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//****************** Logic Gates - August 31st, 2017 ************************//
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- headacheowwww...unsure of cause, but probably stress (diff eq.) related
- Heck, even stats is more of a headache than I predicted (though ultimately not that bad...yet)
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- Just to keep in mind while we're going through electricity:
- Think of electricity like water going through a pipe; the electrons aren't being added, they're being "pushed" around the circuit; resistors are areas of "high-pressure" / compression, etc.
- "Voltage is like pressure, in that it can vary throughout the pipe/wire; we're going to talk about the voltage (potential) in the circuit defining the 1s and 0s"
- We don't talk about current very much in this class (leave that to the ECE people), but don't worry; it's there
- So, there are transistors, P (default closed/wire-like) and N (default open/non-wirelike) types, with a source, drain, and gate
- We can use these 2 transistors to make a very simple logical circuit: the NOT gate
(power)
|
|
________(P-type transistor)
| |
(input)_________| |____________(output)________
| |
________(N-type transistor)
|
|
(ground)
- So, if NO voltage is flowing in the input, the P-type transistor lets electricity flow through...but when voltage is applied to the input, then the P-type transistor shuts off the power and connects the output to the zero-volt ground!
- Now, electrical engineers don't like to draw these gates in this much detail every time, so INSTEAD, they represent NOT-gates with this triangle-with-a-circle-on-the-end sort of symbol:
------ |>o ------ (looks alot better when it's not in ASCII)
- Now, you can draw a NAND gate like this:
(...I can't draw circuits in ASCII this fast, but it's the same diagram as the textbook)
- "Now, I know some of you are saying, 'BILL! I came here to see an AND gate, not a NAND gate!'; well, to get an AND from a NAND, just reverse the output! Just stick a d*** NOT gate on the output, and there you go!"
- AND is represented by this D-like symbol with 2 inputs and 1 output:
==== D --------
- Similarly, we can make a NOR gate like this:
(...diagrams are in Section 3.2, by the way)
-... and turn it into an OR:
(your expectations are far too high of me)
- "And if you're having trouble remembering what the EE symbols for these gates are..." *flashes a picture of the "ANDroid" robot, with different heads for NANDroid, ORdroid, NORdroid, XORdroid*
- As it turns out, you can express ALL of these relationships of true/false with all these different gates with something called "Boolean Algebra"
- AND is expressed with a dot, like multiplication: A*B
- OR is expressed with a plus sign, like addition: A+B
- NOT is expressed with a bar over the letter
- Similarly,
- ...and as Prof. Leahey just pointed out, "This bar works great on blackboards, but is a pain in the neck to type on computers, so we'll usually replace it with PRIME notation" : A' , B'
-...just like complements in set theory!
- Can negate a SET of statements as well: (A*B*(C+D))'
- "ANY expression in boolean algebra CAN BE BUILT as a series of logic gates!"
- Now, in boolean algebra, there's something called "De Morgan's laws", which basically say that you can use an AND to represent an OR operation, or vice versa
- IN OTHER WORDS, (A'B')' = (A+B)
- Literally, you can just prove this via proof-by-exhaustion with a truth table
- "You can actually use this in coding; if you're looking at this complicated 'if(and)' statements, sometimes you can reduce it to an 'or'"
- It's also occasionally useful in manufacturing processes for microchips, but we won't get into that here
- Now, we CAN make gates with 3, 4, or more inputs, but these can all be built using gates with 1 or 2 inputs
- PRACTICALLY, each gate has a nanosecond delay due to the speed of electricity, which means that using 2 gates to build a 3-input gate is HALF as fast...but we'll leave that problem to the EE and CE majors now
- "We basically gloss over all the messy physics of these electronics in this class"
- "Be careful with truth tables for this, ESPECIALLY NOTs! If you hook up 2 NAND gates, you're not going to get a 3-input NAND gate; the 2 NOTs will cancel each other out!"
