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3.2.1 The NOT Gate (Inverter)
Figure 3.4 shows the simplest logic structure that exists in a computer. It is con-
structed from two MOS transistors, one P-type and one N-type. Figure 3.4a is the
schematic representation of that circuit. Figure 3.4b shows the behavior of the
circuit if the input is supplied with 0 volts. Note that the P-type transistor acts
like a short circuit and the N-type transistor acts like an open circuit. | [
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] |
4b shows the behavior of the
circuit if the input is supplied with 0 volts. Note that the P-type transistor acts
like a short circuit and the N-type transistor acts like an open circuit. The output
is, therefore, connected to 1.2 volts. On the other hand, if the input is supplied
with 1.2 volts, the P-type transistor acts like an open circuit, but the N-type tran-
sistor acts like a short circuit. | [
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] |
On the other hand, if the input is supplied
with 1.2 volts, the P-type transistor acts like an open circuit, but the N-type tran-
sistor acts like a short circuit. The output in this case is connected to ground (i.e.,
0 volts). The complete behavior of the circuit can be described by means of a
table, as shown in Figure 3.4c. If we replace 0 volts with the symbol 0 and 1.2 | [
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] |
1.2 volts
.
0 volts
1 0
Figure 3.4
A CMOS inverter.
volts with the symbol 1, we have the truth table (Figure 3.4d) for the complement
or NOT function, which we studied in Chapter 2.
In other words, we have just shown how to construct an electronic circuit that
implements the NOT logic function discussed in Chapter 2. We call this circuit
a NOT gate, or an inverter.
3.2. | [
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] |
3.2.2 OR and NOR Gates
Figure 3.5 illustrates a NOR gate. Figure 3.5a is a schematic of a circuit that
implements a NOR gate. It contains two P-type and two N-type transistors.
Figure 3.5b shows the behavior of the circuit if A is supplied with 0 volts
and B is supplied with 1.2 volts. In this case, the lower of the two P-type transis-
tors produces an open circuit, and the output C is disconnected from the 1. | [
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] |
2 volts. In this case, the lower of the two P-type transis-
tors produces an open circuit, and the output C is disconnected from the 1.2-volt
power supply. However, the leftmost N-type transistor acts like a piece of wire,
connecting the output C to 0 volts.
Note that if both A and B are supplied with 0 volts, the two P-type transistors
conduct, and the output C is connected to 1.2 volts. | [
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] |
Note that if both A and B are supplied with 0 volts, the two P-type transistors
conduct, and the output C is connected to 1.2 volts. Note further that there is no
ambiguity here, since both N-type transistors act as open circuits, and so C is
disconnected from ground.
If either A or B is supplied with 1.2 volts, the corresponding P-type transistor
results in an open circuit. That is sucient to break the connection from C to
the 1. | [
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] |
2 volts, the corresponding P-type transistor
results in an open circuit. That is sucient to break the connection from C to
the 1.2-volt source. However, 1.2 volts supplied to the gate of one of the N-type | [
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] |
1
1
0
1.2 volts
0 volts
1.2 volts
A
B
Figure 3.5
The NOR gate.
transistors is sucient to cause that transistor to conduct, resulting in C being
connected to ground (i.e., 0 volts).
Figure 3.5c summarizes the complete behavior of the circuit of Figure 3.5a.
It shows the behavior of the circuit for each of the four pairs of voltages that can
be supplied to A and B. That is,
A = 0 volts,
B = 0 volts
A = 0 volts,
B = 1. | [
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0.0339096374809742,
0.010325380600988865,
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0.009608160704374313,
0.009898926131427288,
-0.012095815502107143,
0.008703559637069702,
-0.01667051576077938
] |
It shows the behavior of the circuit for each of the four pairs of voltages that can
be supplied to A and B. That is,
A = 0 volts,
B = 0 volts
A = 0 volts,
B = 1.2 volts
A = 1.2 volts,
B = 0 volts
A = 1.2 volts,
B = 1.2 volts
If we replace the voltages with their logical equivalents, we have the truth
table of Figure 3.5d. Note that the output C is exactly the opposite of the logical
OR function that we studied in Chapter 2. | [
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] |
5d. Note that the output C is exactly the opposite of the logical
OR function that we studied in Chapter 2. In fact, it is the NOT-OR function,
more typically abbreviated as NOR. We refer to the circuit that implements the
NOR function as a NOR gate.
If we augment the circuit of Figure 3.5a by adding an inverter at its output, as
shown in Figure 3.6a, we have at the output D the logical function OR. | [
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] |
5a by adding an inverter at its output, as
shown in Figure 3.6a, we have at the output D the logical function OR. Figure 3.6a
is the circuit for an OR gate. Figure 3.6b describes the behavior of this circuit if
the input variable A is set to 0 and the input variable B is set to 1. Figure 3.6c
shows the circuits truth table. | [
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] |
64
A
B
1 0 0 1
1 1 0 1
Figure 3.6
The OR gate.
