# A Novel Optimization Method for Accurate AC resistance prediction of High Frequency Transformer

Sobhi Barg, Souhaib Barg, Kent Bertilsson

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Sobhi Barg, Souhaib Barg, Kent Bertilsson. A Novel Optimization Method for Accurate AC resistance prediction of High Frequency Transformer. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0196. hal-05101131

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# A Novel Optimization Method for Accurate AC resistance prediction of High Frequency Transformer

Sobhi Barg¹, Souhaib Barg², Kent Bertilsson¹

¹Mid Sweden University, Sundsvall, Sweden

²Central University of Tunis, Tunis, Tunisia

E-Mails: sobhi.barg@miun.se, souhaib1.barg1@gmail.com, kent.bertilsson@miun.se

## Keywords

«Area product», «core geometry coefficient», «optimization», «HF transformer», «AC resistance».

## Abstract

Existing methods for high frequency design have two main limitations which significantly affect the design accuracy: (1) the choice of the suitable filling factor and (2) the inaccurate prediction of the Ac resistance. This paper proposes a novel numerical method to determine an accurate estimation of the Ac resistance and avoid the need of using the filling factor in the design process. The performed experimental results are in good agreement with the developed theoretical results. The study shows that the variation of the optimal B is the inverse of the root-mean-square core volume variation.

## Introduction

Optimizing high frequency magnetic components is a necessary design task to increase the efficiency and the power density. Several methods have been developed in the literature. The main objective of these methods is to select the suitable magnetic core which satisfies the application requirements. The developed optimization methods are based on two fundamental methods which are: the area product method and the core-geometry coefficient method [1-2].

The principal idea of the area product and the core geometry coefficient method is to characterize each core with a dimensional factor, as given by Eq (1) and Eq (2) respectively. This factor is equivalent to an energy term expressed as function the electrical and magnetic variables. These variables are principally the frequency, the magnetic flux density B, the voltage, the current and some parameters such as the filling factor.

The optimization Methods aim to define the optimal solution by determining the optimal magnetic flux density. This later is the key variable which determines the optimal core and the optimal winding designs. This approach has been widely applied in several works such as [3-4].

Although the existing optimized methods have brought several advantages to increase the design accuracy, they still suffer from some limitations which can

significantly affect the design accuracy. These limitations result from two main unknowns: the filling factor and the number of layers. The first one is usually chosen arbitrarily by the designer, and it usually varies from 0.2 to 0.8 depending on the user expertise and the application requirements. The second one, despite it has a significant effect on the Ac resistance, it is neglected or calculated based on inaccurate assumptions in most of the existing methods.

In the work of Hurly [3], the Ac resistance was estimated by assuming the optimal value of the Ac resistance which is slightly independent on the winding structure. A proportional factor was multiplied to the DC resistance can gives the optimal Ac resistance. While this assumption can improve the design compared the original methods, it does not give the exact value of the Ac resistance, and its estimation is far from the practical and real case. In fact, the optimal Ac resistance depends also on the number of layers which is a missing data or unknown variable.

Regarding the filling factor, the choice of this parameter is always made by the designer and its impact on the design solution and the optimization method were not discussed in the literature. In practice, the right choice is made through expertise. The effect of this parameter will be discussed in the next section.

This paper proposes a novel practical optimization technique which allows to accurately predict the Ac resistance by knowing the exact number of layers. This later is the critical variable of the proximity effect affecting the Ac resistance. Therefore, knowing the exact number of layers is necessary to determine a correct optimal design. The paper also proposes a method which exclude the filling factor in the design process. This factor is ambiguous, and a wrong selection of the right value leads to a completely wrong solution. It has a significant impact on the design process and the final output.

