AC Source
An AC source can be characterized by its series impedance between terminals. The impedance between each terminal and the source is defined in the impedance matrix \(\mathbf{Z}\), and the admittance matrix \(\mathbf{Y}\) is derived from \(\mathbf{Z}\) by inverting the impedance values.
Mathematical Modeling
Let \(Z_{ii}\) be the impedance at terminal \(i\). The self-admittance at terminal \(i\) is defined as \(Y_{ii} = \frac{1}{Z_{ii}}\). This represents the relationship between the voltage and current at terminal \(i\). For an AC source with \(n\) terminals (pins), we can generalize this for each terminal.
If the impedance is uniform for all terminals, i.e., \(Z_{ii} = Z\) for all \(i\), the self-admittance becomes \(Y_{ii} = \frac{1}{Z}\) for all \(i\).
Y Parameters
For an AC source with \(n\) terminals, we first construct the diagonal elements of the Y matrix using the series impedance. This means setting:
\[\begin{equation} Y_{ii} = \frac{1}{Z} \end{equation}\] for each terminal \(i\). The matrix starts with the form: \[\begin{equation} \mathbf{Y} = \begin{bmatrix} Y_{11} & 0 & 0 & \cdots & 0 \\ 0 & Y_{22} & 0 & \cdots & 0 \\ 0 & 0 & Y_{33} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & Y_{nn} \end{bmatrix} \end{equation}\]
where \(Y_{ii} = \frac{1}{Z_{ii}}\).
To represent the interaction between the terminals, we must introduce off-diagonal terms. These terms are generally set to zero initially, as the mutual impedance is typically not considered for a simple AC source with independent terminals. However, in more complex scenarios, the mutual admittance could be included.
Additional rows and columns are added to account for the terminals’ connections to the source to complete the Y matrix. The generalized Y matrix is formulated as: \[\begin{equation} \mathbf{Y} = \begin{bmatrix} Y_{11} & 0 & 0 & \cdots & 0 & -Y_{11} & 0 & \cdots & 0 \\ 0 & Y_{22} & 0 & \cdots & 0 & 0 & -Y_{22} & \cdots & 0 \\ 0 & 0 & Y_{33} & \cdots & 0 & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & Y_{nn} & 0 & 0 & \cdots & -Y_{nn} \\ -Y_{11} & 0 & 0 & \cdots & 0 & Y_{11} & 0 & \cdots & 0 \\ 0 & -Y_{22} & 0 & \cdots & 0 & 0 & Y_{22} & \cdots & 0 \\ 0 & 0 & 0 & \cdots & -Y_{nn} & 0 & 0 & \cdots & Y_{nn} \end{bmatrix} \end{equation}\]
The off-diagonal negative terms (e.g., \(-Y_{ii}\)) are introduced to account for the interaction between the terminals and ensure that the system satisfies Kirchhoff’s Current Law (KCL), which states that the sum of currents entering and leaving a node must equal zero.
Code Explanation
For detailed code information, see the Harmony manual.