3. NETWORK REDUCTION AND PRINCIPLES OF CIRCUIT ANALYSIS, Electrical Engg 3rd semester notes EE 3002

 

3.1 Source Transformation

Source Transformation is a technique used to simplify circuits by replacing voltage sources and current sources with their equivalent forms. It helps in reducing the complexity of circuits to make analysis easier.

The Two Main Transformations:

1. Voltage Source to Current Source:

   • A voltage source in series with a resistor can be replaced by an equivalent current source in parallel with the same resistor.
   • Transformation formula:
     • $V \text{ (Voltage Source)}$ in series with $R$ becomes:
       $I = \frac{V}{R} \text{ (Current Source)}$ in parallel with $R$.

2. Current Source to Voltage Source:

   • A current source in parallel with a resistor can be replaced by an equivalent voltage source in series with the same resistor.
   • Transformation formula:
     • $I \text{ (Current Source)}$ in parallel with $R$ becomes:
       $V = I \times R \text{ (Voltage Source)}$ in series with $R$.

Example:

• A voltage source $V = 10V$ in series with a resistor $R = 5 \Omega$ can be transformed into a current source:
  • $I = \frac{10V}{5 \Omega} = 2A$, in parallel with $R = 5 \Omega$.

Diagram Suggestion:

• A simple diagram showing the transformation of a voltage source in series with a resistor to a current source in parallel with the resistor.

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3.2 Star / Delta and Delta / Star Transformation

Star-Delta and Delta-Star Transformations are used to simplify complex resistor networks, especially when three resistors are involved in a triangular or Y-shaped (star) configuration.

Star-Delta Transformation:

• In a star (Y) configuration, there are three resistors connected at a common point.
• In a delta (Δ) configuration, the three resistors form a triangular connection.

To convert a star network to a delta network:

• Let the three resistors in the star be $R_1, R_2, R_3$ and the resistors in the equivalent delta network be $R_A, R_B, R_C$.

The following equations are used:

• $R_A = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_3}$
• $R_B = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_1}$
• $R_C = \frac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_2}$

Delta-Star Transformation:

• To convert a delta network to a star network, you use the following equations:
• $R_1 = \frac{R_A R_B}{R_A + R_B + R_C}$
• $R_2 = \frac{R_B R_C}{R_A + R_B + R_C}$
• $R_3 = \frac{R_C R_A}{R_A + R_B + R_C}$

Diagram Suggestion:

• A diagram showing both the star and delta configurations, along with the equations used for conversion between the two.

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3.3 Mesh Analysis

Mesh Analysis is a method used to solve circuits with multiple loops or meshes. It involves writing Kirchhoff’s Voltage Law (KVL) equations for each mesh in the circuit to find the unknown currents.

Steps for Mesh Analysis:

1. Identify the meshes: Label each independent loop in the circuit.
2. Assign mesh currents: Assign a current to each mesh, usually in the clockwise direction.
3. Apply Kirchhoff’s Voltage Law (KVL): For each mesh, write down the sum of the voltage drops (product of current and resistance) around the loop. Set this sum equal to zero.
4. Solve the system of equations: Solve the system of simultaneous equations obtained from the KVL for the mesh currents.

Example:

• For a circuit with three resistors $R_1, R_2, R_3$ and two voltage sources, the mesh current equations could look like:
  • $(R_1 + R_2) I_1 - R_2 I_2 = V_1$
  • $-R_2 I_1 + (R_2 + R_3) I_2 = V_2$

Where $I_1$ and $I_2$ are the mesh currents.

Diagram Suggestion:

• A simple circuit with multiple meshes, with labels for mesh currents and voltage sources, and an indication of how the KVL equations are written.

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3.4 Node Analysis

Node Voltage Analysis (Nodal Analysis) is a method used to determine the voltage at different nodes in a circuit. This method applies Kirchhoff’s Current Law (KCL) at each node to form a system of equations.

Steps for Node Analysis:

1. Identify the nodes: Label each junction or node where two or more elements meet.
2. Choose a reference node: This node is typically chosen as the ground, and its voltage is set to zero.
3. Apply KCL to each non-reference node: For each node, the sum of currents flowing into the node must equal the sum of currents flowing out.
4. Solve the system of equations: The unknowns in the system are the node voltages.

Example:

• For a circuit with resistors and voltage sources, the node equation might look like:
  • $\frac{V_1}{R_1} + \frac{V_1 - V_2}{R_2} = 0$
  • $\frac{V_2 - V_1}{R_2} + \frac{V_2}{R_3} = 0$

Where $V_1$ and $V_2$ are the node voltages.

Diagram Suggestion:

• A simple circuit with labeled nodes, with the KCL equation written for each node.

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Summary Diagrams:

1. Source Transformation: Show a simple circuit with a voltage source and resistor, and its transformation to a current source in parallel with the same resistor.
2. Star-Delta Transformation: Diagram of star and delta connections, with formulas for transforming between them.
3. Mesh Analysis: A multi-loop circuit with mesh current labels and the application of KVL equations.
4. Node Analysis: A circuit with labeled nodes, showing the KCL equations at each node.