 Welcome, now we shall enter into a completely new topics which is fluidics and fluid logic. And in this lecture, lecture 26, we shall discuss about the fluid logic. This will be very introductory lecture on fluid logic. Now, you know that fluid in case of automation, there are valves which are called fluidics valves which are used for just on of operations in a automation. Now, first of all if you look into the very fundamentals of fluidics, what it is, then the what fluidics also known as fluid logic is the use of fluid or compressible medium to perform analog or digital operations similar to those performed with electronics. This it might be fluid incompressible fluid or compressible medium that means compressible fluid. The physical basis of fluidics is pneumatics and hydraulics both are used based on the theoretical foundation of fluid dynamics. The term fluidics is normally used when the devices have no moving parts. In such valves which is known as fluidics valve, you will find that there is no moving parts. This does not mean that fluid is not flowing, fluid is flowing, but no say spool or other things are not there. Therefore, ordinary fluid power components such as hydraulic cylinders, spool valves are not referred as fluidic devices. Now, in 1960s the application of fluidics to sophisticated control system with the introduction of the fluidic amplifier. The first invention of you can say discovery in the which could be named as fluidics device was the fluidic amplifier and that is in 1960s. I must say that time just after the world war there was the researchers were very much interested to explore the fluid power, because they found the lot of applications was there in the machinery. So, that there was maybe everywhere it was going on wherever the engineering was there the research application engineering of fluid power, but at that time the electronic devices were not that much developed. It has started, but not that much developed. So, we will we are thinking of the as analog and digital operation using this fluidic devices. Analogues to electrical and electronic circuits, fluid power circuits can be constructed to provide logic control of the synth systems. Now, what is logic control that will come later? The word logic used in the technological context means predictions and decision taking ability. The prediction and decision takings we call logic, but it really means we are predicting something and on that basis we are taking some decisions only then we should call it is logic. In many engineering situations the problem boils down to designing a system that will give an output only when certain preconditions are fulfilled and the system must be capable of discerning whether those conditions have been fulfilled or not and then passes signal to actuate or not actuate itself as the case may be. So, it is like that it will ask it will ask a question single questions that to do or not to do and we will do these operations. Many operations you will find that you can discretize such operation into just on and off, on and off. If you go through few on and off you will arrived into the same result. For an example in a machine shop for press operation it may be necessary to ensure that the component is in position, that the safety guard is down and that both the start buttons one for each hand are pressed. The press will not operate if these conditions remain unfulfilled that means let us consider a paper cutting machine, shearing machines or sheet metal shearing machine. In that case what normally you will do you will put the paper in the positions, but while you are putting the paper in positions your both hand might be on the paper and it might be just under the cutter. So, if it operates then there will be an accident. So, for the safety purpose what is done first of all you are putting the job in positions then there is no operations. Then you are putting a safe guard in front of that there is a double safety this you are taking out your hand from there and putting the guard at that condition you can operate the switch. Again for the further safety there is not one switch there are two switch and this two side if you operate one it will not operate. If you operate this separately it will not operate you have to put the switch on both switch by two hands then you are sure that your hands are not under the cutter and surely you are not going to put your leg there. So, these operations can be done easily by electronic devices now, but that earlier stage fluidic devices was being used for such sequential operations. You can one can do some mechanical say unless you do not put this guard on that means you do not put the guard you cannot operate that switch even if one switch you cannot operate, but to be in safe side better operate by two switches. The sequences can be operated and controlled by two valued logic device this is look at this term two valued logic device two valued means yes or no, but what is logic and how it is used in engineering let us examine that. Is an action right or wrong a motive good or bad a conclusion true or false much of our thinking and logic involves trying to find the answers to two valued question like this. Yes or no if you can think in this way you are trying to do something then you first step is yes or no then you have forwarded with yes or no answer then another yes or no in that way you can find the logic. In fact, the those who are strongly believing in this logical sequences they always say that each and every action each and everything can be explained by simply no yes or no yes or no by this any you can arrived into a decision for whatever may be the matter just simply answering that yes or no yes or no say for example, you have to go to market and it is raining then first answer what is that is in is it essential to go to market and now come yes or no say say it is yes then if yes then how to go it will come the next questions shall I take an umbrella or some