TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES - TAGUIG

Mission: Be a center of excellence for technology and engineering education in the country.

Vision: Provide continous training, development and advancement of highly qualified & competent technical manpower for national and global socio-economic & industrial growth, progress, & prosperity. To pursue this mission, TUPT shall:

1. Develop globally competitive technicians, engineers, and technology-oriented managers.

2. Conduct researches to support technology and engineering education and existing technical knowledge.

3. Provide quality extension services for the community and industry.

4. And implement realistic resource generations.

COURSE INTRODUCTION

Department: ELECTRICAL AND ALLIED DEPARTMENT (BS EE)

Course Code: CoE 2L Course

Title: LOGICAL CIRCUITS AND SWITCHING THEORY (LEC)

Course Description: Review of the number systems, coding and Boolean Algebra; inputs and outputs; combinational circuits; state and machine equivalence; asynchronous sequential circuits; face conditions; algorithmic state; design and digital subsystems.

Course Objectives:

At the end of course, the students should be able to:

1. Understand the general principles of digital sysytem.

2. Demonstrate skill in constructingcounting and sequential circuitry that will apply Memory circuits.

3. Applies concepts and principle of a Interfacing digital techniques into Analog world.

INTRODUCTION

Definition of Terms

Here are some common terms used in digital electronics:

• Digital systems are used in almost all aspects of life, incommunications, business transactions, traffic control, medical treatment, the internet…etc.
• Digital logic is the first course in studying digital systems, which is an essential part of almost all branches of Electrical Engineering.
• Logical–refers to a signal or device in terms of its meaning, such as "TRUE" or "FALSE"
• Physical–refers to a signal in terms of voltage or current or a device’s physical characteristics
• HIGH–the signal level with the greater voltage • LOW–the signal level with the lower voltage
• TRUE or 1–the signal level that results from logic conditions being met
• FALSE or 0–the signal level that results from logic conditions not being met.
• Active High–a HIGH signal indicates that a logical condition is occurring
  
• Active Low–a LOW signal indicates that a logical condition is occurring
  
• Truth Table–a table showing the logical operation of a device’s outputs based on the device’s inputs
• State–a stable combination of signals or of a set of inputs and outputs, can refer to a device, a circuit, or a set of signals

Number Systems

Digital logic may work with “1s and 0s”, but it combines them into several different groupings that form different number systems. You’re familiar with the decimal system, of course. That’s a base-10 system in which each digit represents a power of ten. In the field of logic you’ll also encounter…

• Binary–base two (each bit represents a power of two), digits are 0 and 1, numbers are denoted with a ‘B’ or ‘b’ at the end, such as 01001101B (77 in the decimal system)
• Hexadecimal or ‘Hex’–base 16 (each digit represents a power of 16), digits are 0 through 9 plus A-B-C-D-E-F representing 10-15, numbers are denoted with ‘0x’ at the beginning or ‘h’ at the end, such as 0x5A or 5Ah (90 in the decimal system) and require four binary bits each. A dollar sign preceding the number ($01BE) is sometimes used, as well.
• Binary-coded decimal or BCD–a four-bit number similar to hexadecimal, except that the decimal value of the number is limited to 0-9.
• Decimal–the usual number system. When used in combination with other numbering systems, decimal numbers are denoted with ‘d’ at the end, such as 23d.
• Octal–base eight (each digit represents a power of 8), digits are 0-7, and each requires three bits. Rarely used in modern designs.

Conversion Bet. Diff. Number Systems

Positional number systems

Our decimal number system is known as a positional number system, because the value of the number depends on the position of the digits. For example, the number 123 has a very different value than the number 321, although the same digits are used in both numbers.

(Although we are accustomed to our decimal number system, which is positional, other ancient number systems, such as the Egyptian number system were not positional, but rather used many additional symbols to represent larger values.)

In a positional number system, the value of each digit is determined by which place it appears in the full number. The lowest place value is the rightmost position, and each successive position to the left has a higher place value.

