Problems in Digital Logic

Problem 1:  Write a boolean expression for the output, Q, in terms of the inputs A, B, and C.

(a)

(b)

(c)

 

Problem 2. Draw a circuit to realize each of the expressions using AND gates, OR gates and Invertors.

 

Problem 3.  Make a truth table, and then a Karnaugh map for the expression indicated.  Then develop the minimum sum of products form.

  1. Do this for W, the output of problem 2a.
  2. Do this for Z, the output of problem 2d.

 

Problem 4. 

  1. Show using truth tables :
  2. Show using truth tables:
  3. Draw circuits for the right hand side of parts a and b.

 

Problem 5.  In the truth table below, the inputs are A, B, C, and D.  Use a Karnaugh map to come up with a minimum sum of products form when:

  1. the output is W
  2. the output is X
  3. the output is Y
  4. the output is Z
  5. Input   Output
    D C B A W X Y Z
    0 0 0 0 0 0 1 1
    0 0 0 1 0 1 1 0
    0 0 1 0 0 1 0 1
    0 0 1 1 0 0 0 0
    0 1 0 0 1 1 1 0
    0 1 0 1 1 0 1 1
    0 1 1 0 1 0 0 0
    0 1 1 1 1 1 0 1
    1 0 0 0 1 1 0 1
    1 0 0 1 1 0 0 0
    1 0 1 0 1 0 0 1
    1 0 1 1 1 1 0 0
    1 1 0 0 0 0 X X
    1 1 0 1 0 1 X X
    1 1 1 0 0 1 X X
    1 1 1 1 0 0 X X

Problem 6.  For part b of the previous problem:

  1. Draw the circuit using AND gate, OR gates and NOT gates.
  2. Draw the circuit using NAND gates and NOT gates
  3. Try to come up with a simpler way to draw the circuit using other types of gates.

 

Problem 7. There is an integrated circuit called a BCD-seven segment decode that takes 4 inputs and has seven output.  The inputs represent a number between 0 and 9, and each of the seven outputs corresponds to one of seven LED's in a seven-segment display.  A typical seven segment display is shown below.

For example the number zero would be represented by lighting segments a, b, c, d, e, and f, as shown below.

The other digits are formed as described in this table. 

Digit Inputs Output
(7 Segments)
D C B A
0 0 0 0 0 wpeE.jpg (872 bytes)
1 0 0 0 1 wpeF.jpg (777 bytes)
2 0 0 1 0 wpe18.jpg (868 bytes)
3 0 0 1 1 wpe11.jpg (868 bytes)
4 0 1 0 0 wpe12.jpg (844 bytes)
5 0 1 0 1 wpe13.jpg (873 bytes)
6 0 1 1 0 wpe14.jpg (875 bytes)
7 0 1 1 1 wpe15.jpg (821 bytes)
8 1 0 0 0 wpe17.jpg (890 bytes)
9 1 0 0 1 wpe16.jpg (891 bytes)

So, for example, segment "a" is used in number 0, 4, 5, 6, 8, and 9.

  1. Write the truth table for segment "a" with inputs A, B, C, and D.
  2. Draw the Karnaugh map for segment "a"
  3. Write the minimum sum-of-products form for segment "a".

 

Problem 8.

  1. Repeat the previous problem for segment b.
  2. Repeat the previous problem for segment c.
  3. Repeat the previous problem for segment d.
  4. Repeat the previous problem for segment e.
  5. Repeat the previous problem for segment f.
  6. Repeat the previous problem for segment g.

 

Problem 9.  The flip-flops in the drawing below are positive edge triggered D flip-flops.  Let Q2, Q1, Q0 = 0,0,0 initially.  

  1. Plot the clock, Q2, Q1 and Q0 until the outputs begin to repeat.  
  2. Show that the circuit acts as a counter.

 

 

Problem 10.  The flip-flops in the drawing below are negative edge triggered J-K flip- flops.  Let Q2, Q1, Q0 = 0,0,0 initially.  

  1. Plot the clock, Q2, Q1 and Q0 until the outputs begin to repeat.  
  2. Show that the circuit acts as a counter.  
  3. What advantage does this circuit have over the previous one?  (It has nothing to do with being negative edge triggered rather than positive edge triggered)

 

Problem 11.  The flip-flops in the drawing below are positive edge triggered D flip flops.  Let Q2, Q1, Q0 = 1,0,0 initially.  Plot the clock, Q2, Q1 and Q0 until the outputs begin to repeat.