raincoat say suppose you are thinking of umbrella yes I would take an umbrella then next question how will go I will takes the scooter then you will find if the scooter with umbrella this is not a combinations so one will be eliminated you and then think of no this path is not possible then next part take a raincoat in that way gradually you will find that with this answer you take the decision but in normal cases you will find this morning it is raining you have to you may try to go not to go to the market if you have to go then what you will do you will take your raincoat and you will take your car and or whatever it may be you will go simply may be car and umbrella car will keep here and with the umbrella it will go but that decision you take within a few seconds but in fact all such logics works and with the yes and no answer you arrive there one so that is called two valued questions the binary or two valued nature of logic and a major influence on had a major influence on Aristotle it is said that Aristotle first talked about this logic who worked out precise methods for getting to the truth given a set of true assumptions he was a philosopher Aristotle you must know he was a philosopher he wanted to always arrive a truth behind anything and then he was thinking of how we can arrived into that truth he gave the idea of logical thinking logic next attracted mathematicians then mathematicians who are doing mathematics they thought that in many cases if they think in this binary way probably they will arrived into they can do some mathematical operations to arrived into answer who intuitively sensed some kind of algebraic process running through all thought any thinking process also there must have some algebraic process. Now Boolean algebra is the algebra of logic and it was originally an abstract mathematical form we will come to that Boolean algebra. Now Boolean after the name of George Boole who invented this algebra he published a paper mathematical analysis of logic in 1847 this means that from now it is more than 160 years 165 precisely 165 years from now the George Boole he first showed some mathematical analysis of logic. Now his friend De Morgan who helped him you can make the theory in this way and he also helped in connecting between logic and mathematics connection between logic and mathematics but ultimately Boolean he summed up all such things suggested by De Morgan and he developed the this algebra that is why it is named after Boolean algebra but in that algebra you will find the he honored Morgan and there are few De Morgan the laws theory of De Morgan which are very essential for this mathematical computations. In fact when Boolean invented this logic that time no computer nothing was there later in computer as you know we all follow this binary algebra that is based on Boolean algebra. Now that replaced Aristotle's methods Aristotle proposed some methods but Boolean algebra found to be much easier than Aristotle methods and then onwards the Boolean algebra was being used. However you will find that this Boolean algebra who found a new way of thinking a new way to reason things out was not used for several years I am coming to that but he decided to use symbols instead of words to reach logical conclusions. So he invented also some symbols for the presentations of logic. Now George Boolean saw a pattern in the way we think that allowed him to invent symbolic logic a method of reasoning based on the manipulation of letters and symbols letters and symbols in many ways symbolic logic resembles ordinary algebra. If you go into these Boolean algebra you will find as if at some places it is ordinary algebra but in other places you will find it is not matching with ordinary algebra. Now although originally intended for solving logic problems Boolean algebra now finds its greatest use in the design of digital computers that which I have mentioned by a coincidence the rules of symbolic logic apply to the electronic circuits is in computers and other digital systems. Boolean proved binary or two valued logic is valid for letters and symbols instead of words the advantages of Boolean algebra are simplicity speed and accuracy. You will find that they are using 0 and 1 0 and 1 with this you will find that in many cases particularly connected through the electronic devices it is very easy to arrive into a result rather than solving it mathematically. However the Boolean algebra did not have any impact on digital electronics until almost 1938 look at this when he published a paper that was 1847. So, 1938 almost 100 years it has no applications much applications people are thinking this is not abundant but still they do not found that this algebra can be used in some engineering application but Shannon who was an electrical engineer he first thing thought of applying this algebra in telephone switching circuits by the time 1938 the telephone has come then for switching circuits he was thinking that he can use this Boolean algebra because a switch is a binary device on or off. Shannon was able to analyze and design complicated switching circuits using Boolean Boolean algebra. So, he was the first engineer who applied this Boolean algebra in greater way and then it was followed for computers and many other applications. To express the logical sequence in building the logic circuits various symbols are used in basic form they are same for any field or technology such as electronics electrical or fluid power. However when the circuit is shown using the symbol of actual devices it will be different based on the field just I you have to identify say for example the similar symbols are used but you will find that for electronics or electrical circuits there are a separate set of symbols then the fluid power symbols but circuits may look more or less alike logic is logic will work in the same way but while you are thinking of the fluid power device or electronic device symbols will be different. Nevertheless the logic