Ian Davies

# Digital Electronics

There will be 5 supervisions for this course.

## Supervision Work

To be handed in at least 24 hours in advance. Hand your work in to the main Caius Porters' Lodge or feel free to email your work to me in any sensible format.

### Supervision 1

Hardware exercises from chapters 3, 13, 20 & 38 of Dewdney's New Turing Omnibus.
• Chapter 3 - Systems of Logic
• Show that NOR is a complete base, all by itself.
HINT: Just show that AND and NOT can be implemented using NOR.
• Show that {•,≡,⊕} is a complete base.
• Redraw the multiplexer circuit using only NAND gates, and make it as simple as possible. Is this the simplest NAND circuit which acts as a multiplexer on four input lines?
• Chapter 13 - Boolean Logic
• The conjunctive normal form for a boolean function ƒ on three variables consists of a product of sums, with each sum containing three literals. How would you use the truth table of ƒ to obtain an expression for ƒ in this form?
• Using the axioms for boolean algebra, prove the validity of De Morgan's laws:
• (x + y)′ = x′ • y
• (xy)′ = x′ + y
• Obtain a conjunctive normal form for the multiplexer, and draw the corresponding circuit. How does it compare, in terms of number of gates, with the circuit in Figure 13.3?
• Chapter 20 - Karnaugh Maps
• Design a vending machine control circuit. A soft drink costs 45¢, and your circuit records the nickels (5¢), dimes (10¢), and quarters (25¢) fed to the machine. It issues a command to dispense a can when the appropriate payment has been equalled or exceeded. Assume that at least one quarter is used and that coin counters make available binary counts for each kind of coin.
• Devise a five-variable Karnaugh map by putting two four-variable maps together in a certain manner. You may have to introduce a new kind of cell adjacency.
• For each kind of map presented here, find a function which cannot be simplified at all. In each case, what proportion of all possible functions cannot be simplified?
• Chapter 38 - Sequential Circuits
• Design circuitry for the clocked RS flip-flop to implement the CLEAR function. This resets the flip-flop to zero.
• Representing a 4-bit register by a long box with four input and four output lines, attach these to the appropriate lines in the 4-bit memory described here so that one may store the contents of this register in a specified word of memory or load the contents of that word into the register.

### Supervision 3

• 1995 Paper 2 Question 21 (Karnaugh Maps)
For some reason, people find it hard to understand what this question is asking them to do. You are given a partially complete Karnaugh map; some squares are shaded to show a value of 1, some are blank to show a value of 0 and some have an X to show that they have not yet been determined. You have to replace the Xs with 0s and 1s in four different ways to make four functions with the specified properties.
• 2002 Paper 2 Question 2 (Sum of products)
• 2001 Paper 2 Question 3 (Adders and Multiplier)
• 2002 Paper 2 Question 3 (FSM using JK flip-flops)

### Supervision 4

• © 2012 Computer Laboratory, University of Cambridge
Information provided by Ian Davies