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4

Experiment I : Ohm's Law









and Not Ohm's Law




I.
Purpose of this Experiment

The main purpose of this experiment is to review the measurement of voltage

(V), current (I),
and resistance (R)

in

dc circuits
.

In the first part, you will measure the

internal resistance of a
battery and
examine the relationship between V and I in a resistor which obeys Ohm's law
. In
second part of the lab, you will measure the resistance of some electrical devices that do not obey
Ohm’s law.


II.
References



Halliday
, Resnick and Krane, Physics, Vol. 2, 4th Ed., Chapters 32, 33

Purcell, Electricity and Magnetism, Chapter 4

Taylor, An Introduction to Error Analysis, Second Edition


II. Equipment


Digital multimeters

Resistor board with 4, 5 10, and 20 ohm resistors

6
-
volt battery

knife switch

10 V Power Supply

diode board with
switching

diode
s, LED

and 40 ohm resistor


III. Introduction



This section contains background material on current
,

voltage

and resistance

that you
should already know from

your prior
physics

classes
, including high
-
school physics,

Physics
174, and Physics 272.
A few days before your lab class starts, at the latest, take a quick look over
this introduction. If you find you are already

familiar with the material, then skip to the next
section a
nd go over the experiment.

You probably will need to read the section on diodes, since
this is

usually
not
covered in beginning physics classes.

If you

are

not familiar with th
e

other

material

in the introduction
, then you have missed or forgotten some ver
y important and basic
physics

and you

need to give
this introduction

a thorough and careful reading
. You also should
dig out your Physics 272 text, or one of the references above, and read over chapters on dc
circuits.



Voltage


W
hen an electric charge

m
ove
s

between two points
that have an electric potential
difference between them
, work is done

on the charge by the source that is creating the potential
.


5

The amount of work that is done is equal to the
decrease in
the potential energy of the charge.

Th
e

di
fference in potential energy

is
equal to

the product of the difference in the electrical
potential between the points and the magnitude of the electric charge.

In the SI system of units,
the unit of

electrical potential

difference

is the

volt

(
written as V
). For this reason,
almost
everyone

who work with circuits say "
voltage

difference
" instead of "electrical potential
difference".

Voltages are measured using a
voltmeter
.
Voltage differences are always measured
between two points, with one lead of the
volt
meter c
onnected to one point and the second lead
connected to another point.
On the other hand, diagrams of circuits almost always show the
voltage at individual locations in the circuit.
If the voltage is given at one point, then this means
that the secon
d point was at "
ground
" potential or "zero volts" and this ground point is labeled
on a circuit schematic using a special symbol

(see Figure 1.2)
.



Current


The rate at which charge passes through
a

surface

is called the electrical
current
.
Current

is me
asured in Ampères
, commonly called
amps
, with units written as A
. One amp of
current is defined as one Coulomb of charge passing through a cross
-
sectional area per second.
Since an electron has a charge of
-
1.609x10
-
19

C,
This is equivalent to about 6×10
18

electrons
passing per second. Current is measured using an
a
mmeter

which is placed in a circuit

so

that
the current flows into the positive terminal of the ammeter and out the negative terminal.

Since
the current flows through
the

ammeter, and we do not

want
the ammeter

to disrupt the current
that is ordinarily flowing through a circu
it
,

an ammeter has a low resistance. Never connect an
ammeter directly across a battery (or other voltage source), since this will result in a large current
flowing through
the ammeter, possibly damaging it or the battery.

Note that in contrast a
voltmeter has a high resistance.


Resistance

and Ohm's Law


When

current

is driven through
an ordinary
electrically
conducting material
, such as a
metal or semiconductor at room temp
erature,

it
encounters
resistance
. You can think of
resistance as a
sort of frictional drag.

In
a sample made of a
good conducto
r
,

the current is
directly proportional to the potential difference,

i.e.



This relationship is

called
O
hm's Law

and is us
ually written:



V=IR

In this relationship,

I

is the current

flowing through the sample. The potential difference
V

is
the
difference

in voltage between one end of the sample (where the current enters) to the other end
(where the current
leaves). Finally,
R

is the
resistance of the sample
. In the SI system of units,
resistance has unit of
ohms
, which is written as


.



In many materials

the resistance does not change with the amount of voltage applied or
the current passing through it, o
ver a large range of both parameters, so it is a constant

to a very
good approximation.

