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electrical_engineering_1:network_analysis [2023/07/17 02:34] mexleadmin |
electrical_engineering_1:network_analysis [2023/11/28 00:45] (aktuell) mexleadmin |
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- | ====== 4. Analysis of Networks ====== | + | ====== 4 Analysis of Networks ====== |
< | < | ||
Zeile 400: | Zeile 400: | ||
===== Exercises ===== | ===== Exercises ===== | ||
- | <panel type=" | + | # |
+ | |||
+ | Imagine you want to develop a circuit that conditions a sensor signal so that it can be processed by a microcontroller. The sensor signal is in the range $U_{\rm sens} \in [-15...15~\rm V]$, and the microcontroller input can read values in the range $U_{\rm uC} \in [0...3.3~\rm V]$. The sensor can supply a maximum current of $I_{\rm sens, max}=1~\rm mA$. For the internal resistance of the microcontroller, | ||
+ | |||
+ | For conditioning, | ||
+ | |||
+ | The following simulation shows roughly the situation (the resistor values are not correct). | ||
< | < | ||
- | Imagine you want to develop a circuit that conditions a sensor signal so that it can be processed by a microcontroller. The sensor signal | + | Questions: |
+ | |||
+ | 1. Find the relationship between $R_1$, $R_2$, and $R_3$ using superposition. \\ | ||
+ | * Determine suitable values for $R_1$, $R_2$, and $R_3$. | ||
+ | * What values for $R^0_1$, $R^0_2$, and $R^0_3$ from the [[https:// | ||
+ | |||
+ | # | ||
+ | Using superposition, | ||
+ | For these two circuits, we calculate $U_\rm A^{(1)}$ and $U_\rm A^{(2)}$. \\ | ||
+ | To make the calculation simpler, the resistors $R_3$ and $R_{\rm s}$ will be joined to $R_4 =R_3 +R_{\rm s}$. | ||
+ | |||
+ | < | ||
+ | === Circuit 1 : only consider $U_{\rm S}$, ignore $U_{\rm I}$ === | ||
+ | {{drawio> | ||
+ | |||
+ | \begin{align*} | ||
+ | U_{\rm O}^{(1)} | ||
+ | = U_{\rm S} \cdot {{ {{R_2 R_4}\over{R_2 + R_4}} }\over{R_1 + {{R_2 R_4}\over{R_2 + R_4}} }} \\ | ||
+ | & | ||
+ | & | ||
+ | \end{align*} | ||
+ | </ | ||
+ | |||
+ | < | ||
+ | === Circuit 2 : only consider $U_{\rm I}$, ignore $U_{\rm S}$ === | ||
+ | {{drawio> | ||
+ | |||
+ | \begin{align*} | ||
+ | U_{\rm O}^{(2)} | ||
+ | = U_{\rm I} \cdot {{ {{R_1 R_2}\over{R_1 + R_2}} }\over{R_4 + {{R_1 R_2}\over{R_1 + R_2}} }} \\ | ||
+ | & | ||
+ | & | ||
+ | \end{align*} | ||
+ | </ | ||
+ | |||
+ | < | ||
+ | === Superposition: | ||
+ | \\ | ||
+ | These two intermediate voltages for the single sources have to be summed up as $U_{\rm O}= U_{\rm O}^{(1)} + U_{\rm O}^{(2)}$. \\ | ||
+ | When deeper investigated, | ||
+ | We can also simplify further when looking at often-used sub-terms (here: $R_2$) | ||
+ | |||
+ | \begin{align*} | ||
+ | U_{\rm O} & | ||
+ | U_{\rm O} \cdot (R_4 R_1 + R_4 R_2+ R_1 R_2 ) & | ||
+ | U_{\rm O} \cdot ({{R_1 R_4}\over{R_2}} + R_4 + R_1 ) & | ||
+ | \end{align*} | ||
+ | |||
+ | The formula $(1)$ is the general formula to calculate the output voltage | ||
+ | </ | ||
+ | |||
