Harmonic Analysis of Sliding-Mode-Controlled Buck Converters Imposed by Unmodeled Dynamics of Hall Sensor

Abstract

DC–DC buck converters have become prominent components for energy optimization in power systems, and how to improve control performances is a challenging issue to be addressed. In this paper, we aim to investigate the harmonic problem of sliding mode (SM) controlled buck converters imposed by the often-ignored unmodeled dynamics of the Hall sensor. The unified mathematical model of the whole system is established by combining the SM controller, the buck converter, and the Hall sensor, where the signal loss in the transmission process of the whole closed-loop control system is considered. Based on the Lyapunov stability theorem, the SM controller is designed to guarantee system stability, as well as to deduce the stable working areas and the tuned controller parameters. Furthermore, we introduce the descriptive function (DF) approach to investigate the influence of the unmodeled dynamics of the Hall sensor on the system harmonics in the frequency domain, which can deduce the relationship between the amplitude-frequency characteristics of the output signal and the Hall sensor. Simulations and experiments validate this paper.

1. Introduction

With the rapid development of new energy, distributed power systems, and the wide use of uninterrupted power systems (UPS), DC–DC buck converters have played an important role in energy optimization. How to improve its conversion efficiency and performance is an important but challenging issue to be considered [1,2]. It is a well-known fact that DC–DC buck converters are a typical class of nonlinear systems due to the contained nonlinear circuit components such as the capacitors, inductors, switches, and the external disturbances imposed by the possible measurement sensors, the signal loss in the transmission process and some unmodeled dynamics [3,4], which are often ignored in modeling and controlling. However, it is worth noticing that all these factors can reduce the performance of the buck converters, or even lead to instability.

An appropriate control approach has a significant function to realize better performances. Compared with the traditional control approaches such as PI and linear feedback control, sliding mode (SM) control is characterized by nonlinear and robust control due to its contained switching function sgn(.). Much research found in the literature has proved that SM control is more suitable for the switching control of buck converters; a review of recent SM applications in power converters can be found in [5]. On the one side, SM is inherently nonlinear and differs from the approximate nonlinearity of the substitutes as fuzzy control [6,7], neural network [8,9], and other intelligent control approaches [10,11,12]. Therefore, the control performances under SM control are better; furthermore, the amount and time of the calculation can also be saved. On the other side, thanks to its natural switching control property, the SM controller can replace the traditional pulse-width modulation (PWM) and realize the direct ON/OFF switching control of the converters by triggering the gate of the controllable power switches. As a result, it is simple and no extra modulation circuits are needed [13,14]. At present, various types of SM have been applied for the control of buck converters, including linear sliding mode (LSM), terminal sliding mode (TSM), and non-singular terminal sliding mode (NTSMC). For example, in [15], an LSM controller was proposed to realize the asymptotic convergence of the output voltage; in [16], a TSM controller was designed to improve the steady accuracy and response speed due to its finite-time convergence; in [17], an NTSMC controller was proposed to realize the wide-range voltage regulation of the buck converters due to its global finite-time convergence. Attention should be paid to the fact that the design of SM controllers is commonly focused on the stability in the time domain, and it is difficult to investigate the control performances in the frequency domain [18,19,20,21], especially for the harmonics, which is a key performance index in the practical applications of buck converters.

In this paper, the harmonic analysis of SM-controlled buck converters is carried out in frequency. Compared with the time-domain analysis based on the Lyapunov stability theorem, frequency-domain analysis is based on the Fourier transform and can make it easier to obtain the detailed information of the concerned signals [22,23,24,25], including the visual representation and the corresponding values of the amplitude and frequency. At present, the Poincaré map and descriptive function are the two main approaches adopted in frequency-domain analysis. For example, in [22], the Poincaré map and linear matrix were combined to investigate the stability and the periodic solutions for a class of relay systems, and then in [23], the Poincaré map was applied in an SM control system to analyze its inherent chattering phenomenon, proving the fact that the control nonlinearity of SM can induce harmonics in practical systems. In [24], the descriptive function (DF) was adopted to analyze the frequency characteristics of DC–DC converters. Compared with the Poincaré map, DF is easier to implement and more suitable for practical systems. The idea of DF is to utilize the characteristic curves of the amplitude, phase, and frequency to visually observe the performance of the system, following the improvement by adjusting the controller parameters. In [25], a boost converter was taken as an example to solve the problem of switch linearization by utilizing DF; in [26], the two problems of the stability and limit cycle were investigated for a class of LSM-controlled uncertain system, and further deduced the relationship between parameter uncertainties, the position, and distribution characteristics of zeros and poles. All of these show the effectiveness of DF approaches applied in SM-controlled systems.

