Download Topic 4 – Switching Circuits

Topic 4 – Switching Circuits
Serial vs. Parallel Transmission
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Circuit elements can be connected in
either a serial or parallel manner.
Serial implies a “one path through”
approach, where a signal would pass
through one element, then the next,
then the next, and so forth.
Serial vs. Parallel Transmission
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Parallel implies a “multiple elements at
once approach,” where a signal passes
through multiple elements
simultaneously.
Switching Circuits
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Switching circuits (digital logic circuits) are
composed of combinations (serial or parallel)
of elements known as gates.
A gate is a high-speed electronic switching
element which is capable of turning on or off
within a few nanoseconds.
In such a system, switching variables are
associated with the input signals to the gates
and switching functions are associated with
the output signals…the switching function
describes the input/output relationship.
Electronic Circuits
and Logic Values
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Truth tables that describe the operation of
gates are presents in terms of high (H) or low
(L) voltage (physical) values.
The designed may choose to use these values
to represent the logic values 0 and 1 in
different ways.
Positive logic uses voltage high (H) to
represent 1 and voltage low (L) to represent
0. Negative logic uses voltage high (H) to
represent 0 and voltage low (L) to represent
1. Mixed logic uses a combination of both.
Electronic Circuits
and Logic Values
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Mixed logic uses positive logic for some
signals and negative logic for others…never
different conventions for the same signals.
A signal that is set to logic 1 is said to be
asserted, active, or true. An active-high
signal is asserted when it is high (H) and an
active-low signal is asserted when it is low
(L).
If not asserted (set to logic 0), the signal is
said to be deasserted, negative, or false.
Logic Gates
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Now, let’s look at the seven types of
digital logic gates.
The roles of these gates are true and
valid for active-high logic. For example,
an AND gate will produce the logic AND
of its two active-high inputs as an
active-high output. Later, we will see
what happens with active-low logic.
The AND Gate
A
B
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Y
This is an AND gate. Its truth table
straightforward…
So, if the two inputs signals A
are asserted (high) the
0
output will also be asserted. 0
Otherwise, the output will
1
be deasserted (low).
1
is
B
0
1
0
1
Y
0
0
0
1
The OR Gate
A
B
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Y
This is an OR gate. Its truth table is also
straightforward…
So, if either of the two
A
B
input signals are
0
0
asserted, or both of
0
1
them are, the output
1
0
will be asserted.
1
1
Y
0
1
1
1
The NOT Gate (Inverter)
A
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Y
This is a NOT gate or an inverter. It has only
one input and is used to implement the
A Y
complement concept of switching algebra.
0 1
A bubble (seen here) at the output of any
logic gate indicates that an internal logic 1
1 0
produces an external logic 0 and vice versa.
Since the inverter has a bubble and does not
implement any additional logic function, the
output is simply the complement of the input.
The NOT Gate (Inverter)
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The inverter can be seen as changing the polarity
of a signal from active-high to active-low. As
such, the bubble can be drawn at either the input
or the output.
By convention, the bubble is always drawn with
the active-low signal. If the input is active-high,
and the inverter is changing it to active-low, the
bubble is drawn on the output. If the input is
active-low and the inverted is changing to to
active-high, the bubble is drawn on the input.
The NAND Gate
A
B
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Y
This is a NAND gate. It is a combination of
an AND gate followed by an inverter. It’s
truth table shows this…
A B
NAND gates have several interesting
0 0
properties…
 NAND(a,a)=(aa)’ = a’ = NOT(a)
0 1
 NAND’(a,b)=(ab)’’ = ab = AND(a,b)
1 0
 NAND(a’,b’)=(a’b’)’ = a+b = OR(a,b)
1 1
Y
1
1
1
0
The NAND Gate
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These three properties show that a NAND gate
with both of its inputs driven by the same signal
is equivalent to a NOT gate, a NAND gate whose
output is complemented is equivalent to an AND
gate, and a NAND gate with complemented
inputs acts as an OR gate.
Therefore, we can use a NAND gate to
implement all three of the elementary operators
(AND,OR,NOT). Therefore, ANY switching
function can be constructed using only
NAND gates. Such a gate is said to be primitive
or functionally complete.
NAND Gates into Other Gates
(what are these circuits?)