3.2.3 Why We Cant Simply Connect P-Type to Ground
Some bright students have looked at our implementation of the OR gate (a NOR-
gate followed by an inverter) and asked the question, why cant we simply connect
the transistors as shown in Figure 3.7a?
Logically, it looks very tempting. Four transistors instead of six. | [
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] |
7a?
Logically, it looks very tempting. Four transistors instead of six. Unfortu-
nately, the electrical properties of transistors make this problematic. When we
connect a P-type transistor to 1.2 volts or an N-type transistor to ground, there is
no voltage across the transistor, resulting in outputs as shown in Figure 3.5, for
example, of 0 volts or 1.2 volts, depending on the input voltages to A and B. | [
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] |
5, for
example, of 0 volts or 1.2 volts, depending on the input voltages to A and B. How-
ever, when we connect a P-type transistor to ground or an N-type transistor to 1.2
volts, because of the electrical characteristics of the transistors, we get what is
usually referred to as a transmission voltage of approximately 0.5 volts across the
transistor. This results in the output of the transistor circuit of Figure 3. | [
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] |
5 volts across the
transistor. This results in the output of the transistor circuit of Figure 3.7 being
0.5 volts + 0.5 volts, or 1.0 volt if A and B are both 0, and 0.7 volts (1.2 volts
minus 0.5 volts) otherwise. Figure 3.7b shows the actual voltages in the resulting
truth table, rather than 0s and 1s. That is, even though the transistor circuit looks
like it would work, the transmission voltages across the transistors would yield
an output voltage of 1 volt for a logical 0 and 0. | [
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] |
7b shows the actual voltages in the resulting
truth table, rather than 0s and 1s. That is, even though the transistor circuit looks
like it would work, the transmission voltages across the transistors would yield
an output voltage of 1 volt for a logical 0 and 0.7 volts for a logical 1. Not what
we would like for an OR gate! | [
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] |
Figure 3.7
An OR gate (not really!).
3.2.4 AND and NAND Gates
Figure 3.8 shows an AND gate. Note that if either A or B is supplied with 0 volts,
there is a direct connection from C to the 1.2-volt power supply. The fact that C
Figure 3.8
The AND gate. | [
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] |
(e) NOR gate
(d) NAND gate
Figure 3.9
Basic logic gates.
Again, we note that there is no ambiguity. The fact that at least one of the two
inputs A or B is supplied with 0 volts means that at least one of the two N-type
transistors whose gates are connected to A or B is open, and that consequently, C
is disconnected from ground. Furthermore, the fact that C is at 1.2 volts means
the P-type transistor whose gate is connected to C is open-circuited. | [
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] |
Furthermore, the fact that C is at 1.2 volts means
the P-type transistor whose gate is connected to C is open-circuited. Therefore,
D is not connected to 1.2 volts.
On the other hand, if both A and B are supplied with 1.2 volts, then both
of their corresponding P-type transistors are open. However, their corresponding
N-type transistors act like pieces of wire, providing a direct connection from C
to ground. | [
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] |
2 volts, then both
of their corresponding P-type transistors are open. However, their corresponding
N-type transistors act like pieces of wire, providing a direct connection from C
to ground. Because C is at ground, the rightmost P-type transistor acts like a
closed circuit, forcing D to 1.2 volts.
Figure 3.8b is a truth table that summarizes the behavior of the circuit of
Figure 3.8a. | [
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] |
8b is a truth table that summarizes the behavior of the circuit of
Figure 3.8a. Note that the circuit is an AND gate. The circuit shown within the
dashed lines (i.e., having output C) is a NOT-AND gate, which we generally
abbreviate as NAND.
The gates just discussed are very common in digital logic circuits and in
digital computers. There are billions of inverters (NOT gates) in Intels Skylake
microprocessor. | [
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] |
The gates just discussed are very common in digital logic circuits and in
digital computers. There are billions of inverters (NOT gates) in Intels Skylake
microprocessor. As a convenience, we can represent each of these gates by stan-
dard symbols, as shown in Figure 3.9. The bubble shown in the inverter, NAND,
and NOR gates signies the complement (i.e., NOT) function. From now on, we
will not draw circuits showing the individual transistors. | [
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] |
, NOT) function. From now on, we
will not draw circuits showing the individual transistors. Instead, we will raise
our level of abstraction and use the symbols shown in Figure 3.9.