Unmodeled dynamics commonly exist in practical systems, which usually originate from sensors, actuators, and higher-order state terms. They are generally unmeasurable and characterized by small time constants, so that they are often ignored in modelling and controlling. Some of the literature concerning the overcoming and observation of unmodeled dynamics can be found in [27]. In this paper, the influence of the often-ignored unmodeled dynamics of the Hall sensor on SM-controlled buck converter systems is another challenging issue to be addressed. In practical circuits, various Hall sensors, such as ACS712 (https://www.alldatasheet.com/datasheet-pdf/pdf/168326/ALLEGRO/ACS712.h-tml, accessed on 18 August 2023), ACS706 (https://www.alldatasheet.com/datasheet-pdf/pdf/174113/ALLEGRO/ACS-706ELC-05C.html, accessed on 18 August 2023) and ACS758 (https://www.alldatasheet.com/datasheet-pdf/pdf/53346-8/ALLEGRO/ACS758.html, accessed on 18 August 2023) are generally used to measure currents, which can be further adopted for the design of controllers. The idea of Hall sensors is based on the Hall effect, which can realize contactless current measurement [28]. Furthermore, it can prioritize the safety of both individuals and equipment, and maintains the structural integrity of the original circuit, whereby the measurement accuracy can be improved. At present, if the Hall sensor is used in an open-loop control system, its measurement error can be less than 1%, while if it is used in a closed-loop control system, the measurement error can be less than 0.2% [29]. Due to the high accuracy of Hall sensors, the signal loss in the transmission process is often ignored. However, the model of the Hall sensor can be categorized as the unmodeled dynamics in the sense of control, which are characterized by rapidity and/or small scale and are generally described as a first-order or second-order transfer function [30]. In practice, the unmodeled dynamics are caused by system complexity, model inaccuracy, and measurement errors. Furthermore, in [31], it has been proved that the influence of unmodeled dynamics can not be avoided, especially in SM control systems. Since the system order is increased artificially by introducing the unmodeled dynamics, the chattering problem (called the harmonic problem in real systems) occurs due to the contained switching function sgn(.), which can reduce the dynamic and static performances or even destroy the system stability [11,32].

In order to overcome the influence of unmodeled dynamics, much of the literature is mainly concentrated on observation and compensation, seen for example in [33,34,35]. In [33], an adaptive neural network control approach is adopted for the information recognition of the unmodeled dynamics of the actuator. In [34], the unmodeled dynamics caused by the nonideal constant power loads were considered and their information was achieved by designing an SM observer. In [35], a second-order SM controller was designed to compensate for the influence of unmodeled dynamics and external disturbances, where the upper boundary was only needed. However, how the unmodeled dynamics affect the performance index, especially in the frequency domain, is seldom studied.

In this paper, the harmonic problem of SM-controlled buck converters imposed by the often-ignored unmodeled dynamics of the Hall sensor is investigated. The unified mathematical model of the SM controller, the buck converter, and the Hall sensor is established. The stability in the time domain and frequency domain is investigated and compared based on the Lyapunov stability theorem and Nyquist stability criterion, respectively. Furthermore, by utilizing the DF approach, the relationship between the amplitude and frequency of the output signal can be deduced for the SM-controlled buck converter system. To be specific, the main contributions of this paper can be summarized as follows:

• The unified mathematical model of the whole system is established by combining the SM controller, the buck converter, and the Hall sensor, and the signal loss caused by the non-ideal Hall sensor is considered;
• Investigations of the stability and the tuning range of the SM controller are carried out in the time domain and frequency domain for comparison;
• By utilizing the DF approach, the harmonics induced by the sensor unmodeled dynamics can be proved;
• The relationship between the amplitude frequency characteristics of the output signal and the Hall sensor is determined.

This paper is organized as follows. In Section 2, the SM-controlled buck converter is described, including the description of the buck converter and the Hall sensor, and the design of the SM controller based on Lyapunov stability in the time domain. In Section 3, the stability analysis in the frequency domain is carried out by utilizing the DF approach and Nyquist stability criterion. In Section 4, the influence of the sensor’s unmodeled dynamics on the harmonics is investigated quantitatively. Finally, the simulations, experiments, and concluding remarks are given in Section 5 and Section 6, respectively.

2. System Description and Modelling

In Figure 1, the system diagram of an SM-controlled buck converter is given [15,16,17,18,19,20,21], which is composed of three parts, i.e., the buck converter, SM controller, and Hall sensor. Where E is the input DC voltage source, S_w is a controllable power switch (N-MOSFET is often adopted), and its ON/OFF is controlled by the output signal u of the SM controller; VD is a continuity diode, and the types of fast recovery diode or Schottky diode are often selected; L and C are the filter inductor and capacitor, respectively, and v_L is the voltage of inductor L; i_L, and i_C are the currents flowing through L and C; R is the load resistor, and v_C is the output voltage and can be achieved by using voltage divider resistor R₁ and R₂ with ration β = R₁/(R₁ + R₂).

In practice, it is known that the current i_C is usually measured by the Hall sensor, but is often ignored. However, its influence on the control system should be ignored, as seen in the literature [27,28,29,30,31,32]. Therefore, this paper innovatively considers it in modelling and controlling. In the following, the description of the buck converter, the Hall sensor, and the design of the SM controller is given.

2.1. Buck Converter

In this paper, the buck converter is assumed to work in continuous conduction mode (CCM), i.e., i_L > 0. Based on the structural characteristics of the converter, the differential equation describing its ON/OFF operation can be given as [15]

$$\{\begin{array}{l}\frac{d{i}_L}{dt}=-\frac{1}{L}{v}_C+\frac{E}{L}u\\ \frac{d{v}_C}{dt}=\frac{1}{C}{i}_L-\frac{1}{R_{O}C}{v}_C\end{array}$$

(1)

where {0, 1} is the output pulse of the controller, R_O = R(R₁ + R₂)/(R + R₁ + R₂).

