# Chapter 3: Combinational Logic Design

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1 Chapter 3: Combinational Logic Design 1

2 Introduction We have learned all the prerequisite material: Truth tables and Boolean expressions describe functions Expressions can be converted into hardware circuits Boolean algebra and K-maps help simplify expressions and circuits Now, let us put all of these foundations to good use, to analyze and design some larger circuits 2

3 Introduction Logic circuits for digital systems may be Combinational Sequential A combinational circuit consists of logic gates whose outputs at any time are determined by the current input values, i.e., it has no memory elements A sequential circuit consists of logic gates whose outputs at any time are determined by the current input values as well as the past input values, i.e., it has memory elements 3

4 Introduction Each input and output variable is a binary variable 2^n possible binary input combinations One possible binary value at the output for each input combination A truth table or m Boolean functions can be used to specify input-output relation 4

5 Design Hierarchy A single very large-scale integrated (VLSI) processos circuit contains several tens of millions of gates! Imagine interconnecting these gates to form the processor No complex circuit can be designed simply by interconnecting gates one at a time Divide and Conquer approach is used to deal with the complexity Break up the circuit into pieces (blocks) Define the functions and the interfaces of each block such that the circuit formed by interconnecting the blocks obeys the original circuit specification If a block is still too large and complex to be designed as a single entity, it can be broken into smaller blocks 5

6 Divide and Conquer 6

7 Hierarchical Design due to Divide and Conquer 7

8 Hierarchical Design A hierarchy reduce the complexity required to represent the schematic diagram of a circuit In any hierarchy, the leaves consist of predefined blocks, some of which may be primitives. No need to design a predefined block! A primitive block is the one with a logic symbol, but no logic schematic Primitive blocks such as gates are of predefined blocks More complex structures can also be defined as predefined blocks The blocks can be reused; for a block reused, only one design is required Instance: Appearance of a design within a block Instantiation: Using a block in the design 8

9 Designing Complex Circuits Computer-Aided Design (CAD) tools Schematic capture tools: Support the drawing of blocks and interconnections at all levels of the hierarchy Libraries of graphic sysmbols Logic Simulator Hardware Description Languages (HDLs) VHDL and Verilog, both are the IEEE standard VHDL: Very High Speed Integrated Circuits (VHSIC) HDL Like programming langueages, but tuned to describe hardware structures and behavior Alternative to schematics (structural description) Behavioral description also possible Logic synthesis: RTL of a system -> Netlist (structural description) 9

10 Levels of Integration Digital circuits are constructed with integrated circuits An integrated circuit (IC) is a silicon semiconductor crystal (informally a chip) containing the electronic comonents for the digital gates and storage elements Small-scale integrated (SSI): Primitive gates, # of gates < 10 Medium-scale integrated (MSI): Elementary digital functions (4-bit addition), 10 < # of gates < 100 Large-scale integrated (LSI): Small processors, small memories, programmable modules, 100 < # of gates < a few thousand Very large-scale integrated (VLSI): Complex microprocessors and digital signal processing chips, several thousand to tens of millions of gates 10

11 Design Procedure The design of a combinational circuit involves the following steps: Specification: How the circuit operates is clearly expressed Formulation: Derivation of the truth table or the Boolean equations that define the relationship between inputs and outputs Optimization: Algebraic or K-map optimization of the truth table and draw the corresponding logic diagram Technology Mapping: Tranform the logic diagram to a new diagram using the available implementation technology Verification: Verify the correctness of the final design 11

12 Comparing 2-bit Numbers - Specification Let s design a circuit that compares two 2-bit numbers, A and B. The circuit should have three outputs: G ( Greater ) should be 1 only when A > B E ( Equal ) should be 1 only when A = B L ( Lesser ) should be 1 only when A < B Make sure you understand the problem Inputs A and B will be 00, 01, 10, or 11 (0, 1, 2 or 3 in decimal) For any inputs A and B, exactly one of the three outputs will be 1 12

13 Comparing 2-bit Numbers - Specification Two 2-bit numbers means a total of four inputs We should name each of them Let s say the first number consists of digits A1 and A0 from left to right, and the second number is B1 and B0 The problem specifies three outputs: G, E and L 13

