1 前言

处理器通过RTD采集电路（芯片）精确获得当前RTD电阻值后，再结合RTD与温度线性关系表，换算出当前环境温度。阻值与温度换算，通常是对“阻值—温度”进行查表匹配获得。因为，RTD“阻值—温度”在整个变化范围内不一定是标准线性关系，只在某一温度区间呈现线性或者接近线性。因此，首先将“阻值—温度”划分为一定数量的区间，然后通过“二分法查找算法”查找测量的阻值处于某一区间，再通过“线性插值法”计算出实际温度值。

相关文章：

【RTD】铂电阻测温原理与具体方法
【RTD】AD7793三线式铂电阻PT100/PT1000应用
【RTD】AD7793四线式铂电阻PT100/PT1000应用
【RTD】AD7793两线式铂电阻PT100/PT1000应用
【RTD】AD7793驱动程序
【RTD】二分法查找和分段线性插值算法在RTD中应用

2 二分法查找

二分法查找，或者成为折半法查找，是一种用于有序数组中查找特定元素的查找算法。二分法查找比较易于理解，查找步骤大致分为：

【1】将待查找数组分为两部分

【2】待查找值与数组中间值比较，确保待查找值处于哪一半部分

【3】将该半部分重复【1】【2】步骤，直至查找到目标值或者查找结束为止

2.1 复杂度

时间复杂度

【1】最坏情况查找最后一个元素（第一个元素），此时时间复杂度为T(n) = O(log2n)

【2】最理想情况首次查找中间元素即为目标值（奇数长度数组的正中间值，偶数长度数组的中间靠左的元素），此时时间复杂度为T(n) = O(1)

空间复杂度

S(n)=logn

2.2 实现

二分法查找算法，可以使用递归和非递归方式实现；递归方式对于查找长度比较长的数组需要更大的栈空间。

递归方式C语言实现

int binary_search(int *buf, uint32_t index_bottom, uint32_t index_top, int s_value)
{uint32_t index_mid = 0;if (index_bottom > index_top){return -1;}index_mid = index_bottom + (index_top-index_bottom)/2;if (buf[index_mid] == s_value) {return index_mid;}else if(buf[mid]  > s_value){return binary_search(buf, index_bottom, index_mid-1, s_value);}else{return binary_search(buf, index_mid+1, index_top, s_value);}
}

非递归C语言实现

int binary_search(int *buf, uint32_t buf_size, int s_value)
{uint32_t index_bottom = 0;uint32_t index_top = buf_size-1;uint32_t index_mid = buf_size/2 - 1;while (index_top-index_bottom > 1){if (s_value < buf[index_mid]){index_top=index_mid;  }else if(s_value>buf[index_mid]){index_bottom=index_mid;    }else{return index_mid;}index_mid=(index_bottom+index_top)/2;}return -1;
}

3 分段线性插值

分段线性插值法，将目标样本点区间划分为多个区间[x，x+1]，分别在每个区间上作一次线性方程，计算x点的插值时，只需用到左右的两个坐标点，计算量与坐标点个数n无关。

假设某一区间两个节点为（x1，y1）和（x2，y2），则该区间上的一次线性方程可以这样表示：

F1=x−x2x1−x2f(x1)+x−x1x2−x1f(x2)F1 = \frac{x-x2} {x1-x2} f(x1)+\frac{x-x1} {x2-x1}f(x2) F1=x1−x2x−x2f(x1)+x2−x1x−x1f(x2)

线性坐标图

以上图为例，利用方程式证明。

斜率：k=(y2−y1)(x2−x1)斜率：k = \frac{(y2-y1)} {(x2-x1)} 斜率：k=(x2−x1)(y2−y1)

