 # 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. Figure 5.50 Comparison between the measured and simulated results (Idc = 0.426 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 : 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. ) 