**1. Introduction**

The LLC-RC has received a great deal of attention due to its high power density, soft switching, and high-efficiency operation. It has been used in many industrial applications, such as on-board battery chargers, panel TVs, adapters for electronic equipment, server power supplies, light-emitting diode drivers, and so on [1–5]. This converter has the benefit of accomplishing ZVS for a wider input voltage and load range, allowing it to run at high frequencies without sacrificing efficiency due to switching losses, resulting in smaller component sizes and higher power density. By placing an additional inductor in parallel with the series resonant converter (SRC), an LLC-RC is formed as shown in Figure 1. Light load regulation concerns in SRC can be overcome by adding this third resonant element. Therefore, it permits the converter to be operated in boost mode (i.e., voltage gain > 1) and increases the efficiency. Furthermore, at no additional cost, the added inductor may be combined with the transformer as the magnetizing inductance. Nonetheless, this topology

**Citation:** Geddam, K.K.; Devaraj, E. Real Time Hardware-in-Loop Implementation of LLC Resonant Converter at Worst Operating Point Based on Time Domain Analysis. *Energies* **2022**, *15*, 3634. https:// doi.org/10.3390/en15103634

Academic Editor: Anna Richelli

Received: 6 April 2022 Accepted: 3 May 2022 Published: 16 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

is challenging to evaluate due to its various resonant components and different operation modes [6].

The LLC-RC operates in three subintervals, namely P, O, and N. When the magnetizing inductor voltage is a positive output voltage, then that subinterval is called the P-subinterval. Similarly, when the magnetizing inductor voltage is a negative output voltage, then that subinterval is called the N-subinterval. In both the above subintervals, the current runs in the secondary rectifier, while in the O-subinterval, the output voltage will not appear across the magnetizing inductor. Therefore, in the O-subinterval, no current flows through the secondary rectifier. These three subintervals form the basis of LLC-RC's 11 major operating modes, which are PO, PN, PON, O, and OPO modes below resonance and O, P, OP, NP, OPO, and NOP modes above resonance [7]. The O and OPO operating modes occur over the whole switching frequency range with no load and low output power, respectively. In the PO operation mode, for example, during half of the switching period, the LLC-RC initially operates in the P-subinterval and then in the O-subinterval. Similarly, for the PON operation mode, the LLC-RC initially operates in the P-subinterval, followed by O, and finally operates in the N-subinterval [8].

Analysis techniques have a significant impact on the precision and efficiency of the parameter design in LLC-RC design. There are four major analytical techniques for the LLC-RC based on the current literature, a list of which is provided below:


FHA is a popular frequency-domain resonant converter analysis approach that considers voltage and current waveforms as purely sinusoidal at the fundamental frequency and ignores additional high-order harmonics [17,18]. Although FHA offers a simple approach to calculating the DC gain, the precision diminishes when the switching frequency moves away from the resonant frequency as the voltage and current waveforms become nonsinusoidal. In practice, however, FHA can be enhanced by taking account of high-order distortions [19,20] or integrating parasites into the analysis [21]. Furthermore, harmonic analysis approaches fail to uncover the LLC converter's different operating modes.

Numerous design techniques based on FDA have been recommended because of the ease of FDA. For instance, in [22], the magnetizing inductor value *L<sup>m</sup>* is governed by comparing the loads. The magnetizing inductance must satisfy *Req* = 2*π* × *f<sup>s</sup>* × *L<sup>m</sup>* for the operating point to be at its most efficient, where *f<sup>s</sup>* is the switching frequency. A voltage gain and power factor requirement must be taken into account when selecting an inductor's inductance.

The LLC resonant tank characteristics can be calculated by iteratively setting the operational switching frequency upper and lower bounds. Using the ZVS operation constraints of the primary switches, the magnetizing inductor is calculated in [23]. After that, the resonant tank parameters can be calculated using the quality factor "Q" and inductor ratio "K" established using the voltage gain curve. The following are some of the drawbacks of frequency-domain analysis-based design techniques.


