A Comprehensive Analysis of Losses and Efficiency in a Buck ZCS Quasi-Resonant DC/DC Converter

Abstract

As power electronics continue to advance, the demand for highly efficient and low-loss DC/DC converters has grown significantly. This article comprehensively analyses ZCS quasi-resonant switch cell losses and efficiency in buck L-type zero-current switching (ZCS) quasi-resonant DC/DC converters. The main part of the study includes a comparative analysis of conduction losses in semiconductor switches of conventional PWM buck converters and zero-current switching (ZCS) quasi-resonant buck converters (L-type), utilizing both specific and generalized design equations. Novel coefficients are introduced that enable the evaluation of static power losses in the classical buck converter compared to those in L-type ZCS buck quasi-resonant converters under identical conditions. The article also discusses design considerations aimed at minimizing static losses. An L-type half-mode zero-current switching (ZCS) buck quasi-resonant DC/DC converter (QRC) is implemented to verify the analytical results. Various simulations were conducted using PSpice in the Texas Instruments simulation environment, along with experimental studies at different switching frequencies and load conditions. The proposed methodology integrates both analytical and simulation approaches to analyze energy losses and key parameters influencing the converter’s efficiency. The obtained results show that the relative error between the analytical, simulation, and experimental results is below 5%.

1. Introduction

The enhancement of secondary power sources, based on high-frequency voltage pulse converters, is intended to optimize their energy characteristics and ensure their performance, thereby meeting the outlined requirements of the design specification. In modern power electronics, the efficiency of power DC/DC converters is crucial for minimizing losses and enhancing reliability, which is significantly influenced by the operating temperature [1,2]. A methodology that has been employed to mitigate the switching losses of semiconductor switches involves the utilization of resonant circuits to achieve soft switching in current and/or voltage. This class of DC/DC converters is designated as quasi-resonant in the existing literature [3,4,5,6,7,8,9,10,11].

Without a detailed discussion of the advantages and disadvantages of these converters, since these issues have been comprehensively reviewed in numerous publications [9,10,11,12,13,14], the focus here is on analyzing the quasi-resonant converter (QRC) with power switching at zero current. The integration of resonant circuits within the power transistor circuit, where switching at zero current can be achieved, represents a promising area for the advancement of power conversion equipment, thereby reducing the dynamic losses in power semiconductor switches to a negligible level. The current and voltage shapes become close to sinusoidal, which contributes to the reduction in losses in the magnetic circuits of the inductances of the output filters and power transformers in the circuits.

The main applications benefiting from quasi-resonant converters with zerocurrent switching (ZCS) are in the following industrial applications: consumer electronics (e.g., laptops, tablets, and smartphones) [15], server application [16]; LED lighting systems [17]; battery management systems for electric vehicles (EVs) and storage [18,19,20]; renewable energy applications [21]; medical devices requiring low EMI and high reliability [22,23,24]. Buck ZCS (zero-current switching) quasi-resonant converters are particularly well suited to low to medium power applications.

Despite the aforementioned advantages over classical pulse converters with rectangular current and voltage waveforms, the maximum current and voltage values during the resonant process in QRC circuits can significantly exceed the analogous values in classical circuits. Furthermore, the inherent complexity of QRCs necessitates a comprehensive investigation of the various parameters inherent to the power part of the converter, where the switching devices are turned off/on at zero-current values within a designated control range. This necessity gives rise to the investigation of electromagnetic processes within QRC circuits.

Despite the availability of numerous studies related to the analysis, modeling, and design of quasi-resonant ZCS DC/DC converters [25,26,27,28,29,30,31], there is a paucity of research regarding the estimation of static power losses in classical buck converters and ZCS buck QRC circuits. The objective is to determine the selection of such parameters that will result in minimal static losses for a given output power control range.

The objective of this paper is to establish a theoretical framework and to provide practical recommendations for the performance evaluation of this converter. This is achieved by addressing the fundamental principles of operation, loss modeling, and optimization options. This analysis is of critical importance for electronic system designers pursuing efficient and reliable energy management solutions in a range of applications, including industrial systems and consumer electronics.

This article presents a comparative analysis of the conduction losses in semiconductor switches of conventional PWM buck converters and zero-current switching (ZCS) quasi-resonant buck converters (L-type). Novel coefficients are introduced that allow for evaluating the static power losses in the classical buck converter circuits vs. static power losses in L-type ZCS buck QRCs (under identical conditions). These coefficients can be used as a methodology of predicting the static losses in the switching devices of the classical PWM and ZCS QRCs buck circuits. The study also examines resonant tank losses. While they are generally smaller than switching losses, they have the potential to significantly affect the overall efficiency of ZCS buck QRCs, especially at high frequencies or light loads. Proper design and component selection are essential to minimize these losses and ensure optimum converter performance.

The work is organized as follows: In Section 2, the approximate analysis of the electromagnetic processes in quasi-resonant L-type ZCS buck QRCs are presented. The converter’s main characteristics are analytically and graphically received. The obtained dependencies are in a general form, i.e., without including the internal parameters of the investigated converters, which allows for some generalized estimates to be made for the static losses of the switching devices for any conditions and parameters within the realization conditions of the corresponding ZSC buck QRC mode. Design considerations to receive minimum static losses are discussed. In Section 3, the efficiency for the L-type ZCS buck QRC due to the resistive resonant tank losses are studied. A discussion on choosing the circuit’s parameters is presented. In Section 4, a simulation and experimental study of half-mode L-type ZCS QRC are presented.

2. Approximate Analysis and DC Voltage Conversion Ratio for the L-Type ZCS QRC

The circuits of the quasi-resonant L-type ZCS buck QRCs (half-mode and full-mode) are shown in Figure 1.

The L-type configuration entails the connection of the resonant capacitor $C_r$ in parallel with diode D₂, and the resonant inductor $L_r$ in series with the transistor Q₁. In the converter depicted in Figure 1a (half-mode), the diode D₁ is connected in series with transistor Q₁. Consequently, the switch Q₁–D₁ is closed when the current of the inductor $i_{Lr}$ first passes through zero.