The rest of the paper is structured as follows. A review of the existing methods, with a detailed examination and discussion of their principles, their advantages and shortcomings, is presented in section II. The proposed method is developed in section III. A design example and a comparison with existing methods is presented in section IV. Finally, the conclusion of the work is presented in section V.
## Review of the existing methods

The subject of HF transformer optimization is deeply discussed in several research papers aiming at improving the design accuracy and solving the limitations and issues existing in the fundamental area product Ap (1) and core geometry coefficient Kgc (2) methods [1-2].

$$Ap = A_c A_w = \frac{VA}{k_u k_v B_m J} \quad (1)$$

$$K_{gc} = \frac{A_c^2 A_w}{l_m} = A_p \frac{A_c}{l_m} = \frac{VA}{k_u k_v B_m J} \quad (2)$$

where A_c, A_w and l_m are the core cross-section, the core window-aera and the core length. VA, K_u, k_v, B_m and J are the electric power in the primary and the secondary, the filling factor, the converter waveform factor, the maximum magnetic flux density and the current density respectively.

Hurley and Petkov have proposed a clear and detailed approach to determining the optimal magnetic core. The optimal condition was given to defined by the ratio between the core loss and the winding loss. This ratio is given by (3) [3-4].

$$P_c = \frac{2}{\beta} P_w \quad (3)$$

where Pc, Pw and β are the core loss, the winding loss and the Steinmetz parameter respectively. This condition (3) is very useful for the verification of the design solution and the simplification of the design process. It can, for example, be used in the initialization to approximate the unknown winding loss. One limitation of the design methods in references [3] and [4] is the inaccurate calculation of the Ac resistance. It was estimated either by considering only the DC resistance as in [4] or by assuming the optimal Ac resistance value as in [3]. However, in the realized designs, the real value is completely different and the impact of the Ac resistance in the design solution is very significant as discussed in [5].

The results of [3] and [4] were confirmed in [5] which also showed, based on experiments, that the increase in the switching frequency has a limited impact on the power density within the operating frequency (f) range [100-1000] kHz. This result contradicts with the result given in [6] which theoretically shows that the power density increases at a rate of f1/2.

An optimized form of the core geometry coefficient was developed in [7]. In contrast to the previous studies, the optimization variable in [7] is the number of turns (N) instead of the magnetic flux density (B). This choice was also made in [8] where it was mentioned that N is a better choice than B for the optimization of HF magnetic components. In [8], the effect of the Ac resistance was considered using an iterative approach. The value of the Ac resistance was updated in every iteration until the algorithm converges to the optimal solution.

A similar optimized form of the core geometry coefficient formula, which does not require iteration to determine the optimal solution, was developed for HF transformer of LLC converter in [9]. The main

drawback of the formula given in [9] is its complexity and the lack of its physics implication. In fact, it is difficult to interpret the role and the significance of each variable on the solution. As an example, the product of the square of the resonant and the magnetizing current is naturally confusing because current can only sum up and not multiplied. In addition to that, the presence of the magnetizing inductance is also confusing as this variable is a physical dimensional variable in principle and not an energy variable.

Advanced methods which principally used to deal with multi-objective optimization problem were also applied for the optimization of the efficiency, the power density and the cost of the magnetic components.

A geometric programming method was applied in [10] for the optimization of HF transformer. An iterative approach was used to optimize the design of MHz coupled inductors in [11]. Genetic algorithms and Pareto front concept have been applied in several works [12-15].

Regarding the filling factor, it is the most critical design parameter. A small variation in this parameter leads to significant changes in the optimal solution. It affects the calculation of the optimal magnetic flux density and eventually the number of turns and the core size. Fig. 1 and Fig. 2 shows the effect of the filling factor on the normalized: area product, core volume, magnetic flux density and number of turns. As it can be understood from equations (1 and 2), the area product and the core volume change inversely to the filling factor k_u. For instance, the area product, the core volume and the optimal B decrease by 60%, 50% and approximately 45% when the filling factor increases from 0.2 to 0.8. In contrast, the optimal number of turns increases by 80% within the same filling factor range. The increase in k_u allows to use more winding turns which in turn reduces the optimal B and the core size.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Effect of the filling factor on the normalized: optimal B, area product and core volume.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2. Effect of the filling factor on the normalized optimal number of turns.</figcaption></figure>

The issue of the filling factor can be solved through iteration; however, this solution is a demanding task for the designer. A wise and optimal solution, as adopted in this paper, consists in avoiding the need for using this parameter and its complete exclusion.