An analytical approach combining the partial time-domain corrections and frequency domain is proposed in [9]. In this technique, the equivalent load in DCM is altered by using TDA. This method's accurateness is considerably enhanced over FHA, but it still makes a large number of assumptions, which reduces its accuracy. For the LLC-RC, in [10], the authors developed an approach in which both the resonant factor and the equivalent load are adjusted, although the method of derivation is difficult and the accuracy increment is not apparent. The above-mentioned issues still remain despite efforts to increase accuracy through the methods of approaches based on FDA with partial and complete corrections in the time domain.

State space investigation is an additional option to be employed, which can describe the current and voltage waveforms accurately [24,25]. However, the interpretation and usage are convoluted and challenging. The literature available in [26,27] is based on operational modes and is mainly concentrated on analyzing the resonant voltage and current behavior according to different modes rather than calculating the DC gain. More valuable in directing the design is an exact DC gain characteristic rather than precisely stated current and voltage characteristics.

The LLC-RC has not been subjected to any additional assumptions in the TDA. There is a strong correlation between theoretical and actual results. The fundamental drawback of time-domain analysis is that it is difficult to solve nonlinear equations because of its complexity. Design techniques established on TDA have been developed to make maximum use of the LLC resonant tank. An automated computer-aided design technique based on the LLC converter power loss model is presented in [15], where the optimum design result may be reached by setting the parameters for the design variables to their limits. System voltage gain operating points are designed as the peak gain operating points of the LLC-RC in [8,14] depending on whether they are operating in PN or PON modes. The LLC resonant tank may be used to its full potential with this design technique; however, the ZVS operation for the primary switches may be compromised at the operating point of maximum gain. Additionally, the text fails to explicitly identify the optimization goal, which is mostly up to the designer. Because of the high processing requirements of these design methods, they are difficult to implement. It is necessary to solve all of the LLC-RC operation modes and boundary conditions in [15], which makes the design process more difficult. We need to find out about the PN and PON operating modes, as well as the boundary conditions that exist between them. Furthermore, because there is no set beginning point, there are a plethora of design options.

In this paper, a simplified analysis of the LLC resonant tanks' DCMs has been thoroughly investigated in PO mode under the worst case instead of PN or PON modes. Due to the possibility of several resonant frequencies, DCM modeling for three or more resonant element converters based on FHA and prolonged descriptive function is highly approximate in nature. Numerous studies have attempted to solve analytical problems for multi-element resonant converters such as LLC, LCC, and LCLC using a state-space time domain method. Few authors have investigated the TDA operating above the resonant

point of the LLC-RC. The majority of these publications make certain assumptions, such as a linear increasing magnetizing current, sinusoidal output current, and complete output diode conduction. The majority of these studies have focused solely on estimating the maximum voltage gain at or around the resonance point. In the current literature, there has not been much consideration devoted to the examination and derivation of closed-form solutions for ZVS angle, component stress, active power, RMS current, switch turn-off current, and other circuit design parameters in DCM mode. As a result, the circuit parameters are incorrectly selected.

This paper strives, by offering a precise model, to bridge this gap. LLC-RCs have demonstrated the exact derivation of the tank RMS current, tank capacitor voltage, converter voltage gain, peak stress, and ZVS. This research provides researchers with userfriendly technologies that allow them to quickly specify parameters, examine trade-offs, prototype the final product design quickly, and perform precise magnetic examination. As an action of the frequency, load, and other circuit parameters, closed-form solutions are developed for converter peak stress, tank capacitor voltage, voltage gain, ZVS angle, and tank RMS current. The rest of this article is structured in the following manner: the time-domain analysis introduction is presented in Section 2. Section 3 discusses the steady state time domain analysis. A complete step-by-step design procedure for LLC-RC is presented in Section 4. Then, the simulation and experimental results are presented in Section 5. Finally, the conclusions are provided in Section 6.

