Parameter Design Method of Variable Frequency Modulation for Grid-Tied Inverter Considering Loss Optimization and Thermal and Harmonic Constraints

Abstract

Electromagnetic interference (EMI) rectification of grid-tied inverters is crucial for their practical application, and the variable frequency modulation (VFM) technique is a low-cost and simple way for EMI reduction. However, changes in loss and harmonic behaviors make it hard for parameter determination of VFM. In this paper, the parameters required for switching frequency (SF) function are determined for loss optimization of MOSFETs and inductors, while total harmonic distortion (THD) and temperature rise in MOSFETs and inductor core are constrained to guarantee the feasibility of the calculated parameters. Current transient is derived through multidimensional Fourier decomposition (MFD) and characteristics of Bessel function for loss estimation of MOSFET and inductor. Modified Steinmetz equation (MSE) is applied for core loss estimation and AC resistance is considered for copper loss estimation. With the constraints of THD and temperature, the loss optimization problem is solved by the augmented Lagrangian (AL) method. With the assistance of the proposed method, total loss optimization can be realized in feasible regions while the temperature rise in essential components can be restricted to the preset values.

1. Introduction

With the revolutionary development of renewable energy power generation worldwide, power electronic devices are almost ubiquitous in power systems nowadays [1]. Grid-tied inverters, which act as medium for power conversion and transmission [2], are highly distributed in the power system network. However, these inverters introduce significant harmonics and disturbances to the power grid because of various operating conditions and switching actions. In terms of the former case, those harmonics and disturbances can be addressed by advanced control strategies for better transient performance or low THD [3,4,5]. However, when it comes to those disturbances generated because of inherent high-frequency switching characteristic of power electronics, these inverters give rise to severe EMI noise [6], which pollutes the electromagnetic environment and affects the normal operation of sensitive equipment and cannot be handled only at the control level. The mechanism of the EMI noise propagation process is shown in Figure 1. Switching the devices of the inverter generates abundant high-frequency noises because of high-speed pulse width modulation (PWM). If the filters are ideal, these noises are supposed to be attenuated to an acceptable level because of their low-pass features. However, parasitic characteristics of passive components like inductors or capacitors provide a low-impedance path for noise propagation [7]. The power loop parasitic characteristics offer the path for the differential mode (DM) noise and the common mode (CM) noise propagates through the grounding system [8]. These noises can amplify their influence with the assistance of power lines and thus must be handled strictly [9].

Several typical methods can be applied for EMI reduction in inverters, such as layout optimization, active or passive filter, soft switching, shielding, VFM, and so on [10,11,12,13]. A rational PCB layout is essential for better EMI behavior, but it is usually constrained by cost, temperature rise, and other relevant limitations, and requires additional modification of the original circuit [14]. An EMI filter is the most effective way to reduce EMI, but it features high cost and occupies a large space [15]. Soft switching can shape the switching process and achieve EMI and loss optimization, yet the added passive or active components increase the cost and instability of the inverter. Shielding is sufficient for radiated emission reduction but has little effect when it comes to conducted emission [16]. VFM reduces the peak value of noise by dispersing its energy over a wider frequency range. Although the effect of VFM cannot be compared with passive filter in terms of EMI reduction, it can be easily implemented in the digital controller and does not require any additional hardware devices; thus, it attracts the attention of more researchers, and a number of forms targeting different objectives have been developed [17].

Different from constant frequency modulation (CFM), VFM changes the frequency of switching devices for specific purposes. It can be separated into three categories according to the frequency variation mode: random PWM (RPWM), periodic PWM (PPWM), and programmed PWM (Pro-PWM). When using RPWM, SF varies randomly to realize uniform distribution of EMI noise. However, it is hard for the digital controller to generate a true random number especially when the control cycle is very short, thus the random number is usually generated with the help of a peripheral circuit [13,18]. Pro-PWM varies SF according to the sampled feedback signals to realize specific optimizations like current harmonic minimization, switching loss or CM voltage reduction [19]; this method occupies more time of controller and might influence the closed-loop control. PPWM varies SF according to a fixed function periodically to realize uniform distribution of EMI noise. Because SF function is preset, the carrier frequency required for the current control cycle can be obtained by simply calculating the value of the function, which is easy to implement, and its EMI performance is close to RPWM, thus, arousing more researchers’ interest.

Although PPWM exhibits great EMI performance, how the parameters required for SF function are determined is rarely mentioned in the current literature. Conventionally, three types of function were applied for PPWM in [20]: triangular, sinusoidal, and exponential function. However, the principle for the selection of SF ranges was not mentioned. A combination of three functions was taken into account in [21] for minimum voltage spikes, but the author ignored the induced variation in loss and THD, which means the effect of the method is full of unknowns when frequency variation range is large. In [22,23], the relationship between distribution characteristic and SF function is studied, and the SF functions to realize uniform distribution and other distribution characteristics of SFs were derived. However, this paper only determined the shape of SF function, and how to get the practical parameters required for the SF function was not investigated in depth. In [24,25,26], the shape of SF function was not preset; it was obtained through calculus of variations for switching or total loss optimization with constant THD constraint. However, the topology applied in these papers is a L-filtered full-bridge inverter, which is nearly obsolete nowadays. Furthermore, because the shape of SF function is dependent on circuit parameters, the method is not suitable for occasions with operating condition variations, and merely reducing total loss without considering respective thermal resistances of MOSFET and inductor might lead to one of them overheating.

Driven by the drawbacks existing in the current techniques of PPWM, this paper proposes a new method for parameter determination of the SF function. Current transients of inverter side and grid side are analyzed with the assistance of MFD and characteristics of Bessel function. The current transient on the inverter side is applied for switching and conduction loss estimation with the help of a linearized loss model. The current transient on the grid side is applied for THD calculation. Core loss is estimated through MSE. Copper loss is evaluated through the characteristics of VFM and an average AC resistance model. The optimization objective is minimum total loss with temperature rise and THD constraints, and the problem is solved by the AL method. With the help of the proposed method, total loss optimization within the feasible region is realized, and the case where the optimal value obtained from the algorithm leads to overheating of the MOSFET or inductor core can be avoided. The main contributions of this paper are listed below.

(1)

Current dynamics of grid side and inverter side are derived through MFD and characteristics of Bessel function.

(2)

Analytical approximation of conduction and switching loss is derived with the assistance of current dynamics and forward Euler method.

(3)

Analytical approximation of core and copper loss is derived through MSE and average AC resistance model of winding.

