Introduction to Logic Circuits

Introduction
Logic circuits deal with digital signals or digital inputs and produce digital outputs. They
manipulate information that is in a digital format. This is useful because digital signals can be
used to represent binary numbers and so logic circuits can manipulate binary mathematics -
this makes them very powerful. Many modern devices use digital electronics and logic
circuits are fundamental to using digital electronics.

There are two distinct types of digital circuit:

Combinational Logic Circuits are circuits where the output only depends on the inputs. If
the inputs change, the output immediately changes too. The combination of the inputs
determines what the output(s) will be. Examples of combinational logic circuits include:

● Simple logic gates

● Logic circuits containing multiple logic gates
● Logic circuits with multiple inputs
● Logic operators such as Full Adders, Half Adders and Multiplexers

Synchronous Logic Circuits are circuits where the output depends on the inputs and also
what the inputs were previously. When the inputs change the outputs do not necessarily
change immediately. The output depends on the state of the inputs and how the inputs have
changed some time before. Examples of synchronous logic circuits include:

● Monostable circuits
● Bistable circuits, latches and flip flops
● Astable circuits
● Counters
● Shift registers

Digital and Analogue

Digital signals have only two possible states, they are either ON or OFF.

ON is called Logic 1. OFF is called Logic 0.

In electrical circuits these states are represented by voltages. Logic 1 is a high voltage and is
usually understood to be +5 V. Logic 0 is a low voltage and is usually understood to be 0 V.
However, this is not always the case. If a circuit is power by a 9 V battery then Logic 1 might
be anything above 6 V and Logic 0 might be anything below, say, 3 V depending on exactly
what components are being used so Logic 1 and Logic 0 are not fixed in stone. The value in
the middle, between 3 V and 6 V in this case, is undefined and is neither Logic 1 or Logic 0.
Analogue signals can have any value between two limits. An analogue signal could be
between 0 V and 9 V when a circuit is run off a 9 V battery or between −15 V and +15 V in a
circuit which is run off a ±15 V power supply. Analogue signals can be both positive and
negative and take any value between the maximum and minimum values.

The diagrams illustrate the difference between analogue and digital signals:

Testing logic circuits

When does logic 0 become logic 1? At exactly what voltage a logic circuit decides that the
input is no longer a '0' and is now a '1' depends on the circuit. To be safe we usually try to
use a value close to zero volts for logic 0 and some voltage close to the supply voltage for
logic 1. For logic gates using a 5 V supply it is usually the case that anything below 2 V is a
Logic 0 and anything above 3 V is a Logic 1 but it is always a good idea to check.

The circuit shown can be used to investigate the

exact transition point of a logic gate. The
potentiometer provides a variable input voltage
and voltmeter measures the input voltage. A
second voltmeter measures the output voltage.
 A graph of Input voltage against
Output voltage is called the
transfer characteristic of the logic
gate - it shows how the input and
output voltages are related. The
diagram shows a typical transfer
characteristic for a Logic NOT
gate but it should be noted that all
logic gates have their own transfer
characteristics and, if the exact
values are needed, then the logic
gate must be tested.

Inputs and Outputs

How do we provide an input to our logic circuit? If the input to a logic circuit is derived from
another logic circuit then there is no problem as it will always be either fully Logic 0 or fully
Logic 1.

If we are using switches as our inputs then we must be careful not to leave any wires
'unconnected' (technically such an unconnected input is called 'floating'). An input that is not
connected to anything is not at Logic 1 or Logic 0, it is Logic whatever it wants to be ... and
this will cause all sorts of problems and erratic behaviour.

Therefore, the rule is inputs should always be connected to something. Pull-up or

pull-down resistors are used to ensure that the input to the logic circuit is always connected
to something.

Pull-up and pull-down resistors are nominally 10 kΩ but can be any high value between 1 kΩ
and 100 kΩ.
Pull-down
When the push button is not pressed, the logic circuit is
connected to 0 V through the resistor. The input to the logic circuit
is Logic 0.

When the push button is pressed, the logic circuit is connected

directly to the positive power supply. The input to the logic circuit
is Logic 1.

The resistor is necessary so that, when the button is pressed, the

positive power supply is not simply connected to 0 V resulting in a
short circuit.

The circuit is effectively a potential divider circuit where R1 is

either 0 Ω or virtually infinite resistance.

Pull-Up
When the push button is not pressed, the logic circuit is connected
to positive through the resistor. The input to the logic circuit is
Logic 1.

When the push button is pressed, the logic circuit is connected

directly to 0 V. The input to the logic circuit is Logic 0.

Again, the resistor is necessary so that, when the button is

pressed, the positive power supply is not simply connected to 0 V
resulting in a short circuit.

