problems

ANALOG AND DIGITAL ELECTRONICS
18CS33
MODULE – 2
THE COMBINATIONAL LOGIC CIRCUITS
THE BASIC GATES
PREREQUISITES:
Electronic circuits and systems can be divided into two broad categories – analog and digital. Analog
circuits are designed for use with small signals and are used in a linear fashion. Digital circuits are
generally used with large signals and are considered nonlinear. Any quantity that changes with time can
be represented as an analog signal or it can be treated as digital signal.
Digital electronics involves circuits that have exactly two possible states. A system having only
two states is said to be binary. The binary number system is widely used in digital electronics.
Hexa-Decimal Decimal Binary Hexa-Decimal Decimal Binary
0 0 0000 8 8 1000
1 1 0001 9 9 1001
2 2 0010 A 10 1010
3 3 0011 B 11 1011
4 4 0100 C 12 1100
5 5 0101 D 13 1101
6 6 0110 E 14 1110
7 7 0111 F 15 1111


The operation of electronic circuits can be described in terms of its voltage levels – high (H) level and low
(L) level. This could be related to the binary number system by assigning L = 0 = F (false) and H = 1 = T
(true).




Symmetrical Signal & Asymmetrical Signal


The frequency is defined as, f = 1 / T where, T is the period of the signal.
Duty Cycle is a convenient measure of how symmetrical or how unsymmetrical a waveform is.
𝑇𝑇𝑜𝑜𝑜𝑜 𝑇𝑇𝑜𝑜𝑜𝑜 𝑇𝑇𝑜𝑜𝑜𝑜𝑜𝑜
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶, 𝐻𝐻 = 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶, 𝐿𝐿 =
𝑇𝑇𝑜𝑜𝑜𝑜 + 𝑇𝑇𝑜𝑜𝑜𝑜𝑜𝑜 𝑇𝑇𝑜𝑜𝑜𝑜 + 𝑇𝑇𝑜𝑜𝑜𝑜𝑜𝑜 𝑇𝑇𝑜𝑜𝑜𝑜 + 𝑇𝑇𝑜𝑜𝑜𝑜𝑜𝑜




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, ANALOG AND DIGITAL ELECTRONICS
18CS33
REVIEW OF LOGIC GATES:


Truth Table
Circuit Symbol VHDL IC Details
A B X

NOT Gate: 0 - 1
X = ~A
X <= not A;
1 - 0
𝑋𝑋 = 𝐴𝐴̅

OR Gate: 0 0 0
0 1 1 X=A|B
1 0 1 X <= A or B;

𝑋𝑋 = 𝐴𝐴 + 𝐵𝐵 1 1 1

AND Gate: 0 0 0
0 1 0 X=A&B
1 0 0 X <= A and B;
𝑋𝑋 = 𝐴𝐴. 𝐵𝐵 1 1 1

NOR Gate: 0 0 1
0 1 0 X = ~(A | B)
1 0 0 X <= A nor B;

𝑋𝑋 = 𝐴𝐴̅. 𝐵𝐵� 1 1 0
NAND Gate: 0 0 1
0 1 1 X = ~(A & B)
1 0 1 X <= A nand B;

𝑋𝑋 = 𝐴𝐴̅ + 𝐵𝐵� 1 1 0
XOR Gate: 0 0 0
0 1 1 X=A^B
1 0 1 X <= A xor B;
𝑋𝑋 = 𝐴𝐴⨁𝐵𝐵
1 1 0
= 𝐴𝐴̅𝐵𝐵 + 𝐴𝐴𝐵𝐵�
XNOR Gate: 0 0 1
0 1 0
X = ~(A ^ B)
1 0 0
X <= A xnor B;
𝑋𝑋 = 𝐴𝐴⨀𝐵𝐵
1 1 1
= 𝐴𝐴̅𝐵𝐵� + 𝐴𝐴𝐴𝐴




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, ANALOG AND DIGITAL ELECTRONICS
18CS33
Universality of NOR Gate:




Universality of NAND Gate:




Bubbled AND Gate:




Bubbled AND gate and NOR gate are equivalent


De Morgan’s First Theorem:
The complement of a sum equals the product of the complements. 𝐴𝐴 + 𝐵𝐵 = 𝐴𝐴̅. 𝐵𝐵�
��������



3

, ANALOG AND DIGITAL ELECTRONICS
18CS33
Proof:
A B A+B ��������
𝐴𝐴 + 𝐵𝐵 𝐴𝐴̅ 𝐵𝐵� 𝐴𝐴̅. 𝐵𝐵�
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0
NOR Gate Bubbled AND Gate
Bubbled OR Gate:




Bubbled OR gate and NAND gate are equivalent


De Morgan’s Second Theorem:
The complement of a sum equals the product of the complements. ���� = 𝐴𝐴̅ + 𝐵𝐵�
𝐴𝐴𝐴𝐴
Proof:
A B AB ����
𝐴𝐴𝐴𝐴 𝐴𝐴̅ 𝐵𝐵� 𝐴𝐴̅ + 𝐵𝐵�
0 0 0 1 1 1 1
0 1 0 1 1 0 1
1 0 0 1 0 1 1
1 1 1 0 0 0 0
NAND Gate Bubbled OR Gate


Duality Theorem: Starting with a Boolean relation, you can derive another Boolean relation by –
1. Changing each OR sign to an AND sign
2. Changing each AND sign to an OR sign
3. Complementing any 0 or 1appearing in the expression.
Example: 1. We say that, A+0 = A; the dual is, A.1 = A
2. Consider, A(B+C) = AB + AC
By changing the OR and AND operation, we get the dual relation:
A + BC = (A+B)(A+C)


Laws of Boolean Algebra:
 The following laws are of immense use in the simplification of Boolean expressions.




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