(4)

Frequency range for optimal total loss restricted by current THD and devices’ temperature is obtained through AL method.

This paper is organized as follows: In Section 2, the topology is introduced and current transients of inverter and grid side are analyzed. In Section 3, how the losses of MOSFETs and inductors are calculated will be explained, and temperature rise in MOSFET and inductor core will be estimated. In Section 4, the process of solving constrained optimization problems using the AL method will be introduced. The parameters obtained from the proposed method are applied and compared with its neighborhood to prove the effectiveness of the proposed algorithm in Section 5. The conclusions will be drawn in Section 6.

2. Topology Introduction and Current Transient Analysis

Topology of LCL-filtered full-bridge grid-tied inverter is shown as Figure 2. u_dc is DC voltage supply; S₁~S₄ are MOSFETs; L₁, L₂ and C₁ are inductors and capacitor of LCL filter and they are symmetrically distributed to reduce mutual transformation between CM and DM noise; u_g is voltage of power grid; u_AB is the output voltage of full bridge; u_C1 is the voltage of C₁; i_L1 is current of L₁; and i_g is current of i_L2.

When using unipolar modulation for driving signal generation, the modulation process is shown as Figure 2. u_m is modulation wave generated from controller; u_c is the triangular carrier wave ranging from −1 to 1; u_m is applied as modulation signal for bridge A and −u_m is applied for bridge B. According to [21], exponential function performs uniformly distributed SFs, and its conducted emission is lower than that of triangular or sine function, thus it is discussed as an example in this paper. The SF function is shown in Equation (1).

$$f_{\mathrm{sw}}\left(t\right)=A^{\alpha t}$$

(1)

Because of symmetry, the curve of SF function can be expressed as Figure 3, the period of which is twice that of the fundamental wave. The phase of f_sw (t) is the same as the absolute value of the fundamental wave.

In terms of inductor loss, L₁ has a greater impact when compared with L₂ since its current ripple is much higher than L₂. In general, both loss of MOSFETs and inductors mainly depend on the current of L₁. As a result, the analysis of the current transient process of L₁ is the foundation for estimating loss of the system.

When considering current dynamic of L₁, voltage fluctuation of C₁ is much smaller than u_AB. In terms of harmonic components resulting from switching frequency, i_L2 achieves an attenuation of −60 dB/dec, while that of i_L1 is −20 dB/dec for LCL filter. Furthermore, the value of L₂ is typically ranging from a few tens of microhenries to several millihenries, which means voltage drop of L₂ resulting from the fundamental components of i_L2 is extremely small. Since u_AB changes between u_dc and −u_dc, voltage drop of L₂ resulting from both harmonic components and fundamental components of i_L2 can be neglected compared with u_AB [27]; thus, the topology in Figure 2a can be simplified as circuit in Figure 4.

Because of high-frequency application, u_g can be considered constant in each switching period. In most cases, the power factor of the inverter is set to 1. In that condition, fundamental component of u_AB, u_g and i_L1 can be expressed as Equation (2), where M is modulation index, u_AB0 is the fundamental component of u_AB, i_L10 is the fundamental component of i_L1, U_g is magnitude of grid voltage, and I_g is magnitude of output current. In fact, the phase of u_AB0 is not equal to u_g, but it is be neglected since the impedance of L₁ is quite a small value.

$$\left\{\begin{array}{l}{u}_{\mathrm{AB}0}=M{u}_{\mathrm{dc}}\sin \left({\omega }_{0}t\right)\\ u_{\mathrm{g}}=U_{\mathrm{g}}\sin \left({\omega }_{0}t\right)\\ i_{L10}=I_{\mathrm{g}}\sin \left({\omega }_{0}t\right)\end{array}\right.$$

(2)

To analyze the transient process of i_L1, the harmonic components of u_AB should be obtained. Conventionally the spectrum of u_AB is obtained through MFD. For each bridge, midpoint voltage can be expressed as Equation (3), where ω_c and φ_c are the angular frequency and phase of carrier, u_m (t) is a cosine function ranging from −M to M, u_c (t) is a triangular wave ranging from −1 to 1, and ε (t) is a step function.

$$u_{\mathrm{bridge}}=u_{\mathrm{dc}}\epsilon \left[u_{\mathrm{m}}\left({\omega }_{0}t-{\displaystyle \frac{\pi }{2}}\right)-u_{\mathrm{c}}\left({\omega }_{\mathrm{c}}t+{\phi }_{\mathrm{c}}\right)\right]$$

(3)

Considering the angle of modulation and carrier wave as variables independent from t, namely x = ω_c t + φ_c, y = ω₀t − π/2, then u_bridge can be considered as a binary function, as shown in Equation (4).

$$u_{\mathrm{bridge}}\left(x,y\right)=u_{\mathrm{dc}}\epsilon \left[u_{\mathrm{m}}\left(y\right)-u_{\mathrm{c}}\left(x\right)\right]$$

(4)

The surface of u_bridge is shown as Figure 5.

As can be seen in Figure 5, u_bridge is a periodic function regarding both x and y; thus, it can be two-dimensional Fourier decomposed as shown in Equation (5), where J_n(x) represents Bessel function of the first kind of order n.

$$\begin{array}{l}{u}_{\mathrm{bridge}}(t)={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=-\infty }^{+\infty }\frac{2u_{\mathrm{dc}}}{m\pi }\sin \left(\frac{m+n}{2}\pi \right)J_n\left(\frac{\pi }{2}Mm\right)\cos \left[m\left({\omega }_{\mathrm{c}}t+{\phi }_{\mathrm{c}}\right)+ny\right]}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+M{\displaystyle \frac{u_{\mathrm{dc}}}{2}}\cos \left(y\right)\end{array}$$

(5)

When using unipolar modulation, u_m of bridge B lags that of A π; thus, u_AB can be expressed as Equation (6). When m is an odd number, whatever n is, it can be derived from Equation (6) that the corresponding harmonic is zero, and that is why unipolar modulation has the function of doubling frequency.

$$\begin{array}{l}{u}_{\mathrm{AB}}(t)={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=-\infty }^{+\infty }\frac{4u_{\mathrm{dc}}}{m\pi }\cos \left(\frac{m}{2}\pi \right){\sin }^2\left(\frac{n\pi }{2}\right)J_n\left(\frac{\pi }{2}Mm\right)\sin \left[m\left({\omega }_{\mathrm{c}}t+{\phi }_{\mathrm{c}}\right)+n{\omega }_{0}t\right]}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+M{u}_{\mathrm{dc}}\sin \left({\omega }_{0}t\right)\end{array}$$

