\begin{equation}\label{eq:cond} I_{ij}=g (V_i-V_j) = g V_i - g V_j, \end{equation}
\begin{equation}\label{eq:cond2} I_{ji}=-I_{ij}=g (V_j-V_i) = -g V_i + g V_j \end{equation}
\begin{equation}\label{eq:Kirchoff} \sum_{j \neq i}^n I_{ij}=0. \end{equation}
\begin{equation}\label{eq:MatrixR1} \begin{bmatrix}\vdots & & \vdots & & \vdots \\ \dots & g & \dots & -g & \dots \\ \vdots & & \vdots & & \vdots \\ \dots & -g & \dots & g & \dots \\ \vdots & & \vdots & & \vdots \end{bmatrix} \begin{bmatrix} \vdots \\ V_i \\ \vdots \\ V_j \\ \vdots \end{bmatrix} =\begin{bmatrix} \vdots \\ 0 \\ \vdots \\ 0 \\ \vdots \end{bmatrix} \end{equation}
\begin{equation}\label{eq:cond3} V_i-V_j - R I_{ij} = 0. \end{equation}
\begin{equation}\label{eq:MatrixR2} \begin{bmatrix}\vdots & & \vdots & & \vdots & & \vdots \\ \dots & & \dots & & \dots & 1 & \dots \\ \vdots & & \vdots & & \vdots & & \vdots \\ \dots & & \dots & & \dots & -1 & \dots \\ \vdots & & \vdots & & \vdots & & \vdots \\ \dots & 1 & \dots & -1 & \dots & -R & \dots \\ \vdots & & \vdots & & \vdots & & \vdots \end{bmatrix} \begin{bmatrix} \vdots \\ V_i \\ \vdots \\ V_j \\ \vdots \\ I_{ij} \\ \vdots \end{bmatrix} =\begin{bmatrix} \vdots \\ 0 \\ \vdots \\ 0 \\ \vdots \\ 0 \\ \vdots \end{bmatrix} \end{equation}
\begin{equation}\label{eq:ccc} j\omega C (V_i-V_j) = j\omega C V_i - j\omega C V_j, \end{equation}
\begin{equation}\label{eq:MatrixC} \begin{bmatrix}\vdots & & \vdots & & \vdots \\ \dots & j\omega C & \dots & -j\omega C & \dots\\ \vdots & & \vdots & & \vdots \\ \dots & -j\omega C & \dots & j\omega C & \dots \\ \vdots & & \vdots & & \vdots \end{bmatrix} \begin{bmatrix} \vdots \\ V_i \\ \vdots \\ V_j \\ \vdots \end{bmatrix} =\begin{bmatrix} \vdots \\ 0 \\ \vdots \\ 0 \\ \vdots \end{bmatrix} \end{equation}
\begin{equation}\label{eq:cL} \frac{1}{j\omega L} (V_i-V_j) = \frac{1}{j\omega L} V_i - \frac{1}{j\omega L} V_j, \end{equation}
\begin{equation}\label{eq:Matrixspoel} \begin{bmatrix}\vdots & & \vdots & & \vdots \\ \dots & \frac{1}{j\omega L} & \dots & -\frac{1}{j\omega L} & \dots\\ \vdots & & \vdots & & \vdots \\ \dots & -\frac{1}{j\omega L} & \dots & \frac{1}{j\omega L} & \dots \\ \vdots & & \vdots & & \vdots \end{bmatrix} \begin{bmatrix} \vdots \\ V_i \\ \vdots \\ V_j \\ \vdots \end{bmatrix} =\begin{bmatrix} \vdots \\ 0 \\ \vdots \\ 0 \\ \vdots \end{bmatrix} \end{equation}
\begin{equation}\label{eq:gm} I_{kl}= g_m (V_i-V_j) , \end{equation}
