Impact of Parameter Mismatch on Three-Phase Dual-Active-Bridge Converters

Abstract

Three-phase dual active bridge converters (DAB3) are a widely used topology in battery charging applications thanks to their numerous advantages, such as bidirectional power flow, galvanic isolation, low output current ripple, and inherent soft-switching. In such applications, three single-phase transformers are commonly employed as the AC-link to simplify manufacturing and reduce costs. These transformers’ leakage inductance can be utilized instead of the external leakage inductance to achieve high power density. However, the assumption of uniformity in these inductances is not always accurate as they can vary significantly during fabrication. This study presents a comprehensive analysis of the impact of transformer leakage inductance variation, which can deviate by up to 24% from the desired value. The effects of this variation are investigated from different perspectives, including power transfer, soft-switching range, root-mean-square (RMS) current, and the temperature rise of the transformer winding. Although the power transfer and total copper loss of transformers are changed insignificantly even under highly mismatched leakage inductance, the currents and thermal distribution among phases are considerably impacted. Based on statistical probability, a maximum leakage inductance variation threshold of 10–15% compared to the desired value is recommended to ensure the maximum acceptable temperature rise among phases. Experimental results are presented to validate the analysis.

1. Introduction

The recent proliferation of electric vehicles (EVs) has created significant interest in developing quick charging solutions. EVs can be recharged using charging stations, and their large battery capacity can improve the stability of a weak grid by providing power back to the grid via a vehicle-to-grid (V2G) interface [1,2]. To achieve this, V2G charging stations must support specific protocols such as CHAdeMO and CCS/Combo, etc. [3,4]. Bidirectional converters are required for V2G charging stations to function, and the Dual-Active-Bridge (DAB) topology is emerging as a promising option due to its bidirectional power transmission, soft-switching capability, galvanic isolation, and other advantages [5,6,7,8,9,10]. DAB converters can be constructed in single-phase (DAB1) or three-phase (DAB3) forms. DAB1 suffers from high output current ripple and is suitable for low-to-mid power range applications, while DAB3 is capable of outputting a lower amplitude and higher frequency current ripple [11,12], making it more suitable for high-power applications, such as battery charging stations. Additionally, DAB3 has a higher power density compared to DAB1, further making it an attractive choice for high-power applications [13,14]. In this study, DAB3 has been chosen as the designated topology for high-power applications, such as battery charging stations.

One of the most critical parts of the DAB3 converter is the transformer system, which can be constructed by either a three-phase transformer or three single-phase transformers. To simplify the design of the transformer, three single-phase transformers are frequently employed [15,16], with each phase connected to its corresponding transformer. In the DAB converter family, series inductance plays an essential role in power transmission. During the phase overlapping period, energy is charged into the series inductance and then released in the rest of the half switching cycle. The inductance can be integrated into the transformer as leakage inductance to save some space and to improve the power density of the converter system. Therefore, this integration is discussed in detail below.

The value of the leakage inductance of transformers is influenced by the core geometry and winding technique, as reported in [17,18,19,20]. In DAB transformer applications, two commonly used winding techniques are shell-type and core-type. Although the shell-type technique can result in higher AC resistance due to the proximity effect, it enables the high integration of leakage inductance. Based on that reason, in this study, only leakage inductance resulted by shell-typed winding is discussed. The same analysis and discussion manners can be applied for the inductance integration of the core-typed winding technique.

Traditionally, it is usually assumed that the series inductances of all three phases of the DAB3 converter are symmetric to simplify the analysis. However, according to [19], these inductances are dependent on the core parameters and the distance between the primary and secondary windings. In practice, the geometry tolerance of the core can be varied up to 5% as regulated by the IEC 63090:2017 standard from the International Electrotechnical Commission [21]. Additionally, the distance between two windings can also be displaced due to fabrication, further contributing to the mismatch of inductances among phases. Consequently, it is difficult to completely avoid inductance mismatch among phases. The next section demonstrates that the tolerance of the resulting inductance can reach up to 24%, corresponding to the presence of only a 5% core geometry parameter mismatch.

The presence of mismatched leakage inductances in the transformer of the DAB3 converter results in unbalanced phase impedances, leading to an uneven distribution of phase currents. Consequently, several converter characteristics, such as power transfer, soft switching range, RMS current, and thermal distribution, may deviate from their symmetrical case values. Therefore, it is imperative to investigate the effects of such mismatched inductances on the converter’s operation. However, the conventional analysis method is inadequate in this scenario, and there are limited studies reported in the literature to address this issue.

Several research reports have addressed the issue of leakage inductance mismatch, including [22,23,24]. However, these studies only discuss the transient unbalance caused by pulsating load changes. In [25], a simple phase compensation technique is proposed to balance the power of three phases when highly mismatched inductances are present. Nevertheless, there has been limited investigation into the impact of leakage inductance mismatch on the characteristics of the DAB3 converter. Therefore, the effects of inductance mismatch on the converter’s power transfer, soft switching range, RMS current, thermal distribution among phases, and other characteristics have yet to be fully understood.