(6)

When only considering harmonic component i_L1H, it can be expressed as Equation (7). The equation cannot be used for transient analysis of i_L1H because it is an infinite series. However, some special points of i_L1H can help determine characteristic of i_L1 preliminarily. At the start and end of a switching period, ω_ct + φ_c is 0. Considering harmonics m ω_c + n ω₀ and m ω_c − n ω₀, their transient value are almost the opposite since n only takes even number and impedance of L₁ is almost the same at two frequencies. At midpoint of each switching period the same conclusion can be derived either. As a result, at start, end, and middle point of every switching period, i_L1H is 0 and i_L1 is equal to its fundamental value i_L10.

$$i_{L1\mathrm{H}}(t)={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=-\infty }^{+\infty }\frac{4u_{\mathrm{dc}}}{m\pi }\frac{-1}{\left(m{\omega }_{\mathrm{c}}+n{\omega }_0\right)L_1}\cos \left(\frac{m}{2}\pi \right)J_n\left(\frac{\pi }{2}Mm\right)\cos \left[m\left({\omega }_{\mathrm{c}}t+{\phi }_{\mathrm{c}}\right)+n{\omega }_{0}t\right]}}$$

(7)

In each switching period, u_g can be considered constant. Combined with start, end, and middle point characteristics, the current dynamic in each switching period can be described as the linear process in Figure 6.

Duration and current slope of each stage can be calculated through principle of unipolar modulation, as shown in Equation (8), where Δt₁~Δt₃ are duration of each stage and k₁~k₃ are their current slopes.

$$\left\{\begin{array}{cc}\begin{array}{l}\Delta t_1={\displaystyle \frac{T_{\mathrm{sw}}}{4}}\left[1-M\sin \left({\omega }_{0}t\right)\right]\\ \Delta t_2={\displaystyle \frac{T_{\mathrm{sw}}}{2}}M\sin \left({\omega }_{0}t\right)\\ \Delta t_3={\displaystyle \frac{T_{\mathrm{sw}}}{4}}\left[1-M\sin \left({\omega }_{0}t\right)\right]\end{array}& \begin{array}{l}{k}_1=-{\displaystyle \frac{U_{\mathrm{g}}}{L_1}}\sin \left({\omega }_{0}t\right)\\ k_2={\displaystyle \frac{u_{\mathrm{dc}}-U_{\mathrm{g}}\sin \left({\omega }_{0}t\right)}{L_1}}\\ k_3=-{\displaystyle \frac{U_{\mathrm{g}}}{L_1}}\sin \left({\omega }_{0}t\right)\end{array}\end{array}\right.$$

(8)

According to Equation (8), current transient can be divided into two identical parts. Because L₁ and L₂ are quite small for fundamental components, U_g is close to M u_dc, then some relationships between stage duration and slope, as shown in Equation (9).

$$\left\{\begin{array}{l}{k}_1=-2{\displaystyle \frac{u_{\mathrm{dc}}}{L_1{T}_{\mathrm{sw}}}}\Delta t_2\\ k_2=4{\displaystyle \frac{u_{\mathrm{dc}}}{L_1{T}_{\mathrm{sw}}}}\Delta t_1\end{array}\right.$$

(9)

Once the current transient of L₁ is obtained, the current of each MOSFET can be analyzed and its conduction and switching loss can be derived.

The current of L₂ directly influences grid-connected power quality and its transient should be analyzed to obtain the THD model of the inverter. In terms of the LCL filter, its transfer function is shown in Equation (11).

$$i_{L2}\left(s\right)={\displaystyle \frac{u_{\mathrm{AB}}\left(s\right)-\left(1+L_1{C}_1{s}^2\right)u_{\mathrm{g}}\left(s\right)}{L_1{L}_2{C}_1{s}^3+\left(L_1+L_2\right)s}}$$

(10)

Since u_g(s) contains few high-frequency harmonic components and the SF of the inverter should keep a certain distance from the resonant frequency, transfer function of i_L2 can be simplified as Equation (11), where i_L2H and u_ABH are harmonic components of i_L2 and u_AB, respectively.

$$i_{L2\mathrm{H}}\left(s\right)={\displaystyle \frac{1}{L_1{L}_2{C}_1{s}^3}}{u}_{\mathrm{ABH}}\left(s\right)$$

(11)

As can be seen from Equation (11), i_L2 can be considered as the third-order integral of u_AB. Combined with Equation (6), harmonic components of i_L2 can be expressed as an infinite series like Equation (12). When m is odd, cos(mπ/2) is zero; when n is even, sin(nπ/2) is zero, thus only when m is even and n is odd should the corresponding harmonic be considered, and this is the basic working principle of unipolar frequency-doubled modulation.

$$i_{L2\mathrm{H}}(t)={\displaystyle \sum _{\begin{array}{c}m=1\\ m \mathrm{is} \mathrm{even}\end{array}}^{+\infty }{\displaystyle \sum _{\begin{array}{c}n=-\infty \\ n \mathrm{is} \mathrm{odd}\end{array}}^{+\infty }\frac{4u_{\mathrm{dc}}}{m\pi }\frac{1}{L_1{L}_2{C}_1{\left(m{\omega }_{\mathrm{c}}+n{\omega }_0\right)}^3}\cos \left(\frac{m}{2}\pi \right){\sin }^2\left(\frac{n}{2}\pi \right)J_n\left(\frac{\pi }{2}Mm\right)\cos \left[m\left({\omega }_{\mathrm{c}}t+{\phi }_{\mathrm{c}}\right)+n{\omega }_{0}t\right]}}$$

(12)

Equation (12) cannot be used for transient of i_L2H because it is an infinite series, but characteristics of Bessel function and impedance of LCL filter can be applied to simplify Equation (12). In terms of LCL filter, its impedance for the fourth harmonic and its sideband is much higher than that for the second harmonic; thus, only m = 2 needs to be considered when analyzing the transient process of i_L2. In terms of J_n(πM), the curves of Bessel function for different n when M ranging from 0 to 1 are shown in Figure 7a. It can be seen that J_n(πM) is almost negligible when n > 3. Current harmonic spectrum in Figure 7a,b reflects that only when m = 1 and n ∈ {±1, ±3} should the corresponding harmonic be considered.