+ | Now, we can use the requested boundaries: | ||
+ | - For the minimum input voltage $U_{\rm I}= -15 ~\rm V$, the output voltage shall be $U_{\rm O} = 0 ~\rm V$ | ||
+ | - For the maximum | ||
+ | |||
+ | This leads to two situations: | ||
+ | |||
+ | < | ||
+ | === Situation I : $U_{\rm I,min}= -15 ~\rm V$ shall create $U_{\rm O,min} = 0 ~\rm V$ === | ||
+ | \\ | ||
+ | We put $U_{\rm A} = 0 ~\rm V$ in the formula $(1)$ : | ||
+ | \begin{align*} | ||
+ | 0 & | ||
+ | - U_{\rm I,min} \cdot R_1 & | ||
+ | | ||
+ | \end{align*} | ||
+ | |||
+ | So, with formula $(2)$, we already have a relation between $R_1$ and $R_4$. Yeah 😀 \\ | ||
+ | The next step is situation 2 | ||
+ | </ | ||
+ | |||
+ | < | ||
+ | === Situation II : $U_{\rm I,max}= +15 ~\rm V$ shall create $U_{\rm O,max} = 3.3 ~\rm V$ === | ||
+ | \\ | ||
+ | We use formula $(2)$ to substitute $R_1 = k_{14} \cdot R_4 $ in formula $(1)$, and: | ||
+ | \begin{align*} | ||
+ | U_{\rm O,max} \cdot (k_{14}{{ R_4^2}\over{R_2}} + R_4 + k_{14} R_4 ) & | ||
+ | U_{\rm O,max} \cdot (k_{14}{{ R_4 }\over{R_2}} + 1 + k_{14} | ||
+ | | ||
+ | | ||
+ | {{ R_4 }\over{R_2}} | ||
+ | \end{align*} | ||
+ | |||
+ | So, another relation for $R_4$ and $R_2$. 😀 \\ | ||
+ | </ | ||
+ | |||
+ | So, to get values for the relations, we have to put in the values for the input and output voltage conditions. For $k_{14}$ we get by formula $(2)$: | ||
+ | \begin{align*} | ||
+ | k_{14} = {{R_1}\over{R_4}} | ||
+ | \end{align*} | ||
+ | |||
+ | This value $k_{14}$ we can use for formula $(3)$: | ||
+ | \begin{align*} | ||
+ | {{ R_4 }\over{R_2}} &= {{5 ~\rm V + 15 ~\rm V \cdot {{1}\over{3}} }\over{ | ||
+ | & | ||
+ | k_{42} | ||
+ | \end{align*} | ||
+ | |||
+ | We could now - theoretically - arbitrarily choose one of the resistors, e.g., $R_2$, and then calculate the other two. \\ | ||
+ | |||
+ | But we must consider another boundary, | ||
+ | \begin{align*} | ||
+ | R_{\rm S} & | ||
+ | &= 15 ~\rm k\Omega | ||
+ | \end{align*} | ||
+ | |||
+ | Therefore, | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | |||
+ | The sensor | ||
+ | \begin{align*} | ||
+ | R_S &= 15 {~\rm k\Omega}\\ | ||
+ | \end{align*} | ||
+ | |||
+ | We can choose $R_3$ arbitrarily. Here I choose a nice value to get integer values for $R_3$ and $R_1$: | ||
+ | \begin{align*} | ||
+ | R_3 &= 45 {~\rm k\Omega}\\ | ||
+ | R_1 &= {{1}\over{3}} | ||
+ | R_2 &= {{1}\over{5.09}}(R_3 + 15 {~\rm k\Omega}) = 11.8 {~\rm k\Omega} | ||
+ | \end{align*} | ||
+ | |||
+ | Based on the E24 series, the following values are next to the calculated ones: | ||
+ | \begin{align*} | ||
+ | R_3^0 &= 43 {~\rm k\Omega}\\ | ||
+ | R_1^0 &= {{1}\over{3}} | ||
+ | R_2^0 &= {{1}\over{5.09}}(R_3 + 15 {~\rm k\Omega}) = 12 {~\rm k\Omega} | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | 2. Find the relationship between $R_1$, $R_2$, and $R_3$ by investigating Kirchhoff' | ||
+ | |||
+ | # | ||
+ | |||
+ | The potential of the node is $U_\rm O$. Therefore the currents are: | ||
