# Lecture 3: Boolean Algebra I * the basics ```Lecture 7: Basic Logic Design
In this lecture, we demonstrate how to apply our knowledge of Boolean algebra, expression
minimisation, and basic gates to the design of relatively simple logic circuits. We also
introduce two practical dimensions to the design problem: speed and complexity.
Learning Outcomes:
On completing this lecture, you will be able to:
 design a simple logic circuit from specification to implementation using NAND/NOR
gates;
 investigate the possibility of a multilevel solution as distinct from a two-level sum-ofproducts or product-of-sums formulation;
 compute and assess complexity and speed metrics.
7.1
Logic Design Procedure
We illustrate the steps involved in the design process by means of a worked example.
Generally, a logic design problem begins with a verbal or functional statement of what the
circuit is required to do.
Example 7.1:
A logic circuit has three inputs, labelled X2, X1, and X0. The single
output, labelled F, is required to be 1 if a majority of the inputs are at 1.
X2
X1
F
X0
The first step in the design process is to convert the problem statement into a formal truth
table:
X2
0
0
0
0
1
1
1
1
X1
0
0
1
1
0
0
1
1
X0
0
1
0
1
0
1
0
1
F
0
0
0
1
0
1
1
1
The second step is to determine a simplified a Boolean expression for the required function.
X1
1
X2
1
1
X0
7-1
1
The Karnaugh map step results in the equation
F  X 2 X1  X1 X 0  X 0 X 2
The next step is to give the corresponding logic diagram. Note that with the Boolean
equation in sum-of-products form, the logic diagram comprises two levels of gates: a socalled level of AND gates, one for each product term, and an OR gate to realise the sum.
X2
X1
X1
X0
F
X0
X2
7.2
NAND Gate Constraint
While it is clearly very convenient to implement the Boolean equation using the above
AND/OR logic architecture, we have earlier noted that a better electrical performance might
be derived from the use of NAND gates (or NOR gates). Thus the question arises: how might
we convert the above logic diagram to one based as much as possible on the use of NAND
gates?
Recall from De Morgan’s laws that the NAND function can expressed:
 A.B   A  B 
If we sketch the corresponding logic diagrams for each side of the equation, we find that
there are in effect two equivalent ways of representing the NAND gate:
Now, noting that inserting two series inversions into a logic signal path does not disturb the
overall logic function, we can amend our previous AND/OR logic diagram by adding two
series inversions into each of the signal paths between AND output and OR input. In doing
so, we in effect change each of the gates to a NAND gate.
7-2
X2
X1
X2
X1
X1
X0
X1
X0
F
F
X0
X2
X0
X2
The resulting logic circuit is termed a two-level NAND architecture. We see that any Boolean
equation in sum-of-product form can be implemented with two levels of NAND gates.
Additional inverters may be required if any of the inputs are required to be in complemented
form.
7.3
Multilevel NAND Implementation
The Karnaugh map gives a sum-of-product type of Boolean expression. Sometimes further
grouping of common terms can produce a more simple implementation than that from the
two levels of NAND gates. Consider, by way of example, the following Boolean function
F  ABC  ABD  ABC  ABD
The two level NAND implementation is shown as LCircuit V1.
A
B'
C
LCircuit V1
A
B'
D'
F
A'
B
C
A'
B
D'
Now, re-writing F in the form
F  ABC  D  ABC  D
7-3
LCircuit V2
C'
D
A
B'
F
A'
B
Further re-writing F gives
F   AB  ABC  D
which in turn leads to LCircuit V3
A
Lcircuit V3
B'
A'
B
F
C'
D
The circuits designated LCircuit V2 and V3 are said to be multilevel logic circuits  they
have more than the standard two levels of logic. A visual inspection of the three circuits
would appear to reveal that V2 and V3 are simpler than V1 and that maybe even V3 is more
simple than V2. However, what we clearly now require is a means for quantifying the
complexity of the three designs.
7.4
Complexity/Area
To put a measure on the complexity of a logic circuit what we basically need is an estimate
of the silicon area the circuit will occupy when fabricated as an integrated circuit. Silicon
area is taken up by active devices, ie transistors, and regions of metal required to
interconnect the active devices. For the present discussion we will omit the interconnect area
 although for current integrated circuits interconnect area is becoming significant.
Today’s standard integrated circuit technology is known as CMOS, standing for
Complementary Metal Oxide Semiconductor, a form of field effect transistor. Since CMOS
