Looking at logic gates

Latency

transistors in series are slower than ones in parallel because there is more resistence on the wire.

[look up]
1.7.8 of H&H psuedo-nMOS logic

Power Consumption

Dynamic Power Consumption is the power used to charge capacitance as signals change, 0 to 1.

capacitance is a measure of how much electric charge can be stored per volt across 2 conductors seperated by an insulator. [look up] look more into the capacitance and electrical side of things

C*V**2 * f

• C = capacitance of the circuit (wires and gates)
• V = supply voltage
• f = charging frequency of the capacitor

When increasing f we want to also increase V to achieve the oputput for f. Voltage has a cubical affect on power.

Static Power Consumption is the power consumed when the circuit is idle. V * I leakage

energy consumption is

power * time

need to include both dynamic and static power

Larger gates

We can extend gates to more than 2 inputs but this leads to increased latency. It would be better to just seperate the larger gate into smaller gates and combine the output.

How do we keep Moore’s Law?

1. manufacturing smaller transistors/structures
   • some structures are already a few atoms in size
2. using materials with better properties
   • copper instead of aluminium (better conductor)
   • hafnium oxide, air for insulators
   • making sure all materials are compatible is the challenge
3. enable precision manufacturing
   • extreme ultraviolet (UEV) light to pattern <10nm structures
4. creating new device technologies
   • FinFET, gate all around transistro, single electron transistor

[look up]

• microelectronic design
• precision manufacturing
• fabrication technologies
• IEDM
• extreme ultraviolet litography
• Moores law paper

Innovation at the bottom enables computing.

How do we make a logic circuit?

A logic circuit is composed of:

• inputs
• ouputs
• specifications

Functional specification describes relationship between inputs and outputs Timing specification describes the delay between inputs changing and outputs responding

Types of a logic circuit

combinational logic:

• memoryless (we don’t remember what we have done with the inputs to make the outputs)
• outputs are strictly dependant on the combination of input values that are being applied to the circuit right now

sequential logic:

• has memory
• outputs are determined by previous and current value of inputs

Boolean Algebra

Boolean algebra is important because if we were to implement a circuit for a very complex boolean formula, the resulting circuit could be extremely expensive. Instead we can simplify the boolean formula, thus reducing the circuits complexity as well.

[look up]
axioms + proofs george boole - the mathematical analysis of logic go explore boolean algebra

duality in boolean algebra means that every axiom opposite form is true as well.

[look up]
logic/hardware synthesis course - simplify circuits DeMorgans law: enabling transformation

Why care about Boolean Algebra?

• It helps us represent the function of a combinational logic block
• transform the function into simpler functions
  • leads to different hardware realizations
  • logic minimization or logic simplification
  • we can automate the process with computer aided design or electronic design automation
• different boolean expressions lead to different logic gate implementations
  • different hardware, cost, latency, energy properties

Standardized function representations

A universally agreed way of representing these boolean functions.

Sum of Products (SOP)
SUM(OR) of all products (minterms) that cause output to be 1.

• AND = product

  OR = sum

• create a truth table for the inputs
• find all the input combinations (minterms) where the output is TRUE
• the function is 1 if any of the input combinations is 1, we do not care for the rest

If 0 then complement the variable
if 1 then keep as is

We can shorten this by enumerating each row with the decimal value of the binary number made with the inputs. This way we do not need the truth table and instead can create an expression that represents it.

But then we always have to get the minimal form for the canonical form of the boolean expression.

Product of Sums (POS)

• find all the input combinations where the output is FALSE
• the function is 0 if any of the input combinations is 0, we do not care for the rest

If 1 then complement the variable
if 0 then keep as is

Combinational logic

• It is often grouped into larger building blocks to build more complex systems
• hides the unnecessary gate level details to emphasise the function of the building block, creates abstraction for easier building.

Decoder

• input pattern detector
• n inputs and 2^n outputs
• exactly one of the outputs is 1 and all the rest are 0 - one hot encoding
• used in memory, instruction decoding, I/O selection Example 2:4 decoder

Multiplexer (MUX)

• selects one of the n inputs to connect it to the output Example 2-to-1 MUX has 2 data inputs and 1 select input.
  If select input is 0 then selects D0, if select input is 1 then selects D1
• can be used as look up tables to perform logic functions

Programmable logic array (PLA) enables the 2 level SOP implementation of any N-input M-output function.

[look up]
ripple carry adder
lookahead adder

Resources

• DDCA lecture 2