6.1.2 Numerical Inductance and Resistance Extraction

The electro-magnetic power in the domain $\mathcal{V}$ can be expressed in two different ways. The first one is by volume integration over the power density distribution [103] in the region $\mathcal{V}$ . The second one is by the current flowing trough the resulting resistance and inductance. As aforementioned, the quasi-magnetostatic case is considered

$$\int_{\mathcal{V}}\left(\vec{J}\cdot\vec{E} + \vec{H}\cdot\partial_t\vec{B}\right)\mathrm{d}V = LI\frac{dI}{dt} + RI^2.$$ (6.13)

In the frequency domain using the constitutive relations (4.7) and (4.8) one obtains

$$\int_{\mathcal{V}}\left(\frac{J^2}{\gamma} + \jmath\omega\mu{}H^2\right)\mathrm{d}V = \jmath\omega{}LI^2 + RI^2.$$ (6.14)

The left hand side of (6.14) can be denoted in the following way:

$$P_1 = \int_{\mathcal{V}}\frac{J^2}{\gamma} \mathrm{d}V, P_2 ... ...mega\int_{\mathcal{V}}\mu{}H^2 \mathrm{d}V, \mathrm{and} P = P_1 + P_2.$$ (6.15)

Consequently the resistance $R$ and the inductance $L$ arising in the domain $\mathcal{V}$ are calculated by

$$R = \frac{\mathrm{\mathcal{R}e\{P\}}}{I^2}, L = \frac{\mathrm{\mathcal{I}m\{P\}}}{\omega{}I^2}.$$ (6.16)

The domain $\mathcal{V}$ is discretized and the linear equation system (5.37) is assembled as described in Section 5.2 and solved to obtain the solution vector $\{c\}$ . The indexes $j$ of the coefficients $c_j$ are arranged as shown in Section 5.2. The fields $\vec{H}_1$ and $\psi$ are constructed as in (5.27) and (5.28). $\vec{H}$ is obtained by (5.23). These quantities are used to determine $P$ . Inserting (5.13) in the expression for $P_1$ from (6.15) gives

$$\begin{split}P_1 & = \int_{\mathcal{V}}\frac{1}{\gamma}\left(... ...\nabla}\times\vec{N}_j\right) \right]^2 \mathrm{d}V \end{split}$$ (6.17)

or

$$\begin{split}P_1 & = \sum_{i=1}^m c_i \sum_{j=1}^m \int_{\m... ...vec{\nabla}\times\vec{N}_j\right) \mathrm{d}V c_j. \end{split}$$ (6.18)

With (5.23) $P_2$ is modified to read

$$P_2 = \jmath\omega\int_{\mathcal{V}}\mu\left(\vec{H}_1 - \vec{\nabla}\psi\right)^2 \mathrm{d}V.$$ (6.19)

Expressing $\vec{H}_1$ from (5.27) and $\psi$ from (5.28) one obtains

$$P_2 = \jmath\omega\int_{\mathcal{V}}\mu \left( \sum_{j=1}^{m}c_j\... ...la}\lambda_j - \sum_{j=M+1}^{N}c_j\vec{\nabla}\lambda_j \right)^2 \mathrm{d}V,$$ (6.20)

which is written in the more convenient form

$$\begin{split}\frac{P_2}{\jmath\omega} & = \left.\sum_ic_i\sum... ... [m+1;n]\cup[M+1;N], j \in [m+1;n]\cup[M+1;N]}. \end{split}$$ (6.21)

Using (6.18) and (6.21) for $P_1$ and $P_2$ the very suitable form for the power $P$ in the simulation domain is derived

$$P = P_1 + P_2 = \left\{c\right\}^T \left[\begin{array}{llll} A_1 ... ...2 & A_3 & B_3 B_2^T & C_2 & B_3^T & C_3 \end{array}\right] \left\{c\right\},$$ (6.22)

where the sub-matrices $A_k$ , $B_k$ and $C_k$ with $k\in[1;3]$ are calculated using the mathematical expressions given in (5.38), (5.39), and (5.40), respectively. The indexes $k$ of $A_k$ , $B_k$ and $C_k$ indicate the different ranges of the sub-matrix entries global indexes $i$ and $j$ . The associated global index ranges are specified in (6.18) and (6.21) and can be given more clearly as follows:

$$\renewedcommand{arraystretch}{1.3} \begin{array}{l\vert l\vert l\... ... C_2 & B_3^T & C_3 \hline \end{array}$$

In (5.37) only the $n\times{}n$ part of the matrix for the unknowns is used. The remaining part is assembled with the known Dirichlet values directly to right hand side vector $\{b\}$ . For the power calculation (6.22) the whole $N\times{}N$ matrix is used. Expression (6.22) can be simplified by involving (5.37)

$$P = \left\{c_{n+1} ... c_N\right\} \begin{array}{\vert llll\ver... ..._2 & B_2 & A_3 & B_3 B_2^T & C_2 & B_3^T & C_3 \end{array} \left\{c\right\}.$$ (6.23)

Figure 6.4: Current density distribution at low frequency.

Figure 6.5: Current density distribution at $3$ GHz.

Figure 6.6: Current density distribution at $30$ GHz.

Figure 6.7: Current density distribution at $300$ GHz.

Figure 6.8: Magnetic field distribution at low frequency.

Figure 6.9: Magnetic field distribution at $3$ GHz.

Figure 6.10: Magnetic field distribution at $30$ GHz.

Figure 6.11: Magnetic field distribution at $300$ GHz.