The resistors used in this lab are made of thin metal films or carbon (a
semiconductor). You should find that they obey Ohm’s Law very well.

Metals are examples of
good

conductor
s. They have a high density of electrons that are relatively free to move around,
so that

connections made with metal tend to have a
low resistance
. In an electrical
insulator
,
the
electrons are more tightly bound and cannot move freely.

In a

sem
iconductor
, most of the
electrons are tightly bound, but there is a small fraction (compared to a metal) that are free to

6

conduct current. The small density of carriers in semiconductors makes them more resistive than
metals, and much more conducting than
insulators. It also gives them many other unusual
properties, some of which we will see in this lab.



Batteries and EMF


There are a variety of ways to generate a voltage difference.
Batteries

produce an
electrical potential difference through chemical r
eactions. If the plus (+) and minus (
-
) leads of a
battery are connected across a resistor, a current will flow out of the positive terminal of the
battery (which has a more positive potential

than the negative terminal
), through the resistor and
into the
negative terminal. In other words, the
positive
current flows from the positive to the
negative
terminal of the battery
.

Inside the battery,
chemical reactions drive a

current flow

from
the more negative region to the more positive regio
n. As a result, a

battery can be thought of

as a
charge pump

that is trying to push positive charge out of the + terminal and suck positive charge
into the
-

terminal
.


In physics and EE textbooks, one also encounters the terms
electromotive force

or
EMF
.
The term

EMF

come
s from the idea that a force needs to be exerted on charges to move them
through a wire (to overcome the resistance of the piece of wire to the flow of the current). T
he
battery
can be

thought
of as

the source of this force.
However, the EMF of a battery
is just the
voltage difference generated across the terminals of the battery and
is measured in volts
. So EMF
is not actually a force, despite its name
.

In
P
hysics 276, we will not

make distinctions between
the EMF, the voltage difference,
and
the electric
al potential difference
, but use these terms
interchangeably
.



Part of this experiment is to measure the EMF and
internal resistance

of a battery.
When a current flows inside a battery it is also

encounters resistance and

the battery
is said to
have an
i
nternal resistance
.

Batteries with low internal resistance, such as

the 12 V lead
-
acid
batteries
commonly found in cars
,
can deliver a lot of current
. They

need to be treated with
caution
; shorting together the terminals of a battery (or other voltage sour
ce) with a low internal
resistance could lead to melted wires, a fire, or the battery exploding
.

On the other hand, batteries
with high internal resistance cannot deliver much current and show significant loss of voltage
when current is supplied.


Electri
cal Symbols


Components used in electrical experiments have standard symbols. Those required in
this experiment are shown in Fig. (
1
.
2). You should understand what each symbol represents and
use them when drawing schematics of your own circuits.







electrical


ground

( V = 0 )


7

Figure

1.2. Some common symbols used in electrical circuits.







Figure 1.3 Simple circuit with a battery and two resistors showing direction of positive current
flow I.



Electrical Circuits


An electrical circuit is formed by using wires to con
nect together resistors, batteries,
switches, or other electrical components into one or more connected closed loops. Where three or
more wires meet, the current will split between the different paths. However each new path for
current flow that is create
d at these junctions must rejoin another channel at some other point, so
that all loops close. All loops that are created must be closed so that current can flow.


Kirchhoff's Rules


There are two very useful rules for

analyzing electrical circuits and

f
inding the currents
and voltages at different points in a circuit
.



Rule 1
:
In going round a closed loop the total change in
voltage

must be zero.



Rule 2:

The current flowing into any junction where wires meet is equal to the current
flowing out of th
e junction.




For example, a
pplying
the first
rule to Fig.
1.3

and assuming that the conductors joining
the components have zero resistance,
we find
the potential differences between the lettered
points in the circuit are given by:




Summing all the differences we get
:




which can be rewritten
:



.


8


As another example, we can apply rule 2 to Fig. 1.3. Considering the nodes at points P
and Q in the circuit, we get


At P:


Current in =

I



At Q:


Current in =
I
1

and
I
2




Current out =
I
1

+ I
2




Current out =

I


Both points yield the equation
I = I
1

+ I
2
.




Figure I
-
3





Figure I
-
4:
(a) Connecting resistors R
1

and R
2

in series produces a resistance
. (b)
Connecting resistors R
1

and R
2

in series produces a resistance
.


