使用電子表格和一些簡單的數(shù)學方法，將線性化接口設計成非線性RTD。
RTD(電阻溫度檢測器)是高精度設計的首選傳感器。RTD雖然在0到100°C的有限溫區(qū)內(nèi)近似為線性，但隨著測量溫度的范圍逐漸加寬，這些傳感器的電阻非線性呈現(xiàn)出輕微但逐漸增大的趨勢。因此，在擴大的溫度范圍里，如果為使系統(tǒng)實現(xiàn)高精度，曲線擬合是必要的。避免RTD傳感器非線性特征的一個方法是，在其他信號處理之前設計類似硬件實現(xiàn)曲線擬合運算。這個方法特別適用于價格低和器件數(shù)少，和微處理器設計不可行的情況。器件數(shù)少對小PCB(印制板)管腳有額外的好處。
最常用的RTD由0°C 100Ω的鉑電阻制造，金屬純度導致它們符合標準帶正溫系數(shù)α為0.00385Ω/Ω/°C的歐洲曲線。不常用但仍可用的稍高金屬純度RTD。這些RTD有0.00392Ω/Ω/°C的α，符合美國曲線。圖1電路使用標準RTD測量0到350°C的擴展溫區(qū)，輸出0到3.5V的電壓，全程系統(tǒng)精度高于0.5°C。下面的線性方程表示了這個傳感器系統(tǒng)：
IC1為管腳可配置型，通過接地傳感器T1驅(qū)動400µA的恒定電流。用這個水平的電流
驅(qū)動T1——“零功率”工作——保持傳感器中電路消耗的最差功率小于40µW，減少對二階效應的自熱誤差(參考文獻1)。用電流源驅(qū)動RTD也保持其固有的非線性，允許表示傳感器的輸出電壓VS為400µA×RS，在這里RS為傳感器電阻。
IC2最初通過先縮放輸出電壓，再將結果偏置的方法，實現(xiàn)傳感器輸出的信號調(diào)理，所以VT稍大于350°C下的3.5V輸出，VT在0°C下為0V。在線性化處理之前加入增益和偏置，減少了曲線擬合電路的負擔，且有助于滿足系統(tǒng)的精度指標。C1與R5結合實現(xiàn)了單極約10Hz的低通濾波，消除電源噪聲。下面的公式描述了IC2的性能和其伴隨電路：VT=75VS–3V。
其次，Excel表格建立了電壓VT和系統(tǒng)輸出VO之間的非線性數(shù)學關系(表1)。表格有17個溫度條目，從0°C到400°C，以25°C遞增。使用數(shù)組擴充超過預設的350°C測量溫度范圍，減少非線性系統(tǒng)的端誤差。RS值——來源于標準RTD電阻隨溫度表——方程用于計算VS和VT。VT和VO的列對于線性化電路分別為輸入和輸出信號;使用Excel的XY分散特性制圖。使用Excel的趨勢特征建立了下面的公式，所需曲線擬合電路的數(shù)學表達將傳感器輸出線性化：VO=0.0005V+0.8597VT+0.0123VT2。IC3和四個1%誤差的電阻或隨意五個電阻實現(xiàn)二階多項式：VO=a+bVT+cVT2，在這里a為偏置量，b為線性系數(shù)，C為二次項系數(shù)。
曲線擬合電路的設計從畫出IC3的四個輸入建立二次項開始，也就是說通過內(nèi)部1/10V的比例縮放芯片輸出。然后，經(jīng)比較發(fā)現(xiàn)系數(shù)C必須為0.0123。因為R6和R7的分壓器削弱了信號VT，可用下面的公式表示系數(shù)：
選擇R7的值為——本設計為10 kΩ——然后使用上述公式計算R6。
電阻R8、R9和隨意無源加法器R10建立偏置項a和線性系數(shù)b。將無源加法器的輸出直接連到IC3的管腳6，Z輸入，其將偏置和線性項相加到二次項形成管腳7的系統(tǒng)響應。再次比較這些項，注意到偏置項必須等于0.0005V。偏置項僅為0.5mV，消除它會增加約0.05°C的誤差，所以可在一開始忽略它。隨后，因為線性項的系數(shù)必須等于0.8507，首先選擇R9合適的值，使用下面的公式計算R8：b=R9/(R8+R9)。
如果希望設計可選電路和包含偏置項，其為無源加法器的一部分，選擇穩(wěn)定的2.5V參考為VREF，計算并聯(lián)結果R8//R9=REQ(R8與R9并聯(lián)的等效電阻)，使用下面的分壓器公式計算R10：a=(REQ/(R9+REQ))VREF。
為校準電路，用精密十進制電阻箱取代傳感器。設置電阻箱模擬0°C，調(diào)整IC3管腳7的R2偏置端使輸出為0V。其次，設置電阻箱模擬350°C，調(diào)整R3的增益端使輸出為3.5V。重復這個端調(diào)整步驟的順序直到兩個點都確定。如圖1電路——包括可選電路——顯示了250°C時最差測量誤差為2.504V的0.16%，或0.4°C。沒有可選電路時測試電路——參考電壓和R10——顯示精度上沒有顯著改善。
英文原文：
Design an RTD interface with a spreadsheet
Using a spreadsheet and some simple math, you can design a linearizing interface to a non-linear RTD.
