0 Dual-beam oscilloscope - Metrology Research Institute

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Nov 15, 2013 (3 years and 6 months ago)

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Dual
-
beam oscilloscope


Goals of the work




Learn basic operation principles of a dual
-
beam oscilloscope



Learn how to use a

dual
-
beam oscilloscope



Learn to measure periodic signals



Learn the meaning of a probe and to use it



Learn how to make a document of the measurements


For what purposes an oscilloscope is used?


The oscilloscope is very commonly utilized measuring device.
By utilizing
oscilloscope, very high
-
speed periodic phenomena can be observed in an illustrative
way. In addition to the voltage, the oscilloscope can be utilized to measure other
electrical and non
-
electrical quantities. It is utilized in the maintenance
and the
testing of the electrical equipment as well in the research, the educational purposes
etc.

0.1

Operational principle of the analog oscilloscope

For historical reasons the operational principle of the oscilloscope is ap
proached
through an analog oscilloscope. The screen is in significant role in analog
oscilloscopes since it affects outstandingly to the performance of the oscilloscope.
Whereas in digital oscilloscopes the conversion of the continuos signal to the
discre
te samples is matter that is in focus. Analog oscilloscopes can’t offer as varied
characteristics as the digital oscilloscopes. The introduction of the analog
oscilloscope however brings out the basic characteristics of the oscilloscope.


The measured peri
odic voltage
-
signal is seen on the screen. The graph on the
oscilloscopes screen represents the voltage (vertical direction, y) as a fuction of the
time (horizontal direction, x). The signal is amplified in the vertical deflection block
to the deflection v
oltage to the cathode ray tube, CRT (fig.
6
) The trigger block
defines the moment when the vertical deflection block starts up the vertical
deflection voltage.


Understanding these basic block is important since the adjusters are groupped into
the blocks o
n the front
-
panel of the device. In the following, these blocks and all
adjusters and connectors are considered in details. It is not a description of any
particular device, only a description of the basic functions. In the simpliest devices
all these func
tions doesn’t even exist, and in the advanced devices there are much
more functions and adjusters that are described below.

0.1.1

Screen

In a cathode
-
ray tube a deflection voltage controlled elec
tron beam is bombarding a
phosphorized screen. As a consequence a visible point is composed on the screen.
While the electron beam is moving, the afterglow duration of the phosphorized
surface make it easier to see the beam. The screen is divided with vert
ical and
horizontal lines into square shaped parts.


Adjusters required for the beam control


Intensity


The intensity of the beam has to be adjustable, since illumination circumstances
varies in different places. If the phenomena is very fast and its freq
uency is very
low, the beam travels throgh the screen only a fractional part of the total time. In
that case the intensity of the beam needs to be increased. Obs. Do not ”light” the
laboratory by an oscilloscope. Too intense beam wears down the phosphorize
d
surface of the screen.



Vertical deflection

Trigger

source

Coupling

HF
-
reject

x
-
amplifier

y
-
amplifier

Beam intensity

amplifier

Comp
-


arator

Level

Slope

Mode


Holdoff

Ch. 1

Ch. 2



Alt

Chop

Add

Ch. 1

Ch. 2

Ext.

Line

Volts/div

Cal.

XY

Time/div

ramp

gen.

Returning
beam cutoff

CRT

Ch. 1 input

Ch. 2 input

preamplifier

Z
-
input

mains

frequnecy

Ext.

trigger

input

AC

DC

Couplin
g

AC

DC

level control

GND

Electronic switch /
summing


Invert

Channel 2

invert

Intensity

Focus

Beam

finder

Triggering


Horizontal deflection



Figure
6
. Block diagram of an oscilloscope .
Texts in oscilloscope front panel are
italicized.

Beam finder


If the beam is missing, the beam finder

button helps you to do the correct
adjustments. The
beam finder adjuster pass the intensity control and decrease
horizontal and vertical deflection voltages so that the beam certainly come into
view. The point in the screen shows in what direction it has to be adjusted to make it
normally appear on the scre
en.


Focus, astigmatism


These adjusters have affect on the shape of the point. In general, the point or the line
is adjusted to be as sharp as possible.

0.1.2

Vertical deflection block


The measured signal is amplified
and the vertical deflection voltage is formed from
it. This voltage moves the beam in vertical direction (y
-
axis). The amplification can
be controlled and therefore signals of very different size can be measured with the
device. In dual
-
beam oscilloscopes
there are two y
-
channels. Behind the pre
-
amplifiers there are a fast switch. With this switch the y
-
channels are chosen by
turns to control the deflection voltage and thus it seems that there are beams for both
y
-
channels on the screen. Normally these chan
nels are called Ch1 and Ch2 (or Y1
and Y2)


Upper frequency limit and sensitivity


Upper frequency limit is typically called the bandwidth of the oscilloscope, the
lower frequency

limit is normally DC
-
voltage. The characteristics of the vertical
deflection bolck define two charcteristics of the oscilloscope: the upper frequency
limit and the sensitivity. The sensitivity of the vertical deflection defines the
maximum amplification p
er spacing (volts/div).
Usual maximum sensitivity per
spacing is 1 mV.


x

y


Figure
7
. Graticule of the oscilloscope

The bandwidth of the amplifier specifies the upper frequency limit. The frequency
with which the amplification is dec
reased 3 dB from the normal is generally
determined as the upper frequency limit. (the power of the constant resistance is
decreased ½, and the voltage has decreased
1
2
0
708

,

of the initial value). The
upper frequency limit is not any absolute l
imit to the oscilloscope functioning.
Frequencies above the upper frequency limit seems to be more damped. While
measuring the digital signals, it has to be taken into account that for example the
rectangular wave of 10 MHz might include frequency componen
ts as high as 100
MHz and even over. Thus, the real shape of this kind of rcctangular wave can’t be
seen with an oscilloscope of 30 MHz. The usual frequency limit of an oscilloscope
is 10…100 MHz.


Adjusters of the vertical deflection block


Amlification (
volts/div)


Vertical deflection sensitivity is chosen separately for channel 1 and channel 2.
Sensitivity is expressed per spacing and it varies between few millvolts to several
volts. Sensitivity can also be adjusted continuous (variable
-
adjustment), if t
here is
need to fit the signal to some certain spacing to enable the studying the signal shape.
In many oscilloscopes there are an ’uncal’
-
light, which is switched on while using
the variable
-
control, to indicate that the sensitivity chosen by volts/div

a
djuster is
not true.


Vertical position


Beams from both cannels can be moved independently in vertical direction.



Input coupling


The measured signal is coupled straight to the vertical deflection amplifier in DC
-
position. Although it is called DC
-
posit
ion, it can be used as well to measure AC
-
voltages. In AC
-
position the DC
-
component is filtered off with a high
-
pass
-
filter.
The limiting frequency is usually 1
-
3 Hz. With frequencies under 10 Hz the
oscilloscope is damping the signal considerably in AC
-
po
sition. AC
-
position is used
when there is a need to distinguish a small AC
-
component from the DC
-
voltage. In
GND
-
position the voltage of 0 V is feeded to the input of the amplifier instead of the
measured signal, thus the zero level of the signal can be ad
justed with the vertical
position control to the desired level. If the sensitivity (volts/div) is changed, the zero
level needs to be recontrolled.


Mode


In dual
-
beam oscilloscopes the proper function mode needs to be chosen. In ALT
-
mode the beams from y
-
channels are swept by turns over the screen. With high
sweeping velocities the rotation can’t be seen, but with low velocities the rotation is
disturbing and the ALT
-
mode is not useful. Another alternative is the rotation of the
beams in fast tempo determi
ned by the internal chopper of the oscilloscope. With
low sweeping velocities it seems like the both beams are travelling on the screen at
the same time, but with high velocities the rotation is visible and disturbs the
measurement.Some oscilloscopes are c
hoosing the mode automatically on grounds
of the sweeping velocity adjustment. In addition, the beams from both channels can
be switched on or off (Ch1 on/off, Ch2 on/off) and the sum signal can be formed.
Usually at least one of the two channels can be in
verted, thus it is pobbible to obtain
the difference signal.