3.2.5 Gates with More Than Two Inputs
Before we leave the topic of logic gates, we should note that the notion of AND,
OR, NAND, and NOR gates extends to larger numbers of inputs. One could build
a three-input AND gate or a four-input OR gate, for example. | [
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5 Gates with More Than Two Inputs
Before we leave the topic of logic gates, we should note that the notion of AND,
OR, NAND, and NOR gates extends to larger numbers of inputs. One could build
a three-input AND gate or a four-input OR gate, for example. An n-input AND
gate has an output value of 1 only if ALL n input variables have values of 1. If
any of the n inputs has a value of 0, the output of the n-input AND gate is 0. | [
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] |
An n-input AND
gate has an output value of 1 only if ALL n input variables have values of 1. If
any of the n inputs has a value of 0, the output of the n-input AND gate is 0. An
n-input OR gate has an output value of 1 if ANY of the n input variables has a
value of 1. That is, an n-input OR gate has an output value of 0 only if ALL n
input variables have values of 0. | [
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] |
1 1 0 0
1 1 1 1
Figure 3.10
A three-input AND gate.
Figure 3.10 illustrates a three-input AND gate. Figure 3.10a shows its truth
table. Figure 3.10b shows the symbol for a three-input AND gate.
Question: Can you draw a transistor-level circuit for a three-input AND gate?
How about a four-input AND gate? How about a four-input OR gate?
3.3 Combinational Logic Circuits
Now that we understand the workings of the basic logic gates, the next step
is to build some of the logic structures that are important components of the
microarchitecture of a computer. | [
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] |
Question: Can you draw a transistor-level circuit for a three-input AND gate?
How about a four-input AND gate? How about a four-input OR gate?
3.3 Combinational Logic Circuits
Now that we understand the workings of the basic logic gates, the next step
is to build some of the logic structures that are important components of the
microarchitecture of a computer.
There are fundamentally two kinds of logic structures, those that include the
storage of information and those that do not. | [
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] |
3 Combinational Logic Circuits
Now that we understand the workings of the basic logic gates, the next step
is to build some of the logic structures that are important components of the
microarchitecture of a computer.
There are fundamentally two kinds of logic structures, those that include the
storage of information and those that do not. In Sections 3.4, 3.5, and 3.6, we
will deal with structures that store information. | [
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] |
5, and 3.6, we
will deal with structures that store information. In this section, we will deal with
structures that do not store information. These structures are sometimes referred
to as decision elements. Usually, they are referred to as combinational logic struc-
tures because their outputs are strictly dependent on the combination of input
values that are being applied to the structure right now. | [
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] |
These structures are sometimes referred
to as decision elements. Usually, they are referred to as combinational logic struc-
tures because their outputs are strictly dependent on the combination of input
values that are being applied to the structure right now. Their outputs are not at
all dependent on any past history of information that is stored internally, since no
information can be stored internally in a combinational logic circuit. | [
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] |
Usually, they are referred to as combinational logic struc-
tures because their outputs are strictly dependent on the combination of input
values that are being applied to the structure right now. Their outputs are not at
all dependent on any past history of information that is stored internally, since no
information can be stored internally in a combinational logic circuit.
We will next examine three useful combinational logic circuits: a decoder, a
mux, and a one-bit adder. | [
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] |
Their outputs are not at
all dependent on any past history of information that is stored internally, since no
information can be stored internally in a combinational logic circuit.
We will next examine three useful combinational logic circuits: a decoder, a
mux, and a one-bit adder.
3.3.1 Decoder
Figure 3.11 shows a logic gate implementation of a two-input decoder. A decoder
has the property that exactly one of its outputs is 1 and all the rest are 0s. | [
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11 shows a logic gate implementation of a two-input decoder. A decoder
has the property that exactly one of its outputs is 1 and all the rest are 0s. The one
output that is logically 1 is the output corresponding to the input pattern that it is
expected to detect. In general, decoders have n inputs and 2n outputs. We say the
output line that detects the input pattern is asserted. That is, that output line has
the value 1, rather than 0 as is the case for all the other output lines. | [
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] |
We say the
output line that detects the input pattern is asserted. That is, that output line has
the value 1, rather than 0 as is the case for all the other output lines. In Figure 3.11,
note that for each of the four possible combinations of inputs A and B, exactly one
output has the value 1 at any one time. In Figure 3.11b, the input to the decoder
is 10, resulting in the third output line being asserted. | [
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] |
In Figure 3.11b, the input to the decoder
is 10, resulting in the third output line being asserted. | [
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] |
A
B
1, if A, B is 11
0
Figure 3.11
A two-input decoder.
The decoder is useful in determining how to interpret a bit pattern. We will
see in Chapter 5 that the work to be carried out by each instruction in the LC-
3 computer is determined by a four-bit pattern that is the part of the instruction
called the opcode, A 4-to-16 decoder is a simple combinational logic structure
for identifying what work is to be performed by each instruction. | [
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] |
The decoder is useful in determining how to interpret a bit pattern. We will
see in Chapter 5 that the work to be carried out by each instruction in the LC-
3 computer is determined by a four-bit pattern that is the part of the instruction
called the opcode, A 4-to-16 decoder is a simple combinational logic structure
for identifying what work is to be performed by each instruction.