In conclusion, the main harmonics of i_L2 can be approximated as Equation (13). As can be seen from the equation, the transient process of i_L2H can be considered as product of high-frequency sinusoidal component related to carrier wave and low-frequency sinusoidal component related to modulation wave. Because the current transient of i_L2H is 0 at the start and end of each switching period, it is applicable for both CFM and VFM.

$$i_{L2\mathrm{H}}(t)={\displaystyle \frac{4u_{\mathrm{dc}}}{\pi L_1{L}_2{C}_1{\left(2{\omega }_{\mathrm{c}}\right)}^3}}\sin \left[2\left({\omega }_{\mathrm{c}}t+{\phi }_{\mathrm{c}}\right)\right]\left[J_1\left(\pi M\right)\sin \left({\omega }_{0}t\right)+J_3\left(\pi M\right)\sin \left(3{\omega }_{0}t\right)\right]$$

(13)

The simulated waveform in a fundamental period and the profile described through Equation (13) are revealed in Figure 8. It can be seen that the analytical profile is almost equal to the simulation result, which means using Equation (13) for THD estimation is reasonable.

In each switching period, the low-frequency part of i_L2H can be considered constant, and, thus, Equation (13) can be considered as a sine function and integration of the square of i_L2H and can be expressed as Equation (14), where t_m represents midpoint of the mth switching period and T_c is the corresponding switching period.

$${\int }_{t_m-T_{\mathrm{c}}/2}^{t_m+T_{\mathrm{c}}/2}{i_{L2\mathrm{H}}}^2(t)=\frac{8{u_{\mathrm{dc}}}^2}{{\left(\pi L_1{L}_2{C}_1\right)}^2{\left(2{\omega }_{\mathrm{c}}\right)}^6}{[J_1\left(\pi M\right)\sin \left({\omega }_{0}t\right)+J_3\left(\pi M\right)\sin \left(3{\omega }_{0}t\right)]}^2\frac{2\pi }{{\omega }_{\mathrm{c}}}$$

(14)

For integration of the square of i_L2H in T/4, it can be considered as summary of Equation (14), because the summation terms vary slowly and 2π/ω_c is the time interval of a switching period, the summation can be simplified as integration process as expressed in Equation (15).

$${\displaystyle {\int }_0^{T/4}{i_{L2\mathrm{H}}}^2(t)=\frac{{u_{\mathrm{dc}}}^2}{2{\left(2\pi \right)}^8{\left(L_1{L}_2{C}_1\right)}^2}}{\displaystyle {\int }_0^{T/4}{f_{\mathrm{sw}}}^{-6}\left(t\right){\left[J_1\left(\pi M\right)\sin \left({\omega }_{0}t\right)+J_3\left(\pi M\right)\sin \left(3{\omega }_{0}t\right)\right]}^2\mathrm{d}t}$$

(15)

SF function f_sw is a constant value for CFM and a time-varying function for VFM. With the help of Equation (15), root mean square (RMS) value of harmonic component can be derived and thus current THD can be obtained either.

3. Loss and Thermal Characteristics of MOSFET and Inductor

When deciding the frequency variation range for total loss optimization, the thermal performance of MOSFET and inductor must be analyzed since their heat dissipation capabilities are different. Chances are that the calculated unconstraint loss optimal working condition results in the overheating of one of them.

3.1. Thermal Characteristics of MOSFET

To obtain thermal characteristics of MOSFET, switching and conduction loss of MOSFETs should be analyzed. As can be seen from Figure 6, there are four switching points in each switching period, and the corresponding current can be derived according to Equation (8), as shown in Equation (16), where i_{sw_upper} is current of two switching points in the upper contour of i_L1 and i_{sw_lower} is the current of two switching points in the lower contour.

$$\left\{\begin{array}{l}{i}_{\mathrm{sw}\_\mathrm{upper}}=I_{\mathrm{g}}\sin \left({\omega }_{0}t\right)+{\displaystyle \frac{u_{\mathrm{dc}}{T}_{\mathrm{sw}}}{4L_1}}M\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\\ i_{\mathrm{sw}\_\mathrm{lower}}=I_{\mathrm{g}}\sin \left({\omega }_{0}t\right)-{\displaystyle \frac{u_{\mathrm{dc}}{T}_{\mathrm{sw}}}{4L_1}}M\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\end{array}\right.$$

(16)

At each upper switching point, the current is transferred from MOSFET bodies (S₁, S₄) to body diodes (S₂, S₃) since upper contour of i_L1 > 0, which corresponds to two turn-off losses of MOSFET body and two turn-on losses of body diode. At each lower switching point, the loss characteristic depends on load current direction. Since the fundamental wave starts from 0, there exists a zero-crossing point t_c for lower contour of i_L1. When t < t_c, current is transferred from MOSFET body to body diode at each lower switching point, which corresponds to two turn-off losses of MOSFET body and two turn-on losses of body diode; when t > t_c, the case is the opposite. In conclusion, at each switching point, the generated switching loss can be considered as combination of turn-on loss of a MOSFET body and turn-off loss of a body diode or the opposite case as shown in Figure 9.

Since u_dc is usually stabilized by former stage, switching loss can be considered as a function regarding to load current. The combination of turn-on loss of MOSFET and turn-off loss of body diode (or the opposite case) can be considered linear to load current, as expressed in Equation (17), where q_on is the combination of turn-on loss of MOSFET and turn-off loss of body diode and q_off is the contrary case; ρ_on, q_on0, ρ_off, q_off0 are their separate linearization parameters.

$$\left\{\begin{array}{l}{q}_{\mathrm{on}}={\rho }_{\mathrm{on}}\left|i\right|+q_{\mathrm{on}0}\\ q_{\mathrm{off}}={\rho }_{\mathrm{off}}\left|i\right|+q_{\mathrm{off}0}\end{array}\right.$$

(17)

After loss model linearization, the total switching loss in T/4 can be expressed as Equation (18), where t_r represents midpoint of each switching period, R is switching period number in T/4, and R_c is switching period number before t_r.