- Now, there's something called COMBINATIONAL LOGIC, where given the set of inputs to the circuit, no matter HOW the circuit is internally combined / made, it will ALWAYS give the same output
- This is NOT always true in SEQUENTIAL logic...but we'll get to that later
- "Combinational logic is kinda like a bike lock; it's this idea that no matter what order we turn the different numbers in, if we turn them all to the right spot, it'll still open"
- Let's say you work for the Whack-a-Mole company, happily as an intern, when all of a sudden the factory breaks! The manager WHISKS into the room, grabs YOU (just because you happen to be the closest person), and screams, "FIX IT!"
-You have a black-box "Whack-a-mole control unit," with 2 output wires; we want to control 4 components based on the output
- If the output is a 00, we want to turn on the "Play WAM!" sign
- If it's a 01, we want to raise the 1st mole
- If it's a 10, we want to raise the 2nd mole
- If it's a 11, we want to raise the 3rd mole
- So, how do we hook up these 4 states in an encoded 2-bit signal to our 4 separate components? Well, we can use these 4 AND gates and some NOT gates to properly split the output to ONLY turn on for their specific code!
- (Prof. Leahey blows our minds by showing how you can hook up the NOT gates to the AND inputs, and it looks like (if "0" represent each NOT gate) "00", "01", "10", "11", in-order, with 1 AND gate hooked up to each component)
- Then the ACTUAL engineering department shows up, sees what you did, and says "Well, yeah, that's a standard circuit called a DECODER"
- Standard Decoders have "n" inputs, "2^n" outputs, and for a given input only turn on ONE of the output wires
- There's an algorithm for how to create this circuit properly for "n" outputs, but we'll skip that for now
- "Wonderful as being a Computer Scientist is, I have some bad news: it's not the most fun job in the world."
-...what is the most fun job? A SUPERVILLAIN!
- "Good LUCK getting people to tremble in fear before a COMPUTER SCIENTIST!"
-...and where do supervillains live? In hollowed-out volcanos, of course!
- "Of course, the job has one problem: a perpetual fear of British secret agents, ESPECIALLY if they're serial number starts with...0"
- So, you decide (awesome Supervillain that you are) to install a sensor to detect approaching British secret-agents
- So, you put one on the top of the volcano, one in the underwater sumbarine hole, one by the laser-equipped shark tank...
- You hook up an alert box with 4 input wires (one for each of your sensors, with 2 switches on the box to switch between each sensor (00, 01, 10, 11) (1 switch per digit, if it isn't clear); when the selected sensor detects the agent, then the light on the box turns off
- You hire someone to stare at the box 24 hours a day, which is GUARANTEED to put your guard to sleep, which is how the secret agent slips in, which is how the movie gets made
-----------------
00-----| |
01-----| |------------(out)----
10-----| |
11-----| |
-----------------
- Well, each input is just 1 bit from the sensors: "agent"(1)/"no agent"(0)
- NOTE: You must NEVER allow the 2.9 volt "1" bit into the ground, because then you'll get like 2.9 volts/ 0.00001 ohm = stupid-high amperage, which will result in an instant short-circuit
- You then hook each input up to a three-input AND gate, 1 input is the sensor and the others are the 2 switches (full circuit diagram in textbook, Section 3.3.2) (sadly, not for an ACME Corp. Supervillain-protection device)
- This circuit is called a MULTIPLEXER (which basically everyone calls a MUX); a device with 1 output, 2^n inputs, and n "control lines"/switches that lets you route the selected input to the output
- You can also have multi-bit multiplexer's, with multi-bit output, by hooking them together
- WHY am I rambling on about this, and all these crazy different componenets? Because ALL of these things are component's that we're going to use to put together a memory, processor, and eventually, a very, very basic computer
- So, on the test, you're going to PANIC when you see this terrifying diagram with all these components, and then you'll realize WAIT! That part's an Adder! That part's a Mux! That's a...well, you get the picture.