1 ECNG 1014: Digital Electronics Combinational Circuits.

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ECNG 1014: Digital ElectronicsECNG 1014: Digital Electronics

Combinational Circuits

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Logic Device ClassificationCombinational Circuit output is a function only of the current inputs eg., AND gates, decoders

SequentialCircuit output is a function only of the current inputs AND past inputs i.e. the circuit has memory. Eg. counters

Small Scale Integration (SSI) Integrated circuit uses only a few gates (20 or less). Typically provides only basic gate functions

Medium Scale Integration (MSI) IC uses 20-200 gates to provide common higher level functions such as decoding, multiplexing, counting etc.

Large Scale Integration (LSI) Ics have 200-200K gates ( 400K transistors) or more to realise still higher functions such as small memories and microprocessors, PLDs, CPLDs etc.

Very Large Scale Integration (VLSI) IC with over 106 transistors or more which realises the highest level of logic functionality. Examples: Pentium level processors (50 mil xsistors!), FPGAs etc.

Device delay (and cost) goes up with complexity

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Design Procedure for Combinational Circuits• State the problem in combinational terms.• Determine the required inputs and outputs• Derive the truth table.• Simplify the output expressions using

Boolean laws or k-maps.• Implement the output expressions with

logic gates.• Design example: Implement a logic circuit

that detects if an unsigned 4-bit number is prime.

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Combinational Logic Using MSI and LSI devices

• Commercial ICs can perform complex functions using a single IC of type MSI or LSI, their characteristics are described in Logic data book (http://www.ti.com, semiconductor logic).

http://www.ti.com/

http://www.ti.com/

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TTL Device Number Description

7483 4-bit adder with fast carry

7485 4-bit magnitude comparator

74139 2-line-to-4-line decoder/demultiplexer



74145 BCD-to-Decimal encoder

74147 Priority encoder

74151 8 x 1 multiplexer

74150 16 x 1 multiplexer

74184 BCD-to-binary converter

74280 9-bit odd/even parity generator

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Decoders (5.4.1-5.4.5) General decoder structure

•A decoder is a MIMO

device that maps an

input code to a

different unique

output code, I.e. the

mapping is 1-to-1• Typically n inputs, 2n

outputs 2-to-4, 3-to-8, 4-

to-16, etc• Most common: Binary

Decoder maps each n-

bit input to assert only 1

of 2n outputs Also: 7-segment and BCD decoders

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Binary decoder applications

• Microprocessor memory systems

– selecting different banks of memory

• Microprocessor input/output systems

– selecting different devices (printer, serial, video etc)

• Microprocessor instruction decoding

– enabling different functional units within the uP.

• Memory chips

– enabling different rows of memory depending on inputted address

• Identifying or selecting various circuit options. Can also be used for logic synthesis since each output actually represents a unique minterm of the inputs…….

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Binary 2-to-4 decoder

Enable input

Once Enabled, ONLY the k’th logic output (k:0, 1,..2n-1) is asserted when the input binary word satisfies: I1I0 = k … the k’th minterm!!

=> ¡¡ Y0 = mo, Y1=m1, … Yi=mi !!

x = don’t care.

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SO… can use to synthesise all logic functions of n variables!!

E.g. Realise F=X,Y(0,3) using a 74x139 2-4 decoder

FYX

0

Bubble <=> Active low enable

Select inputs: B is the MSBin the select word i.e. BA

The 1/2 => the 74139 has TWO (independent) decoders. Only using 1 of them

Most decoders use Active Low outputs => faster

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Complete 74x139 Decoder

All Inputs buffered to minimise loading

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b2

b0

b1

DEC0_LDEC1_LDEC2_LDEC3_L

DEC4_LDEC5_LDEC6_LDEC7_L

How do we get a larger decoder?

Cascade smaller ones….

