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--- projects:electronics:weller:wxp80_reverse_engineer:analog_measurements [2018/05/12 17:43] – admin
+++ projects:electronics:weller:wxp80_reverse_engineer:analog_measurements [2018/05/15 00:39] (current) – [WXP80 analog measurements] admin
@@ Line 1: / Line 1: @@
 ====== WXP80 analog measurements ======
-This page shows the analog measurements of the WXP80 heating element connected to an operation amplifier. From the results a formula is calculated to model this.
+This page shows the analog measurements of the WXP80 heating element connected to an operational amplifier. From the results a formula is calculated to model this.
 ===== PTC resistance and voltages WXP80 =====
@@ Line 30: / Line 30: @@
 \\
-With the 'show equation' option in Libre office a formula is given:\\
+Based on the observations we assume that the relationship between temperature and resistance is linear. We can thus find a first order formula which approaches the data pretty well. A general function looks like:\\
+$R = aT + b \tag{1}$\\
+where $\begin{align*}a\end{align*}$ is the slope and $\begin{align*}b\end{align*}$ the intercept of the vertical-axis.\\
-$Rv = 0.465T + 98.139 \tag{1}$\\
+The so called normal equations for the estimated slope $a$ and intercept $b$ are:\\
-where T is the temperature in °C and Rv the resistance of TH1 in Ω.
+$$\begin{align*}n\ a + ST\ b = SR \end{align*}$$
+$$\begin{align*}ST\ a + STT\ b = SRT \end{align*}$$
+with $\begin{align*}n\end{align*}$ the number of data\\
+$$\begin{align*}ST = \sum\limits_{i=1}^n T_i \end{align*}$$
+$$\begin{align*}STT = \sum\limits_{i=1}^n T_i^2 \end{align*}$$
+$$\begin{align*}SR = \sum\limits_{i=1}^n R_i \end{align*}$$
+$$\begin{align*}SRT = \sum\limits_{i=1}^n T_iR_i \end{align*}$$
+with solution
+$$\begin{align*}a= \frac{n\ SRT- SR\ ST}{n\ STT - ST^2} \end{align*}$$
+$$\begin{align*}b= \frac{1}{ST} (SR-n\ a) \end{align*}$$
+or alternatively
+$$\begin{align*}b=\frac{STT\ SR - ST\ STR}{n\ STT - ST^2} \end{align*}$$
+With the data from the table above, the least squares trendline is:\\
+$R = 0.465T + 98.17 \tag{2}$\\
+where $\begin{align*}T\end{align*}$ is the temperature in °C and $\begin{align*}R\end{align*}$ the resistance of TH1 in Ω.
+Please note that libre office calc offers an easy method to acquire this equation directly, using the 'show equation' option. It will show equation (2) right away:\\
+The opamp is configured as a DC coupled non-inverting amplifier with a bias. To calculate the gain (G) and bias (Vb), we use following equations:\\
+$G = 1 + \frac{R6}{R4 + \frac{R3 \cdot R5}{R3 + R5}} = 1 + \frac{1.0 \cdot 10^6}{33\cdot10^3 + \frac{33\cdot10^3 \cdot 47}{33\cdot10^3 + 47}} = 31.1 \tag{3}$\\
+where $\begin{align*}R3\end{align*}$ = 4.3kΩ, $\begin{align*}R4\end{align*}$ = 33.0kΩ, $\begin{align*}R5\end{align*}$ = 47Ω and $\begin{align*}R6\end{align*}$ = 1.0MΩ\\
+The bias voltage is a simple voltage divider with $\begin{align*}R3\end{align*}$ and $\begin{align*}R5\end{align*}$:
+$V_b = V_{dd} \frac{R5}{R5 + R3} = 4.66 \frac{47}{47 + 4300} = 0.0501 \tag{4}$\\
+where $\begin{align*}V_{dd}\end{align*}$ = 4.66v\\
+\\
+We now are able to calculate a transfer function using the PTC resistor value to find the opamp output voltage relative to the temperature. (Alternatively we could calculate the transfer function using the least squares method of the output voltage data from the table above.)\\
+\\
+The transfer function is:
-The opamp works both as amplifier and level converter. Based on the schematics the following characteristics have been determined:
-  * Gain: 31.3
-  * Vbias: 0.0504
 ==== Tools and measuring equipment ====
@@ Line 44: / Line 76: @@
   * Fluke 87 multimeter
+==== References ====
+  * [[https://ocw.mit.edu/courses/media-arts-and-sciences/mas-836-sensor-technologies-for-interactive-environments-spring-2011/readings/MITMAS_836S11_read02_bias.pdf]]
+  * least square trendline