This paper introduces a novel and comprehensive approach to analyze the DAB3 converter while considering the effects of mismatched leakage inductances. The analysis utilizes the “relative standard deviation ($\rho$)” in relevant equations to investigate important converter characteristics, such as power transfer, soft-switching range, RMS current, and maximum temperature rise among phases. Additionally, this study provides recommendations for the range of $L_k$ based on a statistical probability perspective to ensure an acceptable maximum temperature rise among phases. These recommendations are developed through the analysis of a large number of samples with specific maximum error mismatches which are formed by the normal distribution rule [26].

The rest of this paper is structured as follows: Section 2 provides an in-depth discussion of the influence of core parameters on the mismatch leakage inductance. Section 3 describes the novel approach used to analyze the DAB3 converter, and the relevant equations are formulated by the standard deviation parameter. The analysis of key characteristics of the DAB3 converter using this approach is presented in Section 4. Section 5 shows the experimental results. Finally, conclusions drawn from the study are discussed in Section 6.

2. Error of the Leakage Inductance by the Parameter Tolerance

Figure 1 shows the shell-type winding technique, which greatly helps in integrating a high leakage inductance ($L_k$) in the DAB3 transformer. Following [19], Equation (1) can be used to estimate $L_k$.

$$L_k=\frac{{\mu }_0\left(MLT\right)}{a}\left(c+\frac{b_1+b_2}{3}\right)N_1^2$$

(1)

where $MLT$ is the mean-length-turn of a core, a and b are the winding geometry dimensions, c is the insulation thickness between the two windings, and $N_1$ is the number of turns of the primary winding. Among these parameters, $MLT$, a, b, and c may have a tolerance that leads to the variation of leakage inductance from the desired value.

In the relationship of the core and wire parameters, Equation (1) can be re-written as Equation (2).

$$L_k=2{\mu }_0{N}_1\pi \frac{F+E}{F-E}\left[2D-\frac{2}{3}\frac{N_1}{m_1}O{D}_1-\frac{2}{3}\frac{N_2}{m_2}O{D}_2\right]$$

(2)

where $O{D}_1$ and $O{D}_2,m_1,m_2$ are the wire diameter and number of layers at primary and secondary sides, respectively, and $E,F,D$ are the diameter of the center leg, inner width, and inner height of core, respectively.

Following the IEC 63090:2017 standard, the information about the dimensional tolerances for reference dimensions of each core shape are given. Specifically, the relative error of the core geometry can be varied up to 5%. From Equation (3), it is noted that $m_1,m_2$, and $N_1$ are always integer numbers with no variation. Let $\Delta F$, $\Delta E$, $\Delta O{D}_1$, and $\Delta O{D}_2$ be the absolute error of F, E, $O{D}_1$, and $O{D}_2$ parameters, respectively. Thus, the absolute error of the estimated leakage inductance ($\Delta L_k$) can be determined by Equation (3).

$$\Delta L_k\approx \frac{\partial L_k}{\partial F}\Delta F+\frac{\partial L_k}{\partial E}\Delta E+\frac{\partial L_k}{\partial O{D}_1}\Delta O{D}_1+\frac{\partial L_k}{\partial D}\Delta D+\frac{\partial L_k}{\partial O{D}_2}\Delta O{D}_2$$

(3)

A case study was conducted to investigate the impact of core geometry tolerance on the value of the integrated leakage inductance using the ETD59/31/22 core [27]. Note that the same investigation manner can also be applied for other core shapes (e.g., EE, ELP, EC, ER, etc.), and the conclusions gained from investigating the ETD core still hold true because only the tolerance of the core geometry is involved in the analysis. The core shaping does not contribute to the analysis and thus does not affect the conclusion.

The dimensions of both the core and wire structures can be adjusted with a tolerance range of up to 5% as mentioned above. Changes are made incrementally, with each step representing one-tenth of the maximum tolerance for each parameter in Equation (3). This process generates over 10,000 distinct samples. Subsequently, the relative errors of the data set were analyzed, and the results were synthesized using a distribution graph as shown in Figure 2. The distribution closely followed the normal distribution, with mean and standard deviation values of 0% and 7.76%, respectively. Leakage inductance values were found to be centered around a 5% mismatch of the estimated value, with 48.08% of the total cases falling within this range. Moreover, 80.26% of the total samples had a relative error of leakage inductance that varied around 10%, while 19.74% of the samples had a relative error greater than 10%. In the worst case, the relative error of $L_k$ could be as high as 22%. Therefore, it is crucial to pay attention to the error caused by the dimensional tolerance of the core to ensure the stability of the system on a large scale.

3. Steady-State Analysis

Figure 3 describes the circuit diagram of a DAB3 converter. Terminal DC voltages are $V_1$ and $V_2$, respectively. Three single-phase high-frequency transformers linking two three-phase inverters are connected in the $Y\text{-}Y$ configuration. All transformers have the same winding ratio of n:1.

