Upload
vudien
View
223
Download
1
Embed Size (px)
Citation preview
Exercise 8 – Measurements of time and frequency.
1. Aim of the exercise
The aim of the exercise is to familiarize students with methods of measuring time parameters of electrical signals such as frequency, period and phase shift between the two signals. Additional error evaluation will be described. 2. Main topics
Measurements of time parameters (time, frequency, and duty cycle) of electrical signals by
using the oscilloscope and Digital Frequency Counter (DFC)
Measurements of the phase shift between the two sinusoidal signals with a frequency
counter and a digital oscilloscope.
3. Gained skills
the ability to connect the circuit according to the measurement diagram
the ability to measure correctly time parameters (period, frequency, duty cycle) and phase
shift using a digital frequency counter and an oscilloscope.
the ability to properly document the experiment results
the ability to determine of measurement errors
4. Quantities, definition and units
Time (t): fundamental physical quantity
"The second is the duration of 9191631770 periods of the radiation corresponding to the transition between the hyperfine levels of the ground state of the ceasium-133” Frequency (f):
"Frequency (f) O of a periodic phenomenon is the number (n) of occurrences of phenomenon in a time unit (τ); frequency is the reciprocal of the time (T) between successive occurrences of the phenomenon.” “"One hertz is the frequency of a period phenomenon of which the periodic time is one second; one hertz is one cycle per second”. Pulsatance, ω:
"Angular frequency (Pulsatance) is the frequency of a periodic process expressed in radians per second; pulsatance is the rate of change of angle in time.”
5. Methods of measurements used in the experiment.
There are many methods for measurements of frequency or time. In our experiment only a
few of them are used: analog methods based on measurement of time with the oscilloscope,
and direct method based on of measurement frequency and time with the multifunction
digital counter.
Oscilloscope method used in the experiments are extremely simple - they implement either
internal (linear) time base, or external reference signal (Lissajous method).
Having known the time base speed time/div (a value which may be read from oscilloscope’s
screen), all we need to do is to measure the length of one or more cycles of the observed
signal.. This method is fast but not very precise.
Sinusoidal (external) time-base method (The Lissajous method).
An other analog frequency measurement method involving the oscilloscope rely on using the
oscilloscope as a kind of null indicator for comparison of a sine signal with unknown frequency
with a reference sine signal whose frequency should be well defined and easily varied. One of
the signals is fed to the Y channel of the oscilloscope, the second one to the X channel. An
interaction of these two signals produces on the display more or less complicated snaky loops,
whose form allows for determining the unknown frequency. A typical Lissajous figure is shown
in Fig. 1.
Fig. 1 Frequency measurement with the Lissajous method a) measurement diagram, b)
Lissajous figure and frequency determination.
If we assume that fy is the unknown frequency signal connected to the Y channel and fx is the
reference signal connected to the X channel then we have
𝑓𝑦 = 𝑓𝑥
𝑛𝑥
𝑛𝑦+
1
2𝜋
𝑑𝜑
𝑑𝑡
where nx, ny denote number of intersections of the Lissajous curve with horizontal and vertical
axes, respectively, and dφ/dt is the rate of phase change (the trace rotation speed). The
reference signal should be adjusted until the displayed figure is possibly stable (dφ/dt ≈ 0). It
is sometimes attainable with difficulty, and needs both signal sources involved having
adequate frequency stability. The method's error is roughly equal to relative calibration error
of the reference source. Due to the mentioned stability problems, usefulness of the method
is limited to rather low frequency applications.
Digital counter method.
Digital counter methods rely on continuing number of events (in this case – the number of
cycles) with the counter open during precisely determined window. The periodic input signal
of any shape, including sine waveform, is formed in an input shaper block to have standard
form of possibly short pulses that are fed to the counter controlled by an accurate and stable
quartz oscillator (see Fig.2).
