Measuring the scattering parameters of a microwave transistor to develop an appropriate small-signal device model, the question may come up about the minimum frequency range needed to extract reliable model parameter values. This issue has been studied intensively for microwave devices based on GaN technology and discussed in detail in my book Parameter Extraction and Complex Nonlinear Transistor Models, published by Artech House in 2020. Similar considerations and results may also be valid for GaAs and other related technologies.
It is immediately understood that the measured set of complex scattering coefficients at a given frequency delivers only eight independent equations that can be used to determine the model parameters of a generic FET pi-type equivalent network with extrinsic reference planes (see Figure 7.6 in my book). Such a simple small-signal model is like a black box. Its “overall” model parameters (two-port parameters) are functions of a combination of the small-signal model parameters. Thus, to solve this problem a larger number of S-parameter measurements is needed.
Equivalent networks have been simplified by consideration of various bias points such as pinch-off and gate-forward. In this case, such as the standard 15-element model, only one set of scattering parameters under different operating conditions may be sufficient to determine the model parameters analytically.
However, in the general case, model parameter extraction relies on an optimization process. It is a great challenge to cope with this task as is discussed broadly in my book, that is, to find physically meaningful model parameter values from multiple S-parameter measurements. In this case it is interesting to know the minimum measurement frequency bandwidth, that is, the minimum upper frequency that is needed to determine the small-signal model elements consistently. A few remarks are made on this in the following.
Figure 1 Nonlinear FET model with 20 model elements. |
Figure 2 Nonlinear FET model under cold pinch-off. |
We start with the three-shell model network in Figure 1 that has proven to be suitable for large-size gate-periphery (multifinger) power HEMTs, but also for single- or two-finger devices at high operating frequencies (e.g., 80 GHz, and above). Figure 2 indicates the simplified model under cold pinch-off.
At sufficiently low frequencies, inductances and resistances can be neglected. Moreover, the distributed capacitive networks at the gate and drain port, as well in the gate-drain feedback branch, condense to a single overall gate, drain, and gate-to-drain capacitance. The book describes alternatives, how to find well-estimated physics-related values for these capacitances. This can be based on the analysis of the device physical data or can be determined from low-frequency (quasi-DC) measurements.
The splitting of the total branch capacitances among the distributed structures is described in detail in the book and will not be further elaborated here. The description of the proposed optimization algorithm is also omitted here and only the extraction results for the distributed gate and drain capacitances are presented in Figure 3.
Figure 3 Extracted model parameter values as function of measurement frequency bandwidth (f_{min} = minimum upper frequency). |
Regarding the distributed capacitive network at the gate side, the extracted values of the individual capacitances remain almost constant at low frequencies up to about 14 GHz, denoted as f_{min} (minimum upper frequency). Beyond that, however, some constant value exchange between the gate-source capacitance C_{gs }and the parasitic extrinsic gate-source capacitance C_{pga} can be observed.
The evolution of the distributed capacitances at the drain side with increasing frequency bandwidth is more meaningful. As can be seen there is a vital value interchange between the drain-source capacitance C_{ds} and parasitic intrinsic capacitance C_{pdi} at lower frequency bandwidths. Above the upper frequency limit f_{min}, however, the extracted model parameters remain constant, independent of the increase of measurement frequency bandwidth.
GaN HEMTs with gate-peripheries between 100 µm and 4 mm were investigated showing similar results, that is, at a lower frequency range the parameter values exhibit strong variation indicating no consistant model parameter extraction. Only above an observed frequency limit f_{min} the extraction results become stable.
Figure 4 summarizes the results, showing the minimum frequency range of S-parameter measurement as function of the total gate width of the device. It is seen that the minimum frequency to reliably extract the model parameter values of a “distributed” model topology depends on the size of the gate periphery. Thus, a 100-µm gate FET would indeed need a frequency range of at least 60 GHz, whereas a multifinger GaN power device with a total gate-length of 4 mm requires a much lower measurement bandwidth of much less than 10 GHz. Although the studies referred to GaN HEMTs, further investigations have shown that the characteristic curve is also applicable to microwave devices based on GaAs and related technologies.
Figure 4 Minimum measurement frequency range for consistent model parameter extraction. |