To achieve full-wave mode operation of the ZCS buck QRC from the half-wave mode, an antiparallel diode (D_S) is added across the switch Q₁–D₁ (Figure 1b). In the full-wave mode, the transistor must be turned off during the interval when the resonant inductor current is negative. A MOSFET can facilitate full-wave mode without external diodes D₁ and D_S, as it contains an internal drain-source body diode. However, this approach is less effective at high frequencies due to the slow reverse recovery of the MOSFET’s body diode. Furthermore, the efficiency of the circuit can be improved by using a Schottky diode as an antiparallel diode or using MOSFET with a monolithic Schottky diode. A further method of enhancing the efficiency in the half-mode of the circuit would be to substitute the diode D₁ with a synchronous rectifier.

In both circuit configurations, the values of the resonant capacitor $C_r$ and resonant inductor $L_r$ are relatively small. Furthermore, the resonant frequency ${\omega }_0$ is higher than the switching frequency of the transistor ${\omega }_s$ (${\omega }_0>{\omega }_s$). The following equation determines the resonant frequency:

$${\omega }_0=1/\sqrt{L_r{C}_r}.$$

(1)

During this study, the static losses in the switching devices in an L-type ZCS buck QRC are evaluated, and a comparison is made with similar losses in a classical PWM buck converter with the same topology under other identical conditions, including input and output voltage, load current, and using semiconductors devices.

In this case of ZCS buck QRC, it is assumed that the main part of the power losses are the conduction losses in the switching devices, which are found to be significantly influenced by the operation mode (half-mode or full-mode), the resonant tank parameters, and the types of semiconductor switches. Consequently, the objective of the analysis can be narrowed down to ascertaining the ratio of the static power dissipated on the switching devices in a classical buck converter and the static power dissipated on the switching devices in an L-type ZCS buck QRC.

2.1. Half-Wave L-Type ZCS Buck QR Converter

The analysis of electromagnetic processes is based on the studies presented in [10]. Semiconductor switches are considered ideal. Thus, they have no forward voltage drop during on-state, no off-state leakage current, and instantaneous switching without delays. The impact of the output current ripple is negligible. Thus, the operation of the output filter and load are considered to be equivalent to a constant current sink with a value equivalent to the steady-state output current $I_0$ (Figure 2).

Figure 3 shows typical circuit waveforms of transistor’s Q₁ control pulse $v_{QG1}$, current $i_{Lr}$ through the inductor $L_r$ and voltage $v_{Cr}$ across the capacitor $C_r$ for one half-period of the converter operation in a half-wave mode.

Each switching interval is divided into four subintervals, denoted by the interval lengths ${\theta }_1$, ${\theta }_2$, ${\theta }_3$, and ${\theta }_4$. The switching interval is considered to start when transistor Q₁ is turned on. The initial conditions for the inductor current $i_{Lr}$ and the voltage across the capacitor $v_{Cr}$ are zero.

The following quantities are introduced:

$$Z_0=\sqrt{L_r/C_r}—\mathrm{characteristic} \mathrm{impedance}.$$

(2)

$$M={\displaystyle \frac{V_0}{V_{in}}}={\displaystyle \frac{I_{in}}{I_0}}—\mathrm{DC} \mathrm{voltage} \mathrm{conversion} \mathrm{ratio}.$$

(3)

$$\nu ={\omega }_S/{\omega }_0—\mathrm{frequency} \mathrm{ratio}.$$

(4)

• Resonant inductor charging interval

In the first subinterval ${\theta }_1$, all switching devices are turned on (Q₁, D₁, D₂). Diode D₂ is forward-biased since the current $i_{Lr}\left(t\right)$ is smaller than the output current $I_0$.

This mode is valid for 0 ÷ ${\omega }_0{t}_1$, where current $i_{Lr}$ is given by

$${\displaystyle \frac{d{i}_{Lr}\left({\omega }_{0}t\right)}{dt}}={\displaystyle \frac{Vin}{L_r}}.$$

(5)

with an initial condition $i_{Lr}\left(0\right)=0$. In this subinterval, the inductor current $i_{Lr}$ rises linearly. The solution of (5) is given by

$$i_{Lr}\left(t\right)={\displaystyle \frac{Vin}{L_r}}={\omega }_{0}t{\displaystyle \frac{Vin}{Z_0}}.$$

(6)

This mode ends at ${\omega }_0{t}_1$ when $i_{Lr}\left({\omega }_0{t}_1\right)=I_0$ or

$$i_{Lr}\left({\omega }_0{t}_1\right)={\theta }_1{\displaystyle \frac{Vin}{Z_0}}=I_0.$$

(7)

Thus, the subinterval ${\theta }_1$ can be expressed as

$${\theta }_1={\omega }_0{t}_1={\displaystyle \frac{I_0{Z}_0}{Vin}}.$$

(8)

The voltage across the resonant capacitor at the end of ${\theta }_1$ is $v_{Cr}\left({\omega }_0{t}_1\right)=0$.

• Resonant interval

This mode is valid for ${\omega }_0{t}_1$ ÷ ${\omega }_0{t}_2$. During the second subinterval ${\theta }_2$, transistor Q₁ and the diode D₁ remain on, while the diode D₂ is reverse-biased.