14 Comparing 2-bit Numbers - Formulation For this problem, it s probably easiest to start with a truth table. This way, we can explicitly show the relationship (>, =, <) between inputs A four-input function has a sixteenrow truth table It s usually clearest to put the truth table rows in binary numeric order; in this case, from 0000 to 1111 for A1, A0, B1 and B0 Example: 01 < 10, so the sixth row of the truth table (corresponding to inputs A=01 and B=10) shows that output L=1, while G and E are both 0. A1 A0 B1 B0 G E L

15 Comparing 2-bit Numbers - Formulation A1 A0 B1 B0 G E L

16 Comparing 2-bit Numbers - Optimization Let s use K-maps. There are three functions (each with the same inputs A1 A0 B1 B0), so we need three K-maps B1 B1 B A A0 A A A A B0 B0 B0 G(A1,A0,B1,B0) = A1 A0 B0 + A0 B1 B0 + A1 B1 E(A1,A0,B1,B0) = A1 A0 B1 B0 + A1 A0 B1 B0 + A1 A0 B1 B0 + A1 A0 B1 B0 L(A1,A0,B1,B0) = A1 A0 B0 + A0 B1 B0 + A1 B1 16

17 Comparing 2-bit Numbers - Optimization G = A1 A0 B0 + A0 B1 B0 + A1 B1 E = A1 A0 B1 B0 + A1 A0 B1 B0 + A1 A0 B1 B0 + A1 A0 B1 B0 L = A1 A0 B0 + A0 B1 B0 + A1 B1 17

18 BCD-to-Excess-3 Code Converter - Specificaiton The excess-3 code for a decimal digit is the binary combination corresponding to the decimal digit plus 3. For example, the excess-3 code for decimal digit 5 is the binary combination for = 8, which is Each BCD digit is four bits with the bits, from most significant to least significant, labeled A, B, C, D. Each excess-3 digit is four bits, with the bits, from most significant to least significant, labeled W, X, Y, Z. BCD digit A B C D BCD-to Excess-3 W X Y Z Excess-3 digit 18

19 BCD-to-Excess-3 Code Converter - Formulation 19

20 BCD-to-Excess-3 Code Converter - Optimization 20

21 BCD-to-Excess-3 Code Converter - Optimization 21

22 BCD-to-Seven-Segment Decoder - Specification Digital readouts found in many consumer electronic products often use Light Emitting Diodes (LEDs). Each digit of the readout is formed from seven LED segments. Each segment can be illuminated by a digital signal. A BCD-to-sevensegment decoder is a combinational circuit that accepts a decimal digit in BCD and generates the appropriate outputs for the segments of the display for the decimal digit. The seven outputs of the decoder (a,b,c,d,e,f,g) select the corresponding segments in the display. BCD-to-seven-segment decoder has four inputs, A, B, C, and D for the BCD digit and seven outputs, a through g, for controlling the segments. 22

23 BCD-to-Seven-Segment Decoder - Formulation 23

24 BCD-to-Seven-Segment Decoder - Optimization a= A C + A BD + B C D + AB C b= A B + A C D + A CD + AB C c= A B + A D + B C D + AB C d= A C^D + A B C + B C D + AB C + A BC D e= A CD + B C D f= A BC + A C D + A BD + AB C g= A CD + A B C + A BC + AB C 24

25 4-Bit Equality Comparator - Specification The inputs to the circuit consist of two vectors: A(3:0) and B(3:0). Vector A consists of four bits, A(3), A(2), A(1), and A(0), with A(3) as the msb. Vector B consists of four bits, B(3), B(2), B(1), and B(0), with B(3) as the msb. The output of the circuit is a single bit variable E. Output E is equal to 1 if A and B are equal and equal to 0 if A and B are unequal. 4 4 A B E 25

26 4-Bit Equality Comparator - Formulation Since there are 8 inputs, using a truth table is impractical!!! Apply divide and conquer design approach Observation: In order for A and B to be equal, the bit values in each of the respective positions, 3 down to 0, must be equal We need four 1-bit comparator We need an additional circuit to combine the outputs of 1-bit comparators 26

27 4-Bit Equality Comparator - Optimization A(i) B(i) E(i) bit Comparator E= (E(0) + E(1) + E(2) + E(3)) 27

