Analytical Modeling and Design of 27.12 MHz Single-Switch DC–DC Converter With PCB Transformer

This paper proposes analytical modeling and design of a 27.12 MHz single-switch dc-dc converter with galvanic isolation using a PCB transformer. The ODE-based analytical approach provides a more accurate circuit design by considering the high-order harmonics and non-ideal components such as the diode capacitances. The proposed dimensionless mathematical model simplifies the design of the desired circuit model and requires less tuning procedure. In addition, the effective analytical model offers the design curves to obtain the optimized circuit parameters at any duty ratio. Meanwhile, the multilayer planar PCB transformer is designed to achieve the isolation and high power density for the high-frequency dc-dc converter. Based on Neumann’s formula, the analytical model of the PCB transformer is proposed to calculate the inductances for the spiral structure accurately in the high-frequency conditions. As a result, the computational burden for calculating the coil inductances is alleviated using the proposed method. The simulation and experimental results using a 3W converter prototype verify the effectiveness and feasibility of the analytical modeling and design of the proposed converter.


I. INTRODUCTION
Achieving high power density to reduce the size of devices is becoming increasingly crucial in power electronics. The passive components take a large volume in power converters, especially the magnetic components occupying a significant space in the system. In order to decrease the volume of the passive components, increasing the switching frequency is one efficient way [1], [2], [3], [4].
However, the high switching loss which is proportional to the frequency affects the system's efficiency under high-frequency conditions. Hence, the resonant converter, especially the class E converter with the single-switch topology, has been widely used to relieve the switching loss by The associate editor coordinating the review of this manuscript and approving it for publication was Kan Liu .
The design procedure of the resonant converter in most of the conventional class E and class 2 researches depends on numerous simplifications and assumptions, such as the common sinusoidal approximations and ideal passive components. In particular, with the approximate design procedures VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ in [11], [12], [13], [14], [15], [16], [17], [18], and [19], only the ZVS operation was satisfied, while the complex tuning processes were required. Moreover, when the switching frequency increases, the decreased values of capacitors and inductors become similar to those of the parasitic elements which means the parasitic values have to be considered in the circuit design and analysis. Therefore, as the assumptions induce the differences between the analysis and real-time experiment, considerable parameter tuning is needed after setting an initial value to obtain the optimal operation point empirically.
To solve this problem, previous research used the analytical approach to design the resonant dc-dc converters using ODEbased analysis or harmonic analysis [20], [21]. Since the analytical design procedure does not require such assumptions, the theoretical model shows highly similar tendencies to the actual circuit. However, the parasitic capacitance of the diodes was not considered, which is critical when the switching frequency increases above 10MHz. Besides, the switch duty ratio is not included in the degrees of freedom during the design procedure in [21], which affects the circuit behavior.
Meanwhile, increasing the switching frequency above 10MHz affects the structure of the transformer for the galvanic isolated converter. The core and copper losses of the conventional wire-wound transformer are increased due to the skin and proximity effects under high-frequency conditions. In order to eliminate the core losses, a coreless PCB transformer has been getting attention for high-frequency applications, providing isolation and power transfer [22], [23], [24], [25], [26], [27]. Furthermore, the planar PCB transformer reduces the volume and vertical dimension of the power electronic circuits, which increases the power density. In addition, using PCB traces makes designing and manufacturing inductors and transformers more precise by achieving well consistency and reducing deviation. Previous research investigated the winding loss, core loss, and stray capacitance to achieve optimal parameters of PCB transformers in [32], [33], [34], [35], and [36]. This paper focuses more on the accurate analytical model instead of achieving high efficiency through loss analysis. Since the previous research often uses 3D FEM tools to design the transformer without an accurate PCB model, they suffer from numerous tuning processes. The 3D Finite Element Method (FEM) simulations are more time-consuming than the analytical calculations due to the computational burden and complexity. Therefore, analytical inductance calculation methods such as Neumann's formula, Grover's formula, and Elliptic integrals are used in the design processes [28], [29], [30]. For the spiral planar PCB transformer, the methods in [28] and [29] are used to calculate the inductances, assuming a complete circle or an octagonal structure. For large radii and a number of winding turns, fairly accurate results are given using the approximations; however, a more accurate method is needed to design the smaller spiral PCB transformer.
This paper proposes the analytical modeling and design of the single-switch high-frequency dc-dc converter. The design model with ODE-based circuit analysis reduces the extensive tuning processes and takes into account the high-order harmonics for achieving the low-voltage stress across the switch. The mathematical analysis indicates that the circuit non-idealities and parasitic elements, including the parasitic capacitance of the diodes, play a critical role in the operation modes of the dc-dc converter. The dimensionless system design curves are presented as a function of the duty ratio. These curves help to simplify the converter design process, including the duty ratio as the degree of freedom. Then, the analytical method is proposed to design and analyze the spiral planar PCB transformer for the magnetically isolated converter topology more precisely. When using the design curves of self and mutual inductances calculated by the analytical method, the desired model of the transformer is obtained quickly without any computational burden of 3D FEM simulations. At last, the simulation and experimental results verify the proposed analytical methods of the converter system and planar PCB transformer.

