Boolean algebra and ladder logic

The Boolean equation form can then be simplified or rearranges, and then converted into ladder logic

If we can describe how a controller should work in words, we can often convert it directly to a Boolean equation, as shown in Figure below. In the example a process description is given first. In actual applications this is obtained by talking to the designer of the mechanical part of the system. In many cases the system does not exist yet, making this a challenging task. The next step is to determine how the controller should work. In this case it is written out in a sentence first, and then converted to a Boolean expression. 

The Boolean expression may then be converted to a desired form. The first equation contains an EOR, which is not available in ladder logic, so the next line converts this to an equivalent expression (2) using ANDs, ORs and NOTs. The ladder logic developed is for the second euation. In the conversion the terms that are ANDed are in series. The terms that are ORed are in parallel branches, and terms that are NOTed use normally closed contacts. 

Process Description:
A heating oven with two bays can heat one ingot in each bay. When the heater is on it provides enough heat for two ingots. But, if only one ingot is present the oven may become too hot, so a fan is used to cool the oven when it passes a set temperature.

Control Description:
If the temperature is too high and there is an ingot in only one bay then turn on fan.

Define Inputs and Outputs:
B1 = bay 1 ingot present
B2 = bay 2 ingot present
F = fan
T = temperature overheat sensor

Boolean Equation:

Ladder Logic for Equation (2):

 Ladder Logic for Equation (3):

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