This module covers the fundamental ideas underlying design
of modern digital systems. This module explains the basis
of circuit elements, how they are interconnected to form
digital circuits and also the non-ideal effects of the design.

**Digital Systems**

The word digital refers to representing any information
in just two voltage levels [logic zero (0) or logic high
(1)]. Logic Zero means zero voltage, logic high means some
level of electric potential voltage, which is generally
a 3.3V or a 5V.

Digital systems are more in use than analog systems for
processing data due to the following reasons:Digital systems
are easier to design as digital system use digital signals
which provides many futures such as being very less prone
to noise, signal can be easily regenerated etc. The term
design reference to the systematic process of working out
how to construct circuits that meet given requirements while
satisfying constraints on cost, performance, power consumption,
size, height and other properties.

Let's start discussing about logic gates that operate on
digital signal.

There are three basic gates namely AND, NOT, OR from this
NAND, NOR, XOR, X-NOR gates are derived. NAND and NOR gates
are usually referred as universal gates. Now let's start
discussing about how this gates work.

**NOT gate: **

In a computer world, the character used to represent NOT
gate is "~" (or) (!)(negation). When any logic high signal
enters NOT gate, output of not will be a low signal and
reverse operation takes place when low signal is entering
the NOT gate. The symbol of NOT gate is depicted in the
figure 1 and its working is represented in truth table.

**Figure1**: (a) inverter symbol (b) truth table (c)
IC for not gate (d) schematic of inverter

IC 7404 is used for NOT gate, six NOT gates are embedded
in IC 7404. The schematic diagram for NOT gate is illustrated
in the figure1

**AND gate**

Unlike NOT gate, AND gate has two input terminals. The character
to represent AND function is "& ". Output of AND gate will
be high only when both the inputs are high. Symbol for AND
gate is as shown in the figure 2 below.

** Figure 2:** AND gate, truth table and inner view of
IC7408 and IC7411

IC 7408 is used for AND operation. It contains 4 AND gates
each having two inputs. We can have 3 input AND gate (IC7411).
Pin Configuration of IC 7408 & 7411 is shown in the figure
2.

We can derive 4 input AND gate by two input AND gate. Let
us see how we can do that

**Figure3:** 4 input AND gate derived from two input
AND gate and its truth .

The figure below shows the schematic (circuit diagram) of
two input AND gate is made using MOSFETs.

**OR gate:**

Character to represent or gate is "|". OR gate performs
addition. when any one input of OR gate is logic high, output
will be high. The Symbol and truth table of OR gate is shown
in the figure 5. IC 7432 is used for OR operation. it contains
four, 2 input OR gates.

**Figure 5:** symbol and truth table of OR gate.

**Figure6:** schematic of OR gate

Note: To learn more on IC numbers for digital circuits visit:
https://en.wikipedia.org/wiki/List_of_7400_series_integrated_circuits

**Switch on the light in a room when the room is dark using
a simple AND gate :**

Here is a real-world example that is always simple, AND
gate can be used to switch on the light in a room when the
room is dark. Please see the figure 7. It said two input
AND gate where one input is connected to a switch which
is either connected to +5 V of high-voltage or grounded
to 0 voltage. To make this device working you have to keep
the switch in such a way that it's connected to +5 V so
that the input to the AND gate is always high. The other
input up AND gate is connected to a light sensor which produces
+5 V whenever there is darkness or no light otherwise zero
voltage that is low-level whenever there is light. The AND
gate is connected to a relay which can switch on a light
bulb. The light bulb will switch on only when both and inputs
are high. That is a case only when there is a dark.

Figure 7: Real world example using AND gate.

Applications are as simple as this, logic gates controls
today's world in the form of complex processor chips which
have millions of such gates. In this course, we will take
you through how so many millions of logic gates or wired
to perform a complex function for real-time usage.

We started with logic gates basically to explain with the
above application where logic circuits can be used to automate
many of our needs.

