Efficiency Optimization in Parallel LLC Resonant Inverters with Current-Controlled Variable-Inductor and Phase Shift for Induction Heating

Abstract

This paper presents a comprehensive analysis of a novel control approach to improve the efficiency of parallel LLC resonant inverters using a combination of a current controlled variable inductor (VI) and phase shift (PS). The proposed control aims to reduce the Root Mean Square (RMS) current, thereby reducing conduction and switching losses, and improving power transfer efficiency, while considering zero-voltage-switching (ZVS) operation, output power variations and load changes. Furthermore, a design methodology for the variable inductor is proposed, and design considerations relevant to this application are discussed. The effectiveness of the proposed approach is evaluated through mathematical modeling and experimental results. The mathematical modeling results show that the proposed approach, utilizing SiC MOSFETs, maintains a maximum efficiency of 98.5% for a 20 kW inverter over a wider range of output power, which is a significant improvement over existing approaches. The experimental results also confirm the effectiveness of the proposed control in improving the efficiency of parallel LLC resonant inverters.

1. Introduction

Induction heating systems, known as resonance transformation (R.T.), are extensively utilized in various industrial processes such as brazing, hardening, annealing, and other heat-treatment applications, due to their high efficiency, precise control, and rapid heating capability [1]. These systems usually employ resonant inverters, such as series resonant inverters or parallel resonant inverters, to convert DC energy into AC energy [2,3].

The traditional method of controlling the output current or power by varying the switching frequency has several disadvantages, such as a wide noise spectrum, complex filtering, and inefficient use of magnetic components [4,5,6]. Fixed-frequency or narrow-frequency-range control techniques, such as pulse density modulation (PDM) [7,8], PS [9,10], or the combination of the PS with frequency modulation (FM-PS) control [2,11,12], can address some of these issues. However, they cannot guarantee the soft-switching operation of the resonant converter for large load or output power variations. Therefore, the switching frequency must be increased to achieve ZVS operation, which results in poor performance across the entire power range [4,13]. Even with complex state-plane analysis, the use of a fixed frequency with phase shift in LLC resonant converters requires limitations on the output current in order not to lose smooth switching [14]. Other converter models use saturated or quasi-saturated inductors working in non-linear regions, which results in improved power density, but these techniques do not solve the problem of working with fixed frequencies or extending the range of ZVS operation [15,16]. To overcome these limitations, a new control technique has been proposed which combines a phase shift and variable inductor control (VI-PS).

In this article, the combined use of both techniques in a parallel LLC resonant inverter is thoroughly investigated to achieve enhanced performance. The combined effect of a phase shift and variable inductor control, as the series inductor, across the entire output power range is to reduce the output RMS current, while maintaining the efficiency of the inverter and achieving good performance under load changes, also ensuring ZVS operation. The impact of these control techniques on both conduction and switching losses are analyzed. The experimental results show that the combined use of PS and VI control can provide improved performance compared to traditional narrow or fixed-frequency control techniques.

The paper is organized as follows. Section 2 of the paper discusses the properties of the parallel LLC resonant circuit and provide analysis of the converter operation principle and soft switching commutation. An analysis of a current-control variable inductor and its application to an LLC resonant inverter is provided in Section 3. A design procedure is proposed in Section 4, for an LLC resonant inverter for induction heating, considering the variable series inductor, phase shift angle, and ZVS conditions. In addition, a power loss analysis and the control principles are also provided. In Section 5, the experimental results using an induction heating module are presented and compared with fixed-frequency and narrow-frequency-range phase shift.

2. Parallel LLC Resonant Converter Topology

The parallel LLC resonant circuit, an R.T. circuit consisting of two inductors and a capacitor, along with the equivalent load resistance, forms a series-parallel hybrid topology, as shown Figure 1. The circuit is configured with the heating inductor in parallel with the capacitor to compensate for the reactive energy, while the series inductor is connected in series with them [17].

This R.T. configuration allows the parallel LLC circuit to exhibit advantageous characteristics of both series and parallel resonant circuits [18]. When connected to a voltage-fed inverter, it simplifies the input section of the converter as in a series resonant inverter [19,20,21], where the input inductance in the resonant circuit acts as a current source. Additionally, through the output inductor near parallel resonance, the amplified current functions like a parallel resonant inverter [20,22,23], eliminating the need for current step-up transformers.