Series and Parallel Resistors


In Physics 174, you measured the resistance of two resistors when they were connected in
series (see Fig. 1.4 a) and in parallel
(see Fig. 1.4 b). For the series connection, one finds
, i.e. the resistance adds. For the parallel connection, one finds
.


These

elementary

results can be derived by applying Kirchoff's rules
. For example,
co
nsider the series connected resistors
.

Since current is conserved, the current I in
R
1

must be the
Q

P

(a)

(b)

O'

O


9

same as the current I in
R
2.

Hence the voltage drop across R
1

is

and the voltage drop
across R
2

is
. Thus we
can write
. This is equivalent to writing
where

and
,

i.e.
two resistors connected in series are equivalent
to one resistor whose value is equal to their sum. T
his argument can be generalized to n resistors
in series, and one finds
.


Next, consider the parallel connected resistors.
The potential difference
V

between
O

and
O

’ must be the same whether we go along
OABO

’ or
OCDO

’. Also co
n
servation of current
requires that

,
where:




is the current through R
1

and



is the current through R
2
. Substituting these expressions for I
1

and I
2

into our equation for I
gives:


.

This is equivalent to

writing

where
. This argument can be
generalized to n resistors connected in parallel and one finds
.


Diodes


N
ot everything obeys Ohm's l
aw,
i.e.

current is not necessarily proportional to voltage.
In
this lab

you will also measure the characteristics of a common type of electrical device called a
diode
. A diode consists of a junction of an “n
-
type” semiconductor and a "p
-
type"
semiconducto
r. The current in n
-
type semiconductors is carried by negative charges (the
electrons), while in p
-
type semiconductors the current is best thought of as being carried by
positive charges (
called "
holes
" that are due to

missing electrons). When
n and p

mate
rials are
brought together, a few electrons will drift from n to p and some holes will drift from p to n
. This
charge transfer between n and p regions generates

an internal electrical potential at the junction
which opposes further transfer of electrons an
d holes between the two sides. It is possible to
drive current from p to n (
i.e.

holes from the p region to n and electrons from the n region to the
p) only if this potential “barrier” is overcome by applying a sufficiently large voltage difference
across
the diode
. For current to flow, t
he p region
must be positive with respect to the n
.
Applying a positive voltage to n
and a negative voltage to

p

produces only a very small

“leakage
current”. Thus the diode acts like a one
-
way valve

with low resistance to
current flowing in the
direction of the arrow, and high resistance to current flowing in the opposite dire
ction
. If too
much voltage is applied in either direction, the diode
will

be destroyed.


The symbol for a diode is shown in Fig. 1.1 (a). The tip o
f the triangle points in the

direction that current

can
flow

with low resistance
.

The characteristics of an IN914
switching
diode

are shown in Table 1.1. This is one of the diodes that you can measure in the lab.
A

light
emitting diode

(
LED
)

ha
s

also
been
provided. In an
LED
, the current flow
causes

emission of
light
with a fairly well
-
defined wave
length or color.
LEDs are
efficient,
reliable and long
-
lived,
provided you don't apply too much voltage across them. A red, yellow or green LED can
typically with
stand
about 3 V and about 5V for a blue LED.


10











Figure 1.1:

(a)
E
lectrical symbol for a diode. (b) When V
b

> V
g
+V
a
, current flows through the
diode, from b

(the p
-
type region or anode
)

to a

(the n
-
type region or cathode)
.

Here V
g

is the
threshold voltage that needs to be reached before significant forward conduction occurs.

When
V
b

< V
g
+V
a

the flow of current is blocked.

I
n

particular, when Vb<Va, the device is said to be
"reverse bias
ed" and only a very small leakage current will flow.

(c) Sketch of the physical
layout of an 1N914 switching diode. The dark black band is on the cathode.


Table 1.1

Some electrical characteristics of the IN914 switching diode.


Peak Reverse Voltage

75 V

Average Forward Rectified Current

75 mA

Peak Surge Current, 1 Second

500 mA

Continuous Power Dissipation at 25°C

250 mW

Operating Temperature Range

-
65 to 175°C

Reverse Breakdown Voltage

100 V

Static Reverse Current

25 nA

Static Forward Voltage

1 V
at 10 mA

Capacitance

4 pF

Typical threshold voltage

0.6 V





b

a

(a)

(c)

(b)

I

anode

cathode


11

VII. Experiment


A. Internal Resistance and EMF of a Battery


The purpose of this
part

is to measure the EMF and internal resistance of
a

battery.