**Example 2:**

Consider a plastic manufacturing plant. Plant should be
maintained below with certain temperature and pressure threshold.
If temperature or pressure exceeds the limit, the plant
should shut down.

**Figure8:** real world example for OR gate

Example 3

In a real-world inverters of connected back-to-back with
odd numbers, then feedback is given from output to input
to get oscillation.

**Figure 9:** ring oscillator

Note: Design should satisfy “Barkhausen Criteria ” refer
digital and mixed mode VLSI by Razavi.

**Universal Gates (NAND and NOR)**

NAND gate is inverse of AND gate. Symbol for NAND gate is
shown in the figure 10. Working of AND gate is explained
in the truth table .

**Figure10:** symbol of NAND gate and its truth table

NOR gate is inverse of OR gate. Symbol for NOR gate is depicted
in figure 11. Working of NOR gate is it explained in the
truth table

**Figure 11:** symbol of NOR gate and its truth table

From NAND and NOR gate, We can derive any other gates, hence
the name universal gates. Let’s see how is it made:

**NAND as AND gate**

**NAND as OR gate**

**NOR as AND gate**

**XOR gate: Pronounced as Exclusive OR gate:**

XOR gate performs module sum operation which does not include
Carry. It contains two inputs and one output. The symbol
of XOR gate is shown in the figure 12.

**Figure 12:** symbol of XOR gate and its truth table

**NAND as XOR gate**

**XNOR ( Sometimes referred as EX NOR, ENOR, NXOR) **

XNOR performs inverse operations on XOR

**Use of universal gates in real world.**

**Example 1:** Consider a scenario Industrial chemicals
are stored in warehouses where chemical reactions takes
place leading to the production of toxic fumes that are
dispensed into atmosphere through exhaust fans. Three exhaust
fans have to be implemented in warehouses so as to exhaust
poisonous air in the course of its operation. If any one
or more exhaust fans fails then an alarm gets activated.

Pic above: A NAND gate based Exhaust fan failure detection
system

Example 2: Washing machine has three sensors to check its
operation. The first operation is the functioning of the
door while the second is the water level and the third is
the weighing of the clothes. If one of the functions fails
such as opening or closing of the door, variation in the
water level or the overload of clothes, generates appropriate
sensor outputs to 1. The outputs of 3 sensors are connected
to 3 inputs NOR gate. Sensors outputs will be ‘0’ during
the normal operation of the washing machine. The NOR gate
output is 0, if an erroneous condition is detected by any
one or more sensors.. The NOR gate output is connected to
a switch of the washing machine.

**Boolean algebra:**

A Boolean algebra is used as a foundation for digital design.
Digital systems can be expressed in terms of Boolean expression
which contains Boolean operators, variable and Boolean values.

Boolean algebra deals with Boolean expression

Example of Boolean expression

In the example above A, B, C are variables , ‘+’& ‘.’ are
operators and ‘0’ & ‘1’ are Boolean values. The axioms for
Boolean algebra is listed below:

Identity laws

X+0=X

X.1=X

Association laws

(X+Y)+Z= X+(Y+Z)

(X.Y).Z=X(Y.Z)

Commutation laws

X+Y=Y+X

X.Y= Y.X

Distribution laws

X+ (Y.Z)= (X+Y).(X+Z)

X.(Y+Z)= (X.Y)+(X.Z)

Complement laws

some of the useful theorems are listed below which can be
used while dealing with Boolean expressions.

Idempotence laws

X+X= X

X.X=X

Absorption Laws

X+(X.Y)=X

X.(X+Y)=X

Further identity laws

X+1=1

X.0=0

De Morgans Laws

Here ‘+’ represents OR Operation

‘.’ represents multiplication i.e. AND Operation

(2) f= AB+A(B+C)+B(B+C)

(3) f=A(B+CD)

Boolean expressions can be expressed in two forms namely
sum of products (SOP) and product of sums (POS):

Sum of Products:- Product terms of Boolean Expressions are
summed together by Boolean addition to form SOP

Example:- 1. AB+BC+CA

2. AB+B(C+D)

Product of Sums: sum of the variables are multiplied with
sum of other terms of the expression.

