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Practice MCQ

Unit 1 - Notes

ELE205

Unit 1: Circuit Elements, Energy Sources and Analysis of network by Kirchhoff's laws, Nodal and Mesh Method

1. Classification of Network Elements

Network elements are the fundamental building blocks of an electrical circuit. They are classified based on their behavior, energy processing, and structural properties.

A. Active vs. Passive Elements

  • Active Elements:
    • Definition: Elements capable of generating energy or amplifying power for an infinite duration. They deliver power to the circuit.
    • Examples: Voltage sources, Current sources, Transistors, Operational Amplifiers.
    • Key Characteristic: The slope of the V-I characteristic curve can be negative (indicating power delivery).
  • Passive Elements:
    • Definition: Elements that absorb, dissipate, or store energy. They cannot generate power or amplify signals.
    • Examples: Resistors (dissipate energy), Inductors (store energy in magnetic field), Capacitors (store energy in electric field).
    • Key Characteristic: They always have a positive equivalent resistance.

B. Unilateral vs. Bilateral Elements

  • Bilateral Elements:
    • Definition: Elements that offer the same impedance (magnitude and phase) to current flow in both directions. The V-I characteristic is symmetrical about the origin.
    • Examples: Resistors, Inductors, Capacitors, Transmission lines.
    • Implication: If the polarity of the voltage source is reversed, the magnitude of the current remains the same, only the direction changes.
  • Unilateral Elements:
    • Definition: Elements where conduction or impedance varies depending on the direction of current flow. The V-I characteristic is not symmetrical.
    • Examples: Diodes, Transistors.
    • Implication: A diode acts as a short circuit (ideally) in forward bias and an open circuit in reverse bias.

C. Linear vs. Non-Linear Elements

  • Linear Elements:
    • Definition: Elements where the relationship between voltage (V) and current (I) is a straight line passing through the origin. They satisfy the principle of Superposition (Additivity and Homogeneity).
    • Equation: (Ohm’s Law applies constantly).
    • Examples: Linear Resistors, Air-core Inductors, Ideal Capacitors.
  • Non-Linear Elements:
    • Definition: Elements where the V-I relationship is not a straight line. The parameters (like resistance) change with voltage or current. They do not obey Superposition.
    • Examples: Diodes (), Iron-core inductors (due to saturation), Incandescent lamps.

D. Lumped vs. Distributed Elements

  • Lumped Elements:
    • Definition: Elements where the physical size is very small compared to the wavelength of the electromagnetic wave operating in the circuit. The voltage and current are considered constant throughout the element at any instant.
    • Application: Standard circuit theory (KCL/KVL) assumes lumped elements.
    • Examples: Physical resistors, capacitors, and inductors in a breadboard circuit.
  • Distributed Elements:
    • Definition: Elements distributed along the length of the circuit. The voltage and current vary with spatial position as well as time.
    • Application: Transmission line theory is required.
    • Examples: Transmission lines, Waveguides.

2. Energy Sources

Sources are the active elements that provide energy to the circuit.

A. Independent Sources

An independent source maintains its value (Voltage or Current) irrespective of the circuit parameters connected to it.

  • Independent Voltage Source: Maintains a specified terminal voltage regardless of the current drawn.
    • Ideal: Internal resistance .
    • Practical: Modeled as a voltage source in series with a small internal resistance ().
  • Independent Current Source: Maintains a specified current flow regardless of the voltage across its terminals.
    • Ideal: Internal resistance .
    • Practical: Modeled as a current source in parallel with a large internal resistance ().

B. Dependent (Controlled) Sources

A dependent source's magnitude is controlled by a voltage or current elsewhere in the circuit. Represented by a diamond shape. There are four types:

  1. Voltage Controlled Voltage Source (VCVS): (where is a dimensionless gain).
  2. Current Controlled Voltage Source (CCVS): (where is transresistance).
  3. Voltage Controlled Current Source (VCCS): (where is transconductance).
  4. Current Controlled Current Source (CCCS): (where is a dimensionless gain).

C. Source Transformation

A technique to simplify circuits by converting a practical voltage source into an equivalent practical current source and vice-versa.

  • Voltage to Current: A voltage source in series with resistance can be transformed into a current source in parallel with resistance .
  • Current to Voltage: A current source in parallel with resistance can be transformed into a voltage source in series with resistance .
  • Note: The resistance remains the same value in both configurations. This applies to both independent and dependent sources (provided the controlling variable is outside the transformation).

