3.5 Transformation to Extrinsic Parameters

As already discussed above, the calculation of extrinsic parameters is provided in order to take the parasitics introduced by the measurement environment into account. Based on a standard parasitic equivalent circuit, the simulator can take all parasitics into account and can provide also extrinsic two-port parameters. First of all, it is necessary to configure this equivalent circuit consisting of two ports (contacts) and one ground port, see Appendix A.5. Based on all data given in the contact descriptions, four intrinsic Y-parameters are extended by the parasitics and are also transformed to extrinsic S-parameters as well as to H-, Z-, and ABCD-parameters. The equivalent circuit consists of a single inductance and a single resistance for each port (or contact), and one capacitance between each port and the ground.

Figure 3.13: Two-Port parasitic equivalent circuit.

Since the intrinsic simulation results are transformed to extrinsic parameters, one can think of embedding the device in an equivalent circuit describing the parasitics. In this section, that embedding strategy is discussed. In turn, de-embedding strategies transforms extrinsic parameters into intrinsic ones, which can be already performed by the measurement equipment. The calculation starts with the intrinsic Y-parameters which are the actual result of the numerical simulation. The embedding strategy consists of the following steps:

1. Convert to Z-parameters and add serial parasitics.
   
2. Convert to Y-parameters and add parallel parasitics.
   
3. Calculate other two-port parameter sets.
   

In order to account for the effects of the serial parasitic elements, a Z-parameter transformation is performed. If the parasitic Z-parameters are known, they can be added to the intrinsic simulation results:

$$\ensuremath{\underline{\ensuremath{\mathbf{Z}}}} = \displaystyle ... ...(R_2 + R_3) + \ensuremath{\mathrm{j}}\omega (L_2 + L_3) \end{array} \right) \ .$$ (3.31)

Including the transformation to Z-parameters, these equations read like this:

$${ \ensuremath{\underline{Z_{11}}} = \ensuremath{\underline{Y_{11}... ...hbf{Y}}}}\vert} \,+\,(R1 + R3) \,+\, \ensuremath{\mathrm{j}}\omega (L1 + L3)\ ,$$ (3.32)

$$\ensuremath{\underline{Z_{12}}} = - \ensuremath{\underline{Y_{12}... ...nsuremath{\mathbf{Y}}}}\vert} \,+\,R3 \,+\, \ensuremath{\mathrm{j}}\omega L3\ ,$$ (3.33)

$$\ensuremath{\underline{Z_{21}}} = - \ensuremath{\underline{Y_{21}... ...nsuremath{\mathbf{Y}}}}\vert} \,+\,R3 \,+\, \ensuremath{\mathrm{j}}\omega L3\ ,$$ (3.34)

$$\ensuremath{\underline{Z_{22}}} = \ensuremath{\underline{Y_{22}}}... ...f{Y}}}}\vert} \,+\,(R2 + R3) \,+\, \ensuremath{\mathrm{j}}\omega (L2 + L3)\ . {$$ (3.35)

The determinants of the $2 \times 2$ matrices are calculated after the standard formula:

$$\ensuremath{\vert\ensuremath{\underline{\ensuremath{\mathbf{Y}}}}... ...{Y_{22}}} - \ensuremath{\underline{Y_{12}}} \ensuremath{\underline{Y_{21}}} \ ,$$ (3.36)

$$\ensuremath{\vert\ensuremath{\underline{\ensuremath{\mathbf{Z}}}}... ...{Z_{22}}} - \ensuremath{\underline{Z_{12}}} \ensuremath{\underline{Z_{21}}} \ .$$ (3.37)

The purpose of the second step is to take the parallel capacitances into account:

$$\ensuremath{\underline{\ensuremath{\mathbf{Y}}}} = \displaystyle ... ...22}}} + \ensuremath{\mathrm{j}}\omega (C_{23} + C_{12}) \end{array} \right) \ .$$ (3.38)

Before the parallel parasitics are added, the Z-parameters have to be re-transformed to Y-parameters.

$$\ensuremath{\underline{Y_{11}}} = \ensuremath{\underline{Z_{11}}}\, /\, \ensuremath{\vert\ensuremath{\underline{\ensuremath{\mathbf{Z}}}}\vert}\ ,$$ (3.39)

$$\ensuremath{\underline{Y_{12}}} = \ensuremath{\underline{Z_{12}}}\, /\, \ensuremath{\vert\ensuremath{\underline{\ensuremath{\mathbf{Z}}}}\vert}\ ,$$ (3.40)

$$\ensuremath{\underline{Y_{21}}} = \ensuremath{\underline{Z_{21}}}\, /\, \ensuremath{\vert\ensuremath{\underline{\ensuremath{\mathbf{Z}}}}\vert}\ ,$$ (3.41)

$$\ensuremath{\underline{Y_{22}}} = \ensuremath{\underline{Z_{22}}}\, /\, \ensuremath{\vert\ensuremath{\underline{\ensuremath{\mathbf{Z}}}}\vert}\ .$$ (3.42)

Finally, Y-parameters are multiplied by the characteristic impedance of the respective port:

$$\ensuremath{\underline{Y_{11}}} = \ensuremath{\underline{Y_{11}}} Z_{1}\ ,$$ (3.43)

$$\ensuremath{\underline{Y_{12}}} = \ensuremath{\underline{Y_{12}}} Z_{1}\ ,$$ (3.44)

$$\ensuremath{\underline{Y_{21}}} = \ensuremath{\underline{Y_{21}}} Z_{2}\ ,$$ (3.45)

$$\ensuremath{\underline{Y_{22}}} = \ensuremath{\underline{Y_{22}}} Z_{2}\ .$$ (3.46)

The embedding process consists of nine free parameters: three resistors, three capacitors, and three inductors. So, an optimization can be performed to minimize the error between measured data and simulation results. Whereas for example the Simplex Method can be employed, also optimizations based on the methods mentioned in Appendix C.3.9 are possible. After finding an appropriate parameter setting, the extrinsic parameters are eventually calculated according to the respective formulae in Appendix D.