$$\begin{array}{l}{q}_{\mathrm{sw}}={\displaystyle \sum _{r=1}^{R}2{\rho }_{\mathrm{off}}\left\{I_{\mathrm{g}}\sin \left({\omega }_0{t}_r\right)+\frac{u_{\mathrm{dc}}{T}_{\mathrm{sw}}}{4L_1}M\sin \left({\omega }_0{t}_r\right)\left[1-M\sin \left({\omega }_0{t}_r\right)\right]\right\}+2q_{\mathrm{off}0}}\\ \hspace{1em}+{\displaystyle \sum _{r=1}^{R_{\mathrm{c}}}2{\rho }_{\mathrm{off}}\left\{-I_{\mathrm{g}}\sin \left({\omega }_0{t}_r\right)+\frac{u_{\mathrm{dc}}{T}_{\mathrm{sw}}}{4L_1}M\sin \left({\omega }_0{t}_r\right)\left[1-M\sin \left({\omega }_0{t}_r\right)\right]\right\}+2q_{\mathrm{off}0}}\\ \hspace{1em}+{\displaystyle \sum _{r=R_{\mathrm{c}}}^{R}2{\rho }_{\mathrm{on}}\left\{I_{\mathrm{g}}\sin \left({\omega }_0{t}_r\right)-\frac{u_{\mathrm{dc}}{T}_{\mathrm{sw}}}{4L_1}M\sin \left({\omega }_0{t}_r\right)\left[1-M\sin \left({\omega }_0{t}_r\right)\right]\right\}+2q_{\mathrm{on}0}}\end{array}$$

(18)

Since summation terms in Equation (18) vary slowly, summation process can be equivalent to an integration process, as shown in Equation (19). For CFM, f_sw is a constant; for VFM, f_sw is a time-varying function.

$$\begin{array}{l}{q}_{\mathrm{sw}}={\displaystyle {\int }_0^{T_0/4}\left[2{\rho }_{\mathrm{off}}{I}_{\mathrm{g}}\sin \left({\omega }_{0}t\right)+2q_{\mathrm{off}0}\right]f_{\mathrm{sw}}\mathrm{d}t}+{\displaystyle \frac{{\rho }_{\mathrm{off}}M{u}_{\mathrm{dc}}}{2L_1}}{\displaystyle {\int }_0^{T_0/4}\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\mathrm{d}t}\\ \hspace{1em}+{\displaystyle {\int }_0^{t_{\mathrm{r}}}\left[-2{\rho }_{\mathrm{off}}{I}_{\mathrm{g}}\sin \left({\omega }_{0}t\right)+2q_{\mathrm{off}0}\right]f_{\mathrm{sw}}\mathrm{d}t}+{\displaystyle \frac{{\rho }_{\mathrm{off}}M{u}_{\mathrm{dc}}}{2L_1}}{\displaystyle {\int }_0^{t_{\mathrm{r}}}\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\mathrm{d}t}\\ \hspace{1em}+{\displaystyle {\int }_{t_{\mathrm{r}}}^{T/4}\left[2{\rho }_{\mathrm{on}}{I}_{\mathrm{g}}\sin \left({\omega }_{0}t\right)+2q_{\mathrm{on}0}\right]f_{\mathrm{sw}}\mathrm{d}t}-{\displaystyle \frac{{\rho }_{\mathrm{on}}M{u}_{\mathrm{dc}}}{2L_1}}{\displaystyle {\int }_{t_{\mathrm{r}}}^{T/4}\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\mathrm{d}t}\end{array}$$

(19)

Simplify Equation (19), then it can be divided into SF-dependent terms and SF-independent terms, as shown in Equation (20), the SF-independent terms cannot be influenced by SF function and thus can be neglected when selecting parameters of f_sw(t) for loss optimization. Switching loss power p_sw = 4 q_sw/T₀.

$$\begin{array}{cc}{q}_{\mathrm{sw}}=& \begin{array}{l}\stackrel{\mathrm{\text{SF-dependent}} \mathrm{terms}}{\overbrace{\begin{array}{l}2q_{\mathrm{off}0}{\displaystyle {\int }_0^{T_0/4}{f}_{\mathrm{sw}}\left(t\right)\mathrm{d}t}+2q_{\mathrm{off}0}{\displaystyle {\int }_0^{t_{\mathrm{r}}}{f}_{\mathrm{sw}}\left(t\right)\mathrm{d}t}+2q_{\mathrm{on}0}{\displaystyle {\int }_{t_{\mathrm{r}}}^{T_0/4}{f}_{\mathrm{sw}}\left(t\right)\mathrm{d}t}\\ \hspace{1em}+2\left({\rho }_{\mathrm{on}}+{\rho }_{\mathrm{off}}\right)I_{\mathrm{g}}{\displaystyle {\int }_{t_{\mathrm{r}}}^{T_0/4}\sin \left({\omega }_{0}t\right)f\left(t\right)\mathrm{d}t}\\ \hspace{1em}-\left({\rho }_{\mathrm{on}}+{\rho }_{\mathrm{off}}\right){\displaystyle \frac{u_{\mathrm{dc}}}{2L_1}}{\displaystyle {\int }_{t_{\mathrm{r}}}^{T_0/4}M\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\mathrm{d}t}\end{array}}}\\ \stackrel{\mathrm{\text{SF-independent}} \mathrm{terms}}{\overbrace{+{\rho }_{\mathrm{off}}{\displaystyle \frac{u_{\mathrm{dc}}}{2L_1}}{\displaystyle {\int }_0^{T_0/4}M\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\mathrm{d}t}+{\rho }_{\mathrm{off}}{\displaystyle \frac{u_{\mathrm{dc}}}{2L_1}}{\displaystyle \frac{4M-M^2\pi }{4{\omega }_0}}}}\end{array}\end{array}$$

(20)

In terms of conduction loss, i_L1 either flows through body diode or through MOSFET. Typically, the on-resistance of body diode is close to that of MOSFET. As a result, conduction loss of a full-bridge can be considered as generated by two on-resistances with i_L1 flowing through. To estimate conduction loss, RMS value of i_L1 should be calculated first. According to Equation (8), current transient can be divided into two identical stages. In each stage, integral of the square of i_L1 can be derived as Equation (21), where t_m is the midpoint of half a switching period.

$${\displaystyle {\int }_{t_m-T_{\mathrm{sw}}/2}^{t_m+T_{\mathrm{sw}}/2}{i_{L1\mathrm{H}}}^2\mathrm{d}t}={I_{\mathrm{g}}}^2{\sin }^2\left({\omega }_{0}t\right){\displaystyle \frac{T_{\mathrm{sw}}}{2}}+{\displaystyle \frac{1}{48}}{\left({\displaystyle \frac{u_{\mathrm{dc}}}{L_1}}\right)}^2{\left\{M\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\right\}}^2{T_{\mathrm{sw}}}^2{\displaystyle \frac{T_{\mathrm{sw}}}{2}}$$

(21)

Because T_sw is the time interval corresponding to each summation term, integral of the square of i_L1 during T₀/4 can be expressed as equivalent integral form, as expressed in Equation (22). With the aid of the equation, loss generated by on-resistances in T₀/4 can be obtained, and conduction loss power p_cond = 4 q_cond/T₀.

