Question

For a symmetrical two-port network, which of the following is NOT necessarily true?

• A. \(z_{11} = z_{22} \)
• B. \(y_{11} = y_{22} \)
• C. \(A = D \)
• D. \(AD - BC = 1 \)

Hint:

Keep these distinct two-port rules clear: - Symmetry Condition: \(A = D\), \(z_{11} = z_{22}\), \(y_{11} = y_{22}\). - Reciprocity Condition: \(AD - BC = 1\), \(z_{12} = z_{21}\), \(y_{12} = y_{21}\).

Answer 1

The Correct Option is D

Understanding the Concept: Two-port networks are characterized using distinct sets of parameters such as Impedance (\(Z\)), Admittance (\(Y\)), and Transmission/ABCD parameters.
• Symmetry: A two-port network is defined as structurally and electrically symmetrical if its electrical characteristics do not change when its input and output ports are interchanged.
• Reciprocity: A network is defined as reciprocal if the ratio of the excitation response at one port to the input signal applied at another port remains identical when the excitation and measurement positions are swapped. The specific mathematical boundary conditions required for these states across parameters are outlined as follows:
• For Impedance Parameters: Symmetry requires \(z_{11} = z_{22}\); Reciprocity requires \(z_{12} = z_{21}\).
• For Admittance Parameters: Symmetry requires \(y_{11} = y_{22}\); Reciprocity requires \(y_{12} = y_{21}\).
• For Transmission (ABCD) Parameters: Symmetry requires \(A = D\); Reciprocity requires \(AD - BC = 1\).

Step 1: Analyzing each statement against the condition of symmetry.
The problem explicitly restricts the network condition solely to a symmetrical two-port network. Let's inspect the choices:
• \(z_{11} = z_{22}\) is the absolute prerequisite definition for Z-parameter symmetry. (True)
• \(y_{11} = y_{22}\) is the absolute prerequisite definition for Y-parameter symmetry. (True)
• \(A = D\) is the absolute prerequisite definition for ABCD-parameter symmetry. (True)
• \(AD - BC = 1\) is the condition defining a reciprocal network, not a symmetrical network. End While many practical passive symmetrical networks are also reciprocal, symmetry itself does not mathematically guarantee or require reciprocity (for instance, a network incorporating internal non-reciprocal active components can be designed to maintain structural port symmetry \(A=D\) without satisfying \(AD-BC=1\)). Therefore, the statement \(AD - BC = 1\) is not necessarily true for a network that is only specified as symmetrical. This aligns with Option (D).