Network synthesis realizations: Identify which pairs are correctly matched—Brune’s realization with ideal transformer; Cauer realization as a ladder network; Bott–Duffin realization using a non-ideal transformer. Choose the correct combination.

Introduction / Context:
In classical network synthesis, several canonical realizations are used to realize positive-real (PR) driving-point functions using passive elements. Brune, Cauer, and Bott–Duffin methods are fundamental. This question asks which description pairs are correctly matched to these realizations.

Given Data / Assumptions:

• Brune’s realization is associated with transformer-based extraction for PR functions.
• Cauer’s realization refers to continued-fraction expansions leading to ladder (L–C–R) networks in canonical forms (Cauer I, Cauer II).
• Bott–Duffin’s realization addresses PR functions without requiring transformers.

Concept / Approach:

Brune (1921) provided the first general synthesis for PR impedances using resistors, inductors, capacitors, and ideal transformers. Cauer (1926–1932) introduced ladder-type canonical forms derived from continued fractions, widely known as Cauer forms. Bott–Duffin (1949) later proved that any PR function can be realized with only R, L, C elements, i.e., without transformers, overturning the notion that transformers are necessary in general.

Step-by-Step Solution:

Verification / Alternative check:

Standard network synthesis texts (e.g., Van Valkenburg, Guillemin) consistently describe Brune’s method with an ideal transformer stage, Cauer ladder forms from continued fractions, and Bott–Duffin transformerless realizations for any PR function, confirming the judgments above.

Why Other Options Are Wrong:

• 1, 2 and 3: Includes the incorrect Bott–Duffin pairing.
• 2 and 3 / 1 and 3: Both rely on the incorrect transformer statement for Bott–Duffin.
• None of these: Rejects two correct pairs.

Common Pitfalls:

• Confusing Bott–Duffin as a refinement of Brune that still needs transformers; in fact, it eliminates that need.
• Using “non-ideal transformer” wording—synthesis theory uses ideal transformers when they are used at all.

Final Answer:

1 and 2