Reducing Boolean Expressions:
As all the logic operators represent a corresponding element of hardware, the designer must reduce all the Boolean equations to a simplified form for minimizing cost.
The following procedure can be employed, in general, for such reduction:
i. Multiply all the variables required to remove parentheses.
ii. Look for the identical terms. One of these can be dropped using equation (5.12).
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A.A = 0
iii. Look for a variable and its negation in the same term. This term can be dropped.
As for example,
BBC = O.C = 0.
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iv. Look for pairs of terms which are identical except for one variable.
AND Circuit and its Truth Table:
An AND circuit is shown in Fig. 5.16(a) while Fig. 5.16(b) gives its symbolic representation. This is a simple logic circuit using diodes. In such a circuit both the inputs must be at an UP level to produce an UP level at the output. In the figure, D1 and D2 are two diodes, each receives an independent input signal voltage (+20 V or – 20 V) through S1 and S2 respectively.
To understand the operation of an AND circuit let us consider first the simple circuit diagram of Fig. 5.17. As shown, when the switch S1 is in -20 V position, circuit is completed from the -20 V source through diode D1 and resistor RL to the + 100 V source. Under such condition, since D1 is forward biased it conducts.
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As the forward resistance of D1 is low compared with the load resistance, a negligible voltage drop appears across D1. Therefore, with respect to common ground the output level exhibits -20 V. Similarly, when the switch is placed in the +20 V position, output voltage exhibits + 20 V.
The use of one diode in a circuit is therefore just to transfer either input level to the output. That is why with only one diode and one input, the circuit has little value as a logic circuit. When the second diode and input are included as in Fig. 5.16(a), a real AND logic circuit is formed.
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Let us now consider the circuit of Fig. 5.16(a). When both S1 and S2 are at -20 V position, the output is – 20 V. But if S1 is placed in the + 20 V position and S2 in the – 20 V position, the output remains at – 20 V, since the diode D2 has a – 20 V level on its cathode. Diode D1 is cut off because its cathode is + 20 V and is more positive than its anode. D1 is thus reverse biased. If next the switches S1 and S2 are both set in + 20V position, the output gives + 20V.
From the measurement of the output of an AND circuit we can, therefore, draw the following logical conclusions:
i. If the output is UP, then both the inputs must be UP.
ii. If the output is DOWN, then either or both the inputs must be DOWN.
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Usually the logic of digital circuits is given in the form of a truth table. A digital UP state is represented in convention by the binary 1 while a DOWN state by the binary number 0. All the necessary conditions in a two-input AND circuit are then given in Table 5.3.
OR Circuit and its Truth Table:
The OR circuit is another logic circuit. Such a circuit provides an UP level at the output when any one of its inputs has an UP level. It is, however, similar to the AND circuit with a difference that the polarity of the diodes is reversed and the load resistance RL is connected to the negative voltage source. Fig. 5.18(a) shows an OR circuit while Fig. 5.18(b) is its symbolic representation.
To understand the operation of an OR circuit, for the time being, let us consider that only the diode D1 is present in the circuit of Fig. 5.18(a). When the input to D1 is – 20 V, the diode becomes forward biased since the anode is less negative than the cathode which is connected through the load resistor to the – 100 V source.
The diode D1 thus conducts and it acts as a low resistance switch. A very small voltage therefore drops across D1 as compared to that across RL. The output level so exhibits – 20 V. Similarly, when the switch S1 is placed to + 20 V position, the output will exhibit + 20 V.
Now, if both the diodes are connected and the inputs are at – 20 V position, the output is – 20 V. But if either D1 or D2 has + 20 V on its input, the output is + 20 V.
From the measurement of the output of an OR circuit we can, therefore, draw the following logical conclusions:
i. If the output is DOWN, then both the inputs must be DOWN.
ii. If the output is UP, then either or both the inputs must be UP.
Truth Table:
All the possible conditions in a two-input OR circuit are given in Table 5.4.
Combined AND-OR Circuits:
In computer technology sometimes a combination of AND and OR circuits is used to perform complex logical operations. A simple example of it in the form of block diagram is shown in Fig. 5.19. Analysis shows that an UP level on inputs 1 and 2 will produce an UP level, on input 3 alone produces an UP level at the output.
AND and OR Gates:
AND and OR circuits are also known as Gates. In digital electronics, a gate is a circuit which has two or more inputs and one output. The nature of the gate and input combinations determine the output, i.e., either UP (1) or DOWN (0).
NOT Circuit:
A NOT circuit is nothing but an inverter, mathematically the characteristic of which is given in equation (5.22).
The bar over A represents NOT. Hence, if the letter A̅ represents DOWN level (0), A represents UP (1) and if A = 1, A̅ = 0, or in other words, 0̅ = 1̅ and 1 = 0.
Fig. 5.20 shows a NOT circuit and its symbolic representation. In the ground emitter configuration in Fig. 5.20(a), the output is taken from the collector. When no signal is applied at the input, the transistor is cut off and the output is at VCC, i.e.; it is UP.
On the other hand, when a positive pulse is applied to the base, the transistor conducts and the collector voltage drops. Thus, for a positive input, the output goes DOWN. We can, therefore, conclude that in a NOT circuit the output is present when an input is not applied. Similarly, an output signal is not present when an input signal is applied.
NAND Gate and its Truth Table:
The NAND gate is a combination of NOT and AND functions. A two-input NAND gate and its symbolic representation is shown in Fig. 5.21. In the absence of an input both Q1 and Q2 are forward-biased and conducting heavily. Thus, when there is no signal on A or B, the output voltage taken from collector to ground is closed to zero. When a positive pulse appears on A but not on B, Q1 cuts off but Q2 still conducts and the output is still UP.
Similarly, if a positive pulse appears on B but not on A, the output is UP. But if positive pulses appear on A and B simultaneously, both Q1 and Q2 are cut off and the voltage at the collector drops to – VCC. So the output is DOWN. Thus, we find that the NAND gate is nothing but an AND gate with its output inverted, mathematically, which can be expressed as-
All the possible conditions in a two-input NAND gate are given in Table 5.5.
NOR Gate and its Truth Table:
The NOR gate is a combination of NOT and OR logic, the characteristic of which is that the output is produced without application of signal to input A, nor to input B, …, nor to input N, nor to any combination of inputs. Fig. 5.22(a) is a NOR gate with two inputs while Fig. 5.22(b) is its logic symbol.
Let us consider the basic circuits of Q1 and Q2. When the input signal is absent both the bases are returned to ground through the input circuits (not shown in the figure). So, Q1 and Q2 are cut off for lack of forward bias. The output, common to both Q1 and Q2, taken from the collector resistor RL is, therefore, equal to + VCC when no input signal is present on A or on B. If a positive pulse is applied at input A or B or both, Q1 or Q2 or both respectively conduct and a negative pulse appears at the output. The output is, therefore, DOWN. So we have an effect of an inverter (NOT) and an OR circuit in the operation of this gate, mathematically, which can be expressed as-
All the possible conditions in a two-input NOR gate are given in Table 5.6.
Theorems in Boolean Algebra:
George Boole (1815-1864), an English mathematician, first developed the logical algebra greatly aided for the analysis of logical networks.
The equations above may be proved by considering the definitions. In the table the De Morgan’s theorems are interesting enough showing a useful relation between AND and OR functions.
The theorems of Boolean algebra are utilized to simplify digital logic networks in a similar way as the mathematical logic is utilized for manipulating ordinary algebraic expressions.