A.1

Before taking data, put a 20


30 seconds with a digital voltmeter. If the voltage remains constant

to within about 5 mV
,
proceed, but if it decreases

by more than this
, the battery should be replaced with a new one

if
they

are available
.


A.2

Connect up the circuit shown in Fig. I
.
5.
Leave the switch open

for now.

Here A is a digital
ammeter and V a digital voltmeter. The variable resistance R can be achieved by choosing
various combinations of resistances on the resistor
board. S is a knife
-
switch that is normally
open and should only be closed when you are taking a measurement. This helps avoid
exhausting the battery. The
internal resistance

of the battery is denoted as
r

and you should not
add

it to the circuit, sinc
e it is already there inside the battery itself.





A.3

You should know
that the

ammeter

A introduces a small series resistance
R
ammeter
into the
circuit and the insertion of

the voltmeter

V introduces a large resistance

R
voltmeter

in parallel to R.

Cons
ult the
appendix to this lab
manual

(
or the instruction manual

for the meter
)

to determine
the

"
internal impedance
"
of the meter

when it is acting either as a voltmeter or as an ammeter.

Notice that the internal impedance for the ammeter depends on whether

you are using the 10 A
plug or the 300 mA plug on the multimeter.
Record the values for R
ammeter
and R
voltmeter


in your
lab notebook.



A.
4

H
ave your instructor check your circuit before closing the switch.


A.
5
.

Measure V and I for
about 6 to 10

values
of R.

You should record this data in a table in your
lab notebook. As you are making these measurements, periodically measure the battery EMF
with the switch open using the voltmeter to be sure that it remains constant. If it isn’t
,

consult
with your inst
ructor.

When you are finished, open up the switch.






Figure I.5

Circuit schem
atic for part A showing battery's internal resistance r and EMF


.

S



12

A
.
6.

Now
record

the
random
uncertaint
y

associated with
each

voltage and current

measurement
.

If the meter reading is steady

(it should be in this case)
,
then the random
uncertainty will just be determined by the limited number of digits displayed on the multimeter.
Y
ou can use the 2/3 rule for digital scales, consult the appendix on
the Fluke 75 meter, or check
the Fluke 75 operating manual. The random uncertainties will be used for making a linear fit in
the next section.


A.7
. Finally, record the

systematic
error

or
accuracy

of each current and voltage
measurement
s
. Since we don't
have a current or voltage standard in the lab for you to calibrate
your meters,
you will need to consult

the appendix on the Fluke 75 meter, or check the Fluke 75
operating manual

for the accuracy of each of your measurements
.
Make sure that your instructo
r
takes a look at your numbers.

T
he systematic uncertainties
will

be accounted for
in your error
analysis

in the following section,

after you have extracted the fit parameters.



B.
Analysis of
Y
our Data

using Ohm's Law










B.
1
.
Put
your

data into E
xcel

and

plot V vs I
.
Be sure to include
the random uncertainties in V
and I as
error bars.


B.
2
.
Does your plot look like a straight line? It should if Ohm's law is correct and your battery is
stable.
Neglecting the resistances of the meters, the equation
s describing the circuit are



and




These can be combined and rewritten as


.

Thus, when you plot V versus I, t
h
e theory says that the slope should be
-
r and the intercept

.



B.
3

Use the spreadsheet "
straight line chi
-
square fit.doc
" to fit your data to a straight line. We
know that after Physics 174 and 275, you should be able to make your own spreadsheet for doing
a

2

fit. To save time, we built one for you with all of
the bells and whistles, and all you need to
do is paste in your data and run the solver to get the best fit slopes
. The spreadsheet is also set up
to calculate the uncertainty in the slope and intercept that is caused by your random errors

in
both the curr
ent and voltage measur
e
ments
; make sure for this part that you are only using the
random errors

in your V and I measurements and not the systematic errors.

Do the fits and a
dd
your fitted result to your plot.

F
ind
the slope
-
r
and
intercept



along
with

their uncertainties

and
record in your lab notebook
.

Make sure you copy and save all of the contents of this spreadsheet
in your own spreadsheet.


