Transcript Introduction to Ohm`s Law and Binary Numbers

Resistors
Ohm’s Law and Combinations of
Resistors
See Chapters 1 & 2 in
Electronics: The Easy Way
(Miller & Miller)
PHY 201 (Blum)
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Electric Charge
 Electric charge is a fundamental property of some
of the particles that make up matter, especially (but
not only) electrons and protons.
 Charge comes in two varieties:


Positive (protons have positive charge)
Negative (electrons have negative charge)
 Charge is measured in units called Coulombs.


A Coulomb is a rather large amount of charge.
A proton has a charge 1.602  10-19 C.
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ESD
 A small amount of charge can build up on one’s
body – you especially notice it on winter days in
carpeted rooms when it’s easy to build a charge and
get or give a shock.
 A shock is an example of electrostatic discharge
(ESD) – the rapid movement of charge from a place
where it was stored.
 One must be careful of ESD when repairing a
computer, since ESD can damage electronic
components.
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Current
 If charges are moving, there is a current.
 Current is rate of charge flowing by, that is, the
amount of charge going by a point each second.
 It is measured in units called amperes (amps) which
are Coulombs per second (A=C/s)

The currents in computers are usually measured in
milliamps (1 mA = 0.001 A).
 Currents are measured by ammeters.
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Ammeter in Multisim
Electronics WorkBench
Ammeters are connected in series. Think of the charge as starting
at the side of the battery with the long end and heading toward the
side with the short end. If all of the charges passing through the
first object (the resistor above ) must also pass through second
object (the ammeter above), then the two objects are said to be in
series.
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Current Convention
 Current has a direction.
 By convention the direction of the current is the direction in
which positive charge flows.
 The book is a little unconventional on this point.
 If negative charges are flowing (which is often the case), the
current’s direction is opposite to the particle’s direction.
(Blame Benjamin Franklin.)
Current moving to right
I
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Negative charges moving to left
ee-
e6
Potential Energy and Work
 Potential energy is the ability to due work, such as
lifting a weight.
 Certain arrangements of charges, like that in a
battery, have potential energy.
 What’s important is the difference in potential
energy between one arrangement and another.
 Energy is measured in units called Joules.
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Voltage
 With charge arrangements, the bigger the charges,
the greater the energy.
 It is convenient to define the potential energy per
charge, known as the electric potential (or just
potential).
 The potential difference (a.k.a. the voltage) is the
difference in potential energy per charge between
two charge arrangements
 Comes in volts (Joules per Coulomb, V=J/C).
 Measured by a voltmeter.
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Volt = Joule / Coulomb
=
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Voltmeter in Multisim EWB
Voltmeters are connected in parallel. If the “tops” of two
objects are connected by wire and only wire and the
same can be said for the “bottoms” , then the two
objects are said to be in parallel.
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Voltage and Current
 When a potential difference (voltage) such as
that supplied by a battery is placed across a
device, a common result is for a current to
start flowing through the device.
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Resistance
 The ratio of voltage to current is known as
resistance
R = V
I
 The resistance indicates whether it takes a lot of
work (high resistance) or a little bit of work (low
resistance) to move charges.
 Comes in ohms ().
 Measured by ohmmeter.
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Multi-meter being used as
ohmmeter in Multisim EWB
A resistor or combination of resistors is removed from
a circuit before using an ohmmeter.
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Conductors and Insulators
 It is easy to produce a current in a material
with low resistance; such materials are called
conductors.

E.g. copper, gold, silver
 It is difficult to produce a current in a
material with high resistance; such materials
are called insulators.

E.g. glass, rubber, plastic
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Semiconductor
 A semiconductor is a substance having a
resistivity that falls between that of
conductors and that of insulators.

E.g. silicon, germanium
 A process called doping can make them
more like conductors or more like insulators

This control plays a role in making diodes,
transistors, etc.
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Ohm’s Law
 Ohm’s law says that the current produced by
a voltage is directly proportional to that
voltage.


