A GaN-HEMT Compact Model Including Dynamic RDSon Effect for Power Electronics Converters

In order to model GaN-HEMT switching transients and determine power losses, a compact model including dynamic RDSon effect is proposed herein. The model includes mathematical equations to represent device static and capacitance-voltage characteristics, and a behavioural voltage source, which includes multiple RC units to represent different time constants for trapping and detrapping effect from 100 ns to 100 s range. All the required parameters in the model can be obtained by fitting method using a datasheet or experimental characterisation results. The model is then implemented into our developed virtual prototyping software, where the device compact model is co-simulated with a parasitic inductance physical model to obtain the switching waveform. As model order reduction is applied in our software to resolve physical model, the device switching current and voltage waveform can be obtained in the range of minutes. By comparison with experimental measurements, the model is validated to accurately represent device switching transients as well as their spectrum in frequency domain until 100 MHz. In terms of dynamic RDSon value, the mismatch between the model and experimental results is within 10% under different power converter operation conditions in terms of switching frequencies and duty cycles, so designers can use this model to accurately obtain GaN-HEMT power losses due to trapping and detrapping effects for power electronics converters.


Introduction
During the last few years, research interest to apply gallium nitride (GaN) highelectron-mobility transistors (HEMT) for power electronics converters has been increasing rapidly for different applications in railway [1] and automotive engineering [2]. In terms of power level, they cover the applications from below 48 V DC-DC converter [3] to above 10 kW three-phase DC-AC inverter [4]. GaN-HEMT can operate at higher switching frequency and efficiency than silicon or silicon carbide (SiC) counterparts for below 600 V electrical energy conversion [5]. As a result, they can be widely applied in different power converter topologies, such as multi-phase converter for integrated motor drive, bidirectional converter for e-mobility and power factor correction (PFC) converter for charger. However, voltage and current switching transients of GaN-HEMT can exceed 100 V/ns and 1 A/ns, respectively, which brings the challenge on power converters design to minimise gate voltage, drain voltage overshoot and stability. Another well-noted challenge is thermal management for GaN-HEMT. As device chip area is reduced, and it is usually surface-mounted in packaging, cooling becomes more difficult than conventional 1.
This compact model is integrated into our developed Power Electronics Virtual Prototyping (PEVP) design tool, which takes consideration of electrical circuit EM analysis by PEEC method. In comparison with the above-discussed approaches, one advantage is that it is applied model order reduction (MOR) for PEEC method, which is able to reduce the numbers of equations for EM analysis so as to obtain device switching transients in the range of minutes. It helps designers to save simulation time as discussed above.

2.
The compact model also includes device dynamic R DSon values, which can be applied into different circuit simulators for power electronics converters. It helps designers to accurately obtain device losses in simulation as most GaN-HEMT model do not include dynamic R DSon effect for power electronics converters.
In comparison with our initial results published in [19], more experimental results on device switching transients are added in this paper.
The paper is structured with the following sections. In Section 2, a GaN-HEMT compact model used in the PEVP tool is presented. Simulation results are then compared with experimental measurements. In Section 3, GaN-HEMT dynamic R DSon is represented by an equivalent circuit, in which the process to extract the required parameters is explained. Then, dynamic R DSon is validated by experimental results. The article is concluded in Section 4.