B.4

Finally,
you need to
compute the
systematic error

in
the best fit values

you found for



and

r.
Your instructor should have discussed how
to handle systematic errors

at the beginning of the
lab. Since you have never had to include the systematic error in your analysis before, we'll give
you the result here:



13



Here

the


an
d r are
your best fit values that you found above,


v

is the accuracy in the
calibration of the voltage scale,


I

is the accuracy in the calibration of the current scale,

b
v

is
the accuracy in the zero of the voltage scale (the voltage "offset"), and

b
I

is the accuracy in the
in the zero of the current scale (the current "offset"). The Table below lists values from the Fluke
75 Operators Manual or the appendix.


Note that


v

and


I

are dimensionless numbers
(typically


v

= 0.003 and


I
= 0.015)

and
we have assumed that you have used the same scale
for all of your measurements
.



Table of unceratinty and accuracy for the Fluke 75 current and voltage scales

function

range

instrumental

uncertainty



accuracy



scale



offset

b

V dc

3.200 V

32.00 V

320.0 V

1000 V

0.0003 V

0.003 V

0.03 V

0.3 V

(0.
4
% of reading
) + (0.001 V)

(0.3% of reading) + (0.01 V)

(0.3% of reading) + (0.1 V)

(0.4% of reading) + (1 V)

A dc

32.00 mA

320.0 mA

10.00 A

0.0
03

mA

0.
03

mA

0.0
03

A

(
1
.5% of reading) + (0.
02

mA)

(
1
.5% of reading) + (
0.2

mA)

(
1
.5% of reading) + (0.
02

A)


B5.

In your lab report, you should give the uncertainties



and

r
,
and also

the systematic errors




and

r
, and determine which is more important for


and r
.
Y
ou can also compute the total
uncertainty in


for example,


by adding in quadrature the

unceratinty and the systematic error in

.



C
. Devices that
Don't Obey Ohm's Law









C.1.
Wire up the circuit shown in
Fig
.

I.6

using one of the diodes on the diode board
.

Record in
your lab notebook which diode you chose.

The
40

“load resistor”

R
L

is used to limit the total
amount of current flowing throug
h the diode so that it does not burn up.

V
0

is a power supply.


C.2.

Measure the voltage across the load resistor, the voltage across the diode, the input voltage,
and the current through the loop for various input voltages between ±10 V (you will need
to
move around the voltmeter).


C.3
.
Optional
E
xercise
:
Verify that Kirchoff’s laws are satisfied by calculating the load
resistance and the input voltage and comparing to your measurements.


C.4
. Put your data into a spreadsheet and plot the current I

through the diode as a function of
voltage V across the diode. Take enough data that so that you can clearly see the “non
-
Ohmic”
behavior of the diode. This type of plot is called an
“I
-
V

characteristic
"
.



14




F
igure I
.6 Circuit schematic for measuri
ng the IV characteristic of a diode.

Use a power supply
instead of a battery for the voltage source

V
o
.

The voltmeter used for measuring the voltage drop
across the diode is not shown in this schematic.



C.5.

Calculate the effective resistance of the diod
e as a function of the voltage across it. Make
sure in you lab report that you describe your results.


C.6.

BEFORE LEAVING THE LAB, TURN IN A COPY OF YOUR SPREADSHEET TO
BLACKBOARD.


For Hotshots Only


H.1.
If you have time, repeat steps 1
-
4 above for a

di
fferent

type
of diode on your diode board.
I
n particular, i
f you did not choose an LED in
part C
, do so now. How are the IV curves
different for the two diodes? Make sure you do not exceed the maximum allowed voltage

(5V for
a blue diode, 3 V for red or y
ellow diodes)
.


H.2.

Still got time left? Ask your instructor for a light

bulb. Wire it up and measure its IV curve.
How is it similar to a diode and how is it different from a diode?




VII.

Additional Topics to be Included in your lab report


1.

Don't
forget that you have one week to do your lab report and they
are due at the start of the
lab.
Lab reports should be submitted to Blackboard.
You must write your own report and are not
allowed to copy anyone else's data, text or figures. You must include on

the first page of your
report a signed copy of the honors pledge.



2.

Review the requirements for your lab report before starting to write it up (see the Introduction
to Physics 276 in this manual

and the handout provided by your instructor on how they w
ill be
graded
).