3. | [
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] |
We will
see in Chapter 5 that the work to be carried out by each instruction in the LC-
3 computer is determined by a four-bit pattern that is the part of the instruction
called the opcode, A 4-to-16 decoder is a simple combinational logic structure
for identifying what work is to be performed by each instruction.
3.3.2 Mux
Figure 3.12a shows a logic gate implementation of a two-input multiplexer, more
Figure 3. | [
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] |
2 Mux
Figure 3.12a shows a logic gate implementation of a two-input multiplexer, more
Figure 3.12
A 2-to-1 mux. | [
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OUT
Figure 3.13
A four-input mux.
The mux of Figure 3.12 works as follows: Suppose S = 0, as shown in
Figure 3.12b. Since the output of an AND gate is 0 unless all inputs are 1, the out-
put of the rightmost AND gate is 0. Also, the output of the leftmost AND gate is
whatever the input A is. That is, if A = 0, then the output of the leftmost AND gate
is 0, and if A = 1, then the output of the leftmost AND gate is 1. | [
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] |
Also, the output of the leftmost AND gate is
whatever the input A is. That is, if A = 0, then the output of the leftmost AND gate
is 0, and if A = 1, then the output of the leftmost AND gate is 1. Since the output
of the rightmost AND gate is 0, it has no eect on the OR gate. Consequently,
the output at C is exactly the same as the output of the leftmost AND gate. The
net result of all this is that if S = 0, the output C is identical to the input A. | [
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] |
Consequently,
the output at C is exactly the same as the output of the leftmost AND gate. The
net result of all this is that if S = 0, the output C is identical to the input A.
On the other hand, if S = 1, it is B that is ANDed with 1, resulting in the
output of the OR gate having the value of B.
In summary, the output C is always connected to either the input A or the
input Bwhich one depends on the value of the select line S. | [
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] |
On the other hand, if S = 1, it is B that is ANDed with 1, resulting in the
output of the OR gate having the value of B.
In summary, the output C is always connected to either the input A or the
input Bwhich one depends on the value of the select line S. We say S selects the
source of the mux (either A or B) to be routed through to the output C. Figure 3.12c
shows the standard representation for a mux. | [
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] |
Figure 3.12c
shows the standard representation for a mux.
In general, a mux consists of 2n inputs and n select lines. Figure 3.13a
shows a gate-level description of a four-input mux. It requires two select lines.
Figure 3.13b shows the standard representation for a four-input mux.
Question: Can you construct the gate-level representation for an eight-input
mux? How many select lines must you have?
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13b shows the standard representation for a four-input mux.
Question: Can you construct the gate-level representation for an eight-input
mux? How many select lines must you have?
3.3.3 A One-Bit Adder (a.k.a. a Full Adder)
Recall in Chapter 2, we discussed binary addition. A simple algorithm for binary
addition is to proceed as you have always done in the case of decimal addition,
from right to left, one column at a time, adding the two digits from the two values
plus the carry in, and generating a sum digit and a carry to the next column. | [
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a Full Adder)
Recall in Chapter 2, we discussed binary addition. A simple algorithm for binary
addition is to proceed as you have always done in the case of decimal addition,
from right to left, one column at a time, adding the two digits from the two values
plus the carry in, and generating a sum digit and a carry to the next column. The
only dierence here (with binary addition) is you get a carry after 1, rather than
after 9. | [
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] |
A simple algorithm for binary
addition is to proceed as you have always done in the case of decimal addition,
from right to left, one column at a time, adding the two digits from the two values
plus the carry in, and generating a sum digit and a carry to the next column. The
only dierence here (with binary addition) is you get a carry after 1, rather than
after 9.
Figure 3.14 is a truth table that describes the result of binary addition on one
column of bits within two n-bit operands. | [
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] |
Figure 3.14 is a truth table that describes the result of binary addition on one
column of bits within two n-bit operands. At each column, there are three values | [
-0.012608361430466175,
0.005596224218606949,
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0.018083222210407257,
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] |
1
1
1
1
1
Figure 3.14
The truth table for a one-bit adder.
that must be added: one bit from each of the two operands A and B and the carry
from the previous column. We designate these three bits as Ai, Bi, and Ci. There
are two results, the sum bit (Si) and the carry over to the next column, Ci+1. Note
that if only one of the three bits equals 1, we get a sum of 1, and no carry (i.e. | [
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] |
Note
that if only one of the three bits equals 1, we get a sum of 1, and no carry (i.e.,
Ci+1 = 0). If two of the three bits equal 1, we get a sum of 0, and a carry of 1. If
all three bits equal 1, the sum is 3, which in binary corresponds to a sum of 1 and
a carry of 1.