Connectors of the vertical block


Input


Both channels (Ch1 and Ch2) have their own connectors to couple the measured
voltage. The connector is usually so called BNC
-
connector. The signal has to

be
connected to the oscilloscopewith as short protected cable as possible to prevent the
coupling of the mains
-
borne disturbance and the rf interference. If high frequrncies
are measured and the impedance of the measured object is not low, it is very usef
ul
to utilize probes. Even while measuring the low frequencu phenomena, the rf
interferences interfere with the measurement, if the protection is not well done.



Horizontal deflection
voltage

time

ramp

Return ramp

Suspension
period

Trigger
holdoff

Triggering moment

Triggering
moment


Figure
8
. Waveform of the horizontal deflection voltage

0.1.3

Horizontal

deflection block

Also the deflection voltage in horizontal direction (x
-
axis) is neededto draw the
beam. A ramp generator is forming the horizontaldeflection voltage which is formed
of a ramp, a retrace, a holdoff

and a wait condition (fig.
8
). The beam is drawed
during the increasing ramp, but it is switced of during the retrace, holdoff and wait
condition. The lenght of the holdoff is normally the same than the lenght of the
retrace but the user can lenghten it b
y controlling the holdoff
-
adjuster. During the
wait condition the ramp generator is ready to start from the triggering of the ramp.
The increasing time of the ramp is adjustable in very large scale and thus the
oscilloscope can be utilized to observe very
slow and very fast phenomena (from
microseconds to few seconds).


Adjusters of the horizontal deflection block


Sweep speed (sec/div)


With this adjuster the travelling speed of the beam in horizontal direction (x
-
axis) is
chosen. The time scale is quanti
fied per division and it is usually scaled 3
steps/decade. For example 1 ms, 2 ms, 5 ms, 10 ms…/scaling. The sweep speed can
also be adjusted variable, if there is need to fit the signal to some certain spacing to
enable the studying the signal shape. In t
his case the ‘uncal’
-
light may switch on for
that the variable
-
control is not forgotten on. In contact with the sweep speed adjuster
there are usually also the xy
-
position. If using this xy
-
position one of the inputs
(Ch1) is coupled to the horizontal defl
ection amplifier instead of the sweep
generator.


x
-
position


The beam can be moved in x
-
direction

and e.g. the start point of a period can be
shifted to start at a division line.


Magnification,

5,

10


The image can be amplified in horizontal direction
by a constant (for example 5 or
10), so that only a part of the normal sweeping scale is on the screen.
The beam can
be moved in x
-
direction.

0.1.4

Trigger block

Triggering moment defines when the oscilloscope begins to draw the beam (fig.
8
).
It is important th
at the signal drawing begins every time at the same phase of the
period. Triggering moment is controlled indirectly with the gate trigger voltage. A
multi
-
functional trigger block enables studying a variety of different signals. Most
of the problems in uti
lizing the oscilloscope deals with the selecting of the trigger
level and the trigger mode/source.


Adjusters of the trigger block


Trigger source


In dual
-
beam oscillators there are several alternatives for the trigge
ring source:
signal from either Ch 1 or Ch 2, external triggering or triggering from the line.


Trigger level

and trigger slope


The trigger level control by hand is important to ach
ieve clear graph on the screen.
If the waveshape is complicated, the graph on the screen needs to be adjusted with
the trigger level control to achieve a stabile and unambiguous graph. The trigger
slope defines if the triggering is done from increasing or
decreasing edge of the
signal.


Trigger mode


In normal mode the triggerin takes place, if the triggerin signal cuts the trigger level
in chosen dir
ection in wait condition (fig. 8
). If the trigger condition is not reali
zed,
the wait condition continues, which usually is the reason why there is anything
visible on the screen. In auto
-
trigger mode the oscilloscope is ready to trigger in wait
condition for a moment, but if the condition is not realized, it triggers automati
cally
even if the condition is not fullfilled. The auto
-
trigger mode is oblicatory for
example to make the DC
-
signal visible on the screen. For signals that are repeated in
low frequency the auto
-
trigger occurs too early, and the normal mode is needed.
Usu
ally it is also possible to utilize the single sweep, in which case the oscilloscope
stays in the holdoff condition after single sweep and continues to the wait condition
when the single sweep is chosen again.


Trigger coupling


The input of the trigger circuit might be coupled straight (DC) or through a high
-
pass filter (AC), in which case it is easier to synchronize to a small ripple voltage.


Trigger holdoff


The holdoff condition ca
n be made longer with the holdoff
-
button. It might be
useful if for example the measured signal fulfills the trigger condition several times
during a period. The triggering has to occur at the same phase of the period than
during previous period. Whit hold
off
-
control some phases that fulfill the trigger
condition can be bypassed.


Connectors of the trigger block


Ext trigger


Triggering is done in time with an external signal coupled to this connector, if the
trigger source

adjuster is in ext
-
position.

0.1.5

Cal
ibration signal

In the oscilloscope there is a calibration signal output. The calibration signal is
typically 1 kHz rectangular wave with amplitude of 1 V.

0.2

Oscilloscope with a delayed time axis

0.2.1

Delayed sweep

In the oscilloscopes which are equipped with a
delayed sweep there are two separate
horizontal deflection ramp generators (fig.
9

and
10
). The B
-
deflection is like the A
-
deflection and it is used a kind of auxiliary time
-
axis in sweep delay. The B
-
deflection differs however from the A in the way that t
here is a delay circuit in the
B
-
ramp generator. The trigger produced by the studied phenomenon release the A
-
ramp, which after reaching some certain voltage release the B
-
ramp with a
particuliar calibration circuit. The reference level and thus delay time

can be
controlled by a potentiometer. The beginning of the B
-
horizontal deflection ramp
can be delayed about 0,1…10 s depending on the oscilloscope.

A-pyyhkäisyn
liipaisutaso
B-pyyhkäisyn
liipaisutaso
A-pyyhkäisyn
B-pyyhkäisyn
B-pyyhkäisyn pituinen kirkastuspulssi
A-säteeseen
mitattava
signaali
ramppi
ramppi
t

Figure
9
.
Scheme for generating a delayed sweep. A
-
sweep beam is brightened for
the ti
me period of B
-
sweep

in

order

to see what part of the A
-
beam the B
-
sweep is
displaying.

kirkastettu
osuus
A-pyyhkäisyn säde
B-pyyhkäisyn säde

Figure

10
. Measured signal of figure
9

on the oscilloscope screen when A and B
beams
are chosen to be display
ed at the same time.

0.3

Digital oscilloscope

Compared to the analog oscilloscopes, digital oscilloscopes have many useful
properties and in consequence the digital oscilloscopes have captured the market
from the analog osci
lloscopes. Such properties are for example possibility to save
measured waveform, print the results with a printer connected to the oscilloscope,
automatic measurements, ability to utilize cursors and controlling the oscilloscope
with a PC in order to auto
matize measurement and processing of data.