Assume that single phase shift modulation is used to handle the power flow in the converter, i.e., six-step modulation with 180 degrees conduction mode, and the output voltage of the secondary bridge is shifted by an angle of $\varphi$ degrees with respect to that of the primary bridge.

Figure 4 shows the equivalent circuit diagram of the converter. In the figure, $V_A,V_B$, and $V_C$ are the phase voltages with respect to the negative pole of the input DC terminal, and $V_a,V_b$, and $V_c$ are the phase voltages with respect to the negative pole of the output DC terminal. Since two sides are isolated, there is a common-mode voltage $V_{cm}$ between the two neutral points. The phase impedance $Z_{kabc}$ is determined by

$$Z_{kabc}=\sqrt{r_{abc}^2+X_{kabc}^2}$$

(4)

where $X_{kabc}=2\pi f·L_{kabc}$; f is the harmonic frequency and $L_{kabc}$ is the leakage inductance of a phase; and $r_{abc}$ is the series equivalent resistance of a phase. $r_{abc}$ includes the winding resistance, stray resistance of wires and PCB, and the ON-state resistance of the switches. Since at high frequency, $r_{abc}$ is usually far smaller than $X_{kabc}$, the phase impedance can thus be approximated by the phase reactance:

$$Z_{kabc}\approx X_{kabc}=2\pi f_{sw}·L_{kabc}$$

(5)

The phase currents can then be calculated by

$$\left\{\begin{array}{cc}\hfill L_{ka}{\displaystyle \frac{d{i}_a\left(t\right)}{dt}}& =V_A\left(t\right)-V_a\left(t\right)-V_{cm}\left(t\right)\hfill \\ \hfill L_{kb}{\displaystyle \frac{d{i}_b\left(t\right)}{dt}}& =V_B\left(t\right)-V_b\left(t\right)-V_{cm}\left(t\right)\hfill \\ \hfill L_{kc}{\displaystyle \frac{d{i}_c\left(t\right)}{dt}}& =V_C\left(t\right)-V_c\left(t\right)-V_{cm}\left(t\right)\hfill \end{array}\right.$$

(6)

By solving (6) in each switching state, transition currents can be obtained. However, in (6), while ${\overrightarrow{V}}_{ABC}\left(t\right)=V_A\left(t\right),V_B\left(t\right),V_C\left(t\right)$ and ${\overrightarrow{V}}_{abc}\left(t\right)=V_a\left(t\right),V_b\left(t\right),V_c\left(t\right)$ can easily be derived from the switching state, the determination of $V_{cm}\left(t\right)$ is not that easy. The analysis below considers two circumstances when transformers parameters are identical and mismatched.

3.1. Case I: Transformer Parameters Are Identical

If the leakage inductances of all phases are identical, taking the summation of three equations in (6) and noticing that

$$i_a\left(t\right)+i_b\left(t\right)+i_c\left(t\right)=0,$$

since the transformer is Wye–Wye connected, $V_{cm}\left(t\right)$ can be solved as

$$V_{cm}\left(t\right)={\displaystyle \frac{1}{3}}\left(\left(V_A\left(t\right)-V_a\left(t\right)\right)+\left(V_B\left(t\right)-V_b\left(t\right)\right)+\left(V_C\left(t\right)-V_c\left(t\right)\right)\right)$$

(7)

Substituting (7) into (6), according to [28], the transition currents of phase A can be formulated as

$$\left\{\begin{array}{cc}\hfill I_0& =-I_M\left(2(1-M)+6M{D}_{\varphi }\right)\hfill \\ \hfill I_1& =I_M\left(-2(1-M)+6D_{\varphi }\right)\hfill \\ \hfill I_2& =I_M\left(-(1-M)+6M{D}_{\varphi }\right)\hfill \\ \hfill I_3& =I_M\left(-(1-M)+12D_{\varphi }\right)\hfill \\ \hfill I_4& =I_M\left((1-M)+12M{D}_{\varphi }\right)\hfill \\ \hfill I_5& =I_M\left((1-M)+6D_{\varphi }\right)\hfill \end{array}\right.$$

(8)

where $I_M={\displaystyle \frac{V_1}{18f_s{L}_k}}$; M is the voltage conversion ratio, $M={\displaystyle \frac{n{V}_2}{V_1}}$, and $D_{\varphi }$ is the normalized phase shift, $D_{\psi }={\displaystyle \frac{\varphi }{2\pi }}$. The normalized phase shift $D_{\varphi }$ is usually designed as for $D_{\varphi }<{\displaystyle \frac{1}{6}}$ (or 60 degrees in electric angle scale) for better reactive power reduction.

3.2. Case II: Transformer Parameters Mismatch

As described in the previous section, the core parameters can be varied around 5%, which leads to the variation of leakage inductance from the desired value. When the leakage inductances of phases are non-identical, $V_{cm}\left(t\right)$ cannot be solved as presented in the previous section, which makes the analysis and design of the converter more complicated. This section proposes a formulation for the case where parameter mismatch is present.