Doubling the voltage, doubles the current.
Then, resistance is independent of voltage or
current
I
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Slope=I/V=1/R
V
16
V=IR
=
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Ohmic
 Ohm’s law is an empirical observation



“Empirical” here means that it is something we
notice tends to be true, rather than something that
must be true.
Ohm’s law is not always obeyed. For example, it
is not true for diodes or transistors.
A device which does obey Ohm’s law is said to
“ohmic.”
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Resistor
 A resistor is an Ohmic device, the sole
purpose of which is to provide resistance.

By providing resistance, they lower voltage or
limit current
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Example
 A light bulb has a resistance of 240  when
lit. How much current will flow through it
when it is connected across 120 V, its normal
operating voltage?
V=IR
 120 V = I (240 )
 I = 0.5 V/ = 0.5 A
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Binary Numbers
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Why Binary?
 Maximal distinction among values 
minimal corruption from noise
 Imagine taking the same physical attribute of
a circuit, e.g. a voltage lying between 0 and 5
volts, to represent a number
 The overall range can be divided into any
number of regions
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Don’t sweat the small stuff
 For decimal numbers, fluctuations must be less than 0.25
volts
 For binary numbers, fluctuations must be less than 1.25
volts
5 volts
0 volts
Decimal
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Binary
23
Range actually split in three
High
Forbidden
range
Low
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It doesn’t matter ….
 Some of the standard voltages coming from a
computer’s power are ideally supposed to be
3.30 volts, 5.00 volts and 12.00 volts
 Typically they are 3.28 volts, 5.14 volts or
12.22 volts or some such value
 So what, who cares
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How to represent big integers
 Use positional weighting, same as with decimal
numbers
 205 = 2102 + 0101 + 5100

Decimal – powers of ten
 11001101 =
127 + 126 + 025 + 024
+ 123 + 122 + 021 + 120
=
128 + 64 + 8 + 4 + 1
= 205

Binary – powers of two
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Converting 205 to Binary
 205/2 = 102 with a remainder of 1, place the
1 in the least significant digit position
1
 Repeat 102/2 = 51, remainder 0
0
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27
Iterate
 51/2 = 25, remainder 1
1
0
1
1
0
1
1
0
1
 25/2 = 12, remainder 1
1
 12/2 = 6, remainder 0
0
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28
Iterate
 6/2 = 3, remainder 0
0
0
1
1
0
1
1
0
1
1
0
1
 3/2 = 1, remainder 1
1
0
0
1
 1/2 = 0, remainder 1
1
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1
0
0
1
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Recap
1
1
0
0
1
1
0
1
127 + 126 + 025 + 024
+ 123 + 122 + 021 + 120
205
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Finite representation
 Typically we just think computers do binary math.
 But an important distinction between binary math
in the abstract and what computers do is that
computers are finite.
 There are only so many flip-flops or logic gates in
the computer.
 When we declare a variable, we set aside a certain
number of flip-flops (bits of memory) to hold the
value of the variable. And this limits the values
the variable can have.
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Same number, different
representation
 5 using 8 bits
 0000 0101
 5 using 16 bits
 0000 0000 0000 0101
 5 using 32 bits
 0000 0000 0000 0000 0000 0000 0000 0101
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Adding Binary Numbers
 Same as decimal; if the sum of digits in a
given position exceeds the base (10 for
decimal, 2 for binary) then there is a carry
into the next higher position
+
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3
3
7
9
5
4
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Adding Binary Numbers
carries
1
+
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1
1
1
0
1
0
0
1
1
1
0
1
0
0
0
1
1
1
0
0
1
0
1
0
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Uh oh,
*
overflow
 What if you use a byte (8 bits) to represent an integer
1
1
1
0
1
0
1
0
1
0
+
1
1
0
0
1
1
0
0
1
0
1
1
1
0
1
1
0
 A byte may not be enough to represent the sum of two
such numbers.
*The
End of the World as We Know It
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Biggest unsigned integers
 4 bit: 1111  15 = 24 - 1
 8 bit: 11111111  255 = 28 – 1
 16 bit: 1111111111111111  65535= 216 – 1
 32 bit:
11111111111111111111111111111111 
4294967295= 232 – 1
 Etc.
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Bigger Numbers
 You can represent larger numbers by using
more words
 You just have to keep track of the overflows
to know how the lower numbers (less
significant words) are affecting the larger
numbers (more significant words)
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Negative numbers
 Negative x is the number that when added to x
gives zero
1 1 1 1 1 1 1
1
0
0
1
0
1
0
1
0
1
1
0
1
0
1
1
0
0
0
0
0
0
0
0
0
 Ignoring overflow the two eight-bit numbers above
sum to zero
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Two’s Complement
0
0
1
0
1
0
1
0
0
1
 Step 1: exchange 1’s and 0’s
1
1
0
1
0
1
 Step 2: add 1 (to the lowest bit only)
1
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1
0
1
0
1
1
0
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Sign bit
 With the two’s complement approach, all
positive numbers start with a 0 in the leftmost, most-significant bit and all negative
numbers start with 1.
 So the first bit is called the sign bit.
 But note you have to work harder than just
strip away the first bit.
 10000001 IS NOT the 8-bit version of –1
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Add 1’s to the left to get the same
negative number using more bits