As the circuit is composed of three reactive elements, two resonant frequencies exist between them. The parallel resonant angular frequency defined between the parallel inductor and parallel capacitor is expressed as

$${\omega }_{op}=\frac{1}{\sqrt{L_p{C}_p}}$$

(1)

while the series resonant angular frequency between the series inductor and the parallel capacitor is

$${\omega }_o=\frac{1}{\sqrt{\left(\frac{L_p{L}_s}{L_p+L_s}\right)C_p}}$$

(2)

and their respective quality factors are

$$Q_p=\frac{L_p{\omega }_{op}}{R}$$

(3)

and

$$Q=\frac{L_p{\omega }_o}{R} .$$

(4)

The ratio between the inductances is given by

$$\beta =\frac{L_s}{L_p} .$$

(5)

The expression for the input impedance as a function of the angular frequency is as follows [24]:

$$Z\left(\omega \right)=L_p{\omega }_o\left(\beta +1\right)\frac{\frac{1}{Q}\left(\frac{1}{\beta +1}-{\left(\frac{\omega }{{\omega }_o}\right)}^2\right)+j\frac{\omega }{{\omega }_o}\left(1-{\left(\frac{\omega }{{\omega }_o}\right)}^2\right)}{\left(1-{\left(\frac{\omega }{{\omega }_{op}}\right)}^2\right)+j\frac{\omega }{Q_p{\omega }_{op}}} .$$

(6)

By substituting (1) and (2) in (6), the following is obtained:

$$Z\left({\omega }_o\right)=\frac{L_p{\omega }_o{\beta }^2}{Q-j\left(\beta +1\right)} .$$

(7)

and from this argument, the phase between voltage and current in resonance is determined as

$$\alpha =\arg \left(Z\left({\omega }_o\right)\right)=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{\beta +1}{Q}\right) .$$

(8)

The magnitude and phase of the input impedance of the circuit are shown as a function of angular frequency for a high Q, greater than 6, which is typical for LLC circuits [18,25], in Figure 2. In this figure, the effect of the two resonance frequencies of the circuit can be observed. The parallel resonance frequency is situated at the peak of impedance magnitude, whereas the series resonance frequency is located at the point of minimum impedance. Regarding the phase behavior concerning these points, it is observed that when operating above the series resonance frequency and below the parallel resonance frequency, the phase is positive. However, between these two resonance frequencies, the phase is negative. As the circuit must operate with ZVS, the operating point in the circuit should have a positive phase, and it will be close above the series resonance frequency. This is because the circuit impedance is minimized, thereby optimizing the reactive elements of the LLC resonant circuit when operating with a voltage-fed inverter.

3. Analysis and Application of Current-Controlled Variable Inductor in Parallel LLC Resonant Circuits

The design of a current-controlled variable inductor has its origins in prior studies [26]. Its operating principle involves saturating the outer part of the inductor core with direct current to vary the effective permeability. As shown in Figure 3, this inductor consists of two overlapped ferrite cores in the shape of an “E”. The central branch has an air gap and is wound to form the main inductor (L_ac). On the other hand, the two side arms form the saturation control inductor (L_bias), which is wound in a series with opposite polarity to decouple L_ac from L_bias, nullifying the influence of magnetic flux from the L_ac winding and preventing induced voltages in L_bias.

For the design of this inductance, using the formula for inductance based on its geometry for an inductor with a ferrite core, the value of the inductance of the central branch is expressed as follows:

$$L_{int}=\frac{{\mu }_0{\mu }_i{A_{int}{(n}_{int})}^2}{l_{int}}.$$

(9)

where μ_i is the initial permeability of the core, μ₀ is the permeability of air, $A_{int}$ is the effective cross-sectional area of the central branch, $n_{int}$ is the number of turns in the L_ac winding, and lint is the length of the central branch.

The value of the inductance due to the air gap is expressed by

$$L_{gap}=\frac{{\mu }_0{A_{int}{(n}_{int})}^2}{l_{gap}}$$

(10)

where $l_{gap}$ is the length of the air gap.

The leakage inductance represents the magnetic flux that closes through the air and not through the central body of the ferrite,

$$L_{leakage}=\frac{{\mu }_0{16A_{int}{(n}_{int})}^2}{l_{int}}$$

(11)

and the inductance of the two outer branches is given by

$$L_{ext}=\frac{2{\mu }_0\tilde{\mu }{A_{ext}{(n}_{int})}^2}{l_{ext}}$$

(12)

where $\tilde{\mu }$ is the relative permeability without being subjected to DC bias.

