對於繼電器線圈奇怪的H-B曲線的實驗測量

 

■ 問題的提出

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在昨天的 電磁鐵的磁芯實驗 中出現了繼電器線圈施加的電壓與測量得到的磁芯磁感應強度的奇怪的曲線，也就是和通常的磁滯曲線不同的8字繞行的情況。見下面的線圈電壓與Hall輸出之間的關係。

那麼爲什麼會出現這種情況呢？

■ 猜測可能的原因

這種情況極有可能是線圈在施加電壓（實驗中的電壓爲±24V，遠大於線圈的額定工作電壓）所引起的線圈溫度的變化，從而使得線圈的電阻也出現了變化。這樣在相同的外部施加電壓下，實際的線圈電流與外部施加的電壓之間不再呈現線性關係。

那麼通過以下兩個方式來進行驗證：

• 驗證方案1 ： 直接在線圈上串聯一個電流採樣電阻，來測量流過線圈的電流值，重新繪製出H-B曲線，看是否符合傳統的磁滯曲線；
• 驗證方案2： 測量線圈的溫度與施加電壓之間的關係。

 

01實驗研究

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1.測量電流與磁芯的磁感應強度B

測量方式，就是在繼電器線圈上串聯一個10歐姆的電阻作爲電流取樣電阻。通過測量電流取樣電阻上的電壓來反映流過的電流。最後繪製電流與磁芯的磁感應強度之間的關係。

下面是重新實驗所測量得到的施加電壓與磁芯的磁感應強度之間的曲線。與昨天是相同的。

^{▲ 施加電壓與Hall輸出}

下面是測量施加電壓與流過的電流（從電流取樣電阻的電壓換算而得到的）之間的關係。可以看到他們之間呈現奇怪的“8”字曲線特性。

^{▲ 施加電壓與電流的變化關係}

繪製出電流與HALL所得到的線圈磁芯的磁感應強度之間的關係，可以看到該曲線就呈現比較標準的軟磁性的特點，沒有了奇怪的“8”字曲線的特性了。

^{▲ 線圈電流與磁芯磁場強度之間的關係曲線}

下面是將測量的曲線進行拉長，將爲清晰展示他們之間的關係。

^{▲ 線圈電流與磁芯的HALL測量的信號關係曲線}

2.測量繼電器線圈溫度變化

■ 實驗方法

使用熱電偶粘貼在繼電器的鐵芯上，直接測量鐵芯的溫度。

這個溫度與繼電器線圈內部的溫度具有一定的延遲，但是它們的變化趨勢應該呈現相似的。

■ 數據分析
同樣在繼電器線圈上施加正弦變化的緩變電壓，採集供100個數據點，每個採樣點之間的時間間隔大約爲2秒鐘。下面是測量得到的隨着時間變化，線圈上的電壓與鐵芯溫度 的變化曲線。

^{▲ 線圈施加電壓以及磁芯溫度隨着時間的變化曲線}

^{▲ 施加電壓與溫度曲線之間的關係}

t0=27.5e-6
tt0 = 28.5
t1 = 84.33e-6
tt1 = 48.7
def v2t(v):
    return (v-t0)*(tt1-tt0)/(t1-t0) + tt0