0.3.1

Operational principle of the digital oscilloscope


Measured signal is sampled and converted to digital form with
a fast AD
-
converter
1
,
typically with 8
-
bit resolution.
These byte
s are saved at sampling frequency to
memory, from which the data is collected for microprocessor system. Fast single
phenomena or very slowly varying signals can be easily detected.
A typical
oscilloscope display provides a VGA
-
resolution.
Because

the sign
a
l is stored in the
microprocessor system many different kind of signal processing and signal
analyzing activitiess can be carried out

to support the measurement
. In addition
,

printing the results
with an ordinary printer
and transferring data to computer
for the
further processing is possible.


channel 1
channel 2
external
triggering
vertical
deflection amplifier
vertical
deflection amplifier
AD
-
converter
AD
-
converter
AD
-
memory
trigger
comparator
microprocessor
system
TV
-
screen
AD
-
memory
crystal oscillator
delay counter
recording
stop


Figure

11
. Block diagram of the digital oscilloscope

0.3.2

Peculiarities
of the sampling

Some issues concernig the sampling
that
highlight the specialities of the

d
igital
system:









1

Voltage is transferre
d to numerical values

Progressive scanning


The bandwidth of the vertical deflection
amplifier
of the oscillator used in this

work
is 100

MHz. Shannon sampling theorem says that there must be at least two samples
per
period in order to be able to perfectly construct a sinusoidal signal (samplig
must
happen

at
least
at Nyquist frequency
)
.

In practice the oscilloscope needs more
samples per period.
400 million samples per second would be enough in order to get
enough s
amples from a signal the bandwidth of which reaches up to 100 MHz.
However, the oscilloscope
digidizing

speed is only 20 million samples per second
(20 MS/s
)

for one channe and
10 MS/s
for two channels
.
The price of the
oscilloscope can be muc
h lower if the sampling frequency is kept slow if compared
to a
n

oscilloscope that has a sampling frequency of
400

Ms/s.

With a progressive
scanning a bandwidth of
100 MHz
can be reached.


A

repeated signal
at
100 MHz can be detected by utilizing the progr
essive scanning.
The functioning of a
n

oscilloscope is based on an occasional repeated sampling.

Repeated sampling means that samples are taken from the signal during several
scannings. Sampling moment relative to the signal phase alters and finally a sign
al
can be constructed, figure
12
.

1
3
4
5
6
7
8
9
10
11
12
2
liipaisutaso

Figure

12
.
Digidizing

of the signal during several scanning

(liipaisutaso=triggering
level)
.

Asynchronous means that signal is continuously digidized with the phase of the
oscilloscope own clock regar
dl
ess of the triggering moment.
When signal cuts the

triggerin
g

level in chosen direction
,

the oscilloscope put
s

the stored points as well as
those
point
s

coming after triggering to the places
where
they belong. Because the
measured signal frequency and samp
ling frequency are not syncronized, the
sampling moment relative to the triggering moment
is varying
from one scanning to
other
and, finally,

there exists measured points dense enough to construct the whole
signal.


There
are

two be
n
efits
in

the asynchron
ous system
.
First, signal can be displayed
already before
the
triggering moment, which would be
very difficult to carry out
wit
h

analog oscilloscopes. Second, asynchronous system makes the aliasing
2

in



2
When oscilloscope is used at slow scanning speed, the sampling frequency is typically decreased
due to the limited memory capacity. Because the preamplifier passes frequencies up to over
which the aliase
d signal is stable on the scree
n
very unlikely.
Sometimes there is a
fast rolling aliased signal on the screen and there is a pos
s
ibility to have a wrong
idea ab
o
ut the
signal frequency. When the triggering is functioning so that the
re is
no rolling on the screen, the real signal should be displayed.



One disadvantage of digital oscilloscopes is the bottleneck due to the
microprocessor system.
The rate at which the microprocessor system can update the
screen is typically very slow co
mpared to the sampling frequency.
The screen can be
updated few tens of times per second although the scanning speed requires screen
updating 20

000
times per second.

E.g.

if there is an

occasionally 50 times per
minute existing noise peak
that
occurs only

every 20000th scan in the measurement
signal
, it may take one hour to get the noise peak visible on the screen.
Analog
oscilloscope updates the screen with every sweep and thus there
are

realistic
pos
s
ibilities

to find out the noise peak.


Single scan


Single scan is one of the most significant advantage in the digital oscilloscopes in
comparison with the analog oscilloscopes. (Even if before the digital oscilloscopes
there were analog oscilloscopes which were capable
to the single scan.) Many
phenomena are non
-
recurring or so slow, that they can’t be observed with an
oscilloscope without data storage. Remember that there might exist aliasing while
utilizing single scan.


Due to the low sampling frequency of the used os
cilloscope, the manufacturer has
reported that in the single scan mode the oscilloscope is capable to work only under
frequencies of 2 MHz while utilizing one channel and under 1 MHz while utilizing
two channels. The oscilloscope does however not filter ou
t higher frequencies, thus
there exist aliasing with frequencies which are over half of the sampling frequency.
While utilizing the single scan even the asynchronous sampling can not save the
situation if the signal frequency is near to the sampling freque
ncy or its multiple. For
example sinusoidal wave of 20,05 MHz appears in the screen like sinusoidal wave
of 50 kHz.

0.4

Probe

There exist a variety of probes for different purposes; to convert non
-
electrical and
electrical quantities to the voltage which can b
e measured with the oscilloscope
(current probe, pressure converter etc.) The probe can be either active or passive.







100

MHz to the AD
-
converter
,
there may exist a
liasing at the frequencies that exceed half of the
sampling frequency.
E.g. if the signal frequency is exactly the same as sampling frequency, the
sample will be taken
always
in

the same phase of the signal and there is a DC
-
signal displayed on
the screen.

If the signal frequency is 1 Hz over or below the sampling frequency,

there will be a
1

Hz signal

on the screen
,

the amplitude of which is the same as the original signal. So, a high
frequency signal is aliased to low frequency.

0.4.1

High
-
impedance probe

If an oscilloscope is coupled to the measured circuit without the probe, the
oscilloscope become a part of the circuit
, which interfere with the original
functioning of the circuit. The finite input impedance of the oscilloscope is loading
the circuit.

Oscilloscope input impedance is a transversal resistance, typically 1

M

,
and there is a
capacitor of

20 pF

parallel to i
t
.

At low frequencies the impedance is
normally high enough, but at high frequencies the capacitance lowers the impedance
which may dramatically alter the functioning of the circuit.
The may also exist bit
errors in the fast digital circuits due to the ref
lections in the signal cable.
E.g. a
signal entering a 1,5

m long signal cable reflects from the oscilloscope and return
back

to the circuit

after ca.
10

ns. So, there exist additional pulses in the circuit.


Most common probe type is a passive high
-
impeda
nce probe
and

input impedance
can be increased at the cost of sensitivity.
Figure

0

shows the connection of a probe
to the oscilloscope input.
The resistance of the probe is close to probe tip in order to
let the reflec
tion due to the probe to return
back
as fast as in
ca.
100 ps

and even a
fast circuit can not interpret the reflection as a new pulse.

Basically, a probe is a volt
divider,
although

a resistance is not enough but also a compensating capacitance is
needed p
arallel to the resistance in order to guarantee a constant attenuation ratio at
a
vast frequency
range
.
Capacitor value depends on the voltage division ratio and
input

capacitances of measurement cable and oscilloscope.

Thus the probe
capacitance must be a
djustable.
Sometimes the capacitance value is fixed and an
additional capacitance is placed in a connector that connects the probe to the
oscilloscope.

This capacitance is electrically parallel to the oscilloscope input.


Resistive voltage division ratio o
f the probe is


1
1
:
m
R
R
R
i
i


,








(0.1)


where the resistances are as in the fig. 8. The attenuation of the probe is
m
-
fold.

In
order to keep the voltage division ratio of the probe constant regardless of the
frequency, the capacitive voltage
division must be the same.


1
1
1
1
2
2
1
1
1
2
:
m
C
C
C
C
C
C
C
C
C
i
i
i













(0.2)



Connection to e
arth

Probe

Cable

Oscilloscope

Metal
shielding

Earth conductor
works as shielding

Metal shielding

C

i

C

1

C

2

R

1

R

i

Mains earth

head


Figure
13
. Probe, cable and amplifier input. Oscilloscope amplifier input sees a
voltage that is generated over
R
i

,
C
i

and

ca
ble capacitance

C
2

(ca
. 80 pF/m).
Note
that oscilloscope earth and thus also the earth connection of the probe are
connected to mains earth through the mains cable of the oscilloscope.