In our decimal number system, the rightmost position represents the "ones" column, the next position represents the "tens" column, the next position represents "hundreds", etc. Therefore, the number 123 represents 1 hundred and 2 tens and 3 ones, whereas the number 321 represents 3 hundreds and 2 tens and 1 one.

The values of each position correspond to powers of the base of the number system. So for our decimal number system, which uses base 10, the place values correspond to powers of 10:

...	1000	100	10	1
...	10^3	10^2	10^1	10^0

Converting from other number bases to decimal

Other number systems use different bases. The binary number system uses base 2, so the place values of the digits of a binary number correspond to powers of 2. For example, the value of the binary number 10011 is determined by computing the place value of each of the digits of the number:

1	0	0	1	1	the binary number
2^4	2^3	2^2	2^1	2^0	place values

So the binary number 10011 represents the value

	(1 * 2^4)	+	(0 * 2^3)	+	(0 * 2^2)	+	(1 * 2^1)	+	(1 * 2^0)
=	16	+	0	+	0	+	2	+	1
=	19

The same principle applies to any number base. For example, the number 2132 base 5 corresponds to

2	1	3	2	number in base 5
5^3	5^2	5^1	5^0	place values

So the value of the number is

	(2 * 5^3)	+	(1 * 5^2)	+	(3 * 5^1)	+	(2 * 5^0)
=	(2 * 125)	+	(1 * 25)	+	(3 * 5)	+	(2 * 1)
=	250	+	25	+	15	+	2
=	292

Converting from decimal to other number bases

In order to convert a decimal number into its representation in a different number base, we have to be able to express the number in terms of powers of the other base. For example, if we wish to convert the decimal number 100 to base 4, we must figure out how to express 100 as the sum of powers of 4.

100	=	(1 * 64)	+	(2 * 16)	+	(1 * 4)	+	(0 * 1)
	=	(1 * 4^3)	+	(2 * 4^2)	+	(1 * 4^1)	+	(0 * 4^0)
Then we use the coefficients of the powers of 4 to form the number as represented in base 4:
100	=	1 2 1 0	base 4

One way to do this is to repeatedly divide the decimal number by the base in which it is to be converted, until the quotient becomes zero. As the number is divided, the remainders - in reverse order - form the digits of the number in the other base.

Example: Convert the decimal number 82 to base 6:

82/6	=	13	remainder 4
13/6	=	2	remainder 1
2/6	=	0	remainder 2

The answer is formed by taking the remainders in reverse order: 2 1 4 base 6.

LOGIC OPERATIONS AND TRUTH TABLES

Digital logic circuits handle data encoded in binary form, i.e. signals
that have only two values, 0 and 1.

Binary logic dealing with “true” and “false” comes in handy to describe
the behaviour of these circuits: 0 is usually associated with “false” and
1 with “true.”

Quite complex digital logic circuits (e.g. entire computers) can be built
using a few types of basic circuits called gates, each performing a single
elementary logic operation : NOT, AND, OR, NAND, NOR, etc..

Boole’s binary algebra is used as a formal / mathematical tool to
describe and design complex binary logic circuits.

A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. The logic normally performed is Boolean logic and is most commonly found in digital circuits. Logic gates are primarily implemented electronically using diodes or transistors, but can also be constructed using electromagnetic relays, fluidics, optics, molecules, or even mechanical elements.

In electronic logic, a logic level is represented by a voltage or current, (which depends on the type of electronic logic in use). Each logic gate requires power so that it can source and sink currents to achieve the correct output voltage. In logic circuit diagrams the power is not shown, but in a full electronic schematic, power connections are required.