Use MSB(‘s) to select one decoder (for one part of the Truth Table)

Use lower SB’s to select the output

The “_L “ is a standard nomenclature for an active low signal line

Upper half of TT (b2=0)

Lower half of TT (b2=1)

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Or.. We can use an off-the-shelf device, if available

74x138 3-to-8-decoder

C is MSB

EN = (G1)(G2A)’(G2B)’

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MC6809uP

Data(8)

Address (16)

Control

16K RAMDataAddressCS




A15, A14

2 2-4 Y0Address Y1Decoder Y2

Y3

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To all memorydata pins

To all memoryaddress pins

A0-13

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R/W:Connected to Memory R/W to control data directionThe decoder selects

only 1 of the 4 banks at a time for data bus connection.

Specific words are determined by the lower 14 address lines connected to the memory device’s address inputs

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8

8

8

A0-13

A0-13

A0-13

A0-13

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Address Bit: 15 1413 1211 10 9 8 7 6 5 4 3 2 1 0000016 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 03FFF16 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1400016 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 07FFF16 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1800016 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0BFFF16 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1C00016 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0FFFF16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Bank0: A14=A15=0

Bank1: A15=0.A14=1

Bank2: A15=1.A14=0

Bank2: A15=1.A14=1

FIXED for each bank

Internal Memory Address Lines

Decoder Design Work Sheet

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Comparators

• The basic function of a comparator is to compare the magnitude of two binary quantities to determine the relationship of those quantities.

• In its simplest form, a comparator circuit determines whether two numbers are equal.

• A 1-bit comparator is implemented as:

B

X = 1 if A = B

X

UC1A

7433

2

31A

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•A complex comparator can be implemented taking into account inequality. For two numbers A and B, we will have to consider ( A = B, A < B, and A > B).

• The following figure gives a block diagram of a 4-bit comparator:

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4A[3:0]

B[3:0]

A>B

A=B

A>B

4-bit Comparator

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For the Output A > B to be active, we must have these conditions:- If A3 = 1 and B3 = 0, (A3'B3) then A >B- If A3 = B3 and A2 = 1(condition x3 for equality between A3 and B3) and A2 = 1 and B2 = 0, (x3A2B2'), then A >Betc

The 74X85 (X = LS, HC, or ALS) is a 4-bit comparator.

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Encoders

• An n-input binary encoder is a logic circuit that, given an n-bit input word X that contains one active signal xi, generates and output word Z, which is a binary representation of i, the index of the active input signal.

• Thus an encoder is the inverse of a decoder, and typically has n = 2k input lines of X and k output lines for Z.

• A four-input encoder, for instance, has n = 4 and k = 2, and maps the input combinations 1000, 0100, 0010, and 0001 onto the output combinations 00, 01, 10, and 11 respectively.

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A Binary Encoder

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An encoder is intended to identify a single active input signal. However there is nothing to prevent several of its X input lines from being active at the same time, because they may be driven from independent external sources.

To deal with this situation, most encoders are designed as priority encoders, which have the property that when several inputs are active at the same time, the output number I that appears on Z is the index of the input line xi with the highest priority.

The following figures give the block diagram and logic equations of a priority encoder.

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Block Diagrams

Logic equations

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Combinational Devices: Pt 2 - Multiplexers

Section 5.7, Wakerly

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A What?

A multiplexer (mux) is a device that allows us to select one of n-sets of b-bit data for transmission to its b-bit output. It is a digital switch.

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E.g. 74151 8input x 1 bit MUX

EN ABCD0 YD1 YD2D3D4D5D6D7

74x151

Select Inputs

Data Inputs

If EN = 1Then Y=0Else

k = val(CBA)Y=Dk

Endif

Y=ENmkDk, where mk is the k’th minterm of select word CBA.

