Sufﬁcient Condition-Based Stability Analysis of a Power Converter Applied Switching Transient Waveform Modiﬁcation Using Kharitonov’s Theorem

: The rapid switching action of power metal-oxide-semiconductor ﬁeld-effect transistor (MOSFET) causes high-level electromagnetic interference (EMI) in power converters. The switching transient waveform modiﬁcation method realized by closed-loop gate drive has been recognized as an effective high-frequency EMI reduction approach. However, feedback control of power MOSFET in the saturation region would introduce stability problems. This paper presents a sufﬁcient condition-based stability analysis of all the operating points during turn-off using Kharitonov’s theorem. Firstly, a small-signal MOSFET model during turn-off was used to derive the closed-loop system transfer function. The nonlinear capacitances and the rest constant parameters of the small-signal model were determined based on the device characteristics and the expected outcome of the drain-source voltage. Then we split the turn-off switching transient into several subintervals, during which the system characteristic equation became an interval polynomial due to the nonlinear capacitances. Finally, Kharitonov’s theorem was applied in each subinterval to evaluate the stability, thereby achieving the overall system stability analysis during turn-off. Experiments were conducted to investigate the system’s stability and the results conﬁrmed the validity of the proposed analysis. This work presents an implementable design guideline for the applied switching transient waveform modiﬁcation of power converters via closed-loop gate drive.


Introduction
The high switching speeds of power metal-oxide-semiconductor field-effect transistor (MOSFET) result in serious interference in power converters. Generally, interference suppression can be achieved by applying electromagnetic interference (EMI) filters at a cost of additional space for installation [1][2][3]. The closed-loop gate drive method has been successfully applied to regulate the switching trajectory of insulated-gate bipolar transistor (IGBT)/MOSFETs in power converters [4][5][6][7]. Feedback control of switching devices in the saturation region is employed to modify the switching transient waveform according to a Gaussian S-shaped reference for electromagnetic interference suppression [7,8]. The effect of reduced EMI generation is remarkable. However, the stability issues introduced by the feedback control should be seriously considered in circuit design.
In [9], the stability of the drive system was investigated using standard control theory methods. The enlightening contribution of the work was the application of the IGBT small-signal model for the system open-and closed-loop transfer functions. The parameters of the feedback circuit and IGBT variation on the stability of the drive system were researched by root locus plots. However, Most power converters can be reduced to the standard chopper cell to investigate the transient response of power MOSFETs/IGBTs. Therefore, a standard chopper cell was chosen for the illustration of the drive circuit modeling. Figure 1 shows the chopper in detail, including a freewheeling diode, an inductive load, a power MOSFET, and the closed-loop gate drive. The drive circuit followed the same structural outline with that proposed in [5]. A feedback control was used to modify the drain-source voltage according to the reference waveform. The drain voltage feedback formed with resistors and capacitors was applied to provide a scaled drain voltage. An op-amp with a high bandwidth and a wide voltage swing was applied to obtain the error signal by comparing the scaled drain voltage with the reference waveform. The drive stage was an emitter follower formed with two transistors and provided the necessary current gain for MOSFET driving. The gate resistor Rg played an important role in stability control.