$$\left\{\begin{array}{l}{\displaystyle {\int }_0^{T/4}{i_{L1\mathrm{H}}}^2\mathrm{d}t}={\displaystyle {\int }_0^{T/4}{I_{\mathrm{g}}}^2{\sin }^2\left({\omega }_{0}t\right)+\frac{1}{48}{\left(\frac{u_{\mathrm{dc}}}{L_1}\right)}^2{\left\{M\sin \left({\omega }_{0}t\right)\left[1-M\sin \left({\omega }_{0}t\right)\right]\right\}}^2{f_{\mathrm{sw}}}^{-2}\mathrm{d}t}\\ p_{\mathrm{cond}}=2R_{\mathrm{on}}{\displaystyle {\int }_0^{T_0/4}{i_{L1\mathrm{H}}}^2\mathrm{d}t}\end{array}\right.$$

(22)

In terms of MOSFET, its heat source is concentrated in a relatively small area, so its junction temperature is estimated through a simplified thermal network like Figure 10, where p_mos is total thermal power of four MOSFETs, R_θja is thermal resistance from junction to heat dissipation surface, R_θsink is thermal resistance of heat sink, T_j, T_c, and T_a are temperatures of junction, case and ambient, where T_j is the center temperature of semiconductor junction in the MOSFET and T_c is the center temperature of heat dissipation surface of MOSFET.

3.2. Thermal Characteristics of Inductor

Inductor loss can be divided into two parts: core loss and copper loss. In terms of core loss, MSE should be used for higher precision since magnetic flux density is non-sinusoidal. Core loss estimation process using MSE is shown as Equation (23), where p_{core_sw} is specific core power loss of each switching period, C_m, α, β are Steinmetz coefficients, B is magnetic flux density of the core.

$$\left\{\begin{array}{l}{p}_{\mathrm{core}\_\mathrm{sw}}=C_{\mathrm{m}}{f_{\mathrm{eq}}}^{\alpha -1}{\left({\displaystyle \frac{\Delta B}{2}}\right)}^{\beta }\\ f_{\mathrm{eq}}={\displaystyle \frac{2}{{\left(\Delta B\right)}^2{\pi }^2}}{\displaystyle {\int }_0^{T\mathrm{sw}}{\left(\frac{\mathrm{d}B}{\mathrm{d}t}\right)}^2\mathrm{d}t}\\ \Delta B=B_{\max }-B_{\min }\end{array}\right.$$

(23)

Combing with current transient of i_L1, core loss (q_core) in T/4 can be expressed as Equation (24), where N is number of winding turns and S is cross-sectional area of the inductor, V is volume of the inductor.

$$q_{\mathrm{core}}=V{\displaystyle \sum _{r=1}^R{C}_{\mathrm{m}}\frac{2^{2\alpha -2\beta -1}}{{\pi }^{2\alpha -2}}{\left(\frac{u_{\mathrm{dc}}}{NS}\right)}^{\beta }{\left\{M\sin \left({\omega }_0{t}_{\mathrm{r}}\right)\left[1-M\sin \left({\omega }_0{t}_{\mathrm{r}}\right)\right]\right\}}^{-\alpha +\beta +1}{f_{\mathrm{sw}}}^{\alpha -\beta }{T}_{\mathrm{sw}}}$$

(24)

Since T_sw is the time interval of each switching period, Equation (24) can be equivalent to Equation (25), and core loss power p_core = 4 q_core/T₀.

$$q_{\mathrm{core}}=V{\displaystyle {\int }_0^{T/4}{C}_{\mathrm{m}}\frac{2^{2\alpha -2\beta -1}}{{\pi }^{2\alpha -2}}{\left(\frac{u_{\mathrm{dc}}}{NS}\right)}^{\beta }{\left\{M\sin \left({\omega }_0{t}_{\mathrm{r}}\right)\left[1-M\sin \left({\omega }_0{t}_{\mathrm{r}}\right)\right]\right\}}^{-\alpha +\beta +1}{f_{\mathrm{sw}}}^{\alpha -\beta }}\mathrm{d}t$$

(25)

In this paper the inductor adopts a toroidal core and has single-layer winding, thus winding temperature is not critical and it has little impact on core temperature. As a result, when considering core temperature, it can be assumed to be free of windings, and its surface can be regarded as a natural commutation.

For toroidal core, the heat source is the core itself. Considering the heat source is evenly distributed in the core, the tangential derivative of temperature is zero in the cylindrical coordinate system; thus, the boundary condition can be described as Figure 11. Because the size of the core is relatively small and thermal conductivity of most core materials is close to natural convective heat transfer coefficient, the temperature of anywhere inside the core can be considered as the same. Temperature rise in inductor core mainly results from natural commutation.

Temperature of core is significantly affected by the surrounding environment and the winds and hard to predict. In this paper thermal resistance of convective and radiative heat transfer is obtained through simulation, and the thermal model of inductor core can be simplified as Figure 12. p_core is core loss power and R_θsur is the equivalent thermal resistance between surface of inductor core and the environment.

In terms of copper loss, using Dowell equation to estimate AC resistance is valid only when skin effect is not severe and its accuracy cannot be guaranteed. In this paper, the AC resistance of copper at frequencies ranging from 10 kHz to 500 kHz is derived through finite element simulation. Consider F_R as ratio of R_AC to R_DC, the simulated F_R of a 50-turn copper wound on a 39.9 mm OD magnetic core is shown in Figure 13.

When estimating copper loss, only second-order sideband harmonics need to be considered, and the total copper loss can be expressed as Equation (26), where I_f is harmonic current with frequency f.

$$p_{\mathrm{cu}}={\left({\displaystyle \frac{I_{\mathrm{g}}}{\sqrt{2}}}\right)}^2{R}_{\mathrm{DC}}+{{\displaystyle \sum \left(\frac{I_f}{\sqrt{2}}\right)}}^2{F}_{\mathrm{R}}\left(f\right)R_{\mathrm{DC}}$$

(26)

When using VFM, I_f in second-order sideband can be assumed as the same, and Equation (26) can be equivalent to Equation (27), where I_HRMS is RMS of total harmonics of i_L1. F_R can be approximated as a polynomial by fitting.

$$p_{\mathrm{cu}}={\left({\displaystyle \frac{I_{\mathrm{g}}}{\sqrt{2}}}\right)}^2{R}_{\mathrm{DC}}+{\left({\displaystyle \frac{I_f}{\sqrt{2}}}\right)}^2{R}_{\mathrm{DC}}{\displaystyle \sum F_{\mathrm{R}}\left(f\right)}\approx \left[{\left({\displaystyle \frac{I_{\mathrm{g}}}{\sqrt{2}}}\right)}^2+{\left(I_{\mathrm{HRMS}}\right)}^2{\displaystyle \frac{{\displaystyle {\int }_{f_{\min }}^{f_{\max }}{F}_{\mathrm{R}}\left(f\right)\mathrm{d}f}}{f_{\max }-f_{\min }}}\right]R_{\mathrm{DC}}$$

(27)

4. Loss Optimization with Temperature Constraints

In this section, the parameter determination method will be introduced. The parameters of SF function should meet the requirements that total loss of the inverter is optimized, while THD and working temperature of key components are lower than the preset limit; furthermore, the distance between minimum SF and resonant frequency is restricted to a safe value.

