Magnetic equivalent circuit of MF transformers: modeling and parameter uncertainties

Abstract

Medium-frequency (MF) transformers are extensively used in power electronic converters. Accordingly, accurate models of such devices are required, especially for the magnetic equivalent circuit. Literature documents many different methods to calculate the magnetizing and leakage inductances of transformers, where, however, few comparisons exist between the methods. Furthermore, the impact of underlying hypotheses and parameter uncertainties is usually neglected. This paper analyzes nine different models, ranging from simple analytical expressions to 3D detailed numerical simulations. The accuracy of the different methods is assessed by means of Monte Carlo simulations and linearized statistical models. The experimental results, conducted with a $100\,{\hbox {kHz}}$/$20\,{\hbox {kW}}$ MF transformer employed in a $400\,{\hbox {V}}$ DC distribution system isolation, are in agreement with the simulations (below 14% inaccuracy for all the considered methods). It is concluded that, considering typical tolerances, analytical models are accurate enough for most applications and that the tolerance analysis can be conducted with linearized models.

Appendices

Appendices

These appendices review the theoretical background behind the different transformer equivalent circuits and the corresponding pitfalls (cf.  “Appendix A”). The procedure for extracting equivalent circuits from analytical (cf. “Appendix B”) and numerical (cf. “Appendix C”) computation is also presented. Finally, some approximations are given for transformers with high magnetic coupling factors (cf. “Appendix D”).

1.1 A Transformer equivalent circuits

The magnetic equivalent circuit of a (lossless) linear transformer with two windings is fully described by the following inductance matrix (cf. Fig. 1b) [6, 11, 43]:

$$\begin{aligned} \begin{bmatrix} v_{\mathrm {p}} \\ v_{\mathrm {s}} \end{bmatrix}&= \begin{bmatrix} \frac{\partial \varPsi _{\mathrm {p}}}{\partial t} \\ \frac{\partial \varPsi _{\mathrm {s}}}{\partial t} \end{bmatrix} = \begin{bmatrix} L_{\mathrm {p}}&M \\ M&L_{\mathrm {s}} \end{bmatrix} \begin{bmatrix} \frac{\partial i_{\mathrm {p}}}{\partial t} \\ \frac{\partial i_{\mathrm {s}}}{\partial t} \end{bmatrix} \text {,} \end{aligned}$$

(12)

where $L_{\mathrm {p}}$ is the primary self-inductance, $L_{\mathrm {s}}$ the secondary self-inductance, and M the mutual inductance. The inductance matrix features three independent parameters. The energy, W, stored in the transformer (quadratic form of the inductance matrix) can be computed as

$$\begin{aligned} W&= \frac{1}{2} L_{\mathrm {p}} i_{\mathrm {p}}^2 + \frac{1}{2} L_{\mathrm {s}} i_{\mathrm {s}}^2 + M i_{\mathrm {p}} i_{\mathrm {s}} \text {.} \end{aligned}$$

(13)

The energy stored in the transformer is always positive (i.e., the inductance matrix is positive definite), which leads to the condition $M^2 < L_{\mathrm {p}} L_{\mathrm {s}}$. Therefore, the mutual inductance can be expressed with a normalized parameter, the magnetic coupling,

$$\begin{aligned} k&= \frac{M}{\sqrt{L_{\mathrm {p}} L_{\mathrm {s}}}}, \quad \text {with } k \in [0,1] \text {.} \end{aligned}$$

(14)

During open-circuit operation, the inductance matrix has the following physical interpretation. The self-inductances ($L_{\mathrm {p}}$ and $L_{\mathrm {s}}$) represent flux linkages of the two windings themselves (defined as magnetizing flux linkages). The mutual inductance (M) describes the flux linkage between the windings (defined as coupled flux linkages). The differences between the self- and mutual inductances ($L_{\mathrm {p}}-M$ and $L_{\mathrm {s}}-M$) represent the flux linkage differences (defined as leakage flux linkages), which can, for some designs, be negative (especially if $N_{\mathrm {p}} \ne N_{\mathrm {s}}$.

