A power transformer is one of the most significant and expensive equipment employed in a power grid. As a result, safe operation of the power transformer is crucial for a power system [1]. Transformer core materials generally feature hysteresis and saturation characteristics [2], due to which an unknown residual flux (RF) may be generated in their iron core after a DC resistance test, a transformation ratio measurement, or a no-load closing operation [3]. If this RF is large, a large amount of inrush current may form when the transformer is turned off, which can lead to winding deformation, current imbalance, and harmonic pollution [4]. Such situations render the transformer protection function invalid. Therefore, research on RF detection mechanisms and methods has important academic significance for reducing inrush current [5].

At present, an empirical method is often used to estimate magnetic flux, according to which the RF is estimated based on the 0.2–0.7 saturation flux [6–8]. However, this estimation error depends on historical experience, which is not conducive to on-site detection. The second method is the induced voltage method, which is based on the electromagnetic induction law for detecting magnetic flux changes in the magnetic core [9]. It calculates the RF by recording the induced voltage waveform when the transformer is turned off. However, since the RF calculated on starting the transformer is always different from the stabilized RF [10], the feasibility of this method is limited in practical application. The third method is the pre-magnetization method [11], which involves pre-charging the iron core with a known RF to then use phase-selective closing technology or the demagnetization method to suppress the inrush current. However, this method cannot pre-charge a certain amount of RF in advance for the actual transformer core, which limits its feasibility in practice.

The RF formation process involves complex changes in the magnetic domain structure of ferromagnetic materials [12]. Therefore, some previous studies have also studied indirect detection methods. In [13], RF was measured by analyzing the magnetic leakage around the iron core. Although this method was able to detect RF, it was limited by the influence of different measuring points on the measurement results, ultimately achieving a measurement accuracy as high as 20%. In [14], RF was estimated by analyzing the magnetized inductance. However, its measurement results were limited by the accuracy of the iron core model In addition, an RF detection method based on even the harmonics of the induced voltage is presented in [15], which was able to detect the RF of the current transformer, but its accuracy was limited by the accuracy of the simulation model. In [16], an RF detection method using small loop energy is proposed, but this method ignores the energy change under small RF, thus increasing the error of the extraction relationship. In [17], an RF detection method based on phase difference was analyzed, but the phase difference obtained by this method under different RFs was small, which is not conducive to its detection in practice. Furthermore, [18–20] studied RF detection methods considering transient variables, but the test time of the measured waveform was difficult to extract, rendering the accuracy of the detection results unstable and not conducive to on-site testing. Based on the above research methods, it is evident that there is still no fast and effective RF detection method available.

This paper presents an RF detection method based on the different polarities of response currents to obtain an accurate RF value and polarity. First, when the DC voltage in different directions is loaded in sequenc, the RF detection mechanism is analyzed, and a method for determining the RF direction is proposed. Following this, the relationship between the RF and the response current is established, and the corresponding parameters are obtained by employing the field-circuit coupling method. Finally, an RF test platform for a square core is built to verify the feasibility of the theoretical analysis and the accuracy of the proposed method.

The RF formation mechanism based on different magnetized states is depicted in Fig. 1, indicating that when the magnetic field strength H gradually increases to reach its maximum value, the magnetic flux increases along 0ab. Furthermore, when the external magnetic field disappears at point a or b due to hysteresis characteristics, a different RF is generated in the iron core, such as B_r or Br′. Therefore, it is evident that RF generation is primarily decided by changes in the domain structure at the RF, which can be reflected by changes in the relative differential permeability (μ_rd), expressed as follows:

where μ₀ is a constant, referring to the vacuum permeability of air, and ΔB and ΔH are the increments of B and H, respectively.

To analyze the variation trend of differential permeability at RF, this paper analyzed the material characteristics of the iron core using the silicon-steel sheets (model B30P105). The hysteresis loop of the core at low frequencies was measured, after which the differential permeability trend caused by saturation to the RF state was analyzed, as shown in Fig. 2. It is observed that when the frequency is 5 Hz, the B_m reaches 1.8 T, while the coercive force H_c is 13 A/m, indicating the difficulty involved in magnetizing the material. Furthermore, when ΔH is greater than 5% of H_c, the difference between the positive relative permeability μ_rp and negative relative permeability μ_rn is quite obvious. At this juncture, the relationship between μ_rp and μ_rn can be expressed as follows:

Based on the above relationship, the RF direction can be judged by comparing μ_rp with μ_rn. However, measuring μ_rp and μ_rn is not an easy task, due to the complex changes occurring in the magnetic domain structure. Therefore, it is necessary to analyze an indirect variable that can reflect μ_rd. Notably, when a DC voltage is loaded on one side of the winding of a transformer, the quantitative detection of RF can be realized by analyzing the relationship between the response current and the μ_rd in the circuit.