Fig. 2. Principle of direct digital measurement of frequency
If the counter is opened for e.g. 1 second, then the number of pulses counted during this time
directly gives the measured frequency.
If we denote open gate time as τ, input signal period as Tx, and number of cycles counted as
N, then
𝜏 = 𝑁 ∙ 𝑇𝑥
and the unknown frequency is equal to
𝑓𝑥 = 𝑁/𝜏
It is intuitively obvious that the accuracy of this method mainly depends on the accuracy of
gate timing. It may be shown that the limiting error of direct frequency measurement method
is equal to
𝛿𝑔𝑓𝑥 = 𝛿𝑔𝑓𝑟𝑒𝑓 +1
𝑁= 𝛿𝑔𝑓𝑟𝑒𝑓 +
1
𝑓𝑥 ∙ 𝜏
where δgfref is the limiting error of quartz oscillator frequency.5gfk is the limiting.
AND
gate Counter Display
Input
shape
r
τ
fx
In conclusion, one may see that the limiting error of direct frequency measurement method
decreases
- as the number N increases, i.e. measured frequency rises,
- gating time τ is longer.
The accuracy of the method is getting worse if measured frequency goes down. The way out
of this problem is to switch to digitally measure the period of the signal Tx instead of directly
measure the frequency. In this method, the gate is opened for the time of one period of the
input signal, and during this time pulses with the reference frequency fw produced by quartz
oscillator are counted. The reference frequency fref may be changed by adjusting the
frequency dividing factor m, so that fw = fref/m.
Finally, the frequency we are looking for is equal to fx = 1/Tx. The limiting error of measured
period is equal to
𝛿𝑔𝑇𝑥 = 𝛿𝑔𝑓𝑟𝑒𝑓 +1
𝑁= 𝛿𝑔𝑓𝑟𝑒𝑓 +
𝑚𝑓𝑥
𝑓𝑟𝑒𝑓
and
𝛿𝑔𝑓𝑥 = 𝛿𝑔𝑇𝑥
The limiting error of this method of frequency measurement may be further decreased if we
measure not the one period of input signal but time of few periods, say k periods. In this case,
the limiting error is expressed as
𝛿𝑔𝑓𝑥 = 𝛿𝑔𝑓𝑟𝑒𝑓 +1
𝑘𝑁= 𝛿𝑔𝑓𝑟𝑒𝑓 +
𝑚𝑓𝑥
𝑓𝑟𝑒𝑓 ∙ 𝑘
The universal frequency/time counter may also be used for measurement of time interval. In
this case the first event opens the gate and the second one stops it. In between the counter
count pulses with reference frequency coming from quartz oscillator.
It is worth knowing that modem time-frequency counters are capable to select appropriate
mode of operation automatically, thus minimizing the quantization error. Moreover, the
timer-counter Keysight 53220A measure only the period of input signal. So, measured
frequency is always calculated.
A part of the limiting error of discussed measurements comes from the finite time of gate
switching. The detailed discussion of that problem is beyond the scope of this text.
6. Instrumentation
Laboratory module description
Fig. 1. Module F01. G1 – sinusoidal waveform output, G2 – rectangular waveform output, PF – phase shifter
7. Measurements and tasks
Table 1. Measurement functions of 53220A Keysight digital frequency counter
Function Symbol Settings Period T FREQ/PERIOD PERIOD Frequency f FREQ/PERIOD FREQ Pulse counting n TOTALIZE GATED Duty cycle TIME INTERVAL DUTY CYCLE Pulse width TIME INTERVAL PULSE WIDTH Pulse distance t TIME INTERVAL PULSE WIDTH Phase shift
TIME INTERVAL PHASE PHASE 1-2
Table 2. Measurement functions of 1052 Rigol oscilloscope
Function Symbol Settings Period T MEASURE TIME PERIOD Frequency f MEASURE TIME FREQ Duty cycle MEASURE TIME +DUTY Pulse width MEASURE TIME +WIDTH Pulse distance t MEASURE TIME -WIDTH Phase shift MEASURE TIME PHAS 12
Note: Before the measurements:
- connect power supply to the F01 Module.