-5 using 8 bits
11111011
-5 using 16 bits
1111111111111011
-5 using 32 bits
11111111111111111111111111111011
When the numbers represented are whole numbers
(positive or negative), they are called integers.
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Biggest signed integers
 4 bit: 0111  7 = 23 - 1
 8 bit: 01111111  127 = 27 – 1
 16 bit: 0111111111111111  32767= 215 – 1
 32 bit:
01111111111111111111111111111111 
2147483647= 231 – 1
 Etc.
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Most negative signed integers
 4 bit: 1000  -8 = - 23
 8 bit: 10000000  - 128 = - 27
 16 bit: 1000000000000000  -32768= - 215
 32 bit:
10000000000000000000000000000000  2147483648= - 231
 Etc.
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Riddle
1





1
0
1
0
1
1
0
Is it 214?
Or is it – 42?
Or is it Ö?
Or is it …?
It’s a matter of interpretation

How was it declared?
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3-bit unsigned and signed
7
6
5
4
3
2
1
0
1
1
1
1
0
0
0
0
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1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
3
2
1
0
-1
-2
-3
-4
0
0
0
0
1
1
1
1
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
Think of an
odometer
reading
999999
and the car
travels one
more mile.
45
Fractions
 Similar to what we’re used to with decimal
numbers
3.14159 = 3 · 100 + 1 · 10-1 + 4 · 10-2
+ 1 · 10-3 + 5 · 10-4 + 9 · 10-5
11.001001 =
1 · 21 + 1 · 20 + 0 · 2-1 + 0 · 2-2
+ 1 · 2-3 + 0 · 2-4 + 0 · 2-5
+ 1 · 2-6
(11.001001  3.140625)
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Places
 11.001001
Two’s place
One’s place
Half’s place
Fourth’s place
Eighth’s place
Sixteenth’s place
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Decimal to binary
 98.61

Integer part








98 / 2
49 / 2
24 / 2
12 / 2
6/2
3/2
1/2
= 49
= 24
= 12
= 6
= 3
= 1
= 0
remainder
remainder
remainder
remainder
remainder
remainder
remainder
0
1
0
0
0
1
1
1100010
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Decimal to binary
 98.61

Fractional part







0.61  2 = 1.22
0.22  2 = 0.44
0.44  2 = 0.88
0.88  2 = 1.76
0.76  2 = 1.52
0.52  2 = 1.04
.100111
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Decimal to binary
 Put together the integral and fractional parts
 98.61  1100010.100111
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Another Example (Whole number part)
 123.456