Model Principle
In Figure 1, a simplified structure of packaged GaN-HEMT and its compact model formed by equivalent circuit is presented. Note that GaN-HEMT is a lateral device, where the channel can be modelled by a V ds and V gs controlled bidirectional current source I ch . Device interelectrode capacitance C gd , C ds and C gs can be modelled by nonlinear capacitance, which is dependent on the voltage across each terminal. Finally, parasitic resistance and inductance are added for interconnection between device terminals and packaging pads. Even though this is a compact model, GaN-HEMT semiconductor physics is still included in the equivalent circuit.
A 650 V/7.5 A GaN-HEMT from GaNSystem (GS66502B) is modelled. I ch is represented by Equation (1) for both forward and reverse conduction. Required parameters are given in Table 1. Note that Equation (1) and the parameters are the same as those used in the GaNSystem GS66502B LTspice model. The comparison between model and datasheet values (Rev180420 [20]) for device ON-state characteristics is illustrated in Figure 2, where it is shown that device I D − V DS characteristics are presented precisely by model in both 1st and 3rd quadrant. Note that in order to obtain V DS voltage, the voltage drop across parasitic resistance R d and R s need to be added to V ds voltage of Equation (1). R d = 0.17 Ω and R s = 0.009 Ω are used, which are the same as device manufacturer LTspice model.   Interelectrode capacitance C gs , C gd and C ds are represented by Equation (2) [21], where the used parameters are given in Table 2. Those equations are inspired by Sigmoidtype functions used to model power semiconductor devices static characteristics and capacitance-voltage characteristics [22,23]. We modified tangent hyperbolic functions to represent strong nonlinearity voltage dependence of each interelectrode capacitance, where we can define the variation boundary (parameters b n , e n and h n ), variation trend (parameters c n , f n and i n ) and variation centre (d n , g n and j n ). Three terms are used to represent the observed nonlinear capacitance values in several voltage segments: The comparison between the model and datasheet for device C iss , C oss and C rss variation in respect to V DS voltage (V GS = 0 V) is shown in Figure 3a, and that between model and experiment for C gs -V GS variation is presented in Figure 3b, where the experiment results are obtained by 2-port vector network analyser characterisation using S-parameters [10]. It can be concluded that device interelectrode capacitance values are represented precisely in the compact model. The presented GaN-HEMT model will then be implemented into our developed PEVP software.

Model Implementation into Power Electronics Virtual Prototyping Software
The used simulation techniques in PEVP software are summarised in Figure 4. There are in general two models that can be used for different components in power electronics converters. For example, for the represented half-bridge leg, it is formed by two power semiconductor devices S1 and S2 with their gate drivers, decoupling capacitor C decp , power supply V and PCB tracks for interconnections. For components such as power semiconductor devices, gate driver, C decp and V, power electronics engineers are interested in their characteristics. Therefore, the compact model described by equivalent circuits is suitable in simulation to obtain results quickly. For components like the PCB track or other interconnections, their layout and material properties are important for the design, as they directly influence power electronics converters parasitic inductance (L para ) values. Therefore, the physical model described by component geometry is suitable in simulation to obtain results accurately.
In PEVP, the physical model used for the electromagnetic simulation is determined by PEEC method. Meshing is first applied to generate large numbers of original equations. Afterwards, a Model Order Reduction (MOR) method based on the PRIMA algorithm is used to reduce the numbers of equations; therefore, simulation speed can be accelerated. As compact models are represented by equivalent circuits, they are linked with physical model via electrical boundaries for time domain co-simulation in PEVP. Electrical waveforms can then be obtained. More details on the structure of PEVP and its use for electromagnetic and electro-thermal simulation can be found in our previous work [24,25].

Circuit Equation Generation
Lpara Lpara  As shown in Figure 5a, a GaN-HEMT (GS66502B) half-bridge circuit is built to measure device switching waveforms. The circuit is also modelled in PEVP, in which the presented modelling approach is used. As illustrated in Figure 5b, the PCB track is physically modelled with its real geometry, and then meshed in PEVP. Devices S1 and S2 are modelled by the presented compact model. The device gate driver, C decp , and load inductor, L out , are modelled by their equivalent circuits.
As shown in Figure 5b, gate loop inductance L gate and power loop inductance L power have an important impact on device switching transients. Their values are first extracted in PEVP via frequency domain simulation, and their variation with frequency is then presented in Figure 5c. It can be seen that both L gate and L power slightly decrease with frequency, illustrating the impact of skin effect and proximity effect. The obtained average L gate and L power between 10 and 100 MHz are then compared with another commercial software ADS [10] in Table 3, where it validates the accuracy of PEVP for electromagnetic simulation. Note that the simulation time to obtain these results in PEVP (60 points/decade from 1 kHz to 100 MHz) is within 1 min (25 s for meshing and 17 s for MOR), which is much reduced compared to ADS. We understand that full EM simulation in ADS give comprehensive results including capacitive coupling between each node, and this comparison in Table 3 is only to demonstrate the advantage of using MOR in PEVP, which meets our objective to obtain device switching waveforms quickly. In the next subsection, those reduced order models for L gate and L power (rather than one single value) will be used in time domain simulation to obtain GaN-HEMT switching waveforms.

