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Resistors

Ohm’s Law and Combinations of Resistors

See Chapters 1 & 2 in

Electronics: The Easy Way

(Miller & Miller)

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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.)

Ie-

e-

e-

Negative charges moving to leftCurrent moving to right

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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

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.

R = V

I

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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

V

I Slope=I/V=1/R

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V = I R

=

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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 = I R 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 voltsDecimal Binary

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Range actually split in three

High

Low

Forbidden range

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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 = 2102 + 0101 + 5100

Decimal – powers of ten

11001101 = 127 + 126 + 025 + 024

+ 123 + 122 + 021 + 120 = 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

Repeat 102/2 = 51, remainder 01

0 1

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Iterate

51/2 = 25, remainder 1

25/2 = 12, remainder 1

12/2 = 6, remainder 0

1 0 1

1 1 0 1

0 1 1 0 1

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Iterate

6/2 = 3, remainder 0

3/2 = 1, remainder 1

1/2 = 0, remainder 1

0 0 1 1 0 1

1 0 0 1 1 0 1

1 1 0 0 1 1 0 1

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Recap

1 1 0 0 1 1 0 1

127 + 126 + 025 + 024

+ 123 + 122 + 021 + 120

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

1

3 9

+ 3 5

7 4

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Adding Binary Numbers

1 1 1 1

0 1 0 0 1 1 1

+ 0 1 0 0 0 1 1

1 0 0 1 0 1 0

carries

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Uh oh, overflow*

What if you use a byte (8 bits) to represent an integer

A byte may not be enough to represent the sum of two such numbers.

*The End of the World as We Know It

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

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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

Ignoring overflow the two eight-bit numbers above sum to zero

1 1 1 1 1 1 1

0 0 1 0 1 0 1 0

1 1 0 1 0 1 1 0

1 0 0 0 0 0 0 0 0

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Two’s Complement

Step 1: exchange 1’s and 0’s

Step 2: add 1 (to the lowest bit only)

0 0 1 0 1 0 1 0

1 1 0 1 0 1 0 1

1 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 left-most, 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

Is it 214? Or is it – 42? Or is it Ö? Or is it …? It’s a matter of interpretation

How was it declared?

1 1 0 1 0 1 1 0

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3-bit unsigned and signed

7 1 1 1

6 1 1 0

5 1 0 1

4 1 0 0

3 0 1 1

2 0 1 0

1 0 0 1

0 0 0 0

3 0 1 1

2 0 1 0

1 0 0 1

0 0 0 0

-1 1 1 1

-2 1 1 0

-3 1 0 1

-4 1 0 0

Think of an odometer reading 999999 and the car travels one more mile.

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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)

Places

11.001001

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Two’s placeOne’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 remainder 0 49 / 2 = 24 remainder 1 24 / 2 = 12 remainder 0 12 / 2 = 6 remainder 0 6 / 2 = 3 remainder 0 3 / 2 = 1 remainder 1 1 / 2 = 0 remainder 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

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 .456 27 = 58.368

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

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 C 9

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

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 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 (48) 1 00110001 (49) … A 01000001 (65) B 01000010 (66) … a 01100001 (97) b 01100010 (98) …