First, let $L_k$ be the average leakage inductance. $L_k$ is defined as follows:

$$L_k={\displaystyle \frac{L_{ka}+L_{kb}+L_{kc}}{3}}$$

(9)

Let $\rho$ is the relative standard deviation of the leakage inductance of transformers. It can be calculated as:

$$\begin{array}{cc}\hfill \rho & =\sqrt{{\displaystyle \frac{1}{3}}\left[{\left({\displaystyle \frac{L_{ka}}{L_k}}-1\right)}^2+{\left({\displaystyle \frac{L_{kb}}{L_k}}-1\right)}^2+{\left({\displaystyle \frac{L_{kc}}{L_k}}-1\right)}^2\right]}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & =\sqrt{2\left(1-{\displaystyle \frac{L_{ka}{L}_{kb}+L_{ka}{L}_{kc}+L_{kb}{L}_{kc}}{3L_k^2}}\right)}\hfill \end{array}$$

(11)

Now, let us define $L_{\sigma }$ as follows:

$$L_{\sigma }={\displaystyle \frac{L_{ka}{L}_{kb}+L_{ka}{L}_{kc}+L_{kb}{L}_{kc}}{L_{ka}+L_{kb}+L_{kc}}},$$

(12)

From (11) and (12), $L_{\sigma }$ is thus determined by

$$L_{\sigma }=\left(1-{\displaystyle \frac{{\rho }^2}{2}}\right)L_k$$

(13)

The value of $L_{\sigma }$ depends on the deviation of phase inductance from the mean value of $L_k$. When there is no deviation, $L_{\sigma }=L_k$. When there is a deviation, $L_{\sigma }$ is smaller than $L_k$. Now, let ${\sigma }_a$, ${\sigma }_b$, and ${\sigma }_c$ be ${\displaystyle \frac{L_{ka}}{L_{\sigma }}}$, ${\displaystyle \frac{L_{kb}}{L_{\sigma }}}$, and ${\displaystyle \frac{L_{kc}}{L_{\sigma }}}$, respectively. Obviously, ${\sigma }_a$, ${\sigma }_b$ and ${\sigma }_c$ are real and positive and equal to unity when transformer parameters are identical. Besides, ${\sigma }_a$, ${\sigma }_b$, and ${\sigma }_c$ have the following characteristics:

$$\left\{\begin{array}{cc}\hfill {\sigma }_a+{\sigma }_b+{\sigma }_c& ={\sigma }_a{\sigma }_b+{\sigma }_b{\sigma }_c+{\sigma }_c{\sigma }_a={\displaystyle \frac{6}{2-{\rho }^2}}\hfill \\ \hfill {\displaystyle \frac{1}{{\sigma }_a}}+{\displaystyle \frac{1}{{\sigma }_b}}+{\displaystyle \frac{1}{{\sigma }_c}}& ={\displaystyle \frac{1}{{\sigma }_a{\sigma }_b}}+{\displaystyle \frac{1}{{\sigma }_b{\sigma }_c}}+{\displaystyle \frac{1}{{\sigma }_c{\sigma }_a}}={\displaystyle \frac{6}{2-{\rho }^2}}·{\displaystyle \frac{1}{{\sigma }_a{\sigma }_b{\sigma }_c}}\hfill \end{array}\right.$$

(14)

Dividing the first equation of (6) by ${\sigma }_a$, the second by ${\sigma }_b$, and the last by ${\sigma }_c$, respectively, (6) becomes

$$\left\{\begin{array}{cc}\hfill L_{\sigma }{\displaystyle \frac{d{i}_a\left(t\right)}{dt}}& ={\displaystyle \frac{V_A\left(t\right)-V_a\left(t\right)}{{\sigma }_a}}-{\displaystyle \frac{1}{{\sigma }_a}}{V}_{cm}\left(t\right)\hfill \\ \hfill L_{\sigma }{\displaystyle \frac{d{i}_b\left(t\right)}{dt}}& ={\displaystyle \frac{V_B\left(t\right)-V_b\left(t\right)}{{\sigma }_b}}-{\displaystyle \frac{1}{{\sigma }_b}}{V}_{cm}\left(t\right)\hfill \\ \hfill L_{\sigma }{\displaystyle \frac{d{i}_c\left(t\right)}{dt}}& ={\displaystyle \frac{V_C\left(t\right)-V_c\left(t\right)}{{\sigma }_c}}-{\displaystyle \frac{1}{{\sigma }_c}}{V}_{cm}\left(t\right)\hfill \end{array}\right.$$

(15)

Solving (15) for $V_{cm}\left(t\right)$, the common-mode voltage is obtained as

$$V_{cm}\left(t\right)={\displaystyle \frac{1}{{\displaystyle \frac{1}{{\sigma }_a}}+{\displaystyle \frac{1}{{\sigma }_b}}+{\displaystyle \frac{1}{{\sigma }_c}}}}·\left({\displaystyle \frac{V_A\left(t\right)-V_a\left(t\right)}{{\sigma }_a}}+{\displaystyle \frac{V_B\left(t\right)-V_b\left(t\right)}{{\sigma }_b}}+{\displaystyle \frac{V_C\left(t\right)-V_c\left(t\right)}{{\sigma }_c}}\right)$$

