# HBFEM Matrix Equations, Taking Account of Extended Circuits

Since the three phase transformer model with voltage driven source connected to the magnetic field system, is always coupled to the external circuits [2], the current in the input and output circuits are unknown. The saturation of the current waveform occurs due to the nonlinear characteristic of the magnetic core.

## The Equivalent Circuit of a Three-Phase Transformer

From Kirchhoff s laws, for the equivalent circuit of a three-phase transformer with Y/Y winding connection, the equations can be obtained as:

Considering a three-phase transformer connected in wye, a numerical simulation model with a neutral *N* and external circuits for both primary and secondary windings is obtained using the HBFEM technique. According to the Galerkin procedure, the system matrix equations of HBFEM for the three phase transformer can be obtained through Faraday’s and Kirchhoff’s laws for the external circuit [2]. The matrix equation of input voltage can be defined as:

*V _{ink}* is the input voltage of circuit

*k, V*is the corresponding induced electromotive force,

_{k}*S*and

_{ck}*Z*are the cross areas and impedance of winding

_{k}*k*respectively,

*C*and

_{k}*L*are the capacitance and inductance respectively in circuit

_{k}*k*. The induced electromotive force can be obtained based on Faraday’s law:

*V _{k}* can be rewritten as a compact form, as below:

The input voltage can be defined as:

where matrix [*Z*_{k}] is the circuit impedance, including the resistance of windings and leakage inductance corresponding to the harmonics, and *S _{ck}* is the area of the windings. The input voltage {V

_{ink}}, including all harmonic components which have a known value, is expressed as follows:

Therefore, all input and output circuits related matrix equations can be obtained as below:

where the applied three-phase voltage sources {V_{k},_{in}}, *{V*_{Np}'}, *{Vj _{OUt}}* and {V^} (k = 1,2,3 and

*j*=

*a,b,c)*are the unknown voltages at the neutral point, as expressed below:

and [C_{k}] is obtained from:

*{J _{k}}* and

*{Jj,*are the magnetizing current density and output current density, respectively. It can be expressed in a general form as below:

_{oUt}}The generalized input voltage matrix equation taking account of circuit can be expressed as:

where matrix *[Zk]* is the circuit impedance matrix which includes the resistance and leakage inductance of windings corresponding to the harmonics, and *S _{ck}* is the area of windings.

The input voltage {V_{ki}„}, including all harmonic components which have a known value, is expressed as follows:

where V_{3sk} and V_{3ck} are zero components; only fundamental component exists in the input voltage.

Based on the Galerkin procedure, the system matrix equations of HBFEM for the three-phase transformer, including equivalent circuits, can be obtained through Faraday’s and Kirchhoff’s laws. Therefore, the electric circuit-related matrix and FEM system matrix can be obtained, respectively, as below:

where *{I _{k}} = S_{ck}{J_{k}},* and [G

_{k}] is the current density coefficient obtained from a single element of winding area, that is

*[G*= Д

^{e}]^{е}/3.

The above system matrix, including all input and output currents in the three-phase transformer windings, can be expressed as:

Combining (6-16) and (6-28), the global system matrix equations for multiple input and output are obtained from following system equation:

Finally, the harmonic balance FEM matrix equations, taking into account the external and internal equivalent circuits of the transformer for voltage source excitation, can therefore be expressed as the following system matrix equation:

where *{A* }and {*J*_{k}} are unknown, and can be calculated by solving the system matrix equation, *[H]* is the matrix obtained from ([5] + [M]), [C_{k}] represents the geometric co-efficient related to transformer windings. [Z_{knk}], *[Zj _{OUt}]* and

*S*are external circuit impedances and cross-sectional areas of windings, respectively, [T] is unit matrix,

_{c},_{in}, S_{c},_{out}*V*is the voltage for neutral point when it is not grounded, and

_{NN}*C*and

_{in}*[C*are geometric coefficients related to transformer windings. Current density

_{ou}]*J*can be presented as (no zero component and second harmonic components when transformer in DC-biased case):

_{k}

[H] is the system matrix, and the detailed definitions can be obtained from:

where matrices *D* and *N* are:
and

The compact system matrix can be expressed as below:

where [H] is the FEM system matrix, [G_{k}] is the current density related coefficient matrix, [C_{k}] is the voltage-related coefficient matrix, and [Z_{k}] is the impedance matrix.

*{A _{k}}, {J_{k}}* and {V

_{N}} are unknown, and can be calculated by solving the HBFEM system matrix equation,

*[H]*is the matrix obtained from ([5] + [N]), and:

is the geometric coefficient related to transformer windings.