The resonant circuit $L_r$–$C_r$ is excited by constant sources $V_{in}$ and $I_0$. The system gives the circuit equations in this interval:

$$\left\{\begin{array}{l}{L}_r{\displaystyle \frac{d{i}_{Lr}\left({\omega }_{0}t\right)}{dt}}=V_{in}-v_{Cr}\left({\omega }_{0}t\right)\\ C_r{\displaystyle \frac{d{v}_{Cr}\left({\omega }_{0}t\right)}{dt}}=i_{Lr}\left({\omega }_{0}t\right)-I_0\end{array}\right.$$

(9)

with initial conditions

$$v_{Cr}\left({\theta }_1\right)=0, i_{Lr}\left({\theta }_1\right)=I_0.$$

(10)

Solutions to (9) are written as follows:

$$\left\{\begin{array}{l}{i}_{Lr}\left({\omega }_{0}t\right)=I_0+{\displaystyle \frac{V_{in}}{Z_0}}\sin \left({\omega }_{0}t-{\theta }_1\right)\\ v_{Cr}\left({\omega }_{0}t\right)=V_{in}\left[1-\cos \left({\omega }_{0}t-{\theta }_1\right)\right]\end{array}\right..$$

(11)

The second subinterval ${\omega }_0{t}_2$ ends when the current $i_{Lr}\left({\omega }_0{t}_2\right)$ reaches the first zero value. The duration of the second subinterval can be determined by [10]

$${\omega }_0\left(t_2-t_1\right)={\theta }_2={\displaystyle \frac{\pi +{\sin }^{-1}\left(\frac{I_0{Z}_0}{Vin}\right)}{{\omega }_0}},$$

(12)

where

$${\displaystyle \frac{-\pi }{2}}<{\sin }^{-1}\left({\displaystyle \frac{I_0{Z}_0}{Vin}}\right)<{\displaystyle \frac{\pi }{2}}.$$

(13)

To achieve zero-current switching conditions, the current through the resonant inductor $L_r$ must naturally reach zero. To achieve this condition, it is necessary to provide [10]

$$I_0\le {\displaystyle \frac{Vin}{Z_0}}.$$

(14)

If the load current is too high and Equation (14) is not satisfied, the inductor current will not reach zero, and the transistor will not turn off.

Therefore, for the normalized values of the output current $I_0^{\prime }$, the following relationship applies:

$$1\ge {\displaystyle \frac{I_0{Z}_0}{Vin}}=I_0^{\prime }.$$

(15)

For the normalized value of the output current $I_0^{\prime }$, the following expressions are also valid:

$$I_0^{\prime }={\displaystyle \frac{I_0{Z}_{0}M}{V_0}}={\displaystyle \frac{Z_{0}M}{R_0}}={\displaystyle \frac{M}{R_0^{\prime }}}.$$

(16)

Based on (14)–(16) for the normalized value of the load, it is necessary to satisfy the inequality

$$R_0^{\prime }\ge M.$$

(17)

At the end of the second subinterval, diode D₁ is switched off as a reverse voltage is applied, causing transistor Q₁ to turn off at zero current.

In half-mode, transistor Q₁ is turned off naturally $\left({\omega }_0{t}_3\right)$ when the resonant current $i_{Lr}$ reaches zero.

• Resonant capacitor discharging interval

This mode is valid for ${\omega }_0{t}_2$ ÷ ${\omega }_0{t}_3$. During the third subinterval ${\theta }_3$, transistor Q₁ and diode D₁ are off. The tank capacitor $C_r$ is discharged by the current $I_0$ through the filter inductor $L_0$, causing the voltage $v_{Cr}$ across the tank capacitor to decrease linearly to zero. The equation for the circuit follows:

$$C_r{\displaystyle \frac{d{v}_{Cr}\left({\omega }_{0}t\right)}{dt}}=-I_0.$$

(18)

Solving (18) is performed as follows:

$$v_{Cr}\left({\omega }_{0}t\right)=-{\displaystyle \frac{I_0\left(t-t_2\right)}{C_r}}+V_{in}\left[1-\cos \left({\omega }_{0}t-{\theta }_2\right)\right].$$

(19)

Subinterval ${\theta }_3$ concludes when the voltage across the tank capacitor drops to zero. Diode D₂ becomes forward-biased. Therefore,

$${\omega }_0\left(t_3-t_2\right)={\theta }_3={\displaystyle \frac{C_r{V}_{in}\left[1-\cos \left({\omega }_{0}t-{\theta }_2\right)\right]}{I_0}}.$$

(20)

The angular duration of the switching period is

$${\omega }_0{T}_s\ge {\theta }_1+{\theta }_2+{\theta }_3.$$

(21)

Substituting (8), (12), and (20) into (21), we can obtain the equation for the output characteristics of L-type half-wave ZCS buck QRC. The output characteristics for this half-wave mode can be expressed through the dependence

$$M={\displaystyle \frac{\nu }{2\pi }}\left[0.5I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)+{\displaystyle \frac{1}{I_0^{\prime }}}+\sqrt{{\left({\displaystyle \frac{1}{I_0^{\prime }}}\right)}^2-1}\right]=M_h.$$

(22)

It should be noted that the DC voltage conversion ratio in Equation (22) is also referred to as $M_h$ for the half-wave operation. The dependencies $M=M_h=f\left(I_0^{\prime }\right)$ at different values of the frequency ratio ν are plotted in Figure 4.

The DC voltage gain in the half-wave mode is highly responsive to changes in the load.

Taking into account (16), expression (22) for the DC voltage conversion ratio $M_h$ can also be expressed as follows:

$$M=M_h={\displaystyle \frac{\nu }{2\pi }}\left[\pi +{\sin }^{-1}\left({\displaystyle \frac{M}{R_0^{\prime }}}\right)+{\displaystyle \frac{M}{2R_0^{\prime }}}+{\displaystyle \frac{R_0^{\prime }}{M}}+\sqrt{{\left({\displaystyle \frac{R_0^{\prime }}{M}}\right)}^2-1}\right].$$

(23)

Taking into account (23), in [31] the functions ${\sin }^{-1}\left(M/R_0^{\prime }\right)$ and square-root terms can be approximated as follows:

$$M=M_h\cong {\displaystyle \frac{\nu }{2\pi }}\left[\pi +{\displaystyle \frac{M}{R_0^{\prime }}}+{\displaystyle \frac{M}{2R_0^{\prime }}}+{\displaystyle \frac{R_0^{\prime }}{M}}+{\displaystyle \frac{R_0^{\prime }}{M}}\right].$$

(24)

However, it should be noted that this simplification reduces the accuracy of (24). For example, in the range of $M/R_0^{\prime }\le 0.4$, the relative error of substituting ${\sin }^{-1}\left(M/R_0^{\prime }\right)$ by $\left(M/R_0^{\prime }\right)$ is below 3%; in the range of $0.5\le \left(M/R_0^{\prime }\right)\le 0.7$, the relative error is from 4.8% to 10.7%, and in the range of $0.8\le \left(M/R_0^{\prime }\right)\le 0.9$, the relative error is from 15.9% to 24.4%.