circuit may also be different for the same sequence of operation as making a logic operations in a field one has to use more than one device whereas it is possible with single device for another field. This is again say some of the devices was developed say in electronics that cannot be developed in fluid power. For fluid power the same operation you will find they have used some other devices or may be two in number together to have the same operations which is achieved by a single device in electronics. In case of fluid power it may be common and ordinary devices of fluid power say fluidic devices is not the fluid power component whereas using two fluid power component you can develop a logic element that you should remember which are used for both drive and control or fluidic devices then we call this is a fluidic devices that means you take two ordinary valve or ordinary fluid power components and combine together have a function which is functioning like a fluidic device. Then together you can call it is a fluidic device but ordinarily the moving the fluid power having moving components those are not fluidic device. It is proper to start with a few definitions to understand the text that will follow to understand this what are this fluid logic. Now definitions in definition first of all we will learn about the gate. There is a term which is used in logic gate. What is gate? Gate is a logic circuit with one output and one or more inputs. Now remember gate is not a device gate basically is a circuit which comprises of one output and one or more inputs and output signal occurs only for certain combination of input signals. Now another term is used in logic circuits the truth table. Now while you are thinking of some operations you will think of a circuit. Now to make this circuits you have to go through some truth tables that means there where it is written if this is one this is zero then what will be the output. Sometimes called a table of combinations as I have told you is a list or table that shows all input output possibilities for a logic circuits. The number of horizontal rows in a truth table equals to raised to the power n where n is the number of inputs. Suppose there are n number of inputs in that case in that table there will be 2 to the power n is the number of rows. Let us consider there are 4 inputs. So truth to power 4 means 16. You have to at least 16 row truth table you have to make. We will show you that those truth table. For a 2 input gate for example the truth table has 4 rows a 3 input gate will have a truth table with 8 rows. Ordinary algebra when we solve an equation for its roots we may get a real number positive negative fractional and so forth. In other words the set of numbers in ordinary algebra is infinite. In Boolean algebra when we solve an equation we get either 0 or 1. It is always no means 0, yes means 1 but always you should remember you should not be confused with. There is no signals it might be 0 or might be it is not operating. So a care must be taken to find out real 0 or no function it is not functioning. No other answer is possible because the set of numbers includes only the binary digits 0 and 1. Actually as I have told that whether it is not functioning there are some other signals that it is not functioning. Another starting difference about Boolean algebra is the meaning of plus sign. In ordinary algebra plus means 1 plus 1 means we just 2. In case of Boolean algebra 1 plus 1 is not 2 we will see that what it is. The plus sign symbolizes here the action of an OR gate. This OR gate may be thought of as a device that has 2 inputs say A and B and an output say Y. Now here as I have told that actually answer will come into 0 or 1 but input name we have given AB that output name we have given Y and then we will put its value. Moreover when over A or B or both of them are in the state of 1 the output value of Y becomes 1. If these input combinations are not present the output Y becomes 0. We will see that in the table. Symbolically it is written as Y is equal to A plus B and is read as Y equals A or B. So here the plus sign does not stand for ordinary addition it stands for OR addition. We call this plus sign in Boolean algebra is OR additions. Now these rules can best be understood following its truth table. We have made to understand these equations we have made a truth table. Y is output so we have kept it in right hand side A and the inputs in the left hand side. Now let us consider A is equal to 0, B is equal to 0 then it is 0. Now this is OR additions A is equal to 0 but B is equal to 1 then output is 1. This means that in OR addition if one of that is on then this will be on output will be some output will be there. This means that this is not quantity this is not quantitative only thing there is some signal at input. So output must have signal 1 plus 0 is also 1 and 1 plus 1 is 1 that means if both are giving signals then there is output any of them is giving signals is output none of them giving signals is no output. So this is only this you can think in terms of OR but this quantity actual quantity say quantity of flow quantity of may be in case of that is inputs or power anything that quantity is different this is only thinking of the possibility of on and off. Similarly the multiplications sign that is cross has a new meaning in Boolean algebra this is not an ordinary product when it is written Y is equal to A into B sometimes it is written A dot B or even we write AB then it is it means that it is AND device 2 inputs A and B both must be equal to 1 to get an output Y is equal to 1. If any of them remains 0 the output becomes 0 say this again with a truth table say A dot B is equal to Y now 1 A is 1 that means signal is there but B is no signals output there is no output A is no signal B is a signal still it is no output only there will be output if both are giving signals then there will be output signals. But none of them are giving signals is 0 whereas one