Combination Circuits

Q2: Timing Diagram

Q- Now I make a certain change in the required output. The circuit is same as the previous question but it is not required to be AND gate anymore and output required is a HIGH pulse with width of 2 ns as:

Now one has to choose the BLOCK such that we get the above waveform as output of the whole circuit.

Ans: In the previous question we had the 2 inputs of NOR gate as follow:

By modifying the BLOCK we can only change the 2nd I/P and 1st input would remain the same. Now if we analyze the required O/P also given below to see how we need to change the 2nd I/P 

And we know that output is NOR of two inputs. And we get a ‘1’ only when we have ‘0’ at both inputs.We need high pulse of width 2 ns so we need that both inputs remain ‘0’ for 2 ns. Hence to get the required O/P we need to insert the extra delay of 2 ns to the 2nd pulse. And I/P pulses would be

As we just have to delay the 2nd pulse by 2 ns so we insert an extra buffer or two extra inverters in the BLOCK as shown on next page:





K Maps

K-maps for XOR and XNOR gates

 Even variable map: For 4 variables (even), we have XOR and XNOR compliment of each other and can be represented in K-maps as follow: For XNOR gate we have 2n/2 number of min terms with output as 1 (i.e. we have even number of 0s)

For XOR gate we have 2n/2 number of min terms with output as 1 (i.e. we have odd number of 1s)

From the K-maps also we can we that both the function are compliment of each other. Where we have 1 in one K-map, we have the 0 for the corresponding square in other K-map and vice versa.Odd variable map: For 3 variables (odd), we have XOR and XNOR equal to each other and can be represented in K-maps as follow: 

And equation is While the compliment of the above is represented by 

The above map can be represented by either  which is compliment of 

Boolean Algebra


Q7- We implement the 3-input NAND gate circuit as follow and we are given the gate delay of d ns and

3 inputs are shown as:

And give us the output of the circuit and compare it with the ideal output.

Ans:  Firstly we NAND of A&B is calculated and then it is inverter to get A.B Then we NAND A.B and C to get NAND of A, B & C. The wave forms we get are as:

While the ideal output is

And now we compare the two outputs as:

We observe that following points are to be noticed while comparing:

  • There is a delay of 3d in the actual output
  • Also that actual output has shorter pulse width as compared to ideal.
  • As we have the pulse width as p – 3d, this pulse can even vanish if delay of the gate (d) is increased or pulse width (p) is decreased.
Boolean Algebra


Q5- What is the output of the following circuit?

Ans:  This circuit may look simple to you as one has to just evaluate every NOR gate and get the final output but I say it is even simpler than this. Let’s analyze the truth table of NOR gate:

 And we see that even if only one of the 2 inputs is 1 then we need not find out the other input and we can get the output.

As in the above circuit we have one input as 1 to the last NOR gate so we can say that output of the above circuit is ‘0’ without calculating the outputs of other NOR gates.

Boolean Algebra


Q4- Draw the output waveform of the NAND gate when we have the two inputs as follow and delay of the gate is equal to 12.5 ns when output goes from LOW to HIGH and delay is 17.5 ns when output goes from HIGH to LOW.

Ans: To get the exact output waveform we firstly draw the output waveform without considering any delay and hence we get the output waveform as:

Now to get the actual output we’ll delay the output HIGH to LOW edge by 17.5 ns and LOW to HIGH edge by 12.5 ns.


We can directly check the output of the gate and delay the output accordingly if out is HIGH to LOW or LOW to HIGH and hence no need to draw a waveform without considering the delay.

Boolean Algebra

Question 3: TIMING DIAGRAM (Diff Units)

Q3- Sometimes delay of gates and the waveforms are given in different units. So let’s take such example with same circuit and delay as above but waveforms as below:

Ans: So we convert the delay into ps (Pico seconds)

So delay for NOR= 3ns = 3000ps

Delay for BUFFER =1ns=1000ps

Delay for AND=4ns=4000ps

 So the waveforms of Output are as follow:

Boolean Algebra


Q2- Draw the output of the circuit when waveforms of inputs X & Y are as given.

Take initial output values as zero. We’ll first calculate output of NOR gate and delay the output by 3ns. Then we draw output of BUFFER with delay of 1ns and finally we take AND of 2 outputs and induce delay of 4 ns as shown below:

We have marked change of level with time for clarity.