After the transformer is turned off, the method proposed in this paper only needs to connect the designed DC measurement circuit to a winding located close to the iron core for response current sensing. Subsequently, by analyzing the relationship between the response current and the RF, quantitative detection of the RF can be realized. Fig. 3 presents the DC detection circuit, where u_s(t) is the DC voltage and R_s denotes the series resistance. In an actual operation, when a transformer is reconnected to the power system, the RF direction remains unknown. To address this issue, the current paper analyzed the relationship between the RF and the response current under different excitation directions of the applied voltage. The main waveforms generated by applying the DC excitation directions are presented in Fig. 4. First, a positive DC voltage with the same polarity as the initial RF density was applied, after which a negative DC voltage was introduced—this waveform is abbreviated as PN. Similarly, a negative DC voltage with a polarity opposite to the initial RF density was applied, after which a positive DC voltage was implemented—this waveform is abbreviated as NP.

As shown in Fig. 4, in the case of PN, when the positive voltage is loaded first at t₁, the positive respond current i_p(t) is generated. When the voltage is removed after t₂, the RF in the iron core becomes a new B_rp. Subsequently, when the negative voltage with a polarity opposite to the RF is applied at t₃, a negative respond current i_n(t) is generated. After t₄, when the voltage is removed, the initial RF changes from B_rp to B_rn. However, in the case of NP, as shown in Fig. 4(b), i_n(t) is produced first and i_p(t) is produced second, due to which the initial RF first changes to B_rn and then to B_rp. According to Eq. (2), μ_rp at RF is less than μ_rn, due to which the changes in RF are different in the case of PN and NP. Thus, the waveforms of the measured positive and negative respond current are also different. Consequently, the quantitative detection of the RF was realized by means of this difference.

When a voltage is loaded, the transformer core becomes equivalent to an RL series-parallel circuit. Fig. 5 shows an equivalent circuit model for the measurement circuit, where R_s represents the total resistance in the circuit, primarily including the series resistance and winding resistance. Meanwhile, L_eq stands for the equivalent magnetized inductance of the iron core, which is related to the change in magnetic flux. R_Fe stands for hysteresis loss in transient processes, also called iron loss resistance. Notably, when the turns of the winding (N), the cross-sectional area S, and the average magnetic circuit length l are known, the magnetization inductance L_eq at the RF can be related to the differential permeability μ_rd.

According to the magnetic circuit analysis, the relationship between L_eq and μ_rd can be expressed as:

Furthermore, since the applied excitation is DC excitation, the time constant in the measurement circuit can be expressed as L_eq/R. Therefore, the respond current i(t) in the circuit can be formulated as follows:

where R represents the parallel connection between R_Fe and R_s. Furthermore, according to Eq. (4)i_p(t) and i_n(t) are represented as follows:

Based on Eqs. (2) and (5), the relationship between i_pt) and i_n(t) can be obtained from the following formula:

Regardless of whether the positive or negative excitation was loaded first, the change rate of i_p(t) was found to be faster than that of i_n(t). Therefore, the RF direction can be judged by comparing the waveforms i_p(t) and i_n(t). Notably, to obtain the relationship between RF and the response current, μ_rd can be expressed as follows:

where t_s is the best measurement time for the respond current. According to Eq. (7), the change in the magnetic domain structure at RF maintains a pertinent relationship with respond current. Therefore, this study proposes a method for calculating the RF by establishing a relationship between B_r and i(t_s). This relationship can be expressed as follows:

where a and b are constant coefficients whose values are determined by the finite element method, and I_o represents the current value when the circuit reaches a steady state, which can be expressed as V_DG/R.

In the following sections, the field-circuit coupling method is applied to analyze the above relationship, and an experimental verification is carried out.

This paper used the finite element method to extract empirical formulas for calculating RF. First, the iron core was modeled. To simulate the initial RF in the iron core, the hysteresis model of the iron core was used during modeling. In particular, the J-A hysteresis model presented in [21] was employed to simulate the hysteresis characteristics of the iron core and achieve an accurate simulation of its initial RF. The five parameter values of the hysteresis model obtained through parameter identification were M_s = 1.58 × 10⁶ A/m, a = 4.56 A/m, alpha = 5.67 × 10⁻⁶, k = 8.95 A/m, and c = 0.18. Fig. 6 depicts the research objects selected for this study. As shown, S of the iron core is 0.0016 m², l of the iron core is 1.92 m, and N of the measuring winding is 50.

Notably, before DC excitation loading, the initial RF needs to be preset in the core. In this study, the initial RF was simulated by applying a short-term large current to the core. First, a large current was applied to the winding so that the magnetic flux of the core quickly reached its maximum value. Subsequently, the large current was removed, with the static flux in the core being the initial RF of the simulation. As shown in Fig. 6, when the initial RF is 0.845 T, the magnetic flux inside the core is larger than that outside the core. This can be attributed to the shorter magnetic circuit length of the inner side and the smaller magnetic resistance of the core, which makes the magnetic flux larger. The error between the internal and external magnetic flux with regard to the average magnetic flux was less than 0.12%, indicating that the distribution of magnetic flux at the RF was almost uniform.

