Digital electronics book 1 fundamentals
Why the NAND and NOR gates are universal
- Because first, since any circuit can be dualled, then any gate capable of being used (singly or in combinations of itself) to implement either the AND or OR and the NOT operator must be universal (since the AND OR & NOT operators are really all there are) - and second, the NAND and NOR gates fulfill these requirements because tying the inputs of the NAND and NOR gate together produces a NOT gate - so voilá, we can build any digital circuit with NAND and NOR gates only -
The 2 universal gates
- Gates from which ANY digital circuit DC can be built - the NAND and NOR gate
VLSI design
- VLSI: very large scale integration - the process of creating an integrated circuit IC by combining hundreds of thousands of transistors or devices into a single chip - the microprocessor is an example of a VLSI device - VLSI lets a CPU, ROM and RAM be added together into one single chip
IC
- an integrated circuit - that is, an electronic circuit which is constructed on a small piece of semiconducting material and performs the same function as a larger circuit made from discrete, separate components
What digital electronics is about
- analysing and designing circuits - this can be done in many ways: 1. using the mathematical language of Boolean algebra 2. with circuit diagrams using logic gate symbols 3. With truth tables = they show how the output of a circuit, whether it's 0 or 1, varies according to the inputs, whether 0 or 1 (Note: the rows of truth tables are ordered in / show the binary code that would go into the component & produce the output 4. With a timing diagram = they show how the outputs is a circuit change in response to the inputs, which vary as a function of time
Positive & negative assertion level logic
- arises from duality - based on the fact that since Boolean algebra describes a 2 state system, specifying the the input conditions for an output of 1 also gives the input conditions (all other input combinations) for the other output, an output of 0 - looking at the "other side" of the equation, when does it give 1 and when does it give 0
Rules for dualling
- based on De Morgan's Theorem —> Y = compl of (A + B) = A c • B c - ALL input & output variables must be complemented (bubbled) - type of operation must be swapped (AND to OR, OR to AND)
Principle of duality
- based on De Morgan's theorem - and/or gates can be switched to or/and gates - as long as we have inverters, NOT gates, we can convert any circuit composed of AND and OR gates to be composed of OR and AND gates - arises because Boolean algebraic variables can be one of 2 values, 0 or 1 - has wide ranging implications in digital electronics because every circuit must have a dual: a replica of the circuit made by swapping the circuit bits
When assertion level logic is useful 1
- because it's useful to consider the inputs and outputs of logic circuits in terms of when they are "active", could be either 1, active-HIGH or 0, active-LOW - and that is because it's useful to design circuits so that their function is as clear as possible from the circuit diagram
Boolean algebra idempotent laws
- describe what happens when a variable operates upon itself - in practice: same variable goes to all inputs of a logic gate —> for a 2 input AND gate, Y = A • A —> an AND gate produced 1 if both inputs are 1, and a 0 otherwise —> so whatever A is, A is too, so if A is 1, then Y = 1 • 1 which produces one, if A = 0, then Y = 0 • 0, so produces a 0 - so the whole expression is equal to whichever A is, hence Y = A • A = just A - the OR operation does the same
Boolean algebra law of inverse elements
- describes what happens if we operate on a variable with its complement - in an AND gate, if we have Y = A and A(c), one of them is not 1 so Y = A • A(c) = 0 - in an OR gate, Y = A + A(c) = 1, since one is bound to be 1
A gate array
- eg in VLSI design, when IC (integrated circuit) manufacturers supply a "sea" of universal gates (NAND and NOR gates) - then the universal gates are connected as necessary to create the required digital circuit
B.A. properties of identity elements
- identity elements = constants, 0 & 1 - show what happens when we operate on a variable with the constants 0 or 1, what the output of a logic gate will be when a variable is operated on by a constant —> so shows output when one input is a variable and the other input is a constant
Product terms and sum terms
- product terms = variables AND'd together - sum terms = variables OR'd together
Boolean algebra distributive law
- shows how to expand out Boolean expressions with brackets - A • (B + C) = A • B + A • C - A + (B • C) = (A + B) • (A + C)
Boolean Algebra absorption law
- shows how to take an expression with 2 variables and reduce it to a single variable —> A • (A +B) = A —> A + ( A • B) = A In practice: often used to simplify Boolean expressions
Boolean algebra associative law
- shows how variables can be grouped together - the order is again insignificant —> A • ( B • C) = ( A • B) • C = A • B • C —> A + (B + C) = (A + B) + C = A + B + C In practice: becomes significant when we want to operate with 3 variables and we only have gates with 2 inputs - so if we want to or together A + B + C = Y we can have two 2-input gates, once with A + B connected to the second one, the second input of which receives C - and the order this is done in is arbitrary - so we could group A & C, B &A, etc...
Boolean algebra De Morgan's Theorem
- shows that if we have the complement of a group of variables together such as complement of (A + B) (A and B) it's equal to the complement of each individual variable OR'd together, so A complement individually & B complement individually - works with both OR & AND just flip the operators In practice: useful for simplifying Boolean logic expressions In practice: any AND/OR operation can be considered an OR/AND operation as long as NOT gates are used
Boolean algebra & ordinary algebra
- the 2 are similar, thus AND operations can be represented by multiplication, product symbol • and OR operation by addition or summing symbol + - many Boolean algebra laws are also true in ordinary algebra - in ordinary algebra, multiplication is done first: this is the same in Boolean algebra
Boolean algebra commutative law
- the order of operation when ANDing or ORing is insignificant —> A • B = B • A —> A + B = B + A In practice: it doesn't matter which inputs of a logic gate, top or bottom, the variables A, B, etc. are connected to
Example of using assertion level logic
- the output from a circuit being 1 to turn ON an alarm and 0 to turn off a light
Boolean algebra other identities law
- valuable when trying to simplify complicated Boolean expressions - so you can convert a complex Boolean expression into a simpler, other identity it has
When assertion level logic is useful 2
- when interfacing components such as microprocessors which often, due to the circuits from which they are constructed, initiate communication with others ICs by sending signals LOW
Laws of operating variables with constants
A and 0, A • 0 = 0 A and 1, A • 1 = A A or 0, A + 0 = A A or 1, A + 1 = 1
Timing diagrams appear for example
In the data sheets of components like analogue to digital converters and solid state memory, which give the timing information needed to design circuits containing them
Remember: dualling a circuit always ...
Inverts the output! (Y)
In a digital circuit, there are...
Multiple ways of representing certain information - you must be able to choose the most suitable representation for a given situation - and convert between different representations
Any digital circuit can be constructed using only...
NAND and NOR gates They are the 2 universal gates
Symbols for 3 logic gates / Boolean operators
Not = triangle with dot, - minus And = a D, • times Or = a rocket + plus
NAND and NOR gates
The NOT'd versions of AND and OR gates - simply means their outputs are inverted (indicated by bubbles on the outputs, Y side) - same as AND and OR gates whose outputs are passed through a NOT gate (inverter)
A sum of products expression
Y = (A•B) + (A(c)•B)