EN_L

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2-input, 4-bit-wide 74x157

For I=1 to 4

If G=0

Then iY = 0

Else

If S=0

Then iY=iA

Else iY=iB

Endif

Endif

Endfor

G S1A 1Y1B2A 2Y2B3A 3Y3B4A 4Y4B

74x157

E.g.

iY = G(S’•iA + S • iB), i=12,3,4

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As for decoders, can use MUXes to synthesise logicE.g. Implement F=ab’+a’d+abc’ on a 2-1muxSolution: With 2 inputs, D0 and D1, say, the MUX will have 1 select input, SEL, say. Step 1: Choose variables for direct connection to the select inputs. This can be arbitrary but note that, as in all things, the actual choice will affect the amount of work required to complete the design

Let SEL = a, say

Step 2: Using expression for F, which will be the MUX output, select appropriate logic functions of remaining variables to drive the MUX data inputs. This may be derived by comparison of the MUX expression and the target logic function or from the function’s truth table

SEL

D0 Y

D1

a

d

b’+c’

FF= Y = SEL’D0 + SEL D1 => F = D0 if SEL=0 OR F=D1 if SEL=1 <=> D0 = F|SEL=0 <=> D0 = F|a=0 = dSimilarly, D1 = F|a=1 = b’+bc’ =b’+c’

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E.g. Use a Kmap to realise F= WXZ’ + WX’Y + XZ on an 8-1 MUX.

Step 1: There will be 3 select inputs (why?) SEL0, SEL1, SEL2.Let: W=SEL2, X=SEL1, Y=SEL0.

0 0 0 0

0 1 1 0

1 1 1 1

0 0 1 1

00

01WX

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10

YZ00 01 11 10

D0 = 0 D1 = 0

D3 = Z

D7 = 1

D5 = 1

D4 = 0

D2 = Z

D6 = 1

Step 2: From the K-map determine each input as a logic function of the remaining variable (Z):

e.g.: To determine the function to be applied to D3 note that D3 is selected when W=0, X=1, Y=1.

Now, identify all those cells for which WXY=011. Write the logic values in these cells as a function of the remaining variable, Z.

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EN ABCD0 YD1 YD2D3D4D5D6D7

74x151

0YXW00ZZ0111

F= WXZ’ + WX’YZ + XZ

On a 74151 …..

Note: No logic gates required!!

Exercise: Verify that this works by referring to the operation of the 74151 MUX

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0 0 0 0

0 1 1 0

1 1 1 1

0 0 1 1

00

01WX

11

10

YZ00 01 11 10

S0S1

D0 YD1D2D3

Y

Z

F

Same Function on a 4-1 MUX

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Theorem on MUX realisations:

A 2n- to-1 MUX can be used to realise ALL (n+1)-variable logic functions without the use of logic gates (save for inverters).

Proof:

A 2n- to-1 MUX will have n select lines. Take any n of the

variables, X0 - Xn-1, and connect them individually to these n

select lines (step 1!). For each of the 2n- select combinations, the

function value will depend only on the value of the last variable,

Xn (step 2!) and could therefore assume one of the 4 possible

functions of this one variable: 0, 1, Xn, Xn’. The function is

therefore realised by appropriately connecting any one these 4

functions of the last variable to the data inputs.

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MUX uses

As a data selector (original purpose)

Not generally used for logic realisation using MSI components but… used as the core of combinational logic in some LSI and VLSI devices such as Field Programmable Gate Arrays (FPGAs). FPGAs exploit the MUX Theorem to allow general users to implement logic functions at VLSI density at a very low cost.

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Demultiplexers

• A demultiplexer basically reverses the multiplexing function. It takes data from one line and distributes them to given number of output lines.

• A demultiplexer has N = 2k output lines, k address selection and one data line. The number of the output line active is specified by the k bits address.

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2n outputs

n-bit selections

input

1 x 2n DEMUX

Demultiplexer

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Combinational Devices: ROMs

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Why “ROM”?

• Program storage– Boot ROM for personal computers– Complete application program storage for

embedded microprocessor systems.

• Actually, a ROM is a combinational circuit, basically a truth-table lookup.– Can perform any combinational logic function– Address inputs = function inputs– Data outputs (stored data) = function outputs

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ROM Internal Structure• Internal array of (address) word and (data) bit lines• Each intersection of word and bit lines interconnected by an element that can be

“programmed” to be open or short circuited• Each word line is activated from the address via decoder• Each bit line outputs a logic level corresponding to the state of the

interconnection with an activated word line

Memory array

E.g. A ROMModern memories use MOS transistors as interconnection elements. In this example, a fully connected transistor at the intersection of an activated word line and a bit line results in a stored 0 (actually a 1 since the bit lines are usually buffered via an inverter to the corresponding data output).