In conclusion, the existing methods suffer from two main limitations. The first is the dependence of the solution on the filling factor. This dependence is physically incorrect because the filling factor reflect an empty space, which should not have any effect on the solution. The second limitation is the inaccurate prediction of the Ac resistance, which has a significant impact of the performance of the final solution. In the following, we propose a practical and efficient solution that solve the former issues of the existing methods.

## Principle and Steps of the Optimization Method

The principle of the proposed method is to use the number of turns as the main optimization variable instead of the magnetic flux density as used in the existing methods. This change in the choice of the optimization variable can allow to avoid the necessity to the filling factor as a design parameter. For every core, stored in the database, the optimal number of turns can be swept within a well-defined interval and thereafter the optimal core can be determined. The use of the number of turns has one more important advantage, which is the accurate calculation of the Ac resistance through an exact determination of the unknown number of layers. Additionally, unlike the process used in the existing methods, the magnetic flux density will be determined from the number of turns using Faraday's law and not from the solution of Ap or Kgc.

The design process of the optimization method is detailed as follows:

### Step 1: Input data

The first step in the design process consists in constructing the core database, the design specification and the required design parameters. The required design variables are the cross-section A_c, the window

area A_w, the window height h_w, the core mean length l_m. The invariable design-data are the switching frequency, the input voltage (V_in), the output voltage, the output power.

### Step 2: Wire selection

The conductor diameter (mm) is selected based on the following empirical equation (4).

$$d = \sqrt[4]{\frac{I_{max}}{7.2 \pi}} 10^{-3} \quad (4)$$

where I_max is the maximum current flowing in the conductor.

### Step 3: Interval of number of turns

This step consists in determining the interval of the number of turns for each core by calculating the maximum and the minimum number of turns that can be allocated in the transformer window. It is given by the following dimensional condition:

$$k_p W_a \geq S_p N_p + S_s N_s \quad (5)$$

Where k_p is the winding packing factor equals to 0.78 for round conductors and can be adjusted according to the shape of the used conductors. S_p and S_s are the wires cross section of the transformer primary and the secondary calculated from (4).

The secondary number of turns (N_s) can be expressed as function of the turn ratio and the primary number of turns.

$$N_s = \frac{N_p}{n} \quad (6)$$

Substituting Eq (6) in Eq (5), we get the maximum number of turns for the primary winding (N_pmax) (7):

$$N_{pmax} \leq \frac{k_p W_a}{(S_p + \frac{S_s}{n})} \quad (7)$$

The minimum number of turns can be calculated from Faraday's law.

$$N_{pmin} = \frac{V_{in} f_D}{2 f A_c B_{max}} \quad (8)$$

where f_D is function of the duty cycle (D) which depends on the converter topology and V_in is the input voltage. B_max is the maximum allowable magnetic flux density, which is determined for each core from the maximum allowable power loss (P_max). This later is given by Eq (9) as follows:

$$P_{max} = \frac{T_{max} - T_{amb}}{R_{th}} \quad (9)$$

Where R_th is the thermal resistance of the magnetic core provided by the manufacturer and T_max is the maximum allowable temperature imposed by the application requirements. By equating P_max to Steinmetz core loss equation, we can derive B_max as follows:

$$B_{max} = \left( \frac{P_{max}}{k f \alpha V_c} \right)^{\frac{1}{\beta}} \quad (10)$$

