Analysis of Hysteretic Characteristics Under Sinusoidal and DC-Biased Excitation

Globally Convergent Fixed-Point Harmonic-Balanced Method

As is shown in Equation (5-68), the magnetizing current can be termed as the sum of odd harmonics when the power transformer works under sinusoidal excitation:

where n represents the harmonic number.

Comparison between the measured and simulated results (I = 0.426 A)

Figure 5.50 Comparison between the measured and simulated results (Idc = 0.426 A)

Comparison between the measured and simulated results (7 = 0.847 A)

Figure 5.51 Comparison between the measured and simulated results (7dc = 0.847 A)

Table 5.17 Simulated results of hysteresis loops under sinusoidal excitation (7dc = 0.426 A)

Simulated results H(A/m)

Measured results H(A/m)

-30.064333624243545

-29.912999999999784

-30.926884954766138

-31.109419354838565

-31.888931847504182

-32.163677419354826

-33.02925267913588

-32.7272903225803

-34.44387577305497

-34.8370645161292

-36.24604559236536

-37.087741935483564

-38.56065278624533

-38.70687096774191

-41.5273260176607

-42.082258064516054

-45.3370977251127

-45.811161290322616

-50.34138167711467

-49.82061290322554

The DC bias phenomenon in power transformers occurs due to solar storms or high- voltage direct current transmission. In this case, the magnetizing current includes not only odd harmonics but also even harmonics and direct current. Consequently, the current i can be expressed by Equation (5-69) in the harmonic domain:

According to the fixed point theorem, the fixed-point reluctivity vFP is introduced to overcome the discontinuity of traditional reluctivity in the numerical computation of the

Table 5.18 Simulated results of hysteresis loops under sinusoidal excitation (Idc = 0.847 A)

Simulated results H(A/m)

Measured results H(A/m)

713.775907309398

711.422903225806

884.992348113111

883.35

1104.40012666554

1103.41064516129

1345.18705763059

1344.5564516129

1633.42085469195

1632.99290322579

1935.53725734772

1935.05419354838

2283.28204724313

2282.58193548386

2630.21964951644

2629.35483870967

3016.73501769168

3015.83225806451

3396.60075913058

3395.89354838709

magnetic field involving hysteresis effects. Therefore, a new relationship in the harmonic domain is presented in the two-dimensional nonlinear magnetic field:

in which a specific parameter is usually assigned to the fixed-point reluctivity vFP, and expressions of the harmonic vector Hx and Hy are similar to I in Equation (5-69).

The vector potential equation based on vFP can be used to describe the twodimensional nonlinear magnetic field:

where M is a magnetization-like quantity and о is the electric conductivity.

When electromagnetic devices are excited by voltage, the coupling between the k-th external circuits and the magnetic field should be taken into account in the system equations, so that the magnetizing current and magnetic field can be solved simultaneously by:

where Uin and I are the harmonic vectors of external input voltage and magnetizing current, C and Z represent the coupling matrix and impedance matrix, respectively, P is derived from M, while Q is related to hysteretic nonlinearity and eddy current.

Harmonic solutions in Equation (5-72) can be computed iteratively, according to the following procedure (p represents the iterative step): [A(p), Ik(p)] —? [Bx(p), By(p)] —t

[Hx(p), Hy(p)] —? P(p)—? [A(p + 1), Ik(p + 1)]. If the anisotropy of the iron core is neglected, the magnetic field intensity can be computed, based on the scalar hysteresis model and the following relationship:

The convergence of harmonic solutions of magnetizing current and magnetic vector potential depends on the strategy to determine vFP in the iterative process. However, the proposed method in time-stepping finite element analysis is not applicable in harmonic computation. An alternative scheme is recommended as follows [54]: vFP is initially set to be 3v0 ~ v0, and is reset in the range v0/10 ~ v0/40 after a few initial iterations. When the convergence criterion e in Equation (5-74) is small enough, the iteration is stopped.

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