Combined with the current transient model and thermal model, the constrained optimization problem can be expressed as Equation (28) and the terms independent of SF function in total loss estimation are neglected.

$$\left\{\begin{array}{l}\min : p_{\mathrm{sw}}+p_{\mathrm{cond}}+p_{\mathrm{core}}+p_{\mathrm{copper}}\\ \mathrm{s}.\mathrm{t}. : \left(1\right)\left.\begin{array}{l}{h}_1=I_{\mathrm{g}}\sin \left({\omega }_0{t}_{\mathrm{r}}\right)-{\displaystyle \frac{u_{\mathrm{dc}}{T}_{\mathrm{sw}}}{4L_1}}\mathrm{g}\left(t_{\mathrm{r}}\right)=0\\ g(t) = M\sin \left({\omega }_0{t}_{\mathrm{r}}\right)\left[1-M\sin \left({\omega }_0{t}_{\mathrm{r}}\right)\right]\end{array}\right\}\hspace{1em}\hspace{1em} \mathrm{\text{zero-crossing}} \mathrm{point}\\ \hspace{1em}\hspace{1em} \left(2\right) g_1=\sqrt{{\displaystyle \frac{4{\displaystyle {\int }_0^{T_0/4}{i_{L1\mathrm{H}}}^2\mathrm{d}t}}{T_0}}}-\mathrm{THD}{\displaystyle \frac{I_g}{\sqrt{2}}}\le 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{THD} \mathrm{constraint} \\ \hspace{1em}\hspace{1em} \left(3\right)\left.\begin{array}{l}{g}_2=\left(p_{\mathrm{sw}}+p_{\mathrm{cond}}\right)\left(R_{\theta \mathrm{jc}}+R_{\theta \sin \mathrm{k}}\right)-\Delta T_{\mathrm{mos}\_\max }\le 0\\ g_3=p_{\mathrm{core}}{R}_{\theta \mathrm{sur}}-\Delta T_{\mathrm{core}\_\max }\le 0\end{array}\right\}\begin{array}{c}\mathrm{Temperature} \mathrm{rise}\\ \mathrm{constraint}\end{array}\\ \hspace{1em}\hspace{1em} \left(4\right) g_4=-f_{\min }+\gamma f_{\mathrm{res}}\le 0\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Resonant} \mathrm{frequency}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{constraint}\end{array}\end{array}\right.$$

(28)

Most integral, Equation (28) can be directly solved since VFM is an exponential function, but analytical solution of core loss term cannot be derived due to non-integer power. Because g (t) is a function that varies in a small range, non-integer power term can be approximated as Equation (29).

$$g^{-\alpha +\beta +1}\left(t\right)\approx \tau g^2(t)$$

(29)

The comparison between the original function and the fitted curve is shown in Figure 14. It can be seen that using a quadratic function to fit g^−α+β+1(t) yields very small error.

Because the optimization objective is the total loss of the inverter, it is impossible for the optimal point to lie on the boundaries of both inequalities in constraint (3), so the optimization problem can be separated into two problems that each of them is subject to only one temperature constraint. To facilitate the solution of the optimization problem, slack variables s_i are introduced to inequality constraint, as shown in Equation (30).

$$\left\{\begin{array}{l}\min : f_{\mathrm{loss}}=p_{\mathrm{sw}}+p_{\mathrm{cond}}+p_{\mathrm{core}}+p_{\mathrm{copper}}\\ \mathrm{s}.\mathrm{t}. : h_1=0 \\ \hspace{1em}\hspace{1em} h_{i+1}=g_i+{s_i}^2\le 0 i=1,2,3,4\end{array}\right.$$

(30)

Having introduced slack variables, all inequality constraints are converted into equality constraints, and the optimization problem can be solved through the augmented Lagrangian (AL) method. Because t_r is close to zero when using unipolar modulation, it is set to a series of fixed values to simplify the solution process of the optimization problem. The AL function is expressed as Equation (31), where λ_i is the Lagrange multiplier for constraint h_i. μ is the penalty factor.

$$L_{\mathrm{A}}\left(A,\alpha ,{\lambda }_i,{\mu }_i\right)=f_{\mathrm{loss}}+{\displaystyle \sum _{i=1}^5{\lambda }_i{h}_i\left(A,\alpha \right)}+\mu {\displaystyle \sum _{i=1}^5{h_i}^2\left(A,\alpha \right)}$$

(31)

Solution process of the optimization problem using AL method can be described as Figure 15. At the beginning, t_r, λ_i and μ are initialized, and then optimal point (A^(k+1), α^(k+1), s_i^(k+1)) for contemporary L_A can be derived through Quasi-Newton method, and then Lagrange multiplier λ_i^(k+1) required for next period can be calculated. Calculate constraint violation degree, and μ should be multiplied by ρ if the degree exceeds the limit τ. Repeat the loop until all h_i are sufficiently close to zero, and then the optimal point for t_r^(j) will be obtained. When optimal point for every t_r^(j) is derived, the global optimal point (A_opt, α_opt) will be acquired. For grid-tied inverter, its DC and AC voltage are stabilized by former and post stage, and its output current might be variable, which can be settled by interpolation of optimal points under different current levels. Because the proposed algorithm requires using iteration to solve several non-constraint optimization problems, it is hard to be executed in a digital controller and thus the optimal points under different load current conditions can be calculated in the computer and the results can be saved in a table in the controller. The controller only needs to look up the table and change the count number of PWM module to vary the switching frequency, thus imposing little impact on the ordinary control of the inverter.