If we take the second derivative of the variable v_c, and substitute the term of di_L/dt in (1) into it, the dynamics of the output voltage can be rewritten as a second-order differential equation as

$${\ddot{v}}_C+\frac{1}{R_{O}C}{\dot{v}}_C+\frac{1}{LC}{v}_C=\frac{uE}{LC}$$

(2)

Furthermore, we define V_ref as the DC reference value of the output voltage v_c; x₁ and x₂ are the output voltage error and its derivative, i.e.,

$$\{\begin{array}{l}{x}_1=\beta v_c-V_{ref}\\ x_2={\dot{x}}_1=\beta {\dot{v}}_c\end{array}$$

(3)

By combining (2), (3) can be changed into the state space form as

$$\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{b}u+\mathit{f}$$

(4)

Where the state vector x = [x₁, x₂]^T, the matrix A, b, and f are listed as

$$\mathit{A}=\left[\begin{array}{cc}0& 1\\ -\frac{1}{LC}& -\frac{1}{R_{O}C}\end{array}\right], \mathit{b}=\left[\begin{array}{c}0\\ \frac{\beta E}{LC}\end{array}\right], \mathit{f}=\left[\begin{array}{c}0\\ -\frac{V_{ref}}{LC}\end{array}\right]$$

2.2. Hall Sensor

In this paper, the signal loss of the Hall sensor located in the converter circuit is considered, as seen in Figure 1, i.e., $i_C\ne {\widehat{i}}_C$, $x_2\ne {\widehat{x}}_2$, where ${\widehat{i}}_C$ and ${\widehat{x}}_2$ are the measured values corresponding to the variables i_C and x₂. Meanwhile, the transmission characteristics of the Hall sensor are taken into account. Based on the sensor theory, the unmodeled dynamics of the Hall sensor can be expressed as a standard second-order system as [36]

$${\ddot{\widehat{i}}}_C+2\zeta {\omega }_n{\dot{\widehat{i}}}_C+{\omega }_n^2{\widehat{i}}_C=K{\omega }_n^2{i}_C$$

(5)

where i_C and ${\widehat{i}}_C$ are the input and output of the Hall sensor, respectively; ω_n is the angular frequency, K is the amplification coefficient, and the damp ratio 0 < ζ < 1.

From (5), the transfer function G_h(s) of the Hall sensor can be accordingly deduced in terms of the Laplace transformation as

$$G_h(s)=\frac{K{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$$

(6)

The step response c(t) of the Hall sensor can also be expressed as

$$c(t)=K-\frac{K}{\sqrt{1-{\zeta }^2}}{e}^{-\zeta {\omega }_{n}t}\sin \left({\omega }_n\sqrt{1-{\zeta }^2}t+\arccos \zeta \right)$$

(7)

Therefore, the rise time ψ can be obtained from (5) as

$$\psi =\frac{\mathsf{\pi }-\arccos \zeta }{{\omega }_n\sqrt{1-{\zeta }^2}}$$

(8)

By combining Figure 1 and (1) in the frame of Kirchhoff’s circuit law, there is i_c = C${\dot{v}}_C$ and ${\widehat{x}}_2$ = ${\widehat{i}}_C$β/C. By defining $\hat{\mathit{x}}$ = [${\widehat{x}}_2$, ${\dot{\widehat{x}}}_2$]^T as the measured state vector, (5) can be changed as

$$\dot{\hat{\mathit{x}}}=\mathit{C}\hat{\mathit{x}}+\mathit{D}{x}_2$$

(9)

where the matrix C and D are denoted as

$$\mathit{C}=\left[\begin{array}{cc}0& 1\\ -{\omega }_n^2& -2\zeta {\omega }_n\end{array}\right],\mathit{D}=\left[\begin{array}{c}0\\ K{\omega }_n^2\end{array}\right]$$

2.3. SM Controller

Much of the literature has proved the effectiveness of SM applied for the control of converters by replacing the traditional PWM approach [13,14]. At present, various types of SM approaches have been utilized, as seen in the recent survey in [5]. In this paper, we take LSM as an example to design an SM controller in Figure 1. In practice, LSM is the most widely used type due to is easy implementation. Based on the principle of the SM controller, its design includes an SM surface and a robust switching control law [15].

In this paper, the sliding surface s_L can be designed as the linear combination of the variables x₁ and ${\widehat{x}}_2$ as

$$s_L=\lambda x_1+{\widehat{x}}_2$$

(10)

where the design parameter λ > 0 and c = [λ, 1]. From Figure 1, it is worth noticing that the measured signal ${\widehat{x}}_2$ is adopted directly for the controller design, where ${\widehat{x}}_2={\widehat{i}}_C\beta /C$. By combining (3), it can be seen that the crucial role of the Hall sensor is to measure the capacitor current i_C for constructing the SM controller, so that its measurement accuracy affects the control performance of the system.

For the robust switching control law, it is designed to trigger the gate pole of the controllable N-MOSFET S_w in Figure 1 as

$$u=-0.5\left[\mathrm{sgn}\left(s_L\right)-1\right]$$

(11)

The power switch driven by the control u exhibits two states, i.e., ON (u = 1) and OFF (u = 0). In other words, (11) can be described as

$$u=\{\begin{array}{c}1=‘\mathrm{ON}’ \mathrm{when} s_L<0\\ 0=‘\mathrm{OFF}’ \mathrm{when} s_L>0\end{array}$$

(12)

In order to investigate the influence of the unmodeled dynamics of the Hall sensor on the system stability, we first give the traditional SM controller design based on the time-domain stability. Meanwhile, we temporarily assume the transmission characteristics of the Hall sensor are ideal for performance comparison later, i.e., $x_2={\widehat{x}}_2$. Therefore, based on the Lyapunov stability theorem, the condition $s_L{\dot{s}}_L<0$ should be satisfied [15]. By combining (1) and (10), there is

$$s_L{\dot{s}}_L=s_L\left[\left(\lambda -\frac{1}{R_{O}C}\right)x_2+\frac{1}{LC}\left(\beta Eu-V_{ref}-x_1\right)\right]<0$$

(13)

According to the two statuses of the power switch in (12), the stability condition in (13) can be changed as

$$\{\begin{array}{l}{l}_1:\left(\lambda -\frac{1}{R_{O}C}\right)x_2+\frac{1}{LC}\left(\beta E-V_{ref}-x_1\right)=0 \mathrm{when} s_L<0\\ l_2:\left(\lambda -\frac{1}{R_{O}C}\right)x_2-\frac{1}{LC}\left(V_{ref}+x_1\right)=0 \mathrm{when} s_L>0\end{array}$$

(14)

where the lines l₁ and l₂ are the boundaries of the stable regions with the same slope R_O/[L(λR_OC – 1)]. And according to the two cases 0 < λ < 1/R_OC and λ > 1/R_OC, the stable working areas can be derived from (14) as the following Figure 2.

