Question

Consider the two-port network as shown in the figure.

Which of the following options provides the correct set of values of A, B, C and D parameters?

• A. \(A=\dfrac{1}{2}\), \(B=3\times10^3\ \Omega\), \(C=10^{-3}\ \Omega^{-1}\), \(D=\dfrac{1}{2}\)
• B. \(A=2\), \(B=3\ \Omega\), \(C=1\ \Omega^{-1}\), \(D=2\)
• C. \(A=2\), \(B=6\times10^3\ \Omega\), \(C=2\times10^{-3}\ \Omega^{-1}\), \(D=2\)
• D. \(A=2\), \(B=3\times10^3\ \Omega\), \(C=10^{-3}\ \Omega^{-1}\), \(D=2\)

Hint:

Treat the network as a T-section and use the standard T-to-ABCD conversion formulas.

Answer 1

The Correct Option is D

Step 1: Identify the network topology.
The circuit is a symmetric T-section: a \(1\ k\Omega\) resistor in series on the input side, a \(1\ k\Omega\) resistor as the shunt (middle) arm, and a \(1\ k\Omega\) resistor in series on the output side. Call these \(Z_1=Z_3=1000\ \Omega\) (the two series arms) and \(Z_2=1000\ \Omega\) (the shunt arm).

Step 2: Recall the ABCD parameters of a T-network.
For a general T-section with series arms \(Z_1,Z_3\) and shunt arm \(Z_2\),
\[ A=1+\frac{Z_1}{Z_2},\qquad B=Z_1+Z_3+\frac{Z_1Z_3}{Z_2},\qquad C=\frac{1}{Z_2},\qquad D=1+\frac{Z_3}{Z_2} \]

Step 3: Substitute the resistor values.
\[ A=1+\frac{1000}{1000}=2,\qquad D=1+\frac{1000}{1000}=2 \] \[ C=\frac{1}{1000}=10^{-3}\ \Omega^{-1} \] \[ B=1000+1000+\frac{1000\times1000}{1000}=1000+1000+1000=3000\ \Omega=3\times10^3\ \Omega \]

Step 4: Analyze the options.

(A) \(A=1/2,\ B=3\times10^3\ \Omega,\ C=10^{-3}\ \Omega^{-1},\ D=1/2\): The \(B\) and \(C\) values match but \(A\) and \(D\) are inverted. Incorrect.

(B) \(A=2,\ B=3\ \Omega,\ C=1\ \Omega^{-1},\ D=2\): \(A\) and \(D\) are right but \(B\) and \(C\) are off by a factor of \(1000\), a common unit slip. Incorrect.

(C) \(A=2,\ B=6\times10^3\ \Omega,\ C=2\times10^{-3}\ \Omega^{-1},\ D=2\): \(B\) and \(C\) are exactly double the correct values. Incorrect.

(D) \(A=2,\ B=3\times10^3\ \Omega,\ C=10^{-3}\ \Omega^{-1},\ D=2\): Matches every computed value. Correct.

Step 5: Final conclusion.
\[ \boxed{A=2,\ B=3\times10^3\ \Omega,\ C=10^{-3}\ \Omega^{-1},\ D=2} \]