The circuit is effectively a potential divider circuit where R2 is

either 0 Ω or virtually infinite resistance.
Logic Gates
Introduction
Logic gates are digital devices. They have one or more inputs (labeled A, B, C etc) and one
output (always labeled Q). The output only depends on the inputs.

Logic gates are active devices which means they need a power supply although, for clarity,
this is never shown on the circuit diagrams. The output is 0 V for Logic 0 and the supply
voltage for Logic 1.

The output of a logic gate can supply a small amount of current, nominally 10 mA although
some can provide more. The output can drive a low power LED directly (with a suitable
series resistor), however, for larger output currents a transducer driver should be used.

Logic gates all have symbols and these should be learned. The action of a logic gate is
described by a truth table. A truth table lists all the possible combinations of inputs (Low = 0
and High = 1) and the corresponding output. Truth tables should also be learned.

Most logic gates can be modeled (using switches and resistors or light bulbs) which suffice
as thinking tools but these models do not necessarily represent how a logic gate actually
works.

The NOT gate

The NOT gate shown has a single input and is
represented by a triangle and a small circle.

The circle is the NOT gate, the big triangle just

means you can find it.

On other gates, adding a small circle to the symbol is all that is required to add the NOT
function.

The output is Logic 1 when input A is Logic 0. The output is Logic 0 when the input A is
Logic 1.

In the equivalent circuit shown, A = 0 when switch A is open (as in the

diagram) and A = 1 when switch A is closed.

When the switch is open (A = 0) then the resistor ensures Q is

connected to +Ve and the output is Logic 1. When the switch is closed
(A = 1) then Q is connected directly to 0 V and so the output is Logic 0.
The AND gate
The AND gate shown has two inputs. The output is
only Logic 1 when input A AND input B are both
Logic 1, otherwise the output is Logic 0.

If there are more than 2 inputs then all the inputs

have to be Logic 1 to make the output Logic 1.

If any of the inputs are Logic 0, the output is Logic 0.

In the equivalent circuit shown A = 0 when switch A is open (as in the

diagram) and A = 1 when switch A is closed.

Both the switches need to be closed (A = 1 and B = 1) to connect Q to

+Ve and make the output Logic 1.

If either switch is open then the resistor ensures Q is connected to 0 V

and the output is Logic 0.

The OR gate
The OR gate shown has two inputs. The output is
Logic 1 when input A OR input B OR both inputs are
Logic 1, otherwise the output is Logic 0.

The output is Logic 0 when both inputs are Logic 0.

If there are more than 2 inputs then all the inputs have to be Logic 0 to
make the output Logic 0, if any, or all, of the inputs are Logic 1, the
output is Logic 1.

In the equivalent circuit shown A = 0 when switch A is open (as in the

diagram) and A = 1 when switch A is closed.

Either switch needs to be closed (A = 1 or B = 1) to connect Q to +Ve

and make the output Logic 1.

If both switches are open then the resistor ensures Q is connected to

0 V and the output is Logic 0.
The NAND gate
The NAND gate shown has two inputs. The output is
Logic 0 when input A AND input B are both Logic 1,
otherwise the output is Logic 1. This is the exact
opposite of an AND gate.

The NAND gate is effectively an AND gate followed by a NOT gate,

represented on the symbol as a small circle.

NAND is therefore Not AND, hence the name.

If there are more than 2 inputs then all the inputs have to be Logic 1 to
make the output Logic 0. If any, or all, of the inputs are Logic 0, the
output is Logic 1.

In the equivalent circuit shown A = 0 when switch A is open (as in the

diagram) and A = 1 when switch A is closed.

Both switches need to be closed (A = 1 and B = 1) to connect Q to 0 V

and make the output Logic 0. If either switch is open then the resistor
ensures Q is connected to +Ve and the output is Logic 1.

The NOR gate

The NOR gate shown has two inputs. The output is
Logic 0 when input A OR input B OR both inputs are
Logic 1, otherwise the output is Logic 1. This is the
exact opposite of the OR gate.

The NOR gate is effectively an OR gate followed by a NOT gate, shown on the NOR gate
symbol as a small circle.

NOR is therefore Not OR, hence the name.

If there are more than 2 inputs then all the inputs have to be Logic 0 to
make the output Logic 1, if any, or all, of the inputs are Logic 1, the
output is Logic 0.

In the equivalent circuit shown A = 0 when switch A is open (as in the

diagram) and A = 1 when switch A is closed.

Either switch needs to be closed (A = 1 or B = 1) to connect Q to 0V and

make the output Logic 0. If both switches are open then the resistor
ensures Q is connected to +Ve and the output is Logic 1.
The Exclusive OR gate
The EOR gate shown has two inputs and the symbol
is very similar to the OR gate except for the extra
parallel line near the inputs.

The output is Logic 1 when input A is Logic 1 OR

input B is Logic 1 BUT not both at the same time.

