1. Transistors, Logic Gates and Karnaugh Maps References: http://www.st-and.ac.uk/~www_pa/Scots_Guide/info/comp/active/BiPolar/page1.html Lecture 4 from last semester Introduction to Digital Systems (J.Palmer and D. Perlman)

2. Transistors There are various kinds of transistors: bipolar, field-effect, etc. They differ in stability, energy usage, and so on, but they serve a similar purpose They are used to amplify a signal or to act as a switch It is as switches that they are used in computers

3. Diode Review Recall that a pn junction — the joining together of p-doped (“too few” electrons) and n-doped (“extra” electrons) — makes a diode A diode is a circuit element that allows current to flow in one direction (forward bias) but not in the other (reverse bias)

4. Diode Review (Cont.) Some of the “extra” electrons from the n side fill the empty levels in the p side, forming a region in which the valence band is filled and conduction band is empty This region (called the depletion zone) is a poor conductor

5. Bipolar transistors A bipolar transistor starts with two back-to-back diodes (pn junctions) There are two kinds NPN and PNP The middle region is usually smaller N-doped P-doped N-doped P-doped N-doped P-doped

6. Third lead So far the device seems useless; two back-to-back diodes wouldn’t conduct in either direction But we add a third lead (connection) directly to the middle portion

7. Not symmetric The transistor would seem to be symmetric with the two N-doped regions being the same, but actually these regions differ in their amount of doping and serve different purposes in the transistor

8. Collector, base, emitter Collector Base Emitter

9. Connecting the transistor Imagine applying a potential difference (voltage) across the base-collector leads with the collector higher, this reverse biases that pn junction so there would be no current flow CBECBE

10. No flow There is no flow because of the depletion zone (the region in which the valence band is filled and the conduction band empty) Reverse bias voltages tend to make the depletion zone a bit larger

11. Connecting the transistor (Cont.) Now consider applying a (smaller) voltage across the base-emitter leads with the base higher, this forward biases that pn junction so current will flow CBECBE

12. Flow Forward biasing a pn junction tends to eliminate the depletion zone (in this case putting electrons into the conduction band) Because the transistor has one shared depletion zone that has been eliminated by the base-emitter forward bias, both currents (collector-base and base-emitter) can flow

13. NPN in a circuit The arrow on an NPN points from base to emitter indicating the forward-bias direction that turns the transistor “on” CBECBE

14. Off

15. Base-emitter circuit Little to no current flowing Most of the voltage dropped across the base- emitter as opposed to the resistor in the circuit Collector-emitter circuit Little to no current flowing Most of the voltage dropped across the collector- emitter as opposed to the resistor in the circuit

16. On

17. Base-emitter circuit Current flowing Most of the voltage dropped across the resistor as opposed to the base-emitter in the circuit Collector-emitter circuit Current flowing Most of the voltage dropped across the resistor as opposed to the collector-emitter in the circuit

18. Unusual feature One feature that people find unusual when learning about transistors is that when the transistor is “on” the collector-emitter voltage is less than the base-emitter voltage This is because in the collector-emitter, one is going from n-doped material to n-doped material, whereas in the base-emitter, one is going from p-doped to n-doped

19. Logic gates As seen in lab, the on-off nature of diodes and transistors make them ideal for building logic gates Logic gates have input which is interpreted as a logic value (0 or 1, low or high, false or true) and have output which can also be interpreted logically

20. Logic gates LogicCircuit Symbol NOT AND OR NAND NOR

21. A Truth Table ABCOut 0000 0010 0101 0111 1000 1010 1101 1111

22. Simplifying A´BC´ + A´BC + ABC´ + ABC A´B (C´ + C) + AB (C´ + C) A´B + AB (A´ + A) B B ABC means A and B and C A + B means A or B A’ means not A

23. Simplifying made easy Simplifying Boolean expressions is not always easy So we introduce next a method that is supposed to make simplification more visual

24. Gray code In addition to binary numbers, there is another way of representing numbers using 1’s and 0’s It is not useful for doing arithmetic, but has other purposes In gray code the numbers are ordered such that consecutive numbers differ by one bit only

25. Gray code (Cont.) 000 001 011 010 110 111 101 100

26. Constructing Gray code 0 1

27. Reflect lower bits and 0’s then 1’s in front 00 01 11 10 Lower bits Reflect through red line

28. Reflect lower bits and 0’s then 1’s in front (again) 000 001 011 010 110 111 101 100 Reflect through red line

29. An important property In gray-code order, two consecutive rows of a truth table differ by one bit only If two consecutive rows contain a 1, then a simplification of the Boolean expression is possible

30. A Truth Table Revisited ABCOut 0000 0010 0111 0101 1101 1111 1010 1000

31. Improving Some combinations that differ only by a single bit are not in consecutive rows Thus we might miss such a simplification So we put some of the inputs in as columns

32. ABCOut 0000 0010 0111 0101 1101 1111 1010 1000

33. A row-column version AB\C01 0000 0111 1111 1000

34. Karnaugh-map This way of arranging truth tables combined with the rules for simplifying Boolean expressions goes under the name Karnaugh map

35. The rules One identifies blocks (as large as possible) containing 1’s The blocks must contain all 1’s The number of 1’s should be a power of 2 (1, 2, 4, 8, …) A given 1 can belong to more than 1 block

36. Wrapping Imagine that the rows wrap around, so for instance, a block can include the top and bottom rows (without intermediate rows) Similarly for columns

37. Example WXY’Z + W’XY’Z + WX’Y’Z’ + W’X’Y’Z’ + WXYZ’ + WXY’Z’ + W’XY’Z’ + W’XYZ’

38. Example in Karnaugh Z0110 WX\Y0011 001 W’X’Y’Z’ 000 011 W’XY’Z’ 1 W’XY’Z 01 W’XYZ’ 111 WXY’Z’ 1 WXY’Z 01 WXYZ’ 101 WX’Y’Z’ 000

39. Result Y’Z’ + XY’ + X Z’ A block of size two eliminates one Boolean variable; a block of four eliminates two Boolean variables; and so on For a block identify the elements in the block that don’t change, AND them together, that’s your expression for the block Obtain an expression for each block and OR them together