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Lecture 5: Combinational Components in Digital Design

This lecture covers topics such as products-of-sums form examples, 5-variable and larger Karnaugh maps, decoders, multiplexers, and more in digital design. Readings for this lecture were on September 21, 2015.

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Lecture 5: Combinational Components in Digital Design

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  1. Lecture 5 Combinational Components CSCE 211 Digital Design • Topics • Products-of-Sums Form examples • 5 variable and larger Karnaugh Maps • Components: Decoders, Multiplexers • Readings September 21, 2015

  2. Overview • Last Time: • Boolean Algebra Continued • Combinational Circuit Analysis • Sums-of-Products Form • Karnaugh Maps 3,4 variable maps • Don’t Care Conditions • Products-of-Sums Form • New: • Review Products-of-Sums Form • (5, 6, … variable maps) • Decoders • Multiplexers • Circuits kits on paper

  3. Karnaugh Map Simplification • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  4. Karnaugh Map Simplification • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  5. Products-of-Sums Simplification • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  6. 5 Variable Map Simplification • F(V, W,X,Y,Z) = X X WX 00 01 11 10 YZ WX 00 01 11 10 00 01 11 10 YZ 00 01 11 10 Z Z Y Y W W

  7. 5 Variable Map Simplification • F(V, W,X,Y,Z) = X X WX 00 01 11 10 YZ WX 00 01 11 10 00 01 11 10 YZ 00 01 11 10 Z Z Y Y W W

  8. 6 Variable Map 0 4 12 8 9 1 5 13 • F(U,V,W,X,Y,Z) = 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14

  9. 6 Variable Map 0 4 12 8 9 1 5 13 • F(U,V,W,X,Y,Z) = 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14

  10. Combinational Circuits • A combinational circuit is one that • The outputs are functions strictly of the inputs • There are no feedback loops

  11. 3x8 Decoder

  12. 4x16 decoder from 2x4s

  13. Multiplexers • A multiplexer selects one of its inputs to route to its outputs.

  14. BreadBoard

  15. Wiring an LED To wire an led • Hook the positive to Vcc • Hook the negative to a 330 ohm resistor • Hook the resistor to Gnd • Check for loose wires • Check for shorts - + • See section 3.7.5 page 129-130 for more details • I LED = 10 mA needed to light the LED • Voltage drop is about 1.6V • 303 Ohms

  16. 74LS00 – Quad 2 input NAND

  17. 74LS04 Hex Inverter

  18. Half adder • How many inputs? • How many outputs?

  19. 6 Variable Map 0 4 12 8 9 1 5 13 • F(U,V,W,X,Y,Z) = 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14 0 4 12 8 9 1 5 13 11 3 15 7 10 2 6 14

  20. Analyze This! 0 0 1 1 F1 = ? F2 = ? What are the delays?

  21. Quick What’s This?

  22. What’s This?

  23. 8 to 1 Mux from 4x1 Muxes

  24. Big Multiplexers from smaller ones • Show the design of a 32-to-1 Mux from 8-to-1’s and smaller muxes

  25. BreadBoard

  26. Wiring an LED To wire an led • Hook the positive to Vcc • Hook the negative to a 330 ohm resistor • Hook the resistor to Gnd • Check for loose wires • Check for shorts - + • See section 3.7.5 page 129-130 for more details • I LED = 10 mA needed to light the LED • Voltage drop is about 1.6V • 303 Ohms

  27. 74LS00 – Quad 2 input NAND

  28. 74LS04 Hex Inverter

  29. Two Bit adder • How many inputs? • How many outputs? • Do we have enough chips?

  30. Implementing a Binary Adder Using a Decoder PC X Y 3x8 Decoder

  31. 74LS139 DecoderDual 2x4 decoder

  32. Using a 74LS139 to implement a Half-adder X Y S C

  33. 74LS157 Dual 4 input Mux

  34. Hardware Description Languages • Hardware description language or HDL is any language from a class of computer languages for formal description of electronic circuits • Boolean Algebra was applied to circuits by Shannon 1948. • http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html • Current HDLs include: • Verilog HDL • VHDL – VHSIC HDL • VHSIC – Very High Speed Integrated Circuits • ABEL HDL - Advanced Boolean Expression Language • http://en.wikipedia.org/wiki/Hardware_description_language

  35. Seven Segment Display • Common anode

  36. Functions for 74LS47 with don’t cares • a(D,C,B,A) = D + A.C + A.B + A’.C’ • b(D,C,B,A) = D + (D'*C') + (A'*B') + (A*B) • c <= • d = • e = A(bar) and (B or C(bar)) • f = D + A'B' + B'C + A'BC • g=D + B'C + C'B + A'B

  37. Karnaugh Map Simplification • On a real 74LS47 the outputs for 10, …15 are not don’t cares. • They would indicate errors in BCD input. We could use the period for that. • period(D,C,B,A)=SUM( ) • dc(D,C,B,A) = SUM( ) C DC 00 01 11 10 BA 00 01 11 10 period(D,C,B,A) = A B D

  38. Transistors • History • 1790s Ben Franklin “assigns” negative charge to electrons • 1898 Thompson discovers the electron • 1947 Shockley, Bardeen and Brattain “invent” transistor • 1958 first Integrated Circuit, Texas Instruments • 1971 Intel 4004, microprocessor, Ted Hoff • Timeline • http://www.pbs.org/transistor/

  39. Hot Batteries • You should regularly check your batteries “slightly warm” is OK but hot indicates that your circuit has a short circuit. • Unplug quickly and check. • Look for direct lines Vcc to GND. • Remember you need 330 ohm resistors in series with LEDs and that includes segments of the seven segment display. • Recheck sections of the breadboard.

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