线性方程：y−y1=k(x−x1)线性方程：y-y1 = k (x-x1) 线性方程：y−y1=k(x−x1)

y=(y2−y1)(x−x1)(x2−x1)+y1y = \frac{(y2-y1)(x-x1)} {(x2-x1)}+y1 y=(x2−x1)(y2−y1)(x−x1)+y1
y=x−x1x2−x1y2+(1+x−x1x1−x2)y1y = \frac{x-x1} {x2-x1}y2+(1+\frac{x-x1} {x1-x2})y1 y=x2−x1x−x1y2+(1+x1−x2x−x1)y1
y=x−x2x1−x2y1+x−x1x2−x1y2y = \frac{x-x2} {x1-x2} y1+\frac{x-x1} {x2-x1}y2 y=x1−x2x−x2y1+x2−x1x−x1y2

很显然，如果测量的RTD阻值刚好处于某一区间内，则需要使用插值法计算出当前温度值。

4 RTD实例

以PT00、PT1000为例，分别生成一个【-70℃ ~ +110℃】、【温度步进为0.5℃】的表（数组）。X轴表示阻值，y轴表示温度值，根据线性插值法，温度计算公式表示为：

y=[x−x2x1−x2y1+x−x1x2−x1y2]×s−hy = [\frac{x-x2} {x1-x2} y1+\frac{x-x1} {x2-x1}y2]\times {s}-h y=[x1−x2x−x2y1+x2−x1x−x1y2]×s−h
其中：
斜率 s = 0.5
初值 h = 70