A disconnected transistor results in a stored 1 (a 0 at the output).

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Two-dimensional decoding• Accommodates large address widths in a more compact form

• Uses a decoder and MUX to select bits

S0S1

A0

A1

Y

D0

A2

An-1

2D decoding in a 2n x 1 ROM

2n bits of storage

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Larger example, 32Kx8 ROM: Stack for word storage

215=32K address space

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• Programmable and erasable using floating-gate MOS transistors

• More reliable than fuse based PROMs

EPROMs, EEPROMs, Flash PROMs

• Floating gate is isolated from rest of circuit but can be charged/discharged

• When discharged, transistor can be turned on by word line

• When charged transistor cannot be turned on by word line

• EPROM: floating gate discharged via UV light

• EEPROM/Flash PROMs: floating gate can be discharged electrically

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Typical commercial EPROMs

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EEPROM programming

• Apply a higher voltage to force bit change– E.g., VPP = 12 V

• Various bit erase procedures– Byte-byte– Entire chip (“flash”)– One block (typically 32K - 66K bytes) at a time

• Programming and erasing are a lot slower than reading (milliseconds vs. 10’s of nanoseconds)

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Microprocessor EPROM application

Be cool!

Do NOT panic!

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Full featrured ROM structure

(except for programming)

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Logic-in-ROM example:2-4 Decoder with polarity control

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4x4 multiplier example

Y

HexadecimalEPROM Listing

Row Starting AddressXY

X

Data at Address Location_0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _A _B _C _D _E _F

2*F16=2*1510

=3010=1E16

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Exercise

Determine all the data words stored in the 8x4 PROM of figure 10-5, Wakerly

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Exercise

Determine all the data words stored at A6-0= 0, 5, 29, 120 in the PROM of figure 10-7, Wakerly

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Combinational Devices: PLDs

Wakerly Section 5.3

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Programmable Logic Arrays (PLAs)

• Any combinational logic function can be realized as a sum of products.

• Idea: Build a large AND-OR array with lots of inputs and product terms, and programmable connections.– n inputs

• AND gates have 2n inputs -- true and complement of each variable.

– m outputs, driven by large OR gates• Each AND gate has a programmable connection to each

output’s OR gate.– p AND gates where (p<<2n= no. minterms)

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Example: 4x3 PLA, 6 product terms

Programming selectively breaks hardwired connections. These are all fully committed at manufacture

AND Array

OR Array

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Compact representation• Actually, closer to

physical layout (“wired logic”).

• In this model Default output is a “1”. Must programme “0’s”

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Some product termsI1’.I2’.I3’.I4’

I1’.I3.I4

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PLA Electrical Design• See Section 5.3.5 -- wired-AND logic

Fuse

Fuse intact => line can be pulled low when xsistor ONFuse blown => line remains in default HIGH state when attempt to put xsistor ON

Vcc

Default HIGH

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Programmable Array Logic (PALs)• PLAs share product terms. How beneficial is product

sharing?

– Not enough to justify the amount of programmable interconnections

• PALs ==> fixed OR array

– Each AND gate is permanently connected to a certain OR gate.

• Example: PAL16L8

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• 10 primary inputs• 8 outputs, with 7 ANDs per output• 1 AND for 3-state enable

• 6 outputs (IO1-6)available as inputs – more inputs, at expense of

outputs

– two-pass logic, helper terms

• Note inversion on outputs– output is complement of sum-

of-products– newer PALs have selectable

inversion• EPROM as well as original OTP

(1-time programmable) versions available

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Designing with PALs

• Compare number of inputs and outputs of the problem with available resources in the PAL.

• Write equations for each output using ABEL or VHDL.

• Compile the program, determine whether minimized equations fit in the available AND terms.

• If no fit, try modifying equations or providing “helper” terms.

1 ECNG 1014: Digital Electronics Combinational Circuits.

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Transcript of 1 ECNG 1014: Digital Electronics Combinational Circuits.