For a two port symmetric bilateral network, if A= 3Ω and B = 1Ω, the value of parameter C will be 

Explanation:

Two-Port Symmetric Bilateral Network

Definition: A two-port symmetric bilateral network is a type of electrical network that has two pairs of terminals, referred to as "ports." The network is termed symmetric when certain parameters are equal, and it is bilateral when the network behaves identically when the input and output ports are interchanged. These networks are often analyzed using transmission parameters (A, B, C, D), where the relationships between input and output voltages and currents are described as:

Equations:

    \( V_1 = A \cdot V_2 + B \cdot I_2 \)

    \( I_1 = C \cdot V_2 + D \cdot I_2 \)

For symmetric networks, the parameters satisfy the following conditions:

• \( A = D \) (Symmetry condition)
• \( A \cdot D - B \cdot C = 1 \) (Reciprocity condition)

Given:

• \( A = 3 \, \Omega \)
• \( B = 1 \, \Omega \)

We need to determine the value of the parameter \( C \).

Step 1: Use the Reciprocity Condition

From the reciprocity condition, we know:

    \( A \cdot D - B \cdot C = 1 \)

Since the network is symmetric, \( A = D \). Substituting \( A = 3 \, \Omega \) and \( B = 1 \, \Omega \), we get:

    \( 3 \cdot 3 - 1 \cdot C = 1 \)

    \( 9 - C = 1 \)

Step 2: Solve for \( C \)

Rearranging the equation:

    \( C = 9 - 1 \)

    \( C = 8 \, \text{s} \)

Thus, the value of the parameter \( C \) is \( 8 \, \text{s} \).

Step 3: Evaluate the Correct Option

From the options provided, the correct answer corresponds to:

• Option 1: \( 8 \, \text{s} \)

Important Information:

To analyze why the other options are incorrect, let us revisit the reciprocity condition:

    \( A \cdot D - B \cdot C = 1 \)

For a symmetric network:

• \( A = D = 3 \, \Omega \)
• \( B = 1 \, \Omega \)

Substituting these values into the equation, we calculated \( C = 8 \, \text{s} \). Any other value for \( C \) would violate the reciprocity condition. Let’s examine the incorrect options:

Option 2: \( C = 6 \, \text{s} \)

If \( C = 6 \, \text{s} \), substituting into the reciprocity condition:

    \( 3 \cdot 3 - 1 \cdot 6 = 9 - 6 = 3 \)

Here, the result is \( 3 \), which does not satisfy the reciprocity condition (\( A \cdot D - B \cdot C = 1 \)). Thus, this option is incorrect.

Option 3: \( C = 4 \, \text{s} \)

If \( C = 4 \, \text{s} \), substituting into the reciprocity condition:

    \( 3 \cdot 3 - 1 \cdot 4 = 9 - 4 = 5 \)

Here, the result is \( 5 \), which does not satisfy the reciprocity condition. Thus, this option is incorrect.

Option 4: \( C = 16 \, \text{s} \)

If \( C = 16 \, \text{s} \), substituting into the reciprocity condition:

    \( 3 \cdot 3 - 1 \cdot 16 = 9 - 16 = -7 \)

Here, the result is \( -7 \), which does not satisfy the reciprocity condition. Thus, this option is incorrect.

Option 5: \( C = 1 \, \text{s} \)

If \( C = 1 \, \text{s} \), substituting into the reciprocity condition:

    \( 3 \cdot 3 - 1 \cdot 1 = 9 - 1 = 8 \)

While the arithmetic is correct, the parameter \( C \) is defined as a reciprocal value in seconds (\( \text{s} \)), and the units here do not align with the requirements of the problem. Thus, this option is also incorrect.

Conclusion:

The correct value of \( C \) in the given two-port symmetric bilateral network is \( 8 \, \text{s} \), satisfying the reciprocity condition. This corresponds to Option 1. The other options fail to meet the necessary conditions or have incorrect units, as demonstrated in the analysis above.