An RF detection platform comprising a square iron core was built to verify the feasibility of the theoretical analysis and the accuracy of the proposed method. Fig. 11 shows a photograph of the experimental platform, where the tested core (“1”) is composed of Baosteel B30P105 silicon steel sheets. The material type was kept consistent with the simulation to ensure the accuracy of the test results. Furthermore, a signal generator (“2”) is employed to generate different square wave voltage signals, which can help realize the control of voltage polarity. A power amplifier (“3”) is used to amplify the applied voltage signal, while a switch (“4”) is employed to control the turn-on and turn-off actions of the detection circuit. The resistor (“5”) is used to block the current. An oscilloscope (“6”) is added to observe the flow of the current through the windings (“7”). Furthermore, the high-precision current probe N2782B (“8”) is used to acquire current signals since it can realize accurate current detection in the ms or even the μs range. Moreover, Fluxmeter480 (“9”) is included in the setup, since it is capable of tracking real-time changes in magnetic flux density, with the measured magnetic flux being the flux based on the voltage integration principle.

The initial RF direction of offline transformers usually remains unknown. At this moment, the same positive voltage as the initial RF polarity can be applied to measure the positive response current i₊(t). Subsequently, when applying negative voltage, the negative current i₋(t) can be measured. Subsequently, if the change rate of the i₊(t) is found to be greater than i₋(t), i₊(t) can be determined to be i_p(t), and i₋(t) as i_n(t), which would prove that the iron core has positive RF. If the opposite result is achieved, the iron core can be presumed to have negative RF. Finally, when t=t_s, the value of the negative current can be integrated using Eq. (13) to calculate the RF in the iron core.

Fig. 12 shows the voltage and respond current waveforms when the preset B_r is 0.846 T. This shows that when the positive voltage is loaded, i_p(t) is generated, whereas when the negative voltage is loaded, i_n(t) is generated. Fig. 13 shows the i_p(t) and i_n(t) at different RFs. It is evident that the change rate of i_p(t) is faster than that of i_n(t), as a result of which the iron core is determined to have a positive RF. This conclusion also indicates that i_p(t) represents the positive RF direction and i_n(t) represents the negative RF direction. Furthermore, the above analysis proves the accuracy of the theoretical analysis conducted in this paper.

As shown in Fig. 13, when the time is 0.1 seconds, the voltage is removed. At this time, the negative current was measured to calculate the RF in the iron core. Therefore, when t_s = 0.1 seconds, the negative current was measured and then substituted into Eq. (13) to calculate the RF in the iron core, represented as B_r1. In the case of PN, as shown in Fig. 13(a), when the preset RF is 0.922 T, the current value reaches 0.0311 A, with the calculated B_r1 value being 0.929 T. Compared to the preset B_r, the relative error was found to be 0.76%. Meanwhile, in the case of NP, as shown in Fig. 13(b), when the preset B_r is 0.935 T, the calculated B_r1 value reaches 0.946 T, with the relative error being 1.18%. This indicates that the change rate of RF is greater for NP than for PN. As a result, this paper primarily analyzed the measurement error in the case of NP.

Based on [6–8], RF is usually approximated as 0.2–0.7 times the saturation magnetic density. Since the saturated magnetic density of the iron core material selected for study in this paper was about 1.8 T, the RF detection range was determined to be 0.4 T to 1.2 T. Table 1 shows the experiment results at NP, with B_r2 indicating the measured RF obtained using the voltage integration method [15, 16]. The relative error between the pre-set RF and B_r1 is expressed as ɛ₁%, while the relative error between the preset RF and B_r2 is expressed as ɛ₂%. The maximum ɛ₁% is within 5%, which is less than the ɛ₂% obtained using a different preset B_r. This result highlights that the accuracy of the proposed method is higher than that of the voltage integration method.

Fig. 14 presents a comparative analysis of the simulation and experimental results at different RFs. It is observed that the variation trend of the experimental results is largely the same as that of the simulation results. Furthermore, the measurement error indicated that the proposed detection method exhibits a certain feasibility. Compared to existing detection methods [15–20], the proposed method was able to not only judge the RF direction, but also accurately detect the RF value. In addition, since the RF generation mechanism is closely related to variations in the differential permeability at the RF, the differential permeability may be significantly reflected by the measured response current. Furthermore, the proposed method can be applied for RF detection in other power equipment composed of iron cores, indicating its wide applicability.

In this paper, an indirect RF detection method for power transformer cores is proposed based on the different polarities of respond currents. The proposed method was able to accurately identify RF polarity and conduct a quantitative detection of the RF value. In particular, RF polarity was determined by comparing the change rates of different polarities of response currents. Furthermore, the relationship between the RF and the negative response current was constructed using the field-circuit coupling method to realize the quantitative detection of the RF value. The experimental results exhibited an accuracy that reached 5%, which is higher than the accuracy achieved by the existing methods. Notably, since transformer cores composed of different structures use different empirical formulas for RF calculation, their corresponding empirical formulas need to be extracted based on the presented method. The method proposed in this paper can also be applied to detect RF in other power equipment cores characterized by closed magnetic circuit structures.