Figure 3.15 shows a logic gate implementation of a one-bit adder. Note that
each AND gate in Figure 3.15 produces an output 1 for exactly one of the eight
input combinations of Ai, Bi, and Ci. | [
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] |
Note that
each AND gate in Figure 3.15 produces an output 1 for exactly one of the eight
input combinations of Ai, Bi, and Ci. The output of the OR gate for Ci+1 must be 1
in e
pro
Figure 3.15
Gate-level description of a one-bit adder. | [
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] |
S0
S1
S2
S3
Figure 3.16
A circuit for adding two 4-bit binary numbers.
are the outputs of the AND gates corresponding to those input combinations.
Similarly, the inputs to the OR gate that generates Si are the outputs of the AND
gates corresponding to the input combinations that require an output 1 for Si in
the truth table of Figure 3.14.
Note that since the input combination 000 does not result in an output 1 for
either Ci+1 or Si, its corresponding AND gate is not an input to either of the two
OR gates. | [
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] |
14.
Note that since the input combination 000 does not result in an output 1 for
either Ci+1 or Si, its corresponding AND gate is not an input to either of the two
OR gates.
Figure 3.16 shows a circuit for adding two 4-bit binary numbers, using four
of the one-bit adder circuits of Figure 3.15. Note that the carry out of column i is
an input to the addition performed in column i + 1. | [
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] |
15. Note that the carry out of column i is
an input to the addition performed in column i + 1.
If we wish to implement a logic circuit that adds two 16-bit numbers, we can
do so with a circuit of 16 one-bit adders.
We should point out that historically the logic circuit of Figure 3.15 that
provides three inputs (Ai, Bi, and Ci) and two outputs (the sum bit Si and the
carry over to the next column Ci+1) has generally been referred to as a full adder
to dierentiate it from another structure, which is called a half adder. | [
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] |
We should point out that historically the logic circuit of Figure 3.15 that
provides three inputs (Ai, Bi, and Ci) and two outputs (the sum bit Si and the
carry over to the next column Ci+1) has generally been referred to as a full adder
to dierentiate it from another structure, which is called a half adder. The dis-
tinction between the two is the carry bit. Note that the carry into the rightmost
column in Figure 3. | [
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] |
The dis-
tinction between the two is the carry bit. Note that the carry into the rightmost
column in Figure 3.16 is 0. That is, in the rightmost circuit, S0 and C1 depend only
on two inputs, A0 and B0. Since that circuit depends on only two inputs, it has been
referred to as a half adder. Since the other circuits depend on all three inputs, they
are referred to as full adders. We prefer the term one-bit adder as a simpler term
for describing what is happening in each column. | [
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] |
Since the other circuits depend on all three inputs, they
are referred to as full adders. We prefer the term one-bit adder as a simpler term
for describing what is happening in each column.
3.3.4 The Programmable Logic Array (PLA)
Figure 3.17 illustrates a very common building block for implementing any col-
lection of logic functions one wishes to implement. The building block is called
a programmable logic array (PLA). | [
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] |
17 illustrates a very common building block for implementing any col-
lection of logic functions one wishes to implement. The building block is called
a programmable logic array (PLA). It consists of an array of AND gates (called
an AND array) followed by an array of OR gates (called an OR array). The num-
ber of AND gates corresponds to the number of input combinations (rows) in
the truth table. | [
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] |
It consists of an array of AND gates (called
an AND array) followed by an array of OR gates (called an OR array). The num-
ber of AND gates corresponds to the number of input combinations (rows) in
the truth table. For n-input logic functions, we need a PLA with 2n n-input AND | [
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72
Figure 3.17
A programmable logic array.
gates. In Figure 3.17, we have 23 three-input AND gates, corresponding to three
logical input variables. The number of OR gates corresponds to the number of
logic functions we wish to implement, that is, the number of output columns in
the truth table. The implementation algorithm is simply to connect the output of
an AND gate to the input of an OR gate if the corresponding row of the truth table
produces an output 1 for that output column. | [
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The number of OR gates corresponds to the number of
logic functions we wish to implement, that is, the number of output columns in
the truth table. The implementation algorithm is simply to connect the output of
an AND gate to the input of an OR gate if the corresponding row of the truth table
produces an output 1 for that output column. Hence the notion of programmable.
That is, we say we program the connections from AND gate outputs to OR gate
inputs to implement our desired logic functions. | [
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] |
Hence the notion of programmable.
That is, we say we program the connections from AND gate outputs to OR gate
inputs to implement our desired logic functions.