Modeling of Closed-Loop Gate Drive
The drive circuit was partitioned into the blocks displayed in Figure 2. The drive stage had a negligible effect on the input voltage and was ignored when modeling. The drain voltage feedback term composed of resistors and capacitors was modelled as a low-pass filter with a cut-off frequency of fFB and a gain of α. The transfer function HD is given by: Likewise, the summing op-amp is modelled as a low-pass filter, taking into account the bandwidth fOP and the gain AOP. The transfer function TOP is given by:  The drive circuit was partitioned into the blocks displayed in Figure 2. The drive stage had a negligible effect on the input voltage and was ignored when modeling. The drain voltage feedback term composed of resistors and capacitors was modelled as a low-pass filter with a cut-off frequency of f FB and a gain of α. The transfer function H D is given by: Most power converters can be reduced to the standard chopper cell to investigate the transient response of power MOSFETs/IGBTs. Therefore, a standard chopper cell was chosen for the illustration of the drive circuit modeling. Figure 1 shows the chopper in detail, including a freewheeling diode, an inductive load, a power MOSFET, and the closed-loop gate drive. The drive circuit followed the same structural outline with that proposed in [5]. A feedback control was used to modify the drain-source voltage according to the reference waveform. The drain voltage feedback formed with resistors and capacitors was applied to provide a scaled drain voltage. An op-amp with a high bandwidth and a wide voltage swing was applied to obtain the error signal by comparing the scaled drain voltage with the reference waveform. The drive stage was an emitter follower formed with two transistors and provided the necessary current gain for MOSFET driving. The gate resistor Rg played an important role in stability control.
The drive circuit was partitioned into the blocks displayed in Figure 2. The drive stage had a negligible effect on the input voltage and was ignored when modeling. The drain voltage feedback term composed of resistors and capacitors was modelled as a low-pass filter with a cut-off frequency of fFB and a gain of α. The transfer function HD is given by: Likewise, the summing op-amp is modelled as a low-pass filter, taking into account the bandwidth fOP and the gain AOP. The transfer function TOP is given by:  Likewise, the summing op-amp is modelled as a low-pass filter, taking into account the bandwidth f OP and the gain A OP . The transfer function T OP is given by: A small-signal MOSFET model, as shown in Figure 3, takes into account the internal gate resistance R G , source resistance R S , drain resistance R D , output resistance R ds , the terminal stray inductances L G , L S , L D , the drain-source C ds , drain-gate C dg , and gate-source C gs capacitances, as well as the transconductance g m . In the following analysis, R G should consist of both the external and internal gate resistances. Additionally, the terminal stray inductances should include the drive circuit stray inductances. The small-signal conditions are different for the turn-on and turn-off process [10]. In this study, we illustrated the stability investigation based on the turn-off process. The same analysis procedure can be performed for the turn-on process.  A small-signal MOSFET model, as shown in Figure 3, takes into account the internal gate resistance RG, source resistance RS, drain resistance RD, output resistance Rds, the terminal stray inductances LG, LS, LD, the drain-source Cds, drain-gate Cdg, and gate-source Cgs capacitances, as well as the transconductance gm. In the following analysis, RG should consist of both the external and internal gate resistances. Additionally, the terminal stray inductances should include the drive circuit stray inductances. The small-signal conditions are different for the turn-on and turn-off process [10]. In this study, we illustrated the stability investigation based on the turn-off process. The same analysis procedure can be performed for the turn-on process.
The chopper under closed-loop gate drive (Figure 2), had an inductive load that can be regarded as a current source at turn-off. Therefore, the connection between the drain terminal and the inductive load was an open-circuit in small signal analysis. At turn-off, the node voltage equations are given by: where m1 represents 1/(RG + s × LG), m2 represents (1/Rds + s × Cds), and m3 represents 1/(RS + s × LS).
Then the transfer function from VGS to VDS can be derived from (3), (4), and (5), as given by: with the coefficients: The chopper under closed-loop gate drive (Figure 2), had an inductive load that can be regarded as a current source at turn-off. Therefore, the connection between the drain terminal and the inductive load was an open-circuit in small signal analysis. At turn-off, the node voltage equations are given by: where m 1 represents 1/(R G + s × L G ), m 2 represents (1/R ds + s × C ds ), and m 3 represents 1/(R S + s × L S ). Then the transfer function from V GS to V DS can be derived from (3), (4), and (5), as given by: with the coefficients: a 0 = −R ds g m a 1 = R S C dg + C gs + R ds C dg (1 + R S g m ) With all the blocks are modelled in the small-signal domain, we get the system closed-loop transfer function, as given by: with the coefficients: where V REF represents the reference voltage.

Determination of the Parameters
Most of the small-signal model parameters can be directly extracted from the device datasheet and the device SPICE model. However, the nonlinear drain-source C ds and drain-gate C dg capacitances that depend on the drain-source voltage need to be seriously treated. It is noteworthy that the drain-source voltage has an expected trajectory under closed-loop gate drive, which provides an advantage when evaluating the voltage-dependent capacitances. The determination procedure of small signal parameters is illustrated with the following conditions. The power MOSFET used in this paper was an IRL2703 module. The DC voltage source was set to 30 V. The reference waveform was a PWM signal with Gaussian S-shaped switching transients that can be expressed by convolution between a Gaussian function and a rectangular signal g(t), as given by: where τ represents the switching time and σ t = τ/4. The Gaussian function is infinitely differentiable and hence the reference waveform could provide a remarkable EMI suppression [7,13]. The switching frequency, switching time, and duty cycle of the reference waveform are 400 kHz, 450 ns, and 0.5, respectively. According to the device SPICE model and datasheet, R G , R D , R S , L G , L D , and L S can be directly estimated to be 4.72 Ω, 0.0001 Ω, 0.03 Ω, 15 nH, 4.5 nH, and 7.5 nH, respectively. A parametric curve tracer was used to obtain the output characteristics and the transfer characteristics. The curves of the input capacitance C iss , output capacitance C oss , and reverse transfer capacitance C rss versus V DS were also measured using the curve tracer. Note that R ds and g m can be approximated by dV DS /dI D and dI D /dV GS , respectively. Therefore, R ds can be estimated from the output characteristics and can be regarded as a constant with a value of 10 kΩ. The transfer characteristics of the MOSFET can be used to estimate g m to be 13 S in the saturation region. C gs , C ds , and C dg can be estimated by the capacitance curves (C gs = C iss -C rss , C dg = C rss , and C ds = C oss -C rss ). Therefore, C gs can be regarded as a constant during turn-off with a value of 350 pF. The nonlinear capacitances C ds and C dg mainly depend on the drain-source voltage and can be approximated via curve-fit, as given by: C ds = 144.79e −V DS /1.53 + 149.69e −V DS /9.52 + 87.48 (11) C dg = 331.65e −V DS /1.61 + 1171.25e −V DS /12.11 + 87.69 (12) where the drain-source voltage V DS increases from 0 to 30 V according to (10). The measured nonlinear capacitances and the fitted functions are shown in Figure 4.   (12) where the drain-source voltage VDS increases from 0 to 30 V according to (10). The measured nonlinear capacitances and the fitted functions are shown in Figure 4.