It follows that:


C
R
C
C
R
i
i
1
1
2


(
)
.







(0.3)

0.4.2

Calibrating the pr
obe

The probe can be calibrated with a rectangular wave. The rectangular wave is
commonly taken from the calibration output of the oscilloscope and it is coupled to
the input through the probe. When the sweep time and the oth
er controls have been
adjusted so that the rectangular wave is clearly visible in the screen, the rectangular
wave is adjusted as right
-
angled as possible with the capacitor of the probe.


There might be a switch on the probe, which affect to the damping p
roperties. In 1


position the damping resistance and capacitance are passed (normally by a small
resistance), in 10


position the damping is on and in ref

position
the switch short
circuits the central cable to earth and position of neutral (zero) level c
an be checked
on the oscilloscope screen.



is too large

is too small

right value


C

1

C

1


Figure

14
.
Calibrating the probe with an adjustable capacitor.

0.5

The usual measurement with the oscilloscope

In most cases it is enough to see the waveshape on the screen and it is not ne
cessary
to specify numerical values.

0.5.1

Measuring the amplitude, the frequency and the period lenght

When reading signal values from the screen
,

the
scale
adjusters must be put on cal
-
position
(volts/div
and

sec/div).

0.5.2

Measuring the

rise time

and the

pulse width

An ideal step
-
function response is composed of unlimited amount of frequency
components. Due to the limited frequency band the step
-
function response has
always a finite rise time. Because in practise the step
-
fu
nction response can be very
complicated, it has been agreed that the rise time is between 10 % and 90 % of the
initial and the final level of the step. A continuos adjuster of the vertical deflection
facilitate
s

the reading of the rise time. The length of
the positive pulse is defined
from the midpoint of the increasing edge to the midpoint of the decreasing edge.

0.5.3

Measuring the phase difference

The phase between two equifrequent sinusoidal signal can be measured as follows.
Adjust

the amplitude of the both signal to be equal. Read the time difference from
the screen. The relation between the time difference and the period length multiplied
by 360° is the phase difference.




10%

50%

90%

droop

Pulse width

Rise time

crossing

Setting time

50%

Underswing

before rising

edge


Figure
15
. Definitions of pulse characteristics

0.6

Measurement

Equipment




Dual
-
beam oscilloscope



Probe



Signal generator



RC
-
circuit and rectifier circuit


The experimental work in the laboratory is

not only filling the answering forms.
Making clear notes about the used equipment, the measuring system and the
measurement results is as important as the data itself.

0.6.1

Becoming acquainted with the oscilloscope

What are the type identification markings, th
e input resistance and the input
capacitance, the most sensitive voltage range and the shortest sweep time of the used
oscilloscope? What kind of functional blocks there are in the oscilloscope? Label
them and list adjusters and connections related to each

block. Is there some adjuster
or connection that is not mentioned in the summary above? Mention as well if in
your opinion some fundamental adjuster or connection is missing.

0.6.2

Adjusters of the display and the calibration

Switch on the oscilloscope and wai
t a while till the CRT warms up. Adjust the beam
intensity and sharpen the line or the point. If you can’t find the signal, ask for help
for the assistant. Couple probes to the oscilloscope and calibrate them. What is the
attenuation ratio and the magnitud
e of the damping resistance of the used probe?
Find out the frequency and the amplitude of the internal calibration signal.

0.6.3

Measurement of amplitude, phase and period

Couple sinusoidal signal to the input of the RC

low
-
pass filter. Connect probes
from the

oscilloscope to the input an output of the RC
-
circuit. Measure the complex
transfer function
U
U
Out
In

(amplitude/phase) with the frequencies given in the
answering form. Measure the period length with the oscilloscope and calculate the
freq
uency of the signal.


R
C
U
in
U
out


Figure

16
. RC
-
low
-
pass filter

Compare measured transfer function values with those calculated in the pre
-
laboratory exercises and explain what might be the reason for the dif
ferences
between these two values.


With what frequency the signal level has decreased 3 dB? What is the phase
difference with this frequency?

0.6.4

Rectifier circuit

Couple 50 Hz sinusoidal signal ( 8 V peak
-
to
-
peak ) from the signal generator to the
input of
the rectifier circuit. Measure the magnitude of the rectified output voltage
and the ripple (AC
-
component). Measure the ripple
as well with the frequency of
1

kHz.

0.6.5

Measurement of pulse, rise time

Couple 10 kHz rectangular wave (
amplitude 5 V
) from the sign
al generator to the
other input of the oscilloscope. Measure the rise time and the pulse width, in other
words the time that the signal is over 50

% of its maximum value (fig. 15
). Measure
as well the rise time of the decreasing part of the signal (from de
creasing edge) and
the pulse width (time that the signal is under 50 % of its maximum value). Compare
these results.


1

Multimeter measurements



Goal of the laboratory work




To learn the basics of the structure of the multimeters



To learn the capability of

the studied multimeters



To learn the restrictions of the studied multimeters



To learn to effectively use the usual multimeters.

1.1

Operation principles of a digital multimeter

Multimeters can usually be utilized to measure
voltage, current and resistance.
Voltage and current measurements can be carried out with AC
-

or DC
-
signals. In
addition to these basic properties it can be possible to measure a lot of other
measurands such as capacitance, frequency or transistor current
amplification. The
digital multimeters studied in this work are Fluke 8050 and Metex M
-
4650.

1.1.1

Block diagram of a multimeter

Figure
17

illustrates a block diagram of a typical multimeter. There is a attenuator in
the front part of the multimeter so that desi
red voltage, current or resistance range
can be selected. When resistances are measured the altering of the range adjusts the
current value of the current generator. The attenuator is followed by a rectifier
which usually is an idealized diode rectifier ba
sed on a linear circuit. The last stage
is an analog
-
to
-
digital converter and display.

1.1.2

Analog
-
to
-
digital converter

The basic component of a digital multimeter is the analog
-
to
-
digital converter
(ADC). The performance of
the
ADC determines the fastness, accuracy and
interference immunity of the multimeter.


ADC of the digital multimeter used in this work is basically a dual
-
slope ADC.
Dual
-
slope ADC is the most common converter type i
n digital voltmeters,
multimeters and in other slow measurement applications. The operation is slow
compared to the other converter types but a very good linearity and interference
immunity can be reached. The operation principles of the converter is shown

in

figures 18 and 19
.


Voltage
Current
Current generator
AC/DC
DC
AC
Ohm.
A/D
Display
Voltage
Current
Resistance


Figure
17
. Block diagram of a digital multimeter

In the beginning of the conversion the measured voltage
U
X

is connected to the input
of the integrator. Unknown voltage is integrated a constant time
T
1
. Int
egration time
is determined by the clock, counter and control logic of the converter. After the
integration a negative reference voltage
U
R

is connected to the input of the
integrator. The reference voltage is integrated until the output voltage of the
int
egrator
U
I

has reduced to zero.


n
R
C
U
U
R
reference
voltage
Clock
control logic
counter
integrator
comparator
X
digital output
U
I


Figure
18
. Dual
-
slope analog
-
to
-
digital converter



U
I
t
T
1
2
T
1
2
3

Figure
19
. The output voltage of the integrator U
I

at three different voltage levels.

Clock, counter and control logic measu
re the integration time until a comparator
detects zero voltage. The measured time
T
2

is proportional to the measured voltage.


T
T
U
U
X
R
2
1






(1.1)


Because of the operation principle the accuracy of the converter is independent of
the stabil
ities of the clock frequency and integrator time constant (however, they
must be stable and unchanged the duration of the integration).