Boolean Algebra

Special Characteristics of an IC

A gate IC can be realized using different logic families (explained later) and hence every gate have the following characteristics which may vary with different logic family. The most important parameters are the following

Fan-out: It specifies the maximum number of standard gates that output of a gate can rive without affecting its own working. This normally depends upon the amount of current needed by a gate of same IC family. We generally connect output of a given gate to the inputs of other gates but there is a limit on the number of gates to which the output of a given gate can be inputted and this limit is called fan out of the given gate. If we exceed this limit then this may cause the malfunctioning of the circuit because the given gate may not be able to provide enough power to the gates which are connected to the output of the given gate.

When we have logical 0 at the output current is drawn into the circuit while when we have output as logical 1 then current is supplied to the load. So number of loads that can be connected to the output varies whether we have output as logical 1 or output as logical 0 but we take worst of the two as fan-out e.g. A gate in the TTL family can take load of 10 gates at logical 1 while 5 gates at logical 0. So we take fan-out as 5.

If we have fan out of he given gate as N, then we can connect the output of the given gate to N different gates of same family as shown below.

If we have fan out of he given gate as N, then we can connect the output of the given gate to N different gates of same family as shown below.

Some gates like XOR, BUFFER, INVERTER consume loading factor of 2 i.e. their load is equivalent to 2 usual gates.

If we consider this then a gate with loading factor of 5 can drive 2 XOR gate inputs and 1 OR or AND gate input.

BUFFER has fan-out of 25

Fan-in: It is basically the number to which we can extend the inputs of a gate. Generally this is equal to 8. This means that we can have a gate with maximum of 8 inputs as shown below

Power dissipation:  It is the average of powers dissipated in a gate with all inputs at logic 1 and the gate with all inputs at logic 0 and expressed in mW. If we have a IC with 4 gates then we need power supply which is equal to 4 times the power dissipation of the gate. It is an important factor. Industry is working continuously to decrease this factor further and further.

Propagation delay:  It is the average transition time for a signal to propagate from input to output. If we make a change in the binary signal at the input at time t=0 and we observe the corresponding change in the output at time t sec. Then the propagation delay for gate is taken as t. it is generally expressed in ns. The gate must have the minimum propagation delay. Propagation delay depends on the number of levels we have in a circuit. If we have 3 levels of gates in a circuit and each gate provides a delay of t ns then we have the total delay of the circuit as 3t ns.

Noise margin:  It is maximum noise which can be tolerated at the input of the circuit i.e. the maximum amount of noise which when added to the input does not give an undesirable change in the output. The noise if generally of two types: DC noise-that leads to the shift in the voltage level of the signal and the AC noise which is a random signal that can be created by any other switching signal and this signal s superimposed over the input signal. Noise margin should be as high as possible.

Boolean Algebra

Division of the circuit in different levels

We have some eternal inputs which are used directly or indirectly by the elements of the circuit to obtain output. The elements or the gates which receive all their input from the external inputs directly constitute the logical level one of the circuit while the gates which receive at least one input from the output of the gates of level one and not higher are under the logical level two of the circuit. Similarly we get 3rd and 4th level and so on.

From the above circuit we can see that only gate 1 is receiving all the input from external inputs derectly hence gate 1 is in logic level1

As gate 2 is receiving one input directly while the other input from gate 1 so gate 2 is in level 2 of the circuit.

Gate 3 receives the1st input directly from external input, 2nd input from output of gate 1 but it is not a level 2 gate as it is also receiving its third input which is from gate 2 (a level 2 gate), hence gate 3 is a level 3 gate.

Boolean Algebra

Question: Implement XOR using NAND only

Q- Implement 2 variable XOR gate using NAND only in minimum number of gates.

F= x XOR y = x’y+xy’ = x’y+xy’+xx’+yy’ = (x+y) (x’+y’)

Now we need to implement this circuit using NAND gates

F= (x+y)(xy)’ = x. (xy)’ + y. (xy)’

Take compliment

F’= ( x. (xy)’ + y. (xy)’ )’ = (x. (xy)’)’. (y. (xy)’)

Take compliment again

F=( (x. (xy)’)’. (y. (xy)’) )’

Now we can implement this using NAND gates

So we get that we need minimum of 4 NAND gates to implement XOR gate and if we are to implement XNOR gate then we’ll use 5 NAND gates with 5th gate used as inverter and placed in-front of 4th NAND gate in above circuit.