Therefore, the following expression is used to calculate the maximum value of L_ac:

$$\frac{1}{L_{{ac}_{max}}}=\left(\frac{1}{L_{ext}+L_{leakage}}+\frac{1}{L_{int}}+\frac{1}{L_{gap}}\right).$$

(13)

Because $\tilde{\mu }$ decreases when exposed to continuous magnetic field intensity (H_bias), to which the L_bias winding is subjected, the relationship between $L_{{ac}_{max}}$ and $L_{{ac}_{min}}$, with the second being the minimum value of L_ac for the minimum $\tilde{\mu }$ due to the maximum H_bias, can be obtained using the following expression:

$$\frac{L_{{ac}_{max}}}{L_{{ac}_{min}}}=\left(\frac{\frac{2{\mu }_e\frac{l_{int}}{l_{ext}+l_{int}}}{\tilde{\mu }+32}+\frac{{\mu }_e\frac{l_{int}}{l_{ext}+l_{int}}}{{\mu }_i}+1}{\frac{2{\mu }_e\frac{l_{int}}{l_{ext}+l_{int}}}{{\mu }_i+32}+\frac{{\mu }_e\frac{l_{int}}{l_{ext}+l_{int}}}{{\mu }_i}+1}\right)$$

(14)

where μ_e is the effective relative permeability of the core with a gap, from the elementary equation of an inductor:

$$L=\frac{{\mu }_0{\mu }_e{A_{e}n}^2}{l}.$$

(15)

The effect of using this variable inductance in the L_s of the LLC circuit can be observed in Figure 4. If the operating point of the circuit is set near and above the series resonance frequency at the minimum impedance value, it can be observed how both the magnitude and phase of the impedance increase as the value of the series inductance is increased. This effect leads to two substantial advantages.

The first one is that for a fixed frequency, the inverter highest output current and power will occur at the point of minimum series inductance, where both the magnitude and phase of the impedance are minimized. However, when maximum output power is not required, the value of the series inductance can be increased to raise the magnitude of impedance, thereby reducing the output current.

The second advantage is that by varying both magnitude and phase, a better response to load changes is achieved, enabling the maintenance of ZVS even when working very close to the series resonance frequency. Previous studies, when operating with a fixed frequency near the series resonance frequency [4,18,24,27], showed poor inverter performance in response to load changes, leading to the loss of ZVS. In another previous study [28], a good response to load changes was achieved when working near the parallel resonance frequency because the input impedance was higher. However, this resulted in poorer optimization of reactive elements.

Therefore, working with a variable inductance adds a degree of freedom to the circuit, which, when combined with PS, allows for the optimization of the output current of the inverter.

These observations will be demonstrated both theoretically and experimentally later in the article.

Given that a ferrite core will be used for the high frequency LLC series inductance, and considering that induction heating applications operate with high input and output voltages [18], it is necessary to obtain the voltage across the series inductance ($V_{L_s})$ in order to calculate the magnetic flux density and avoid operating near the saturation point of the L_ac core.

From the circuit analysis, the following equation is obtained:

$$V_{L_s}\left(\omega \right)={V_d\left(\omega \right)-V}_o\left(\omega \right)$$

(16)

here, V_d(ω) and V_o(ω) represent the input and output voltages of the LLC circuit, respectively, with V_d(ω) lying within the range of

$$V_d\left(\omega \right)=\left\{\begin{array}{c}{V}_d, 0\le \omega <\pi \\ {-V}_d, \pi \le \omega <2\pi .\end{array}\right.$$

(17)

The voltage gain H_v(ω) is given by

$$H_v\left(\omega \right)=\frac{V_o\left(\omega \right)}{V_d\left(\omega \right)}=\frac{Z\left(\omega \right)-j{L}_s\omega }{Z\left(\omega \right)}$$

(18)

and the phase difference between the voltages V_d(ω) and V_o(ω) is obtained from the argument

$$\varphi =\arg \left(H_v\left(\omega \right)\right).$$

(19)

Hence, through the integration of (16) and the subsequent solution of the equation, the expression for the voltage across the series inductor is formulated as follows:

$$V_{L_s}\left(\omega \right)=\left\{\begin{array}{c}{V}_d\left(\frac{\omega }{2}+ncos\left(\omega -\varphi \right)\right), 0\le \omega <\pi \\ {-V}_d\left(-\frac{\omega }{2}-ncos\left(\omega -\varphi \right)\right), \pi \le \omega <2\pi \end{array}\right.$$

(20)

where n represents the voltage ratio

$$n=\frac{V_p}{V_d}$$

(21)

for sufficiently large values of the quality factor Q_p, the value of V_o(ω) is sinusoidal of the peak output voltage value V_p [24],

$$V_o\left(\omega \right)=V_{p}sin\left(\omega \right).$$

(22)

Ergo, the maximum value of the series inductor voltage is expressed as

$$V_{L_s}=V_d\left(\frac{\pi }{2}+\frac{V_p}{V_d}cos\left(\pi -\varphi \right)\right).$$

(23)

And the expression of the magnetic flux can be expressed as

$$B_{pk}=\frac{\sqrt{3}{V}_{L_s}}{{8n_{int}A}_{e}f} .$$

(24)

where

A_e is the effective area (m²);

f is the applied frequency (Hz).

