc (V)
Vac (V)
Vrms (V)
Make sure to include a numbered heading for the table and show a sample calculation for
each formula.
5. Using the measured Vdc and Vac values, compute the measured rms values of the four built-in
waveforms (sine, square, triangular, sawtooth) and the arbitrary waveform and fill out the
following table for each instrument.
Waveform
Sine
Square
Triangular
Sawtooth
Arbitrary
Oscilloscope
Vrms (V)
Benchtop Multimeter
Vrms (V)
6
Handheld Multimeter
Vrms (V)
Make sure to include a numbered heading for the table and show a sample calculation for
each formula.
6. Compare the relative error between the theoretical and computed rms values of the four builtin waveforms (sine, square, triangular, sawtooth) and those of the arbitrary waveform using
oscilloscope and multimeter measurements by filling out the following table.
Waveform
Sine
Square
Triangular
Sawtooth
Arbitrary
Oscilloscope
∆Vrms /Vrms (%)
Benchtop Multimeter
∆Vrms /Vrms (%)
Handheld Multimeter
∆Vrms /Vrms (%)
Note that
∆Vrms
Vrms
=
|Vrms,theoretical − Vrms,measured |
× 100%
Vrms,theoretical
(22)
Make sure to include a numbered heading for the table and show a sample calculation.
7. (Discussion) Assuming that the function generator is 100% accurate, rank the measuring
instruments according to their accuracy.
7
Boise State University
Department of Electrical and Computer Engineering
ECE 212L – Circuit Analysis and Design Lab
Experiment #1: Waveform Measurements
Date:
Data Sheet
Recorded by:
Equipment List
Equipment Description
Agilent DSO5014A Digital Storage Oscilloscope
Agilent 33220A Function/Arbitrary Waveform Generator
HP/Agilent 34401A Benchtop Multimeter
Fluke 110/111 True RMS Multimeter
Oscilloscope
Waveform
Sine
Square
Triangular
Sawtooth
Arbitrary
Vavg
(DC+AC)Vrms
BSU Tag Number or Serial Number
Benchtop
Multimeter
(AC)Vrms
DC V
AC V
Handheld
Multimeter
V̄
Ṽ