This reduction in complexity, however, permits an explicit solution for M, as demonstrated by the following expression:

$$M=M_h={\displaystyle \frac{0.5\nu +\sqrt{\frac{{\nu }^2}{4}+\frac{4\nu R_0^{\prime }}{\pi }-\frac{3{\nu }^2}{{\pi }^2}}}{2-\frac{3\nu }{2\pi R_0^{\prime }}}}.$$

(25)

Three-dimensional plots showing $M=M_h=f(\nu ,R_0^{\prime })$ dependencies in the DC voltage conversion ratio as a function of frequency ratio ν and normalized load resistance $R_0^{\prime }$ are illustrated in Figure 5.

Based on (24), the dependence of the frequency ratio ν from the normalized load resistance and DC voltage conversion ratio is given by

$$\nu ={\displaystyle \frac{4\pi M{R}_0^{\prime 2}}{3M^2+2\pi M{R}_0^{\prime }+4R_0^{\prime 2}}}.$$

(26)

The approximate results of the characteristics (26) ${\left.\nu =f(R_0^{\prime })\right|}_{M=const}$ are plotted in Figure 6a at the different values of the DC voltage conversion ratio $M=M_h$. Three-dimensional plots showing $\nu =f(R_0^{\prime },M)$ dependencies for frequency ratio ν as a function of normalized load resistance $R_0^{\prime }$ and DC voltage conversion ratio are illustrated in Figure 6b.

The results in Figure 5 demonstrate the output voltage regulation range in the half-mode depending on the load variation.

• Freewheeling interval

During this fourth subinterval ${\theta }_4$, the output current flows through diode D₂.

2.2. Full-Wave L-Type ZCS Buck Quasi-Resonant Converter

The circuit of the full-mode L-type ZCS buck QRC is presented in Figure 1b. The resonant inductor current can reverse direction, as shown in the waveforms in Figure 7.

For the full-mode L-type ZCS buck QRC, approximate equations for estimating DC voltage gain are given in

Table 1. L-type ZCS buck QRC full-mode dependencies.

Operation Mode	Expressions	Equations
0 ÷ ${\omega }_0{t}_1$	$i_{Lr}\left({\omega }_0{t}_1\right)={\theta }_1{\displaystyle \frac{Vin}{Z_0}}=I_0$	(27)
	${\theta }_1={\omega }_0{t}_1={\displaystyle \frac{I_0{Z}_0}{Vin}}$	(28)
${\omega }_0{t}_1$÷${\omega }_0{t}_3$	$\begin{array}{l}\left\{\begin{array}{l}{i}_{Lr}\left({\omega }_{0}t\right)=I_0+{\displaystyle \frac{V_{in}}{Z_0}}\sin \left({\omega }_{0}t-{\theta }_1\right)\\ v_{Cr}\left({\omega }_{0}t\right)=V_{in}\left[1-\cos \left({\omega }_{0}t-{\theta }_1\right)\right]\end{array}\right.\end{array}$	(29)
	${\omega }_0\left(t_3-t_1\right)={\theta }_2={\displaystyle \frac{2\pi -{\sin }^{-1}\left(\frac{I_0{Z}_0}{Vin}\right)}{{\omega }_0}}$	(30)
	${\theta }_{2a}={\displaystyle \frac{\left[\pi -2{\sin }^{-1}\left(I_0^{\prime }\right)\right]}{{\omega }_0}}$	(31)
${\omega }_0{t}_3$÷${\omega }_0{t}_4$	$v_{Cr}\left({\omega }_{0}t\right)=V_{in}\left[1-\sqrt{1-{\left(Z_0{I}_0/V_{in}\right)}^2}\right]-{\displaystyle \frac{I_0\left(t-t_3\right)}{C_r}}$	(32)
	${\omega }_0\left(t_4-t_3\right)={\theta }_3={\displaystyle \frac{Vin\left[1-\sqrt{1-{\left(I_0{Z}_0/Vin\right)}^2}\right]}{I_0{Z}_0{\omega }_0}}$	(33)
Full-mode DC voltage conversion ratio	$M=M_f={\displaystyle \frac{\nu }{2\pi }}\left[2\pi -{\sin }^{-1}\left(I_0^{\prime }\right)+0.5I_0^{\prime }+{\displaystyle \frac{1}{I_0^{\prime }}}-\sqrt{{\left({\displaystyle \frac{1}{I_0^{\prime }}}\right)}^2-1}\right]\approx \nu$	(34)

[10]. It has been established that in contrast to the behavior exhibited by a half-wave mode in a full-mode, the resonant current $i_{Lr}$, upon reaching a state of zero (${\omega }_0{t}_2$), continues to oscillate as stored energy is fed back into the power source through diode D₁ until ${\omega }_0{t}_3$.

It should be noted that the DC voltage conversion ratio in Equation (34) is also referred to as $M_f$ for the full-wave operation. It was evident from (34) that in a full-wave mode, the circuit’s ability to regulate output voltage is relatively insensitive to load variations because the energy dynamics of the tank circuit self-adjust with changing loads, or the full-wave mode naturally balances energy transfer between the source and load.

