Computer Science 1000
Digital Circuits

Digital Information
computers store and process information
using binary
 as we’ve seen, binary affords us similar
advantages as other representations


e.g. we saw how we could add binary numbers,
just as we add decimal numbers

Binary Representation

recall that the way in which information is represented depends on
the component




for our purposes, the most important point to remember is that there
are exactly two states




RAM: capacitor charge
hard drive: magnetic fields
processor: voltages
binary numbers: 1 0
logic: true false
gate input/outputs: on off
your text refers to such a binary system as a PandA representation,
where information is made up of states that are Present and Absent

PandA

from pg. 198 of your text

Processing
your CPU processes information
 what does that mean?

one possible definition: create a new state
(output) from an existing state (input)
 by state, I am referring to a binary sequence

Input
5
120
Processing
Output

Processing – Example

consider our addition example from the
previous lecture
1011
+0010
1101
Input (Existing State)
Output (New State)

Processing – Example

what we require is a device that can:


1011
0010
take two binary numbers as input
produces their sum as output
1011
0010
1101
Binary Adder


in fact, this is (part of) what the arithmetic logic unit (ALU)
in your processor does
the question is: how?

Logic Gates


at a very basic level, processors are constructed
using gates
a logic gate is a very simple device:



has a number of inputs
produces a single output
think of each input as a wire that is either on or off


on represents a 1 in binary, or a true in logic
off represents a 0 in binary, or a false in logic

Logic Gates - Continued


the output of a logic gate depends on (and only on)
its inputs
in other words, for each possible combination of its
inputs, there is a specific output


we represent this using a truth table
the truth table of a gate depends on its type

Not Gate




one of the simplest gate types
also known as an inverter
takes one input, and produces a single output
symbol:
Input
Output

Not Gate


a not-gate simply inverts its input
that is



if its input wire is on, its output is off
if its input wire is off, its output is on
0
1
1
0
hence, the truth table is constructed as follows:
Input
Output
on
off
off
on

Truth Table Representation



in the interest of space, on and off in truth tables
are often represented as 1 and 0
hence, our previous truth would be written as
follows:
Input
Output
0
1
1
0
unless otherwise stated, this will be the
representation that we adopt

AND Gate


an example of a gate with two inputs
its output is on only if both of its inputs are on

Symbol:
otherwise, the output is off
Truth Table:
A
B
Output
0
0
0
0
1
0
1
0
0
1
1
1

Truth Table – Multiple Inputs

note that our truth table now has four rows


one for each possible configuration of inputs
suppose we had a gate with three inputs

how many rows would that require?
A
B
C
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Output
In general, any circuit
with n binary inputs will
require 2n rows in its
truth table.

Digital Circuits

by themselves, gates are quite limited in
what they do


there are only so many uses for an AND gate
the real power of gates is when we begin to
combine them

that is, the output of one gate becomes the input
to another gate

Digital Circuits

Example:




suppose we have a NOT gate and an AND gate
we could connect them in the following configuration
notice that we still have two inputs (A,B) and one output
the output of the not gate is attached to the first input of
the AND gate

Digital Circuits

Example:

what does the truth table of our circuit look like?

to determine this, let’s consider each input state individually
A
B
0
0
0
1
1
0
1
1
Output

Digital Circuits

Example:

when A is 0 and B is 0



0
the output of the not gate is 1
hence, the input to our AND gate is 1 and 0
the output of our AND gate is 0
1
0
0
A
B
Output
0
0
0
0
1
1
0
1
1

Digital Circuits

Example:

when A is 0 and B is 1



0
the output of the not gate is 1
hence, the input to our AND gate is 1 and 1
the output of our AND gate is 1
1
1
1
A
B
Output
0
0
0
0
1
1
1
0
1
1

Digital Circuits

Example:

when A is 1 and B is 0



1
the output of the not gate is 0
hence, the inputs to our AND gate are 0 and 0
the output of our AND gate is 0
0
0
0
A
B
Output
0
0
0
0
1
1
1
0
0
1
1

Digital Circuits

Example:

when A is 1 and B is 1



1
the output of the not gate is 0
hence, the inputs to our AND gate are 0 and 1
the output of our AND gate is 0
0
0
1
A
B
Output
0
0
0
0
1
1
1
0
0
1
1
0

Digital Circuit

previous example demonstrated a digital circuit


like logic gates, each digital circuit has:




an assembly of logic gates
input(s)
output(s)
a truth table
like logic gates, digital circuits can be used in the
construction of other circuits

output from one circuit connected to another digital circuit

Digital Circuit - Rendering


circuits are often rendered in an enclosed shape
this makes it clear what the inputs and outputs to
the system are

Digital Circuit - Rendering

in fact, if we know the inputs, outputs, and truth
table, then we often are not concerned with how the
circuit is built


becomes a “black box”
handy when this circuit is used as part of a larger circuit
A
B
Output
0
0
0
0
1
1
1
0
0
1
1
0

Digital Circuit – Example 2

suppose I rearrange my previous circuit into
the following configuration

what would the truth table of this circuit be?

Digital Circuit – Example 2

Answer (discussed in class):
A
B
Output
0
0
1
0
1
1
1
0
1
1
1
0

Digital Circuit – Solving

our circuits so far are reasonably small, and
therefore easy and quick to derive truth table


step through each individual configuration of the inputs
however, as circuits grow, this task becomes more
difficult and time consuming

Digital Circuit – Solving

as a consequence, it is often helpful to label
intermediate points in the circuit, and solve for
those points explicitly in the truth table


usually, these intermediate points are the output from a
gate that are being input into another gate
consider our previous example

let’s label the output of the NOT gate as point C
C

Digital Circuit – Solving


now, when we create our truth table, we will create
an “output” column for C as well
although this might initially seem like more work, it
will ultimately make computing the output easier
C
A B C
0
0
0
1
1
0
1
1
Output

Digital Circuit – Solving

compute the values for column C
C
A B C
0
0
1
0
1
1
1
0
0
1
1
0
Output

Digital Circuit – Solving

now, compute the values for the output column

notice that the output depends only on C and B


we need not even consider A
the output is 1 if C and B are 1, and 0 otherwise
C
A B C
Output
0
0
1
0
0
1
1
1
1
0
0
0
1
1
0
0

OR Gate


another common example of a gate with two inputs
its output is off only if both of its inputs are off


Symbol:
otherwise, the output is on
notice how this contrasts with and
Truth Table:
A
B
Output
0
0
0
0
1
1
1
0
1
1
1
1

Digital Circuit


suppose we replace the AND gate in our previous
circuit with an OR gate
compute the resulting truth table for the output
A B Output
0
0
0
1
1
0
1
1

Digital Circuit

first, to simplify our computation, let’s add an
intermediate point at the NOT gate


call it C
C will require a column in the truth table
A BB COutput
Output
C
0 00
0 11
1 00
1 11

Digital Circuit

now, let’s calculate the values for column C
A B C Output
C
0 0 1
0 1 1
1 0 0
1 1 0

Digital Circuit

now, we can calculate the values for the output
based on columns B and C

recall our definition for the OR gate – the output is off if both of its inputs
are off, and on otherwise
A B C Output
C
0 0 1 1
0 1 1 1
1 0 0 0
1 1 0 1