Furthermore, a good attenuation of power line disturbance (at 50 Hz) can be reached
if the integration time
T
1

is chos
en to correspond to the period of the disturbance
signal or its multiple.

1.1.3

Multimeter readout at AC
-
range

Effective value (RMS
-
value) of an AC
-
voltage is determined as follows. Effective
value of the AC
-
voltage (U
RMS
) is equal to the DC
-
voltage value, whic
h has the
same average power to a resistive load as the studied AC
-
voltage. In case of AC
-
signals, effective value (U
RMS
) can be used to calculate different parameters in stead
of DC
-
values. For example, the current consumption of a 60 W bulb can be
calcul
ated with I=P/U
RMS
.

The instantaneous power of AC
-
voltage is


P
t
U
t
R
(
)
(
)

2
,






(1.2)


where
U
(
t
) is instantaneous voltage and
R

is load resistance. Average power is


P
U
t
R
dt
T
Ave
T


(
)
2
0
,







(1.3)


where
T

is period. In case the signal is not periodic, the integration time
T

is chosen
so that there will be no large deviation between many individual measurements.
Average power can be used to calculate the effective value of the voltage (U
RMS
):


U
P
R
RMS
Ave







(1.4)


For example the effective value (U
RMS
) of a sinusoidal wave is
1
2

times its
maximum value or
11
,
1
2
2



times the rectified average value of the voltage
signal. The response of the multimeter is usually b
ased on the average value or on
the effective value. Because the object is most often the effective value (U
RMS
), an
average value
-
based multimeter is adjusted to display effective value of sinusoidal
signal by multiplying the average value by number 1,11
. This kind of multimeters
display correct effective value only for sinusoidal signals. However, in case of other
wave types, the correct effective value can be calculated if the wave form is known.
For example moving
-
coil multimeters and the most simple c
ommercial multimeters
(Metex M
-
4650) are typically average value based multimeters. Effective value
based meters are for example moving iron multimeter (in electric power
measurements) and specific digital multimeters that calculate the effective values
wi
th suitable integrated microcircuits. Fluke 8050 studied in this work displays the
correct effective value. When measuring AC
-
signals, many multimeters do not take
into account the DC
-
component of the signal (when AC
-
range of multimeter is
selected). This
means that multimeter is AC
-
coupled. Both multimeters in this work
are AC
-
coupled. However, multimeters that measure the summed value of both AC
-
component and DC
-
component (AC + DC) does exist.

1.1.4

Measurement of small resistances

When using a typical multimeter

(Figure

17
) the resistance of measuring cable is
added to the measured resistance. When measuring small resistances (< 1


) can
cable and junction resistances have a remarka
ble influence. 4
-
wire measurement can
be used to effectively eliminate the influence of unwanted resistances (Figure

20
).
The current through the measured resistance (
R
) is constant and independent of
cable and junction resistances and the voltage meter me
asures only the voltage over
the studied resistance. In this work HP

3468A or HP 34401A multimeter with 4
-
wire measurement option is used to measure a small resistance.


V

Junction and conductor resistances

R

I

Voltage
meter

Measurement

current source


Figure
20
. 4
-
wire measurement of a small resistance

1.1.5

Studied

multimeters

The studied Fluke is a three
-
and
-
half digit multimeter (maximum displayed value is
1999) and Metex M
-
4650 is a four
-
and
-
half digit multimeter. Both multimeters are
based on the dual
-
slope ADC. In case of AC
-

and DC
-
signals, Fluke 8050 measures

directly the correct effective value (true root mean square, TRMS), so that the
displayed value is correct despite of the measured wave form. Metex, on the other
hand, represents simpler technology. The value Metex displays is proportional to the
rectifie
d average value which is corrected by multiplying with the number 1,11 (in
order to show the effective value of the sinusoidal signal). So, in case of other wave
forms, the value Metex displays is different from the correct effective value. Both
Fluke and
Metex are AC
-
coupled, so when measuring in AC
-
mode, both
multimeters display the value of AC
-
component only. The effective value of
summed AC
-

and DC
-
signal can be calculated as


2
2
ACRMS
DC
RMS
U
U
U


.




(1.5)

1.2

Measurements

Devices




Multimeter Fluke 805
0



Multimeter Metex M
-
4650



Multimeter HP 3468A tai HP 34401A



Voltage source Mascot 0
-
30 V



Oscilloscope



Potentiometer



Function generator



"Black box", (gray in color)



Resistor test board


1.2.1

Measurement of current and voltage

Connect the voltage source, resistor

and two multimeters so that you can measure
the voltage over the resistor and the current through the resistor. Adjust the voltage
source to ca. 5 V. Calculate the resistance with the help of measurements.

1.2.2

Measurement of DC
-
voltage of a high
-
impedance ci
rcuit

The measurement of the voltage of a high
-
impedance circuit is problematic because
the multimeter introduces a load to the circuit. In our "black box" case (Fig
21
.), the
high
-
impedance circuit is modelled by a voltage source with high internal resis
tance
R
S
. The voltage source is connected to the operational amplifier which has much
higher input impedance
Z
in

compared to the input impedance of a usual multimeter.
The operational amplifier has unity gain and very small output impedance which is
not ef
fected by the internal resistance of the multimeter. Measure the output voltage
of the operational amplifier with Metex. After that, measure the voltage of the
"black box" voltage source with the Fluke meanwhile the Metex is still connected to
the output o
f the operational amplifier. What value does the Metex display now?
Use the voltage difference measured with Metex and calculate the
R
S

.

A
=1
R
S
E
Z
in

Figure
21
. Black box

1.2.3

Measurement of AC
-
voltage

Measure sinusoidal, triangular and square w
ave signals (f=100 Hz,
unloaded

V
pp
=5

V) generated by function generator with both multimeters. (Do the
measurement results change if both multimeters are connected simultaneously
instead of one multimeter?) Add +2 V (DC
-
offset) to function generator outpu
t
signal and repeat the measurements.

1.2.4

Current measurement, voltage drop of multimeter

Measure the voltage over multimeters at the current of 100 mA. Use the range 0,2
A. Use one of the multimeters to measure current and the other one to measure
voltage. Th
e connections
are shown in Figure

22
. Change the multimeters and repeat
the measurement. ATTENTION! Set the potentiometer to its maximum value before
switching on the voltage source (in order to avoid short
-
circuit). Suitable voltage
level is 5V.

Voltage

source

V

A

100 R


Figure
22
. Measurement of the voltage drop of the current meter (ammeter)

1.2.5

4
-
wire measurement for small resistance

Measure the small resistance with Fluke, Metex and HP multimeters (HP 4
-
wire
measurement). How large is the junction resist
ance in the case of normal 2
-
wire
measurement?

1.3

Questions

1.3.1

How large is inaccuracy of Fluke and Metex at range 200 V (in accordance
with manufacturer data sheets) when the measured voltage is:


-

50 % of full scale, i.e. 100 V


-

25 % of full scale,
i.e. 50 V


-

5 % of full scale, i.e. 10 V


1.3.2

Why does the voltage change (in section
1.2.2
) when another multimeter is
connected to the input of the operational amplifier? Calculate the internal
resistance
R
S
. I
nput impedance of the operational amplifier is ca. 10
12


.


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ㄮ㈮1

by䵥瑥砠慮搠捡汣畬慴攠捯牲散c

RMS

values. Take into account that Metex uses correl
ation coefficient which
is suitable only for sinusoidal signal.


1.3.4

Compare the results in section
1.2.4

to the results given by the manufacturer .


1.3.5

Is it possible to find out (with Metex or Fluke) if the m
easured signal is DC
-
voltage, AC
-
voltage or the sum of those.


Appendix:


Notation


0.05% of reading + 3 digits means:




0.05%
of reading means 0,05% of the measured value.



3 digits

means change of three digits in the last displayed number. E.g. in Metex

at 200

mV range 3 digits is equal to 30

V.


