Making other logic gates using only NAND Gates

This is pretty cool. So, since NAND chips (74LS00) are much cheeper to buy than the other logic gates, it is convenient to know how to make other gates with only NAND gates. As my teacher said in class "If I had to choose which chip to take with me onto a deserted island, I'd take the NAND chip" :)

NAND Gate

As mentioned in my previous post titled "Logic Gates" a NAND gate is a combination of a NOT gate and an AND gate (NOT + AND = NAND).

You can make any other logic gate with just NAND gates.

NOT Gate

A NOT gate is also known as an inverter gate. It inverts the values sent in. Since a NAND gate is a NOT gate and an AND Gate, by making A = B, we create a NOT gate.

AND Gate

To create an AND Gate using only NAND Gates, we must combine a NAND Gate and a NOT Gate made with NAND Gates (above). This way we can cancel out the NOT part of the NAND gate. 

OR Gate

An or gate returns 1 if any of the inputs is 1, a NAND gate will return 1 if one or more of the inputs is 0. To convert a NAND gate to an OR gate, we must invert the inputs. 

Its kind of magical! Click here for more gates, information and to see where I got the images above!

Gain

Gain is defined as the ability of a circuit to amplify the power or amplitude of a signal from the input to the output. It is usually calculated by finding the ratio between the output and the input. Gain can refer to voltage, current, or power. But, if the question does not otherwise specify it is asking for voltage gain.

P-N Junction

What is a P-N junction and how does it work?

Well, a P-N Junction is the boundary between the two semiconductor metals known as the p type and the n type. The n-type is created by doping a silicon atom with small amounts of Antimony, and a p-type is made by doping another silicon atom with small amounts of Boron. When they make contact they behave differently than they do by themselves and create one semiconductor that has been doped differently on either side. In particular, the current will only flow through it in one direction (forward or reversed bias).  The lattice between the two metals (diffusion region) have holes in them that when filled with electrons cancel each other and vanish. This causes an area which is depleted of any moving charges called the "depletion zone" and this transfer of electrons occurs on both sides creating an equilibrium. When a free charge enters the depletion zone it does not see any other free charges, instead it sees positive charges on the n side and negative charges on the p side. The force that these charges place on the free charge cause it to return to its original spot and leave the depletion zone. As a result the junction acts like a barrier and regulates the current. In order for charges to move from one type to the other they need to use extra energy and "climb uphill". The extra energy required can be supplied by a voltage applied between the two ends of the junction diode. If voltage is applied the wrong way the free charges are pulled away from the junction, causing it to not work.

Helpful Links: 

http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/info/comp/passive/diode/pn_junc/pn_junc.htm

http://www.electronics-tutorials.ws/diode/diode_2.html

DeMorgan's Theorem

This theorem is a very important logic theorem for digital electronics and states that any logical binary expression will not change if we change all the variables to their complements, change all the AND operations to OR operations (and vice versa), or take the complement of the whole expression.

http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/ietron/demorgan2.gif

More Info

Static Discipline

There are two ways to present data and information: analog and digital. The analog system typically represents the data in the form of a continuous wave which is easily interrupted and polluted by external noise. The digital system shows the data in two distinct states: 1 and 0. This creates a square wave which is much more resistant to noise.

Since digital devices are made by a variety of different manufacturers a set of development criteria have been created to ensure that devices made by different manufacturers will be able to communicate smoothly and efficiently. This criteria is called "Static Discipline" and states that if inputs meet the valid input thresholds (VIL --> VIH) then it is guaranteed that the outputs will meet the valid output thresholds (VOL ---> VOH). The "noise margin" is defined as the absolute value of the difference between the forbidden region voltage threshold for the reciever and the prescribed output voltage for the logic value.

Click here for example problems

Logic Gate Simulation Arduino Activity

I did a really fun simulation activity to help cement my understanding of Logic Gates:

AND Gates

OR Gates

NOT Gates

NAND Gates

Boolean Algebra

Hi all, we have been learning some Boolean Algebra in my Computer Architecture class and I thought I would share what I have learned so far. As always, if you have any questions or comments please share them below.

Boolean Algebra is the subset of Algebra formulated by English mathematician George Boole. In "normal Algebra" the variables are numbers and the main operations are addition, subtraction, multiplication and division. But, in Boolean Algebra the values of the variables are true and false (1 and 0) and the main operations are "and" (^), "or" (v) and "not". Now, why is this significant? How is Boolean Algebra useful and why couldn't we just use Algebra? Well, the laws of Boolean Algebra are mainly used to describe circuits whose state can either be true or false (1 or 0). Boolean Algebra differs from Algebra because the values of true and false are represented by binary digits (bits) that do not behave like normal integers (think about the NOT, AND, and OR gates in my previous post). Don't get confused, in Boolean Algebra the 1 and 0 do NOT behave like normal integers, they represent true and false.

Boolean Postulates:

1. 0 . 0  = 0
2. 0 + 0 = 0
3. 1 . 1 = 1
4. 1 + 1 = 1
5. 1 . 0 = 0 . 1 = 0
6. 0 . 1 = 1 . 0 = 1

Boolean Laws:

AND ( . )
OR ( + )

Every law has a part (a) and a part (b) this is called duality and is done by switching every AND for an OR and every 1 for a 0 (inverting it).

1. Commutative Law
• (a) A + B = B + A
• (b) A . B = B . A
2. Associate Law
• (a) (A + B) + C = A + (B +C) 
• (b) ( A . B ) . C = A + (B . C)
3. Distributive Law
• (a) A (B + C) = A . B + A. C 
• (b) A + (B . C) = (A + B) . (A +C) 
4. Identity Law
• (a) A + A = A
• (b) A . A = A
5. Redundance Law
• (a) A + A . B = A
• (b) A . (A + B) = A