With the aforementioned inductance matrix, the inductances, voltage transfer ratios, and current transfer ratios can be expressed for short-circuit and open-circuit operations (cf. Fig. 2b):

$$\begin{aligned} L_{\mathrm {oc,p}}&= L_{\mathrm {p}} \text {,}&\frac{v_{\mathrm {s}}}{v_{\mathrm {p}}}&= +k \sqrt{\frac{L_{\mathrm {s}}}{L_{\mathrm {p}}}} \text {,}&\text {with }&i_{\mathrm {s}} = 0 \text {,} \end{aligned}$$

(15)

$$\begin{aligned} L_{\mathrm {oc,s}}&= L_{\mathrm {s}} \text {,}&\frac{v_{\mathrm {p}}}{v_{\mathrm {s}}}&= +k \sqrt{\frac{L_{\mathrm {p}}}{L_{\mathrm {s}}}} \text {,}&\text {with }&i_{\mathrm {p}} = 0 \text {,} \end{aligned}$$

(16)

$$\begin{aligned} L_{\mathrm {sc,p}}&= \left( 1-k^2 \right) L_{\mathrm {p}} \text {,}&\frac{i_{\mathrm {s}}}{i_{\mathrm {p}}}&= -k \sqrt{\frac{L_{\mathrm {p}}}{L_{\mathrm {s}}}} \text {,}&\text {with }&v_{\mathrm {s}} = 0 \text {,} \end{aligned}$$

(17)

$$\begin{aligned} L_{\mathrm {sc,s}}&= \left( 1-k^2 \right) L_{\mathrm {s}} \text {,}&\frac{i_{\mathrm {p}}}{i_{\mathrm {s}}}&= -k \sqrt{\frac{L_{\mathrm {s}}}{L_{\mathrm {p}}}} \text {,}&\text {with }&v_{\mathrm {p}} = 0 \text {.} \end{aligned}$$

(18)

Fig. 14

Transformer (lossless) linear equivalent circuits. a T circuit, b PI circuit, c series–parallel circuit, and d parallel–series circuit. The circuits are referred to the primary side of the transformer

The terminal behavior (cf. (12)) and the stored energy (cf. (13)) of the transformer can be represented with different equivalent circuits. The circuits shown in Fig. 1b are directly related to the inductance matrix. On the contrary, Fig. 14 depicts equivalent circuits, which do not provide a direct insight on the magnetic flux linkages. The following important remarks can be given about the different equivalent circuits of transformers:

• All the presented equivalent circuits (cf. Figs. 1b, 14) model perfectly the terminal behavior and the stored energy.

• The equivalent circuits with more than three degrees of freedom (cf. Fig. 14a, b) are underdetermined and do not have a physical meaning, without accepting restrictive hypotheses [11].

• The turns ratio of the transformer, $N_{\mathrm {p}}:N_{\mathrm {s}}$, is not clearly defined for some transformer geometries (e.g., inductive power transfer coils) [47, 48]. This implies that the turns ratio is not always directly related to the flux linkages and to the magnetic parameters.

• The magnetizing and leakage fluxes cannot be spatially separated. In other words, it is not always possible to sort the magnetic field lines into leakage and magnetizing field lines [48].

• In a transformer, a phase shift is present between the primary and secondary currents, which originates from the modulation scheme, the load, and/or the losses. This implies that the distribution of the magnetic field lines is time dependent [11, 48]. Therefore, the leakage and magnetizing flux linkages only have a clear interpretation for a lossless transformer during open-circuit and short-circuit operations.

• The magnetizing and leakage inductances are usually defined as the parallel and series inductances in the equivalent circuit, respectively. However, the values of these inductances depend on the chosen equivalent circuit (cf. Fig. 14) and therefore do not have a clear and/or unique physical interpretation. This also implies that the associated magnetizing current ($i_{\mathrm {m}}$) and leakage voltage ($v_{\sigma }$) only represent virtual parameters, which are not directly measurable [11].

It can be concluded that only the equivalent circuits shown in Fig. 1b feature a clear physical interpretation and therefore should be preferred. The circuits with more than three degrees of freedom (cf. Fig. 14a, b) should be avoided since they are unnecessarily complex. The circuits depicted in Fig. 14c, d are interesting for designing transformers with high magnetic coupling factors as explained in “Appendix D.”

1.2 Equivalent circuit from analytical computations

The analytical methods are based on several assumptions (cf. Sect. 3.1). The inductances are accepted to scale quadratically with the number of turns. The inductances are extracted for $i_{\mathrm {p}} = 0 \vee i_{\mathrm {s}} = 0$ (the energy is confined inside the core and air gaps) and for $+N_{\mathrm {p}} i_{\mathrm {p}} = -\,N_{\mathrm {s}} i_{\mathrm {s}}$ (the energy is confined inside the winding window). Then, the following inductances can be extracted for a virtual 1 : 1 transformer:

$$\begin{aligned} L_{\mathrm {m}}'&= 2 \frac{W}{i_{\mathrm {p}}^2 N_{\mathrm {p}}^2} = 2 \frac{W}{i_{\mathrm {s}}^2 N_{\mathrm {s}}^2} \text {,}&\text {with }&i_{\mathrm {p}} = 0 \vee i_{\mathrm {s}} = 0 \text {,} \end{aligned}$$