Model Validation
A comparison between the experimental measurement and simulation for switching current I D and voltage V DS under 200 V/4 A is presented in Figure 6. They have been measured by a 1 GHz current shunt and a 4 kV/400 MHz passive voltage probe to have sufficient large bandwidth and voltage-frequency attenuation. In terms of turn-OFF switching, the model matches well with the measurement for both di/dt and dv/dt. In terms of turn-ON switching, the observed mismatch is mainly on attenuation of a 270 MHz resonance and V DS voltage drop due to L para di dt . This is supposedly due to having an inaccurate model of effective resistance at such high frequency (270 MHz). A finer mesh size can be used in the simulation to improve model accuracy, but at the cost of computation time. Another factor which may lead to this mismatch is the measurement probe. Both current shunt and passive voltage probe bring the ground connection to the measurement circuit, which creates a complex common-mode loop that may interfere with device switching loop, bringing common-mode noise in the measurement results. Despite that, the model still matches well with the measurement for both di/dt and dv/dt. Based on the above waveforms, device turn-ON and turn-OFF switching energy (E sw,ON , E sw,OFF ) are obtained and compared in Table 4. The difference between simulation and measurement is around 1 µJ, which confirms the accuracy of the model to estimate device switching losses. Table 4. Comparison between simulation and measurement for switching energy.
The above I D and V DS transient waveforms are then converted into frequency domain by FFT to compare their magnitude. A Blackman window function was used to extract I D and V DS magnitude for turn-ON and turn-OFF. As turn-ON and turn-OFF transient time (τ) is between 5 and 10 ns, the magnitude above 10 MHz is therefore compared, which corresponds to the switching transients ( 1 τπ ). The results are then illustrated in Figure 7. For turn-ON switching, the magnitude obtained in the simulation matches well with the measurement above 100 MHz, and resonance frequency at 270 MHz was accurately modelled in the simulation. For turn-OFF switching, the magnitude obtained in the simulation matches well with the measurement until 100 MHz. Therefore, it can be concluded that both GaN-HEMT switching transients and switching loop L para are accurately modelled.
In the next section, a compact model will be presented to model device dynamic R DSon effect due to current collapse.

GaN-HEMT Dynamic R DSon Compact Model
It can be summarised from the literature that there are two origins of GaN-HEMT dynamic R DSon [26]: the V DS bias effect, which is associated with device operation when it is in OFF-state, and the hot electron effect, which is associated with device switching transients (overlapping of I D and V DS ). As commercial GaN-HEMT is driven by V GS = 6 V in power electronics converters, the impact of threshold voltage V th shift for dynamic R DSon can be almost neglected [27]. In this section, GaN-HEMT dynamic R DSon characterisation and modelling when the device is operated under zero voltage soft switching (ZVS) will be presented. Soft turn-ON is used in preference to hard switched turn-ON for the following reasons: (1) as presented in Table 4, GaN-HEMT E sw,ON is higher than E sw,OFF . Therefore, device total switching losses can be greatly reduced under soft switching, so as to operate device in higher switching frequency, which is close to device application for high power density power converters. (2) Due to hot electron effect, dynamic R DSon can be greatly reduced and device junction temperature can be easily controlled in soft switching. Therefore, the V DS bias effect can be studied with little interference.