As mentioned earlier, the operating point where the output power is maximized is when the value of the series inductance is minimized, thus saturating the outer branches of the inductor. At this point, the inductor voltage is at its highest, and therefore, the maximum flux density that the core can withstand is obtained under this assumption.

Considering the relationship B = µH and taking into account that the relative permeability of the core will change with exposure to H_bias to which the core will be subjected, the relationship between the maximum magnetic flux and the relative permeability for a constant magnetic field (H) is obtained [29].

4. LLC Resonant Inverter Design

4.1. Design Procedure

In the design procedures of the LLC inverter, the inductance value of the heating inductor (L_p) is typically provided from the geometric design parameters [30,31]. And the required power (P_o), frequency, and quality factor is obtained from the heating time, penetration depth into the workpiece, and thermal treatment [32,33]. With these, the capacitance of the capacitor is adjusted to available values, considering its maximum voltage V_p, using

$$C_p=\frac{2Q_p{P}_o}{\omega {V_p}^2}.$$

(25)

With the DC input voltage (V_d) of the inverter, the minimum value of the series inductance is calculated as

$$L_{S_{min}}=L_{{ac}_{min}}=\frac{2{V_d}^2}{\pi P_o\omega }.$$

(26)

After obtaining the series inductor value, the switching frequency ${(\omega }_{sw})$ is adjusted to operate with a small phase angle by

$$\alpha =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{{2Q_p\omega }_{sw}}{{\omega }_o}-\sqrt{4{Q_p}^2}\right)$$

(27)

then, the maximum voltage across the capacitor is recalculated using (25) and also the minimum series inductance (26).

The output current is obtained from

$$Irms=\frac{V_d\sqrt{8+{\left(\pi n\right)}^2+2\pi n\left(2\mathrm{c}\mathrm{o}\mathrm{s}\left(\gamma \right)\right)}}{{\pi L}_s{\omega }_{sw}}$$

(28)

where the voltage phase γ represents the inverter commutation phase relative to the phase of the output voltage [24],

$$\gamma =\pi -arc\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{1}{n}\right).$$

(29)

At this point a ferritic material is selected and the most restrictive value of the flux density is obtained. To do this, the core-loss curves at the operating frequency are compared with the obtained value of B = µH using the maximum permeability variation obtained from the relative permeability variation curves under DC bias.

Applying the effective area relationship for maximum flux density (24) to the winding area yields the expression for the area product as follows:

$$A_p=A_e{A}_w=\frac{\sqrt{3}{V}_{L_s}Irms}{{8B}_{pk}{f}_{sw}JK}$$

(30)

where

A_e is the effective area of the core given by the manufacturer;

J represents current density (A/m²), and in this case, when working with high-output currents from the inverter and aiming to minimize losses due to current carried in L_bias, only L_ac is considered for the design since the L_bias winding will be negligible;

K is the fill factor, which will be assumed to have a value between 0.5 and 0.9.

Using the obtained value for the area product, a core size is selected to maintain a higher ratio, and the number of turns for L_ac is calculated through

$$n_{int}=\frac{\sqrt{3}{V}_{L_s}}{{8A}_e{f}_{sw}{B}_{pk}} .$$

(31)

With the number of turns and the characteristics of the selected core, using the equation (13), $L_{{ac}_{max}}$ is calculated, and with the previous value of $L_{{ac}_{min}}$, the relationship between inductances is determined to obtain the necessary $\tilde{\mu }$ from Equation (14). With this value and the curve on the datasheet regarding the relative permeability subjected to DC bias, the value of H_bias is determined to achieve the value of $L_{{ac}_{min}}$. If the $L_{{ac}_{min}}$ value is not reached, iteration with a different ferritic material is performed. In the event of obtaining a value within the range, optimization is carried out to reduce the maximum DC current value (I_bias) of L_bias through the relationship given by

$$I_{bias}{2n}_{ext}=H_{bias}{2l}_{ext}$$

(32)

where

n_ext represents the number of turns of an external branch;

l_ext is the length of the external branch from center to center of the core (m).

Table 1. Initial requirements and results of the design.

Magnitude	Symb.	Eq.	Value	Unit
Output power	Po	(36)	20	kW
Quality factor	Qp	(3)	10
Switching frequency	fsw	(27)	157	kHz
DC input voltage	Vd	(17)	500	V
Parallel inductor	Lp	(1)	2	µH
Parallel capacitor	Cp	(25)	0.66	µF
Minimun series inductor	$\begin{array}{c}{L}_{s_{min}}\\ L_{{ac}_{min}}\end{array}$	(26)	8	µH
Maximun series inductor	$\begin{array}{c}{L}_{s_{max}}\\ L_{{ac}_{max}}\end{array}$	(13)	14.5	µH

presents the initial requirements for an annealing induction heating application [32], and also the results of applying the design procedure discussed earlier, where a ferrite core of N27 material with reference E 70/33/32 from TDK Electronics was ultimately selected.