(19)

$$\begin{aligned} L_{\sigma }'&= 2 \frac{W}{i_{\mathrm {p}}^2 N_{\mathrm {p}}^2} = 2 \frac{W}{i_{\mathrm {s}}^2 N_{\mathrm {s}}^2} \text {,}&\text {with }&+N_{\mathrm {p}} i_{\mathrm {p}} = -\,N_{\mathrm {s}} i_{\mathrm {s}} \text {.} \end{aligned}$$

(20)

The equivalent circuit (cf. (12) and (14)) of the transformer is extracted such that the stored energy (cf. (13)) matches, which leads to

$$\begin{aligned} L_{\mathrm {p}}&= N_{\mathrm {p}}^2 L_{\mathrm {m}}' \text {,}&L_{\mathrm {s}}&= N_{\mathrm {s}}^2 L_{\mathrm {m}}' \text {,} \end{aligned}$$

(21)

$$\begin{aligned} M&= N_{\mathrm {p}} N_{\mathrm {s}} \left( L_{\mathrm {m}}' - \frac{1}{2} L_{\sigma }' \right) \text {,}&k&= 1-\frac{1}{2} \frac{L_{\sigma }'}{L_{\mathrm {m}}'} \text {.} \end{aligned}$$

(22)

The following expressions can be extracted for the open-circuit and short-circuit operations of the transformer:

$$\begin{aligned} L_{\mathrm {oc,p}}&= N_{\mathrm {p}}^2 L_{\mathrm {m}}' \text {,}&\frac{v_{\mathrm {s}}}{v_{\mathrm {p}}}&= +k \frac{N_{\mathrm {s}}}{N_{\mathrm {p}}} \text {,}&\text {with }&i_{\mathrm {s}} = 0 \text {,} \end{aligned}$$

(23)

$$\begin{aligned} L_{\mathrm {oc,s}}&= N_{\mathrm {s}}^2 L_{\mathrm {m}}' \text {,}&\frac{v_{\mathrm {p}}}{v_{\mathrm {s}}}&= +k \frac{N_{\mathrm {p}}}{N_{\mathrm {s}}} \text {,}&\text {with }&i_{\mathrm {p}} = 0 \text {,} \end{aligned}$$

(24)

$$\begin{aligned} L_{\mathrm {sc,p}}&= N_{\mathrm {p}}^2 \left( \frac{1+k}{2} \right) L_{\sigma }' \text {,}&\frac{i_{\mathrm {s}}}{i_{\mathrm {p}}}&= -k \frac{N_{\mathrm {p}}}{N_{\mathrm {s}}} \text {,}&\text {with }&v_{\mathrm {s}} = 0 \text {,} \end{aligned}$$

(25)

$$\begin{aligned} L_{\mathrm {sc,s}}&= N_{\mathrm {s}}^2 \left( \frac{1+k}{2} \right) L_{\sigma }' \text {,}&\frac{i_{\mathrm {p}}}{i_{\mathrm {s}}}&= -k \frac{N_{\mathrm {s}}}{N_{\mathrm {p}}} \text {,}&\text {with }&v_{\mathrm {p}} = 0 \text {.} \end{aligned}$$

(26)

1.3 Equivalent circuit from FEM simulations

Different methods exist for the extraction of the magnetic equivalent circuit of a transformer from numerical simulations (e.g., FEM): integration of the magnetic flux, computation of the induced voltages, extraction of the energy, etc. The energy represents a numerically stable parameter which is easy to extract. Therefore, the energy is extracted for the following cases: $i_{\mathrm {p}} \ne 0 \wedge i_{\mathrm {s}} = 0$, $i_{\mathrm {p}} = 0 \wedge i_{\mathrm {s}} \ne 0$, and $+N_{\mathrm {p}} i_{\mathrm {p}} = -\,N_{\mathrm {s}} i_{\mathrm {s}}$. This last solution can be obtained by the superposition of the two first solutions. This leads to

$$\begin{aligned} L_{\mathrm {p}}&= 2 \frac{W}{i_{\mathrm {p}}^2} \text {,} \quad \text {with }&i_{\mathrm {p}} \ne 0 \wedge i_{\mathrm {s}} = 0 \text {,} \end{aligned}$$