From (14) and Figure 2, we can see that the control performance of the system is affected by the parameters λ. Generally, the design of the SM controller in the time domain is merely based on stability. Restricted by the stability condition (14), we can see from Figure 2 that, as the output voltage v_C varies from 0 to the given value E, the system phase trajectory can only navigate between the two dashed lines l₁ and l₂, where the intersections of two lines with s_L = 0 are defined as A and B, respectively. In other words, only when the phase trajectory reaches the sliding surface between points A and B, can the system stability be guaranteed.

In practice, the controller parameter λ is typically selected as the critical value 1/R_OC, seen in [15]. Meanwhile, the influence of the Hall sensor is often ignored. Therefore, in the following, we consider the signal loss in the transmission process of the Hall sensor to investigate its influence on the system performance in the frequency domain.

3. Design of SM Controller in Frequency Domain

Differing from the traditional design of the SM controller in the time domain, seen in Figure 2, we give the alternative controller design in the frequency domain for comparison. In the following, we still temporarily assume the transmission characteristics of the Hall sensor are ideal as the traditional time-domain SM controller in (13), i.e., $x_2={\widehat{x}}_2$. For the switching control law in (11), we introduce a virtual control v as

$$v=u-0.5=-0.5\mathrm{sgn}\left(s_L\right)$$

(15)

By substituting (15) into (4), we can obtain

$$\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{b}v+{\mathit{f}}_v$$

(16)

where the matrix f_v= [0, (βE − 2V_ref)/(2LC)]^T. Therefore, if we temporarily ignore the Hall sensor, the closed-loop block diagram of the SM-controlled buck converter can be obtained by combining (10), (15), and (16), as in Figure 3, where σ = −s_L, the variable y is defined to represent the SM variable s_L, N(A) is the descriptive function of the nonlinear unit (15), and G(s) represents the transfer function of the linear unit of the buck converter (16) and SM surface s_L.

In order to realize the controller design in the frequency domain, it is worth noticing the fact that the SM controller is inherently nonlinear due to the contained switching control law in (15). Therefore, DF is introduced in the following to realize the approximate linearization of the switching function sgn(.).

Based on the application of DF [24,25,26], it is assumed that the input of the nonlinear unit σ(t) is a sinusoidal signal Asin(ωt), with A and ω as the amplitude and angular frequency, respectively. By utilizing the Fourier transform for the nonlinear unit v in (15), we can obtain

$$v(t)=\frac{a_0}{2}+a_1\cos (\omega t)+b_1\sin (\omega t)$$

(17)

where a₀, a₁, and b₁ are the corresponding coefficients of the Fourier series. Since the relay function sgn(.) is odd, there is a₀ = a₁ = 0 so that (17) can be changed as

$$v(t)=b_1\sin (\omega t)$$

(18)

Here we define

$$\{\begin{array}{c}{b}_1=\frac{\omega }{\pi }{\displaystyle {\int }_{-\frac{\pi }{\omega }}^{\frac{\pi }{\omega }}v\left(t\right)\sin \omega t}dt=-\frac{2}{\pi }\hfill \\ B=\sqrt{a_1^2+b_1^2}=\frac{2}{\pi }\hfill \\ \phi =\arctan \frac{a_1}{b_1}=0\hfill \end{array}$$

(19)

Therefore, the DF of the switching control law v in (15) can be deduced from [24,25,26] as

$$N\left(A\right)=\frac{B}{A}{e}^{j\phi }=\frac{2}{\pi A}$$

(20)

and the transfer function of the linear unit of buck converter (16) can also be deduced from (16) as

$$\begin{array}{ll}G\left(s\right)& =\mathit{c}{\left(\mathit{I}s-\mathit{A}\right)}^{-1}\mathit{b}\\ & =\beta E{R}_O\frac{s+\lambda }{R_{O}LC{s}^2+Ls+R_O}\hfill \end{array}$$

(21)

In the frequency domain, the stability of the closed-loop control system in Figure 3 should be satisfied by the Nyquist stability criterion. Furthermore, by substituting jω with s in (21), we can obtain

$$G\left(j\omega \right)=\beta E{R}_O[\frac{\lambda R_O-\lambda R_{O}LC{\omega }^2+L{\omega }^2}{{\left(R_O-R_{O}LC{\omega }^2\right)}^2+L^2{\omega }^2}+j\frac{-{\omega }^3{R}_{O}LC+\omega R_O-\omega \lambda L}{{\left(R_O-{\omega }^2{R}_{O}LC\right)}^2+{\omega }^2{L}^2}]$$

(22)

Based on the Nyquist stability criterion, it is known that the system stability is determined by the relationship of G(jω) and −1/N(A), where the three types can be seen in Figure 4. In other words, the system is stable when the trajectory of the linear part G(jω) does not enclose the trajectory of −1/N(A), and conversely, it is unstable. For the third case, self-excited oscillation will occur when the trajectory of G(jω) intersects with −1/N(A).