The output is Logic 0 when the inputs are either both Logic 0 or both Logic 1. The output is
Logic 0 when the inputs are the same as each other.

The output is Logic 1 when the inputs are different. The EOR gate is therefore useful for
detecting when two inputs either are, or are not, in the same state.

In the equivalent circuit the switches are SPDT types - single pole,
double throw. This means they are not simply on or off but connected to
the left hand side or connected to the right hand side as the diagram is
drawn.

When the switches are both to the left (as in the diagram) then A = 0
and B = 0. The output Q is connected to 0 V and the output is Logic 0.
Similarly, if both the switches are in the right hand position then A = 1
and B = 1 and again the output Q will be connected to 0 V giving an
output of Logic 0.

If, however, the switches are in opposite orientations such that A = 0

and B = 1 or vice versa then the output Q is not connected to 0 V and so
the resistor connected to +Ve ensures the output is Logic 1.

Testing Logic Gates

Logic gates can be tested using simple push buttons with pull down resistors as the inputs
and an LED and series resistor to show the state of the output. A suitable circuit is shown
below:
Logic Circuit Analysis
Introduction
It is important to be able to analyse a logic circuit that contains several logic gates and has
two or three inputs.

To analyse the behaviour of a logic circuit a truth table can be used. As well as the inputs
and the output, extra columns are added for the intermediate points in the logic circuit.

Note: Logic circuits can also be analysed using Boolean Algebra or Karnaugh Maps as well
as using truth tables.

Example Circuit 1

A and B are the inputs.

The aim is to find the output Q for all possible combinations of A and B.

Step 1: Fill in columns A and B with all possible combinations of 0 and 1 - in this case there
are only four possible combinations.

Step 2: Add intermediate columns X and Y for the outputs of the first two logic gates.

Step 3: Fill in column X where X is the output of an AND gate and so X=1 when A=1 AND
B=1, otherwise X=0.

Step 4: Fill in column Y where Y is the output of an OR gate and so Y=1 when A=1 OR B=1,
otherwise Y=0.

Step 5: Fill in the column for the output Q where Q is the output of an OR gate that has X
and Y as the inputs. Therefore Q=1 when X=1 OR Y=1, otherwise Q=0.

Note: This whole circuit can be simplified to a single OR gate with A and B as the inputs.
Example Circuit 2

A and B are the inputs.

The aim is to find the output Q for all possible combinations of A and B.

Step 1: Fill in columns A and B with all possible combinations of 0 and 1 - there are only four
possible combinations.

Step 2: Add intermediate columns X as the output of the AND gate and Y as the output of
the NOT gate.

Step 3: Fill in column X where X is the output of an AND gate and so X=1 when A=1 AND
B=1, otherwise X=0.

Step 4: Fill in column Y where Y is the output of the NOT gate so when B=0, Y=1 and when
B=1, Y=0.

Step 5: Fill in the column for the output Q where Q is the output of the OR gate that has X
and Y as the inputs. Therefore Q=1 when X=1 OR Y=1, otherwise Q=0.
Example Circuit 3

A, B and C are the inputs.

The aim is to find the output Q for all possible combinations of A, B and C.

Step 1: Fill in columns A, B and C with all possible combinations of 0 and 1 - there are eight
possible combinations.

Step 2: Add intermediate columns X, Y and Z as the outputs the two OR gates and the NOT
gate.

Step 3: Fill in column X where X is the output of an OR gate with A and B as the inputs so
X=1 when A=1 OR B=1, otherwise X=0.

Step 4: Fill in column Y where Y is the output of the NOT gate so when B=0, Y=1 and when
B=1, Y=0.

Step 5: Fill in column Z where Z is the output of the OR gate with Y and C as the inputs so
Z=1 when C=1 OR Y=1, otherwise Z=0.

Step 6: Fill in the column for the output Q where Q is the output of the AND gate that has X
and Z as the inputs. Therefore Q=1 when X=1 AND Z=1, otherwise Q=0.
Example Circuit 4

A, B and C are the inputs.

The aim is to find the output Q for all possible combinations of A, B and C.

Step 1: Fill in columns A, B and C with all possible combinations of 0 and 1 - there are eight
possible combinations.

Step 2: Add intermediate columns X as the output of the first NOR gate and Y as the output
of the NAND gate.

Step 3: Fill in column X where X is the output of a NOR gate and so X=0 when A=1 OR B=1,
otherwise X=1.

Step 4: Fill in column Y where Y is the output of the NAND gate so Y=0 when B=1 AND
C=1, otherwise Y=1.

Step 5: Fill in the column for the output Q where Q is the output of the NOR gate that has X
and Y as the inputs. Therefore Q=0 when X=1 OR Y=1, otherwise Q=1.

Note: This whole circuit can be simplified to a single AND gate with B and C as the inputs,
the value of input A is irrelevant.
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August 2021

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