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Fail-Safe Autopilot Logic
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This example aims to illustrate the signi cance of De Morgan s laws and of the duality of the sum-of-products and product-of-sums forms Suppose that a fail-safe autopilot system in a commercial aircraft requires that, prior to initiating a takeoff or landing maneuver, the following check must be passed: two of three possible pilots must be available The three possibilities are the pilot, the co-pilot, and the autopilot Imagine further that there exist switches in the pilot and co-pilot seats that are turned on by the weight of the crew, and that a self-check circuit exists to verify the proper operation of the autopilot system Let the variable X denote the pilot state (1 if the pilot is sitting at the controls), Y denote the same condition for the co-pilot, and Z denote the state of the autopilot, where Z = 1 indicates that the autopilot is functioning Then, since we wish two of these conditions to be active before the maneuver can be initiated, the logic function corresponding to system ready is: f =X Y +X Z+Y Z
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This can also be veri ed by the truth table shown below
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Pilot 0 0 0 0 1 1 1 1 Co-pilot 0 0 1 1 0 0 1 1 Autopilot 0 1 0 1 0 1 0 1 System ready 0 0 0 1 0 1 1 1
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The function f de ned above is based on the notion of a positive check; that is, it indicates when the system is ready Let us now apply De Morgan s laws to the function f , which is in sum-of-products form: f = g = X Y + X Z + Y Z = (X + Y ) (X + Z) (Y + Z) The function g, in product-of-sums form, conveys exactly the same information as the function f , but it performs a negative check; in other words, g veri es the system not ready condition You see then that whether one chooses to implement the function in one form or another is simply a matter of choice; the two forms give exactly the same information
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EXAMPLE 134 Realizing Logic Functions from Truth Tables
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Realize the logic function described by the truth table below
A 0 0 0 0 1 1 1 1
B 0 0 1 1 0 0 1 1
C 0 1 0 1 0 1 0 1
y 0 1 0 1 1 1 1 1
Solution
Known Quantities: Value of function y(A, B, C) for each possible combination of logical variables A, B, C Find: Logical expression realizing the function y
13
Digital Logic Circuits
Analysis: To determine a logical expression for the function y, we rst need to convert the truth table into a logical expression We do so by expressing y as the sum of the products of the three variables for each combination that yields y = 1 If the value of a variable is 1, we use the uncomplemented variable If it s 0, we use the complemented variable For example, the second row ( rst instance of y = 1) would yield the term A B C Thus,
y =A B C+A B C+A B C+A B C+A B C+A B C = A C(B + B) + A B (C + C) + A B (C + C) = A C + A B + A B = A C + A (B + B) = A C + A = A + C
A + C = y or A C OR y
Thus, the function is a two-input OR gate, as shown in Figure 1319
Comments: The derivation above has made use of two rules from Table 1311: rules 4
and 18 Could you have predicted that the variable B would not be used in the nal realization Why
EXAMPLE 135 DeMorgan s Theorem and Product-of-Sums Expressions
Problem
Realize the logic function y = A + B C in product-of-sums form Implement the solution using AND, OR, and NOT gates
Solution
Known Quantities: Logical expression for the function y(A, B, C) Find: Physical realization using AND, OR, and NOT gates Analysis: We use the fact that y = y and apply DeMorgan s theorem as follows:
y = A + (B C) = A (B C) = A B + C y =y =A B +C The above sum-of-products function is realized using complements of each variable (obtained using NOT gates) and is nally complemented as shown in Figure 1320
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