Equation defined models

Often it will happen that a user needs to implement his own model. Therefore, it is useful to supply devices that are defined by arbitrary equations.

Models with Explicit Equations

For example the user must enter an equation $i(V)$ describing how the port current $I$ depends on the port voltage $V=V_1-V_2$ and an equation $q(V)$ describing how much charge $Q$ is held due to the voltage $V$. These are time domain equations. The most simple way then is a device with two nodes. Defining

$$I = i(V) \qquad\textrm{and}\qquad g = \dfrac{\partial I}{\partial V} = \underset{h\rightarrow 0}{\textrm{lim}}\dfrac{I(V+h) - I(V)}{h}$$ (10.271)

as well as

$$Q = q(V) \qquad\textrm{and}\qquad c = \dfrac{\partial Q}{\partial V} = \underset{h\rightarrow 0}{\textrm{lim}}\dfrac{Q(V+h) - Q(V)}{h}$$ (10.272)

the MNA matrix for a (non-linear) DC analysis writes:

$$\begin{split}\begin{bmatrix}+g^{(m)} & -g^{(m)}\\ -g^{(m)} & ... ... g^{(m)}\cdot (V_1^{(m)} - V_2^{(m)}) \end{bmatrix} \end{split}$$ (10.273)

For a transient simulation, equation (6.89) on page has to be used with $Q$ and $c$.

For an AC analysis the MNA matrix writes:

$$(\underline{Y}) = (g + j\omega\cdot c)\cdot \begin{bmatrix}+1 & -1\\ -1 & +1 \end{bmatrix}$$ (10.274)

And the S-parameter matrix writes:

$$S_{11} = S_{22} = \frac{1}{2\cdot Z_0\cdot Y + 1}$$ (10.275)

$$S_{12} = S_{21} = 1-S_{11}$$ (10.276)

$$Y = g + j\omega\cdot c$$ (10.277)

The simulator needs to create the derivatives $g$ and $c$ by its own. This can be done numerically or symbolically. One might ask why the non-linear capacitance is modeled as charge, not as capacitance. Indeed this may be changed, but with a computer algorithm, creating the derivative is easier than to integrate.

The component described above can be expanded to one with two ports (two pairs of terminals: terminal 1 and 2 and terminal 3 and 4). That is, the currents and charges of both ports depend on both port voltages $V_{12}=V_1-V_2$ and $V_{34}=V_3-V_4$. Thus, the defining equations are:

$$I_1 = i_1(V_{12}, V_{34}) \qquad\textrm{and}\qquad g_{11} = \dfra... ...V_{12}} \qquad\textrm{and}\qquad g_{12} = \dfrac{\partial I_1}{\partial V_{34}}$$ (10.278)

$$I_2 = i_2(V_{12}, V_{34}) \qquad\textrm{and}\qquad g_{21} = \dfra... ...V_{12}} \qquad\textrm{and}\qquad g_{22} = \dfrac{\partial I_2}{\partial V_{34}}$$ (10.279)

as well as

$$Q_1 = q_1(V_{12}, V_{34}) \qquad\textrm{and}\qquad c_{11} = \dfra... ...V_{12}} \qquad\textrm{and}\qquad c_{12} = \dfrac{\partial Q_1}{\partial V_{34}}$$ (10.280)

$$Q_2 = q_2(V_{12}, V_{34}) \qquad\textrm{and}\qquad c_{21} = \dfra... ...V_{12}} \qquad\textrm{and}\qquad c_{22} = \dfrac{\partial Q_2}{\partial V_{34}}$$ (10.281)

The MNA matrix for the DC analysis writes:

$$\begin{bmatrix}+g_{11}^{(m)} & -g_{11}^{(m)} & +g_{12}^{(m)} & -g... ...g_{21}^{(m)}\cdot V_{12}^{(m)} - g_{22}^{(m)}\cdot V_{34}^{(m)}\\ \end{bmatrix}$$ (10.282)

For a transient simulation, the DC equations have to be extended by the non-linear (trans-) capacitances, e.g. for backward Euler:

$$I_{C11}^{n+1,m} = \underbrace{\dfrac{c_{11}(V_{12}^{n+1,m})}{h^n}}_{g_{eq,11}}\cd... ...n+1} \underbrace{- \dfrac{c_{11}(V_{12}^{n})}{h^n}\cdot V_{12}^{n}}_{I_{eq,11}}$$ (10.283)

$$I_{C12}^{n+1,m} = \underbrace{\dfrac{c_{12}(V_{12}^{n+1,m})}{h^n}}_{g_{eq,12}}\cd... ...n+1} \underbrace{- \dfrac{c_{12}(V_{12}^{n})}{h^n}\cdot V_{12}^{n}}_{I_{eq,12}}$$ (10.284)

$$I_{C21}^{n+1,m} = \underbrace{\dfrac{c_{21}(V_{34}^{n+1,m})}{h^n}}_{g_{eq,21}}\cd... ...n+1} \underbrace{- \dfrac{c_{21}(V_{34}^{n})}{h^n}\cdot V_{34}^{n}}_{I_{eq,21}}$$ (10.285)

$$I_{C22}^{n+1,m} = \underbrace{\dfrac{c_{22}(V_{34}^{n+1,m})}{h^n}}_{g_{eq,22}}\cd... ...n+1} \underbrace{- \dfrac{c_{22}(V_{34}^{n})}{h^n}\cdot V_{34}^{n}}_{I_{eq,22}}$$ (10.286)

So with $g_{tr} = g + g_{eq}$ it is:

$$\begin{split}&\begin{bmatrix}+g_{tr,11}^{(m)} & -g_{tr,11}^{(... ...} + I_{eq,21}^{(n)} + I_{eq,22}^{(n)} \end{bmatrix} \end{split}$$ (10.287)

For an AC analysis the MNA matrix writes:

$$(\underline{Y}) = \begin{bmatrix}+g_{11} + j\omega\cdot c_{11} & ... ...} & -g_{22} - j\omega\cdot c_{22} & +g_{22} + j\omega\cdot c_{22} \end{bmatrix}$$ (10.288)

As can bee seen, this scheme can be expanded to any number of ports. The matrices soon become quite complex, but fortunately modern computers are able to cope with this complexity. S-parameters must be obtained numerical by setting equation 10.288 into equation 15.7.

