With the continuous development of RF wireless communication system，the demand for high-performance and low-cost RF solutions is increasing ^［1］. Compare to traditional Si and Ge-based CMOS devices，InP HEMT transistors have better frequency response characteristics，power density and breakdown voltage，which makes InP HEMT transistors become an excellent candidate for many monolithic microwave integrated circuits（MMICs）working at GHz frequency range ^［2］. As an important link between transistors and circuits，a large signal model which can accurately simulate dc，S-parameters and large signal characteristics determines the accuracy of circuits. Meanwhile，small signal model is the approximation of the large signal model at a single bias point ^［3］. Therefore，the accuracy of the large signal model depends mostly on the small signal equivalent circuit model which can reflect the physical and electrical properties of the device ^［4］.

In the past decades，various extraction techniques，including numerical optimization method ^［5］ and direct extraction method ^［6-7］have been developed. The numerical optimization method uses the numerical method to find the optimal parameter value. Although it is very accurate，it depends largely on the initial value of the parameter and may not converge. The direct extraction method uses the analytical equations to obtain the expression of each parameter without optimization，but it is difficult to be applied to small signal model with complex topology.

In order to solve these problems，the method combining numerical optimization and direct extraction method is used to extract the equivalent circuit parameters. This method takes the parameter value obtained by the direct extraction method as the initial value of the optimization method，which not only maintains the accuracy of the optimization method，but also avoids the convergence problem.

A distributed small signal equivalent circuit model shown in Fig. 1 is used for 4×75 µm gate width，0.15 µm gate length of InP HEMT device. The 19-element model includes 12 bias-independent extrinsic parameters and 7 bias-dependent intrinsic parameters. Bias independent extrinsic elements consist of C_pg，C_pd，C_pgd（pad parasitic capacitances），L_g，L_d，L_s（pad parasitic inductances），R_g，R_d，and R_s（extrinsic resistances of gate，drain，and source，respectively）. In addition，the distributed capacitance effect of the InP HEMT becomes more and more significant as the frequency gets higher ^［8］. In this model，C_dg，C_dd and C_dgd are used to describe the distributed capacitance effect between the gate fingers，while C_pg，C_pd and C_pgd are used to characterize the parasitic capacitance between PAD and ground. Bias dependent intrinsic elements mainly include R_is（channel resistance），G_ds（drain conductance），g_m （transconductance），τ（time delay），C_gs，C_gd，and C_ds（gate to source，gate to drain and drain to source capacitances，respectively）.

The key of the small signal extrinsic parameter’s extraction method is to simplify the equivalent circuit at a specific bias point in Fig. 1. Under cold pinch-off condition（V_ds=0，V_gs<-V_th），the drain source current source and output conductance are negligible，so the depletion region can be characterized by three capacitors C_ig，C_id and C_igd，as shown in Fig. 2. Usually，the parasitic capacitances are extracted first，which cannot eliminate the effect of parasitic inductances. Consequently，the parasitic inductances L_g，L_d and L_s must be de-embedded before extracting the parasitic capacitances，which is also the difference between the method in this paper and Gao's method ^［9］.

Since parasitic resistances and inductances are sensitive at low frequencies，it is necessary to extract parasitic inductances at high frequencies（>25 GHz）. In addition，the parasitic resistances don’t affect the imaginary value of the Y parameters，so they can be excluded when extracting the parasitic inductances.

The Z-parameters of the simplified circuit in Fig. 2 can be expressed as：

where

Multiplying the Z-parameters by ω and the taking the imaginary parts gives：

Consequently，the values of L_g，L_d and L_s can be extracted from the slope of Im（Z_ij）verse ω² as shown in Fig. 3.

After de-embedding the parasitic inductances，parasitic capacitances C_pg，C_pd，C_pgd and distributed capacitances C_dg，C_dd and C_dgd can be determined using device gate width scaling method ^［2］.

Y-parameters of the simplified circuit shown in Fig. 2 can be written as：

The intrinsic capacitances C_ig，C_id and C_igd are directly proportional to the gate-finger width，and the relationship between them can be described as：

By substituting Eqs. 14-16 into Eqs. 11-13，the total capacitances of gate source，gate drain and drain source branch can be obtained.

The capacitance C_xg，C_xd and C_xgd can be calculated form Eqs.17-19 with the Y-parameters measurements of 4×25 μm，4×50 μm，4×75 μm，4×100 μm and 4×150 μm InP HEMT. As shown in Fig. 4，the intercepts of Im（Y_ij）/ω（i，j=1，2）are the extracted values of C_xg，C_xd and C_xgd.