R
L

= 40


癯汴age

獯畲se†

†

o


IN914


15

3
.

Among other things, your lab report should have a clear
discussion
of the analysis you did in
part B and what you found as a result of that analysis. Be sure to compare your deduced EMF
with your direct measurements of the battery outp
ut voltage.


4
. Your report
must

include a discussion of the
random uncertainties

in your measur
e
ments of
current and voltage, and how much these
a
ffect your values for r and

.


5
.
Your report should

include a discussion of the

systematic uncertainties

i
n your
measur
e
ments of current and voltage, and how much these
a
ffect your values for r and

.


6
. You should also have a section that i
nclude
s

the resistances of the meters in your analysis by

including their effects in the circuit equations.

Don't forge
t to look in the appendix to get these
parameters.

Determine whether an appreciable error has been made in the determination of
either


or
r

by neglecting the resistances of the meters.



16

Appendix C.
The Fluke 77 series Multimeter


The following table gi
ves specifications for the Fluke 7
5

series II multimeter. Data used in
constructing the table was obtained from the
Fluke 7
5

Users Manual
.

For each listed function of
the

multimeter
, the Table gives the possible range scales available. For each possible ra
ng
e scale
of each listed function
, the table gives the

resolution, instrumental uncertainty (using the 2/3
rule), the accuracy of a reading and the input impedance of the meter.
For full analysis of the
systematic errors, the accuracy of each scale is give
n in terms of the accuracy


of the scale
factor and the accuracy

b of the zero (the offset error).



Table of Selected Specifications for the Fluke 77 series II Multimeter

function

range

resolution

instrumental

uncertainty



accuracy


scale



offset

b

input
impedance

V dc

3.200

V

32.00 V

320.0 V

1000 V

0.001 V

0.01 V

0.1 V

1 V

0.0003 V

0.003 V

0.03 V

0.3 V

(0.
4
% of reading) + (0.001 V)

(0.3% of reading) + (0.01 V)

(0.3% of reading) + (0.1 V)

(0.4% of reading) + (1 V)


R>10 M


C

50 pF

mV dc

320.0 mV

0.1 mV

0.03 mV

(0.3% of reading)
+ (0.1
m
V)

R>10 M


C

50 pF

V ac

3.200

V

32.00 V

320.0 V

750 V

0.001 V

0.01 V

0.1 V

1 V

0.0003 V

0.003 V

0.03 V

0.3 V

(
2
% of reading) + (
0.0
02

V)

(
2
% of re
ading) + (
0.
02

V)

(
2
% of reading) + (
0.2
V)

(
2
% of reading) + (1
0

V)


R>10 M


C

50 pF



320.0


3.200 k


32.00 k


320.0 k


3.200 M


32.00 M


0.1


0.001 k


0.01 k


0.1 k


0.001 M


0.01 M


0.03


0.0003
k


0.003 k


0.03 k


0.0003 M


0.003 M


(0.
5
% of reading) + (
0.2


)

(0.
5
% of reading) + (1

)

(0.
5
% of reading) + (1
0


)

(0.5% of reading) + (10
0


)

(0.5% of reading
) + (1

k

)

(
2
% of reading) + (
10 k

)

not
applicable

A ac

32.00 mA

320.0 mA

10.00 A

0.01 mA

0.1 mA

0.01 A

0.0
03

mA

0.
03

mA

0.0
03

A

(
2.5
% of reading) + (
0.
02

mA
)

(2.5% of reading) +
(
0.2

mA)

(2.5% of reading) + (0.
02

A)

6


6


0.
05



A dc

32.00 mA

320.0 mA

10.00 A

0.01 mA

0.1 mA

0.01 A

0.0
03

mA

0.
03

mA

0.0
03

A

(
1
.5% of reading) + (0.
02

mA)

(
1
.5% of reading)
+ (
0.2

mA)

(
1
.5% of reading) + (0.
02

A)

6


6


0.
05






Example:

If a reading of 20 V is measured on the 32.00 V

dc

range setting, the resolution is
0.01 V, the instrumental uncertainty is

v

=
0.003 V, and the accuracy is 0.3%
of the
measurement plus
0.01 V
,
i.e.

the accuracy

(systematic error) of the measurement is:




0.3%*20 V + 0.01 V =
(0.3/100)*20 V + 0.01 V =
0.06 V +

0.01 V = 0.07 V.