(27)

$$\begin{aligned} L_{\mathrm {s}}&= 2 \frac{W}{i_{\mathrm {s}}^2} \text {,} \quad \text {with }&i_{\mathrm {p}} = 0 \wedge i_{\mathrm {s}} \ne 0 \text {,} \end{aligned}$$

(28)

$$\begin{aligned} M&= \frac{W}{i_{\mathrm {p}} i_{\mathrm {s}}} - \frac{1}{2} L_{\mathrm {p}} \frac{i_{\mathrm {p}}}{i_{\mathrm {s}}} - \frac{1}{2} L_{\mathrm {s}} \frac{i_{\mathrm {s}}}{i_{\mathrm {p}}} \text {,} \quad \text {with }&+N_{\mathrm {p}} i_{\mathrm {p}} = -\,N_{\mathrm {s}} i_{\mathrm {s}} \text {.} \end{aligned}$$

(29)

It should be noted that these expressions are general since no assumptions are required for the geometry, the turns ratio, the coupling factor, etc.

Fig. 15

Voltage and current waveforms for the considered MF transformer ($k = 0.986$, series-resonant LLC converter, cf. Fig. 2). a Terminal voltages, b terminal currents, c voltage across the leakage inductance, and d current through the magnetizing inductance. The equivalent circuits shown in Fig. 14c, d are considered and compared

1.4 Approximations for high magnetic coupling factors

For a transformer with a high magnetic coupling factor ($k>0.95$), the parameters of the equivalent circuits shown in Fig. 14 are converging together. Then, it is possible to define the transformer with the following parameters:

$$\begin{aligned} L_{\sigma }&\approx N_{\mathrm {p}}^2 L_{\sigma }' \approx L_{\mathrm {sc,p}} \approx L_{\mathrm {sc,s}} \frac{N_{\mathrm {p}}^2}{N_{\mathrm {s}}^2} \text {,} \nonumber \\&\approx 2 L_{\sigma ,\mathrm {T,p}} \approx 2 L_{\sigma ,\mathrm {T,s}} \approx L_{\sigma ,\mathrm {PI}} \approx L_{\sigma ,\mathrm {SP}} \approx L_{\sigma ,\mathrm {PS}} \text {,}\end{aligned}$$

(30)

$$\begin{aligned} L_{\mathrm {m}}&\approx N_{\mathrm {p}}^2 L_{\mathrm {m}}' \approx L_{\mathrm {oc,p}} \approx L_{\mathrm {oc,s}} \frac{N_{\mathrm {p}}^2}{N_{\mathrm {s}}^2}\text {,} \nonumber \\&\approx L_{\mathrm {m},\mathrm {T}} \approx \frac{1}{2} L_{\mathrm {m},\mathrm {PI,p}} \approx \frac{1}{2} L_{\mathrm {m},\mathrm {PI,s}} \approx L_{\mathrm {m},\mathrm {SP}} \approx L_{\mathrm {m},\mathrm {PS}} \text {,} \end{aligned}$$

(31)

$$\begin{aligned} \ddot{u}&\approx \frac{N_{\mathrm {p}}}{N_{\mathrm {s}}} \approx \ddot{u}_{\mathrm {T}} \approx \ddot{u}_{\mathrm {PI}} \approx \ddot{u}_{\mathrm {SP}} \approx \ddot{u}_{\mathrm {PS}} \text {,} \end{aligned}$$

(32)

where $L_{\sigma }$ is the leakage inductance, $L_{\mathrm {m}}$ the magnetizing inductance, and $\ddot{u}$ the voltage transfer ratio. These three parameters, which are nearly independent of the chosen equivalent circuit, are typically used for the design process of transformers.

Figure 15 illustrates the leakage voltage and magnetizing current obtained with the equivalent circuits depicted in Fig. 14c, d for the considered MF transformer ($k = 0.986$, series-resonant LLC converter, cf. Fig. 2). It can be seen that the aforementioned approximations (cf. (31), (30), and (32)) are valid.

With these assumptions, a clear and unique definition of the leakage and magnetizing fluxes is achieved. The leakage field, which is linked to the magnetizing inductance ($L_{\sigma }$ and $v_{\sigma }$), is located inside the winding window and is related to the load current (series inductance). The magnetizing field, which is linked to the magnetizing inductance ($L_{\mathrm {m}}$ and $i_{\mathrm {m}}$), is located inside the core and air gaps and is related to the applied voltage (parallel inductance). During rated operating condition, the leakage and magnetizing magnetic fields feature ${90}^{\circ }$ phase shift (cf. Fig. 15).