# 气隙铁心的磁特性及对电流互感器的影响分析Analysis of Magnetic Properties about the Air-Gapped Core and Its Influence on Current Transformer

• 全文下载: PDF(1827KB)    PP.71-82   DOI: 10.12677/JEE.2018.61009
• 下载量: 727  浏览量: 2,498

This paper constructs the dynamic magnetization model of air-gapped core by fitting curve func-tion, and analyzes the influence of air gap on core magnetic properties. On this basis, the current transformer of different air gap ratio is simulated. The difference between the air-gapped core current transformer and the closed core current transformer is obtained, and the influence of air gap on current transformer error is analyzed theoretically.

1. 引言

2. 气隙铁芯的磁特性曲线模型

2.1. 闭合铁芯的磁化性能模拟

2.2. 极限磁滞回环的拟合曲线函数

$B=a\mathrm{arctan}b\left(H-c\right)+dH+e{H}^{2}$ (1)

$B=a\mathrm{arctan}b\left(H+c\right)+dH-e{H}^{2}$ (2)

$B=a\mathrm{arctan}b\left(H-c-m\right)+d\left(H-m\right)+e{\left(H-m\right)}^{2}+n$ (3)

$B=a\mathrm{arctan}b\left(H+c+m\right)+d\left(H+m\right)-e{\left(H+m\right)}^{2}-n$ (4)

2.3. 气隙铁芯的磁化模型

1) 气隙铁芯的截面平行且垂直于磁力线方向

2) 气隙周围没有凸出的磁力线

${H}_{iron}=f\left(B\right)$ (5)

${i}_{0}={H}_{iron}{l}_{iron}+{H}_{air}{l}_{air}$ (6)

 (7)

${H}_{equ}\left({l}_{iron}+{l}_{air}\right)={H}_{iron}{l}_{iron}+{H}_{air}{l}_{air}$ (8)

Figure 1. The air-gapped core structure

${H}_{equ}={H}_{iron}+{H}_{air}\frac{{l}_{air}}{{l}_{iron}+{l}_{air}}=f\left(B\right)+\lambda \frac{B}{{u}_{0}}$ (9)

${H}_{equ}=f\left(B\right)+\lambda \frac{B}{{u}_{0}}$ (10)

3. 气隙铁芯的磁特性分析

3.1. 气隙铁芯的磁性能特点

1) 铁芯的剩磁显著降低；

2) 铁芯的抗饱和能力增强；

3) 铁芯的磁导率减小；

4) 铁芯线性度变好。

3.2. 气隙影响磁性能的理论分析

1) 气隙对剩磁特性的影响

Figure 2. The comparison curve of core magnetization characteristic

${H}_{闭合铁芯}=f\left(B\right)$ (11)

${H}_{闭合铁芯}=mB+n$ (12)

${H}_{闭合铁芯}=f\left(B\right)+\lambda \frac{B}{{u}_{0}}$ (13)

${H}_{气隙铁芯}=mB+n+\lambda \frac{B}{{u}_{0}}$ (14)

${B}_{气隙铁芯}=\frac{{B}_{闭合铁芯}}{1+\frac{\lambda }{m{u}_{0}}}$ (15)

2) 气隙对饱和特性的影响

${H}_{气隙铁芯}={H}_{闭合铁芯}+\lambda \frac{{B}_{s}}{{u}_{0}}$ (16)

3) 气隙对磁导率的影响

$y={k}_{\mathrm{max}}x+{a}_{1}$ (17)

$y={k}_{\mathrm{min}}x+{b}_{1}$ (18)

$y=\frac{1}{1+\frac{\lambda }{{u}_{0}}{k}_{\mathrm{max}}}{k}_{\mathrm{max}}x+{a}_{2}$ (19)

Figure 3. Differential graph of magnetization curve

$y=\frac{1}{1+\frac{\lambda }{{u}_{0}}{k}_{\mathrm{min}}}{k}_{\mathrm{min}}x+{b}_{2}$ (20)

$1-\frac{1}{1+\frac{\lambda }{{u}_{0}}{k}_{\mathrm{max}}}=\frac{\Delta {k}_{\mathrm{max}}}{{k}_{\mathrm{max}}}>\frac{\Delta {k}_{\mathrm{min}}}{{k}_{\mathrm{min}}}=1-\frac{1}{1+\frac{\lambda }{{u}_{0}}{k}_{\mathrm{min}}}$ (21)

4) 气隙对非线性的影响

$\delta =\left(1-\frac{{k}_{\mathrm{min}}}{{k}_{\mathrm{max}}}\right)×100%$ (22)

$\delta =\left(1-\frac{{k}_{\mathrm{min}}}{{k}_{\mathrm{max}}}\cdot \frac{{u}_{0}+\lambda {k}_{\mathrm{max}}}{{u}_{0}+\lambda {k}_{\mathrm{min}}}\right)×100%$ (23)

$\left(1-\frac{{k}_{\mathrm{min}}}{{k}_{\mathrm{max}}}\cdot \frac{{u}_{0}+\lambda {k}_{\mathrm{max}}}{{u}_{0}+\lambda {k}_{\mathrm{min}}}\right)×100%<\left(1-\frac{{k}_{\mathrm{min}}}{{k}_{\mathrm{max}}}\right)×100%$ (24)