+ | - the current $I_2$ over $R_2$ is flowing to ground: $I_2 = - {{U_\rm O}\over{R_2}} $ | ||
+ | - the current $I_1$ over $R_1$ is coming from the supply voltage $U_{\rm S}$ to the nodal voltage $U_{\rm O}$: $I_1 = {{U_{\rm S} - U_{\rm O}}\over{R_1}}$ | ||
+ | - the current $I_4$ over $R_4$ is coming from the input voltage | ||
+ | |||
+ | This led to the formula based on the Kirchhoff' | ||
+ | |||
+ | \begin{align*} | ||
+ | \Sigma I = 0 &= I_1 + I_2 + I_3 \\ | ||
+ | 0 &= {{U_{\rm S} - U_{\rm O}}\over{R_1}} + {{U_{\rm I} - U_{\rm O}}\over{R_4}} - {{U_\rm O}\over{R_2}} | ||
+ | \end{align*} | ||
+ | |||
+ | The formula can be rearranged, with all terms containing $ U_{\rm O}$ on the left side: | ||
+ | \begin{align*} | ||
+ | {{U_{\rm O}}\over{R_1}} + {{U_{\rm O}}\over{R_2}} + {{U_{\rm O}}\over{R_4}} | ||
+ | U_{\rm O}\cdot \left( {{1}\over{R_1}} + {{1}\over{R_2}} + {{1}\over{R_4}} \right) & | ||
+ | \end{align*} | ||
+ | |||
+ | Both sides can be multiplied by $\cdot R_1$, $\cdot R_2$, $\cdot R_4$ - in order to get rid of the fractions : | ||
+ | \begin{align*} | ||
+ | U_{\rm O}\cdot \left( {{R_1 R_2 R_4 }\over{R_1}} + {{R_1 R_2 R_4 }\over{R_2}} + {{R_1 R_2 R_4 }\over{R_4}} \right) & | ||
+ | U_{\rm O}\cdot \left( R_2 R_4 + R_1 R_4 + R_1 R_2 \right) & | ||
+ | U_{\rm O} &= {{R_2}\over{R_2 R_4 + R_1 R_4 + R_1 R_2 }} \left( R_4 \cdot U_{\rm S} + R_1 \cdot U_{\rm I} \right)\\ | ||
+ | \end{align*} | ||
+ | |||
+ | The last formula was just the result we also got by the superposition but by more thinking. \\ | ||
+ | So, sometimes there is an easier way... | ||
+ | * Unluckily, there is no simple way to know before, what way is the easiest. | ||
+ | * Luckily, all ways lead to the correct result. | ||
+ | |||
+ | # | ||
+ | |||
+ | 3. What is the input resistance | ||
+ | |||
+ | # | ||
+ | |||
+ | \begin{align*} | ||
+ | R_{\rm in}(R_1, R_2, R_3) &= R_3 + R_1 || R_2 \\ | ||
+ | &= R_3 + {{R_1 R_2}\over{R_1 + R_2}} | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | 4. What is the minimum allowed input resistance ($R_{\rm in, min}(R_1, R_2, R_3)$) for the sensor to still deliver current? | ||
+ | |||
+ | # | ||
+ | |||
+ | \begin{align*} | ||
+ | R_{\rm in, min} &= {{U_{\rm sense}}\over{I_{\rm sense, max}}} \\ | ||
+ | &= \rm {{15 V}\over{1 mA}} \\ | ||
+ | &= 15 k\Omega \\ | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
- | For conditioning, | ||
- | - Find the relationship between $R_1$, $R_2$ and $R_3$ using superposition. | ||
- | - Find the relationship between $R_1$, $R_2$ and $R_3$ using star-delta transformation. | ||
- | - What is the input resistance $R_{\rm in}(R_1, R_2,R_3)$ of the circuit (viewed from the sensor)? | ||
- | - What is the maximum allowed input resistance $R_{\rm in}(R_1, R_2,R_3)$ for the sensor to still deliver current? | ||
- | - Determine suitable values for $R_1$, $R_2$ and $R_3$. | ||
- | - What values for $R^0_1$, $R^0_2$, and $R^0_3$ from the [[https:// | ||
- | </ | ||
{{page> | {{page> | ||
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