# Nonlinear Magnetic Material and its Saturation Characteristics

As discussed in Chapter 3, when a nonlinear magnetic system is excited by a sinusoidal waveform, a number of harmonics will be generated in this nonlinear magnetic system.

Figure 4.11 Comparison between computation and measurement. (a) Input voltage source; (b) Current caused by voltage source

Figure 4.12 A circuit diagram of a magnetic frequency tripler

Figure 4.13 B-H curve with (a) hysteresis; and (b) without hysteresis characteristics

For the non-DC biased case, only odd harmonics can be generated in the magnetic field. Figure 4.13 shows the B-H curve with and without hysteresis characteristics. The magnetizing characteristics of the magnetic core used in a tripler can be expressed by the following arbitrary function of flux density B:

where the B-H curve with hysteresis is obtained from Equation (4-22). Without hysteresis characteristics, it can be obtained from the first two terms of Equation (4-22).

# Voltage Source-Driven Connected to the Magnetic Field

The computation model of magnetic frequency tripler can be built as a voltage source- driven CEM model, where input voltages *V*_{in} in primary are given. The output voltages

Figure 4.14 A configuration of the magnetic frequency tripler with a voltage driven source connected to the magnetic system

V_{out} in secondary winding, current at both primary and secondary sides, will be unknown. Since the magnetic frequency tripler usually works in the magnetic saturating state, as shown in Figure 4.13, the harmonic components will be generated in the magnetic core. Figure 4.14 shows a magnetic frequency tripler with three input magnetizing coils, and two secondary coils connected in a series as an output winding.

The model has a voltage-driven source connected to the magnetic system, which is always coupled to the external circuits [9]. The current in the input circuits will be unknown, but saturation of the current waveform occurs due to the nonlinear characteristic of the magnetic core. From Kirchhoff’s laws, the following system equation for the circuit can be obtained:

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

where the applied three-phase voltage sources {*V*_{u}}, {*V*_{v}}, {*V*_{w}}, and {*V*_{NN}'} and {*V*_{out}} are the unknown voltage at the neutral point as expressed below:

and [C_{ck}] is obtained from:

{J_{u}}, {J_{v}}, {J_{w}} and {J_{out}} are magnetizing current density and output current density, respectively. It can be expressed in a general form as below:

The generalized input voltage matrix equation taking account of circuit can be expressed as:

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

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

where V_{0}, *V _{3sk}* and

*V*are zero components; only fundamental component exists in the input voltage.

_{3ck}The above system matrix, including all input and output circuits, can be expressed as:

where [Gk] is obtained from a single element, that is [G^{e}] = Д73.

Combining Equations (4-24) and (4-35), the global system matrix equations for multiple input and output are obtained from following system equation.

The harmonic balance FEM matrix equations 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

_{ink}],

*[Z*and

_{out}]*S*are external

_{c},_{in}, S_{c},_{out}circuit impedances and cross-sectional areas of windings, respectively, [T] is unit matrix, *V _{NN}* is the voltage for neutral point when it is not grounded, and

*[C*and [C

_{in}]_{OM}J are geometric coefficients related to transformer windings. Current density

*J*can be presented as:

_{k}

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

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

Figure 4.15 A three-phase magnetic tripler problem as a voltage-driven source connected to the magnetic system

Figure 4.15 shows a real application model with the physical structure shown in Figure 4.16. This is a three-phase circuit with magnetic field-coupled problem, which can be solved by HBFEM taking account of electric circuit.

The output voltage waveform can be calculated from numerical result *i (V* = iR), and the simulation result is compared with the experimental result, as illustrated in Figure 4.17.

The magnetic flux distribution for each harmonic component calculated from HBFEM is presented as below. From Figure 4.18, we can see that only the third harmonic component can pass through the lag 2 and lag 4, which will generate the induced voltage with three times fundamental frequency on secondary windings. The input and output currents with harmonic distortion in each winding can be calculated from the HBFEM matrix equation. The calculation results are also compared with experimental results. A good agreement has been achieved in this calculation.

The waveforms of input and output currents, and input voltage (phase U) and neutral voltage *V _{NN}-* are calculated from HBFEM. The simulation results compared with experiment results are presented in Figure 4.19.

The waveforms of the magnetic flux density distribution for each phase of magnetic leg, and output side of magnetic legs, are calculated from HBFEM. The simulation results, compared with experiment results, are presented in Figure 4.20.

Figure 4.16 Geometric size * ^{l}A,* a configuration of the magnetic frequency tripler

*for numerical computation model*

^{l}AFigure 4.17 Output voltage waveform: (a) experimental result, (b) simulation result

Figure 4.21 illustrates characteristics of output current against load, and phase U input current against input voltage, respectively. The output current of fundamental component is very small, as shown in Figure 4.21(a). When the input voltage reaches to the rated voltage (100 V), the input current will be increased significantly.