of them giving signal is 0. So these two are ok but when these two are 0 there is no signals there is also a dangerous because it might be whole system is not working. So there should have some precautionary measure for that but here as you see this when both are on in that case this is one. In case of OR also if both are one then this is one. So you will find that there is a some similarity in AND and OR for some operations but if you come to say these operations in case of OR the output will be 1 whereas in case of AND this output is 0. This means that depending on the requirement we will think of such device either AND or we are going to use OR. Even though the cross or dot sign does not mean multiplication in the ordinary sense the result of AND multiplications are the same as ordinary multiplications. If you look into this these are like an ordinary multiplications. So there is a similarity with the ordinary algebra. Another operation of Boolean algebra is the NOT operations NOT. This NOT implies inversion and it is written as a bar is equal to NOT a. This means that this device if we this in this gate if we put a device so if you put a signal that will after the device that will become into no signal. So that operation is required say one signal is coming we have to put into no signal so that combining with other signals we can we can arrived into a desired signal. So this NOT a is also a device thus if the variable a is equal to 1 of a particular instant of time a bar is equal to 0 and vice versa in a particular instant of time a bar is equal to 0 and vice versa that is if a bar is equal to 0 sorry this if a is equal to 0 then a bar will be 1. If a is equal to 0 then a bar is equal to 1 NOT a bar here. The following rule can be can be easily checked by truth table 3 that is a plus a bar is equal to 1 how say which follows from the identity that a plus 0 is equal to 0 plus a and we can have from this truth table what it is a plus a bar is equal to y then this plus means or so 1 plus 0 is equal to 1 0 plus 1 is equal to 1 that means a plus a bar a or a bar this combination always will give a positive signal if y is equal to a plus a bar then a plus a bar is always 1 several basic rules and theorems of Boolean algebra as follows it is presented in the tabular form so this is the description or and and now there are identities that means if you combine with a or that is or addition then a plus 0 is equal to a and a dot 1 is equal to a we to arrived into a we can use a plus 0 or a dot 1 is equal to a so these are called identities now I would say it is very difficult to just remember all this thing looking into just one time if you want to be expert in an Boolean algebra you have to remember these things because only with such identities and other combination which I will show you can simplify this Boolean algebra we will come to that next the identity laws this is simply called identities then next come identities laws a plus a is equal to a and a dot a is equal to a say this these combinations is called identities whereas these combinations is called identities laws now coming to the third one the commutative laws a plus b can also can be achieved by b plus a and a dot b also can be achieved by b dot a fourth one the associative laws in that case a plus b plus c that this means this is or addition of a and b then this is one output that output is a input and that is further being combined with an or we will give you a and then b plus c and in case of and function a b into c is a into b into c now the distributive laws are a into b plus c is equal to a b plus a c you will have same output with such combinations also you will have the same output is such combinations this is I do not know may be you can may you can remember these things apparently it is a difficult but if you can remember these while you are simplifying it will become easy the second relation underline is not valid in ordinary algebra this you will find they are following ordinary algebra but this is not following ordinary algebra however it can easily be shown by truth table in next slide that it is valid for Boolean algebra now in this table in this truth table what we have considered there are three variables a b c we have taken three variables then first we have made this or additions and then we have made these combinations and then b c and a plus b c we are trying to prove this one now let us consider that this is we have taken one this is one and this is also one so definitely a plus b will be one a plus c will be one a plus b one into one is equal to one b dot c will be also one and this is one so hence this you will find this is proved now if you take a b both one but c is equal to zero in that case a plus b is equal to one a plus c is equal to one this is one but b dot c will be zero whereas a plus b c will be one because a is equal to one now still you can find this with a is equal to one c is equal to one b is equal to zero so this is one this is one this is one this is zero and this is one now how many rows will be there again there are three so three to the power two is not it or two to the power three so there will be eight rows let us see how many rows are there so now then we find one zero zero this is one this is one this is one this is zero this is one again it can be proved zero one one this will be one this will be one this will be one this all are one and zero one zero then one zero zero zero zero and zero zero one this is zero one zero zero zero and lastly if three of them zero all will become zero so this will be also proved but this is dangerous one so in that way if you consider any such theory or mathematical expression Boolean algebra then you can prove it by using the truth table also this can be you can simplify this but then you have to apply some laws which you should remember but it might not be difficult because if you put one value then you go for the simplification by algebraic method also you will arrive