Introduction to State Equations , State Tables & State Diagram

STATE EQUATIONS , STATE TABLES & STATE DIAGRAM:

Step 1 :

How to make State Equations ? 

Consider the Sequential circuit given below ,

 Make State Equation of Next State of Flip Flop with the help of basic gates as ,

                                         A(t+1) = A(t)x(t) + B (t) x (t)

Description :

As A is the output of first D Flip Flop , we make Next State equation of A(t+1) . t+1 represent the Next State . As A is connected with 2 AND Gates and an OR Gate . The inputs of first AND Gate are present state of A and x . The inputs of second AND Gate are present state of B and x . Therefore, these inputs are written in equation as A(t)x(t) and B(t)x(t)  . Then both outputs of AND Gates are connected with an OR Gate. Then , the final equation of first D flip flop is ,

                                     A (t+1) = A (t) x (t)  + B (t) x (t)

You can also write as ,

                                   A (t+1) = A . x  + B . x

{Multiplication due to AND GATE and addition symbol because of OR Gate }

Now do the same for B . Make State Equation of Next State of Flip Flop B as ,

                                    B(t+1) = A'(t)  x(t) 

Description :

Because the second D Flip Flop is connected with a single AND Gate . The  inputs of AND Gate are invert of A (A') and x . So the Next State Equation of second D flip flop is ,

                                    B ( t+1) = A'(t) x (t) 

You can also write as ,

                                    B(t+1) = A' . x

{ Multiplication because of AND Gate }

Output Equation :

If the Sequential circuit have Output (As this circuit has y output). Then you have to make output equations using the same method . y is connected with an OR Gate (B and A) or  B(t)+A(t) . Then connected with a NOT Gate x' . Then both outputs of OR Gate and AND Gate are connected with an AND Gate. So, the final output equation will become ,

                              y ( t+ 1) = [ A(t) + B(t) ] x'(t)

You can also write as ,

                             y( t+1 ) = (A + B) x'

What will you do if there is JK - Flip Flop instead of D - Flip Flop? Is there any change ?

The only change is you  will now put the next state of A and B in the general equation of JK- Flip Flop 

i-e :            JQ' + K'Q

For example :

Consider the sequential circuit given below ,

Now, make State Equation of A using the above method.

The State Equation of A is J =B(t) and K = x' B(t)

Now put these values in  J A' + K'A { HERE OUTPUT IS A therefore Q = A }

                          A(t+1) = A' B(t) + (x'B(t))' A

                           A(t+1) =A'B (t) + (xB'(t))A

                            A(t+1)=A'B(t) + Ax+AB'(t)

or you can also write as,

                            A(t) = A' B  + Ax +AB'

Now, make State Equation of B using the above method.

 The State Equation of A is J=x'  and K= A' x' + A x 

Now put these values in  J B' + K'B { HERE OUTPUT IS B therefore Q = B }

                           B(t+1) = x' B' + (A' x' + A x)' B

                          B(t+1) = x' B' + A' B x + A B x'

Notice that this circuit has no output therefore no output equation . 

What will you do if there is T- Flip Flop instead of JK-Flip Flop? Is there any change ?

No , You will just put the Next State of A and B in the general equation of T-Flip Flop 

i-e      T Q' + T'Q

Step 2 :

How to make State Tables ?

 State Tables (sometimes called transition Table) can be made by two methods ,
Now consider the circuit with D-Flip Flop ,

First Form of State Table :

First form of State Table includes Present State , Input , Next State and Output (if present in the circuit) . Now! There are three inputs A , B and x.Therefore there are 8 possible combinations. The Next State of   State table will fill by State Equation. For example for the Next State of A x must be equals to 1.So, where x=0 you can simply put the Next State of A = 0 in the State Table. The next condition for Next State of A is either A or B is equals to 1 then the Next State of is 1 . Otherwise Next State of A is 0. So you can see in the Table given below where x is 0 Next State of A is 0 . And when A and B both are 0 , Next State of A is again 0. Except these cases , the next state  of A is 1 .

Next State of B and Output in the State Table can be filled by the same method.  

Second Form of State Table :

The second form of State Table has no input column but it will divide the inputs in x=0 and x=1 in the Next State & Output as given below. The second form of State Table has less number of rows than First Form of State Table and it is easier to fill. Second form of State Table has greater number of Columns if there are more than one inputs. 
In the Table given below there is only two inputs A and B. Therefore only 4 possible combinations .
Then fill the Next State and Output by the method described above. 

Step 3 :

How to make State Diagram ?

State Diagram can be made by using two models ,

• Mealy Model
• Moore Model                             

 Mealy Model :

Description :

State Diagram is made with the help of State Table. The Binary number inside the circle represent the Present State of Flip-Flop . The arrows represent the Next State of Flip-Flops. The binary number before the slash represent the input and the binary number after the slash represent the output state at the time of conversion of Flip-Flop from Present State to the Next State. 

(Mealy Model is suitable for this State Table)

Is there any change in the State Table and State Diagram of JK and T - Flip Flop ?

No , the State Table and State Diagram of JK and T- Flip Flop can be created by using the above method.

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