Question

What are the y-parameters of two-port networks?

Answer 1

The **Y-parameters**, also known as **admittance parameters**, describe the behavior of electrical networks (specifically **two-port networks**) in terms of input and output current and voltage relationships. These parameters are particularly useful in analyzing networks where admittance (the inverse of impedance) is a natural measure, such as in low-frequency or high-frequency circuits.

In a two-port network, we have four Y-parameters: \( Y_{11} \), \( Y_{12} \), \( Y_{21} \), and \( Y_{22} \), which relate the port voltages and currents as follows:

### Two-Port Network Representation:
Let:
- \( I_1 \) and \( I_2 \) be the currents entering ports 1 and 2, respectively.
- \( V_1 \) and \( V_2 \) be the voltages at ports 1 and 2, respectively.

The **Y-parameter** matrix describes the relationship between the currents and voltages using the following equations:

\[
\begin{pmatrix}
I_1 \\
I_2
\end{pmatrix}
=
\begin{pmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{pmatrix}
\begin{pmatrix}
V_1 \\
V_2
\end{pmatrix}
\]

In expanded form:

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

### Meaning of Each Y-Parameter:
1. **\( Y_{11} \) (Input Admittance):**  
   The ratio of \( I_1 \) to \( V_1 \) when \( V_2 = 0 \) (output port is short-circuited). It describes how the input current responds to the input voltage with the output shorted.  
   \[
   Y_{11} = \left( \frac{I_1}{V_1} \right)_{V_2=0}
   \]

2. **\( Y_{12} \) (Reverse Transfer Admittance):**  
   The ratio of \( I_1 \) to \( V_2 \) when \( V_1 = 0 \) (input port is short-circuited). It describes the influence of the output voltage on the input current.  
   \[
   Y_{12} = \left( \frac{I_1}{V_2} \right)_{V_1=0}
   \]

3. **\( Y_{21} \) (Forward Transfer Admittance):**  
   The ratio of \( I_2 \) to \( V_1 \) when \( V_2 = 0 \) (output port is short-circuited). It shows how the input voltage affects the output current.  
   \[
   Y_{21} = \left( \frac{I_2}{V_1} \right)_{V_2=0}
   \]

4. **\( Y_{22} \) (Output Admittance):**  
   The ratio of \( I_2 \) to \( V_2 \) when \( V_1 = 0 \) (input port is short-circuited). It describes how the output current responds to the output voltage with the input shorted.  
   \[
   Y_{22} = \left( \frac{I_2}{V_2} \right)_{V_1=0}
   \]

### Applications of Y-Parameters:
- Y-parameters are typically used in **high-frequency circuits** (like RF circuits) and **amplifiers**, where admittances are more straightforward to measure or calculate than impedances.
- They are used in the analysis of **filters**, **matching networks**, and **communication circuits**.

### Key Points:
- Y-parameters are convenient for modeling networks when the input/output behavior is primarily related to **currents** rather than voltages.
- The Y-parameters assume **linear, bilateral** circuits unless specified otherwise.

Answer 2

Y-parameters, or admittance parameters, are used to describe the behavior of two-port networks in terms of their admittance. They provide a way to represent how voltages and currents are related in a network. For a two-port network, the Y-parameters are defined as follows:

1. **Y11**: Input admittance when the output port is open (i.e., \( I_2 = 0 \)).
2. **Y12**: Reverse transfer admittance, representing the ratio of input current to the output voltage when the output port is open.
3. **Y21**: Forward transfer admittance, representing the ratio of output current to the input voltage when the input port is open.
4. **Y22**: Output admittance when the input port is open (i.e., \( I_1 = 0 \)).

Mathematically, they can be expressed in the following matrix form:

\[
\begin{bmatrix}
I_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\begin{bmatrix}
V_1 \\
V_2
\end{bmatrix}
\]

Here, \( I_1 \) and \( I_2 \) are the currents entering the input and output ports, respectively, and \( V_1 \) and \( V_2 \) are the voltages at the input and output ports, respectively.