(16)

Substituting the obtained common-mode voltage into (15), we have

$$L_{\sigma }{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_a\left(t\right)\\ i_b\left(t\right)\\ i_c\left(t\right)\end{array}\right]={\displaystyle \frac{1}{{\sigma }_a+{\sigma }_b+{\sigma }_c}}\left[\begin{array}{ccc}{\sigma }_b+{\sigma }_c& -{\sigma }_c& -{\sigma }_b\\ -{\sigma }_c& {\sigma }_c+{\sigma }_a& -{\sigma }_a\\ -{\sigma }_b& -{\sigma }_a& {\sigma }_a+{\sigma }_b\end{array}\right]·\left[\begin{array}{c}{V}_A\left(t\right)-V_a\left(t\right)\\ V_B\left(t\right)-V_b\left(t\right)\\ V_C\left(t\right)-V_c\left(t\right)\end{array}\right]$$

(17)

Now, considering (17) in each switching state then solving for transition currents, phase A, B, and C currents can be represented by (18), (19), and (20), respectively:

$$\begin{array}{cc}\hfill I_a& ={\displaystyle \frac{V_1}{18f_s{L}_k}}\times \left\{\begin{array}{c}-\left({\sigma }_b+{\sigma }_c-{\sigma }_{b}M-{\sigma }_{c}M+6D_{\varphi }{\sigma }_{b}M\right)\hfill \\ \left(6D_{\varphi }{\sigma }_c-{\sigma }_c-{\sigma }_b+{\sigma }_{b}M+{\sigma }_{c}M\right)\hfill \\ \left({\sigma }_{b}M-{\sigma }_b+6D_{\varphi }{\sigma }_{c}M\right)\hfill \\ \left(6D_{\varphi }{\sigma }_b-{\sigma }_b+6D_{\varphi }{\sigma }_c+{\sigma }_{b}M\right)\hfill \\ \left({\sigma }_c-{\sigma }_{c}M+6D_{\varphi }{\sigma }_{b}M+6D_{\varphi }{\sigma }_{c}M\right)\hfill \\ \left({\sigma }_c+6D_{\varphi }{\sigma }_b-{\sigma }_{c}M\right)\hfill \end{array}\right.\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill I_b& ={\displaystyle \frac{V_1}{18f_s{L}_k}}\times \left\{\begin{array}{c}(1-M){\sigma }_c-6M{\sigma }_a{D}_{\varphi }\hfill \\ (1-M){\sigma }_c-6({\sigma }_a+{\sigma }_c)D_{\varphi }\hfill \\ -(1-M){\sigma }_a-6M({\sigma }_a+{\sigma }_c)D_{\varphi }\hfill \\ -(1-M){\sigma }_a-6{\sigma }_c{D}_{\varphi }\hfill \\ -(1-M)({\sigma }_a+{\sigma }_c)-6M{\sigma }_c{D}_{\varphi }\hfill \\ -(1-M)({\sigma }_a+{\sigma }_c)+6{\sigma }_a{D}_{\varphi }\hfill \end{array}\right.\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill I_c& ={\displaystyle \frac{V_1}{18f_s{L}_k}}\times \left\{\begin{array}{c}(1-M){\sigma }_b+6M({\sigma }_a+{\sigma }_b)D_{\varphi }\hfill \\ (1-M){\sigma }_b+6{\sigma }_a{D}_{\varphi }\hfill \\ (1-M)({\sigma }_a+{\sigma }_b)+6M{\sigma }_a{D}_{\varphi }\hfill \\ (1-M)({\sigma }_a+{\sigma }_b)-6{\sigma }_b{D}_{\varphi }\hfill \\ (1-M){\sigma }_a-6M{\sigma }_b{D}_{\varphi }\hfill \\ (1-M){\sigma }_a-6({\sigma }_a+{\sigma }_b)D_{\varphi }\hfill \end{array}\right.\hfill \end{array}$$

(20)

Equation (18) represents the transition current of phase A considering the parameter mismatch of transformers. The mismatch is denoted as ${\sigma }_a$, ${\sigma }_b$, and ${\sigma }_c$ for each phase, respectively. Obviously, when there are no mismatches, the parameters are the same for all three phases, ${\sigma }_a={\sigma }_b={\sigma }_c=1$, and then (18) becomes identical to (6).