Figure 3.15 shows seven AND gates connected to two OR gates since our
requirement was to implement two functions (sum and carry) of three input vari-
ables. Figure 3.17 shows a PLA that can implement any four functions of three
variables by appropriately connecting AND gate outputs to OR gate inputs. | [
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] |
Figure 3.17 shows a PLA that can implement any four functions of three
variables by appropriately connecting AND gate outputs to OR gate inputs. That
is, any function of three variables can be implemented by connecting the outputs
of all AND gates corresponding to input combinations for which the output is 1
to inputs of one of the OR gates. Thus, we could implement the one-bit adder by
programming the two OR gates in Figure 3. | [
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] |
That
is, any function of three variables can be implemented by connecting the outputs
of all AND gates corresponding to input combinations for which the output is 1
to inputs of one of the OR gates. Thus, we could implement the one-bit adder by
programming the two OR gates in Figure 3.17 whose outputs are W and X by
connecting or not connecting the outputs of the AND gates to the inputs of those
two OR gates as specied by the two output columns of Figure 3. | [
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] |
Thus, we could implement the one-bit adder by
programming the two OR gates in Figure 3.17 whose outputs are W and X by
connecting or not connecting the outputs of the AND gates to the inputs of those
two OR gates as specied by the two output columns of Figure 3.14.
3.3.5 Logical Completeness
Before we leave the topic of combinational logic circuits, it is worth noting an
important property of building blocks for logic circuits: logical completeness. | [
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] |
3.5 Logical Completeness
Before we leave the topic of combinational logic circuits, it is worth noting an
important property of building blocks for logic circuits: logical completeness. We
showed in Section 3.3.4 that any logic function we wished to implement could
be accomplished with a PLA. We saw that the PLA consists of only AND gates,
OR gates, and inverters. That means that any logic function can be implemented,
provided that enough AND, OR, and NOT gates are available. | [
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] |
We saw that the PLA consists of only AND gates,
OR gates, and inverters. That means that any logic function can be implemented,
provided that enough AND, OR, and NOT gates are available. We say that the set
of gates {AND, OR, NOT} is logically complete because we can build a circuit
to carry out the specication of any truth table we wish without using any other | [
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kind of gate. That is, the set of gates {AND, OR, and NOT} is logically complete
because a barrel of AND gates, a barrel of OR gates, and a barrel of NOT gates are
sucient to build a logic circuit that carries out the specication of any desired
truth table. The barrels may have to be big, but the point is, we do not need any
other kind of gate to do the job.
Question: Is there any single two-input logic gate that is logically complete?
For example, is the NAND gate logically complete? Hint: Can I implement a
NOT gate with a NAND gate? If yes, can I then implement an AND gate using a
NAND gate followed by a NOT gate? If yes, can I implement an OR gate using
just AND gates and NOT gates?
If all of the above is true, then the NAND gate is logically complete, and I
can implement any desired logic function as described by its truth table with a
barrel of NAND gates. | [
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] |
The barrels may have to be big, but the point is, we do not need any
other kind of gate to do the job.
Question: Is there any single two-input logic gate that is logically complete?
For example, is the NAND gate logically complete? Hint: Can I implement a
NOT gate with a NAND gate? If yes, can I then implement an AND gate using a
NAND gate followed by a NOT gate? If yes, can I implement an OR gate using
just AND gates and NOT gates?
If all of the above is true, then the NAND gate is logically complete, and I
can implement any desired logic function as described by its truth table with a
barrel of NAND gates.
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] |
Question: Is there any single two-input logic gate that is logically complete?
For example, is the NAND gate logically complete? Hint: Can I implement a
NOT gate with a NAND gate? If yes, can I then implement an AND gate using a
NAND gate followed by a NOT gate? If yes, can I implement an OR gate using
just AND gates and NOT gates?
If all of the above is true, then the NAND gate is logically complete, and I
can implement any desired logic function as described by its truth table with a
barrel of NAND gates.
3.4 Basic Storage Elements
Recall our statement at the beginning of Section 3. | [
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] |
3.4 Basic Storage Elements
Recall our statement at the beginning of Section 3.3 that there are two kinds of
logic structures, those that involve the storage of information and those that do
not. We have discussed three examples of those that do not: the decoder, the mux,
and the full adder. Now we are ready to discuss logic structures that do include
the storage of information.
3.4. | [
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] |
3.4.1 The R-S Latch
A simple example of a storage element is the R-S latch. It can store one bit of
information, a 0 or a 1. The R-S latch can be implemented in many ways, the
simplest being the one shown in Figure 3.18. Two 2-input NAND gates are con-
nected such that the output of each is connected to one of the inputs of the other.
The remaining inputs S and R are normally held at a logic level 1. | [
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] |
Two 2-input NAND gates are con-
nected such that the output of each is connected to one of the inputs of the other.
The remaining inputs S and R are normally held at a logic level 1.
The R-S latch
ets its name from the old desi nations for settin
the latch
R
b
Figure 3.18
An R-S latch.
The Quiescent State
We describe the quiescent (or quiet) state of a latch as
the state when the latch is storing a value, either 0 or 1, and nothing is trying to. | [
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] |
18
An R-S latch.
The Quiescent State
We describe the quiescent (or quiet) state of a latch as
the state when the latch is storing a value, either 0 or 1, and nothing is trying to | [
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change that value. This is the case when inputs S and R both have the logic value
1. In Figure 3.18 the letter a designates the value that is currently stored in the
latch, which we also refer to as the output of the latch.