4. 气隙铁芯电流互感器的建模与仿真

4.1. 气隙铁芯电流互感器的数学建模

Figure 4. The equivalent circuit of air-gapped core current transformer

${n}_{2}\frac{\text{d}\psi }{\text{d}t}={i}_{2}R+L\frac{\text{d}{i}_{2}}{\text{d}t}$ (25)

${n}_{1}{i}_{1}-{n}_{2}{i}_{2}={n}_{1}{i}_{0}$ (26)

${n}_{2}{s}_{iron}\frac{\text{d}B}{\text{d}t}=\frac{{n}_{1}}{{n}_{2}}\left({i}_{1}-{i}_{0}\right)R+\frac{{n}_{1}}{{n}_{2}}L\frac{\text{d}\left({i}_{1}-{i}_{0}\right)}{\text{d}t}$ (27)

${n}_{2}s\frac{\text{d}{H}_{equ}}{\text{d}t}\frac{\text{d}B}{\text{d}{H}_{equ}}=\frac{{n}_{1}}{{n}_{2}}\left({i}_{1}-{i}_{0}\right)R+\frac{{n}_{1}}{{n}_{2}}L\frac{\text{d}\left({i}_{1}-{i}_{0}\right)}{\text{d}t}$ (28)

$\frac{{n}_{1}{n}_{2}{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{equ}}\frac{\text{d}{i}_{0}}{\text{d}t}=\frac{{n}_{1}}{{n}_{2}}\left({i}_{1}-{i}_{0}\right)R+\frac{{n}_{1}}{{n}_{2}}L\frac{\text{d}\left({i}_{1}-{i}_{0}\right)}{\text{d}t}$ (29)

4.2. 气隙铁芯电流互感器的仿真

Figure 5. Waveform simulation when the air gap exists or not

Figure 6. The waveform simulation of different air gap ratio

5. 气隙铁芯电流互感器的误差分析

5.1. 气隙铁芯电流互感器的传递函数模型

${W}_{\left(s\right)}=\frac{{I}_{2\left(s\right)}}{{I}_{1\left(s\right)}}=\frac{{n}_{1}}{{n}_{2}}\cdot \frac{{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{equ}}s}{\frac{R}{{n}_{2}}+\frac{L}{{n}_{2}}s+{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{equ}}s}$ (30)

${E}_{\left(s\right)}=\frac{{n}_{1}{I}_{1\left(s\right)}-{n}_{2}{I}_{2\left(s\right)}}{{n}_{1}{I}_{1\left(s\right)}}=\frac{\frac{R}{{n}_{2}}+\frac{L}{{n}_{2}}s}{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}+{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{iron}}s}$ (31)

5.2. 气隙铁芯电流互感器的误差

${E}_{\left(jw\right)}=\frac{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}}{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}+jw{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{iron}}}$ (32)

$e=f+j\delta$ (33)

$f=-\frac{{\left(\frac{R}{{n}_{2}}\right)}^{2}+{w}^{2}\frac{L}{{n}_{2}}\left(\frac{L}{{n}_{2}}+{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{iron}}\right)}{{\left(\frac{R}{{n}_{2}}\right)}^{2}+{w}^{2}\frac{L}{{n}_{2}}{\left(\frac{L}{{n}_{2}}+{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{iron}}\right)}^{2}}$ (34)

Figure 7. The control block of air-gapped core current transformer

$\delta =\frac{wR\frac{{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{iron}}}{{\left(\frac{R}{{n}_{2}}\right)}^{2}+{w}^{2}\frac{L}{{n}_{2}}{\left(\frac{L}{{n}_{2}}+{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}\frac{\text{d}B}{\text{d}{H}_{iron}}\right)}^{2}}$ (35)

5.3. 气隙对误差的影响

5.4. 气隙对误差非线性的影响

$\frac{\text{d}B}{\text{d}{H}_{iron}}={u}_{\mathrm{min}}$$\frac{\text{d}B}{\text{d}{H}_{iron}}={u}_{\mathrm{max}}$ 时，误差的变化量为：

Figure 8. The relation between ratio difference of current transformer and air gap

Figure 9. The relation between angle difference of current transformer and air gap

Figure 10. The relation between the relative variation of angle difference and air gap

$\Delta e=\frac{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}}{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}+jw{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}{u}_{\mathrm{min}}}-\frac{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}}{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}+jw{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}{u}_{\mathrm{max}}}$ (36)

Figure 11. The relation between the relative variation of ratio difference and air gap

$\frac{\Delta e}{e}=1-\frac{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}+jw{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}{u}_{\mathrm{min}}}{\frac{R}{{n}_{2}}+jw\frac{L}{{n}_{2}}+jw{n}_{2}\frac{{s}_{iron}}{{l}_{iron}}{u}_{\mathrm{max}}}$ (37)

6. 总结

 [1] 杨鹏, 赵伟, 黄松岭, 等. 铁心开气隙电流互感器原理表述新探[J]. 电测与仪表, 2007, 44(10): 44-47. [2] 谭丽中. 电流互感器误差分析与校验[J]. 内蒙古石油化工, 2013(5): 33-34. [3] 曹团结, 张剑, 尹项根, 等. 电流互感器的误差分析与工程计算[J]. 电力自动化设备, 2007(1): 53-56. [4] 刘继发. 校验用标准双级电流互感器[J]. 电测与仪表, 1982(10): 43-44. [5] JJG 313-2010. 测量用电流互感器检定规程[Z].