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Network Analysis

Table of Contents

4. Network Analysis
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Network analysis deals with determining currents, voltages, and power in electrical networks.
The fundamental tools are Kirchhoff’s Laws, mesh (loop) analysis, node analysis, and simplifications such as the Thevenin and Norton equivalents.

4.1 Kirchhoff’s Laws
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Kirchhoff’s Current Law (KCL)
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At any junction (node) in an electrical network, the sum of currents entering equals the sum of currents leaving:

$$ \sum I_{in} = \sum I_{out} $$

or equivalently:

$$ \sum I = 0 $$

Interpretation:
Electric charge is conserved — no charge is lost at a node.

Kirchhoff’s Voltage Law (KVL)
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In any closed loop, the algebraic sum of all voltages equals zero:

$$ \sum U = 0 $$

Interpretation:
The total energy supplied by sources equals the energy consumed by resistors and loads.

4.2 Mesh (Loop) Analysis
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Mesh analysis applies KVL to determine the unknown currents in independent loops of a network.

Steps:

  1. Identify independent loops.
  2. Assign a loop current (usually clockwise).
  3. Apply KVL to each loop.
  4. Solve the resulting system of equations.

Example:

For a two-loop network:

$$ \begin{aligned} U_1 &= R_1 I_1 + R_3 (I_1 - I_2) \\\\ 0 &= -R_3 (I_1 - I_2) + R_2 I_2 - U_2 \end{aligned} $$

Solving yields the loop currents \( I_1, I_2 \).

4.3 Node Voltage Analysis
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Node voltage analysis is based on KCL and expresses all branch currents in terms of node voltages.

Steps:

  1. Choose a reference (ground) node.
  2. Label the remaining node voltages relative to ground.
  3. Write current equations for each node (using KCL).
  4. Substitute branch currents via Ohm’s Law.
  5. Solve the resulting linear equations for the node voltages.

Example:
At a node connected by resistors \( R_1, R_2, R_3 \):

$$ \frac{U_A - U_1}{R_1} + \frac{U_A - U_2}{R_2} + \frac{U_A}{R_3} = 0 $$

4.4 Superposition Principle
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If multiple independent sources act within a linear network, the resulting voltage or current at any point is the sum of the effects of each source acting alone.

Procedure:

  1. Consider one source active, deactivate all others:
    • Voltage sources → short circuit
    • Current sources → open circuit
  2. Compute the desired quantity.
  3. Repeat for all sources and add the individual effects algebraically.

Example:
A resistor network with two voltage sources \( U_1 \) and \( U_2 \):

$$ I_{total} = I_{U1} + I_{U2} $$

4.5 Thevenin’s Theorem
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Any linear two-terminal network can be represented by an equivalent voltage source \( U_{Th} \) in series with a resistance \( R_{Th} \).

Determination
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  1. Open-circuit voltage:

    $$ U_{Th} = U_{open} $$

  2. Internal resistance:
    Deactivate all independent sources and calculate the equivalent resistance seen from the terminals:

    $$ R_{Th} = R_{eq} $$

Equivalent Circuit:

Load current:

$$ I_L = \frac{U_{Th}}{R_{Th} + R_L} $$

4.6 Norton’s Theorem
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Alternatively, any two-terminal network can be represented by a current source \( I_N \) in parallel with a resistance \( R_N \).

Determination
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  1. Short-circuit current:

    $$ I_N = I_{short} $$

  2. Internal resistance:

    $$ R_N = R_{Th} $$

Equivalent Circuit:

Load current:

$$ I_L = I_N \times \frac{R_N}{R_N + R_L} $$

4.7 Relationship Between Thevenin and Norton Equivalents
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Both representations are interchangeable:

$$ U_{Th} = I_N \times R_N \\\\ I_N = \frac{U_{Th}}{R_{Th}} \\\\ R_N = R_{Th} $$

4.8 Maximum Power Transfer
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Maximum power is delivered to a load when the load resistance equals the internal resistance of the source:

$$ R_L = R_i $$

Proof:
For a source with \( U_q \) and \( R_i \):

$$ P = I^2 R_L = \left( \frac{U_q}{R_i + R_L} \right)^2 R_L $$

The derivative \( \frac{dP}{dR_L} = 0 \) leads to \( R_L = R_i \).

Maximum Power:

$$ P_{max} = \frac{U_q^2}{4 R_i} $$

💡 Summary:
Kirchhoff’s laws form the backbone of network analysis.
Mesh and node methods allow systematic calculation of unknown currents and voltages, while Thevenin and Norton theorems simplify complex networks into easily analyzable equivalents.