Type	Distinctive shape	Rectangular shape	Boolean algebra between A & B	Truth table
AND				INPUT OUTPUT A B A AND B 0 0 0 0 1 0 1 0 0 1 1 1
OR			A + B	INPUT OUTPUT A B A OR B 0 0 0 0 1 1 1 0 1 1 1 1
NOT				INPUT OUTPUT A NOT A 0 1 1 0
In electronics a NOT gate is more commonly called an inverter. The circle on the symbol is called a bubble, and is generally used in circuit diagrams to indicate an inverted (active-low) input or output.[1]
NAND				INPUT OUTPUT A B A NAND B 0 0 1 0 1 1 1 0 1 1 1 0
NOR				INPUT OUTPUT A B A NOR B 0 0 1 0 1 0 1 0 0 1 1 0
XOR				INPUT OUTPUT A B A XOR B 0 0 0 0 1 1 1 0 1 1 1 0
XNOR				INPUT OUTPUT A B A XNOR B 0 0 1 0 1 0 1 0 0 1 1 1

KARNAUGH MAP METHOD

The Karnaugh map (K-map for short), Maurice Karnaugh's 1953 refinement of Edward Veitch's 1952 Veitch diagram, is a method to simplify Boolean algebra expressions. The Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability, permitting the rapid identification and elimination of potential race hazards.

In a Karnaugh map the boolean variables are transferred (generally from a truth table) and ordered according to the principles of Gray code in which only one variable changes in between squares. Once the table is generated and the output possibilities are transcribed, the data is arranged into the largest possible groups containing 2ⁿ cells (n=0,1,2,3...) and the minterm is generated through the axiom laws of boolean algebra.

Size of map

The size of the Karnaugh map with n Boolean variables is determined by 2ⁿ. The size of the group within a Karnaugh map with n Boolean variables and k number of terms in the resulting Boolean expression is determined by 2^nk. Common sized maps are of 2 variables which is a 2×2 map, 3 variables which is a 2×4 map, and 4 variables which is a 4×4 map.

2-variable k-map

3-variable k-map

4-variable k-map

Rules of Simplification

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The Karnaugh map uses the following rules for the simplification of expressions by grouping together adjacent cells containing ones

• Groups may not include any cell containing a zero
  

• Groups may be horizontal or vertical, but not diagonal.
  

• Groups must contain 1, 2, 4, 8, or in general 2ⁿ cells.
  That is if n = 1, a group will contain two 1's since 2¹ = 2.
  If n = 2, a group will contain four 1's since 2² = 4.
  

• Each group should be as large as possible.
  

• Each cell containing a one must be in at least one group.
  

• Groups may overlap.
  

• Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.
  

• There should be as few groups as possible, as long as this does not contradict any of the previous rules.
  

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Summmary:

1. No zeros allowed.
2. No diagonals.
3. Only power of 2 number of cells in each group.
4. Groups should be as large as possible.
5. Every one must be in at least one group.
6. Overlapping allowed.
7. Wrap around allowed.
8. Fewest number of groups possible.

CONTINUATION OF KARNAUGH METHOD

3-Variable Karnaugh Maps

Here is the truth table for a 3-person majority voting system:

input C	input B	input A	output
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

This is converted into a Karnaugh map, as follows:

Look carefully at the variables at the top of the Karnaugh map. These are not written in binary order 00, 01, 10, 11 etc. Instead, each column differs from the previous one by just one bit. This is called Grey code and it is essential for your Karnaugh map to work that you enter the column values in this order. (You will find out more about Grey code later.)

Within the K-map, you can identify three groups of two, as indicated. The left hand horizontal group combines the cells and A.B.C. Within this group, the value of B does not affect the cell values. This means that B can be eliminated from the expression, leaving A.C.

Work through the other groups to confirm that you understand how the remaining terms in the Boolean expression were derived.

With a little practice, this method is going to be quicker than the alternative, simplfiying the Boolean expression derived from the truth table:

4-Variable Maps

A 4-variable map will contain 2⁴ = 16 cells. It is important to write the variable values along the columns and rows in Grey code:

To simplify the equation:

The Karnaugh map becomes:

To give the simplest Boolean statement, you should put a circle round the maximum number of terms.

