# HBFEM for Electromagnetic Field and Electric Circuit Coupled Problems

## HBFEM in Voltage Source-Driven Magnetic Field

In most cases, HVDC transformer, voltage tripler, pulse width modulation (PWM) and zero-current switched resonant converters, including line level control (LLC) converters, as shown in Figure 3.8, can be considered as a voltage source to the magnetic system. They are always coupled to the external circuits. Therefore, the current in the input

Figure 3.7 Size of the system matrix. (a) Static FEM; (b) HBFEM

Figure 3.8 Simulation model of LLC converter with resonant tank with idea transformer

circuits will be unknown, and the saturation of the current waveform occurs because of the nonlinear characteristic of the magnetic core [10-12].

## Generalized Voltage Source-Driven Magnetic Field

When electrical devices are excited by voltage source, such as LLC converter transformers and HVDC transformer under DC bias, the excitation current is unknown and

Figure 3.9 Coupling between the electric circuit and the magnetic field

Equation (3-68) is no longer applicable. In this case, the coupling between the electric circuit and the magnetic field should be taken into account [10]. The generalized electric circuit coupled with magnetic field can be illustrated as Figure 3.9.

According to Kirchhoff’s law, the applied voltage on the external port can be defined

as follows:

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

*k,*V

_{k}is the corresponding induced electromotive force, S

_{ck}and Z

_{k}are the cross areas and impedance of winding

*k*respectively, C

_{k}and

*L*are the capacitance and inductance respectively in circuit k. The induced electromotive force can be obtained based on Faraday’s law:

_{k}

where *d _{0}* is the depth in the z-direction,

*N*is the turn number of the k-th winding.

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

_{k}where

In the series connection, the impedance of external circuit or equivalent circuit of transformer at nth harmonic can be expressed as

The expression of the impedance matrix is given by Equations (3-80) through (3-82), respectively,

The generalized impedance matrix can be expressed as 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 windings. The input voltages

*{V*including all harmonic components which have a known value, are expressed as follows:

_{ink}},

where *V _{in},_{0k}* (=0) is a DC component.

The circuit-related matrix and FEM system matrix can be obtained as below respectively,

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

_{e}] = Д

_{е}/3.

Combining Equations (3-86) and (3-87), the global system matrix equations for multiple input and output are obtained. The harmonic balance FEM matrix equations for voltage source excitation can therefore be expressed as:

where {A} and {*J _{k}}* are unknown and can be calculated by solving the system matrix equation, [H] is the matrix obtained from ([S] + [N]), and:

is the geometric coefficient related to transformer windings.

From Equation (3-68), Equation (3-87) can be rewritten as:

However, *D* and *N* have different details when the DC component is considered, where *D* is:

and the harmonic matrix, *N*, is:

The compact system matrix can be expressed as below:

where *[H]* is the FEM system matrix, [G] is the current density-related coefficient matrix, [C] is the voltage-related coefficient matrix, and [Z] is the impedance matrix.