const float PT100_CODE[361] =
{ /* -70 ~ + 110C */
72.335,     72.535,     72.735,     72.935,     73.134,     73.334,     73.534,     73.734,     73.934, 74.133,
74.333,     74.533,     74.732,     74.932,     75.131,     75.331,     75.53,      75.73,      75.929, 76.129,
76.328,     76.527,     76.726,     76.926,     77.125,     77.324,     77.523,     77.722,     77.921, 78.12,
78.319,     78.518,     78.717,     78.915,     79.114,     79.313,     79.512,     79.71,      79.909, 80.108,
80.306,     80.505,     80.703,     80.902,     81.1,       81.299,     81.497,     81.695,     81.894, 82.092,
82.29,      82.488,     82.687,     82.885,     83.083,     83.281,     83.479,     83.677,     83.875, 84.073,
84.271,     84.469,     84.666,     84.864,     85.062,     85.26,      85.457,     85.655,     85.853, 86.05,
86.248,     86.445,     86.643,     86.84,      87.038,     87.235,     87.432,     87.63,      87.827, 88.024,
88.222,     88.419,     88.616,     88.813,     89.01,      89.207,     89.404,     89.601,     89.798, 89.995,
90.192,     90.389,     90.586,     90.783,     90.98,      91.177,     91.373,     91.57,      91.767, 91.963,
92.16,      92.356,     92.553,     92.75,      92.946,     93.143,     93.339,     93.535,     93.732, 93.928,
94.124,     94.321,     94.517,     94.713,     94.909,     95.106,     95.302,     95.498,     95.694, 95.89,
96.086,     96.282,     96.478,     96.674,     96.87,      97.066,     97.261,     97.457,     97.653, 97.849,
98.044,     98.24,      98.436,     98.631,     98.827,     99.023,     99.218,     99.414,     99.609, 99.805,
100,        100.195,    100.391,    100.586,    100.781,    100.977,    101.172,    101.367,    101.562,    101.758,
101.953,    102.148,    102.343,    102.538,    102.733,    102.928,    103.123,    103.318,    103.513,    103.708,
103.903,    104.097,    104.292,    104.487,    104.682,    104.876,    105.071,    105.266,    105.46,     105.655,
105.849,    106.044,    106.238,    106.433,    106.627,    106.822,    107.016,    107.211,    107.405,    107.599,
107.794,    107.988,    108.182,    108.376,    108.57,     108.764,    108.959,    109.153,    109.347,    109.541,
109.735,    109.929,    110.123,    110.316,    110.51,     110.704,    110.898,    111.092,    111.286,    111.479,
111.673,    111.867,    112.06,     112.254,    112.447,    112.641,    112.835,    113.028,    113.221,    113.415,
113.608,    113.802,    113.995,    114.188,    114.382,    114.575,    114.768,    114.961,    115.155,    115.348,
115.541,    115.734,    115.927,    116.12,     116.313,    116.506,    116.699,    116.892,    117.085,    117.278,
117.47,     117.663,    117.856,    118.049,    118.241,    118.434,    118.627,    118.819,    119.012,    119.205,
119.397,    119.59,     119.782,    119.975,    120.167,    120.359,    120.552,    120.744,    120.936,    121.129,
121.321,    121.513,    121.705,    121.898,    122.09,     122.282,    122.474,    122.666,    122.858,    123.05,
123.242,    123.434,    123.626,    123.818,    124.009,    124.201,    124.393,    124.585,    124.777,    124.968,
125.16,     125.352,    125.543,    125.735,    125.926,    126.118,    126.309,    126.501,    126.692,    126.884,
127.075,    127.649,    127.458,    127.649,    127.84,     128.032,    128.223,    128.414,    128.605,    128.796,
128.987,    129.178,    129.37,     129.561,    129.752,    129.942,    130.133,    130.324,    130.515,    130.706,
130.897,    131.088,    131.278,    131.469,    131.66,     131.85,     132.041,    132.232,    132.422,    132.613,
132.803,    132.994,    133.184,    133.375,    133.565,    133.755,    133.946,    134.136,    134.326,    134.517,
134.707,    134.897,    135.087,    135.277,    135.468,    135.658,    135.848,    136.038,    136.228,    136.418,
136.608,    136.798,    136.987,    137.177,    137.367,    137.557,    137.747,    137.936,    138.126,    138.316,
138.506,    138.695,    138.885,    139.074,    139.264,    139.453,    139.643,    139.832,    140.022,    140.211,
140.4,      140.59,     140.779,    140.968,    141.158,    141.347,    141.536,    141.725,    141.914,    142.103,
142.293,
};const float PT1000_CODE[361] =
{/* -70 ~ + 110C */
723.345,    725.346,    727.346,    729.345,    731.344,    733.343,    735.341,    737.339,    739.337,    741.334,
743.331,    745.327,    747.324,    749.319,    751.315,    753.309,    755.304,    757.298,    759.292,    761.285,
763.278,    765.271,    767.263,    769.255,    771.247,    773.238,    775.229,    777.219,    779.21,     781.199,