Robert S Villanucci, Wentworth Institute of Technology, Boston; Edited by Charles H Small and Fran Granville -- EDN, 2/7/2008
RTDs (resistance-temperature detectors) are the preferred sensor choices for designs requiring precision. Although RTDs are approximately linear over the limited temperature range of 0 to 100°C, these sensors exhibit a slight but progressively more nonlinear temperature-versus-resistance characteristic as the measurement range widens. Consequently, over an extended span, curve fitting is necessary if the system is to achieve a high level of precision. One way to obviate the nonlinear characteristic of an RTD sensor is to design analog hardware to perform the curve-fitting mathematics before any additional signal processing occurs. This approach is especially attractive if you can keep both cost and component count low and if a microprocessor-driven design is not feasible. With low component count comes the added benefit of a small PCB (printed-circuit-board) footprint.
The most popular RTDs are made from platinum with a resistance value of 100Ω at 0°C and a metal purity that allows them to follow a standard European curve with a positive-temperature coefficient, α, equal to 0.00385Ω/Ω/°C. Less popular but still common are RTDs with a slightly higher metal purity. These RTDs have α of 0.00392Ω/Ω/°C and follow the US curve. The circuit in Figure 1 uses a standard RTD to measure temperature over the extended range of 0 to 350°C, an output voltage of 0 to 3.5V, and overall system accuracy greater than 0.5°C. The following linear equation expresses this sensor system:
IC1 is pin-configured to drive a constant current of 400 µA through the grounded sensor, T1. Driving T1 with this level of current—“zero-power” operation—keeps the worst-case power that the circuit dissipates in the sensor to less than 40 µW and reduces the self-heating errors to a second-order effect (Reference 1). Also, driving the RTD with a current source preserves its intrinsic nonlinearity and allows you to express the sensor’s output voltage, VS, as: 400 µA×RS, where RS is the resistance of the sensor.
IC2 initially signal-conditions the sensor’s output by first scaling the output voltage and then offsetting the result so that VT is slightly larger than the 3.5V output at 350°C and that VT equals 0V at 0°C. Adding gain and offset before linearization places less of a burden on the curve-fitting circuitry and helps to meet the system’s precision specification. The combination
of C1 and R5 implements a lowpass filter with a pole at approximately 10 Hz to remove power-supply noise. The following term describes the performance of IC2 and its accompanying circuitry: VT="75VS"–3V.
Next, an Excel spreadsheet creates the nonlinear-mathematical relationship between the voltage, VT, and the system output, VO (Table 1). The spreadsheet features 17 temperature entries—starting at 0°C, increasing in increments of 25°C, and ending at 400°C—for the measured temperature. Using a data set that extends beyond the intended measurement range of 350°C can reduce end errors in nonlinear systems. Values for RS—which you derive from a standard RTD-resistance-versus-temperature table—and the equations allow you to compute VS and VT. The VT and VO columns are the input and output signals, respectively, for the linearization circuitry; you chart them using Excel’s XY-scatter feature. You can use Excel’s Trendline feature to create the following equation, the mathematical representation of the curve-fitting circuitry you need to linearize the sensor’s output: VO="0".0005V+0.8597VT+0.0123VT 2. IC3 and four 1%-tolerant resistors or, optionally, five resistors implement a second-order polynomial: VO="a"+bVT+cVT 2, where a is the offset term, b is the linear coefficient, and c is the square-term coefficient.
The curve-fitting-circuit design begins by first wiring the four inputs of IC3 to create a positive square term that is scaled at the chip’s output by an internal scale factor of 1/10V. Then, comparing terms, you find that the coefficient, c, must equal 0.0123. Because R6 and R7 form a voltage divider that attenuates the signal, VT, you can express the coefficient with the following equation:
Select a value for R7—10 kΩ for this design—and then use the preceding equation to find the value for R6.
Resistors R8, R9, and, optionally, R10 form a passive adder to create the offset term, a, and the linear coefficient, b. You apply the output of the passive
adder directly to the Z input, Pin 6 of IC3, which adds the offset and linear terms to the square term to form the system response at Pin 7. Again comparing these terms, note that the offset term must equal 0.0005V. The offset term is only 0.5 mV, and eliminating it would add an error of approximately 0.05°C, so you can initially neglect it. Then, because the linear term’s coefficient, b, must equal 0.8507, you first select a suitable value for R9 and use the following equation to solve for R8: b="R9/"(R8+R9).
If you wish to design the optional circuitry and include the offset term, which is part of the passive adder, choose a stable 2.5V reference for VREF, calculate the parallel combination of R8//R9=REQ (the equivalent resistance of R8 in parallel with R9), and solve for R10 using the following voltage-divider equation: a=(REQ/(R9+REQ))VREF.
To calibrate this circuit, replace the sensor with a precision decade box. Set the decade box to simulate 0°C and adjust the offset trim of R2 for an output of 0V at Pin 7 of IC3. Next, set the decade box to simulate 350°C and adjust the gain trim of R3 for an output of 3.5V. Repeat this sequence of trim steps until both points are fixed. The circuit in Figure 1—which includes optional circuitry—exhibits a worst-case measurement error at 250°C and 2.504V of 0.16%, or 0.4°C. Testing the circuit without the optional circuitry—the reference voltage and R10 —shows no discernible improvement in precision.
Reference
“IC Generates Second-Order Polynomial,” Electronic Design, Aug 5, 1993.
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