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The use of reluctance networks has been a conventional practice to analyze transformer structures. Basic transformer structures can be well analyzed by using the magnetic-electric analogues discovered by Heaviside in the 19
^{th} century. However, as power transformer structures are getting more complex today, it has been recognized that changing transformer structures cannot be accurately analyzed using the current reluctance network methods. This paper presents a novel method in which the magnetic reluctance network or arbitrary complexity and the surrounding electrical networks can be analyzed as a single network. The method presented provides a straightforward mapping table for systematically linking the electric lumped elements to magnetic circuit elements. The methodology is validated by analyzing several practical transformer structures. The proposed method allows the analysis of coupled inductor of any complexity, linear or non-linear.

Magnetic circuits are commonly represented by equivalent electrical circuits [

These electric-magnetic analogies constitute a basis to analyze basic transformer structures. However, as transformer structures of today are getting more and more complex, a need has arisen to come up with a more systematic treatment of coupled inductors so that their dynamics could be modeled effectively in situations, which are considered to be challenging. The methodologies presented in transformer literature [

Most circuit analysis tools, such as Simulation Program with Integrated Circuit Emphasis (SPICE), provide methods to include transformers into transient and steady state analysis [

Power system analysis softwares such as Electro Magnetic Transients Program/ Alternative Transients Program (EMTP/ATP) [

The motivation of the development of the methodology presented in this paper arose from practical complex transformer design problems, which could not be easily addressed using the methods available today. For example, the magneto-electric circuit in

with 6 equations and 6 state variables, which would correspond to magnetic fluxes in the limbs and leakage paths. While this is doable, there is complexity in it, which increases the error possibilities and cannot be generalized to arbitrary transformer structures and the system of differential equations must be formed again for every new case as there is no systematic way to derive them automatically. The analysis method presented in this paper enables to make the analysis automatically for a coupled inductor of any complexity.

In Section 2 mapping rules are introduced in order to transform the equivalent electrical circuits of transformer’s windings of a complex coupled inductor into magnetic domain. Section 3 presents the methodology of analyzing the non- linear magnetic cores. Further in Section 4, the proposed methodology is applied to analyze the performance of two types of practical transformers under different load conditions while conclusion is presented in Section 5.

This section presents the primary elements of transformers. A methodology is presented, where the electrical circuits surrounding the coupled inductor can be mapped to magnetic domain using defined rules of transformation as described below.

Rule 1: Transformation of voltage source into current source

A voltage source connected to the windings of an ideal single-phase magnetic core can be seen in

Faraday’s law of induction Equation (1) illustrate that the magnetic flux created by the voltage source is an integral of the voltage and can be represented in Laplace space according to Equation (2). Considering the electric-magnetic analogies we can thereby represent voltage source in electrical domain as current source in magnetic domain as shown in

Rule 2: Transformation of current source into voltage source

When there is a current source connected to reactor windings, this can be represented in magnetic circuit as a voltage source using the traditional analogy:

current I is mapped to MMF of NI, when N is the number of turns as shown in

Rule 3: Transformation of resistive load into inductance

Considering a transformer limb in which there is constant flux (_{LOAD} connected, there is a relationship between flux (

On the other hand, the magnetomotive force is_{m} can be written as,

This described that resistive load connected to transformer windings can be represented as a reluctance given in (5). Considering the nature of the expression it can be seen, that a resistive load in electrical domain can be represented as an inductor in magnetic domain.

Rule 3: Transformation of inductive load into resistance

When the impedance connected to the winding is an inductor, similar analysis (as described above) can be done to get that the reluctance presented is not frequency dependent as can be seen in Equation (6). Thus, it can be represented as a resistor in magnetic domain.

Rule 4: Transformation of capacitance into non-linear negative resistance

When the impedance in the winding is purely capacitive, the resulting reluctance becomes as described in (7). In frequency space the reluctance is negative, real and squarely dependent on frequency (8).