28 4-Bit Equality Comparator - Optimization 28

29 Techology Mapping NAND Gate Implementation The circuit is defined in Sum-of-Products form Goal: Use only NAND gates to implement the circuit F= XY + X Y + Z F= (F ) = [ (XY + X Y + Z) ] = [ (XY ). (X Y). (Z ) ] X Y F Z 29

30 Techology Mapping NOR Gate Implementation The circuit is defined in Product-of-Sums form Goal: Use only NOR gates to implement the circuit F= (A + B)(C + D)E F= (F ) = [ { (A + B)(C + D)E } ] = [ (A + B) + (C + D) + E ] A B C D F E 30

31 Multilevel NAND (NOR) Implementation 1. Replace each AND and OR gate with the NAND (NOR) gate and inverter equivalent circuits shown below 31

32 Multilevel NAND (NOR) Implementation 2. Cancel all inverter pairs 3. Without changing the logic function, (a) Push all inverters (b) Replace inverters in parallel with a single inverter that drives all of the outputs of the parallel inverters (c) Repeat a and b until there is at most one inverter between the circuit input or driving NAND(NOR) gate output and the attached NAND(NOR) gate inputs 32

33 Example F= AB + (AB) C + (AB) D + E 33

34 Example F= AB + (AB) C + (AB) D + E 34

35 Summary Functions can be represented with expressions, truth tables or circuits. These are all equivalent, and we can arbitrarily transform between them Designing a circuit requires you to first find a (simplified) Boolean expression for the function you want to compute. You can then convert the expression into a circuit 35

36 Summary NAND and NOR are universal gates which can replace all others There are two representations for NAND gates (AND-NOT and NOT- OR), which are equivalent by DeMorgan s law Similarly, there are two representations for NOR gates too You can convert a circuit with primitive gates into a NAND or NOR diagram by judicious use of the axiom (x ) = x, to ensure that you don t change the overall function 36

37 Summary Circuits made up of gates, that don t have any feedback, are called combinatorial circuits No feedback: outputs are not connected to inputs If you change the inputs, and wait for a while, the correct outputs show up Why? Capacitive loading ( fill up the water level analogy) So, when such ckts are used in a computer, the time it takes to get stable outputs is important For the same reason, a single output cannot drive too many inputs Will be too slow to fill them up May not have enough power So, the design criteria are: Propagation delay (how many gets in a sequence from in to out) Fan-out Fan-in (Number of inputs to a single gate) 37

38 Verification - Circuit Analysis Circuit analysis involves figuring out what some circuit does Every circuit computes some function, which can be described with Boolean expressions or truth tables So, the goal is to find an expression or truth table for the circuit The first thing to do is to figure out what the inputs and outputs of the overall circuit are Inputs: x, y,z Output: f 38

39 Verification - Circuit Analysis Write expressions for the outputs of each individual gate, based on that gate s inputs Start from the inputs and work towards the outputs It might help to do some algebraic simplification along the way 39

40 Verification - Circuit Analysis It s also possible to find a truth table directly from the circuit. Once you know the number of inputs and outputs, list all the possible input combinations in your truth table A circuit with n inputs should have a truth table with 2 n rows x y z f

41 Verification - Circuit Analysis You can simulate the circuit by hand to find the output for each possible combination of inputs x y z f

42 Verification - Circuit Analysis Doing the same thing for all the other input combinations yields the complete truth table This is simple, but tedious x y z f

43 Verification - Circuit Analysis Remember that if you already have a Boolean expression, you can use that to easily make a truth table For example, since we already found that the circuit computes the function f(x,y,z) = xz + y z + x yz, we can use that to fill in a table: x y z xz y z x yz f

44 Verification - Circuit Analysis The opposite is also true: it s easy to come up with an expression if you already have a truth table Convert a truth table into a sum of minterms expression x y z f f(x,y,z) = x y z + x yz + xy z + xyz = m 1 + m 2 + m 5 +m 7 You can then simplify this sum of minterms if desired using a K-map, for example 44

45 Circuit Analysis Summary After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does You can easily convert between expressions and truth tables Find the circuit s inputs and outputs Find a Boolean expression for the circuit Find a truth table for the circuit 45

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