Consider rst the case where the value stored and therefore the output a is
1. Since that means the value A is 1 (and since we know the input R is 1 because
we are in the quiescent state), the NAND gates output b must be 0. | [
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] |
Consider rst the case where the value stored and therefore the output a is
1. Since that means the value A is 1 (and since we know the input R is 1 because
we are in the quiescent state), the NAND gates output b must be 0. That, in turn,
means B must be 0, which results in the output a equal to 1. As long as the inputs
S and R remain 1, the state of the circuit will not change. That is, the R-S latch
will continue to store the value 1 (the value of the output a). | [
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] |
As long as the inputs
S and R remain 1, the state of the circuit will not change. That is, the R-S latch
will continue to store the value 1 (the value of the output a).
If, on the other hand, we assume the output a is 0, then A must be 0, and
the out ut b must be 1. That, in turn, results in B equal to 1, and combined with
t S equal to 1 (again due to quiescence), results in the output a equal to
n, as long as the inputs S and R remain 1, the state of the circuit will not. | [
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] |
If, on the other hand, we assume the output a is 0, then A must be 0, and
the out ut b must be 1. That, in turn, results in B equal to 1, and combined with
t S equal to 1 (again due to quiescence), results in the output a equal to
n, as long as the inputs S and R remain 1, the state of the circuit will not
. In this case, we say the R-S latch stores the value 0.
the Latch to a 1 or a 0
The latch can be set to 1 by momentarily setting
provided we keep the value of R at 1. | [
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] |
In this case, we say the R-S latch stores the value 0.
the Latch to a 1 or a 0
The latch can be set to 1 by momentarily setting
provided we keep the value of R at 1. Similarly, the latch can be set to 0
entarily setting R to 0, provided we keep the value of S at 1. In order for
the R-S latch to work properly, both S and R must never be allowed to be set to 0
at the same time.
We use the term set to denote setting a variable to 0 or 1, as in set to 0 or
set to 1. | [
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] |
In order for
the R-S latch to work properly, both S and R must never be allowed to be set to 0
at the same time.
We use the term set to denote setting a variable to 0 or 1, as in set to 0 or
set to 1. In addition, we often use the term clear to denote the act of setting a
variable to 0.
If we set S to 0 for a very brief period of time, this causes a to equal 1, which
in turn causes A to equal 1. | [
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] |
In addition, we often use the term clear to denote the act of setting a
variable to 0.
If we set S to 0 for a very brief period of time, this causes a to equal 1, which
in turn causes A to equal 1. Since R is also 1, the output at b must be 0. This causes
B to be 0, which in turn makes a equal to 1. If, after that very brief period of time,
we now return S to 1, it does not aect a. Why? Answer: Since B is also 0, and
since only one input 0 to a NAND gate is enough to guarantee that the output of
the NAND gate is 1, the latch will continue to store a 1 long after S returns to 1. | [
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If, after that very brief period of time,
we now return S to 1, it does not aect a. Why? Answer: Since B is also 0, and
since only one input 0 to a NAND gate is enough to guarantee that the output of
the NAND gate is 1, the latch will continue to store a 1 long after S returns to 1.
In the same way, we can clear the latch (set the latch to 0) by setting R to 0
for a very short period of time. | [
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] |
Why? Answer: Since B is also 0, and
since only one input 0 to a NAND gate is enough to guarantee that the output of
the NAND gate is 1, the latch will continue to store a 1 long after S returns to 1.
In the same way, we can clear the latch (set the latch to 0) by setting R to 0
for a very short period of time.
We should point out that if both S and R were allowed to be set to 0 at the same
time, the outputs a and b would both be 1, and the nal state of the latch would
depend on the electrical properties of the transistors making up the gates and
not on the logic being performed. | [
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] |
In the same way, we can clear the latch (set the latch to 0) by setting R to 0
for a very short period of time.
We should point out that if both S and R were allowed to be set to 0 at the same
time, the outputs a and b would both be 1, and the nal state of the latch would
depend on the electrical properties of the transistors making up the gates and
not on the logic being performed. How the electrical properties of the transistors
would determine the nal state in this case is a subject we will have to leave for
a later semester. | [
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] |
We should point out that if both S and R were allowed to be set to 0 at the same
time, the outputs a and b would both be 1, and the nal state of the latch would
depend on the electrical properties of the transistors making up the gates and
not on the logic being performed. How the electrical properties of the transistors
would determine the nal state in this case is a subject we will have to leave for
a later semester | [
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] |
We should point out that if both S and R were allowed to be set to 0 at the same
time, the outputs a and b would both be 1, and the nal state of the latch would
depend on the electrical properties of the transistors making up the gates and
not on the logic being performed. How the electrical properties of the transistors
would determine the nal state in this case is a subject we will have to leave for
a later semester. :-(
Finally, we should note that when a digital circuit is powered on, the latch
can be in either of its two states, 0 or 1. | [
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] |
How the electrical properties of the transistors
would determine the nal state in this case is a subject we will have to leave for
a later semester. :-(
Finally, we should note that when a digital circuit is powered on, the latch
can be in either of its two states, 0 or 1. It does not matter which state since we
never use that information until after we have set it to 1 or 0.
