A complex network consisting of a number of parallel branches, where each parallel branch consists of voltage source with series impedance, can be replaced with equivalent circuit consisting of one voltage source in series with equivalent impedance. This is: 

Explanation:

Milliman’s Theorem

Definition: Milliman’s theorem is a network theorem used in electrical engineering, particularly in the analysis of complex networks. It states that a complex network consisting of multiple parallel branches, where each parallel branch contains a voltage source in series with an impedance, can be simplified to an equivalent circuit with a single voltage source in series with an equivalent impedance.

Mathematical Formulation:

The theorem can be mathematically expressed as:

V_eq = (V₁/Z₁ + V₂/Z₂ + ... + V_n/Z_n) / (1/Z₁ + 1/Z₂ + ... + 1/Z_n)

Z_eq = 1 / (1/Z₁ + 1/Z₂ + ... + 1/Z_n)

Where V₁, V₂, ..., V_n are the voltages of the individual branches, and Z₁, Z₂, ..., Z_n are the impedances of the individual branches.

Working Principle:

Milliman’s theorem simplifies the analysis of complex electrical networks by reducing multiple parallel branches to a single equivalent branch. This equivalent branch consists of an equivalent voltage source (V_eq) in series with an equivalent impedance (Z_eq).

Advantages:

• Simplifies the analysis of complex networks with multiple parallel branches.
• Reduces the computational effort required to solve for voltages and currents in the network.

Disadvantages:

• Applicable only to linear, bilateral networks with voltage sources in series with impedances.
• Not suitable for networks with dependent sources or non-linear components.

Applications: Milliman’s theorem is commonly used in the analysis and design of electrical circuits, particularly in power distribution systems and communication networks where multiple voltage sources and impedances are present in parallel.

Correct Option Analysis:

The correct option is:

Option 3: Milliman’s theorem.

This option correctly describes Milliman’s theorem, which allows for the replacement of a complex network with multiple parallel branches, each containing a voltage source in series with an impedance, with an equivalent circuit consisting of a single voltage source in series with an equivalent impedance.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: The Compensation theorem.

The Compensation theorem states that the effect of a change in the value of an element in a linear bilateral network can be found by considering the change as an independent voltage or current source. This theorem is not related to simplifying parallel branches of voltage sources and impedances.

Option 2: Tellegen’s theorem.

Tellegen’s theorem applies to any lumped network and states that the summation of power in a network is zero. This theorem is more general and does not specifically address the simplification of parallel branches with voltage sources and impedances.

Option 4: The Reciprocity theorem.

The Reciprocity theorem states that in a linear, bilateral network, the current due to a single voltage source is the same if the positions of the voltage source and the resulting current are interchanged. This theorem does not deal with the simplification of networks with parallel branches.

Conclusion:

Milliman’s theorem is a valuable tool in electrical engineering for simplifying complex networks with multiple parallel branches containing voltage sources and impedances. By reducing such networks to an equivalent circuit with a single voltage source and impedance, it simplifies analysis and design processes. Understanding the distinctions between various network theorems is crucial for correctly applying them to solve electrical circuit problems.