circuits are based solely on the use of transistors, and most of the transistors are made as
small as the technology permits, a useful indicator of the area occupied is the number of
transistors. For basic logic circuits, the following transistor counts (T) apply:
7-4
2T
or
4T
or
6T
or
8T
For every input, there are two transistors. Hence, comparing the three circuits of the
previous section, we arrive at the following measures for the complexity:
V1:
V2:
V3:
(3 x 2T) + (4 x 6T) + (1 X 8T) = 38T
(3 x 2T) + (1 x 4T) + (2 x 6T) + (1 x 4T) = 26T
(3 x 2T) + (5 x 4T) + (1 x 2T) = 28T
We indeed find that circuits V2 and V3 are simpler than V1 and that there is little difference
between V2 and V3.
7.5
Delay/Speed
Another very important specification for a logic circuit is its propagation delay t p, which in
turn determines what is known as the switching or clock speed  a quantity which is also
frequently attached to a PC as a performance measure.
VH
Vi
Vi
t
Vo
VL
VH
Vo
t
VL
tp
The basic idea behind propagation delay is that, if the input changes state, the output also
changes state with a small but finite delay; it literally takes the internal circuitry time to
7-5
respond to the changing input. If we consider the case of an inverter, we can represent the
input/output situation with the above waveform diagram.
The significance of tp is that, during tp, the logic circuit may be malfunctioning. In the case of
the inverter, during tp, both the input and the output are in the same logic state so that the
element is not functioning as an inverter. Thus, a key rule for a digital system is that
significance must not be attached to a signal output until after the propagation delay has
elapsed, thus delaying the next event in the system.
Specific values for tp depend on the particular CMOS technology. Typical values for today’s
technology would be as follows:
2 ns (ie nano-seconds, 10-9 s)
or
2.5 ns
or
3 ns
or
3.5 ns
Note that the delay increases with the number of inputs. Thus it is preferable to try to use
gates having as few inputs as possible. Integrated circuit engineers do not normally employ
gates having more than four inputs.
Applying these measures to the circuit designs of section 7.3, and noting that delays add up
as we proceed through logic levels, we arrive at the following worst-case delays:
V1:
V2:
V3:
tp = 3.0 + 3.5 = 6.5ns
tp = 2.5 + 3.0 + 2.5 = 8.0ns
tp = 2.5 + 2.5 + 2.5 + 2.0 = 9.5ns
In this particular instance, the most complex circuit turns out to be the fastest (having the
shortest delay). Of the circuits of roughly equal complexity, we see that V3 will have a small
The example illustrates that in the design world, the engineer has to be clear about resolving
conflicting design requirements; very often, the particular application will help in the
resolution.
7-6
7.6
NOR Gate Implementation
Again when confronting a design problem, it is often worthwhile to investigate as many
solutions as possible. Thus far we have explored complexity and speed for NAND-based
circuits. We now ask the question: can we convert a NAND-oriented design to a NOR
implementation and work out its performance measures?
By way of example, we return to the majority circuit of section 7.2 specified by the sum-ofproducts equation
F  X 2 X1  X1 X 0  X 0 X 2
The first step is to convert the equation to product-of-sums format. To do this, we form the
Karnaugh map for the complement of F and minimise to get
X1
1
X2
1
1
1
X0
 
 
 
F   X 2 X1  X1 X 0  X 0 X 2
Complementing this via De Morgan to get back to F yields

 
 
 
F   X 2 X 1  X 1 X 0  X 0 X 2 





 
 
 
  X 2 X 1   X 1 X 0   X 0 X 2 




  X 2  X 1  X 1  X 0  X 0  X 2 
Clearly this can readily be implemented as a two-level OR/AND logic architecture shown
below.
X2
X1
X2
X1
X1
X0
X1
X0
F
X0
X2
X0
X2
7-7
F
Again, De Morgan can be used to generate two alternative but equivalent forms of the NOR
gate:
 A  B   A B 
Thus, adding in a pair of cancelling inverters in the above logic diagram for F allows the
OR/AND architecture to be converted to a NOR only logic circuit. Again multilevel options
can be explored and complexity and propagation delay measures determined for each
design.
7.6
Conclusion
In this lecture, we have illustrated the basic procedure for designing logic circuits: we go
from formulation to truth table, to Karnaugh map, to Boolean equation, to implementation.
We have treated both NAND and NOR forms of implementation, two levels of logic and
multiple levels. We have introduced two quantitative means for comparing competing
designs: complexity and overall propagation delay.
We now go on to examine a number of important standard logic functions.
7-8
```