4.6 AC Simulation:
Equivalent Circuits and Parameter Extraction

The small-signal response of a two-port network (Fig. 4.21) can be described by several equivalent parameter sets. The $\ensuremath{{\mathbf{Y}}}$-matrix (admittance matrix) gives the relation between input voltages and output currents:

$$\begin{pmatrix}I_1 I_2 \end{pmatrix} = \begin{pmatrix}Y_{11} \q... ...12} Y_{21} \quad Y_{22} \end{pmatrix} \begin{pmatrix}V_1 V_2 \end{pmatrix}$$ (4.82)

with

$$Y_{11} =\left.\frac{I_1}{V_1}\right\vert _{V_2=0} Y_{12} =\left.\frac{I_1}{V_2}\right\vert _{V_1=0}$$ (4.83)

$$Y_{21} =\left.\frac{I_2}{V_1}\right\vert _{V_2=0} Y_{22} =\left.\frac{I_2}{V_2}\right\vert _{V_1=0}.$$ (4.84)

Figure 4.21: Voltages and currents at a two-port network.

Another established relation is the $\ensuremath{{\mathbf{Z}}}$-matrix (impedance parameters), which links the output voltages to the input currents:

$$\begin{pmatrix}V_1 V_2 \end{pmatrix} = \begin{pmatrix}Z_{11} \q... ...12} Z_{21} \quad Z_{22} \end{pmatrix} \begin{pmatrix}I_1 I_2 \end{pmatrix}$$ (4.85)

with

$$Z_{11} =\left.\frac{V_1}{I_1}\right\vert _{I_2=0} Z_{12} =\left.\frac{V_1}{I_2}\right\vert _{I_1=0}$$ (4.86)

$$Z_{21} =\left.\frac{V_2}{I_1}\right\vert _{I_2=0} Z_{22} =\left.\frac{V_2}{I_2}\right\vert _{I_1=0}.$$ (4.87)

Figure 4.22: Incident and reflected waves at a two-port network.

However, measurement of those parameter sets requires open or shortcut conditions, which are difficult to achieve at high frequencies. To avoid this problem, matched loads can be used. Thus, the device is embedded into a transmission line with a specific impedance ($Z_0$). For a traveling wave, the inserted network acts as an impedance, different from the characteristic impedance of the line. S-parameters are the complex valued reflexion and transmission coefficients (Fig. 4.22):

$$\begin{pmatrix}b_1 b_2 \end{pmatrix} = \begin{pmatrix}S_{11} \q... ...12} S_{21} \quad S_{22} \end{pmatrix} \begin{pmatrix}a_1 a_2 \end{pmatrix}$$ (4.88)

with

$$S_{11} =\left.\frac{b_1}{a_1}\right\vert _{a_2=0} S_{12} =\left.\frac{b_1}{b_2}\right\vert _{a_1=0}$$ (4.89)

$$S_{21} =\left.\frac{b_2}{a_1}\right\vert _{a_2=0} S_{22} =\left.\frac{b_2}{a_2}\right\vert _{a_1=0}.$$ (4.90)

The power waves can be expressed as a function of the currents, voltages, and the complex reference impedance [362].

Figure 4.23: Equivalent circuit for a HEMT.

In order to obtain important figures of merit for the frequency characteristics of the devices, such as the cut-off frequency and the maximum oscillation frequency, an equivalent circuit is useful. Here, the one used by Dambrine et al. [363] for FETs is applied (Fig. 4.23). The expressions to calculate the values of its circuit elements are as following [363,364]:

$$\omega = 2 \pi f$$ (4.91)

$$Y_\ensuremath{\mathrm{GS}} = Y_{11}+Y_{12}$$ (4.92)

$$Y_\ensuremath{\mathrm{gm}} = Y_{21}-Y_{12}$$ (4.93)

$$Y_\ensuremath{\mathrm{DS}} = Y_{22}+Y_{12}$$ (4.94)

$$Y_\ensuremath{\mathrm{GD}} = -Y_{12}$$ (4.95)

$$C_\ensuremath{\mathrm{GD}} = \frac{-1}{\ensuremath{\mathrm{Im}}\left(\frac{1}{Y_\ensuremath{\mathrm{GD}}}\right) \omega}$$ (4.96)

$$C_\ensuremath{\mathrm{DS}} = \frac{\ensuremath{\mathrm{Im}}\left(Y_\ensuremath{\mathrm{DS}}\right)}{\omega}$$ (4.97)

$$C_\ensuremath{\mathrm{GS}} = \frac{-1}{\ensuremath{\mathrm{Im}}\left(\frac{1}{Y_\ensuremath{\mathrm{GS}}}\right) \omega}$$ (4.98)

$$R_\ensuremath{\mathrm{GS}} = \ensuremath{\mathrm{Re}}\left(\frac{1}{Y_\ensuremath{\mathrm{GS}}}\right)$$ (4.99)

$$R_\ensuremath{\mathrm{DS}} = \frac{1}{\ensuremath{\mathrm{Re}}\left(Y_\ensuremath{\mathrm{DS}}\right)}$$ (4.100)

$$R_\ensuremath{\mathrm{GS}} = \ensuremath{\mathrm{Re}}\left(\frac{1}{Y_\ensuremath{\mathrm{GD}}}\right)$$ (4.101)

As the cut-off frequency is defined as the frequency at which the gain is unity, based on the given equivalent circuit the following expression for $f_\ensuremath {\mathrm {t}}$ can be obtained:

$$f_\ensuremath{\mathrm{T}} = \frac{g_\ensuremath{\mathrm{m}}}{2 \pi (C_\ensuremath{\mathrm{GS}} + C_\ensuremath{\mathrm{GS}} )}$$

with

$$g_\ensuremath{\mathrm{m}} = \mid Y_\ensuremath{\mathrm{gm}} ( 1 + j \omega R_\ensuremath{\mathrm{GS}} C_\ensuremath{\mathrm{GS}})\mid.$$

In order to separate eventual errors introduced by the measurement equipment it is helpful to embed the device in a parasitic equivalent circuit, independent of the biasing conditions. Such a parasitic equivalent circuit is shown in Fig. 4.24 [365]. The values for those elements are given in Table 4.15.

Figure 4.24: Extrinsic parasitic elements.

Table 4.15: Applied parasitic elements for GaN HEMT simulation.

Element	L $_\ensuremath{\mathrm{s}}$	L $_\ensuremath{\mathrm{g}}$	L $_\ensuremath{\mathrm{d}}$	C $_\ensuremath{\mathrm{pgs}}$	C $_\ensuremath{\mathrm{pgd}}$	C $_\ensuremath{\mathrm{pds}}$
Unit	[pH]	[pH]	[pH]	[fF]	[fF]	[fF]
Value	1	44	46	18	6	9