Integer part








123 / 2 = 61 remainder 1
61 / 2 = 30 remainder 1
30 / 2 = 15 remainder 0
15 / 2 = 7 remainder 1
7 / 2 = 3 remainder 1
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
1111011.
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Checking: Go to All
Programs/Accessories/Calculator
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Put the calculator in Programmer view
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Enter number, put into binary mode
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Another Example (fractional part)
 123.456

Fractional part









0.456  2 = 0.912
0.912  2 = 1.824
0.824  2 = 1.648
0.648  2 = 1.296
0.296  2 = 0.592
0.592  2 = 1.184
0.184  2 = 0.368
…
.0111010…
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Checking fractional part: Enter digits
found in binary mode
Note that the leading zero does not display.
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Convert to decimal mode, then
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Edit/Copy result. Switch to
Scientific View. Edit/Paste
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Divide by 2 raised to the number of digits
(in this case 7, including leading zero)
1
2
3
4
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Finally hit the equal sign. In most
cases it will not be exact
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Other way around
 Multiply fraction by 2 raised to the desired number of
digits in the fractional part. For example
7 = 58.368
 .456  2
 Throw away the fractional part and represent the whole
number
 58 111010
 But note that we specified 7 digits and the result above
uses only 6. Therefore we need to put in the leading 0.
(Also the fraction is less than .5 so there’s a zero in the ½’s
place.)
 0111010
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Limits of the fixed point approach
 Suppose you use 4 bits for the whole number
part and 4 bits for the fractional part
(ignoring sign for now).
 The largest number would be 1111.1111 =
15.9375
 The smallest, non-zero number would be
0000.0001 = .0625
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Floating point representation
 Floating point representation allows one to
represent a wider range of numbers using the
same number of bits.
 It is like scientific notation.
 We’ll do this later in the semester.
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Hexadecimal Numbers
 Even moderately sized decimal numbers end
up as long strings in binary
 Hexadecimal numbers (base 16) are often
used because the strings are shorter and the
conversion to binary is easier
 There are 16 digits: 0-9 and A-F
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Decimal  Binary  Hex








0  0000  0
1  0001  1
2  0010  2
3  0011  3
4  0100  4
5  0101  5
6  0110  6
7  0111  7
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







8  1000  8
9  1001  9
10  1010  A
11  1011  B
12  1100  C
13  1101  D
14  1110  E
15  1111  F
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Binary to Hex
 Break a binary string into groups of four bits
(nibbles)
 Convert each nibble separately
1 1 1 0 1 1 0 0 1 0 0 1
E
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C
9
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Digit grouping and Hex mode
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Addresses
 With user friendly computers, one rarely encounters
binary, but we sometimes see hex, especially with
addresses
 To enable the computer to distinguish various parts,
each is assigned an address, a number





Distinguish among computers on a network
Distinguish keyboard and mouse
Distinguish among files
Distinguish among statements in a program
Distinguish among characters in a string
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How many?
 One bit can have two states and thus distinguish
between two things
 Two bits can be in four states and …
 Three bits can be in eight states, …
 N bits can be in 2N states
0
0
0
1
0
0
0
0
0
0
1
1
1
0
1
1
1
1
0
1
1
1
0
1
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IP(v4) Addresses
 An IP(v4) address is used to identify a
network and a host on the Internet
 It is 32 bits long
 How many distinct IP addresses are there?
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Characters
 We need to represent characters using numbers
 ASCII (American Standard Code for Information
Interchange) is a common way
 A string of eight bits (a byte) is used to correspond
to a character


Thus 28=256 possible characters can be represented
Actually ASCII only uses 7 bits, which is 128 characters;
the other 128 characters are not “standard”
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Unicode
 Unicode uses 16 bits, how many characters
can be represented?
 Enough for English, Chinese, Arabic and
then some.
 (Actually Unicode is extensible)
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ASCII









0  00110000
1  00110001
…
A  01000001
B  01000010
…
a  01100001
b  01100010
…
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(48)
(49)
(65)
(66)
(97)
(98)
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