All the inaccuracies can be added directly together.




Appendix 1.1 Metex M
-
4650

specifications


4. SPESIFICATIONS


Accuracies are ± (% reading + No. of digits) Guaranteed for 1 year, 23°C

± 5°C, less than 75% RH. Warm up
time is 1 minute.


DC Voltage


Range

Accuracy

Resolution


200 mV


10

V


2 V


100

V


20 V

± 0.05 % of rdg + 3 digits

1 mV


200 V


10 mV


1000 V

± 0.1 % of rdg + 5 digits

100 mV

Input impedance: 10 Mohm on all ranges. Overload protection: 1000V dc or peak ac on all ranges.


AC Voltage


Range

Accuracy

Resolution


200 m
V


10




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DC Current


Range

Accuracy

Resolution


200

A


10A


2A

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100A


20䄪




A


200mA


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A


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AC Cu
rrent


Range

Accuracy

Resolution


200

A


10A


2A

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Resistance


Range

Accuracy

Resolution


200 ohm

± 0.2 % of rdg + 5 digi
ts

0.01 ohm


2 Kohm


0.1 ohm


20 Kohm


1 ohm


200 Kohm

± 0.15 % of rdg + 3 digits

10 ohm


2 Mohm


100 ohm


20 Mohm

± 0.5 % of rdg + 5 digits

1 Kohm

Overload protection: 500 V dc/ac rms on all ranges, except 200 ohm range (250 V dc/ac).

Open circuit v
oltage: <1.2 V.

Relative humidity:

0 to 75%, 0°C to 35°C on 2 Mohm, 20 Mohm


0 to 90%, 0°C to 35°C on all other ranges


0 to 70%, 35°C to 50°C




Appendix 1.2 Fluke 8050A

Specification



Appendix 1.3 HP 3468A

s
pecifications


2

Signal spectrum measurements


Goals of the work




To study spectrums of periodic signals



To learn the most essential properties of spectrum analyzer

2.1

Signal spectrum

A convenient way to study sign
als in frequency domain is to consider them as a
superposition of sinusoidal signals of different frequencies. In most cases, the
systems and instruments in nature and techniques are such that only sinusoidal
stimulus gives a response of the same waveform,

although signals are e.g.
attenuated or delayed


the amount of attenuation and delay depending on
signal frequency. Examples of this kind of instruments are electric circuits built
of resistors, capacitors, and coils. Or, walls of buildings that attenuat
e acoustic
waves of different frequencies in different ways, still preserving the sinusoidal
waveform. (However, at higher signal levels walls have nonlinear response that
leads to distortion of the sinusoidal waveform). Transistors and diodes, for
example
, are nonlinear components and thus make an exception. Anyway, in
many cases, spectrum analyzer is a suitable instrument for studying signals in
frequency domain.


Bandpass filter of spectrum analyzer makes it possible to measure signals
within a narrow fr
equency band. Center frequency of the bandpass filter is
controlled by the analyzer. Usually the center frequency is swept over a certain
frequency range repeatedly, and the signal level is measured simultaneously as
a function of frequency. Thus, a spectr
um of the measured signal is obtained,
and displayed on the screen of the analyzer. Mathematically speaking, signals
consist of infinite amount of sinusoidal components that are infinitesimally
close to each other within the frequency band. Of course, when

measuring, it is
not possible to distinguish the frequency components that close to each other.
Instead, the bandpass filter picks up sinusoidal waves within a narrow band,
and all the waves contribute to the measured signal level. In other words,
bandwid
th of the bandpass filter determines the frequency resolution of the
spectrum analyzer.

2.1.1

Sinusoidal wave

This chapter concentrates on the special case of sinusoidal signal.
Mathematically it would be possible to study any signal as a super
position of
also others than sine functions. Then why does it come naturally to divide
signals to sine components? One justification is that, in nature oscillators and
systems reacting to oscillations usually resonate sinusoidally. Some examples
of this ar
e e.g. motion of a mass attached to a spring, harmonic motion of a
pendulum (if the amplitude of the oscillation is not too large), and electric
oscillation in an LC
-
circuit. This is the reason why sinusoidal signal passes the
medium without changing its w
aveform, while other signals are distorted.


Next we will see why harmonic oscillators oscillate sinusoidally. State of
oscillator will be solved as a function of time, after forcing it into a motion. As
an example, let us consider a mass attached to a sp
ring (mass of the spring is
assumed to be zero). The force applied to the mass (by the spring) is given by
F
ky


, where
k

is the elastic constant, and
y

is displacement of the mass
from the equilibrium position. Thus, the acceleration of
mass
m

is

k
m
y
t
(
)

,
where the displacement is given as a function of time. On the other hand, we
may write the acceleration as the second derivative of the displacement
d
y
t
dt
2
2
(
)
. This leads to a differential equation of the f
orm


d
y
t
dt
k
m
y
t
2
2
0
(
)
(
)


.








(2.1)


Equation 2.1 has a general solution
y
t
A
t
B
t
(
)
cos
sin




, where


is
angular frequency of the oscillation, and

2

k
m
/
. Constants
A

and
B

are
obtained from boundary conditions, such as th
e values of velocity and
displacement at time
t

=

0.


2.1.2

Fourier series of periodic signals

A sinusoidal signal that is written in time domain as cos(

0
t
), is described in
frequency domain by a single spectral line at

0
. Non
-
sinusoidal, periodic
function may be written as a superposition of sine functions at harmonic
frequencies
n

0

(
n

= 1,2,3,...). This gives a line spectrum with spectral
components at frequencies
n

0
,
which mathematically correspond to the
Fourier series of the signal. For a non
-
periodic signal, at certain conditions, the
spectrum is given by the Fourier transform of the signal. In the following, only
the case of periodic signal (and thus only the Fouri
er series) is discussed.


The Fourier series gives us spectral components of a periodic signal
u
(
t
). That
is, the signal is written as a sum of sine and cosine functions of different
frequencies.


u
t
a
a
t
a
t
a
t
b
t
b
t
b
t
a
a
n
t
b
n
t
n
n
n
(
)
cos(
)
cos(
)
cos(
)
.
.
sin(
)
sin(
)
sin
(
)
.
.
(
cos(
)
sin(
))















0
1
2
3
1
2
3
0
1
2
3
2
3














(2.2)


a
0

is amp
litude of the DC component, and AC components are given by
a
n
and
b
n

(n > 0). Fundamental, i.e. the lowest, frequency is



2
T

where
T

is the signal period. Cosine functions and sine functions describe

even and
odd parts (in respect of time
t

=

0) of
u
(
t
), respectively. Sum of a cosine term
and a sine term is a sine function that has a phase angle defined by the ratio of
a
n

and
b
n
. In consequence, each spectral
component has not only the
amplitude, but also a phase.

2.1.3

Fourier integral

Values for coefficients
a
0
,
a
n
, and
b
n

are obtained by integrating the studied
function, weight
ed with cosine and sine, over one period.



a
T
u
t
dt
T
T
0
2
2
1



(
)








(2.3)


a
T
u
t
n
t
dt
n
n
T
T





2
1
2
2
(
)
cos(
)
(
.
.


,
)




(2.4)


b
T
u
t
n
t
dt
n
n
T
T





2
1
2
2
(
)
sin(
)
(
.
.


,
)




(2.5)


Terms 1/
T

and 2/
T

preceding the integrals are used for normalizing. That is,
a
n

in equation (2.4) equals to 1 if the f
unction
u
(
t
) is
cos(
)
n
t

. Also, integral may
be considered as an inner product. Using inner product, a period of the function
is projected to sine and cosine components of each frequency. Thus, each value
of inner product is a real number that i
ndicates how much of each frequency
component is contained in the function. Inner product of functions, e.g. the
Fourier transform, corresponds to inner product (dot product) of vectors in
geometry. FFT (Fast Fourier Transform) spectrum analyzers sample si
gnals in
time domain and use the Fourier transform to calculate the signal spectrum
from the saved data.