Model Principle and Parameters Extraction
The V DS bias effect on the GaN-HEMT dynamic R DSon value can be illustrated in Figure 8. At each switching cycle T, during the OFF-state ((1 − D)T), the charge will be trapped by V DS bias voltage in device structure, which will increase its R DSon value when it is in ON-state (DT). This is known as the trapping effect. Then, during the ON-state, those trapped charges will be gradually released, therefore the device R DSon value will decrease towards its static value. This is known as the detrapping effect. In order to model the device dynamic R DSon value for power electronics converters, it is necessary to obtain dynamic R DSon trapping and detrapping time constants. Dynamic R DSon of the presented GaN-HEMT is then measured by a proposed electrical circuit shown in Figure 9a. There are mainly two parts in this circuit: device switching circuit (DSC) and voltage clamping circuit (VCC). DSC and DUT form a standard Hbridge circuit. Under the control signal given in Figure 9b (control signals of T2 and T3 are complementary to those of T1 and DUT), DUT hard or soft switching conditions as well as its OFF-state (t off ) and ON-state time (t on ) can be precisely controlled. t off can be precisely controlled at interval t 1 -t 2 . As DUT is under ZVS soft switching at t 2 by negative drain current, t on includes its reverse (t2-t3) and forward conduction (t3-t4). For VCC, its function is to reduce measurement voltage V DS(m.) when DUT is in OFF-state, so as to increase measurement resolution for its ON-state voltage V DSon by using a 8-12 bit oscilloscope. When DUT is in the OFF-state, depletion MOSFET M 1 voltage ∆V ≈ V DC ; therefore, a small voltage of a few volts which is equal to Zener diode Z 1 Zener voltage is measured. When DUT is in the ON-state under both reverse and forward conduction, ∆V = 0, which guarantees accurate measurement of V DSon (V DSon = V DS(m.) ). Device dynamic R DSon is then obtained by R DSon = V DS(m.) I D . More details on the measurement accuracy of the proposed measurement circuit can be found in our previous publication [28].
In this paper, we focus on the use of this measurement circuit to extract parameters for GaN-HEMT dynamic R DSon model.  Figure 10a, where dynamic R DSon values are quickly obtained within 50 ns after device V GS reaches ON-state gate voltage (6 V). This result illustrates the trapping effect on device dynamic R DSon value. It is observed that dynamic R DSon quickly increased more than 25% after 100 µs bias time. Then, it increases slowly with t off until 1 s. From 1 s to 10 s, dynamic R DSon increase rapidly until 70% more than static R DSon value. After 30 s, dynamic R DSon is stabilized to reach its maximal value R DSon,max .
Device dynamic R DSon values under different t off and t on is presented in Figure 10b. This result illustrates the detrapping effect on dynamic R DSon value. It is observed that dynamic R DSon decreases about 20% until 10 µs, then it is almost stable until 10 ms. After that, it starts to decrease again to static R DSon value until t on = 50 s.
It can be concluded from the above measurements that multiple time constants of trapping and detrapping effect are observed, which can be further expressed by an analytical Equation (3). R 0 represents device static R DSon value, and R i − R 0 represents the increase of resistance value. τ off i and τ on i represent the time constant of the trapping and detrapping effect, respectively. The numbers of the unit n correspond to the observed multiple time constants. Therefore, dynamic R DSon value increases with t off towards R DSon,max and then decreases with t on towards static R DSon . Based on measurement results, R i , τ off i and τ on i , and n can be determined by fitting method and they are given in Table 5.    The comparison between the model and measurement for GaN device dynamic R DSon is illustrated in Figure 10. Five terms are finally used in the model, in which the obtained τ off i and τ on i correspond to the above analysis. It is also shown that the model represents well the evolution of device dynamic R DSon under different bias time t off and ON-state time t on . Note that even though there is no measurement data for t off below 100 µs, the data obtained by the model represent a reasonable dynamic R DSon evolution. The analytical model will be implemented into the GaN-HEMT compact model, which will be presented in the next subsection.