II. DESIGN APPROACH OF THE PROPOSED SYSTEM
FIGURE 1 shows the topology of the isolated dc-dc converter, which consists of two stages: the inverter stage and the rectifier stage. For the rectifier stage, diodes D 1 and D 2 form a half-wave rectifier. The inverter stage is composed of several passive elements and a single-ended switch. The equivalent model of the transformer with the turns ratio of n, is expressed with the magnetizing inductance L m and leakage inductances absorbed by L r and L rec . In this paper, the analysis focuses on selecting the duty ratio of the single switch S and values of passive elements in the resonant network to satisfy the lowvoltage stress with the ZVS and ZVDS operation.
Using the higher-order harmonic components helps to achieve the low voltage stress and ZVS [14]. FIGURE 2 shows that the third-order harmonic component lowers the peak of the fundamental component. On the other hand, the asymmetric waveform of the second-order harmonic component increases the voltage stress.
Focusing on Z ds , the equivalent impedance seen by the drain-source node of the switch in FIGURE 1, the location of the system pole and zero affects the voltage waveform [14]. Locating the zero at the second-order harmonic and two poles near the fundamental and third-order harmonic realizes the   [19], Z ds is given by To place the zero of Z ds at second-order harmonic which is 2ω sw , where ω sw is the switching frequency, ω sw is derived as Defining the first pole near the fundamental as m 1 ω sw and the other pole near the third-order harmonic as m 3 ω sw , the conditions derived from the denominator of K 1 , K 2 , and K 3 in (1) are given by To minimize the number of passive components, Chung [19] replaced L r and L rec with L p,lk and L s,lk which are the leakage inductances of the equivalent model transformer.
Using the coupling coefficient of the transformer k, the relationships between the magnetizing inductance and leakage inductances are as follows:

III. ODE-BASED CIRCUIT ANALYSIS
In this paper, the optimized circuit design method is proposed by using the dimensionless approach of the analytical model while satisfying ZVS and ZVDS operation without tuning procedure. When using dimensionless circuit analysis, the values of passive elements are easily calculated under various conditions such as switching frequency, output power, input voltage, and output voltage.

A. MODE ANALYSIS
In order to analyze the converter accurately with a mathematical model, the parasitic effects of devices are considered.
FIGURE 4 shows the eight operation modes of the converter. Using Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL), the ordinary differential equations (ODEs) that relate V Cr , V Crec , I Lr , and I Lrec are obtained. With a normalized time θ = ω sw t, the normalized current and voltage are as follows:  where n is the turn ratio of the transformer. To normalize the other terms, the dimensionless parameters are defined by The quality factors Q Lr , Q Lrec , Q Lm , Q Cr , and Q Crec are defined as in (5).
In order to incorporate the equations derived from KCL and KVL in FIGURE 4, m on and d on are defined. Here, m on = 1 when the switch is ON, and m on = 0 when the switch is OFF. For the diodes, d 1,on = 1 and d 2,on = 1 when the diode D1 turns ON and D2 turns ON, respectively. Also, TABLE 1. The eight operation modes and θ conditions. d 1,on = 0 and d 2,on = 0 when the diode D1 is OFF and D2 is OFF, respectively. Using the parameters mentioned above, (8) shown at the bottom of the page is obtained.
The system of ODEs in (8) can be formulated in the matrix form, with coefficient matrix A and vector B, as follows: Using eigenvalues and corresponding eigenvectors of matrix A, the solution of (9) can be found.
The eight operation modes' conditions are in Table 1. The number of the operation modes depends on the diodes' parasitic capacitance magnitude. If the parasitic capacitance C D is negligibly small, some modes are removed. In this paper, since the value of C D is assumed to be similar to the values of the other resonant elements in high-frequency conditions, the eight operation modes are analyzed.
Mode 2 operates when m on = 1, d 1,on = 0 and d 2,on = 0. In mode 2, C D1 and C D2 , the parasitic capacitance of the diode D1 and D2, is charged and discharged, respectively. Then, v D2 , the voltage across the diode D2, drops from 1 + v D to −v D . Mode 2 finishes at θ = θ 2 when v D2 reaches −v D and the diode D2 turns ON.