4. Discussion

4.1. Power Characteristics

From (18)–(20), we assume a loss-less converter, and the normalized output power can be calculated by

$$\begin{array}{cc}\hfill P_{out}^{*}& ={\displaystyle \frac{P_{out}}{P_m}}=4M{D}_{\varphi }\left(2-3D_{\varphi }\right)·{\displaystyle \frac{{\sigma }_a+{\sigma }_b+{\sigma }_c}{3}}\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & =\left(4M{D}_{\varphi }\left(2-3D_{\varphi }\right)\right)·{\displaystyle \frac{2}{2-{\rho }^2}}\hfill \end{array}$$

(22)

where $P_m={\displaystyle \frac{V_1^2}{12f_s{L}_k}}$. The first factor in (22) denotes the output power of the converter when parameters are identical. The second factor shows to the effect of parameter deviation on the generated power of the converter. Since ${\rho }^2$ is always positive, (22) implies that output power increases when parameter mismatch is present. However, as shown in Figure 5, even though the relative standard deviation is about 10%, the power increment is only 0.5%. Therefore, it can be concluded that the parameter mismatch does not have a strong effect on the power characteristics of the converter.

4.2. Soft-Switching Conditions

From (18)–(20), soft-switching condition for each phase can be determined as

$$\left\{\begin{array}{cc}\hfill I_{a0}& =-(1-M)({\sigma }_b+{\sigma }_c)-6M{\sigma }_b{D}_{\varphi }\le 0\hfill \\ \hfill I_{a1}& =-(1-M)({\sigma }_b+{\sigma }_c)+6{\sigma }_c{D}_{\varphi }\ge 0\hfill \\ \hfill I_{b4}& =-(1-M)({\sigma }_a+{\sigma }_c)-6M{\sigma }_c{D}_{\varphi }\le 0\hfill \\ \hfill I_{b5}& =-(1-M)({\sigma }_a+{\sigma }_c)+6{\sigma }_a{D}_{\varphi }\ge 0\hfill \\ \hfill I_{c2}& =(1-M)({\sigma }_a+{\sigma }_b)+6M{\sigma }_a{D}_{\varphi }\ge 0\hfill \\ \hfill I_{c3}& =(1-M)({\sigma }_a+{\sigma }_b)-6{\sigma }_b{D}_{\varphi }\le 0\hfill \end{array}\right.$$

(23)

Solving (23) for $D_{\varphi }$, the soft-switching condition can be found as

$$D_{\varphi }\ge max\left\{\begin{array}{c}{\displaystyle \frac{{\sigma }_a+{\sigma }_b}{6{\sigma }_a}}·\left(1-{\displaystyle \frac{1}{M}}\right)\hfill \\ {\displaystyle \frac{{\sigma }_b+{\sigma }_c}{6{\sigma }_b}}·\left(1-{\displaystyle \frac{1}{M}}\right)\hfill \\ {\displaystyle \frac{{\sigma }_a+{\sigma }_c}{6{\sigma }_c}}·\left(1-{\displaystyle \frac{1}{M}}\right)\hfill \\ {\displaystyle \frac{{\sigma }_a+{\sigma }_c}{6{\sigma }_a}}·(1-M)\hfill \\ {\displaystyle \frac{{\sigma }_a+{\sigma }_b}{6{\sigma }_b}}·(1-M)\hfill \\ {\displaystyle \frac{{\sigma }_b+{\sigma }_c}{6{\sigma }_c}}·(1-M)\hfill \end{array}\right.$$

(24)

Figure 6 shows the power characteristics of the converter with regard to its soft-switching boundary. The blue continuous line denotes the conventional soft-switching boundary when there is no parameter mismatch (or relative standard deviation $\rho =0\%$). When the parameters deviate from their mean value, the soft-switching area tends to be narrower, particularly when the voltage conversion ratio is greatly different from unity. For example, soft-switching can be attained even at the voltage conversion ratio of 0.66 when transferring a power of 0.5 p.u. However, if the relative standard deviation of leakage inductance $\rho$ is 12.3%, soft-switching is lost when transferring the same amount of power at a voltage ratio M smaller than 0.75.

4.3. Conduction Loss

The root-mean-squared (RMS) value of the primary current of transformers can be derived as

$$\begin{array}{cc}\hfill I_{a,rms}& =\sqrt{{\displaystyle \frac{{\sigma }_b^2+{\sigma }_b{\sigma }_c+{\sigma }_c^2}{3}}}·I_{RMS}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill I_{b,rms}& =\sqrt{{\displaystyle \frac{{\sigma }_a^2+{\sigma }_a{\sigma }_c+{\sigma }_c^2}{3}}}·I_{RMS}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill I_{c,rms}& =\sqrt{{\displaystyle \frac{{\sigma }_a^2+{\sigma }_a{\sigma }_b+{\sigma }_b^2}{3}}}·I_{RMS}\hfill \end{array}$$

(27)

where $I_{RMS}$ is the nominal RMS current when parameters are identical, $I_{RMS}={\displaystyle \frac{V_1}{18f_s{L}_k}}·\sqrt{216D_{\varphi }^2(1-D_{\varphi })M+5{(M-1)}^2}$.

