ECNG 1014: Digital Electronics

1. ECNG 1014: Digital Electronics Combinational Circuits

2. Logic Device Classification Combinational Circuit output is a function only of the current inputs eg., AND gates, decoders Sequential Circuit 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

3. 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.

4. 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).

6. 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

7. 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…….

8. Binary 2-to-4 decoder Enable input x = don’t care. 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!!

9. 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 The 1/2 => the 74139 has TWO (independent) decoders. Only using 1 of them Bubble <=> Active low enable 0 F Y X Select inputs: B is the MSB in the select word i.e. BA Most decoders use Active Low outputs => faster

10. Complete 74x139 Decoder All Inputs buffered to minimise loading

11. DEC0_L DEC1_L DEC2_L DEC3_L DEC4_L DEC5_L DEC6_L DEC7_L b2 b0 b1 How do we get a larger decoder? Cascade smaller ones…. Use MSB(‘s) to select one decoder (for one part of the Truth Table) Upper half of TT (b2=0) Lower half of TT (b2=1) Use lower SB’s to select the output The “_L “ is a standard nomenclature for an active low signal line

12. Or.. We can use an off-the-shelf device, if available 74x138 3-to-8-decoder EN = (G1)(G2A)’(G2B)’ C is MSB

14. 2-4 Y0 Address Y1 Decoder Y2 Y3 16K RAM Data Address CS 8 A0-13 MC6809uP Data(8) Address (16) Control To all memory data pins 8 16K RAM Data Address CS 8 To all memory address pins A0-13 A0-13 14 2 16K RAM Data Address CS 8 A15, A14 A0-13 R/W: Connected to Memory R/W to control data direction 16K RAM Data Address CS 8 A0-13 The 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

15. Address Bit: 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0000160 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3FFF160 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4000160 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7FFF160 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8000161 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 BFFF161 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 C000161 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 FFFF161 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Decoder Design Work Sheet Internal Memory Address Lines Bank0: A14=A15=0 Bank1: A15=0.A14=1 Bank2: A15=1.A14=0 Bank2: A15=1.A14=1 FIXED for each bank

16. 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:

17. 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: 4 A[3:0] A>B A=B 4-bit Comparator A>B B[3:0] 4

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

19. 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.

20. A Binary Encoder

21. 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.

22. Block Diagrams Logic equations

23. ECNG 1014: Digital Electronics Combinational Devices: Pt 2 - Multiplexers Section 5.7, Wakerly

24. 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.

25. 74x151 EN A B C D0 Y D1 Y D2 D3 D4 D5 D6 D7 E.g. 74151 8input x 1 bit MUX EN_L Select Inputs Data Inputs If EN = 1 Then Y=0 Else k = val(CBA) Y=Dk Endif Y=ENmkDk, where mk is the k’th minterm of select word CBA.

26. 74x157 G S 1A 1Y 1B 2A 2Y 2B 3A 3Y 3B 4A 4Y 4B E.g. 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 iY = G(S’•iA + S • iB), i=12,3,4

27. a d b’+c’ SEL D0 Y D1 F As for decoders, can use MUXes to synthesise logic E.g. Implement F=ab’+a’d+abc’ on a 2-1mux Solution: 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 F= Y = SEL’D0 + SEL D1 => F = D0 if SEL=0 OR F=D1 if SEL=1 <=> D0 = F|SEL=0 <=> D0 = F|a=0 = d Similarly, D1 = F|a=1 = b’+bc’ =b’+c’

28. D0 = 0 D1 = 0 00 01 WX 11 10 0 0 0 0 0 1 1 0 1 1 1 1 0 0 1 1 YZ 00 01 11 10 D2 = Z D3 = Z D7 = 1 D6 = 1 D5 = 1 D4 = 0 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. 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.

29. 74x151 EN A B C D0 Y D1 Y D2 D3 D4 D5 D6 D7 On a 74151 ….. 0 Y X W 0 0 Z Z 0 1 1 1 F= WXZ’ + WX’YZ + XZ Note: No logic gates required!! Exercise: Verify that this works by referring to the operation of the 74151 MUX

30. 00 01 WX 11 10 0 0 0 0 0 1 1 0 1 1 1 1 0 0 1 1 YZ 00 01 11 10 Same Function on a 4-1 MUX Y S0 S1 D0 Y D1 D2 D3 Z F

31. 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.

32. 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.

33. 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.

34. input 2n outputs 1 x 2n DEMUX n-bit selections Demultiplexer

35. ECNG 1014: Digital Electronics Combinational Devices: ROMs

36. 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

37. 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 E.g. A ROM Modern 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). Memory array

38. A2 An-1 A0 A1 S0 S1 Y D0 Two-dimensionaldecoding 2D decoding in a 2n x 1 ROM • Accommodates large address widths in a more compact form • Uses a decoder and MUX to select bits 2n bits of storage

39. Larger example, 32Kx8 ROM:Stack for word storage 215=32K address space

40. EPROMs, EEPROMs, Flash PROMs • Programmable and erasable using floating-gate MOS transistors • More reliable than fuse based 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

41. Typical commercial EPROMs

42. 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)

43. Microprocessor EPROM application Be cool! Do NOT panic!

44. Full featrured ROM structure (except for programming)

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

46. X Row Starting Address XY 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 Hexadecimal EPROM Listing 4x4 multiplier example Y

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

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

49. ECNG 1014: Digital Electronics Combinational Devices: PLDs Wakerly Section 5.3

50. 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)