B. CIRCUIT DESIGN CONDITIONS
In order to ensure the continuity of the capacitor voltages and inductor currents waveform, stationary conditions should be satisfied as follows: In the steady-state conditions, the output current I out is directly related to the average value of current flowing through the diode D1 since the average current of the output capacitor is zero. With the normalization, the equations are derived as To achieve the ZVS characteristics, the switch voltage V ds should satisfy the following condition as v Cr (2π) = 0.  As the ZVDS is expected when the differential value of voltage V ds is zero, the inductor current is expressed as For locating the poles and zero of Z ds in the proper location, (2) and (3) are normalized as 4 = q rec k rec q m (1 − k r k rec ) , In this circuit design procedure, ODEs and design conditions depend on dimensionless parameters: The quality factors of inductors and capacitors are technologically constrained as well as the parasitic components of the diodes and the switch. If the input and output voltage is determined, the value of µ is fixed. For k r and k rec in (7), k r = k rec = k which is the coupling coefficient of the equivalent model transformer. The other parameters in (16) are used to achieve the ZVS and ZVDS.
Equations (11) - (15) show the nine nonlinear equations with ten unconstrained parameters: which means that the degree of freedom is only one. The numerical solutions are obtained using the numerical solver, such as MATLAB or Wolfram Mathematica notebook software.
With the normalized variables and dimensionless parameters, the design curves of the general analysis are obtained without considering any actual circuit parameters. Thus, only the denormalization process with the actual values of V out , I out , n and ω sw is needed to calculate the specific solution of the designed circuit.

IV. PCB TRANSFORMER DESIGN
In the high-frequency resonant converter, the stored energy required for the resonant elements decreases as the switching frequency increases. Then, the design values of inductors are sufficiently reduced to utilize the copper traces on PCB in the air-core transformer model, achieving high power density. Since designing the accurate self and mutual inductance values is required for the planar PCB transformer, the calculation methods such as the analytical modeling and 3D FEM simulation are needed.

A. INDUCTANCE CALCULATION METHOD
In the planar PCB transformers, the leakage and mutual inductances depend on the winding structures. For spiral structure, previous researches calculate the inductances assuming a complete circle or an octagonal structure [28], [29]. Although these approximations give highly accurate results for large radii, more accurate methods are needed when the smaller size of planar PCB transformers are used at higher frequencies. In this paper, the leakage and mutual inductances are calculated using numerical methods with Neumann's formula in the spiral structure. Furthermore, the inductances of the designed transformer are verified by 3D FEM tools (Ansys Maxwell). FIGURE 7 shows the spiral structure model of the PCB transformer. Using the exact  inductance equations for rectangular conductors from [31], (18) is derived at the bottom of the previous page, are derived at the bottom of the page. Here, M ij is the mutual inductance between the i th conductor and j th conductor of the primary and the secondary windings, respectively. H 1 , W 1 , S 1 , H 2 , W 2 and S 2 represent the thickness, width and the separation between the tracks of the primary and the secondary windings, respectively. R i , θ i , r j and ϕ j represent the radial and angular distances of the i th conductor and j th conductor from the central axis, respectively. Also, d represents the central misalignment between the primary and the secondary windings, and H ps represents the separation between the height of the windings.
Since the structure of the windings is spiral, the equations between the radial and angular distances are as follows: where R 1 and R 2 are the inner radius of the primary and the secondary windings, respectively. To solve (18), the numerical method is needed. Gaussian quadrature rule helps to transform the equation in integral form into summation form, which is expressed as where w i is the i th weight. Using (20), M ij becomes (21), as shown at the bottom of the page. The calculations of inductances using the analytical method are implemented by MATLAB. The total mutual inductance is given by: The calculation of the self-inductance is similar to (21). When the i th and j th conductors belong to the same coil, (21) becomes To determine the parameters of the desired transformer from the circuit analysis, the design procedure is required. The values of the inductances of the transformer depend on the parameters such as the number of turns (N ), separation between the tracks (S), separation between the height of the windings (H ps ), conductor width (W ), thickness (H ), inner radius (R), and the central misalignment between the coils (d).
FIGURE 8(a)-8(f) shows the design curves of the self and mutual inductances. As can be seen from the general formula of inductance, the value of inductance forms a proportional relationship with the inner radius R and the number of the turns N . FIGURE 8(b) and 8(c) show that the slope of the larger S and W is small. As the hollow area becomes larger at wide S and W , the variation of the self-inductance reduces. To increase the quality factors of the windings, the conductor width W and conductor separation S should be as small as possible [32]. Also, under the high-frequency conditions, the inner windings in a spiral coil have higher ac resistance compared to outer windings due to the higher flux density. Thus, the ratio of the inner radius to the outer radius, called the hollow factor, is considered to increase the quality factor.
The analytical calculation results are compared with the FEM simulation results. The theoretical values and simulation values of inductances are illustrated in Figure 9. The results show that the proposed method is effective and provides an accurate analytical model without a computational burden for simulation.