783.189,    785.178,    787.166,    789.155,    791.143,    793.13,     795.117,    797.104,    799.091,    801.077,
803.063,    805.048,    807.033,    809.018,    811.003,    812.987,    814.97,     816.954,    818.937,    820.919,
822.902,    824.884,    826.865,    828.847,    830.828,    832.808,    834.789,    836.769,    838.748,    840.728,
842.707,    844.685,    846.663,    848.641,    850.619,    852.596,    854.573,    856.55,     858.526,    860.502,
862.478,    864.453,    866.428,    868.403,    870.377,    872.351,    874.325,    876.298,    878.271,    880.244,
882.217,    884.189,    886.16,     888.132,    890.103,    892.074,    894.044,    896.015,    897.985,    899.954,
901.923,    903.892,    905.861,    907.829,    909.797,    911.765,    913.732,    915.7,      917.666,    919.633,
921.599,    923.565,    925.53,     927.496,    929.461,    931.425,    933.39,     935.354,    937.317,    939.281,
941.244,    943.207,    945.169,    947.132,    949.093,    951.055,    953.016,    954.977,    956.938,    958.899,
960.859,    962.819,    964.778,    966.737,    968.696,    970.655,    972.613,    974.572,    976.529,    978.487,
980.444,    982.401,    984.358,    986.314,    988.27,     990.226,    992.181,    994.136,    996.091,    998.046,
1000,       1001.954,   1003.908,   1005.861,   1007.814,   1009.767,   1011.72,    1013.672,   1015.624,   1017.576,
1019.527,   1021.478,   1023.429,   1025.38,    1027.33,    1029.28,    1031.229,   1033.179,   1035.128,   1037.077,
1039.025,   1040.973,   1042.921,   1044.869,   1046.816,   1048.764,   1050.71,    1052.657,   1054.603,   1056.549,
1058.495,   1060.44,    1062.385,   1064.33,    1066.274,   1068.218,   1070.162,   1072.106,   1074.049,   1075.992,
1077.935,   1079.877,   1081.82,    1083.762,   1085.703,   1087.644,   1089.585,   1091.526,   1093.467,   1095.407,
1097.347,   1099.286,   1101.225,   1103.164,   1105.103,   1107.042,   1108.98,    1110.917,   1112.855,   1114.792,
1116.729,   1118.666,   1120.602,   1122.538,   1124.474,   1126.41,    1128.345,   1130.28,    1132.215,   1134.149,
1136.083,   1138.017,   1139.95,    1141.884,   1143.817,   1145.749,   1147.681,   1149.614,   1151.545,   1153.477,
1155.408,   1157.339,   1159.27,    1161.2,     1163.13,    1165.06,    1166.989,   1168.918,   1170.847,   1172.776,
1174.704,   1176.632,   1178.56,    1180.487,   1182.414,   1184.341,   1186.268,   1188.194,   1190.12,    1192.046,
1193.971,   1195.896,   1197.821,   1199.746,   1201.67,    1203.594,   1205.518,   1207.441,   1209.364,   1211.287,
1213.21,    1215.132,   1217.054,   1218.975,   1220.897,   1222.818,   1224.739,   1226.659,   1228.579,   1230.499,
1232.419,   1234.338,   1236.257,   1238.176,   1240.095,   1242.013,   1243.931,   1245.848,   1247.766,   1249.683,
1251.6,     1253.516,   1255.432,   1257.348,   1259.264,   1261.179,   1263.094,   1265.009,   1266.923,   1268.837,
1270.751,
};/*  */
/*** @brief  二分法查找* @param buf:待查找数据表* @param buf_size:表格大小* @param s_value:查找目标值* @param out_data:返回值* @retval 匹配返回0，否则返回左值*/
int8_t binary_search(const float *buf, uint32_t buf_size, float s_value, uint32_t *out_data)
{uint32_t index_bottom = 0;uint32_t index_top = buf_size-1;uint32_t index_mid = buf_size/2 - 1;while (index_top-index_bottom > 1){if (s_value < buf[index_mid]){index_top=index_mid;  }else if(s_value>buf[index_mid]){index_bottom=index_mid;    }else{*out_data = index_mid;  /* 恰好中值 */return 0x00;}index_mid=(index_bottom+index_top)/2;}*out_data = index_bottom;   /* 处于区间内，返回左值 */return 0x01;
}/*** @brief 插值法阻值-温度换算* @param code:温度-阻值表* @param size:表格大小* @param resi:电阻阻值* @retval 实际温度值(℃)*/
float resi_temp_calc(const float *code, uint32_t size, float resi)
{int8_t ret = 0;uint32_t value = 0;float temp = 0.0f;float x1 = 0.0f;float x2 = 0.0f;float y1 = 0.0f;float y2 = 0.0f;ret = binary_search(&code[0], size, resi, &value);if (0x00 == ret){temp = value*0.5f-70;}else if(0x01 == ret){/* 插值法计算 */x1 = code[value];y1 = value;x2 = code[value+1];y2 = value+1;temp = (y1*(resi-x2)/(x1-x2) + y2*(resi-x1)/(x2-x1))*0.5f - 70;}else{/* tod: error */}return temp;
}#include <stdio.h>
#include <stdint.h>
int main(char argc, char **argv)
{float temp0 = 0.0f;float temp1 = 0.0f;temp0 = resi_temp_calc(&PT100_CODE[0], sizeof(PT100_CODE)/sizeof(float), 102);printf("PT100 temp=%.2fC\r\n", temp0);temp1 = resi_temp_calc(&PT1000_CODE[0], sizeof(PT1000_CODE)/sizeof(float), 1001.954);printf("PT1000 temp=%.2fC\r\n", temp1);return 0;
}

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【RTD】二分法查找和分段线性插值算法在RTD中应用

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