The design and construction of

electrical circuits to implement

logical and arithmetical
operations.
CHAPTER 4
` Identify the basic gates and describe
the behaviour of each;
` Describe how gates are implemented
using transistors;
` Combine basic gates into circuits;
` Describe the behaviour of a gate or
circuit using Boolean expressions,
truth tables, and logic diagrams;
` Compare and contrast a half adder
and a full adder;
` Explain how an S-R latch operates;
` Describe the characteristics of the four
generations of integrated circuits.
Two valued Logic (F/T, 0/1)
Allows formal decision on logical
expressions

Gate
A device that performs a basic
operation on electrical signals.

Circuits
Gates combined to perform more
complicated tasks.
Statements:
A = My grandmother is alive
B = My grandfather is alive

NOT A = My grandmother is dead

A AND B = Both my grandparents are alive
A OR B = At least one of my grandparents is
alive
Statements:
A = My grandmother is alive
B = My grandfather is alive

A XOR B = Exactly one of my grandparents is

alive
A NAND B = My grandparents are not both
alive
A NOR B = My grandparents are both dead
Boolean expressions
Uses Boolean algebra, a mathematical
notation for expressing two-valued
logic.
Logic diagrams
A graphical representation of a circuit;
each gate has its own symbol.
Truth tables
A table showing all possible input values
and the associated output values.
Six types of gates
◦ NOT
◦ AND
◦ OR
◦ XOR
◦ NAND
◦ NOR
A NOT gate accepts one input signal (0 or 1) and
returns the opposite signal as output

Figure 4.1 Various representations of a NOT gate

An AND gate accepts two input signals
if both are 1, the output is 1;
otherwise the output is 0

Figure 4.2 Various representations of an AND gate

An OR gate accepts two input signals
If both are 0, the output is 0;
otherwise, the output is 1

Figure 4.3 Various representations of a OR gate

An XOR gate accepts two input signals
If both are the same, the output is 0;
otherwise, the output is 1

Figure 4.4 Various representations of an XOR gate

Note the difference between the XOR gate
and the OR gate; they differ only in one
input situation

When both input signals are 1, the OR gate

produces a 1 and the XOR produces a 0

XOR is called the exclusive OR

The NAND gate accepts two input signals
If both are 1, the output is 0;
otherwise, the output is 1

Figure 4.5 Various representations of a NAND gate

The NOR gate accepts two input signals
If both are 0, the output is 1; otherwise,
the output is 0

Figure 4.6 Various representations of a NOR gate

` A NOT gate inverts its single input
` An AND gate produces 1 if both input values
are 1
` An OR gate produces 0 if both input values
are 0
` An XOR gate produces 0 if input values are
the same
` A NAND gate produces 0 if both inputs are 1
` A NOR gate produces a 1 if both inputs are 0
Gates can be designed to accept three or
more input values
A three-input AND gate, for example,
produces an output of 1 only if all input
values are 1

Figure 4.7 Various representations of a three-input AND gate

Transistor
A device that acts either as a wire that conducts
electricity or as a resistor that blocks the flow of
electricity, depending on the voltage level of an
input signal
A transistor has no moving parts, yet acts like
a switch
It is made of a semiconductor material, which is
neither a particularly good conductor of electricity
nor a particularly good insulator
 A transistor has three
terminals
◦ A source
◦ A base
◦ An emitter, typically
connected to a ground
wire

If the Base is “on”, then the

Figure 4.8 The connections of a transistor transistor Emits, and is Earthed.

If the Base is “off”, then the

transistor becomes an insulator
and the Output is on.
The easiest gates to create are the NOT, NAND, and
NOR gates

Figure 4.9 Constructing gates using transistors

Combinationall circuit
The input values explicitly determine the
output
Sequential circuit
The output is a function of the input values
and the existing state of the circuit
We describe the circuit operations using
Boolean expressions
Logic diagrams
Truth tables
Gates are combined into circuits by using the
output of one gate as the input for another
(A.B) + (A.C)
Consider the following Boolean expression
A.(B + C)
Circuit equivalence
Two circuits that produce the same output for
identical input
Boolean algebra allows us to apply provable
mathematical principles to help design circuits
A.(B + C) = A.B + A.C (distributive law) so circuits
must be equivalent
+
We can prove these rules by setting out all the
values in a truth table and showing that the
truth table for each side is the same.

For example take the Distributive Law:

A+(BC) = (A+B).(A+C)
A B C A+B A+C (A+B).(A+C) BC A+(BC)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C A+B A+C (A+B).(A+C) BC A+(BC)
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
A B C A+B A+C (A+B).(A+C) BC A+(BC)
0 0 0 0 0
0 0 1 0 1
0 1 0 1 0
0 1 1 1 1
1 0 0 1 1
1 0 1 1 1
1 1 0 1 1
1 1 1 1 1
A B C A+B A+C (A+B).(A+C) BC A+(BC)
0 0 0 0 0 0
0 0 1 0 1 0
0 1 0 1 0 0
0 1 1 1 1 1
1 0 0 1 1 1
1 0 1 1 1 1
1 1 0 1 1 1
1 1 1 1 1 1
A B C A+B A+C (A+B).(A+C) BC A+(BC)
0 0 0 0 0 0 0
0 0 1 0 1 0 0
0 1 0 1 0 0 0
0 1 1 1 1 1 1
1 0 0 1 1 1 0
1 0 1 1 1 1 0
1 1 0 1 1 1 0
1 1 1 1 1 1 1
A B C A+B A+C (A+B).(A+C) BC A+(BC)
0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 1 1 1 1
1 0 0 1 1 1 0 1
1 0 1 1 1 1 0 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1
A B C A+B A+C (A+B).(A+C) BC A+(BC)
0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 1 1 1 1
1 0 0 1 1 1 0 1
1 0 1 1 1 1 0 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1
A B A+B (A+B)' A' B' A'B'
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0
A B A+B (A+B)' A' B' A'B'
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0
At the digital logic level, addition is performed
in binary

Addition operations are carried out

by special circuits called, appropriately,
adders