3. Overview of Kirchhoff's Laws

These laws form the foundation of circuit analysis, based on conservation of charge and energy.

A. Kirchhoff’s Current Law (KCL)

  • Principle: Conservation of Charge.
  • Statement: The algebraic sum of currents entering and leaving a node (junction) is zero. Alternatively, sum of entering currents equals sum of leaving currents.
  • Equation:
  • Convention: Currents entering = Positive (+); Currents leaving = Negative (-).

B. Kirchhoff’s Voltage Law (KVL)

  • Principle: Conservation of Energy.
  • Statement: The algebraic sum of all voltages (drops and rises) around a closed loop is zero.
  • Equation:
  • Convention:
    • Moving from to (Potential Rise) Positive ().
    • Moving from to (Potential Drop) Negative ().

4. Voltage and Current Division Rules

Shortcuts for analyzing series and parallel circuits without writing full loop/node equations.

A. Voltage Division Rule (Series Circuits)

Applied to resistors connected in series across a voltage source . The voltage drops across resistors are proportional to their resistance values.

  • Voltage across resistor in a series chain:

    Where .

B. Current Division Rule (Parallel Circuits)

Applied to resistors connected in parallel supplied by a total current . The current splits inversely proportional to the resistance.

  • Two Resistors () in parallel:
    • Current through :
    • Current through :
  • General Formula (N resistors):

    Where (Conductance).

5. Mesh Analysis (Loop Analysis)

Mesh analysis uses Mesh Currents as variables. It is based on KVL. It is only applicable to planar circuits (circuits that can be drawn without crossing wires).

Procedure:

  1. Identify Meshes: A mesh is a loop that contains no other loops within it.
  2. Assign Currents: Assign mesh currents () to all meshes, usually in a clockwise direction.
  3. Apply KVL: Write KVL equations for each mesh in terms of mesh currents.
    • Self-resistance: Voltage drop is .
    • Mutual-resistance: Voltage drop due to adjacent mesh is .
  4. Solve: Solve the simultaneous linear equations to find .

Handling Independent Sources

  • Voltage Sources: Simply add or in the KVL equation depending on polarity.
  • Current Sources:
    • On perimeter: The mesh current becomes equal to the source current immediately.
    • Between two meshes (Supermesh): If a current source exists between two meshes, remove the source (open circuit) to create a larger "Supermesh." Apply KVL to the Supermesh and use the constraint equation from the current source () to solve.

Handling Dependent Sources

  1. Treat the dependent source as a normal voltage source while writing the KVL equation.
  2. Write a Constraint Equation: Express the controlling variable (e.g., or that controls the source) in terms of the mesh currents.
  3. Substitute the constraint into the KVL equations and solve.

6. Nodal Analysis

Nodal analysis uses Node Voltages as variables. It is based on KCL. It is applicable to both planar and non-planar circuits.

Procedure:

  1. Identify Nodes: Identify all principal nodes (junction of 3+ elements).
  2. Select Reference Node: Choose one node as the Ground/Reference (). Usually, the node with the most connections.
  3. Assign Voltages: Assign variables to the remaining non-reference nodes.
  4. Apply KCL: Write KCL equations for each non-reference node.
    • Assume currents leave the node: .
  5. Solve: Solve the system of linear equations for node voltages.

Handling Independent Sources

  • Current Sources: Directly add to the KCL equation (entering = negative, leaving = positive, or vice versa based on convention).
  • Voltage Sources:
    • Between Node and Reference: The node voltage is simply known ().
    • Between two Non-Reference Nodes (Supernode): If a voltage source exists between two unknown nodes, treat the two nodes and the source as a single "Supernode." Apply KCL to the entire Supernode and use the constraint equation () to solve.

Handling Dependent Sources

  1. Treat the dependent source as a normal current source while writing KCL (or voltage source for Supernode).
  2. Write a Constraint Equation: Express the controlling variable in terms of the Node Voltages.
  3. Substitute the constraint into the KCL equations and solve.

7. Comparison: Mesh vs. Nodal Analysis

Feature Mesh Analysis Nodal Analysis
Fundamental Law Kirchhoff's Voltage Law (KVL) Kirchhoff's Current Law (KCL)
Variables Mesh Currents Node Voltages
Applicability Planar circuits only Planar and Non-planar circuits
Ideally suited for Circuits with many voltage sources Circuits with many current sources
Complexity Dependent on number of meshes Dependent on number of nodes
Equation Structure Sum of voltages = 0 Sum of currents = 0