2.3. Estimation of Static Power Losses in an L-Type Quasi-Resonant Buck ZCS Converter

Despite the numerous advantages of resonant and quasi-resonant converters over traditional pulse width modulation (PWM) converters with rectangular waveforms in the power circuit, the peak current and voltage values during the resonant cycle can significantly exceed those seen in conventional DC/DC converters. Moreover, these values may also greatly exceed those of conventional converters, which increase the demands on the performance limits of the semiconductor power components.

To draw a comparison between the static power losses in the switching transistor of a conventional step-down converter with pulse-width modulation (PWM) and an L-type quasi-resonant buck converter with zero-current switching, it is necessary to analyze the root-mean square (RMS) current values. This is because the RMS current through the transistor is directly proportional to the losses.

Based on (34), the following expression is introduced [10]:

$$I_{0n1}={\displaystyle \frac{1}{2\pi }}\left[\pi +{\sin }^{-1}\left(I_0^{\prime }\right)+0.5I_0^{\prime }+{\displaystyle \frac{1}{I_0^{\prime }}}+\sqrt{{\left({\displaystyle \frac{1}{I_0^{\prime }}}\right)}^2-1}\right].$$

(35)

The transistor’s RMS current of the L-type half-mode ZCS buck QRC is given by

$$I_{rms\_QR\_h}=I_{\max }\sqrt{{\displaystyle \frac{\gamma }{2T_S}}}=\left(I_0+{\displaystyle \frac{V_{in}}{Z_0}}\right)\sqrt{{\displaystyle \frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]M}{4\pi I_{on1}}}},$$

(36)

where

$${\theta }_1+{\theta }_2=\gamma ={\displaystyle \frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]}{{\omega }_0}}.$$

(37)

The following analysis assumes that the switching ripple in the converter’s filter elements, L₀ and C₀, is negligible. Then, the RMS current value of an ideal PWM buck converter is given by

$$I_{rms\_pwm}=I_0\sqrt{V_0/V_{in}}=I_0\sqrt{M}.$$

(38)

In this study, the parameter $K_{rms\_half}$ is introduced to indicate the ratio of the RMS current value of the transistor operating in PWM mode to that of the L-type ZCS buck QRC under identical conditions, thus allowing for a more in-depth analysis of the static losses in the switching transistor. This parameter is defined as a function of the normalized load current, as follows:

$$K_{rms\_half}={\displaystyle \frac{I_{rms\_pwm}}{I_{rms\_QR\_h}}}\approx {\displaystyle \frac{I_0\sqrt{M}}{\left(I_0+\frac{V_{in}}{Z_0}\right)\sqrt{\frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]M}{4\pi I_{on1}}}}}={\displaystyle \frac{1}{\left(1+\frac{1}{I_0^{\prime }}\right)}}\sqrt{{\displaystyle \frac{4\pi I_{on1}}{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]}}}$$

(39)

The graphical representation of (39) is given in Figure 8.

Based on the obtained results, it is evident that the static losses in the switching device of the quasi-resonant converter in half-mode are more significant than in the traditional buck PWM, as at $I_0^{\prime }=1$, $K_{rms\_half}\approx 0.74$.

Following the analysis performed, dependency (36) can be simplified as follows:

$$I_{rms\_QR\_h}=I_{\max }\sqrt{{\displaystyle \frac{\gamma }{2T_S}}}=I_0\left(1+{\displaystyle \frac{1}{I_0^{\prime }}}\right)\sqrt{{\displaystyle \frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]\nu }{4\pi }}}.$$

(40)

To estimate the static losses in the half-wave ZCS buck QRC circuit, it is also necessary to consider the static losses on diode D₁. The following relationship is valid for the static losses $P_{static,ZCS}$ on the Q₁–D₁ switch:

$$P_{static,ZCS}=I_{rms\_QR\_h}^2{R}_{ds\left(on\right)}+I_{avg\_D}{v}_f,$$

(41)

where $R_{ds\left(on\right)}$ is on-resistance of the MOSFET; $v_f$ is a forward voltage drop of the diode; and $I_{avg\_D}$ is an average current through the diode.

For the average current through diode D₁ is received:

$$I_{avg\_D}=I_0\left(1+{\displaystyle \frac{1}{I_0^{\prime }}}\right){\displaystyle \frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]M}{{\pi }^2{I}_{on1}}}.$$

(42)

Thus, make a comparison between the power losses in the conventional buck PWM converter and the half-mode ZSC buck QRC, it is necessary to ascertain the ratio:

$${\displaystyle \frac{P_{static,PWM}}{P_{static,ZCS}}}={\displaystyle \frac{I_{rms\_PWM}^2{R}_{ds\left(on\right)}}{I_{rms\_QR\_h}^2{R}_{ds\left(on\right)}+I_{avg\_D}{v}_f}}.$$

(43)

Substituting (36), (38), and (42) into (43) for the power losses ratio between the two circuits yields:

$${\displaystyle \frac{P_{static,PWM}}{P_{static,ZCS}}}={\displaystyle \frac{1}{{\left(1+\frac{1}{I_0^{\prime }}\right)}^2\frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]}{4\pi I_{on1}}+\left(1+\frac{1}{I_0^{\prime }}\right)\frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]}{{\pi }^2{I}_{on1}}\frac{v_f}{R_{ds\left(on\right)}{I}_0}}}$$

(44)

Expression (44) depends on the parameters of the semiconductor devices used and the load. The graphical representation of (44) at $0.1<v_f/R_{ds\left(on\right)}{I}_0<10$ is given in Figure 9.

To minimize the static losses in the ZCS buck QRC in half-mode, it can be seen that it is necessary to choose parameter $I_0^{\prime }$ close to unity ($0.8<I_0^{\prime }<1$), and the circuit’s parameters $v_f/R_{ds\left(on\right)}{I}_0$ are necessarily smaller.