The next step is how to search for the optimal parasitic and distributed capacitances，which can ultimately minimize the error between the measured and simulated results. Before optimizing the extrinsic capacitances，the intrinsic capacitances C_ig，C_id and C_igd can be determined using Eqs.11~13：C_ig=70 fF，C_id=52.9 fF，C_igd=72 fF. Unlike the small signal model of GaN HEMT in Ref.［10］，the assumption of C_ig = C_igd cannot be used in the optimization process.

During optimization，C_dg is scanned from 0 to C_xg，C_dd is scanned from 0 to C_xd，and C_dgd is scanned from 0 to C_xgd. In order to reduce the optimization difficulty，some specific extrinsic capacitance relations must be assumed ^［11］：

When the error between the measured and simulated results reaches a minimum，the values of parasitic and distributed capacitances can be determined.

Figure 5 is the small signal circuit under V_gs>V_th and V_ds=0，the depletion region can be represented by channel distribution resistance R_c and gate differential resistance R_j.

The Z-parameters of the circuit as shown in Fig. 5 can be written as：

where R_j=nKT/qI_g， I_g is the gate leakage current，n is the ideal factor of the schottky diode，k is the boltzmann constant，T is the kelvin temperature ^［11］. To extract parasitic resistances，the impact of R_j and R_c must be eliminated. Channel distribution resistance R_c is proportional to 1/（V_gs-V_th）^［12］，the Eqs.22-23 can be expressed as：

Therefore，it is necessary to measure the Z-parameters at the bias point of V_ds=0，V_gs=-0.25 V，0 V，0.25 V and 0.5 V，then plot the curve R_e（Z_ij）verse 1/（V_gs-V_th）. The intercept of R_e（Z₂₂）verse 1/（V_gs-V_th）is the extracted value of the sum of R_s and R_d，the intercept of Re（Z₁₂）verse 1/（V_gs-V_th）is the extracted value of R_s as shown in Fig. 6（a）.

Similar to extracting resistances R_s and R_d，the resistance R_j is directly proportional to 1/I_g ^［13］. Consequently，Eq.21 can be expressed as：

When the InP HEMT is biased at V_ds=0 and different I_g，the resistance R_g can be calculated from Eq.26. As shown in Fig. 6（b），the intercept of Im（Z₁₁）/（1/I_g）is the extracted value of the sum of R_s and R_g，then R_g can be obtained.

Before extracting intrinsic parameters，the accuracy of the extrinsic parameters needs to be verified. Complete extrinsic parameters of the pinch-off InP HEMT device（V_ds=0V，V_gs=-2V）small signal model are tabulated in Table 1.

After de-embedding the extrinsic parameters，the Y parameter can be used to determine the intrinsic parameters. Calculate the intrinsic parameters’ values using Eqs. 27-33. Figures7-10 show the extracted intrinsic parameters at V_gs=-0.75V，V_ds=4V.

It can be seen from Figs. 7-10 that the direct extraction method can obtain the average value of intrinsic parameters biased at V_gs=-0.75V，V_ds=4V，which can be optimized as the initial value of the numerical optimization method. Finally，the extrinsic and intrinsic parameters extracting with the proposed extraction method is shown in Table 2.

In order to verify the distributed small signal model of InP HEMT，the simulated results of the model need to be compared with the measured results. Figure 11 shows the simulated S-parameters of the small-signal equivalent circuit of the InP HEMT including the distributed capacitances with the measured data. The comparison demonstrates good agreement from 2 ~ 50 GHz，which also verifies the validity of the model and extraction method.

To further evaluate the accuracy of the small signal model，the modeling error is defined as：

where S_S，ij is the simulated data and S_M，ij is the measured data. As shown in Fig. 12，it is obvious that the modeling error is less than 4% in the frequency range of 2~50 GHz，which mathematically proves the accuracy of the small signal model.

A distributed small signal extraction method for 4×75 µm gate width，0.15 µm gate length InP HEMT is proposed in this paper.Before extracting the parasitic capacitance，the parasitic inductances are first de-embedded to eliminate errors. The extrinsic capacitances including distributed and parasitic capacitances are determined at four different gate widths using algorithmic optimization and a gate-width scalable method. The values of the model parameters obtained by the direct extraction method are used as the initial values of the optimization method for model optimization. Finally，there is good agreement between the measured and simulated S-parameters up to 50 GHz.