Models with Implicit Equations

The above-mentioned explicit models are not useable for all components. If the Y-parameters do not exist or if the equations cannot be analytically transformed into the explicit form, then an implicit representation must be taken. hat is, for a one-port (two-terminal) component the following formulas are defined by the user:

$$0 = f(V, I) \qquad \qquad g_V = \dfrac{\partial f(V, I)}{\partial V} = \underset{h\rightarrow 0}{\text{lim}}\dfrac{f(V+h, I) - f(V, I)}{h}$$ (10.289)

$$\qquad g_I = \dfrac{\partial f(V, I)}{\partial I} = \underset{h\rightarrow 0}{\text{lim}}\dfrac{f(V, I+h) - f(V, I)}{h}$$ (10.290)

The MNA matrix for the AC analysis writes as follows:

$$\begin{bmatrix}. & . & +1\\ . & . & -1\\ +g_V & -g_V & g_I \end{b... ...}V_{1}\\ V_{2}\\ I_{out} \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0 \end{bmatrix}$$ (10.291)

As usual, for the DC analysis the last zero on the right hand side has to be replaced by the iteration formula:

$$g_V\cdot (V_1 - V_2) + g_I\cdot I_{out} - f(V_1-V_2, I_{out})$$ (10.292)

The S-parameters are:

$$S_{11} = S_{22} = \dfrac{g_I}{g_I - 2\cdot Z_0\cdot g_V}$$ (10.293)

$$S_{12} = S_{21} = 1-S_{11}$$ (10.294)

Consequently, for a two-port device two equation are necessary: One for first port and one for second port:

$$= f_1(V_{12}, V_{34}, I_1, I_2)$$ 0 (10.295)

$$= f_2(V_{12}, V_{34}, I_1, I_2)$$ 0 (10.296)

Building the MNA matrix is again straight forward:

$$\begin{split}&\begin{bmatrix}. & . & . & . & +1 & 0\\ . & . &... ...2(V_{12}, V_{34}, I_{out1}, I_{out2}) \end{bmatrix} \end{split}$$ (10.297)

Once more, this concept can easily expanded to any number of ports. It is also possible mix implicit and explicit definitions, i.e. some ports of the device may be defined by explicit equations whereas the others are defined by implicit equations.

The calculation of the S-parameters is not that trival. The Y-parameters as well as the Z-parameters might be infinite. A small trick can avoid this problem, as will be shown in the following 2-port example. First, the small-signal Y-parameters should be derived by using the law about implicit functions:

$$(\underline{J}) = \begin{pmatrix}y_{11} & y_{12}\\ y_{21} & y_{22... ...al f_2}{\partial V_1} & \dfrac{\partial f_2}{\partial V_2} \end{pmatrix}}_{J_v}$$ (10.298)

The equation reveals immediately the difficulty: The inverse of the current Jocobi matrix $J_i$ may not exist. But this problem can be outsourced to one single scalar number by using Cramer's rule for matrix inversion:

$$J_i^{-1} = \dfrac{1}{\Delta J_i}\cdot A_{Ji}$$ (10.299)

The matrix $A_{Ji}$ is built of the sub-determinantes of $J_i$ in the way that $a_{(n,m)}$ is the determinante of $J_i$ without row $m$ and without column $n$ but multiplied with $(-1)^{n+m}$. It therefore always exists, whereas dividing by the determinante of $J_i$ may become infinity. Now parameters can be defined as follows:

$$(\underline{J}') = \begin{pmatrix}y_{11}' & y_{12}'\\ y_{21}' & y_{22}' \end{pmatrix} = -A_{Ji}\cdot J_v$$ (10.300)

Before converting to S-parameters the matrix must be expanded to a 4-port matrix, because the 2-ports are not referenced to ground:

$$(\underline{J}'') = \begin{pmatrix}+y_{11}' & -y_{11}' & +y_{12}'... ...\\ -y_{21}' & +y_{21}' & -y_{22}' & +y_{22}' \end{pmatrix} = -A_{Ji}'\cdot J_v'$$ (10.301)

Finally, equation (15.7) converts the parameters to S-parameters:

$$(\underline{S}) = \left( (E) - Z_0\cdot (\underline{Y}) \right) \cdot \left( (E) + Z_0\cdot (\underline{Y}) \right)^{-1}$$ (10.302)

$$= \left( (E) + Z_0\cdot \dfrac{1}{\Delta J_i}\cdot A_{Ji}'\cdot J... ...\left( (E) - Z_0\cdot \dfrac{1}{\Delta J_i}\cdot A_{Ji}'\cdot J_v' \right)^{-1}$$ (10.303)

$$= \left( \Delta J_i\cdot (E) + Z_0\cdot A_{Ji}'\cdot J_v' \right) \cdot \left( \Delta J_i\cdot (E) - Z_0\cdot A_{Ji}'\cdot J_v' \right)^{-1}$$ (10.304)

The calculations proofs that the critical factor $1/\Delta J_i$ disappears and a solution exists if and only if the S-parameters of this device exist.