3.4.2 The Gated D Latch
To be useful, it is necessary to control when a latch is set and when it is cleared. | [
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] |
4.2 The Gated D Latch
To be useful, it is necessary to control when a latch is set and when it is cleared.
A simple way to accomplish this is with the gated latch.
Figure 3.19 shows a logic circuit that implements a gated D latch. It consists
of the R-S latch of Figure 3.18, plus two additional NAND gates that allow the | [
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] |
R
Figure 3.19
A gated D latch.
latch to be set to the value of D, but only when WE is asserted (i.e., when WE
equals 1). WE stands for write enable. When WE is not asserted (i.e., when WE
equals 0), the outputs S and R are both equal to 1. Since S and R are inputs to the
R-S latch, if they are kept at 1, the value stored in the latch remains unchanged,
as we explained in Section 3.4. | [
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Since S and R are inputs to the
R-S latch, if they are kept at 1, the value stored in the latch remains unchanged,
as we explained in Section 3.4.1. When WE is momentarily set to 1, exactly one
of the outputs S or R is set to 0, depending on the value of D. If D equals 1, then S
is set to 0. If D equals 0, then both inputs to the lower NAND gate are 1, resulting
in R being set to 0. | [
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] |
If D equals 1, then S
is set to 0. If D equals 0, then both inputs to the lower NAND gate are 1, resulting
in R being set to 0. As we saw earlier, if S is set to 0, the R-S latch is set to 1. If R
is set to 0, the R-S latch is set to 0. Thus, the R-S latch is set to 1 or 0 according
to whether D is 1 or 0. When WE returns to 0, S and R return to 1, and the value
stored in the R-S latch persists. | [
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] |
Thus, the R-S latch is set to 1 or 0 according
to whether D is 1 or 0. When WE returns to 0, S and R return to 1, and the value
stored in the R-S latch persists.
3.5 The Concept of Memory
We now have all the tools we need to describe one of the most important struc-
tures in the electronic digital computer, its memory. We will see in Chapter 4
how memory ts into the basic scheme of computer processing, and you will see
throughout the rest of the book and indeed the rest of your work with computers
how important the concept of memory is to computing. | [
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] |
5 The Concept of Memory
We now have all the tools we need to describe one of the most important struc-
tures in the electronic digital computer, its memory. We will see in Chapter 4
how memory ts into the basic scheme of computer processing, and you will see
throughout the rest of the book and indeed the rest of your work with computers
how important the concept of memory is to computing.
Memory is made up of a (usually large) number of locations, each uniquely
identiable and each having the ability to store a value. | [
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] |
We will see in Chapter 4
how memory ts into the basic scheme of computer processing, and you will see
throughout the rest of the book and indeed the rest of your work with computers
how important the concept of memory is to computing.
Memory is made up of a (usually large) number of locations, each uniquely
identiable and each having the ability to store a value. We refer to the unique
identier associated with each memory location as its address. | [
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] |
Memory is made up of a (usually large) number of locations, each uniquely
identiable and each having the ability to store a value. We refer to the unique
identier associated with each memory location as its address. We refer to the
number of bits of information stored in each location as its addressability.
For example, an advertisement for a laptop computer might say, This com-
puter comes with 2 gigabytes of memory. | [
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] |
We refer to the
number of bits of information stored in each location as its addressability.
For example, an advertisement for a laptop computer might say, This com-
puter comes with 2 gigabytes of memory. Actually, most ads generally use
the abbreviation 2 GB (or, often: 2 Gig). This statement means, as we will
explain momentarily, that the laptop includes two billion memory locations, each
containing one byte of information. | [
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Actually, most ads generally use
the abbreviation 2 GB (or, often: 2 Gig). This statement means, as we will
explain momentarily, that the laptop includes two billion memory locations, each
containing one byte of information.
3.5.1 Address Space
We refer to the total number of uniquely identiable locations as the memorys
address space. A 2 GB memory, for example, refers to a memory that consists of
two billion uniquely identiable memory locations. | [
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] |
1 Address Space
We refer to the total number of uniquely identiable locations as the memorys
address space. A 2 GB memory, for example, refers to a memory that consists of
two billion uniquely identiable memory locations. | [
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] |
Actually, the number two billion is only an approximation, due to the way we
specify memory locations. Since everything else in the computer is represented
by sequences of 0s and 1s, it should not be surprising that memory locations are
identied by binary addresses as well. With n bits of address, we can uniquely
identify 2n locations. Ten bits provide 1024 locations, which is approximately
1000. | [
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