The PS control technique involves shifting the switching phases of the two diagonals of the inverter with a phase angle (φ) ranging from 0 to π. Consequently, this modifies the voltage applied to the LLC resonant circuit, thereby affecting the output current and power of the inverter. Hence, by introducing this factor, the relationship between the input and output voltages of the circuit is now expressed as

$$n\left(\phi \right)=n cos\left(\frac{\phi }{2}\right)$$

(33)

and consequently, the phase between the output voltage of the inverter and the output voltage of the resonant circuit is obtained from

$$\gamma \left(\phi \right)=\left|\begin{array}{ccc}\pi -arc\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{1}{n\left(\phi \right)}\right)& \mathrm{i}\mathrm{f}& n\left(\phi \right)>1\\ arc\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\phi -\pi }{2n\left(\phi \right)}\right)-\phi & \mathrm{i}\mathrm{f}& n\left(\phi \right)\le 1\end{array}\right.$$

(34)

where it has been considered that γ is dependent on $\phi$ to maintain soft switching [24] through the following inequality:

$$\gamma \left(\phi \right)\ge \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\phi -\pi }{2n\left(\phi \right)}\right)-\phi .$$

(35)

For controlling the output power of the inverter, it is considered that by combining the phase shift angle through PS control and the ability to vary the series inductance of the resonant circuit, power can be optimized by adjusting both the voltage applied to the resonant circuit and its impedance.

The equation for output power is expressed as a function of these two variables as follows:

$$P_o\left(\phi ,L_s\right)=\frac{{V_d}^{2}n\left(\phi \right)\left(\mathrm{s}\mathrm{i}\mathrm{n}\left(\gamma \left(\phi \right)+\phi \right)+\mathrm{s}\mathrm{i}\mathrm{n}\left(\gamma \left(\phi \right)\right)\right)}{L_s{\omega }_{sw}}$$

(36)

and the same operation is performed with the equation for the RMS value of the output current; thus,

$$\begin{array}{l}{I}_{RMS}\left(\phi ,L_s\right)=\frac{V_d}{{\pi L}_s{\omega }_{sw}}\\ \sqrt{4\cos \left(\phi \right)+{\left(\pi n\left(\phi \right)\right)}^2+2\pi n\left(\phi \right)\left(\mathrm{c}\mathrm{o}\mathrm{s}\left(\gamma \left(\phi \right)+\phi \right)+\mathrm{c}\mathrm{o}\mathrm{s}\left(\gamma \left(\phi \right)\right)\right)+4}.\end{array}$$

(37)

A system of two equations and two variables is obtained. This system is used to find the optimal operating point for the inverter, ensuring that the output current remains minimized across the entire power range, thus reducing conduction, and switching losses in the transistors.

4.2. Losses Analysis

To verify the advantages of this optimization, the losses of the transistors of the inverter are mathematically analyzed to obtain the efficiency across the entire operating range [19], considering the right commutation to improve the power distribution of the transistors [34,35]. The losses due to the conduction current in each channel of the transistors are given by

$$P_{cd}\left(\phi ,L_s\right)={\left(\frac{Irms\left(\phi ,L_s\right)}{\sqrt{2}}\right)}^2{R}_{{DS}_{on}}$$

(38)

where $R_{{DS}_{on}}$ is the ON-state MOSFET channel resistance.

For switching energy losses, only the turn-OFF switching is considered, as the turn-ON switching will be negligible when operating in ZVS throughout the operating range. To calculate it, the polynomial equation of the turn-OFF switching loss curves from the manufacturer is obtained for the drain voltage equal to V_d and the gate resistance used. It is expressed as follows:

$$E_{off}\left(\phi ,L_s\right)=\mathrm{a}{I_c\left(\phi ,L_s\right)}^2+b{I}_c\left(\phi ,L_s\right)+c$$

(39)

where I_c is the switching current, and for the upper transistors, Q1 and Q4, it is derived from the following function:

$$I_{C_{Q1}}\left(\phi ,L_s\right)=\frac{V_d}{L_s{\omega }_{sw}}\left(n\left(\phi \right)\cos \left(\gamma \right)+\frac{\pi -\phi }{2}\right)$$

(40)

while the amplitude of the switching current for the lower transistors, Q2 and Q3, is given by

$$I_{C_{Q3}}\left(\phi ,L_s\right)=\frac{V_d}{L_s{\omega }_{sw}}\left(n\left(\phi \right)\cos \left(\gamma +\phi \right)+\frac{\pi -\phi }{2}\right).$$

