Digital Electronics

By Author: TAHIR LATEEF
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Digital Electronics

Introduction

The entire world of electronics can be divided into two broad groups of circuits analog and digital. In analog circuits the voltage and current waveforms have variations similar to the signal information. The Sine wave of V or I is a perfect example.
A digital signal however is a group of pulses that have the same level and or either on or off. These conditions are also labeled HIGH are LOW and 1 or 0. This is a binary system as it has just two bits of information. Variations are indicated by a HIGH or LOW pulse in the digital signal.

The logical use of digital circuits is for numerical counting as in electronic calculators and test equipment. Digital electronics also includes alpha numeric information though with letters and numbers. The pulse information can be encoded to represent letters in the alphabet, punctuation marks, and instructions.

Digital signals have an advantage over analog signals in that they reduce the effect of noise interference. It is relatively easy for a digital circuit to recognise a specific level of the pulses.

Because of ...
... these features, particularly all modern electronic equipment contains a mixture of analog and digital circuits. Changing from one form of signal to the other is convenient with digital to analog (D/A) converters.

Most Important Terms Are Explained In Following Sections:

1.Binary Numbers:
An analog meter measures quantities over an infinite number of steps. For example the pointer of an analog multimeter can come to rest at any one of an infinite number of points along the scale. The digital meter on the other hand indicates readings in discrete steps of digits. In digital electronics the discrete steps are characterized by high or low voltage, open or closed circuits, and the digits 0 to 1. For this reason digital electronics lends itself to the binary number system.

The binary number system uses only the digits 0 and 1 which represent the two levels in a digital signal. Examples of binary numbers are 0001, 0101, and 1001. Any number of places is possible each with a 0 or 1. Figure 2 illustrates how the two levels of digital waveform correspond to the binary digits 0 and 1.

The base or radix of a number system specifies how many different digits can be used for counting in each place. For binary numbers, the base is two. The base or radix is often indicated in a subscript. For example 1112 represent a base-two number. Read this as 111 to base two.

In conventional decimal system for counting uses all 10 digits from 0 to 9. Therefore, the base or radix for decimal number is ten. The highest digit for any place is one less than the base. In decimal numbers, 9 is the highest digit, while one is the highest digit in binary numbers, for counting in successive places.

How Each Place Is Used For Different Counts: To count numbers more than the base itself, the complete number must have an order of places for larger counts. The order is indicated by the decimal point in decimal numbers or the binary point in binary numbers.

Figure 3 illustrates the decimal orders for numbers greater than 1, the first place to the left of the decimal point is for the count of the digits alone, from 0 to 9. The second place is for counts of ten. Next is the place for counts of 102 or 100's and then for 103 or 1000's. The counts for successive places increase in multiples of 10.
For decimal fractions less than 1, the places are to the right of the decimal point. These values decrease in multiples of 1/10 are 0.1.

Fig.3
103 =
thousands 102 =
hundreds 101 =
tens Digits
0 to 9 Quantity
0 1 1 1 (111)2

The same method is used with binary numbers. However, the first place to the left of the point can be only 0 or 1. As usual higher counts are carried over to the next place. The second place is for counts of 2. Successive places to the left or for counts of 22, or 4's, 23, or 8’s etc. As illustrated in Fig.4,

Fig.4
23=
eights 22=
fours 21=
twos 20=
ones Quantity
0 1 1 1 (7)10

Places to the right of the binary point can be used for fractions less than 1, as in decimal numbers. However the values decrease in multiples of 1/2 instead of 1/10. For the binary example in figure 4 the (111)2 is 4 + 2 + 1 is equal to (7)10. The binary number 111 is actually 111.0, but the point is usually omitted as in decimal numbers.
As another example, (1100)2 is 8+4+0+0=(12)10. Binary numbers need more places for counting because the base is smaller.

Fig.5.
(Base)3 (Base)2 Base Digits Base or
Radix Name of
Number System
1000
8
512
4096 100
4
64
256 Ten
Two
Eight
Sixteen 0 to 9
0 or 1
0 to 7
O to 9 and A, B, C, D, E, F 10
2
8
16 Decimal
Binary
Octal
Hexadecimal

The general method of counting in number systems with a different base or radix is illustrated in figure 5. Included with 10 and 2 as the base or base 8 for octal and base 16 for extra decimal numbers. Binary, octal, and hexadecimal numbers are widely used in digital electronics.

Converting Binary to Decimal: Just make the count for each place and add the values. For example, the seven places in (1010101)2 show the count as

64+0+16+0+4+0+1=(85)10

Converting decimal to Binary: There are several methods, but a common system is as,

a)Divide the decimal number successively by 2 until the quotient is zero.
b)Note whether each remainder is 0 or 1.
c)These remainders of 0 and 1 are the digits in the binary number.

The places of the digits 0 and 1 in the binary number are to the left of binary point. Continue the division until the last quotient is zero. For example, to convert (13)10 to binary form:

13/2=6 with remainder of 1
6/2=3 with remainder of 0
3/2=1 with remainder of 1
1/2=0 with remainder of 1

(13)10 = ( 1 1 0 1)2

The remainders or assembled into the binary number as mentioned above. Therefore, (13)10 is equal to (1101)2. Note that the last division, with a quotient of zero, is most important because it may give a 1 for the highest place in the binary number. This system of dividing by 2 for converting to binary form is called double dabbling.

Shifting The Decimal or Binary Point: In decimal numbers moving the point one place to the right multiplies the number by 10. For instance 10 multiplied by 53.7 and 537.0 are equal. Moving the point one place to the left divides by 10.

In binary numbers, shifting the point one place multiplies are divides by 2. For instance (111.0)2 is (7)10 but (1110.0)2 is (14)10. The 7 is doubled to 14 by moving the binary point one place to the right.

For division by 2, the point is moved one place to the left. For instance (111.0)2 is (7)10, but (11.10)2 is (3.5)10. The binary places in 11.1 have the values of 2 + 1 + 1/2 which equals 3.5. This 3.5 is one half of 7.

Binary Bits. A 0 or 1 is a bit because it has one item of information.
Word . A string of bits with 0s and 1s form a word.
Byte. An 8-bit word is a byte.
Nibble. One half of a byte(four bits) is a nibble. Some examples are

0 or 1 = bit
0110 = nibble
0110 0011 = byte
0101 0011 0010 1010 = two bytes for a 16 bit word.

Bits are in groups of four, for convenience in reading. The binary information is generally processed with 8 bit or 16 bit words.

Binary Coded Decimal(BCD).
In the BCD system. each digit in a decimal number is encoded as a binary number. Four bits take care of digits up to 9. For example, 3 is coded into 0011. Also, 9 becomes 1001. The decimal 39 is 0011 1001 in the BCD form. As an example, when you punch 39 on a calculator keyboard, this number is converted to 0011 1001.

Hexadecimal Values.
This system uses 16 characters with digits 0 to 9 and letters A to F for 10 to 15. hexadecimal values or commonly used in microcomputers because they are much shorter than binary numbers. As an example, hexadecimal 9F corresponds to binary 1001 1111.

Practice Problems:
a). Convert (1111)2 to the decimal form.
b). Convert 10 in decimal form to a binary number.
c). How much is (000)2 as a decimal number?
d). How many bits are in one byte?
e). Give the binary equivalent of B in hexadecimal notation.
Answers. (a). 15, (b). 1010, (c). Zero, (d). Eight, (e). 1011.

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