Having calculated the optimal points, SF function required for the inverter can be obtained. The whole control structure of VFM method can be described as Figure 16, where k_pow determines inverter’s output current and k_damp is active damping coefficient. The closed-loop control of the inverter is realized through simple PI controller and active damping. The grid voltage is sampled to obtain phase-locked loo to determine the phase of output current. When hardware parameters of the inverter are obtained, the optimal parameters of SF function under different load current levels can be derived and saved in the table, and the optimal value A_opt and α_opt for I_g can be obtained through interpolation of data in the table. Combining the optimal value and grid phase, SF for current switching period can be derived, and then count number of PWM module in the controller will be determined. Combing the modulation signal generated by the closed-loop controller and carrier generated by PWM module, the driving signal required for the inverter will be created.

5. Simulation Verification

In this section, simulation verification is conducted to prove the effectiveness of the proposed method. A comprehensive comparison between the proposed VFM and CFM in terms of EMI, losses and temperature rise will be carried out.

The MOSFET adopted in the simulation is C3M0120065D manufactured by Wolfspeed. When u_dc is set to 200 V, the curves of turn-on loss and turn-off loss versus current and their fitting curves are shown in Figure 17. The fitting curves are obtained through least squares method. Although the fitting error is relatively large when load current is small. However, since the exponential SF function performs uniformly distributed frequencies, the summary of these losses using fitting curves is close to that using the simulated curves.

According to the fitting curves, ρ_on, q_on0, ρ_off, q_off0 can be derived.

Thermal simulations of MOSFETs under different thermal power levels are conducted in COMSOL Multiphysics 6.2 and the results are shown in Figure 18. According to the simulated junction and case temperatures, R_θjc and R_θcase can be estimated.

The inductor core adopted in the paper is A77083 manufactured by Magnetics. Thermal simulations of inductor core under different power levels are shown in Figure 19. According to the simulation results, R_θsur can be estimated.

Parameters of the inverter are shown in

Table 1. Parameters of the inverter.

Parameter	Symbol	Value
Inverter parameters	udc	200 V
	ug	110 V (RMS)
	L1	400 μH
	L2	400 μH
	C1	0.37 μF
Switching loss fitting coefficients	ρon	3.989 μJ/A
	qon0	2.582 μJ
	ρoff	0.727 μJ/A
	qoff0	−0.534 μJ
Steinmetz coefficients	Cm	193 mW/cm3
	α	1.29
	β	2.01
On-resistance	Ron	0.124 Ω
DC resistance of copper	Rdc	0.0184 Ω
Thermal resistances	Rθjc + Rθcase	2.6 °C/W
	Rθsur	12 °C/W

.

When output current is set to 10 A and maximum temperature rise in MOSFET and core is set to 50 °C; the operating points derived from iterative process are shown in Figure 20. The calculated optimal point (A = 57504, ψ = −175) is located at minimum loss power point within the constraint boundary.

Switching and conduction loss are simulated in PLECS, and i_L1 is exported to COMSOL for inductor loss and thermal simulation. Waveforms of i_L1, average p_mos and p_inductor under different SF ranges are shown in Figure 21. Because p_mos is more influential than p_inductor, it receives more attenuation by the proposed algorithm to find a balance between loss of MOSFET and inductor.

Total losses under different SF ranges are shown in Figure 22. As can be seen from bar chart, the total loss of the inverter reaches the minimum when the calculated optimal frequency range is adopted. According to copper loss simulation results, it can be concluded that SF has a negligible effect on the copper loss. Although an increased SF will raise the AC resistance, the corresponding current will be attenuated by inductor impedance because of the increased frequency. Compared with frequencies within the optimal frequency range, the total loss power is reduced by up to 3.8%. Compared with adjacent frequencies outside the optimal range, the total loss power is reduced by up to 9.2%.

Temperature distribution of MOSFET and inductor core are shown in Figure 23. Temperature rise in core is higher than MOSFET because of the absence of heat sink, but both of them are less than 50 °C.

EMI simulation is conducted in CST Studio Suite. Parameters of PCB are extracted by ODB++ file, parameters of filters are obtained from data sheet provided by manufacturer, and MOSFET’s behavior is simulated by spice model provided by manufacturer. Structure and parameters of the circuit are provided in Figure 24.

A line impedance stabilization network (LISN) is inserted between the inverter and the power grid to obtain noise signal, impedance between heat dissipation surface of MOSFET and grounded heat sink is approximated as a capacitor drawn from the drain.

Conducted emissions under different SFs when using CFM and VFM are shown in Figure 25. Compared with CFM, conducted emission under optimal SF range is lower than that under 40 kHz, 60 kHz or 80 kHz; compared with VFM, conducted emission under optimal range is lower than that under 40–50 kHz, 60–70 kHz, 80–90 kHz. Conducted emission under the optimal range is lower than the ranges within or above the calculated optimal range. When using CFM, only when SF is lower than minimum value of the optimal SF function will the conducted emission perform better. Compared with frequencies within or near the optimal boundary, conducted emission under optimal frequency range is reduced by up to 4 dBμV.

6. Conclusions

In this paper, a new VFM parameter design method for total loss optimization with THD and thermal constraints is proposed. Current dynamics of inverter side and grid side when using unipolar modulation are obtained through MFD and characteristics of Bessel function; with the help of these, the loss model of MOSFETs and inductors are obtained for total loss and temperature rise estimation. Total loss optimization is realized within the constraint region imposed by temperature rise and THD through the AL method. When using the calculated optimal SF range, minimum total loss is realized while temperature rise in inductor and MOSFET are restricted to certain values. The total loss power is reduced by up to 3.8% compared with frequencies within the optimal range and 9.2% compared with frequencies outside the optimal range. Meanwhile, conducted emission of the inverter when using VFM is lower than using any constant frequency within or above the calculated optimal range. Compared with frequencies within or near the optimal boundary, conducted emission under the optimal frequency range is reduced by up to 4 dBμV.

Although both grid-connected current THD and temperature rise in essential components like MOSFET and inductor are taken into account in the constraints, the AL method used for the constrained optimization problem might converge once all equality constraints are satisfied. However, chances are that the optimal value lies on a contour line of loss where every point on the line generates the same loss, and the point that results in larger frequency range performs better in terms of conducted emission. How can these better operating points be found should be addressed in future research. The accuracy of loss and harmonic models derived in this paper rely on the assumption that the control system is ideal and low-frequency harmonics are negligible. Future work will be carried out on the combination of a low THD control strategy and the proposed VSFM technique to improve the practicality of VSFM method and make it suitable for wider working conditions.