The same approach is also applied to estimate the ratio of the RMS transistor current value in the PWM buck converter to the RMS current value $I_{rms\_QR\_f}$ of a full-wave ZCS buck QRC under other identical conditions. For the full-mode, parameter $K_{rms\_full}$ is introduced as follows:

$$\begin{array}{l}{K}_{rms\_full}={\displaystyle \frac{I_{rms\_pwm}}{I_{rms\_QR\_f}}}\approx {\displaystyle \frac{I_0\sqrt{M}}{I_0\sqrt{M}\sqrt{\frac{{\left(1+\frac{1}{I_0^{\prime }}\right)}^2\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]+{\left(-1+\frac{1}{I_0^{\prime }}\right)}^2\left[\pi -2{\sin }^{-1}\left(I_0^{\prime }\right)\right]}{4\pi }}}}=\\ =\sqrt{{\displaystyle \frac{4\pi }{{\left(1+\frac{1}{I_0^{\prime }}\right)}^2\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]+{\left(-1+\frac{1}{I_0^{\prime }}\right)}^2\left[\pi -2{\sin }^{-1}\left(I_0^{\prime }\right)\right]}}}\end{array}$$

(45)

The coefficient $K_{rms\_full}$ is sketched in Figure 10.

The results demonstrate that the static losses in the transistor of the quasi-resonant converter in full-mode are more significant than in traditional buck PWM ($I_0^{\prime }=1$,$K_{rms\_full}\approx 0.74$), which is analogous to the results in half-mode.

If the input, output voltage, and load current are stable or change insignificantly, the parameters of the resonant tank should be selected so that the parameter $I_0^{\prime }$ is close to unity ($0.8<I_0^{\prime }<1$), which minimizes the static losses. In other words, an L-type quasi-resonant buck ZCS converter operating in half-wave and full-wave modes can be more efficient compared to a conventional buck PWM, provided that the dynamic losses in the buck PWM switching exceed the static losses in the QRC buck ZCS.

The obtained dependencies are in a general form, i.e., without including the internal parameters of the investigated converters, which allows for some generalized estimates for the static losses in the switching devices for any conditions and parameters within the realization conditions of the corresponding ZSC buck QRC mode.

It also should be noted that subinterval 4, characterized by an angular length of ${\theta }_4$, corresponds to the diode conduction phase in the conventional PWM switch network. During this subinterval, diode D₂ carries the filter inductor current $I_0$, while the tank capacitor voltage $v_{Cr}$ remains zero. Transistor $Q_1$ is in the off state, resulting in an input current $i_{Lr}$ of zero.

3. Efficiency for the L-Type ZCS QRC with Resonant Tank Losses

3.1. Resistive Losses in the Circuit with the Resonant Inductance

To ascertain the resistive losses in the circuit of the resonant inductance, the approach proposed in [31] is employed. $R_L$ denotes the resistive losses in the resonant inductance circuit and $R_C$ denotes the resistive losses in the resonant capacitor circuit. The converter efficiency due to each of these losses is calculated separately to gain insight into the impact of each. The efficiency of the two types of losses is determined by the superposition of the individual losses, which is a valid approximation.

The present analysis operates under the assumption that the addition of parasitic resistances remains sufficiently negligible to induce a change in the shape of the current through the resonant inductance or the current through the resonant capacitor. This is a valid assumption since the objective of any design is to maximize the efficiency of the circuit.

• The efficiency is given by the following relationships

$$\eta ={\displaystyle \frac{P_0}{P_{in}}}={\displaystyle \frac{P_0}{P_0+P_{loss}}}.$$

(46)

For the output power can be written

$$P_0=I_0^2{R}_T=V_0^2/R_T.$$

(47)

The losses due to R_L are given by

$$P_{loss}=I_{rms\_QR}^2{R}_L.$$

(48)

The ratio $P_{loss}/P_0$ can then be expressed as

$${\displaystyle \frac{P_{loss}}{P_0}}={\left[1+\left({\displaystyle \frac{1}{I_0^{\prime }}}\right)\right]}^2{\displaystyle \frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]I_0^{\prime }{\xi }_L}{2\pi I_{0n1}}},$$

(49)

where ${\xi }_L=R_L/2Z_0$ is a damping factor related to $R_L$.

Considering (36), (16) (46), and (49), the efficiency ${\eta }_L$ of the half-wave L-type ZCS buck QRC circuit due to $R_L$ can be written as

$${\eta }_L={\eta }_{Lh}={\displaystyle \frac{1}{1+\frac{P_{loss}}{P_0}}}={\displaystyle \frac{1}{1+{\left[1+\left(\frac{1}{I_0^{\prime }}\right)\right]}^2\frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]I_0^{\prime }{\xi }_L}{2\pi I_{0n1}}}}.$$

(50)

It should be noted that the efficiency ${\eta }_L$ in Equation (50) is also referred to as ${\eta }_{Lh}$ for the half-wave operation. Using (50), to evaluate the impact of ${\xi }_L$ on efficiency at different values of $R_0^{\prime }$, the characteristics ${\eta }_L=f({\zeta }_L)$ at ν = const are plotted.

Figure 11a presents the results at $\nu =0.5$. Figure 11b presents 3D dependencies of efficiency ${\eta }_L=f({\zeta }_L,\nu )$ due to resistive losses related to $R_L$ and frequency ratio $0.1<\nu <0.9$ at $R_0^{\prime }=const$.

The results shown in Figure 11 highlight the importance of carefully selecting the components in the resonant circuit, as they significantly impact the overall efficiency of ZCS buck QRCs.

