truth table to boolean expression

Product-Of-Sums expressions lend themselves well to implementation as a set of OR gates (sums) feeding into a single AND gate (product). This last sum term represents a 0 output for an input condition of A=1, B=1 and C=1. What we need in this system is a sure way of detecting the presence of a flame, and permitting waste to be injected only if a flame is “proven” by the flame detection system. Suppose that one of the three sensors were to fail in such a way that it indicated no flame when there really was a good flame in the incinerator’s combustion chamber. For more information contact us at [email protected] or check out our status page at https://status.libretexts.org. A truth table shows the evaluation of a Boolean expression for all the combinations of possible truth values that the variables of the expression can have. It would be nice to have a logic system that allowed for this kind of failure without shutting the system down unnecessarily, yet still provide sensor redundancy so as to maintain safety in the event that any single sensor failed “high” (showing flame at all times, whether or not there was one to detect). To illustrate this procedural method, we should begin with a realistic design problem. The first step in designing this “sensor disagreement” detection circuit is to write a truth table describing its behavior. If we added circuitry to detect any one of the six “sensor disagreement” conditions, we could use the output of that circuitry to activate an alarm. A far better solution would be to design the system so that the valve is commanded to open if and only if all three sensors detect a good flame. It would be nice to have a logic system that allowed for this kind of failure without shutting the system down unnecessarily, yet still provide sensor redundancy so as to maintain safety in the event that any single sensor failed “high” (showing flame at all times, whether or not there was one to detect). To illustrate this procedural method, we should begin with a realistic design problem. Thus, they should either all register “low” (000: no flame) or all register “high” (111: good flame). For a 2-input AND gate, the output Q is true if BOTH input A “AND” input B are both true, giving the Boolean Expression of: ( Q = A and B). All we have to do is examine the truth table for any rows where the output is “high” (1), and write a Boolean product term that would equal a value of 1 given those input conditions. Create one now. Whoever is monitoring the incinerator would then exercise judgment in either continuing to operate with a possible failed sensor (inputs: 011, 101, or 110), or shut the incinerator down to be absolutely safe. Do we want the valve to be opened if only one out of the three sensors detects flame? Suppose we were given the task of designing a flame detection circuit for a toxic waste incinerator. The truth table for such a system would look like this: Here, it is not necessarily obvious what kind of logic circuit would satisfy the truth table. This last sum term represents a 0 output for an input condition of A=1, B=1 and C=1. For instance, in the fourth row down in the truth table for our two-out-of-three logic system, where A=0, B=1, and C=1, the product term would be A’BC, since that term would have a value of 1 if and only if A=0, B=1, and C=1: Three other rows of the truth table have an output value of 1, so those rows also need Boolean product expressions to represent them: Finally, we join these four Boolean product expressions together by addition, to create a single Boolean expression describing the truth table as a whole: Now that we have a Boolean Sum-Of-Products expression for the truth table’s function, we can easily design a logic gate or relay logic circuit based on that expression: Unfortunately, both of these circuits are quite complex, and could benefit from simplification. As its name suggests, a Product-Of-Sums expression is a set of added terms (sums), which are multiplied (product) together. Suppose we were given the task of designing a flame detection circuit for a toxic waste incinerator. Note that the Boolean Expression for a two input AND gate can be written as: A.B or just simply ABwithout the decimal point. Any other output combination (001, 010, 011, 100, 101, or 110) constitutes a disagreement between sensors, and may therefore serve as an indicator of a potential sensor failure. In designing digital circuits, the designer often begins with a truth table describing what the circuit should do. Each sensor comes equipped with a normally-open contact (open if no flame, closed if flame detected) which we will use to activate the inputs of a logic system: Our task, now, is to design the circuitry of the logic system to open the waste valve if and only if there is good flame proven by the sensors. Several different flame-detection technologies exist: optical (detection of light), thermal (detection of high temperature), and electrical conduction (detection of ionized particles in the flame path), each one with its unique advantages and disadvantages. Being that there are much fewer instances of a “low” output in the last truth table column, the resulting Product-Of-Sums expression should contain fewer terms.

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