Model Implementation
GaN-HEMT dynamic R DSon compact model (DCM) is shown in Figure 11. It is constituted by a standard GaN-HEMT compact model (SCM) (e.g., Figure 1) and a behavioural voltage source V b . When GaN-HEMT is in conduction, its R DSon (t) is obtained by Equation (4), where the term V b I D corresponds to the exponential function and the term V b is then expressed by Equation (5), where V C i corresponds the voltage increase of each RC unit. Time constant τ off i and τ on i of each unit is represented by a RC circuit. In order to transform the resistance increase (Ω in Equation (3)) to that of V C i , it is introduced one parameter k (unit: A and k = 1). Figure 11. GaN-HEMT dynamic R DSon compact model.
As proposed, GaN-HEMT DCM is a compact model that can be used in different simulation platforms (e.g., PEVP, LTspice, Pspice, SIMetrix, ADS, etc.), and proposed V b can be also used alone with device manufacturer model. The comparison between DCM and SCM for dynamic R DSon simulation after 10 switching periods (100 kHz, D = 50%) is presented in Figure 12. Note that the device dynamic R DSon value increases to approximately 20% more than its static R DSon value in the proposed DCM, and it decreases slightly with ON-state time, which illustrates the detrapping effect. By contrast, only static R DSon value is obtained in SCM. In the next subsection, proposed DCM will be validated by experimental measurements for dynamic R DSon value at transient and steady state when device switches at different operation conditions.

Transient and Steady State
By using the circuit presented in Figure 9a, the GaN-HEMT dynamic R DSon value at transient and steady state is at first measured under the following condition: V DC = 200 V, I L = 1.3 A, T1 remains OFF-state (T2 remains ON-state) and DUT switching frequency f sw = 100 kHz with D = 50%. Therefore, DUT is operated under soft switching and reverse conduction. Its operation time can be precisely controlled by the numbers of switching cycles. As shown in Figure 12, dynamic R DSon value decreases during ON-state. Therefore, we use R B DSon to represent its average dynamic R DSon value at the beginning of each ONstate cycle (within 100 ns) and R E DSon to represent its average dynamic R DSon value at the end of each ON-state cycle (also within 100 ns). The comparison between the model and the measurement for dynamic R DSon value at transient and steady-state is presented in Figure 13.   In terms of measured dynamic R DSon value, it is shown that both R B DSon and R E DSon gradually increase with power converter operation time until 3 s. After 3 s, they increase rapidly until 30 s and reach steady state after 100 s. This transition conforms to the τ off i observed in Figure 10a, where there is a rapid increase of dynamic R DSon value when t off is between 1 s to 30 s. The model represents good agreement to measured dynamic R DSon values at both transient and steady state. This result also suggests that reported device dynamic R DSon characterisation methods by double-pulse test or multiple pulse test in literature may not reveal device dynamic R DSon value at steady state in power converter.
In terms of dynamic R DSon value at steady state (see Figure 13b), the model shows similar trajectory as the measurement for the decrease of R B DSon towards R E DSon , where the term ∆R = R B DSon − R E DSon obtained in the model is almost the same as measurement. This transition conforms to the τ on i observed in Figure 10b, where the dynamic R DSon value decreases with t on until 10 µs.
The difference between the model and the measurement is within 10%, which validates the accuracy of the model at one operation condition. Differences between the actual, changing junction temperature and the constant value assumed in the model is one reason for this. Future work to include temperature dependency will be discussed in Section 3.3.3.