Kharitonov's Theorem
Consider an interval polynomial, where the value range of aj is [εj, βj]. There exists the following necessary and sufficient condition for stability.
Kharitonov's Theorem [14]: The polynomial family defined by (13) is Hurwitz stable if and only if four polynomials, shown as,

Sufficient Condition-Based Stability Analysis
Four designs of closed-loop gate drive for power MOSFET are provided in Table 1. The DC voltage and reference waveform remain the same as in Section 3. The turn-off switching transient can be divided into five subintervals. The minimum and maximum values of V DS , C dg , and C ds in each subinterval can be calculated by (10), (11), and (12), as shown in Table 2. For different circuit designs, the ranges of the parameters c 0 , c 1 , c 2 , c 3 , c 4 , and c 5 in (8) can be obtained by substituting the parameters in Tables 1 and 2 into (7) and (9). Therefore, the denominator of the system closed-loop transfer function can be regarded as an interval polynomial in each subinterval. Kharitonov's theorem can be performed to assess the stability of the system during every subinterval, thereby achieving the overall system stability analysis during turn-off switching transient. From Kharitonov's theorem, the stability of the system during each subinterval can be checked by testing the stability of the following four polynomials: c 0,max + c 1,min s + c 2,min s 2 + c 3,max s 3 + c 4,max s 4 + c 5,min s 5 (18) c 0,max + c 1,max s + c 2,min s 2 + c 3,min s 3 + c 4,max s 4 + c 5,max s 5 (19) c 0,min + c 1,max s + c 2,max s 2 + c 3,min s 3 + c 4,min s 4 + c 5,max s 5 (20) c 0,min + c 1,min s + c 2,max s 2 + c 3,max s 3 + c 4,min s 4 + c 5,min s 5 where c j,max and c j,min represent the maximum and minimum of c j , respectively. In order to highlight the analysis process, the ranges of the parameters c 0 , c 1 , c 2 , c 3 , c 4 , and c 5 during (0, 100 ns) under Design I are provided in Table 3. It was found that Design I was stable during the first subinterval according to (18), (19), (20), and (21). The theoretical analysis showed that Design I, Design II, Design III, and Design IV were stable during the turn-off switching transients. It should be noted that the proposed stability analysis approach is a sufficient condition. The fewer the number of subintervals, the stronger the sufficient condition.

Experimental Results
A chopper with an inductive load was tested. The MOSFET was an IRL2703. The drain voltage feedback term was realized by resistors and capacitances. A THS3091 operational amplifier was chosen to provide the necessary gains and bandwidths. The DC voltage was set to be 30 V. The reference waveform was formed with an arbitrary waveform generator. It has been proposed in [11] that the stability of the closed-loop gate drive for switching transient modification can be accessed by the appearance of the gate-source voltage with the reference voltage as input. The switching waveforms during turn-off under Design I, Design II, Design III, and Design IV are shown in Figure 5. For Design I, it can be seen that the drain-source voltage had a Gaussian S-shaped switching transient with smooth corners. There was no oscillation in the gate-source voltage waveform. For Design II, though some tracking errors exist at the bottom corner of the drain-source voltage, the gate-source voltage was stably damped. Compared with Design I, Design III was stable with a smaller gate resistance, resulting in smaller tracing errors, as shown in Figure 5c. Compared with Design II, Design IV had a smaller gain in drain voltage feedback. No oscillation was identified in Figure 5d and hence Design IV was stable. All the experimental results confirmed the theoretical analysis.

Conclusions
The closed-loop gate drive was applied for the switching transient waveform modification of power converters. This paper presents an implementable approach to assess the system's stability during the switching transient. Although the approach was a sufficient condition, it still holds great significance for drive circuit design. A stable system guarantees well-damped switching transient waveforms, whereas possible tracking errors may exist and result in a negative impact on the performance of EMI mitigation. Future works will explore the behavior of the drive circuit during switching transients.
Funding: This research received no external funding.

Conclusions
The closed-loop gate drive was applied for the switching transient waveform modification of power converters. This paper presents an implementable approach to assess the system's stability during the switching transient. Although the approach was a sufficient condition, it still holds great significance for drive circuit design. A stable system guarantees well-damped switching transient waveforms, whereas possible tracking errors may exist and result in a negative impact on the performance of EMI mitigation. Future works will explore the behavior of the drive circuit during switching transients.
Funding: This research received no external funding.