(41)

Therefore, the power losses for each transistor are obtained from

$$P_{sw}\left(\phi ,L_s\right)=E_{off}\left(\phi ,L_s\right)\frac{{\omega }_{sw}}{2\pi }.$$

(42)

The losses due to the transistor gate depend on the total gate charge Q_G and the gate-to-source voltage V_G, related through the following expression:

$$P_{gate}=Q_G{V}_G\frac{{\omega }_{sw}}{2\pi }.$$

(43)

On the other hand, the inductor core losses are due to the magnetic flux density, and the influence of the DC bias is negligible [26,36]; therefore, they can be obtained using the following equation:

$$P_{core}\left(\phi ,L_s\right)=P_v{V}_e$$

(44)

where P_v is the core-loss density of the inductor given by the manufacturer for the operating frequency and magnetic flux density, and V_e is its volume.

As a final consideration, for the copper losses P_wire of the inductor, it is necessary to know the effective resistance of the winding R_copper, taking into account the conductivity, the total length, and the skin effect using

$$P_{wire}\left(\phi ,L_s\right)={{I_{RMS}}^{2}R}_{copper}.$$

(45)

To conclude, the total losses are determined by the following equation:

$$P_{tot}\left(\phi ,L_s\right)=4P_{cd}\left(\phi ,L_s\right)+2\left(P_{{sw}_{Q1}}\left(\phi ,L_s\right)+P_{{sw}_{Q3}}\left(\phi ,L_s\right)\right)+4P_{gate}+P_{core}\left(\phi ,L_s\right)+P_{wire}\left(\phi ,L_s\right)$$

(46)

meanwhile, the efficiency is given by

$$\eta \left(\phi ,L_s\right)=\frac{P_o\left(\phi ,L_s\right)}{P_o\left(\phi ,L_s\right)+P_{tot}\left(\phi ,L_s\right)}.$$

(47)

For the losses analysis, the SiC MOSFET G320MT12K was used as the transistor for each of the inverter switches.

Table 2. Transistor characteristics.

Magnitude	Symb.	Eq.	Value	Unit
On resistance	$R_{{DS}_{on}}$	(38)	17	mΩ
Turn-OFF losses	a	(39)	0.0268	µJ/A2
	b		0.2679	µJ/A
	c		18.929	µJ
Gate charge	QG	(43)	180	nC

presents the characteristics obtained from the datasheet for the efficiency analysis of the inverter.

Considering (36), Figure 5 represents the entire possible operating range, for both the phase shift and series inductance variation, expressed as a function of the inverter output power. In the figure, there is a red dotted curve that results from optimizing the output current of the inverter. Therefore, this is the optimal operating point for control, where the losses in the inverter will be minimized. Following the path of the curve, it can be observed that at high-output powers, the increase in series inductance value is prioritized over increasing the phase shift. This is because reducing the output current is achieved by increasing the input circuit impedance and, when switched with a small phase shift, for the same output current, the switching current is lower. However, when a lower output power is demanded, working with a higher series inductance value, which implies a greater input impedance, this results in applying a lower voltage through a larger phase shift. This enables switching with smaller angles and counteracts the effect of the increased phase between voltage and output current due to the elevated series inductance value.

To illustrate the difference in efficiency when operating across the entire power range using only PS control, as opposed to including variable inductance VI-PS, the efficiency and output current of both approaches have been compared as a function of power in Figure 6. The starting point for both controls is the same and corresponds to the maximum power value because, in both controls, at that point, the phase shift is minimized, and in the control with the variable inductor, the inductance is set to the minimum value, which is the same value maintained by PS control across its entire power range. As seen in the figure, the improvement in efficiency occurs as the inverter output power decreases because the output RMS current is reduced. This allows for maintaining a performance above 98.5% across the entire operating range and even reaching 99% when the output current is significantly reduced.

Table 3. Losses results for 12 kW operation power.

Magnitude	Symb.	Eq.	PS	VI&PS	Unit
Conduction losses	$P_{cd}$	(38)	45.61	18.478	W
Switching losses	$P_{{sw}_{Q1}}$	(42)	3.465	7.783	W
Switching losses	$P_{{sw}_{Q3}}$		3.238	6.075	W
Gate losses	$P_{gate}$	(43)	0.508	0.508	W
Inductor core losses	$P_{core}$	(44)	3.57	6.12	W
Inductor wire losses	$P_{wire }$	(45)	15.499	6.279	W
Total losses	$P_{tot}$	(46)	216.952	116.065	W

shows the breakdown of the losses for the optimization of the VI-PS control compared to the PS control operating at 12 kW.