• (A+B+C)A (D+F)

We cannot convert Boolean expression into SOP or POS. Let
us see how we can achieve it:

• AB+B(CD+EF) converted to SOP

Ans. AB+BCD+BEF

• (A+B) (B+C+D) convert to SOP

Ans. AB+AC+AD

Note:- So for we have seen SOP and POS expressions. There
are the examples of non-standard way of expressing Boolean
expressions.

Standard SOP:

Consider an example f= AB+BC+CA, let us convert it to standard
SOP form:

>AB+BC+CA

A Standard SOP expression is one in which all the variables
in the domain appear in each product term in the expression.

Standard POS

Simplification of Boolean expressions using Boolean algebra

• AB+A(B+C)+B(B+C)

=AB+AB+AC+BB+BC

= AB+AC+BC+B

= B+ AC+AB

= B (1+A)+AC

=B+AC

Simplifying the Boolean expression will simplify the logic.

Consider the expression AB+A(B+C)+B(B+C), it is equivalent
logic circuits and simplified circuit diagram is shown below

Simplified logic

Logic simplification can be done by many methods and some
of them are listed below:

1. Karnaugh map

2. Quine Mccluskey algorithm

3. State variable minimization

For more information about logic simplification go through
“digital logic applications and designs” by John M Yarbrough.

Non ideal effects of circuit elements

(1) Propagation delay: Propagation delay is due to internal
switching activity (i.e. the time taken for charging and
discharging nature of parasitic capacitance of transistors)

The signal levels do not change instantaneously due to switching
activity. The time taken for the signal voltage to rise
from low-level to high-level is known as rise time. The
time taken for the signal voltage to fall from high-level
to low-level is known as fall time.

Consider a circuit shown in the figure below

One way to reduce propagation delay is to reduce parasitic
capacitance, if more than one logic block is connected to
the output of previous block, Parasitic Capacitance gets
added up which results to more propagation delay in other
words fanout should be less.

(2)Wire delay

We think that wire acts like a pure conductor without any
resistance, but in reality wires will have resistance inductance
and capacitance. When wire length is too small these values
are very negligible. When wire length is too large, they
become a transmission line which has reasonable propagation
delay. While designing high-speed systems these delays should
be taken into considerations.

(3)Glitches

Glitches are also known as invalid output which is due to
the propagation delay in the inputs entering logic block
for example assume a circuit diagram shown below

In the above example we can say that unexpected zero at
the output of the circuit. This is termed as glitch and
it causes problems for the rest of the circuits as it will
propagate into the next circuits and results to more glitches.

Solution for this problem: adding buffer to the path which
has less propagation delays. These buffers compensates the
delay so that we can expect correct results at the output.

Combinational logic circuits and sequential logic circuits

combinational logic circuits performs the operation on the
inputs entering the circuit and it determine output as a
Boolean function of input.

In other words output is a function of input alone. Figure
below shows general block diagram of combinational circuit.

These combinational logic circuits lacks behind in storing
any previous outputs. In other words it has no memory. This
can be overcome by using sequential circuits, in which output
is determined by both previous and present inputs. Figure
below shows a block diagram of sequential logic circuits.

For more information about sequential logic circuits and
combinational logic circuits, please refer CMOS Digital
integrated circuits by sung mo kang Yusuf leblebic.

Digital circuits are classified into synchronous circuits
and asynchronous circuits

. In synchronous circuits, there will be one global clocks
signal that’ll will be driving entire circuits.

Advantages of synchronous circuits

• Design is simpler

• likelihood of correct operation is higher

disadvantages

routing becomes a problem

In asynchronous circuits there will be two are more clocks,
but they are ideal and synchronous.

Note:Most of the commercial digital design use synchronous
circuits