3. Effect of signal waveshape and frequency on DMM reading
Task
1. Measure the voltage across a load, power of which is regulated by a triac circuit. The
switching phase is 0°, 45° and 90°. Measure with DMMs V1 to V3.
2. Observe the signal shape with an oscilloscope (oscilloscope connected to the output of a
voltage divider). RMS and mean value measure via automatic measurement functions of
the oscilloscope.
3. Decide which of the DMMs measures the true RMS value and estimate the relative error
of the RMS measurement for the other instruments.
4. Determine the signal rectified mean value from the instrument readings where possible.
5. Calculate the rectified mean value and RMS value for switching phase α = 90°
(calculation from definition). Compare the calculated values with the measurements and
explain possible differences.
6. Connect the DMMs to the signal generator. Observe the readings of different DMMs for
sine waveform with RMS value of 0.7V. Measure for following frequencies: 10 Hz, 100
Hz, 1 kHz, 10 kHz, 100 kHz and 1 MHz (maintain the RMS value 0.7 V according to
oscilloscope display).
7. Connect the waveform generator directly to the oscilloscope. Consequently set-up
following voltage waveforms: rectangular, sine, triangle and sin(x)/x. Frequency
100kHz, amplitude 1Vp-p. For rectangular waveform observe the effect of following
FFT settings on presented spectra: number of acquired points, number of acquired
signal periods, data acquisition mode (Sample, HiRes, Average in “Acquire” menu).
Compare the measured FFT spectra with theoretical presumptions. Measure the amplitude
of the first, the second, the third, the fourth and the fifth harmonic of all used waveform
shapes.
8. Load the waveforms saved in internal memory of the generator (1, 2 memory positions),
set the frequency to 100 kHz and amplitude to 1 Vp-p and compare the spectrum with the
rectangular shapes from previous task.
Circuit diagram
A
triac
regulator
B1
~ 48 V,
50 Hz
R1
V1
B2
R2
Fig. 1 Circuit diagram
-1-
V3
OSC
Fig.2 Voltage waveforms for different switching phases
List of used instruments
A
- ampere-meter electromagnetic, accuracy ...
range ...
Z
- a box containing two bulbs B1, B2 and resistor divider R1, R2
V1
- bench-top multimeter, type ...,
V2
- bench-top multimeter, type ...,
- hand-held multimeter, type ...,
V3
A box containing a triac-based power regulation.
Theoretical background
There are two basic principles used for measurement of basic parameters of AC voltage
through its conversion to DC voltage (that can be easily measured):
1) Rectified mean value converter
2) RMS value converter
Low cost instruments usually use rectified mean value converter based on op-amp
rectifier. These instruments measure the rectified mean value (in agreement with its
definition) but they usually show RMS value after reading recalibration using form
factor for sine wave (multiplication of converter output voltage by 1.11, more exactly
by π/2√2). This recalibration can be done either digitally – in MCU, or analogically – by
changing converter gain. While measuring non-harmonic signals (ff ≠1.11) using these
meters and supposing that their reading is the RMS value, there can be a serious error
introduced. It is only possible to calculate the rectified mean value of a non-sinusoidal
voltage by dividing the instrument reading by the form factor of the measured voltage ff.
The more expensive instruments use RMS converters to correctly measure the RMS
value. This fact is usually presented on the instrument (marked RMS – Root Mean
Square or TrueRMS). In this case the presented RMS value is valid also for nonharmonic signals.
Note:
Most of the instruments add a capacitor into the signal path while measuring AC voltage
(or current), thus the measured RMS value does not include the DC part.
-2-
Mathematical calculation of RMS and rectified mean values for AC voltage
RMS value of a periodic AC voltage is given by the following formula:
T
1 2
=
u (t ) dt
T ∫0
U RMS
and the rectified mean value is given by the formula:
T
U RM =
1
2
u (t ) dt =
∫
T0
T
T /2
∫ u (t ) dt
0
where T is the voltage period time. These equations can be used to calculate the RMS
and rectified mean value of the signals in triac regulator for different switching phase α.
For sine wave we have:
2π
u (t ) = U m sin ωt , T =
ω
The value Um (peak voltage) can be determined from the RMS value measured by the
most precise available instrument (Um = √2 * URMS).
Arithmetic mean value (for specific switching phase α):
(while solving the integral we use following substitution: x = ωt, and t1 =
U RM ,α =
2
T
T /2
∫ U m sin ωt dt =
t1
T α
)
2π
π
U
1
U m sin x dx = m [− cos x]απ
∫
πα
π
RMS value:
U RMS ,α =
2
T
T /2
∫
π
U m2 sin 2 ωt dt =
t1
U
1
U m2 sin 2 x dx = m
∫
πα
π
π
∫
α
1 − cos 2 x
dx
2
After substituting α = π/2 is possible to deduce that the arithmetic mean value is:
U RM ,90 =
U m U RM , 0
=
π
2
U RM ,90 =
U m U RM , 0
=
2
2
and for RMS value:
-3-
FFT
Any periodic function can be written as a sum its harmonic components:
x(t ) =
n
ao
+ ∑ ( bk sin kωt + ak cos kωt )
2 k =1
or
n
x (t ) = ∑ X k sin( kω t + ϕ k )
k =1
FFT (Fast Fourier Transform) is an efficient algorithm to compute the Discrete Fourier
Transform (DFT) and it’s inverse.
In the oscilloscope the FFT is implemented in the mathematical processing section (red
button “MATH”). The scope offers advanced functions for the FFT processing and
results presentation (magnifier, cursors, variable bandwidth and resolution).
DFT has some presumptions that are hardly met in the oscilloscope (mainly: integral
number of acquired signal periods in 2n samples). So it is important to use windowing
and setup the measurement correctly. The oscilloscope has 8-bit ADC; it is useful to
scale the input signal in such a way that the quantization noise is minimized.
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