Where α and β are the Steinmetz parameters and V_c is the core volume. Eq (10) neglects the effect of the winding loss and oversize the value of B_max. In practice, the value of B_max is much lower. Hence, it is expected that the real N_pmin to be higher. This choice will not affect the optimization results as long as N_pmin of eq (8) is smaller than the real value, which is always the case.
### Step 4: Core loss calculation

Once the interval of the number of turns [$N_{pmin}$-$N_{pmax}$] is known, the interval of the operating magnetic flux density $B_m$ can be determined for each core using Faraday's law and hence the core loss can be calculated. In our paper, the Improved Steinmetz equation (ISE) is applied [16].

$$P_c = k_1 D \left(\frac{f}{2D}\right)^{\alpha_1} B_m^{\beta_1} + \frac{k_2}{2} D' \left(\frac{f}{D'}\right)^{\alpha_2} B_m^{\beta_2} \quad (11)$$

where $D'$ is 1-D.

### Step 5: Ac resistance calculation

Since the core dimensions are given data, the number of layers of the primary and the secondary can be determined with high accuracy. The number of layer ($N_l$) plays a significant role the Ac resistance as the proximity effect goes to the square of $N_l$ as shown in Dowell's equation (13). $N_l$ is given by the following equation.

$$N_l = \frac{N d}{h_w} \quad (12)$$

where $N$, $d$ and $h_w$ are the number of turns, the conductor diameter and the core window-height, the total Ac resistance is calculated using [17].

$$R_{Ac} = R_{DC} X (R_{skin} + R_{prox}) \quad (13)$$

$$X = d_p \sqrt{\frac{\pi \mu_0 f}{\rho}} \quad (14)$$

$$R_{skin} = \left( \frac{\sinh 2A - \sinh 2A}{\cosh 2A - \cos 2A} \right) \quad (15)$$

$$R_{prox} = \frac{N_l^2 - 1}{3} \left( \frac{\sinh A - \sinh A}{\cosh A + \cos 2A} \right) \quad (16)$$

The total loss is the sum of the primary and secondary winding loss given as follows:

$$P_w = R_{Acp} I_p^2 + R_{Acs} I_s^2 \quad (17)$$

The total loss of each core can be plotted as function of the number of turns within the interval [$N_{pmin}$-$N_{pmax}$] and the optimal solution can be figured out graphically or numerically. A design example is shown in the following section for implementation and verification of the proposed method.

## Design Example and Analysis

The proposed method is used to design a transformer for a full-bridge converter with the specifications given in Tab. I. The core database includes 6 ETD cores as given in Tab. I.

<table><caption>Tab. I. Converter specifications</caption><tr><td>Output power</td><td>250 W</td></tr><tr><td>Input voltage</td><td>80 V</td></tr><tr><td>Output voltage</td><td>24 V</td></tr><tr><td>frequency</td><td>100 kHz</td></tr><tr><td>Magnetic material</td><td>3C92</td></tr><tr><td>Core database</td><td>ETD:29 to ETD54</td></tr></table>

An important step in the design, which is mostly forgotten, is the analysis of the results. Analyzing the interaction between the design variables helps to understand the final output. In the following, we explain step by step how the analysis could be done. The first step is to analyze the interval $N_p$ for the different cores. Fig.3 shows the results of our example design. As it can be noticed, the interval of $N_p$ increases with the core volume, however, $N_{pmin}$ decreases as the core volume increases. Both results are expected because bigger core volume has bigger window resulting in wider $N_p$ interval and big $A_c$ results in less $N_{pmin}$.

