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					Lecture 5                                        Digital Electronics                                                A.D


Topic 4: Digital Electronics

5.1 Logic
What can a digital circuit do?
The simplest task we can think of is a combinational type of logic decision. For example, we
can design a digital electronic circuit to make an instant decision based on some information.
Here we emphasize “instant” in the decision making process. That means, the process has no
time delay.
              X = today‟s weather is good
              Y = today is a holiday
       decision Z = go to picnic

Suppose our rule is Z = X and Y. The circuit is a simple AND gate.


Truth tables
   An easy way to represent a combinational logic result is to tabulate all possible inputs.
    X Y Z = X.Y                              X Y Z = X+Y                             X     Z= X
     0 0         0                           0 0          0                          0        0
    0 1          0                           0 1          1                          0        1
    1 0          0                           1 0          1
    1 1          1                           1 1          1
   Truth Table for AND                      Truth Table for OR                      Truth Table for NOT



     X      Y    Z = X.Y                     X     Y     Z= XY                      X     Y     Z=X Y
     0      0        1                       0     0        1                        0     0        0
     0      1        1                       0     1        0                        0     1        1
     1      0        1                       1     0        0                        1     0        1
     1      1        0                       1     1        0                        1     1        0

   Truth Table for NAND                     Truth Table for NOR                     Truth Table for XOR




Boolean algebra
Logic can also be expressed in algebraic form such as the expression for AND gate: Z = X.Y

Finding expression from truth table
Once we have the truth table, we can find the output expression by adding up all min-terms.Min-
term corresponds to the product term that give a 1 in the output.
  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 5                                       Digital Electronics                                                 A.D


For example, here the min-terms are X .Y , X .Y , and X.Y




Question: This is an OR gate! Can it be simplified down to Z = X + Y? How can we do it?

More examples




Circuit realization




Note: This is same as a NAND gate (see the truth table), and hence should be the same as




The question is HOW TO SIMPLIFY A MIN-TERM
EXPRESSION!
  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 5                                       Digital Electronics                                                 A.D


Boolean algebra simplification
Basic Laws:

                                                                              X.X = X
                                                                              X+X = X
                                                                               X.0 = 0
                                                                               X.1 = X
                                                                              X+0=X
                                                                              X+1=X




Example 1




It‟s a NAND gate!
Example 2




                                                                          It‟s a OR gate!

  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 5                                       Digital Electronics                                                 A.D


Boolean algebra can be tedious.
Is there any easier method to simplify the min-term expression?

Karnaugh Maps (K- Maps)
Karnaugh maps are a graphical way to simplify a Boolean expression and thus simplify the
resulting circuit.

Steps
1. Fill up the karnaugh map table with the output requirement i.e 0 or 1.
2. Apply karnaugh map rules.
3. A karnaugh map provides a logic solution in terms of boolean expression.
4. Since, for each output bit a separate karnaugh map used, the number of boolean expression
   will be equal to the number of output bits in design.
5. Finally, replace your gates as per the expressions, and your design is ready.

Example:




  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 5                                       Digital Electronics                                                 A.D


Tutorial 5
1. What is meant by „binary‟? Give some examples to illustrate this idea.
2. If a system has 3 input terminals how many possible combinations of inputs is possible?
3. Write the truth tables of these logic circuits:




4. Write the truth table for the logic circuit below. To what gate is it equivalent?




5. Draw a circuit that has output Z  A  B  C
6. Minimize the following expressions algebraically:

A  BCB  CABC  BC 
 A  BC A  B C  A  BC 
XYZ  XYZ  XYZ

7. Reduce the following expressions using De Morgon‟s theorem:

             i)        f  A B C  ABC

             ii)                     
                       f  A  B C A B C      
             iii)      f  A  B  C AB  C D ACD
8. Describe a switched logic circuit to control the electric motor of a drilling machine and make
    the machine safer to use.
9. A system is designed to switch on a room heater if the temperature is below 15°C, but not if
    the door has been left open. Draw the system diagram, including the logic required.
10. A system is designed to flash a beacon lamp and sound a siren when any one of three
    windows or the door is opened. But the beacon is not to be flashed at night. Draw the system
    diagram, including the logic required.




  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09

				
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