B. MULTILAYER CORELESS PCB TRANSFORMER
In the high-frequency converter, using only a single layer of PCB conductors for the transformer is not recommended for high current and low voltage applications [24]; thus, the multilayer coreless PCB transformer is employed to increase the current carrying capacity. From [33], [34], [35], [36], the interleaving structure for the primary and the secondary coils minimizes the leakage inductance and ac winding loss. Since the transformer for the proposed circuit requires a high coupling coefficient above 0.7, the interleaving structure of   the multilayer transformer brings about a desirable result.
Considering the cost and the complexity of the analysis, the 4-layer PCB transformer is suitable for the high-frequency dc-dc converter. FIGURE 10 shows the design of the 4-layer transformer. The turns ratio n is 0.5 since the value of µ is determined to be two from the circuit design. Therefore, one turn and two turns are selected for the primary and secondary sides, respectively. The self and mutual inductances of the primary and secondary coils are calculated as follows: From (24), primary leakage, secondary leakage, magnetizing inductances, and coupling coefficient are derived as

V. PROTOTYPE DESIGN AND EXPERIMENTAL RESULTS
The 27.12MHz/3W single-switch converter prototype with V in = 15V and V out = 15V is implemented as a design example. FIGURE 11 shows the prototype of the isolated dc-dc converter with a planar PCB transformer. After the normalization process, the dimensionless input parameters of the design condition are listed in Table 2. Also, as shown in FIGURE 6(b), the normalized output parameters are calculated using Wolfram Mathematica notebook software and listed in Table 3. By denormalization process using (7), the values of capacitors and inductors in the circuit are in Table 4. From the inductance values in Table 4, the 4-layer spiral coreless PCB transformer is designed. By using the analytical method and simulation, the self and mutual inductances are calculated. The design parameters and inductance values are in Table 5.
The system parameters for the power stage are calculated based on the analytical models and design methods of the converter system and PCB transformer. The components used in the prototype are shown in Table 6. Here, a 60-V GaN device (EPC 2035) is used as a switch device. Because of the output junction capacitance of the switch device (C oss ), the external capacitance (C ext ) connected to the switch in parallel is needed to compensate for the resonant capacitance (C r ). In the experiment, C oss is determined as the value at the normal operating condition, and C ext is determined by 240 pF.
The switch voltage V ds and gate voltage V gs waveforms from ODE-based analysis, PSIM, and experiment are shown in FIGURE 12. The experimental result of the drain-source switch voltage waveform in FIGURE 12(c) shows that the switch device achieves ZVS. Also, the peak voltage is reduced to 34.40 V, 2.29 times the input voltage.
The measured output voltage and output current are 14.6 V and 200 mA, respectively. In the rated conditions with a full load, the measured efficiency is 69.5%. Measurement shows a similar result to the theoretical analysis (15 V,200mA, 71.4%). The loss analysis is shown in FIGURE 13. The switch losses include the conduction loss (P S,con ) and the switching loss (P S,sw ). For the rectifier stage, the conduction losses of two diodes take up about 20% of the total losses. The transformer losses take the largest portion of the losses in the converter system. Under the high-frequency condition, the ac resistance of the transformer becomes much higher than the dc resistance because of the skin and proximity effect. The ac losses could be relieved through the design optimization of PCB transformer windings to improve the efficiency of the converter. The remaining 3% loss is due to the conduction loss of copper trace on PCB.  The properties of the proposed converter have been compared with other high-frequency single-switch converter research, as shown in Table 7. The proposed 27.12MHz DC-DC converter significantly reduces normalized switch voltage stress and the inductor number. Also, by utilizing the planar coreless PCB transformer, the proposed converter has a higher power density than the isolated converters in Table 7. As a result, the experiment results verify the accuracy of the analytical models and the design methods of the converter and the PCB transformer.

VI. CONCLUSION
This paper proposes the analytical method and design of the single-switch isolated dc-dc converter and spiral planar PCB transformer. The analytical approach helps to reduce the voltage stress while achieving ZVS and ZVDS at any duty ratio. Also, more exact analysis and results are obtained in the proposed dimensionless system by considering the parasitic elements, such as the diode capacitances. In addition, the planar PCB transformer is designed based on the accurate calculation method to reduce the size and increase the power density. The accurate analytical modeling of the PCB transformer allows for the practical design of the multilayer spiral transformer without the computational burden of the 3D FEM simulation. A 27.12 MHz prototype is implemented and shows that the peak voltage of the single-switch is reduced to 2.29 times the input voltage. The converter achieves an efficiency of 69.5%, and the experimental results verify the analytical model.