# 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 *(I _{dc}* = 0.426 A)

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

Table 5.17 Simulated results of hysteresis loops under sinusoidal excitation (7_{dc} = 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 v_{FP} 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 (I_{dc} = 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 v_{FP}, and expressions of the harmonic vector *H _{x}* and

*H*are similar to

_{y}*I*in Equation (5-69).

The vector potential equation based on v_{FP} 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 *U _{in}* 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),* I_{k}(p)] —? *[B _{x}(p), B_{y}(p)] —t*

*[H _{x}(p), H_{y}(p*)] —?

*P(p)—? [A(p +*1),

*I*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:

_{k}(p +

The convergence of harmonic solutions of magnetizing current and magnetic vector potential depends on the strategy to determine v_{FP} 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]: v_{FP} is initially set to be *3v _{0}* ~

*v*and is reset in the range v

_{0},_{0}/10 ~ v

_{0}/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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