These are the following steps to design 2 bit synchronous up down counter using T flip flop:

Step 1: To design a synchronous up-down counter, we need one extra input called control input. Other than this, in next state column, half of the input must be appeared as up counter and the remaining must be treated as a down counter.

Step 2: After that, we need to construct a state table with excitation table.

Note: To construct excitation table from state table you should know the excitation table of respective flip flop, in this case, it is T flip flop. So check the excitation table for T flip flop Which is:

T Flip Flop Excitation Table

Present state	Next State	T
0	0	0
0	1	1
1	0	1
1	1	0

So, the above table is the excitation table for T Flip Flop.

State Table with  excitation  table

Control input	Present State		Next State		Flip Flop
Ci	Q2	Q1	Q'2	Q'1	T2	T1
0	0	0	0	1	0	1
0	0	1	1	0	1	1
0	1	0	1	1	0	1
0	1	1	0	0	1	1
1	0	0	1	1	1	1
1	0	1	0	0	0	1
1	1	0	0	1	1	1
1	1	1	1	0	0	1

Above the table is created as per follow :

When Q₂ =0 which is the present state and Q₂‘=0 which is next state then T₂ become 0 [As per excitation table, have a look ]
Similarly, if Q₂ is 0 and Q₂‘ is 1 then T_{2 becomes} 1.
In a similar way, it goes on .

Step 3: After making the excitation table the next thing  to do is dig out the equation from the boolean algebra or K map for the design of the counter. So, for T₁ and  T₂we got 1, and Q₁‘.Ci + Q₁.Ci’

K-Map

 

T₂= Q₁‘.Ci + Q₁.Ci’

T₁=1

These are the following steps to design a 2 bit synchronous down counter using T Flip flop:

Step 1: To design synchronous down counter, we just require to change the order of present state and next state, just put 0 where is 1 in synchronous up counter. In other words, start from 11 (3) to 00 (0)

Step 2: After that, we need to construct a state table with excitation table.
Note: To construct excitation table from state table you should know the excitation table of respective flip flop, in this case, it is T flip flop. So check the excitation table forT flip flop Which is:

T Flip Flop Excitation Table

Present state	Next State	T
0	0	0
0	1	1
1	0	1
1	1	0

So, the above table is the excitation table for T Flip Flop.

State Table with  excitation  table

Present State		Next State		Flip Flop
Q2	Q1	Q'2	Q'1	T2	T1
0	0	1	1	1	1
0	1	0	0	0	1
1	0	0	1	1	1
1	1	1	0	0	1

Above the table is created as per follow :

When Q₂ =1 which is present state and Q₂‘=1 which is next state then T₂ become 0 [As per excitation table, have a look ]
Similarly, if Q₂ is 1and Q₂‘ is 0 then T_{2 becomes} 1.
In a similar way, it goes on.

Step 3: After making the excitation table the next thing to do is dig out the equation from the boolean algebra or K map for the design of the counter. So For T₁ and T₂ we got 1 and Q₁‘ .

K-Map

For T_{2  Flip flop,}

 

 

 

 

For T₁ Flip flop,

 

Step 4: Lastly according to the equation got from K map create the design for 2 bit synchronous down counter.

In above design T₁ is getting high input  and T₂ is getting input from the output of the T₁ which is complement of the output.  A clock is attached to it which is always high in blue color.

T₂= Q₁

T₁=1

In the above design, T₁ is getting high input and T₂ is getting input from the output of the T₁ flip flop.  A clock is attached to it which is in blue colour.

These are the following steps to Design a 3 bit synchronous up counter using T Flip flop:

Step 1: To design a synchronous up counter, first we need to know what number of flip flops are required. we can find out by considering a number of bits mentioned in the question. So, in this, we required to make 3 bit counter so the number of flip flops required is 3 [2ⁿ where n is a number of bits].

Step 2: After that, we need to construct state table with excitation table.
Note: To construct excitation table from state table you should know the excitation table of respective flip flop, in this case, it is T flip flop. So check the excitation table for T flip flop Which is:

T Flip Flop Excitation Table

Present state	Next State	T
0	0	0
0	1	1
1	0	1
1	1	0

So, the above table is the excitation table for T Flip Flop.

State Table with  excitation  table

Present State			Next State			Flip Flop
Q3	Q2	Q1	Q'3	Q'2	Q'1	T3	T2	T1
0	0	0	0	0	1	0	0	1
0	0	1	0	1	0	0	1	1
0	1	0	0	1	1	0	0	1
0	1	1	1	0	0	1	1	1
1	0	0	1	0	1	0	0	1
1	0	1	1	1	0	0	1	1
1	1	0	1	1	1	0	0	1
1	1	1	0	0	0	1	1	1

Above  table is created as per follow :

When Q₃ =0 which is present state and Q₃‘=0 which is next state then T₃ become 0 [As per excitation table, have a look ]
Similarly, if Q₃ is 0 and Q₃‘ is 1 then T₃  become 1.
In similar way it goes on .

Step 3: After making the excitation table the next thing  to do is dig out the equation from the boolean algebra or K map for the design of the counter. So, for T₁ , T₂ and T₃ we got 1,  Q₁ and Q₁.Q₂

K-Map

For T₃ Flip flop,

 

T₂= Q₁

T₁=1