The next step in the analysis is to define the efficiency of the full-mode circuit. The transistor’s RMS current of the L-type full-mode ZCS buck QRC is given by:

$$I_{rms\_QR\_f}=I_0\sqrt{M_f}\sqrt{{\displaystyle \frac{{\left(1+\frac{1}{I_0^{\prime }}\right)}^2\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]+{\left(-1+\frac{1}{I_0^{\prime }}\right)}^2\left[\pi -2{\sin }^{-1}\left(I_0^{\prime }\right)\right]}{4\pi }}}.$$

(51)

Then, equating the ratio

$${\displaystyle \frac{P_{loss}}{P_0}}={\displaystyle \frac{\left[{\left(1+\frac{1}{I_0^{\prime }}\right)}^2\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]+{\left(-1+\frac{1}{I_0^{\prime }}\right)}^2\left[\pi -2{\sin }^{-1}\left(I_0^{\prime }\right)\right]\right]I_0^{\prime }{\xi }_L}{2\pi }}.$$

(52)

Considering (46), (51), and (52), the efficiency ${\eta }_L={\eta }_{Lf}$ of the full-wave L-type ZCS buck QRC circuit due to the presence of $R_L$ can be written as

$${\eta }_L={\eta }_{Lf}={\displaystyle \frac{1}{1+\frac{P_{loss}}{P_0}}}={\displaystyle \frac{1}{1+\frac{\left[{\left(1+\frac{1}{I_0^{\prime }}\right)}^2\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]+{\left(-1+\frac{1}{I_0^{\prime }}\right)}^2\left[\pi -2{\sin }^{-1}\left(I_0^{\prime }\right)\right]\right]I_0^{\prime }{\xi }_L}{2\pi }}}$$

(53)

The graphical representation of (53) taking into account (34) at $M_f=0.5$ is given in Figure 12.

3.2. Resistive Losses in the Resonant Capacitor Circuit

To determine these losses under [31], we have in the half-mode ZCS buck QRC

$$I_{Crms}^2\approx {\displaystyle \frac{\nu }{2\pi }}{\displaystyle \underset{0}{\overset{\pi }{\int }}\frac{V_{in}^2}{Z_o^2}{\sin }^2\left(\theta \right)d\theta +}{\displaystyle \frac{\nu }{2\pi }}{\displaystyle \underset{0}{\overset{2R_0^{\prime }/M}{\int }}{I}_0^2\left(\theta \right)d\theta }$$

(54)

Solving (54) yields:

$${\displaystyle \frac{P_{loss}}{P_0}}={\displaystyle \frac{\nu {\xi }_C}{M}}\left({\displaystyle \frac{R_0^{\prime }}{2M}}+{\displaystyle \frac{2}{\pi }}\right),$$

(55)

where ${\xi }_C=R_C/2Z_0$ is a damping factor related to $R_C$.

The efficiency due to $R_C$ is given by [31]:

$${\eta }_C={\eta }_{C_h}={\displaystyle \frac{1}{1+\frac{\nu {\xi }_C}{M}\left(\frac{R_0^{\prime }}{2M}+\frac{2}{\pi }\right)}}.$$

(56)

In Figure 13a, (56) is plotted as a function of damping factor ${\eta }_C={\eta }_{C_h}=f({\zeta }_C)$ at $\nu =0.5$. Figure 13b presents the 3D plot of efficiency ${\eta }_C={\eta }_{C_h}=f({\zeta }_C,\nu )$ due to resistive losses related to $R_C$ and frequency ratio $0.1<\nu <1$ at $R_0^{\prime }=const$.

To determine the overall efficiency ${\eta }_h$ of the half-mode ZCS buck QRC due to resistive losses related to $R_L$ and $R_C$, (49) and (56) can be combined as follows:

$$\eta ={\eta }_h={\displaystyle \frac{1}{1+\frac{P_{loss}}{P_0}}}={\displaystyle \frac{1}{1+\left[{\left[1+\left(\frac{1}{I_0^{\prime }}\right)\right]}^2\frac{\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]I_0^{\prime }{\xi }_L}{2\pi I_{0n1}}+\frac{\nu {\xi }_C}{M}\left(\frac{R_0^{\prime }}{2M}+\frac{2}{\pi }\right)\right]}}.$$

(57)

The 3D plots of (57) at $\nu =0.5$ and different normalized load resistances are illustrated in Figure 14.

To determine the resistive losses in the resonant capacitor circuit in the full-mode according to [31], we have

$$I_{Crms}^2\approx {\displaystyle \frac{\nu }{2\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\frac{V_{in}^2}{Z_o^2}{\sin }^2\left(\theta \right)d\theta }.$$

(58)

Considering (46), the efficiency ${\eta }_{Cf}$ due to $R_C$ is given by [31]:

$${\eta }_C={\eta }_{Cf}={\displaystyle \frac{1}{1+\frac{R_0^{\prime }}{M_f}{\xi }_C}}.$$

(59)

The graphical representation of (59) taking into account (34) at $M_f=0.5$ is given in Figure 15.

To determine the overall efficiency ${\eta }_f$ of the full-mode ZCS buck QRC due to resistive losses related to $R_L$ and $R_C$, (52) and (59) can be combined as follows:

$${\eta }_f={\displaystyle \frac{1}{1+\frac{P_{loss}}{P_0}}}={\displaystyle \frac{1}{1+\left[\frac{\left[{\left(1+\frac{1}{I_0^{\prime }}\right)}^2\left[I_0^{\prime }+\pi +{\sin }^{-1}\left(I_0^{\prime }\right)\right]+{\left(-1+\frac{1}{I_0^{\prime }}\right)}^2\left[\pi -2{\sin }^{-1}\left(I_0^{\prime }\right)\right]\right]I_0^{\prime }{\xi }_L}{2\pi }+\frac{R_0^{\prime }}{M_f}{\xi }_C\right]}}$$

(60)

The 3D plots of (60) at $M=M_f=0.5$ and different normalized load resistances are illustrated in Figure 16.