- set a frequency counter impedance to 50 Ω for inputs 1 and 2:
(1 or 2) Impedance: 50 Ω.
Task 1. Measurements of time parameters of sinusoidal signals
1.1 Use the DFC 53220A (Digital Frequency Counter) to measure time parameters (period and
frequency) of sine signal from generator G1 output in F01 module.
NOTE: Prior to the measurements assess the stability of the G1 generator and adjust the
resolution of the frequency counter. To do this, set the number of digits displayed on the
frequency counter display in a way to achieve observed instability of the result on the least
significant digit ( Digits AutoDigits: Off, and then use the knob to set number of digits).
To make reading the results easier, the manual trigger of measurements can be used. Once you
activate following sequence of commands: Trigger Source Manual, each time you
press the button Trigger, a single measurement will be performed. To go back to automatic
trigger mode execution following sequence of commands: Trigger Source Internal.
Measure appropriate parameters using following functions:
a) PERIOD – measure the period of the signal with gating time of 1 s. If selected gating time is
longer than measured period, the DFC measures an average period.
b) FREQ – measure the frequency of the signal with gating time of 1 s. In this case, the frequency
is measured with the indirect method.
c) TOTALIZE:GATED – counting the number of pulses during 1 s of Gate Time. This case realizes
the direct frequency measurement method (using frequency definition – counting of
phenomena occurrences during reference time interval)
Compare and comment obtained results from PERIOD, FREQ and TOTALIZE:GATED
measurement method.
1.2 Use the oscilloscope to measure time parameters (period and frequency) of sine signal from
generator G1 output in F01 module.
Take measurements of the period and frequency with the use the manual procedure (length
measurements) and the automatic measurement functions. Set the oscilloscope to minimize
measurements errors.
!
Insert the oscillograms into the report. Write down Cx constant and measurement results.
Estimate the limiting error of both measurements.
Compare and comment obtained results of period and frequency measurements with the
corresponding values obtained in point 1.1a and 1.1b. Assume the DFC as a reference
instrument.
Task 2. Measurements of time parameters of rectangular signals. 2.1 Use the DFC 53220A (Digital Frequency Counter) to measure time parameters (period and
frequency) of sine signal from generator G2 output in F01 module.
Prior to the measurements assess the stability of the G2 generator and adjust the resolution of
the frequency counter. To do this, set the number of digits displayed on the frequency counter
display in a way to achieve observed instability of the result on the least significant digit.
a) Measure the period and frequency using the PERIOD and FREQ functions respectively. Set
the gating time of 1 s.
Compare and comment on the results.
b) Measure the pulse width of rectangular waveform (Width: Pos) and time interval between
pulses (Width: Neg).
c) Determine duty cycle of the signal based on the parameters obtained in point a) and b).
d) Measure the duty cycle of the signal using automatic function in the DFC (DutyCycle: Pos).
Compare and comment results obtained in points c) and d).
2.2 Use the oscilloscope to measure time parameters (period and frequency) of sine signal from
generator G2 output in F01 module
a) Take measurements of the period and frequency with the use the manual procedure (length
measurements) and the automatic measurement functions. Set the oscilloscope to minimize
measurements errors.
Insert the oscillograms into the report. Write down Cx constant and measurement results.
Estimate the limiting error of both measurements.
Compare obtained results of period and frequency measurements with the corresponding values
obtained in point 2.1a. Assume the DFC as a reference instrument.
b) Take measurements of the pulse width and time interval with the use the manual procedure
and the automatic measurement functions (WIDTH+ and WIDTH-).
Insert the oscillograms into the report. Estimate the limiting error of the measurements.
c) Based on results in point a) and b), determine value of the duty cycle and estimate the limiting
error of both results.
!
!
!