into this but truth table is very easy one other laws are as follows that a plus a b is equal to a so that you can prove also in a truth table say if this is one then say this is zero then one zero is equal to a say this is zero and this is one then this is zero say is equal to zero so this is one that means zero plus zero must be is equal to a in zero so in that way you can prove this making the truth table though now this already we have shown this is always true a plus a bar is equal to one and a into a bar a dot a bar is always zero isn't it because one of them is zero other will be one so zero into one is always will be zero so which I have shown in eight now this is again true a something into zero is always zero it is like an ordinary algebra then ten a plus one is equal to one now here as you see this suddenly we have used all this letter symbol and why it this is numerical this means that is one positive and another is whatever may be this is unknown we have written in the form of a letter that this might be zero might be one whereas when we are indicating one or zero that means zero means zero one is one right so the equations are written in this form now an important theorem this is proposed by Morgan's but this algebraic form which is not known who develop this may be Boolean or together as you see this far first or operations here and then not of this or operations is equal to not of a dot not of b and not of c or inverse of this similarly these equations also now these two we will prove by using a table last two theorems state that the inverse of function can be obtained by inverting all the variables and then changing the and to or and the or to and so this you can see this combinations this is quite interesting if you look into this this a plus b plus c then inverse of that a inverse dot b inverse dot c dot now simply we have bought this side then automatically say as if you are bringing this one and then joining this three but together and bringing this this right hand side and just you making this three discrete but this will be true we will verify in the next slides now again in this truth table we are trying to prove this true so definitely there will be also 8 rows 1 2 3 4 5 6 7 8 but we have to make the column depending on that for each element then for each combinations so what we have done we have made a b c these three columns then one column for a plus b plus c one column for inverse of that then one column for a inverse another b inverse another c inverse c naught you can say then a bar dot b bar dot c bar dot etc etc so all possible that what we would like to get in the form of equation that we have made the column so column number may be anything but rows number will be 2 to the power n now again in first column we have taken 1 plus sorry a is equal to 1 b is equal to 1 and c is equal to 1 and you just look that for our combination this will be 1 and inverse of that will be definitely 0 not of that will be 0 and as these are 1 each 1 so these will be all 0 0 sorry c bar will be 0 also so there is a mistake so a dot b dot c dot any of them will be 0 say even if for this also it is 0 but this will be 0 a dot b dot c dot will this would be again 1 may be this is a a dot b dot c dot is equal to 0 this will be 1 and this will be 1 but a dot b dot plus c dot may be this sorry may be this combination is something wrong let us see this next table 1 1 0 a plus b plus c is equal to 1 right this obviously a plus b plus this is inverse into 0 this is equal to 0 whereas this will be I think this tables are just exchange then a b c is equal to 1 and this is obviously 0 this is I think this 2 rows are exchanged so we will come to the next 1 1 0 1 so this is 1 and this is 0 right a bar is equal to 0 this is equal to 1 may be so I will give you the correct table possibly that while I were I was just copying from the actual table to this slide form these are the mistakes we have made anyway you please develop this table of your own and you will find that this can be proved let us see this one 1 this is correct this is correct a is equal to 0 1 0 no this is not correct again a b c is equal to 0 so 0 1 1 1 0 that is correct this is correct this is correct but this is not correct anyway this table is wrong what I find so but it can be proved using this theorem may be this and these are interchanged somewhere although these are written but these combinations are not correctly written this is the end of that column 5 and 9 we will find 5 and 9 and 11 are 11 and 12 these are putting proving these two equations one is this one combination of 5 and 9 and other is 11 and 9 although here it is correctly written but perhaps these are not in proper form ok I will give you the correct table while I will send you the note now what we have learned so far on the fluid logics one may still wonder as to how this knowledge that has been for described is going to help one in dealing with logic or logic circuits for automations this query can perhaps best be ensured at once by by saying that even this little bit of knowledge about Boolean algebra enables one to simplify a large Boolean algebraic expression into a much smallest form which means that an elaborate binary switching circuit can be transformed to its absolute minimal functional equivalent from and form and thus saving a lot of complications and cost this means that why such combinations are shown that what we will find to arrived into a solution we will find that many intermediate steps can be omitted because they are combination of few inputs which is giving the output and that can be achieved by much simpler input so in many operations in computer you will find that many intermediate steps are eliminated following this Boolean algebra simplifications and circuit becomes very simple and the operations become much much faster now we have followed this books you will find there but that is much elaborate form here I have presented in very concise form ok thank you.