If the AC resistance of transformers is the same as $r_w$, the total copper loss is

$$\begin{array}{cc}\hfill \Delta P_{Cu}& =\left(I_{a,rms}^2+I_{b_{r}ms}^2+I_{c,rms}^2\right)r_w\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{2({\sigma }_a^2+{\sigma }_b^2+{\sigma }_c^2)+({\sigma }_a{\sigma }_b+{\sigma }_b{\sigma }_c+{\sigma }_c{\sigma }_a)}{9}}·\left(3I_{RMS}^2{r}_w\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{2(2+{\rho }^2)}{{(2-{\rho }^2)}^2}}\Delta P_{Cu,nom}\hfill \end{array}$$

(30)

Figure 7 represents the dependence of relative copper loss with respect to the relative standard deviation of leakage inductance according to (30). As shown, when $\rho$ is 10%, the total copper loss increases about 1.5% compared to that when parameters are identical. Therefore, it could be concluded that the non-identity of leakage inductance causes an insignificant impact on the copper loss of transformers.

4.4. Temperature Rise among Phase Transformers

As discussed in Section 4.3, it was found that the total copper loss of the transformer experienced only a minor increase when the leakage inductance was significantly altered. This suggests that the phase currents were not being distributed evenly, resulting in an unequal distribution of thermal energy among the transformer’s phases. However, despite this observation, there has been a lack of research on the impact of variations in leakage inductance on the temperature rise of phase transformers. Therefore, this section aims to investigate this particular area of interest.

The study comprised six individual cases, in which the leakage inductance was varied from 5% to 30% of the desired value in increments of 5%. As discussed in Section 2, the variation of leakage inductances by the dimensional tolerance followed the normal distribution rule. Accordingly, three different leakage inductance values were randomly chosen around the relative variation, resulting in over 15,000 samples. The generated data sets followed the normal distribution. The RMS current of each phase was then calculated using Equations (25)–(27), assuming equal resistance for each phase. This allowed the determination of the power loss and temperature rise among the phases. The results were summarized using normal distribution graphs (Figure 8) in which the mean and the standard deviation values are denoted as $\mu$ and $\rho$, respectively.

To derive the probability of the maximum temperature rise exceeding 10% compared to the base case from the normal distribution graph, two methods can be employed. The first method is the empirical rule, also known as the 68, 95, 99.7 rule [26], which is enabled to estimate the proportion of observations lying within a specified number of standard deviations from the mean in a normal distribution. For other ranges beyond the empirical rule, the Standard Normal Distribution Table can be utilized to determine the corresponding probability values, as described in the relevant literature [26].

When the percentage mismatch in $L_k$ is less than 5%, as depicted in Figure 8a, the maximum temperature rise ($\Delta T$) among phases consistently remains below 10% in all instances. In scenarios where the maximum deviation of $L_k$ ranges between 10% and 15% (Figure 8b,c), there are only a few $\Delta T$ values that exceed 10% of the average case. This phenomenon is mainly observed at the border region, where the errors in the leakage inductances are at their maximum and minimum values. Thus, with a maximum variation of 15% in $L_k$, it is almost certain that the maximum temperature rise will remain below 10% of the base case in most scenarios.

However, when the maximum variation in $L_k$ attains 20% of the base value (Figure 8d, it is crucial to consider the maximum $\Delta T$. The likelihood of the maximum $\Delta T$ exceeding 10% relative to the base case is approximately 20%. While this probability may not be high, it should serve as the threshold for $L_k$ variation. If the variation exceeds this threshold, it becomes a cause for concern, particularly as over 40% of the case samples exhibit this issue when the maximum variation in $L_k$ surpasses 25%, as shown in Figure 8e. Moreover, when the maximum variation is 30%, over half of the total cases indicate probabilities of $\Delta T$ greater than 10% of the base case (Figure 8f). Therefore, in conclusion, when the $L_k$ mismatch exceeds 20%, it should be a significant cause for concern, and thermal balancing techniques must be implemented to address the issue.

5. Experimental Results

The purpose of the experiments is to verify the impact of leakage mismatch on the characteristics of the DAB3 converter, as discussed in the previous section. To conduct these experiments, the DC-bus voltage at the input side is connected to a programmable power supply (BK Precision MR50040), while the output is linked to a programmable electronic load (BK Precision 8116) configured in constant-voltage (CV) mode. The input and output voltages are set to specific values: $V_1$ = 50 V and $V_2$ = 50 V. The SiC MOSFET of C2M0025120D from CREE was used for both the primary and secondary inverters. Three single-phase transformers were employed, and their key parameters are listed in

Table 1. The key parameters of the DAB3 converter.

Parameters	Symbol	Value
Frequency	$f_s$	25 kHz
Turn ratio	n	1 (10:10)
Magnetizing inductance	$L_m$	0.5 mH
Base leakage inductance	$L_k$	12.5 $\mathsf{\mu }$H
Wire		Lizt 600AWG38
Core		ETD59/31/22 (N87)

. The entire system was controlled by an ST MicroElectronics NUCLEO-STM32G474RE board.

