SYRCSE261: Electronics and Digital Systems (Combinational Logic 1 of ZyBook)

1. Fundamental Electrical Concepts

  • Voltage (V): The electrical potential difference between two points or the potential for electrical charge to move. It is the driving force behind the flow of charge and is measured in volts (V).

  • Current (I): The rate at which electrical charge flows through a conductor (I = \frac{dQ}{dt}). It is measured in amperes (A). In digital systems, current conventionally flows from higher potential to lower potential.

  • Resistance (R): The property of a material to resist the flow of electric current, measured in ohms (\Omega). It is influenced by the material's resistivity (\rho), length (L), and cross-sectional area (A) according to the formula R = \rho \frac{L}{A}.

  • Ohm's Law: A fundamental principle stating that the current through a conductor between two points is directly proportional to the voltage across those points: V = IR. Increasing voltage increases current, while increasing resistance decreases it.

  • Electrical Power (P): The rate at which energy is consumed or produced in a circuit. It is measured in watts (W) and calculated as:

    • P = VI

    • P = I^{2}R

    • P = \frac{V^{2}}{R}

  • Kirchhoff’s Laws:

    • Kirchhoff's Current Law (KCL): Based on the conservation of charge, the total current entering a junction must equal the total current leaving it (\sum I{in} = \sum I{out}).

    • Kirchhoff's Voltage Law (KVL): Based on the conservation of energy, the sum of all electrical potential differences around any closed loop in a circuit is zero (\sum V = 0).

  • Ground: A reference point of 0V in an electrical circuit, often represented graphically by three horizontal line segments of decreasing length.

2. Transistors and CMOS Technology

  • Digital vs. Analog Systems: Digital systems utilize discrete values (typically 0 for low/GND and 1 for high/V_{DD}), while analog systems use a continuous range of values (0.15V, 0.33V, etc.).

  • Switches: Devices that either conduct (on) or do not conduct (off) current. Electronically controlled switches use a control input to determine their state.

  • Transistors: Small electronic switches with no mechanical parts that act as the building blocks of digital logic.

    • nMOS: Conducts current when the control input (gate) is high (1).

    • pMOS: Conducts current when the control input (gate) is low (0).

  • CMOS (Complementary MOS): A technology that pairs nMOS and pMOS transistors to create logic gates that consume very little power when in a steady state.

3. Digital Logic Gates and Truth Tables

  • Logic Gates: Physical building blocks that implement Boolean functions.

    • Primary (Fundamental) Gates:

    • NOT: Inverts the input (if 0, output is 1).

    • AND: Output is 1 only if all inputs are 1 (A \cdot B).

    • OR: Output is 1 if at least one input is 1 (A + B).

    • Universal Gates: NAND and NOR gates. They are called "universal" because any Boolean function can be constructed using only these gates.

    • Derived Gates: XOR (Exclusive OR) and XNOR (Exclusive NOR), primarily used in arithmetic operations like binary addition.

  • Truth Tables: Mathematical tables representing the functional behavior of a logic circuit. For a system with n inputs, the table contains 2^{n} rows representing every possible input combination.

4. Boolean Algebra

  • Boolean Equations: Mathematical expressions used to represent logic circuits, such as y = (a \cdot b) + (c' \cdot d), where prime symbols (') or bars denote negation.

  • Key Identities and Properties:

    • Commutative: A + B = B + A and A \cdot B = B \cdot A

    • Associative: (A + B) + C = A + (B + C)

    • Distributive: A \cdot (B + C) = (A \cdot B) + (A \cdot C)

    • Identity: A \cdot 1 = A and A + 0 = A

    • Idempotent Law: A + A = A and A \cdot A = A

    • De Morgan’s Theorems: Essential for gate simplification; (A \cdot B)' = A' + B' and (A + B)' = A' \cdot B'

5. Circuit Design, Analysis, and Applications

  • Combinational Logic: Circuits where the output is determined solely by the current inputs (e.g., Multiplexers, Decoders, Adders).

  • Sequential Logic: Circuits where the output depends on both current inputs and the history of past inputs (memory). These require storage elements such as Flip-Flops and Latches.

  • Timing Diagrams: Visual representations showing how signals (inputs and outputs) transition between logic states over time, used to identify propagation delays and race conditions.

  • Practical Applications:

    • 7-Segment Display: Uses combinational logic to convert a 2-bit or 4-bit input into signals for segments (a through g) to display numerical characters.

    • Logic Control: Alarm systems (e.g., trigger if [System Enabled] AND [Door Open OR Window Open]) and voting/majority circuits.

  • Digital Circuit Simulators: Software tools like Logisim or SPICE used to verify circuit behavior before physical prototyping.

6. Circuit Equations and Simplification

  • Standard Forms:

    • Sum-of-Products (SOP): A group of ANDed terms (minterms) ORed together (e.g., F = AB + BC').

    • Product-of-Sums (POS): A group of ORed terms (maxterms) ANDed together (e.g., F = (A + B) \cdot (B' + C)).

  • Simplification Tools:

    • Karnaugh Maps (K-maps): A graphical method for simplifying Boolean expressions by identifying redundant terms in a grid layout.

    • Quine-McCluskey Algorithm: A tabular, algorithmic method for logic minimization, preferred for systems with many variables or for computer-based optimization.