From (20), it is obvious that −1/N(A) exists on the negative real axis and the system stability is determined by the intersection point of G(jω). Furthermore, let the imaginary part of the transfer function G(jω) be equal to zero; we can obtain −ω³R_OLC + ω–_O − ωλL = 0. Since the angular frequency satisfies ω > 0, the solution ω₀ can be deduced as

$${\omega }_0=\sqrt{\frac{R_O-\lambda L}{R_{O}LC}}$$

(23)

From the above time-domain SM controller design in Figure 2, we can see that the controller parameters can be chosen as λ ≥ R_O/L and 0 < λ < R_O/L simultaneously. Therefore, by the frequency-domain analysis in Figure 4, the following conclusion can be obtained:

• For the case of λ ≥ R_O/L, (23) has an imaginary root and there is no intersection point for the trajectory G(jω) and the real axis, which means the system can be kept stable;
• For the case of 0 < λ < R_O/L, we substitute (23) into (22) and the intersection point of the trajectory G(jω) and the real axis can be calculated as (βER_O/L, 0), which indicates that G(jω) does not intersect with the negative real axis and the system can also be kept stable.

Based on the above analysis, we can conclude that, if we ignore the influence of unmodeled dynamics of the Hall sensor, the design of the SM controller in the time domain and frequency domain is the same for the choice of controller parameter λ in (10).

4. Influence of the Unmodeled Dynamics of Hall Sensor

In the following, we continue to investigate the influence of the unmodeled dynamics of the Hall sensor. By combining (9) and (16), the unified model of the whole closed-loop control system in Figure 1 can be obtained as

$${\dot{\mathit{x}}}_s={\mathit{A}}_s{\mathit{x}}_s+{\mathit{b}}_{s}v+{\mathit{f}}_s$$

(24)

where the state vector x_s= [x₁, x₂,${\widehat{x}}_2$,${\dot{\widehat{x}}}_2$]^T, and the matrix A_s, b_s, and f_s are listed as

$${\mathit{A}}_s=\left[\begin{array}{cccc}0& 1& 0& 0\\ \frac{-1}{LC}& \frac{-1}{R_{O}C}& 0& 0\\ 0& 0& 0& 1\\ 0& K{\omega }_n^2& -{\omega }_n^2& -2\zeta {\omega }_n\end{array}\right],{\mathit{b}}_s=\left[\begin{array}{c}0\\ \frac{\beta E}{LC}\\ 0\\ 0\end{array}\right],{\mathit{f}}_s=\left[\begin{array}{c}0\\ \frac{\beta E}{2LC}-\frac{V_{ref}}{LC}\\ 0\\ 0\end{array}\right]$$

By using DF, the corresponding closed-loop block diagram of the SM-controlled buck converter can be similarly obtained in Figure 3 as the following Figure 5, where c_s= [λ, 0, 1, 0], G_s(s) is the transfer function of the system (24).

By combining (4) with (24) and Figure 3 with Figure 5, we can see that the insertion of the Hall sensor in (5) increases the converter system order from 2 to 4, which would inevitably destroy the control performance, or even the system stability. Therefore, it is necessary to investigate the influence of the unmodeled dynamics of the Hall sensor.

4.1. Stability Analysis

From (24), the transfer function G_s(s) can be deduced as

$$\begin{array}{ll}{G}_s\left(s\right)& ={\mathit{c}}_s{\left(\mathit{I}s-{\mathit{A}}_s\right)}^{-1}{\mathit{b}}_s\\ & =\beta E{R}_O\frac{a_1{s}^2+a_{2}s+a_3}{b_1{s}^4+b_2{s}^3+b_3{s}^2+b_{4}s+b_5}\hfill \end{array}$$

(25)

where a₁ = λ, a₂ = (Kω_n² + 2λζω_n), a₃ = λω_n², b₁ = R_OLC, b₄ = (ω_n²L + 2ζω_nR_O), b₂ = (2ζω_nR_OLC + L), b₃ = (ω_n²R_OLC + 2ζω_nL + R_O), and b₅ = ω_n²R_O.

Furthermore, by substituting s = jω into (25), there is

$$\begin{array}{ll}{G}_s\left(j\omega \right)& =\beta E{R}_O[A_1{\omega }^6+A_2{\omega }^4+A_3{\omega }^2+A_4\\ & +j(A_5{\omega }^5+A_6{\omega }^3+A_7\omega )]/[{\left(b_1{\omega }^4-b_3{\omega }^2+b_5\right)}^2+{\left(b_2{\omega }^3-b_4\omega \right)}^2]\hfill \end{array}$$

(26)

where A₁ = −a₁b₁, A₂ = a₃b₁ + a₁b₃ − a₂b₂, A₃ = a₂b₄ − a₃b₃ − a₁b₅, A₆ = a₁b₄ + a₃b₂ − a₂b₃, A₅ = a₂b₁ − a₁b₂, A₄ = a₃b₅, and A₇ = a₂b₅ − a₃b₄.