2.1.4

Example: Fourier series of rectangular wave

t
E
u(t)
t
E
u(t)
T/2
-T/2
T
-T
0
T/2
-T/2
T
-T
0
a)
b)

Fig
ure

23
. a) Even and b) odd rectangular wave

Let us first calculate the Fourier series coefficients for the even rectangular
wave shown in figure
Error! Reference source not found.
a). The coefficients,
a
n

and
b
n
, can be calculated using equations (2.4) and (2.5):


a
0

equals to the time average of the signal, i.e. it gives the DC component. In
the case of pure AC signal, its value is 0.


a
T
u
t
n
T
t
dt
n
T
T









2
2
2
2
(
)
cos










dt
t
T
n
T
E
dt
t
T
n
T
E
dt
t
T
n
T
E





























2
4
4
4
4
2
2
1)
(
2
2
2
2
1)
(
2



cos
cos
cos



(2.6)



even

for

,
0









odd

for

,
4
)
1
(
2
sin
2
sin
2
2
sin
2
2
2
1
n
n
n
E
n
n
n
T
Tn
E
n







































(2.7)


Both the rectangular wave

of figure
23
a) and cosine function are symmetric
relative to time t

=

0, i.e. they are even functions
1
. Thus, it is possible to
calculate integral for example for the negative half of the period only, and
multiply it then by factor of two to get the final

result.




1

It should be noted that
even or odd function relative to time
t
=0 is not the same thing as
even or odd harmonic component, which refer to multiple frequencies of fundamental
frequency.


On the other hand,
b
n

equals to zero for every
n
. This comes from the fact that
sine function is an odd function and the rectangular wave in figure
23
a) is even


Integrals of negative and positive half periods thus have equal magnitudes of
diff
erent signs, and they cancel out each other.


The original rectangular wave in figure
23
a) can now be re
-
constructed using
sinusoidal waves given by the Fourier series. Using numerical values
calculated for
a
n
, equation (2.2) gives:


u
t
E
T
t
T
t
T
t
(
)
cos
cos
cos
.
.


































4
2
3
2
3
5
2
5







(2.8)


If the function to be described is odd, the coefficients
a
n

equal to zero while
b
n
:s have non
-
zero values. This gives a Fourier series representation for the
signal of figure
23
b):


u
t
E
T
t
T
t
T
t
(
)
sin
sin
sin
.
.


































4
2
3
2
3
5
2
5




.



(2.9)


If the time origin is chosen in such

a way that the rectangular wave is neither
even nor odd, both the
a
n

and
b
n

coefficients have non
-
zero values when
n

is
odd.


Fourier series coefficients (up to
n

=

5) of the rectangular waves in figure
Error! Reference source not found.

are shown below, in figure

24
.





f
e(f)
0
1/T
2/T
3/T
4/T
5/T
4E

4E


4E




Fig
ure

2
4
. Fourier series components of a rectangular wave build up a line
spectrum where the amplitudes are given by coefficients a
n

and b
n
. To get RMS
values of the compo
nents, the amplitudes must be multiplied with
1
2
.
Spectrum analyzer measures the RMS values of the spectrum components.

2.2

Noise

White noise consists of thermal noise
U

(equations 2.10 and 2.11) and shot
noise
I

(2.12), and it
s power is evenly spread over all the frequencies.


P

=
kTB









(2.10)


U
kTBR

4








(2.11)


where

k

= Boltzmann constant (1,38 x 10
-
23

Ws/K)


T

= temperature (Kelvin)


B

= bandwidth


R

= internal resistance of the noise source (resisto
r)


I
qI
B
DC

2








(2.12)



where

q

is elementary charge

(1,602 x 10
-
19
C) and
I

is current through a
potential gap (pn junction). Thermal noise is produced by random thermal
motion of the charge carriers (e
.g. electrons) that leads to arbitrary varying
current flow between e.g. resistor ends.


Shot noise is caused by statistical variation in the amount of charge carriers at
different time moments, for example in a transistor junction. Also an additional
noi
se contribution, so called 1/f noise, is present. 1/f noise arises from several
different sources, and its power is spread over the frequency domain inversely
proportional to
n
:th power of frequency (
n

varies from 0,9 to 1,3).



Noise spectral density
S
(
f
)

is given as a function of frequency, and it describes
how the spectral components of noise are spread over the frequency domain.
For example, thermal noise of a resistor has a constant noise spectral density
(white noise is evenly spread over all the freq
uencies).



S
f
U
BR
kT
(
)


2
4

[W/Hz]







(2.13)


As an example, let us consider a model of operational amplifier where the noise
sources can be described using voltage and current sources, and the noise
power thus depends on an external resistance. N
oise of the source is given as
voltage noise density
V
Hz

or, as current noise density
A
Hz
.


When the noise spectral density is known as a function of frequency, the total
noise power can be calculated by integrating over th
e frequency range. In case
of noise voltage, the voltage noise density must be squared before integrating.
After that, square root of the integral is calculated to get RMS value of the
noise voltage. This corresponds to sum of squares of the AC voltage
com
ponents. RMS value of noise current is calculated in the same way.


2.3

Measuring spectrum of periodic signal

Depending on the application, measurement equipment of different types is
used to measure spectrum. In this chapter, three measurement instruments wit
h
different operating principles are introduced.


-

Selective voltage meter

-

Each frequency component is measured separately using a tunable
narrow bandwidth filter.


-

Sweeping spectrum analyzer

-

Input signal is mixed with local oscillator that can be swept in
frequency.

-

Contains an intermediate frequency filter at fixed frequency.

-

Spectrum is shown on a display. Y
-
axis indicates the measured voltage
and X
-
axis indicates the

frequency. If the local oscillator frequency is
e.g. 500

MHz, and the center frequency of the intermediate filter is
300

MHz, the analyzer is measuring either the frequency of 800

MHz
or the frequency of 200

MHz.


-

Is typically used at radio frequencies
.


-

Digital spectrum analyzer

-

Takes samples of the signal to be measured.

-

Processor of the analyzer calculates signal spectrum from the samples
using discrete Fourier transform

(DFT). In practice, Fast Fourier
Transform (FFT) is usually used. A minor drawback of the FFT is that
number of the samples must be a power of two.


-

Can be applied also for measuring spectrums of non
-
periodic signals.


-

Is typically used at audio freq
uencies.


Network analyzer is an instrument that resembles spectrum analyzer, except
that it also has an output in addition to the input. Output of the network
analyzer is set to the same frequency as the bandpass filter, and the analyzer
measures the ampl
itude and phase differences between the input and the output.
Therefor
e
, network analyzer is a suitable device for measuring e.g. frequency
and phase responses of an amplifier. Signal reflection at the amplifier input or
output can also be easily measured
using network analyzer.


In this laboratory work, a sweeping spectrum analyzer is used. Frequency
resolution of the spectrum analyzer is set by intermediate frequency filter
bandwidth. Figure
Error! Reference source not found.

describes how the
intermediate frequency filter affects the measurement result.



f
f
a)
b)
c)

Fig
ure 25
. Effect of a bandpass filter to measurement result
: a) response of the
bandp
ass filter, b) ideally measured sinusoidal signal at frequency f, and c)
the measurement result in practice.

If bandwidth of the filter is too large, it is not possible to make difference
between two frequency components that lie close to each other. In
that case, the
bandwidth must be narrowed. However, if the filter bandwidth is about to be
narrowed, the sweep time must be increased in order to allow the filter work
properly. Thus, in practice, a suitable value for the filter bandwidth is given by
the s
weep time. The resolution bandwidth
B
r

that can be obtained is


B
B
t
r
tot
s











(2.14)


where
B
tot

is the frequency band that is swept over, and
t
s

is sweep time.


