L1516

1
Digital Logic

• L15-16
• Guilin Wang
• School of Computer Science
• The University of Birmingham
• adapted from Ata Kaban

2
Topics for This Lecture

• Gates and Boolean logic
• AND, OR, NOT, NAND, NOR
• Integrated circuits
• SSI, MSI, LSI, VLSI
• Memory
• Flip-Flop
• Arithmetic circuits
• Half-adder, Full-adder, ALU

3
Basics of Hardware Design

• Components of a computer
• CPU ALU, adder, multiplier, instruction
  controller, data path,
• Memories registers, cache, main memory,
• I/O devices
• buses
• The lowest logical design level
• Digital circuits are built from gates
• gates were built from transistors
• Device level (physics)
• N/P-type semiconductors (transistors)

4
Digital Circuits

• In a digital circuit, only two logical values are
  present 0 or 1.
• continuous voltage range (within bounds)
• 0 is low signal (voltage range 0 to 1)
• 1 is high signal (voltage range 2 to 5)
• Gates
• electronic devices that compute functions of 0/1
• made from transistors (very fast small switches)

5
How a Transistor Works

• - If Vin lt Vcv, no current flows through So,
  Vout is high.
• If Vin gt Vcv, current flows through So, Vout is
  low.
• As input signal is inverted, this forms a NOT
  gate.

6
The Three Main Gates

Truth Table A, B inputs X output X f(A,B)
NOT invert (negate) single input. AND 1
only if both inputs 1 OR 1 if at least one
input 1
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Two More Gates
NAND Output 0 only if both inputs 1 (inverted
AND) NOR Output 0 if at least one input
1(inverted OR)
Simpler - 2 instead of 3 transistors...
8
Boolean Algebra

• A Boolean function can have N variables, eg.
• M f (A, B, C).
• Write
• AB for A AND B
• AB for A OR B
• for NOT A
• Write functions as as a sum of product terms, e.g
  
• M BC A C AB ABC (majority vote).
• Such a formula leads directly to a possible
  circuit implementation.

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Boolean Algebra

• A circuit for majority function of three
  variables.

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Boolean Algebra

• Some identities of Boolean algebra.

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Circuit Equivalence

• Different circuits may implement the same
  function.
• 3 circuits for XOR (eXclusive OR)

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Circuit Equivalence

• Circuit designers often try to reduce the number
  of gates in their products.
• AND, OR, NOT gates are enough to implement any
  Boolean function.
• NAND itself has the above property. So, we say
  NAND is complete.
• NOR is also complete.
• How to prove the above statements?

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Circuit Equivalence

• For example, we can construct AND and OR gates
  using NAND gate as follows

• Exercises
• Can you do NOT?
• How to construct AND, OR, and NOT gates from NOR
  gate?

14
Integrated Circuits (chips)
SSI chip 5mm x 5mm Standardised Pins
Classification based on number of gates Small
Scale Integrated (SSI) circuit 1-10 gates Medium
Scale Integrated (MSI) circuit 10-100
gates Large Scale Integrated (LSI) circuit
100-100,000 gates Very Large Scale Integrated
(VLSI) circuit gt 100,000 gates
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Types of Chips

• Combinational circuits
• Boolean functions, transform inputs to output
• Control circuits
• data buses, clocks, etc
• Memories
• can store bits contain feedback
• Flip-Flop
• Arithmetic circuits
• Half-adder, Full-adder, ALU

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Clock Signals

• Clock signals are used to provide synchronization
  among events.
• Clock is a circuit that emits a series of pulses
  with a precise pulse width and precise interval
  btw consecutive pulses.
• This precise time interval is called clock cycle
  time.
• Pulse frequencies 1 to 500 MHz.
• Clock cycles 1000 to 2 nsec.
• The clock frequency is usually controlled by a
  crystal oscillator.

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Clock Signals

• Some events may happen during one single clock
  cycle and in a specific order.
• In such cases, the clock cycle must be divided
  into subcycles.
• A common way is to get a phase-shifted clock
  signal from the primary.
• The example below provides four time references,
  i.e., rising (falling) edge of C1 (C2).

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The Flip-Flop

• A bistable device (also called a clocked D latch)
• Has inputs D and clock signal, and output Q
• When control is on, Q D
• When control is off, output is always available
  but cannot change. Hence, one bit value D is
  stored.

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Memories

• Registers
• N-bit flip-flop gives N-bit register
• N bits one word
• Memories
• M registers gives M-word memory

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Memories

• Logic diagram for a 4x3 memory
• (Book 2, p166)
• 8 input lines
• - Data I0, I1, I2
• - Address A0, A1
• - Control CS, RD, OE
• 3 output lines D0, D1, D2
• Function
• - Write 3 bits into a selected word or
• - Read 3 bits from a selected word.

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Arithmetic Circuits

• Addition
• 1-bit addition yields 1-bit result and 1-bit
  carry
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 0 carry 1
• Adders
• half-adder 1-bit adder with 2 bit input
• full-adder 1-bit adder with additional carry
  input
• N-bit adder requires N full-adders

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Half-adder

• 2 bits on input
• 1 bit sum plus 1 bit carry on output
• Half-adder cannot handle carry in the middle of
  the word...

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Full-adder

• 2 bits and carry on input
• 1 bit sum plus 1 bit carry on output
• Built from two half-adders
• N-bit adder can be built from this 1-bit full
  adder. However, there is an ripple effect.
• How to get faster adder without such a delay?

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Arithmetic Logic Units (ALU)

• A 1-bit
• ALU

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Arithmetic Logic Units (ALU)

• The above 1-bit ALU can perform any of the
  following operations A AND B, A OR B, , AB.
• This depends on the values of the function-select
  inputs F0 and F1 , i.e., 00, 01, 10, or 11.
• An 8-bit ALU can be built up from eight 1-bit ALU
  slices.

(For simplicity, the enables and invert signals
are not shown.)
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Summary

• We have discussed
• Boolean function, gates
• Flip-flop, memories
• Adder, ALU
• Further topics
• How to use data buses transferring data/control
  info along wires, one wire per bit.
• How to use control signals (clocks) avoiding
  conflicts.