A
Y
NOT Gate
A
B
Y
AND Gate
A
Y
B
OR Gate
The NOR Gate
A
B
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Y
This is a NOR gate. It is a combination of an
OR gate followed by an inverter. It’s truth
table shows this…
A B
NOR gates also have several
0 0
interesting properties…
0 1
 NOR(a,a)=(a+a)’ = a’ = NOT(a)
 NOR’(a,b)=(a+b)’’ = a+b = OR(a,b)
1 0
 NOR(a’,b’)=(a’+b’)’ = ab = AND(a,b)
1 1
Y
1
0
0
0
Functionally Complete Gates
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Just like the NAND gate, the NOR gate is
functionally complete…any logic function can
be implemented using just NOR gates.
Therefore, both NAND and NOR gates are
very valuable as any design can be realized
using either one. It is easier to build an IC
chip using all NAND or NOR gates than to
combine AND,OR, and NOT gates.
Additionally, NAND/NOR gates are typically
faster at switching and cheaper to produce.
NOR Gates into Other Gates
(what are these circuits?)
A
Y
NOT Gate
A
B
Y
OR Gate
A
Y
B
AND Gate
The XOR Gate (Exclusive-OR)
A
B
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This is a XOR gate.
XOR gates assert their output
when exactly one of the inputs
is asserted, hence the name.
The switching algebra symbol
for this operation is , i.e.
1  1 = 0 and 1  0 = 1.
Y
A
0
0
1
1
B
0
1
0
1
Y
0
1
1
0
The XNOR Gate
A
B
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This is a XNOR gate.
This functions as an
exclusive-NOR gate, or
simply the complement of
the XOR gate.
The switching algebra symbol
for this operation is , i.e.
1  1 = 1 and 1  0 = 0.
Y
A
0
0
1
1
B
0
1
0
1
Y
1
0
0
1
Combinatorial Circuits
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Now that we know of all the logic gates, let’s
look at circuits composed of a combination of
these gates. We will do two things with these
circuits…
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Analysis – Here, we will look at a circuit and derive
an expression which summarizes its operation.
This can be used to verify a circuit does what we
want it to and/or to allow us to convert the circuit
to a different form, either in fewer gates, or a
different type of gates.
Combinatorial Circuits
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Design and Synthesis – Here, we will begin
with an expression or description and then
design and then implement a circuit to
fulfill that function. This circuit can be
designed with gates, programmable logic
devices (more about them later) or other
logic elements.
Now, let’s look at analysis of
combinatorial circuits.
Analysis: Algebraic Method
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Any given switching network can be
completely represented by a switching
expression.
Remember, any switching expression may be
written with AND, OR, and NOT gates. This
also means any circuit can be completely
represented with NAND gates or with NOR
gates.
As an example, let’s analyze a simple
combinatorial circuit.
Example #1 - Analysis
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Example #1 on the board.
This is a combinatorial circuit. To analyze this
circuit using algebraic analysis, we will derive
switching expressions at each point
propagating to the output of the circuit.
Then, we will simplify the resulting expression
and implement the resulting simplified
expression in logic gates.
Analysis: Timing Diagrams
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Another method of analyzing combinatorial
circuits is through the use of timing diagrams.
A timing diagram is a graphical representation
of input and output signal relationships in a
switching network. Often, intermediate
signals are also represented.
From this, if all possible input combinations
and all outputs are shown, a switching
expression for the circuit can be determined.
Example #2 –
Timing Diagrams
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Example #2 on the board.
This is a combinatorial circuit and a
corresponding timing diagram. We can
examine this diagram to find all possible input
combinations, and then learn the resulting
output signal(s) at that point in time.
In this example, let’s determine the truth
table for the circuit and the minterms for the
two outputs.
Propagation Delay
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Ideally logic gates would function
immediately. As soon as one of their input
changed, the output would immediately
change.
In physical devices, the delay between an
input change and the corresponding output
delay is nonzero and is referred to as the
propagation delay.
This value is dependent on many issues such
as the type of gate, logic family, temperature,
and many other factors.
Propagation Delay
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Many gates have different propagation delays
for going high-to-low than for going low-tohigh.
In a datasheet (the manufacture's
specifications for their product), the low-tohigh delay is normally specified as tPLH and
the high-to-low delay to normally specified as
tPHL.
When precise timing is not necessary, the
average propagation delay is used for each
gate.
Synthesis of
Combinatorial Logic Circuits
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Now, let’s turn to designing and
implementing combinatorial logic
circuits.
AND-OR Networks
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As you’ll remember, AND gates implement
product terms and OR gates implement sum
terms.