<figure><img src="image_4.png" /><figcaption>Fig. 3. $N_p$ interval for the different cores indexed from 1 to 6 in the x-axis.</figcaption></figure>

The results noticed for $N_p$ applies same for $B_m$. For same $N_p$, $B_m$ of small cores is higher than the one for bigger cores (Fig.4). It can be concluded that, smaller cores can operate at higher $B_m$ which means they can achieve higher power density than bigger cores. The capability of the small cores to achieve high power density is constrained by the maximum allowable loss and the thermal performance. As it can be seen in Fig.5, for same $N_p$, smaller cores have higher core loss than bigger ones which is due to high $B_m$. As an example, the core loss of ETD34 is about 4 times of ETD54 for 54 turns in the primary.
<figure><img src="image_5.png" /><figcaption>Fig. 4. Variation of B_m with respect to N_p.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Core loss variation with respect to N_p.</figcaption></figure>

In our proposed design approach, the variation of winding loss with respect to N_p can be well analyzed and known, a feature that is not possible in existing methods. Fig.6 depicts P_w for the different cores. As it can be seen, the evolution is non-linear and the loss-curves intersect at some specific N_p. Winding loss depends on the DC resistance and the A_c resistance. This later depends on the number of layers which also depends on the number of turns and the core's window height, however, the DC resistance depends on N_p and the core's cross section. Hence, at low N_p, small cores achieve lower copper losses because within that interval, the copper loss depends more on the DC resistance than the A_c resistance.

For all cores, P_w starts linear with respect to N_p and then we can notice a sudden change in the loss increase, which happens first for small cores (Fig.6). This change is obviously due to step-increase in the number of layers.

Fig.7 shows the dependence of the number of layers on N_p. Bigger cores have a smaller number of layers as they have bigger window height and more importantly because they have lower N_pmin. This later depends on the core's cross section A_c. This is an important point to note as it shows that A_c can indirectly affect the A_c resistance. To understand this point, we can take the example of ETD29 and ETD34. These cores have approximately the same core window height (20 and 22 mm respectively). This basically will give the same number of layers in practice, however, since N_pmin of

ETD34 is less than N_p of ETD29 (10 to 12), we can see their N Ip are different. In addition to that, the slope of N Ip variation is lower for bigger core which allows achieving smaller N Ip with high number of turns. This is a very important characteristic as it helps to reduce significantly the core loss while it linearly increases the winding loss which can be noticed in Fig. 5 and Fig. 6.

The last step is to analyze the total loss and select the optimal design (Fig.8). We can notice that ETD29 does not meet the thermal requirement as its P min is higher than P max. In general, P min decreases as the volume increases, however, the difference is negligible. It can be assumed that all cores have achieved approximately the same P min, however, the safety margin is higher with big cores. All optimal solutions are located in the region where we have number of layers smaller than 2 for the primary, meaning that there is no loss due to the proximity effect. We can also notice that the optimal B decreases with the core volume or more accurately with the cross section as Vc and Ac are proportional. As an example, the optimal B for ETD44 and ETD34 are 0.07 and 0.11 Tesla approximately.

Fig.9 shows that the optimal number of turns varies inversely to the core volume. For instance, ET34 and ETD54, which achieved approximately the same minimum loss, their optimal Np are 17 and 10 respectively. The turn-ratio between these two cores is 1.7 while their volume ratio is about 1/5 meaning that the ETD54 is not economically suitable for the design application. The ideal solution could be ETD39 as it achieves similar minimum loss with enough safety margin for the thermal requirements and acceptable cost and volume.

<figure><img src="image_7.png" /><figcaption>Fig. 6. winding loss variation with respect to N_p</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7. Variation of the number of layers with respect to $N_p$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Total loss variation as function of: (a) $N_p$, (b) $B_m$, the horizontal lines are $P_{max}$ of each core.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Optimal number of turns for the different cores.</figcaption></figure>

## Experiments and Verification

To verify the results of the proposed method, four designed transformers were built. The performance of the transformers is evaluated based on the temperature rise due to core loss and winding loss by performing no-load test and short circuit test.

The characteristics of the tested cores are summarized in Tab. II and the transformers are shown in Fig. 10. The three first cores are ETD34, however they have different ferromagnetic materials, and they have 23, 25 and 21 turns in the primary respectively as given in Tab. II. The first core is made of N87, the second is made of N97 while the third one is made of 3C97. The fourth core is an ETD44 core made of 3C97 ferrite material and has 15 turns in the primary.