\begin{equation}\label{eq:Matrixgm} \begin{bmatrix}\vdots & & \vdots & & \vdots & & \vdots & & \vdots \\ \dots & & \dots & & \dots & & \dots & & \dots\\ \vdots & & \vdots & & \vdots & & \vdots & & \vdots\\ \dots & & \dots & & \dots & & \dots & & \dots\\ \vdots & & \vdots & & \vdots & & \vdots & & \vdots\\ \dots & g_m & \dots & -g_m & \dots & & \dots & & \dots\\ \vdots & & \vdots & & \vdots & & \vdots & & \vdots\\ \dots & -g_m & \dots & g_m & \dots & & \dots & & \dots\\ \vdots & & \vdots & & \vdots & & \vdots & & \vdots \end{bmatrix} \begin{bmatrix} \vdots\\ V_i\\ \vdots \\ V_j\\ \vdots\\ V_k\\ \vdots \\ V_l\\ \vdots \end{bmatrix} =\begin{bmatrix} \vdots\\ 0 \\ \vdots \\ 0 \\ \vdots\\ 0 \\ \vdots \\ 0 \\ \vdots \end{bmatrix} \end{equation}
\begin{equation}\label{eq:gm2} I_{DS}= g_m (V_G-V_S) . \end{equation}
\begin{equation}\label{eq:MatrixMOS} \begin{bmatrix}\vdots & & \vdots & & \vdots & & \vdots \\ \dots & & \dots & & \dots & & \dots \\ \vdots & & \vdots & & \vdots & & \vdots \\ \dots & g_m & \dots & & \dots & -g_m & \dots \\ \vdots & & \vdots & & \vdots & & \vdots \\ \dots & -g_m & \dots & & \dots & g_m & \dots \\ \vdots & & \vdots & & \vdots & & \vdots \end{bmatrix} \begin{bmatrix} \vdots\\ V_G\\ \vdots \\ V_D\\ \vdots \\ V_S\\ \vdots \end{bmatrix} =\begin{bmatrix} \vdots\\ 0 \\ \vdots \\ 0 \\ \vdots \\ 0 \\ \vdots \end{bmatrix} \end{equation}
\begin{equation} V_i-V_j=V_{oc}. \end{equation}
\begin{equation}\label{eq:bron} \begin{bmatrix}\vdots & & & & & \vdots & & & \vdots \\ \dots & & & & & \dots & 1 & & \dots \\ \dots & & & & & \dots & -1 & & \dots \\ \vdots & & & & & \vdots & & & \vdots \\ \dots & 1 & -1& & & \dots & \dots & & \dots\\ \vdots & & & & & \vdots & & & \vdots\end{bmatrix} \begin{bmatrix} \vdots\\ V_i\\ V_j\\ \vdots\\ I_{sc}\\ \vdots\end{bmatrix} =\begin{bmatrix} \vdots\\ 0\\ 0\\ \vdots \\ V_{oc}\\ \vdots \end{bmatrix} \end{equation}
{math}
:label: TransformEq
\begin{aligned}
V_i-V_j - j \omega L_{ij} I_1 - j \omega M I_2 = 0 \\
V_k-V_l - j \omega M I_1 - j \omega L_{kl} I_2 = 0 \\
\end{aligned}
{math}
:label: Transformmatrix
\begin{bmatrix}\vdots & & & & & \vdots & & & \vdots \\
\dots & & & & & \dots & 1 & & \dots \\
\dots & & & & & \dots & -1 & & \dots \\
\dots & & & & & \dots & & 1 & \dots \\
\dots & & & & & \dots & & -1 & \dots \\
\vdots & & & & & \vdots & & & \vdots \\
\dots & 1 & -1& & & \dots & - j \omega L_{ij} & - j \omega M & \dots\\
\dots & & & 1 & -1& \dots & - j \omega M & - j \omega L_{kl} & \dots \\
\vdots & & & & & \vdots & & & \vdots\end{bmatrix} \begin{bmatrix} \vdots\\ V_i\\
V_j\\
V_k\\
V_l\\
\vdots\\
I_1\\
I_2\\
\vdots\end{bmatrix} =\begin{bmatrix} \vdots\\ 0\\
0\\
0\\
0\\
\vdots \\
0\\
0\\
\vdots \end{bmatrix}
Hieronder tonen we een eenvoudig verschilversterker circuit met 5 transistors dat eenvoudig in spice kan geimplementeerd worden.