During the experiments, the leakage inductances referring to the transformer’s primary side are accomplished by shorting the secondary sides of the transformers. The leakage inductance values are then measured using an LCR meter 300 kHz (BK Precision 891) to ensure precise measurements. The phase current waveforms are captured using a Rogowski current probe (PEM CWT Mini50HF), which provides reliable and high-resolution data. Due to limitations in the number of available current measurement tools, the three-phase currents are measured independently for each experiment scenario. Subsequently, the phase current data are exported and processed to combine them on the same plot, allowing for a comprehensive analysis and comparison of the waveforms.

The results of the measurement of the transformers’ leakage inductance using the core parameters listed in Table 1 are presented in Figure 9. It should be noted that the leakage inductance can be influenced by core parameter tolerance and winding technique, as previously mentioned. To address this variability, 20 transformers were fabricated using an EDT59 core and Litz wire (600AWG38). The expected leakage inductance, as per theoretical calculations, is around 12.5 $\mathsf{\mu }$H. However, due to core parameter tolerances and winding techniques, the measured leakage inductance ranged from 11.62 $\mathsf{\mu }$H to 13.75 $\mathsf{\mu }$H. The results indicate that the leakage inductance values deviate from the design value by approximately 10% and fall within the high probability region, as discussed in Section 2.

It should be noted that a small number of samples exceeded the above variation as shown in Figure 2. In the worst-case scenario, the maximum variation of $L_k$ was found to be 24%. To assess the effect of the leakage inductance variation on the DAB3 converter characteristics, three case studies were conducted. Case study 1 represents the symmetrical case, where the $L_k$ values for phases A, B, and C are 12.5 $\mathsf{\mu }$H, 12.4 $\mathsf{\mu }$H, and 12.75 $\mathsf{\mu }$H, respectively. For case studies 2 and 3, the 24% maximum variation of $L_k$ was taken into account, with values of (13.05 $\mathsf{\mu }$H, 10.43 $\mathsf{\mu }$H, and 15.5 $\mathsf{\mu }$H) and (10.43 $\mathsf{\mu }$H, 10.86 $\mathsf{\mu }$H, and 15.5 $\mathsf{\mu }$H) corresponding to phases A, B, and C, respectively.

The standard deviations $\rho$ of case studies 2 and 3 are 27.59% and 30.3%, respectively. The total copper loss of the transformer changes by about 10% compared with the symmetrical case as per Equation (30). These findings were supported by experimental results, as shown in Figure 10. However, the impact of varied leakage inductances on phase current shapes is significant, as evident from Figure 11, in which the phase-shift is 20 degrees. In case study 1, where the mismatch of $L_k$ is just 2%, which is lower than 5%, the phase currents are almost balanced, as shown in Figure 11a. However, in case studies 2 and 3, the maximum variation of leakage inductance is as high as 24%, leading to significant phase current imbalances. Specifically, the peak current of Phase B is approximately 1.3 A (22% compared to the symmetrical case) higher than that of Phase C (Figure 11b). The same results are observed in case study 3 with about 2 A (33.8% compared to the symmetrical case) (Figure 11c). Consequently, the RMS phase currents differ significantly, as demonstrated in Figure 12. For instance, with 15 degrees of phase shift, the phase RMS current of case study 1 is almost balanced due to only a 2% variation in $L_k$. In contrast, in case studies 2 and 3, the difference between the highest and lowest RMS current is large, with 1 A (32.25% of the average RMS value) and 0.6 A (17.6% of the average RMS value), respectively. The same trend is observed with 20 degrees of phase shift.

Figure 13 illustrates the soft-switching boundary under conditions of inductance mismatch. The red curve represents the symmetrical scenario, where the mismatch ratio $\rho$ is approximately 1.2%. Cases 2 and 3 are depicted by the green and black curves, respectively. The results demonstrate that the soft-switching boundaries of cases 2 and 3 are narrower than that of the symmetrical case. For example, at M = 0.8, the conventional case achieves soft switching at a power of approximately 0.32 p.u., while with cases 2 and 3, this power must exceed 0.36 and 0.38 p.u., respectively. This experiment serves as confirmation of the analysis detailed in Section 4.2.

6. Conclusions

This paper presents an investigation of the impact of leakage inductance mismatch on the performance of the DAB3 converter. The analysis considers several important aspects, including power transfer, soft-switching range, phase RMS current, and maximum temperature rise among transformer phases. Both experimental and analytical results demonstrate that power transfer and total copper loss are relatively insensitive to $L_k$ mismatch. However, the phase currents exhibit a strong dependence on $L_k$ variation. A higher degree of $L_k$ mismatch leads to a greater difference in RMS current among phases, resulting in an unbalanced thermal distribution among the transformer phases. Based on a statistical probability analysis of $L_k$ variation, it is recommended that the mismatch threshold should be around 10–15% of the desired value to ensure that the maximum temperature rise between phases remains evenly distributed with less than a 10% deviation compared to the design case. Moreover, the same process can be applied to another multi-phase converter in which a parameter mismatch can have occurred (interleave structure for example). This topic is out of the scope of this work, and it can be considered in future studies.