Rule 5: Transformation of complex impedance into complex reluctance

When the impedance connected to the transformer contains both resistive and reactive parts, the reluctance has a complex value:

Rule 6: Transformation of Thevenin source into Thevenin or Norton source

Thevenin equivalent source is typically connected to transformer windings. This transforms a Thevenin source in magnetic domain. This can be calculated by first writing the induced voltage in the electrical circuit (11) and combining this with the definition of magnetomotive force (12).

Developing this further, an expression for the magnetomotive force is given as,

Therefore, a Thevenin source can be transformed to electrical domain as Thevenin or Norton source, depending which suits better for the analysis following. Thevenin source voltage is in (14) and Norton source current is in (16). The corresponding mappings are presented in

Electric lumped element | Variable | Mapped magnetic element | Transform in Laplace space |
---|---|---|---|

Voltage source | E | Flux source | |

Current source | I | Voltage source | |

Resistance | R | Inductance | |

Inductance | L | Resistance | |

Capacitance | C | Negative impedance with square frequency dependency | |

Thevenin source | E, Z | Thevenin or Norton source with inductance | Voltage: |

The magnetic cores typically present non-linear flux-MMF relationships, which depend on the characteristics of the B(H) curves of the core material. This non- linearity can be taken into account in this analysis by considering the reluctance elements as non-linear resistances. Using the mappings as described in the previous sections together with the non-linear magnetic core reluctances makes it possible to systematically construct system of non-linear differential equations describing the behavior of the electric-magnetic system. Symbol p marks the differentiation operator.

Referring to

Two cases are analyzed by using the proposed method. First one is a three-limb three-phase transformer with asymmetric core, leakage flux and winding resistances. Symmetry behavior of this transformer is studied using the proposed methods. Second one is a single-phase transformer with leakage reluctance and winding resistances and its performance is analyzed in transient and high load situations.

Referring to the transformer in

• The impact of reluctance asymmetry to the output voltage asymmetries in different symmetric load conditions

• The behavior of the voltages in the secondary, when the load is heavily asymmetric.

As discussed in section one, this kind of transformer with asymmetries in core is hard to analyze using traditional methods.

The proposed method is applied by first mapping the electrical circuits surrounding the coupled inductor. The three primaries are Thevenin sources and rule 6 is applied. The secondaries are resistive loads and rule 3 is applied.

After applying the described rules the network to be solved is as shown in

Impact of yoke reluctance asymmetry is studied by providing symmetrical voltage to the primary, symmetrical load at different levels and observing the negative sequence voltage in the secondary. Negative sequence output voltage as a function of load resistance is shown in

This analysis shows that the asymmetry created by yoke reluctances in the magnetic core do induce substantial asymmetries in the output voltages.

Second view is to look at, how the transformer performs, when primary voltages are symmetrical, but load is asymmetrical and resistive.

Second case concerns single-phase transformer including leakage flux and winding resistances as in

To demonstrate the capabilities of the proposed method the system shown in

magnetomotive force. This function is presented in

The differential equations derived using the methods in this paper are described in (19) and (20).

Solving these differential equations numerically in a case, where the load is 10 Ohm and primary winding resistance is 10 Ohm and primary and secondary turns are 1000 and 100 respectively, gives fluxes as in

A systematic method to analyze complex coupled inductors based on straight forward reluctance transformation rules is presented. This method facilitates the analysis of such transformer structures that could not be analyzed using the methods known in current transformer literature. In particular this applies to transformers, which have asymmetric reluctance network. The method makes it

possible also to systematically incorporate the dynamics of the surrounding electric circuits around the coupled inductor in a systematic manner.

Penttonen, J., Shafiq, M. and Lehtonen, M. (2017) Reluctance Network Analysis for Complex Coupled Inductors. Journal of Power and Energy Engineering, 5, 1-14. http://dx.doi.org/10.4236/jpee.2017.51001