In the sense of the Nyquist stability criterion, the influence of the Hall sensor on the system stability in Figure 5 is investigated in the frequency domain. Similar to Figure 3, the system stability is determined by the relationship of G_s(s) and −1/N(A) in (20). Observed from (26), since its denominator is always positive, the intersection of G_s(jω) with the real axis leads to the following equation:

$$A_5{\omega }^5+A_6{\omega }^3+A_7\omega =0$$

(27)

Since the angular frequency ω is typically positive, excluding the scenario with ω = 0, the solution of (27) can further be deduced from Vieta’s theorem as

$$\{\begin{array}{l}{\omega }_1=\sqrt{\frac{R_O-\lambda L}{R_{O}LC}}\\ {\omega }_2=\sqrt{\frac{{\omega }_n^4{R}_{O}C\left(K{R}_O-\lambda L\right)}{\left(R_O-\lambda L\right)\left(K{\omega }_n^2{R}_{O}C-\lambda \right)}}\end{array}$$

(28)

By comparing (23) and (28), we can see the stability influence imposed by the Hall sensor is determined by ω₂ in (28). According to the application of the Hall sensor, the linear types of ACS712, ACS706, and ACS758 are generally adopted [37]. Therefore, considering the signal loss in the Hall sensor, its transmission characteristic property can be described as ${\widehat{i}}_C$ = Ki_C. It implies that K is approximately equivalent to the Hall sensor’s linear amplification factor, where K can be determined by the external auxiliary circuit, seen for example in [38] for the Hall sensor of ACS712. Meanwhile, since ${\widehat{i}}_C\cong i_C$ in practice, it means K $\cong$ 1. Therefore, by substituting K $\cong$ 1 into ω₂ in (28), it approximately yields

$${\omega }_2\cong \sqrt{\frac{{\omega }_n^4{R}_{O}C}{{\omega }_n^2{R}_{O}C-\lambda }}$$

(29)

Furthermore, for the rise time ψ in (8), it is equal to the following form as

$${\omega }_n=\frac{\mathsf{\pi }-\arccos \zeta }{\psi \sqrt{1-{\zeta }^2}}$$

(30)

It should be noted that the reason for the the rise time ψ of the Hall sensor is mainly because the sensor’s dynamic performance is predominantly influenced by ψ, which is a crucial parameter. Furthermore, the rise time ψ depends particularly on the adjustment of the filter capacitance of the auxiliary circuit in typical application systems, seen for example in [39] for the Hall sensor of ACS712.

From (29) and (30), we can see that ω₂ is determined by the parameters of the SM controller λ and the Hall sensor’s rise time ψ simultaneously. Consequently, the stability of the system depends on the two parameters, so the following two cases need to be discussed.

• Case 1: If λ > ${\omega }_n^2$R_OC, ω₂ is an imaginary root. Since there is no intersection point for the trajectory of G_s(s) and the negative real axis, the system still keeps stable even with the consideration of the Hall sensor.
• Case 2: If 0 < λ < ${\omega }_n^2$R_OC, ω₂ is a real root. By substituting (27) into (24), the intersection for the trajectory of G_s(s) and the real axis can be deduced as

  $$G_s\left(j{\omega }_2\right)=-\beta E{R}_O\frac{\zeta {\omega }_n{R}_{O}C+\lambda R_{O}C}{2{\zeta }^2{R}_{O}LC\left({\omega }_n^2{R}_{O}C-\lambda \right)+2{\zeta }^{2}L}$$

  (31)

Since 0 < λ < ω_n²R_OC, G_s(jω₂) yields a negative value. This means that there exists an intersection between G_s(s) and −1/N(A) and the whole system with the Hall sensor in (24) is critically stable, or in other words, the harmonics are contained due to the consideration of the Hall sensor.

4.2. Harmonic Analysis Affected by Hall Sensor

In the following, we continue to investigate the influence of unmodeled dynamics of the Hall sensor and further determine the relationship between the amplitude-frequency characteristics of the output signal and the Hall sensor.

From (29), since ω₂ in (29) is the solution of (31), it corresponds to the angular frequency of the harmonics, i.e., the harmonic frequency can be denoted as f₂ = ω₂/2π. Here we define a variable χ = (π − arccosζ)/(1 − ζ²)^1/2 so that (30) can be changed as

$${\omega }_n=\frac{\chi }{\psi }$$

(32)

By substituting (32) into (29), the relationship between the harmonic frequency f₂ and the rise time ψ of the Hall sensor can be formulated as

$$f_2=\frac{1}{2\mathsf{\pi }\psi }{\left(\frac{{\chi }^4{R}_{O}C}{{\chi }^2{R}_{O}C-\lambda {\psi }^2}\right)}^{{}^{1/2}}$$

(33)

In accordance with the Nyquist stability criterion, G_s(jω₂) in (31) can be evaluated by setting G_s(jω) + 1/N(A) = 0. From (20), we can have −1/N(A) = −πA/2 so that the relationship between the harmonic amplitude A₂ and the rising time ψ can be determined as

$$A_2=\frac{\beta E{R}_O^{2}C\left(\zeta \chi \psi +\lambda {\psi }^2\right)}{\mathsf{\pi }{\zeta }^2{R}_O^{2}L{C}^2{\chi }^2-\mathsf{\pi }{\zeta }^2{\psi }^{2}L\left(\lambda R_{O}C-1\right)}$$

(34)

It is worth noticing that the buck converter is assumed to work in CCM, as seen in (1). In other words, if the induct current i_L is assumed to be critical continuous, the current harmonics ΔI_L in a switching period T can be expressed from [39] as

$$\Delta I_L=\left|\frac{1}{L}{\displaystyle {\int }_0^{DT}{v}_L\mathrm{d}t}\right|=\frac{V_C}{EL{f}_{2\min }}\left(E-V_C\right)=\frac{2V_C}{R_O}$$

(35)

where f_2min is the minimum value of the harmonic frequency f₂ in (33), V_C is the average value of the output voltage v_c in a switch period T, and D represents the duty cycle with D = V_C/E. Therefore, the critical value of the switching frequency f_m is deduced from [39] as

$$f_{\mathrm{m}}=\frac{\left(E-V_C\right)R_O}{2EL}$$

(36)

Since the switching frequency is proportional to the harmonic frequency of the system, it means the switching frequency f_m in (36) corresponds to the minimum value of the harmonic frequency f_2min. Hence, the constraint f₂ ≥ f_m should be guaranteed. Therefore, by combining (33) and (36), the rising time ψ of the Hall sensor can be deduced as

$$0<\psi \le \frac{EL}{\mathsf{\pi }\left(E-V_C\right)R_O}{\left(\frac{{\chi }^4{R}_{O}C}{{\chi }^2{R}_{O}C-\lambda {\psi }^2}\right)}^{{}^{1/2}}$$

(37)

which is helpful for the choice of the Hall sensor.