In this case, you can make two groups of four, one of which wraps around from top to bottom. You identify the two variables which remain constant in each group and eliminate the other two:

DeMorgan’s Theorem

For N variables, DeMorgan’s theorem is expressed as:

and

EX-OR (EXclusive-OR) gate

The output Q is true if either input A is true OR input B is true, but not when both of them are true: Q = (A AND NOT B) OR (B AND NOT A)
This is like an OR gate but excluding both inputs being true.
The output is true if inputs A and B are DIFFERENT.
EX-OR gates can only have 2 inputs.

		Input A Input B Output Q 0 0 0 0 1 1 1 0 1 1 1 0
Traditional symbol	IEC symbol	Truth Table

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EX-NOR (EXclusive-NOR) gate

This is an EX-OR gate with the output inverted, as shown by the 'o' on the output.
The output Q is true if inputs A and B are the SAME (both true or both false): Q = (A AND B) OR (NOT A AND NOT B)
EX-NOR gates can only have 2 inputs.

		Input A Input B Output Q 0 0 1 0 1 0 1 0 0 1 1 1
Traditional symbol	IEC symbol	Truth Table

CHARACTERISTICS OF DIGITAL IC

Digital ICs process only on/off signals. These devices can be found in microprocessors, memory chips, and microcomputers.

o

ICs are categorized by the type of components used in their circuits. nBipolar ICs use NPN and PNP transistors nUnipolar ICs use FET transistors. oThe transistor-transistor logic (TTL) and the complementary metal-oxide semiconductor (CMOS) families will both be examined.

Figure 4-30 (a) TTL INVERTER circuit; (b) CMOS INVERTER circuit.
Pin numbers are given in parentheses.

o

The TTL family consists of subfamilies.

• The 74 series devices are all part of the standard TTL series.
  
• The 74LS series devices are all part of the low power Schottky TTL series.
  
• The differences between devices is limited to electrical characteristics like power dissipation and switching speed. The pin layout and logic operations are the same.
  
• The 7404, 74S04, 74LS04, and 74ALS04 are all hex (six to a chip) inverters.

o

The CMOS family consists of several series.

• CMOS devices perform the same function as, but are not necessarily pin for pin compatible with TTL devices.

o

*VCC for TTL devices is normally +5 V.

*VDD for CMOS devices can be from +3 to +18 V.

*Inputs that are not connected are said to be floating. The consequences of floating inputs differ for TTL and CMOS.

*Floating TTL input acts like a logic 1. The voltage measurement may appear in the indeterminate range, but the device will behave as if there is a 1 on the floating input.

*Floating CMOS inputs can cause overheating and damage to the device. Some ICs have protection circuits built in, but the best practice is to tie all unused inputs either high or low.

Internal Digital IC Faults

o

Malfunction in internal circuitry
-Outputs do not respond properly to inputs. Outputs are unpredictable

Input internally shorted to ground or supply
-The input will be stuck in LOW or HIGH state.

Output internally shorted to ground or supply
-Output will be stuck in LOW or HIGH state.

Open-circuited input or output
-Floating input in a TTL device will result in a HIGH output. Floating input in a CMOS device will result in erratic or possibly destructive output. nAn open output will result in a floating indication.

o
Short between two pins

n-The signal at those pins will always be identical.

Figure 4-35 (a) IC input internally shorted to ground; (b) IC input internally shorted to supply voltage. These two types of failures force the input signal at the shorted pin to stay in the same state. (c) IC output internally shorted to ground; (d) output internally shorted to supply voltage. These two failures do not affect signals at the IC inputs.

Since A is pulsing, Z1-4 should also be pulsing. Z1 is malfunctioning, Z1 has a shorted output, or Z2 has a shorted input

NAND output should be high because Z2-1 is low.

o

TROUBLESHOOTING CASE STUDY

Example 4-28 illustrates the process involved in troubleshooting a fairly simple circuit. oThe reasoning process can be applied to more complex digital circuits.

NAND outputs correct
XOR output should produce a low output

Internal component of Z1 could be bad
External short to Vcc along conductor
Z1-3 internally shorted to Vcc
Z2-5 internally shorted to Vcc
Z2-13 internally shorted to Vcc