Different Operation Conditions
Power converter switching frequency and duty cycles are then varied to validate the model at different operation conditions. In order to increase power converter switching frequency towards megahertz, the switching losses of all four GaN transistors in the measurement circuit need to be reduced. For this reason, an LC load is used and a phase shift between two legs is added (D DUT + D T 1 = 1) to control the load current I L in trapezoidal current mode (TZCM). The advantages of TZCM are that (1) all four GaN-HEMTs can be operated under ZVS soft switching to reduce switching losses and (2) the dynamic R DSon value can be measured under constant current amplitude. Measurement error due to parasitic inductance and probe deskew on measurement results can be eliminated [28].
At steady state, measurement waveforms by proposed TZCM when device switches at 1 MHz is presented in Figure 14. By setting an appropriate phase shift value, I L is in the trapezoidal form. Figure 14a represents the condition that DUT control signal delays that of T2; therefore, device R B DSon = V DSon I D is obtained when it is under constant reverse current conduction (using red curves in Figure 14a). In comparison, Figure 14b represents the condition that T2 control signal delays that of DUT; therefore, device R E DSon = V DSon I D is obtained when it is under constant forward current conduction (using red curves in Figure 14b). The comparison between the measurement and the model for steady-state dynamic R DSon under different operation conditions is presented in Figure 15. When the device switches at 100 kHz, the model is compared with the measurement for different duty cycles. As shown in the results (see Figure 15a), the model matches well with the measurement results. Both R B DSon and R E DSon decrease with duty cycle, because effective device bias time decreases when duty cycle increases. When the duty cycle is 50%, the model is compared with the measurement for different switching frequencies (see Figure 15b), where again the model shows good agreement. The difference between R B DSon and R E DSon decreases with the increase of f sw , which suggests that device dynamic R DSon finally converges in megahertz switching power converter. As the difference between the model and the measurement is still within 10% in the above results, it can be concluded that the accuracy of the model can be validated under different operation conditions.

Discussion
When comparing the model with the measurement for dynamic R DSon values at steady state, it can be noted that model predicts a value smaller than the measurement. Temperature is one factor that may cause this mismatch.
In the model, R i is obtained at room temperature. When the device is operated continuously in power converter by soft switching, the junction temperature increase ∆T j is mainly caused by device conduction losses, and it is estimated to be ∆T j = I 2 L · R DSon · D · R th,j−c . A heat sink is placed at the top side of the device in experiment, so junction-top thermal resistance (R th,j−t = 28 • C/W) is used to estimate T j , which is around 13 • C above ambient when D = 0.8. For future work, R i of different temperature can be obtained by the proposed modelling method to include temperature dependency.
In the paper, R i , τ off i and τ on i are all obtained when bias voltage V DS is 200 V. For future work, the dependency of those parameters with different bias voltage can be also included in the model by the same modelling method.

Conclusions
A GaN-HEMT compact model including dynamic R DSon effect is proposed in the paper, with the objective to finely model GaN-HEMT switching transients and power losses. Device static characteristics and capacitance-voltage characteristics are accurately represented by mathematical equations. The model is then implemented into our developed Power Electronics Virtual Prototyping (PEVP) software, where the parasitic inductance of the power loop and gate driver loop is accurately extracted by PEEC method. A model order reduction technique is also used in PEVP software to reduce the numbers of equations, so as to obtain the above parasitic inductance within 1 min. By comparing with experimental measurement, the model represents well turn-ON and turn-OFF switching waveforms in both time and frequency domain.
For dynamic R DSon modelling, a measurement circuit is at first proposed to extract device dynamic R DSon values as well as multiple time constants for trapping and detrapping effect. Those parameters are then represented by multiple units of RC circuit and form a behavioural voltage source in the compact model. By comparing with the experimental measurement, the model is validated to accurately represent device dynamic R DSon values at transient and steady state under different switching frequencies and duty cycles.
For future work, in order to further expand operation conditions of the model, the dependency on junction temperature and different bias voltage can be included by the same modelling method.