Two-Port Network Fundamentals

Two-port networks are essential models in electrical engineering used to represent and analyze various electronic circuits such as amplifiers, filters, and matching networks. These networks have an input port and an output port, each characterized by a pair of terminals. To describe the behavior of these networks, different parameter sets are used, including Z-parameters (impedance), Y-parameters (admittance), H-parameters (hybrid), and ABCD-parameters (transmission).

Y-Parameters Definition

The admittance parameters, or Y-parameters, relate the currents at the ports to the voltages across the ports. They are particularly useful when a circuit is analyzed under short-circuit conditions, as admittance is the ratio of current to voltage. The standard Y-parameter equations for a linear two-port network are:

• \( I_1 = Y_{11}V_1 + Y_{12}V_2 \)
• \( I_2 = Y_{21}V_1 + Y_{22}V_2 \)

Where:

• \( I_1 \) is the current flowing into port 1.
• \( V_1 \) is the voltage across port 1.
• \( I_2 \) is the current flowing into port 2.
• \( V_2 \) is the voltage across port 2.

The individual Y-parameters are defined under specific short-circuit conditions:

• \( Y_{11} \): Input Admittance when port 2 is short-circuited (\( V_2 = 0 \)). \( Y_{11} = \frac{I_1}{V_1} \Big|_{V_2=0} \)
• \( Y_{12} \): Reverse Transfer Admittance when port 1 is short-circuited (\( V_1 = 0 \)). \( Y_{12} = \frac{I_1}{V_2} \Big|_{V_1=0} \)
• \( Y_{21} \): Forward Transfer Admittance when port 2 is short-circuited (\( V_2 = 0 \)). \( Y_{21} = \frac{I_2}{V_1} \Big|_{V_2=0} \)
• \( Y_{22} \): Output Admittance when port 1 is short-circuited (\( V_1 = 0 \)). \( Y_{22} = \frac{I_2}{V_2} \Big|_{V_1=0} \)

Short-Circuit Condition at Port 1

The question specifies that port 1 of the two-port circuit is short-circuited. This means the voltage across port 1, \( V_1 \), is zero (\( V_1 = 0 \)).

Let's substitute \( V_1 = 0 \) into the general Y-parameter equations:

1. From the first equation: \( I_1 = Y_{11}(0) + Y_{12}V_2 \) This simplifies to: \( I_1 = Y_{12}V_2 \) From this, we can define \( Y_{12} \) as: \( Y_{12} = \frac{I_1}{V_2} \) (when \( V_1 = 0 \))
2. From the second equation: \( I_2 = Y_{21}(0) + Y_{22}V_2 \) This simplifies to: \( I_2 = Y_{22}V_2 \) From this, we can define \( Y_{22} \) as: \( Y_{22} = \frac{I_2}{V_2} \) (when \( V_1 = 0 \))

Calculating Y-Parameters from Given Relations

We are provided with two relationships that hold true when port 1 is short-circuited:

• \( I_1 = 4I_2 \)
• \( V_2 = 0.25 I_2 \)

Calculating Y₁₂

We use the expression for \( Y_{12} \) derived under the \( V_1 = 0 \) condition: \( Y_{12} = \frac{I_1}{V_2} \)

Now, we substitute the given relationships into this expression.

First, substitute \( I_1 = 4I_2 \) into the numerator: \( Y_{12} = \frac{4I_2}{V_2} \)

Next, substitute \( V_2 = 0.25 I_2 \) into the denominator: \( Y_{12} = \frac{4I_2}{0.25I_2} \)

Assuming \( I_2 \) is not zero (which it must be for a meaningful relationship), we can cancel \( I_2 \) from the numerator and denominator: \( Y_{12} = \frac{4}{0.25} \) To simplify the division by 0.25, which is equivalent to \( \frac{1}{4} \): \( Y_{12} = \frac{4}{\frac{1}{4}} \) \( Y_{12} = 4 \times 4 \) \( Y_{12} = 16 \)

So, the value of \( Y_{12} \) is 16.

Calculating Y₂₂ (Optional Verification)

Let's also calculate \( Y_{22} \) to check other options, using its definition under \( V_1 = 0 \): \( Y_{22} = \frac{I_2}{V_2} \)

Substitute the given relation \( V_2 = 0.25 I_2 \) into the denominator: \( Y_{22} = \frac{I_2}{0.25I_2} \)

Cancel \( I_2 \) from the numerator and denominator: \( Y_{22} = \frac{1}{0.25} \) \( Y_{22} = 4 \)

Thus, \( Y_{22} = 4 \).

Evaluating the Given Options

Based on our analysis and calculations when port 1 is short-circuited:

• Option 1: Y₁₁ = 4 The parameter \( Y_{11} \) is defined as \( \frac{I_1}{V_1} \) when port 2 is short-circuited (\( V_2 = 0 \)). The problem specifies port 1 is short-circuited (\( V_1 = 0 \)), which prevents direct calculation of \( Y_{11} \) from the provided conditions. Therefore, this statement cannot be confirmed as true or false with the given information.
• Option 2: Y₁₂ = 16 Our calculation directly yields \( Y_{12} = 16 \). This statement is true based on the given conditions.
• Option 3: Y₂₁ = 16 The parameter \( Y_{21} \) is defined as \( \frac{I_2}{V_1} \) when port 2 is short-circuited (\( V_2 = 0 \)). Similar to \( Y_{11} \), the given condition of \( V_1 = 0 \) does not allow for direct calculation of \( Y_{21} \). This statement cannot be confirmed.
• Option 4: Y₂₂ = 0.25 Our calculation shows that \( Y_{22} = 4 \), which contradicts this option. Therefore, this statement is false.

Based on the detailed analysis and calculations, the only statement that is true given the conditions is \( Y_{12} = 16 \).