Notes on Logic Circuits by LK

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Jean Kelley Foster
7 years ago
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1 Notes on Logi Ciruits y LK These notes introdue an interpretation of logi in terms of swithing iruits. This appliation of logi to swithing iruits is the disovery of Claude Shannon Symoli nalysis of Relay and Swithing Ciruits. y Claude E. The Journal of Symoli Logi, Vol. 18, No. 4 (De., 1953), p. 347 and forms the asis for the design of omputers to this day. Here is an astrat swith: open =F losed =T signal an go from left to right through the swith when it is losed, and no signal an go through when the swith is open. We hoose to designate a swith y a lael suh as aove and we let = T orrespond to the losed swith position (T for "transmit" if you like!) and we let = F when the swith is in the open position (F = ~ T = not t r a n s m i t ). The position of a swith an ontrol a devie suh as a l a m p.

2 attery ul attery ul OFF ON Two asi ways to put swithes together are Series and Parallel Connetion: B Series Connetion: ^ B B Parallel Connetion: v B s you an see, the only way for a signal to get from left to right in the series onnetion of and B is if oth and B are T (losed). Thus the series onnetion orresponds to and B whih we write in logi notation as ^ B. Similarly, the only way for a signal to get from left to right in a parallel onnetion is if one of or B is losed. Hene this orresponds to or B, whih we write as v B. Thus our two asi logial operations are mirrored in the ehaviour of networks that arry signals. What aout negation? n example will show you how we handle this. What we do is, we allow multiple appearanes of a given lael or its negation. We will write either ~ or ' for the negation of. Here we will use ' ok?

3 Then in the mutiple appearanes, all 's will e either losed or open and if you have ' and is losed, then ' will e open. If is open then ' will e losed. Look at this example: ' ^ ' = F ' v ' = T Eah of these iruits has an and an '. In the series onnetion, this means that one swith is always open and so the value of the whole iruit is the same as a simple open iruit. On the other hand in the parallel onnetion of with ' either one line will transmit, or the other line will transmit. So the iruit as a whole ehaves like a single swith that is losed. Thus we see that the series onnetion orresponds to the identity ^ ' = F, while the parallel onnetion orresponds to the identity v ' = T. Terminology: Sine a lael in one of our iruits an appear in many plaes we will still refer to as a 'swith' even though it may e omposed of a numer of elementary swithes. In suh a multiple swith, if you make one, there has to e a mehanial way to make sure that all the different parts of the swith work together. lso these swithing patterns may, in pratie, happen in eleroni iruitry that synronizes ations at separate loations. For our purposes, we shall imagine simple mehanial swithing devies. Prolem 1. Design a swithing iruit with three swith laels a,, suh that eah of a, and individually ontrol transmission of the signal. That is if the iruit is open, then hanging the state of any one of a,, will lose the iruit and if the iruit is losed, then hanging the state of any one of a,, will open the iruit.

4 (We disussed how to do this with in analogous ase of two laels and how it is related to (a ^ ) v ( ^ ) in lass.) Solution to Prolem : The logial ondition for the three swithes orresonds to the following settings a T T T T F F F T F F F T That is, we will have the line transmitting when all the swithes are in the T-state, and whenever any two of them are flipped to the F state. This is expressed y the symoli e x p r e s s i o n E =(a^^)v(a^^)v(^^)v(^^). This expression has orresponding iruit as shown elow. a a Now we fae an interesting prolem of simplifiation. The swithes in this solution are a it ompliated, and it is lear that if we want to generalize to four or more swithes that ontrol one light, then this method will give us inreasingly omplex designs. One way to searh for simplifiation is to algeraially simplify the expression E. We an apply the distriutive law (a^x)v(a^y) = a^(xvy) and redue the numer appearanes of a and of in E. Then E = a^((^)v(v)) v ^((^)v(^)). The iruit for this version of E is shown elow.

5 a Now omes the sutle part! What we have just done suggests a orresponding modfiation on the right hand side of the iruit graph. We an learly redue to just one and one on the right in same pattern y whih we did it through the distriutive law on the left. In order to do this, we will have to ross some wires in the iruit, and we shall understand that this does not mean that these rossed wires interat in any way as far as how they arry urrent. Here is a drawing of the result of the next s i m p l i f i a t i o n. a We have simplified the left and right swithes as muh as possile, and the middle swith has a really nie desription if you look at it. When = T it orressponds to two parallel lines and when =F it orresponds to two rossed lines.

6 = T: = F: In the figure aove we have indiated the middle swith y a ox with the lael, and we have illustrated how the two states of the swith orrespond to parallel and rossed lines. By simplifying the symoli logi solution to the three swith light prolem, we have invented a useful new swith. Lets all this swith, as shown aove, the r o s s o v e r s w i t h. I CLIM THT YOU CN NOW MKE USE OF THE CROSSOVER SWITCH, ND SOLVE THE PROBLEM OF DESIGNING TRNSMISSION NETWORK THT WILL CONTROL SINGLE LIGHT FROM NY ONE OF SET OF N SWITCHES. ENOUGH HINTS. YOU SHOULD THINK BOUT IT BY SEEING WHT HPPENS WHEN YOU CONNECT CROSSOVER SWITCHES TO ONE NOTHER. Prolem 2. Generalize Prolem 1. to an aritrary numer N of swith laels. That is, you would like to e ale to ontrol a single lamp with any one of N swithes. You want to find a design that will work in priniple for (say) a uilding with N floors, so that you an ontrol the

7 entrane light from any of the floors. The swith on any given floor will turn the entrane light on or off. That swith on a given floor will have two positions just as our laels have two states T or F.

1.3 Complex Numbers; Quadratic Equations in the Complex Number System*

1.3 Complex Numbers; Quadratic Equations in the Complex Number System* 04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x - 2x - 22 x - 2 2 ; x