Some other spectrum analyzer properties that must be taken into account when
pe
rforming measurements, are e.g. noise floor of the analyzer, distortions of
preamplifier and mixer, and uncertainties of frequency and voltage references.


2.4

Measurements

Equipment list




Spectrum analyzer HP 8590B



Oscilloscope



Signal generator, output impeda
nce 50


, with variable pulse symmetry


e.g. Hung Chang 8205A (SYM
-
control)



Noise generator



Band
-
reject f
i
lter



The first thing to do is to turn on the spectrum analyzer in order to allow the
analyzer to warm up before the measurements. In this work, the

frequencies to
be measured are relatively small compared to the frequency span of the
analyzer. Thus, while warming up, drifting of the 300

MHz reference oscillator
of the analyzer may cause problems. Push PRESET to reset the analyzer
settings.


The zero

frequency (DC) of the spectrum analyzer may differ from the actual
zero frequency by more than one MHz because it depends on the 300

MHz
reference oscillator and on the center frequency of the intermediate frequency
amplifier. One can verify the actual z
ero frequency e.g. by setting the center
frequency on the display to 0

Hz
(FREQUENCY cen
ter freq) and frequency
span to 5

MHz (SPAN). Even if there is no signal connected to the analyzer,
the actual zero frequency is shown as a high peak on the display. O
ne can
check the frequency of the peak using PEAK SEARCH or MKR keys, and
tuning the knob. Frequency of the DC peak can be set to zero by selecting
FREQUENCY freq offset, and after that inserting the desired frequency offset
using front
-
panel keys. The num
erical value of the frequency offset is obtained
by inverting the frequency reading of the DC peak. After this procedure the DC
peak is shown at zero frequency on the display. Repeat the procedure when
necessary.


2.4.1

Spectrums of sinusoidal and rectangular wa
ves

Use a BNC ‘T’ adapter and two coaxial cables to connect the oscilloscope
input to the signal generator output and to the input of the spectrum analyzer.
Due to the 50



input impedance, spectrum analyzer reduces the signal
generator output voltage to approximately half of its open
-
circuit voltage.
Using the oscilloscope, adjust the output signal of the signal generator
according Fig.

27.





1 V
-1 V

s
U
t


Figure 27. Signal generator output

Measure the frequency components of the signal in figure 27 up to 10

MHz.
Frequency span can be set for example by using FREQUENCY start freq and
stop freq
1
. For the vertical axis of the spectrum analyzer display, u
se setting
AMPLITUDE units = volts. To read out the frequency component values, use
MKR key and the knob. If it appears to be difficult to measure the frequency
components accurately, use AMPLI
TUDE ref lvl and the knob to adjust
sensitivity.


Change the
bandwidth control of the spectrum analyzer to manual mode (BW
res bw auto/
man
) and measure the input signal with four different bandwidth
settings. The analyzer automatically selects the sweep time. Write down on
paper the sweep time, the bandwidth, and th
e frequency span. Use equation
2.14 to calculate the theoretical sweep times (re: Question 2.5.2).


Set the bandwidth control of the analyzer back to the automatic mode using res
bw
auto
/man. Select a sinusoidal waveform from the signal generator and
measu
re its spectrum components up to 10

MHz.

2.4.2

Noise measurement

There is a noise generator on the laboratory table. Apply a voltage of 12 to
15

V from the power supply to the noise generator, and connect the noise
generator output to the spectrum analyzer using

a cable with BNC connectors.
Due to the characteristics of noise, amplitudes of the frequency components
vary after every sweep. Thus, use BW video avg
on
/off to select averaging.
Voltage noise spectral density is used to measure the noise voltage as a
fu
nction of frequency [V/
Hz
], see paragraph 2.2. If the noise voltage at a
certain frequency and the bandwidth of the spectrum analyzer bandpass filter
are known, one can calculate the voltage noise spectral density. In practice, the
res
olution bandwidth well corresponds to the bandwidth of the bandpass filter.


What is the voltage noise spectral density at 2.5

MHz? What would be the
RMS value of the total noise voltage, assuming the voltage noise spectral
density to be constant up to 1

GHz. Apply a band
-
reject filter between the noise
generator and the spectrum analyzer. The filter has several stopbands. Measure
the center frequencies (<

5

MHz) of the stopbands and calculate their
differences. It is possible that one has to optimize the
frequency span of the
analyzer to find the minimums and measure them accurately.

2.5

Questions

2.5.1

Compare the measured spectrum of the rectangular wave to that
calculated in homework. If the measured spectrum differs from the
calculated one, try to explain

the differences.


2.5.2

Is equation (2.14) valid with different resolution bandwidth (res.bw)
settings?


References:

Engelson ja Telewski, Spectrum Analyzer Theory and
Applications, Artech House Inc. USA 1974.





Mittaustekniikan perusteet opetusmoni
steet 1993 (In Finnish)


Appendix 2.1 A paragraph from HP 8590B

user’s guide



3

Frequency Counter


Goals in this work




Learn the structure of

a

frequency counter



Familiarize with capabilities of digit
al meters



Learn to handle measuring results with the help of
mathematical
statistics

3.1

General Information Concerning Digital Meters

Almost all digital meters return
to measuring time (number of pulses)
one way
or another.
The advantages of a digital meter c
ompared to an analog one are:



-

prob
ability
of an

error in reading

decreases


-

position

of the meter does

n
o
t effect on the result
s
3


-

a more accurate result is usually obtained with a digital measuring
method

-

it gives a possibility to automate the m
easurements (
many digital
meters can be connected straight to a computer)


It is to be noticed when using a digital meter, that although the meter gives the
result with many numbers,
not all numbers are necessarily correct
. One of the
advantages of
many digital meters is,
that the measuring time can be modified
.
The measuring time can be chosen so,
that the part of the random error is
smaller than the part of the systematic error
.

3.2

Frequency Counter

3.2.1

General Information

With a frequency counter can
be
measured frequency, period,
time difference
of the pulses

and ratio of the frequencies
.
With t
he frequency counter used in
this measurement can only be measured
frequency and period
. Regardless of
what quantity is measured at the time,
the measurement is b
ased on counting
the pulses
.
Next we take a look at the two most important functions of the
frequency counter
,

direct frequency measurement and measurement of period.




3
In very spec
ific measurements do not believe this.
One very accurate counter was noticed to
g
ive different results in different positions:
The frequency of the clock of

a GSM base station
was measured

13,000 000 01 MHz
when the meter was horizontal and

13,000 000 09 MHz
when vertical
,
which is also a very common position used in measurements
. Turn
ed upside
down the meter showed 13,000 000 17 MHz. The change in results in different positions was
many times greater than
the drifting of the meters clock

between the calibrations
.
The
manufacturer answered to the inquiry, that gravity effects on all cry
stal oscillators, so that th
e

meter needs to be
calibrated in the same position, it will be used
.

3.2.2

Direct Frequency Measurement

The connection of
the

frequency

counter in direct frequency measurement is in
picture
28
.



f

m

T

N

f

osk

Input
Circ
uit

Gate

Counter

Screen

Divider

Oscillator




Picture

28
. The connection of
the

frequency counter in direct frequency

measurement
.

In direct frequency measurement a pulse is created with the help o
f a
frequency
-
change oscillator and a divider
,
that controls the gate so that the
frequency to be measured gets into the counter
,
when the pulse is up
.
Number
of the
pulses that come into the counter is


N
Tf
m

=




1









(3.1)


where
f
m

is the frequency t
o be measured and
T

the measuring time
. The factor
of the uncertainty ±1 in the equation (3.1) comes from the fact, that
the
beginning of the pulse from the divider

is in arbitrary phase

compared to the
frequency
f
m
.
Picture

29

clarifies the uncertainty