Based on the above analysis concerning the SM controller parameter λ and the rise time ψ of the Hall sensor, we conclude the following:

• If the SM controller parameter λ > ω_n²R_OC, the whole closed-loop control system of the SM-controlled buck converter system has no harmonics;
• If the SM controller parameter 0 < λ < ${\omega }_n^2$R_OC, the constraint of the harmonic frequency f₂ ≥ f_m can be deduced, and the rising time ψ of the Hall sensor can also be obtained, which gives guidance to the choice of SM parameter λ and the Hall sensor in practical systems.

5. Simulations and Experiments

In order to validate this paper, the circuit parameters of the buck converter in Figure 1 are listed in

Table 1. Circuit Parameters of the Buck Converter.

Component	Symbol	Value
Input voltage	E	20 V
Inductor	L	1 mH
Capacitor	C	3.2 mF
Load resistor	R	10 Ω
Divider resistor 1	R1	10,000 Ω
Divider resistor 2	R2	50,000 Ω
Reference output voltage	Vref/β	10 V

. The Hall sensor of ACS 712 is chosen, where the measurement characteristics in its chip manual [40] and the corresponding fitting model based on the collected data can be seen in Figure 6.

In Figure 6b, we utilize the GetData software to collect the test date of the chip ACS712 in the datasheet, and then using MATLAB’s Identification Toolbox, the unmodeled dynamics of the Hall sensor in (6) can be described as

$$G_h(s)=\frac{2.235\times {10}^{11}}{s^2+6.691\times {10}^{5}s+2.251\times {10}^{11}}$$

(38)

By comparing (6) and (38), we can deduce the amplification coefficient K = 0.993, not the ideal 1, which indicates the signal loss of the real sensor ACS712 is included. The damping ratio ζ = 0.705, and the angular frequency ω_n = 4.744 × 10⁵.

5.1. Simulation Results

In the following, we test the influences of the unmodeled dynamics of the Hall sensor and SM controller parameter λ on the buck converter system simultaneously. In order to facilitate the expatiation, we abbreviate the model of the buck converter without consideration of the Hall sensor in (4) as the ‘ideal system’, and the model with the Hall sensor in (24) as the “actual system”.

Firstly, we investigate the influence of the unmodeled dynamics of the Hall sensor on the system. The rise time of the Hall sensor ACS712 is chosen as ψ = 6.647 µs and the angular frequency calculated from (30) as ω_n = 4.992 × 10⁵. For the SM controller parameter λ, the traditional time-domain design approach in Section 2.3 differs from the frequency-domain design approach. For the former, it is based on the model of the ideal system with the SM controller parameter λ > 1/R_OC and 0 < λ < 1/R_OC, seen in Figure 2. While for the actual system with consideration of the Hall sensor, the stability region without harmonics is deduced as λ > ω_n²R_OC, seen from (29). Since there is 1/R_OC = 31.25 and ω_n²R_OC = 7.973 × 10⁹, we select SM parameters λ₁ = 31.25 for the ideal system and λ₂ = 8 × 10⁹ for the actual system for comparison. Simulation results are given in Figure 7 and Figure 8.

For the SM controller parameter λ₁ = 31.25 < ω_n²R_OC, Figure 7a shows that the Nyquist curve G(jω) of the ideal system defined in (22), which neither encloses nor intersects with the line of −1/N(A) defined in (20). However, for the Nyquist curve G_s(jω) of the actual system defined in (31), corresponds to the scenario in Figure 4c, which means the existence of an intersection point for the Nyquist curve G_s(jω) with the line of −1/N(A). Since the line of −1/N(A) passes from an enclosed region of G_s(jω) to an unenclosed region, stable self-oscillation occurs, which is just the reason for the occurrence of the harmonic problem. On the other hand, if the SM controller parameter is chosen λ₂ = 8 × 10⁹ > ω_n²R_OC, similarly Figure 7b gives the comparison of the Nyquist curves for the ideal system and the actual system. We can see that neither G(jω) for the ideal system nor G_s(jω) for the actual system encloses the line of −1/N(A), which is in accordance with the scenario in Figure 4a.

Figure 8 shows the time-domain response curve of the SM surface s_L defined in (10). For the case with λ₁ = 31.25 in Figure 8a, we can see the ideal system and the actual system can both keep stable, but there exists an overshoot for the latter, which is caused by the insertion of the unmodeled dynamics of the Hall sensor in (38), which makes the system order increase from 2 to 4. This is in accordance with the explanation in (31). Figure 8b shows the comparison in the case with λ₂ = 8 × 10⁹. As mentioned in Figure 6b, it corresponds to the situation without consideration of the Hall sensor, so that the response curves of the ideal and actual system are almost the same and no harmonics occur.

In the following, we continue to investigate the influence of the SM controller parameter λ on the actual system by choosing λ₁ = 31.25, λ₂ = 10³, λ₃ = 10⁵, λ₄ = 8 × 10⁹, and λ₅ = 9 × 10⁹ for performance comparison. The simulations concerning the inductor current i_L and the output voltage v_c are given in Figure 8.