#### **2. Time-Domain Analysis Introduction**

The LLC-RC's typical circuit is shown in Figure 1. During the first half of the switching cycle, there are three subintervals. As long as the voltage across the magnetizing inductor is held at (+*Vo*)/n, the subinterval is defined as "P". The subinterval is "N" when the voltage is held down at (−*Vo*)/n, and the "O" subinterval occurs when no current runs through the secondary side of the transformer [8]. The LLC-RC operates primarily in the following six modes of operation: PO, PON, PN, NP, NOP, and OPO, which are determined by the sequences of these three subintervals. An LLC-RC running in PO mode first operates in the P-subinterval, followed by the O-subinterval, for half of the switching time.

Figure 2 depicts the significant waveforms produced when the LLC-RC is working in the PO mode. The switch current *iS*<sup>1</sup> is negative before the driving signal *S*<sup>1</sup> is supplied; thus, its anti-parallel diode will turn on and perform the ZVS process on *S*1. In the same way, the remaining switches (*S*2–*S*4) are capable of the ZVS process. The secondary diodes (*D*1–*D*4) can accomplish ZCS functioning based on the waveform of the transformer secondary current *isec*. The PO and OPO modes of the LLC-RC are extremely efficient because the primary switches and secondary diodes operate in ZVS and ZCS modes, respectively [28]. Other operating modes, such as PON or PN modes, do not ensure ZVS functioning for the primary switches, resulting in worse overall system efficiency. The switching frequency in NOP or NP mode is higher than the resonance frequency, and the secondary diodes cannot perform ZCS. Design considerations include making sure it can function in OPO or PO modes across the complete working range. The other three analytical techniques had significant errors; therefore, the resonant tank components were designed using time-domain analysis. The analysis for PO mode is identical to that for the OPO mode, which follows in the next section.

**Figure 2.** LLC converter steady state waveforms in PO mode.

## **3. Steady State Time-Domain Analysis**

The variable-frequency controlled LLC-RC's steady state time-domain analysis is presented in this section. In order to keep the bridge stable, two complementary gating signals are used, each having a duty cycle of 0.5%. Figure 1 depicts the overall configuration of the LLC-RC in PO mode, as well as the corresponding equivalent circuit that results. Figure 3A,B illustrates the analogous circuits for an LLC resonant converter working in the P and O stages, respectively. In order to analyze the converter's steady state performance, the following assumptions are made.


The reasons for choosing the PO operating mode are summarized as follows.


converter's efficiency. As a result, in terms of soft switching, the PO mode is favored above the PN or PON modes;


Figure 2 depicts the LLC converter operational waveforms in boost mode. The resonant tank is driven by a square wave input generated by the full bridge's variable switching frequency control. The ZVS angle is indicated by *φ*, which is a measurement of the exact ZVS and *t<sup>d</sup>* . Differential equations utilizing KCL/KVL have been developed for each mode. For the sake of analysis, the subsequent quantities have been defined:

$$Z\_0 = \sqrt{\frac{L\_s}{C\_s}}\tag{1}$$

$$Z\_1 = \sqrt{\frac{L\_s + L\_m}{C\_s}}\tag{2}$$

$$
\omega\_r = \frac{1}{\sqrt{L\_s \mathbb{C}\_s}} \tag{3}
$$

$$
\omega\_{r1} = \frac{1}{\sqrt{(L\_s + L\_m)C\_s}} \tag{4}
$$

$$
\omega = \frac{\omega\_{sw}}{\omega\_r} \tag{5}
$$

$$K = \frac{L\_m}{L\_s} \tag{6}$$

where *Z*<sup>0</sup> = characteristic impedance, *L<sup>m</sup>* = magnetizing inductance, *L<sup>s</sup>* = resonant inductance, *C<sup>s</sup>* = resonant capacitor, *ω<sup>r</sup>* = series resonant angular frequency, *ωr*<sup>1</sup> = parallel resonant angular frequency, *ω* = angular normalized frequency, and *ωsw* = angular switching frequency.

**Figure 3.** Equivalent circuits in PO mode, (**A**) energy transfer period in P mode, (**B**) freewheeling period in O mode.