An important design consideration for quasi-resonant converters (QRCs) is to select the normalized load resistor $R_0^{\prime }$ equal to the maximum voltage conversion ratio M. This minimizes peak current stress in the transistor and maximizes efficiency, as both are linked to lower RMS current and reduced losses. In ideal conditions, full-wave and half-wave mode resonant waveforms are identical when $R_{0\min }^{\prime }=M_{\max }$, requiring precise transistor turn-off timing for zero-current switching. However, practical designs are often set to $R_0^{\prime }>M$, which allows a margin for transistor on-time. Proper timing for turning off the transistor depends on the mode: in half-wave mode, it occurs after the resonant-inductor current returns to zero but before the resonant-capacitor voltage falls below the input voltage, whereas in full-wave mode, it occurs during the negative resonant-inductor current interval.

4. Simulation and Experimental Results

A 25 W, L-type half-mode ZCS buck QRC DC/DC is implemented to verify the results of the analysis. The resonant circuit elements are set at 660 kHz and evaluated as follows [3,4,5,9,10]: $L_r=2.7\hspace{0.33em}\mathsf{\mu }\mathrm{H}$ and $C_r=22nF$; input voltage $V_{in}$ = 24 V and the switching frequency f_S: 200 kHz ÷ 300 kHz. For the simulation study we also added resistor $R_C=5\hspace{0.33em}\mathrm{m}\mathsf{\Omega }$ in series with $C_r$ to analyze resistive losses in the resonant capacitor circuit. Furthermore, the direct current resistance (DCR) $DCR=5.65\hspace{0.33em}\mathrm{m}\mathsf{\Omega }$ is added in series with $L_r$. The MOSFET used is SQJA96EP (V_DS = 80 V; I_d = 30 A (Tc = 25 °C); R_ds(on) = 17.9 mΩ (T_j = 25 °C)) and the diodes are V15KM100C (V_rrm = 100 V, I_f = 15 A, V_f = 0.59 V). For the semiconductor devices used, manufacturers offer PSpice models. Output filter inductance $L_o=100\hspace{0.33em}\mathsf{\mu }\mathrm{H}$ and output filter capacitance $C_o=100\hspace{0.33em}\mathsf{\mu }\mathrm{F}$.

The measured and simulation data of the circuit under investigation are presented in Figure 17, Figure 18 and Figure 19, respectively. These figures illustrate the operation of the circuit under 25–30% load and different switching frequencies (200 kHz, 230 kHz, and 300 kHz). Simulation waveforms include the average input current, the output current, the output voltage, the voltage across the resonant capacitor $C_r$, and the Q₁ drain to source voltage ($v_{DS}$). Experimental waveforms include Q₁ control pulse $v_{QG1}$, the resonant current, Q₁ drain to source voltage ($v_{DS}$), and the voltage across the resonant capacitor $C_r$. A summary of these data is provided in

Table 2. The measured circuits parameters under different frequencies and load conditions.

Frequency	Measured and Simulated Circuit Parameters Under Different Frequencies and Load Conditions							Calculated	Relative Error
	Vin, V	Iin, A	Pin, W	Vo, V	Io, A	Po, W	η, %	ηC, % (57)	|δη|,%
200 kHz From simul. From experim.
	24	0.26	6.24	16.02	0.356	5.703	91.4	95.3	4.09
	24	0.254	6.1	16.14	0.356	5.75	94.3	95.3	1.05
230 kHz From simul. From experim.
	24	0.308	7.392	17.47	0.3881	6.78	91.7	95.5	3.98
	24	0.299	7.176	17.5	0.388	6.79	94.6	95.5	0.94
300 kHz From simul. From experim.
	24	0.420	10.08	20.61	0.458	9.44	93.6	95.3	1.78
	24	0.415	9.96	20.5	0.456	9.35	93.9	95.3	1.15

.

As illustrated in Table 2, the data relating to the calculated, simulated, and measured efficiency of the circuit are also presented, along with the relative error |δ_η| between them. The obtained results show that the relative error |δ_η| between the analytical, simulation, and experimental results is below 5%. The utilization of the aforementioned method has been demonstrated to yield results that are considered to be satisfactory for engineering calculations. The analyzed zero-current switching (ZCS) buck quasi-resonant converter (QRC) demonstrates a high efficiency, approximately 95%, which highlights its superior performance.

5. Conclusions

Reducing switching losses in power transistors is essential for minimizing power converter losses and enhancing reliability. This work discusses zero-current switching (ZCS) in L-type buck quasi-resonant converters (QRCs). While commutation at zero current offers several advantages, it has been shown that the maximum current and voltage values during the resonant process in QRC circuits can significantly exceed those in traditional circuits.

The paper presents a comparative analysis of the static losses in the semiconductor switches of conventional pulse-width modulation (PWM) buck converters versus L-type ZCS quasi-resonant buck converters. The findings indicate the following results:

• The transistor’s static losses in ZCS buck QRCs are greater than in traditional PWM buck converters, and their ratio has a limit independent of the resonant tank parameters.
• When the input and output voltages and the load current of the designed ZCS buck QRCs are stable or change only slightly, the resonant tank parameters should be selected to ensure that the normalized output current is close to unity. This condition helps achieve minimal static losses in the switching elements.

Additionally, this study highlights that although resonant tank losses are generally lower than switching losses, they can still significantly affect the overall efficiency of ZCS buck QRCs. The analytical study conducted emphasizes that effective design and careful selection of components are crucial for minimizing these losses and optimizing converter performance.

To verify the analytical results, an L-type half-mode zero-current switching (ZCS) buck quasi-resonant (QRC) DC/DC converter was implemented. Various simulations were conducted using PSpice for TI simulation environment, along with experimental studies at different switching frequencies (200 kHz, 230 kHz, and 300 kHz) and load conditions. The analyzed zero-current switching (ZCS) buck quasi-resonant converter (QRC) demonstrates a high efficiency of approximately 95%, which highlights its superior performance.

The relative error between the analytical, simulation, and experimental results is below 5%. This indicates that the method utilized yields satisfactory results for engineering calculations. Therefore, the discussed a comprehensive methodology described for estimating the efficiency of a buck zero-current switching (ZCS) quasi-resonant DC/DC converter can find applications both in engineering practice and in the education of students in power electronics.