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. As you can see, both the Sum-Of-Products and Products-Of-Sums standard Boolean forms are powerful tools when applied to truth tables. Here, Boolean algebra proves its utility in a most dramatic way. Don't have an AAC account? Probably not, because this would defeat the purpose of having multiple sensors. Do we want the valve to be opened if only one out of the three sensors detects flame? They allow us to derive a Boolean expression—and ultimately, an actual logic circuit—from nothing but a truth table, which is a written specification for what we want a logic circuit to do. To be able to go from a written specification to an actual circuit using simple, deterministic procedures means that it is possible to automate the design process for a digital circuit. Create one now. The entire logic system would be the combination of both “good flame” and “sensor disagreement” circuits, shown on the same diagram. 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. An example of a POS expression would be (A + B)(C + D), the product of the sums “A + B” and “C + D”. 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. What are the difference between Sum-Of-Products expressions and Product-Of-Sums expressions? If any one of the sensors were to fail in such a way as to falsely indicate the presence of flame when there was none, a logic system based on the principle of “any one out of three sensors showing flame” would give the same output that a single-sensor system would with the same failure. 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. Watch the recordings here on Youtube! 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. At minimum, this is what we need to have a safe incinerator system. The intense heat of the fire is intended to neutralize the toxicity of the waste introduced into the incinerator. The design task is largely to determine what type of circuit will perform the function described in the truth table. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. A strategy that would meet both needs would be a “two out of three” sensor logic, whereby the waste valve is opened if at least two out of the three sensors show good flame. Sum-Of-Products expressions lend themselves well to implementation as a set of AND gates (products) feeding into a single OR gate (sum), while Product-Of-Sums expressions lend themselves well to implementation as a set of OR gates (sums) feeding into a single AND gate (product). Suppose that due to the high degree of hazard involved with potentially passing un-neutralized waste out the exhaust of this incinerator, it is decided that the flame detection system be made redundant (multiple sensors), so that failure of a single sensor does not lead to an emission of toxins out the exhaust. 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). Thus, our truth table would look like this: It does not require much insight to realize that this functionality could be generated with a three-input AND gate: the output of the circuit will be “high” if and only if input A AND input B AND input C are all “high:”. The first step in designing this “sensor disagreement” detection circuit is to write a truth table describing its behavior. 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. Do we want the valve to be opened if only one out of the three sensors detects flame? We can, however, extend the functionality of the system by adding to it logic circuitry designed to detect if any one of the sensors does not agree with the other two. Sum-Of-Products expressions lend themselves well to implementation as a set of AND gates (products) feeding into a single OR gate (sum). 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). That single failure would shut off the waste valve unnecessarily, resulting in lost production time and wasted fuel (feeding a fire that wasn’t being used to incinerate waste). 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 we were given the task of designing a flame detection circuit for a toxic waste incinerator. While some people seem to have a natural ability to look at a truth table and immediately envision the necessary logic gate or relay logic circuitry for the task, there are procedural techniques available for the rest of us. Also, if the incinerator is shut down (no flame), and one or more of the sensors still indicates flame (001, 010, 011, 100, 101, or 110) while the other(s) indicate(s) no flame, it will be known that a definite sensor problem exists. An example of an SOP expression would be something like this: ABC + BC + DF, the sum of products “ABC,” “BC,” and “DF.”. 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. The steps to take from a truth table to the final circuit are so unambiguous and direct that it requires little, if any, creativity or other original thought to execute them. Published under the terms and conditions of the, Converting Truth Tables into Boolean Expressions. Thus, our truth table would look like this: It does not require much insight to realize that this functionality could be generated with a three-input AND gate: the output of the circuit will be “high” if and only if input A AND input B AND input C are all “high:”. Implemented in a Programmable Logic Controller (PLC), the entire logic system might resemble something like this: As you can see, both the Sum-Of-Products and Products-Of-Sums standard Boolean forms are powerful tools when applied to truth tables. Have questions or comments? 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. As you might suspect, a Sum-Of-Products Boolean expression is literally a set of Boolean terms added (summed) together, each term being a multiplicative (product) combination of Boolean variables. Suppose that due to the high degree of hazard involved with potentially passing un-neutralized waste out the exhaust of this incinerator, it is decided that the flame detection system be made redundant (multiple sensors), so that failure of a single sensor does not lead to an emission of toxins out the exhaust. 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. To begin, we identify which rows in the last truth table column have “low” (0) outputs, and write a Boolean sum term that would equal 0 for that row’s input conditions. Sum-of-Products and Product-of-Sums Expressions Worksheet, Add Voice Commands to Your Next Project with Bluetooth LE, Technology Enablers for Faster, Safer, High-Efficiency EV chargers, Architecture and Design Techniques of Op-Amps. If using relay circuitry, we could create this AND function by wiring three relay contacts in series, or simply by wiring the three sensor contacts in series, so that the only way electrical power could be sent to open the waste valve is if all three sensors indicate flame: While this design strategy maximizes safety, it makes the system very susceptible to sensor failures of the opposite kind. Review. This way, any single, failed sensor falsely showing flame could not keep the valve in the open position; rather, it would require all three sensors to be failed in the same manner—a highly improbable scenario—for this dangerous condition to occur. 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. While some people seem to have a natural ability to look at a truth table and immediately envision the necessary logic gate or relay logic circuitry for the task, there are procedural techniques available for the rest of us. Such combustion-based techniques are commonly used to neutralize medical waste, which may be infected with deadly viruses or bacteria: So long as a flame is maintained in the incinerator, it is safe to inject waste into it to be neutralized. For instance, in the first row of the truth table, where A=0, B=0, and C=0, the sum term would be (A + B + C), since that term would have a value of 0 if and only if A=0, B=0, and C=0: Only one other row in the last truth table column has a “low” (0) output, so all we need is one more sum term to complete our Product-Of-Sums expression. Consider another example. For instance, in the first row of the truth table, where A=0, B=0, and C=0, the sum term would be (A + B + C), since that term would have a value of 0 if and only if A=0, B=0, and C=0: Only one other row in the last truth table column has a “low” (0) output, so all we need is one more sum term to complete our Product-Of-Sums expression. Since we already have a truth table describing the output of the “good flame” logic circuit, we can simply add another output column to the table to represent the second circuit, and make a table representing the entire logic system: While it is possible to generate a Sum-Of-Products expression for this new truth table column, it would require six terms, of three variables each! What we have, is an accurate Boolean expression that describes a truth table, and therefore whatever system the truth table was based on. Implemented in a Programmable Logic Controller (PLC), the entire logic system might resemble something like this: As you can see, both the Sum-Of-Products and Products-Of-Sums standard Boolean forms are powerful tools when applied to truth tables. The design task is largely to determine what type of circuit will perform the function described in the truth table. 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. If all three sensors are operating properly, they should detect flame with equal accuracy. Waste introduced into the incinerator begins with a truth table describing its behavior sensors detects flame one! At minimum, this is what we need to have a safe incinerator system an input condition of A=1 B=1! 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