Regarding the core loss measurement, as it can be seen in Fig. 11, core 1 and core 3 has achieved the least temperature rise within the magnetic flux density range [0.12-0.25] T, however, core 4 has achieved the worst temperature rise at high B. The thermal performance of core 4 is better at low flux range. As it can be seen, it has the least temperature at B lower than 0.12 T. Beyond this limit, its performance degrades, and the temperature rise increases significantly.

It can be concluded that it is more efficient to operate bigger cores at low B less than 0.1 T to achieve good thermal performance. The experimental results agree with the optimization results (Fig. 8), which shows that the optimal B is about 0.07 T for ETD44.

On the other hand, it is suitable to operate small cores at high B, typically higher than 0.11 T. This result is not common for all materials, but as it can be noticed, some materials such as N97 is characterized by poor thermal performance which does not enable to operate the core at high B.

The obtained result for the short-circuit tests are very similar to what is found in the no-load tests (Fig. 12). ETD34 with N87 and 3C97 has the best thermal performance. Despite that core4 has the least number of turns, its temperature rise is the worst.

In general, the obtained experimental and theoretical results are in good agreement, and they are both consistent with the thermal empirical equations of the thermal resistance and temperature rise for magnetic design. Equation (18) for the temperature rise shows that this later is linearly proportional to the thermal resistance and the core volume. However, the thermal resistance is dependent on the core volume and its empirical mathematical law given by eq (19) shows that the thermal resistance is inversely proportional $V_c^{0.54}$. Therefore, the temperature rise is generally proportional to $V_c^{0.5}$. In other terms, for same loss density, the temperature increases with the root-mean-square of the volume increase. For that reason, it is necessary to reduce the loss density by decreasing the operating B when the volume increases.

As an example, the volume of ETD44 and ETD 34 are 35.6 and 15.26 cm³ respectively. The calculated root mean square of the volume ratio is 1.52 which is
approximately equal to the optimal B ratio found for same cores (1.57).

$$T_r = R_{th}(P_{core} + P_w) \approx R_{th} \left( \frac{P_{core}}{m^3} + \frac{P_w}{m^3} \right) V_c \quad (18)$$

$$R_{th} = 53 V_c^{-0.54} \qquad (19)$$

<figure><img src="image_11.png" /><figcaption>Fig. 10. Photos of the tested transformers</figcaption></figure>

<table><caption>Tab. II. Tested designed transformers specifications</caption><thead><tr><th></th><th>reference</th><th>Material</th><th>N<sub>p</sub></th><th>A<sub>c</sub>(mm<sup>2</sup>)</th></tr></thead><tbody><tr><td>Core 1</td><td>ETD34</td><td>N87</td><td>23</td><td>97</td></tr><tr><td>Core 2</td><td>ETD34</td><td>N97</td><td>25</td><td>97</td></tr><tr><td>Core 3</td><td>ETD34</td><td>3C97</td><td>21</td><td>97</td></tr><tr><td>Core 4</td><td>ETD44</td><td>3C97</td><td>15</td><td>173</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>Fig. 11. Cores temperature in no-load test</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. Cores temperature in short-circuit test</figcaption></figure>

## Conclusion

This paper proposes a novel practical optimization method of high frequency transformer. The proposed method gives the designer a more straightforward approach to determine accurately the optimal magnetic core. It allows to determine exactly the number of layers and give the designer better accuracy in the calculation of the Ac resistance. A major feature of the proposed method compared to all existing methods is it completely exclude the unknown filling factor parameter from the design and optimization process. The theoretical and experimental results performed at 100 kHz for different core sizes show that the optimal B and N<sub>p</sub> decreases with core volume. The main reason of the former result is because the temperature rise increases with the core volume. It was found the decreasing rate of the optimal B is approximately equal to the increasing rate of the core volume.

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