4.3. Control Principles

The simplified VI-PS control scheme is depicted in Figure 7. Due to its multivariable nature, a Multiple Inputs Multiple Outputs (MIMO) system is employed, which is nonlinear and coupled. It is solved using a Proportional Integral Differential (PID) control and a decoupled diagonalization network [37,38]. This configuration enables the manipulation of one variable to affect the other and vice versa, generating a control loop that iterates between them. The system inputs include the input power of the inverter compared with the selected reference, and the phase angle $\alpha$ between the output voltage and current of the inverter, which is fixed at 15°, as defined in Equation (27). Consequently, the output variables are the phase angle φ of the phase shift control (PS) and the direct current (Ibias) of the L_bias winding.

Furthermore, a digital observer with a dynamic optimizer is employed to reduce the RMS value of the inverter output current [39]. In this block, by measuring the inverter output current and comparing it with the values of the phase shift angle and I_bias, interventions are made in the outputs to optimize both values. This ensures the inverter operates within the optimal operating range, as represented by the red dashed line in Figure 5. The signal conditioning and triggering logic block, besides generating the triggering signals of the transistors with the corresponding dead times based on the phase shift angle value received from the system, compares the zero-crossings of the current in each cycle to ensure ZVS during switching. If, during the heating process, the angle $\alpha$ shifts significantly, approaching the capacitive region due to changes in the inductance of L_p, modified due to changes in permeability, electrical conductivity of the material, or due to a load jump caused by coupling and decoupling the piece in the inductor, the control system directly decreases the I_bias current to increase the value of L_s. This adjustment maintains soft switching, preventing over-voltages or over-currents at the inverter output. Although the variable inductor has been designed to work far from the saturation limits, in case where the inductor becomes saturated, the angle α would be reduced, as seen in Figure 4; this will also be detected in this part of the control, decreasing the bias current to avoid saturation.

5. Experimental Results with Discussions

To conduct the experimental tests discussed in this section, the variable inductance design was first evaluated. The result is shown in Figure 8, which shows the calculated and experimentally measured value of the inductance as a function of the bias current.

As can be seen in Figure 8, there is a dispersion between the calculated value and the experimentally measured value. This deviation is mainly due to the range of dispersion of the initial permeability provided by the manufacturers and its dependence on temperature. Nevertheless, it can be seen that the mathematical model well describes the behavior of the variable detector as a function of the bias current.

A 20 kW induction heating inverter module was used to test the complete design. This module includes FPGA-based control, sigma–delta modulator sensing stages, isolated power sources, a controllable current source based on the Howland current pump with an OPA548 as a high-current operational amplifier, trigger drivers, transistors, and a water cooler heatsink. The inverter is powered by a 500 VDC supply.

The test bench that was used is shown in Figure 9, and it has been assembled with the design components from Section 4. Each number corresponds to the following:

(1)

Induction heating inverter with four SiC G3R20MT12K;

(2)

300 MHz bandwidth DSO;

(3)

Differential voltage probe and Rogowski current probe;

(4)

Hall effect probe for DC bias current;

(5)

Controllable series inductor (L_s);

(6)

Parallel capacitor (C_p);

(7)

Parallel inductor (L_p);

(8)

Water-cooled load.

Figure 9. Inverter with LLC resonant load test bench. The numbered labels describe the components on the bottom.

Figure 9. Inverter with LLC resonant load test bench. The numbered labels describe the components on the bottom.

To validate the design, the theoretical values obtained from the equations were compared with the measurements made on the inverter at two power levels. The result is shown in

Table 4. Comparison of theoretical and measured results.

Magnitude	Symb.	12 kW		20 kW		Unit
		Theor.	Meas.	Theor.	Meas.
Phase angle	$\alpha$	15	15.1	15	14.8	°
Voltage phase	$\gamma$	52.4	51.8	39.4	40.7	°
Output current	$I_{RMS}$	46.6	49.5	83.5	85.1	A
Upper transistors current	$I_{C_{Q1}}$	39.2	40	55.1	55.6	A
Lower transistors current	$I_{C_{Q3}}$	38.6	38.3	54.3	53.7	A

, where a small dispersion is observed due to the modeling method used and the existence of parasitic elements and losses of other elements are not taken into account in the calculation, such as conductors, connections, temperature dependence components, and the error of the measurement system, voltage, and current sensors.