From Figure 9a,b, we can see harmonics occur in both the inductor current i_L and the output voltage v_c in cases of λ₁ = 31.25, λ₂ = 10³, and λ₃ = 10⁵, while for the other cases of λ₄ = 8 × 10⁹ and λ₅ = 9 × 10⁹, there are no harmonics. This is in accordance with the theoretical analysis in (30). The values of the amplitude and frequency for the cases with λ₁ = 31.25, λ₂ = 10³, and λ₃ = 10⁵ are listed as 1.6 × 10⁻⁵ V and 78.74 kHz, 1.6 × 10⁻⁵ V and 78.77 kHz, and 1.6 × 10⁻⁵ V and 78.77 kHz. In the premise of system stability, we can see that the different values of the SM controller parameter λ have less influence on the amplitude-frequency characteristics of the buck converter system with the Hall sensor.

Thirdly, we investigate the influence of the rising time ψ of the Hall sensor on the system performances. According to the datasheet of ACS712 [40], we choose five values of ψ = 6.647 µs, 32.09 µs, 88.18 µs, 211.3 µs, and 291.26 µs for comparison. The simulations are given in Figure 10 and

Table 2. Comparison of the Output Voltage with Varying ψ.

Rise Time ψ (μs)	Theoretical Value of Harmonic Amplitude A2 (mV)	Simulation Value of Harmonic Amplitude A2 (mV)	Theoretical Value of Harmonic Frequency f2 (kHz)	Simulation Value of Harmonic Frequency f2 (kHz)	Steady Error of Output Voltage vC (mV)
6.647	0.94	0.96	79.45	78.74	0.032
32.09	4.55	4.62	16.46	16.18	0.741
88.18	12.51	12.67	5.99	5.88	5.59
211.3	30.03	29.99	2.50	2.50	31.3
291.26	41.44	32.99	1.81	1.99	48.6

.

From Figure 10, we can see that the influence of the unmodeled dynamics of the Hall sensor cannot be ignored, despite the small value of the rising time ψ in the grade of µs. As observed from Table 2, as the increase of the rise time ψ, the frequency of the harmonics will decrease, while the amplitude and the steady error will increase. It is worth noticing that, if ψ = 211.3 μs, it corresponds to the maximum value to keep the buck converter working in CCM based on the data sheet of ACS 712 in [40] and (36), (37). Therefore, if ψ = 291.26 μs, the inductor current i_L is discontinuous, which can degrade the control performance and increase the steady error, as seen in Table 2.

5.2. Experimental Results

In the following, we further test the influence of the unmodeled dynamics of the Hall sensor by experiment. In Figure 11, the experiment platform based on DSpace 1106 is given, where the sampling time is 0.5 ms and the PWM frequency is 10 k0 Hz. In Figure 12a, the hardware of the buck converter is given, and in Figure 12b, the circuit of the contained Hall sensor ACS712 is given, where the five chosen values 6.647 µs, 32.09 µs, 88.18 µs, 211.3 µs, and 291.26 µs of the rise times ψ in Figure 10 and Table 2 can be obtained by adjusting the capacitor C_F on the basis of its chip manual [40].

Based on the chip manual of the Hall sensor ACS712 in [40], we focus on investigating the main issue of this paper, i.e., the influence of the unmodeled dynamics of the Hall sensor on the control performance of the SM-controlled buck converter system in Figure 1. Here we let the SM controller parameter be λ = 31.25 and vary the rise time ψ as 6.647 µs, 32.09 µs, 88.18 µs, 211.3 µs, and 291.26 µs, where the corresponding values of the capacitor C_F can be seen in

Table 3. Comparisons of the Output Voltage with Varying ψ.

Rise Time ψ (μs)	Adjusted Capacitor CF (nF)	Harmonic Amplitude A2 (mV)	Steady Error of Output Voltage vC (mV)
6.647	0	428	12.3
32.09	10	528	26.1
88.18	47	568	50.1
211.3	70	688	83.4
291.26	100	812	103.7

. The experimental results are given in Figure 13 and Table 3. We can see that, as the rise time ψ increases from 6.647 µs to 291.26 µs, the steady-state error of the output voltage v_C increases from 12.3 mV to 103.7 mV, and the harmonic amplitudes vary from 428 mV to 812 mV. Therefore, it proves the influence of the Hall sensor cannot be ignored.

6. Conclusions

In this paper, we investigate the harmonic problem of SM-controlled buck converters imposed by the often-ignored unmodeled dynamics of the Hall sensor. The advantages of this paper lies on the following three points:

• The unmodelled dynamics of the Hall sensor are included into the modelling of the SM-controlled converter system, and the experimental modeling approach in Figure 6 and (38) guarantees the modelling precision and accuracy;
• For the design of SM controller, this paper replaces the traditional time-domain approach based on the Lyapunov stability theorem with a frequency-domain approach by using DF and Nyquist stability criterion, which innovatively proves the inevitable existence of harmonics caused by the unmodeled dynamics of Hall sensor;
• The quantitative relationship between the amplitude frequency characteristics of the output signal and the Hall sensor is determined, seen in (33) and (34), which gives guidance for the choice of SM parameter λ and the Hall sensor in practical systems.

At the same time, some issues concerning the limitations and potential areas for the further research should also be considered. For example:

• In this paper, the sensor unmodelled dynamics are only considered; what about other or multiple unmodelled dynamics affect the stability and control performance of the system?
• In this paper, the commonly used linear sliding mode control approach is adopted; what about other complex SM types are used for the control of buck converters?
• What is significant and valuable for the theoretical and applied research of SM-controlled buck converter systems.