!
d) Take measurements of the duty cycle with the use of automatic function +DUTY.
Compare values of duty cycle obtained in points 2.1c and 2.1d. Assume the DFC as a reference
instrument
Task 3. Measurements of phase shift of sinusoidal signals.
3.1 Connect the output of the G1 generator to the input of the phase shifter PF in F01 module.
Use the DFC to measure the phase shift between signals on the input and output of the PF
module. To gain measurement result in range of 0 - 360ºset option: Time Interval Phase
Phase Meas: 0–360.
3.2 Use the oscilloscope to measure phase shift between signals on the input and output of the
PF module. Use the sine signal from G1 as an input signal for the phase shifter (PS).
Take measurements with the use the manual procedure (length measurements. Set the
oscilloscope to minimize measurements errors. Insert the oscillogram into the report and mark
all sectors used for ε calculation.
Estimate the limiting error of phase shift measurement.
Compare and comment obtained result of phase shift measurement with the corresponding
value of phase shift obtained in point 3.1.
!
!
8. Control questions
1. Describe how one can measure frequency with the use of digital oscilloscope.
2. Describe how one can measure phase shift between two signals with the use of dual
channel oscilloscope.
3. Is it possible to measure the phase shift between two signals of the same shape,
amplitude and frequency but from two independent sources? Justify your answer.
4. What is the role of the internal generator in the frequency counter?
5. For 1 ms/div time scale a waveform as in figure below has been observed at the
oscilloscope:
Estimate the signal frequency and the limiting error (absolute and relative) knowing that
the formula for the relative limiting error is expressed by formula:
𝛿𝑔𝑓𝑥 = 3% +0,1 𝑑𝑖𝑣
𝑙𝑙𝑥∙ 100%
Where lx is the measured length.
6. At what timescale (1 µs or 2µs) the oscilloscope offers a smaller relative error in
measurement of the frequency of the signal of 1/6 MHz with the use of the method of
the length measurement? The oscilloscope screen has 10 divs in horizontal direction
and the formula for the error is:
𝛿𝑔𝑇𝑥 = 3% +0,1 𝑑𝑖𝑣
𝑙𝑙𝑥∙ 100%
7. For 1 ms/div time scale a waveform as in figure below has been observed at the
oscilloscope:
Estimate the duty cycle and the limiting error of this measurement knowing that the
limiting error formula is:
𝛿𝑔𝜏𝑥 = 50𝑝𝑝𝑚 +0,1 𝑑𝑖𝑣
𝑙𝑙𝑥∙ 100%
8. Derive the formula for the limiting error of the duty cycle of the rectangular signal
measurement with the method of the length measurements with the digital
oscilloscope. The parameter is defined as follows:
a) T
b) t)
9. Derive the formula for the limiting error of the phase shift measurement with the
method of the length measurement with the digital oscilloscope.
10. Describe the direct method of the frequency measurement with the digital frequency
counter and propose the analysis of the measurement error.
11. Describe the indirect method of the frequency measurement with the digital frequency
counter and propose the analysis of the measurement error. Explain the difference
between measuring period and averaged period.
12. Neglecting the instability of the quartz generator estimate the value of the measured
frequency fx for which the measurement error of the direct measurement gets smaller
than the measurement error of the indirect measurement (period measurement).
13. For which frequency range (high of low) one should apply indirect frequency
measurement method and for which a direct method?
14. Digital frequency counter measured 2500 pulses for the gate of 2 s. Calculate the period
of the signal and the limiting error of the measurement. Neglect the quartz oscillator
instability.
15. Digital frequency counter measured 5000 pulses for the gate of 1 s and 50002 pulses
for 10 s. Which measurement is more accurate? Justify your answer.
16. How long should the direct frequency measurement take if the 10 kHz signal measured
with the digital frequency counter in a way that the error component due to the
instability of the internal quartz generator of 10-7 constituted a half of the total limiting
error?