Figure 10 displays oscillograms captured by the DSO for the inverter operating at 60% power with three different control techniques and their response to load changes by rapidly extracting the load from the inductor. The first control strategy is fixed frequency, using the PS control. The second control strategy involves a narrow frequency range, combining frequency modulation with a phase shift (FM-PS). The third strategy, analyzed in this article, is fixed frequency VI-PS control. In all three cases, the LLC resonant circuit remains the same, with the minimum series inductance value. In terms of performance with the same power setpoint, as depicted in the figure, the PS control has been configured at a higher frequency than the resonant design. This adjustment is necessary to operate above the series resonance frequency, ensuring an adequate operating range for the commutation phase α and enabling soft switching [13]. Consequently, for the same power level, the inverter output current is higher with PS control, reaching 64 A_RMS, compared to the other control techniques. In the case of the FM-PS control, the output current of 56 A_RMS is slightly higher than that of the VI-PS control. Despite being able to maintain the same phase angle α of 15°, the frequency is higher due to regulation [12,19], and it commutates with a wider phase shift φ, leading to increased switching losses due to the higher current amplitude during OFF switching compared to the VI-PS control. On the other hand, the VI-PS control maintains a closed-phase shift by desaturating the series inductance, allowing for a lower output current of 50 A_RMS, and switching with less current due to a narrower φ angle, while maintaining the 15° phase angle α, and a lower frequency closer to the series resonance frequency.

As observed in the load changes behavior, the PS control, despite operating at a higher frequency with increased circuit input impedance and phase angle α, fails to maintain ZVS behavior and control over-current despite widening the phase shift angle. Meanwhile, FM-PS control manages to maintain ZVS behavior but at the cost of a sudden frequency variation of 8 kHz, a complex issue to handle in terms of harmonic filtering. Furthermore, it is observed that the current increases slightly despite the control correction, increasing the angle α width. This effect could be mitigated by allowing the control to increase the frequency even further. Lastly, as shown in the VI-PS control oscillogram, soft switching is preserved, and the over-current caused by the load jump is controlled. This is achieved by rapidly desaturating the external branches of inductor L_s, causing the impedance difference from the load jump to attenuate due to the increased impedance resulting from a higher series inductance in the LLC resonant circuit. It is observed that the change in impedance by increasing L_s is so effective that the control has to decrease the angle to maintain the current level, in contrast to the other two control techniques.

Finally, to conclude the comparison between the three control types and validate the design discussed in the article, the inverter efficiency has been measured across the entire power range by the difference of the input power measured directly at the input DC voltage supply and the output power measured by the DSO measurement of the principal value of the instantaneous multiplication of the output voltage and current of the inverter bridge [24], starting from the same frequency and phase shift angle, with the minimum series inductance. As shown in Figure 11, at the starting point, all three controls achieve the same performance. However, as the power decreases, the VI-PS control gradually increases its efficiency, as explained earlier in Section 4. In this figure, in comparison to the theoretical calculation in Figure 6, the measurement for the FM-PS control has been added. Despite maintaining similar performance to the variable inductance control, FM-PS has the disadvantage of losing reactive element optimization. This occurs as the frequency increases with a lower power demand, leading to poorer efficiency and a more complex harmonic filtering. It is crucial to note that in the PS control, in order to handle load jumps or changes in the permeability of the heating inductor, the frequency should be increased, resulting in lower efficiency. As observed in the figure, the measurements closely align with the theoretically calculated values, despite the dispersion in real components and the losses not being accounted for in the theoretical calculation.

6. Conclusions

This article explores the combined effect of variable inductance control with phase shift VI-PS. It analyzes and proposes a design for the variable inductance and the LLC resonant circuit for a high-frequency application in induction heating annealing. The study provides an analysis of the optimal control strategy, reducing the output current, maximizing efficiency across the entire power range, and maintaining ZVS operation even with significant load changes.

The proposed control technique has been implemented in a 20 kW full-bridge inverter using SiC MOSFETs with an LLC resonant load. The experimental results have been validated against theoretical predictions and compared experimentally with two traditional phase shift control techniques: fixed-frequency and narrow-frequency-range control. The comparison was conducted at the same power level, comparing conduction losses in the MOSFET channels and losses due to OFF switching. The behavior for over-currents and soft switching during load jumps was also compared. Both tests were conducted using the same LLC resonant circuit.

Ultimately, it has been demonstrated that at different output power levels, the VI-PS control technique achieves superior efficiencies across the entire operating range compared to traditional control methods. It can maintain efficiency above 98.5% and operate with load jumps while keeping ZVS functionality.

Therefore, this article introduces a novel VI-PS control technique which, by using a variable inductor and a phase shift, achieves higher efficiency over the whole operating range. This is because it allows operation at a fixed frequency close to the series resonance frequency, setting a low-switching angle and reducing the output current value. It also allows operation over a wide range of loads while maintaining ZVS switching. Consequently, this work opens the way for future research on the fixed-frequency control of inverters for induction heating by controlling the reactive elements, whether in two-element, three-element, or multi-element resonant tanks, thus enabling improved efficiency and load range.