v=[0.000000,1.522174,3.038219,4.542030,6.027552,7.488803,8.919899,10.315078,11.668722,12.975380,14.229790,15.426903,16.561896,17.630201,18.627515,19.549823,20.393410,21.154881,21.831168,22.419549,22.917654,23.323478,23.635386,23.852123,23.972816,23.996979,23.924515,23.755715,23.491259,23.132212,22.680020,22.136503,21.503851,20.784610,19.981677,19.098284,18.137990,17.104660,16.002456,14.835816,13.609437,12.328257,10.997437,9.622333,8.208483,6.761581,5.287453,3.792034,2.281345,0.761470,-0.761470,-2.281345,-3.792034,-5.287453,-6.761581,-8.208483,-9.622333,-10.997437,-12.328257,-13.609437,-14.835816,-16.002456,-17.104660,-18.137990,-19.098284,-19.981677,-20.784610,-21.503851,-22.136503,-22.680020,-23.132212,-23.491259,-23.755715,-23.924515,-23.996979,-23.972816,-23.852123,-23.635386,-23.323478,-22.917654,-22.419549,-21.831168,-21.154881,-20.393410,-19.549823,-18.627515,-17.630201,-16.561896,-15.426903,-14.229790,-12.975380,-11.668722,-10.315078,-8.919899,-7.488803,-6.027552,-4.542030,-3.038219,-1.522174,-0.000000]
t=[0.000006,0.000005,0.000005,0.000005,0.000005,0.000005,0.000005,0.000005,0.000005,0.000005,0.000005,0.000006,0.000006,0.000006,0.000006,0.000006,0.000007,0.000007,0.000008,0.000008,0.000009,0.000010,0.000010,0.000011,0.000012,0.000013,0.000014,0.000015,0.000017,0.000018,0.000019,0.000020,0.000022,0.000023,0.000025,0.000026,0.000027,0.000029,0.000030,0.000031,0.000033,0.000034,0.000035,0.000036,0.000037,0.000038,0.000039,0.000040,0.000041,0.000042,0.000043,0.000043,0.000044,0.000045,0.000045,0.000045,0.000046,0.000046,0.000046,0.000047,0.000047,0.000047,0.000048,0.000048,0.000049,0.000049,0.000050,0.000050,0.000051,0.000052,0.000052,0.000053,0.000054,0.000055,0.000055,0.000056,0.000057,0.000058,0.000059,0.000060,0.000061,0.000063,0.000064,0.000066,0.000068,0.000069,0.000070,0.000072,0.000073,0.000075,0.000076,0.000077,0.000078,0.000079,0.000080,0.000081,0.000082,0.000082,0.000083,0.000084]

■ 銅絲溫度與電阻關係

根據 銅的電阻率與溫度的關係 ,銅絲的電阻爲：

其中：

• $R$:在溫度T下銅絲的電阻
• $R_{ref}$：在參考溫度（通常爲20℃，或者0℃）下的銅線的溫度
• $\alpha$：銅線的電阻溫度係數
• $T,T_{ref}$是測量溫度和參考溫度的數值，同時使用攝氏度

銅的溫度係數爲：

根據 電磁鐵的磁芯實驗 中測量電磁線圈的電阻在室溫下爲：$R_{ref} = 90.4\Omega$，室溫：$T_{ref} = 27$℃。
根據前面測量的數據，可以繪製出隨着時間變化，線圈的電阻的變化曲線爲：

^{▲ 根據溫度計算出線圈的電阻隨着時間的變化}

■ 繪製電流變化曲線

根據隨着時間變化的電壓值以及前面線圈的電阻值變化，可以繪製出電壓與線圈電流之間的關係。

考慮到在測量時，線圈上串聯了10Ω測電阻，所以在計算的時候需要考慮到這個關係。已知電壓$V\left[ n \right]$，以及線圈電阻$R\left[ n \right]$，則對應的電流爲：

下面是繪製出電壓與電流的曲線。可以看到與前面實際測量的數值所呈現的**奇怪的“8”**形狀是相似的。

^{▲ 電壓與電流之間的變化曲線}

之所以不嚴格相同，是因爲此處所測量的文檔只是繼電器線圈鐵芯的溫度，與線圈的溫度之間還是有相當大的延遲和差別。但是從趨勢上來看，的確是溫度造成了前面所測量的奇怪的結果。

 

※ 結論

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如果施加在線圈上的電壓超過的它的額定電壓值，就像本文中的實驗那樣，就會造成線圈溫度的上升，從而引起線圈電阻的變化。

在測量過程中所呈現的施加的電壓與磁芯內部磁感應強度之間的奇怪的“8”字型的變化關係，到頭來都是溫度惹的禍。

在線圈的實際使用中，需要遵循線圈的額定電壓的限制。如果過壓過久，高溫就會造成線圈的永久性損壞，包括鐵芯的退磁等。