If a switching expression is expressed in the
sum-of-products (SOP) form, it can be
implemented easily using AND-OR networks.
Each product term is implemented using an
AND gate in the input stage of the network.
Then, each product term is ORed together to
implement the sum term. This will result in
the desired output.
Example #3 –
AND-OR Networks
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Example #3 on the board.
Here, we will implement the switching
function f(p,q,r,s) = pr’ + qrs + p’s
Each product term is implemented
using an AND gate and then the sum
term is implemented using an OR gate.
Therefore, this expression is directly
implemented in AND-OR logic.
NAND Networks
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Any switching function can be implemented
using just NAND gates.
If we take the SOP function, complement it
twice and apply DeMorgan’s Theorem, we will
arrive at the NAND form of the expression…
f(p,q,r,s) = (pr’ + qrs + p’s)’’
= ((pr’)’ (qrs)’ (p’s)’)’
= ((pr’)(qrs)(p’s))’
= NAND((pr’),(qrs),(p’s))
Thus, we can implement this directly in NAND
logic.
OR-AND Networks
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If a switching expression is expressed in the
product-of-sums (POS) form, it can be
implemented easily using OR-AND networks.
Each sum term is implemented using an OR
gate in the input stage of the network. Then,
each product term is ANDed together to
implement the product term. This will result
in the desired output.
Example #4 –
AND-OR Networks
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Example #4 on the board.
Here, we will implement the switching
function f(a,b,c,d) = (a’+b+c)(b+c+d)(a’+d)
Each sum term is implemented using an OR
gate and then the product term is
implemented using an AND gate.
Therefore, this expression is directly
implemented in OR-AND logic.
NOR Networks
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Any switching function can be implemented
using just NOR gates.
If we take the SOP function, complement it
twice and apply DeMorgan’s Theorem, we will
arrive at the NAND form of the expression…
f(a,b,c,d) = ((a’+b+c)(b+c+d)(a’+d))’’
= ((a’+b+c)’+(b+c+d)’+(a’+d)’)’
= NOR((a’+b+c)’, (b+c+d)’,(a’+d)’)
Thus, we can implement this directly in NOR
logic.
Two-Stage Networks
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Networks like the AND-OR and OR-AND networks
we have just seen are referred to as two-level
networks.
Inputs pass through two levels of gates before
reaching the output.
The first level is defined as the level that contains
the gate that produces the output. The gates
that receive the circuit inputs are on the second
level.
In general, a network has n levels if at least one
input signal must pass through n gates to arrive
at the output.
Three-Stage Networks
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Switching functions in the POS or SOP form
can be implemented directly in two-level
networks when the inputs are available in
both complemented and uncomplemented
forms.
A three-level network is required when only
one form of the inputs is available. In this
case, only NOT gates are needed on level 3
of the network.
Two and Three
Stage Networks
Two-Stage Network
p
Three-Stage Network
p
U1A
U1A
U4A
r
r’
U2A
U3A
U1B
Level 2
U2A
U4B
Level 1
U3A
U1B
Level 2
Level 3
Level 1
NAND Gate Networks
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At this point, you now have all of the
tools necessary to take a switching
function in minterm form and
implement it in NAND logic.
The implementation procedure is easy…
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(1) Express the function in minterm list
form.
(2) Write out the minterms in algebraic
form.
NAND Gate Networks
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(3) Simplify the function in SOP form using
switching algebra.
(4) Use switching algebra theorems to
transform the expression into NAND
formulation.
(5) Draw the NAND logic diagram.
As an example, let’s implement f(X,Y,Z)
= Σ m(0,3,4,5,7) in NAND logic
(Example on the board).
Factoring
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In some cases, gates with a number of inputs
required may not be available (i.e. we need a
5-input NAND gate and only have 3-input
NAND gates).
In such cases, factoring can be used to
reduce the number of literals in large product
or sum terms to values less than or equal to
the number of available gate inputs.
Factoring normally involves using the
distributed law. There is no step-by-step way
to perform it…it’s more of an art.
Factoring Example #1
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Given the expression
f(A,B,C,D) = AB’ + AD’ + AC’
we can factor it…
f(A,B,C,D)
= A(B’ + D’ + C’)
= A(BCD)’
Therefore, we can now implement this
with one 3-input NAND gate and one 2input AND gate, instead of three 2-input
NAND gates and one 3-input NOR gate.
Factoring Example #2
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The function f(a,b,c,d) = Σ m(8,13)
needs to be implemented using only 2input AND and OR gates.