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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
@@ Zeile 1: / Zeile 1: @@
-====== 4. Analysis of Networks ======
+====== 4 Analysis of Networks ======
 <callout> <WRAP> <imgcaption imageNo1 | examples for networks> </imgcaption> {{drawio>Beispiele Netzwerke.svg}} </WRAP>
@@ Zeile 400: / Zeile 400: @@
 ===== Exercises =====
-<panel type="info" title="Exercise 4.5.1 Converting a bipolar signal to a unipolar signal"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+#@TaskTitle_HTML@#4.5.1 Converting a bipolar signal to a unipolar signal <fs medium>(not from written test)</fs>#@TaskText_HTML@#
+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, input applies: $R_{\rm uC} \rightarrow \infty$
+For conditioning, the input signal is to be fed via the series resistor $R_3$ to the center potential of a voltage divider $R_1 - R_2$ with $R_1$ to a supply voltage $U_{\rm s}$.
+The following simulation shows roughly the situation (the resistor values are not correct).
 <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCDMCmC0DsIBskB08CcGkA4AsYGADEQKxFIYjnUh6TVxhgBQATiLHjiAEy95kg-oKLgSJdkL4DpkIqOoSiUpMNlrwSMWN4SWAczkK5ekDpYAlaWG03ePHXTGlzboqlIsAbp0K8oE1h-PjwnCDC+Jx1PQz8MAPlBYISoMwsAd3iAgTEUnNkVDnyZZOxStzB9SFxpEQr6iwAjTl5SJHA8ZOZEgnMWAA9W0kSkRFgMCEgMRGFBJoBLAAcAewAbAEM2AB0AZwBlFYBXNgBjaD3t7YA7AApoVANUPY293ehr3ZW2AEpBkFShAYVSohA6cxAln+zF0REQYF4DF4VTooXA0IRfHgwLaWPBaN4-wwwlIoOYfCQAQhUKGwR4enGrmRYghkH+sBq4AwDBSXPxshp4Bwoymtigk1RsiO12W6y2bwW1wA1lc7g8nnsjgBhH5-SDwPLlepcemyCAWWlgHA8K3deAQK2zNGC4KULk8sCkYHlCH7FgrAHiQrODD0sRIVCKAKuFRAA noborder}} </WRAP>
-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 $U_{sens} \in [-15...15~\rm V]$, 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, input applies: $R_{\rm uC} \rightarrow \infty$
+Questions:
+. 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://de.wikipedia.org/wiki/E-Reihe|E24 series]] can be used to do this?
+#@HiddenBegin_HTML~1,Solution~@#
+Using superposition, we create two separate circuits where one source is considered.
+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 $R_4 =R_3 +R_{\rm s}$.
+<callout>
+=== Circuit 1 : only consider $U_{\rm S}$, ignore $U_{\rm I}$ ===
+{{drawio>electrical_engineering_1:exc541circ1.svg}}
+\begin{align*}
+U_{\rm O}^{(1)}  &= U_{\rm S} \cdot {{R_2||R_4}\over{R_1 + R_2||R_4}}
+                  = 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}} }} \\
+                 &= U_{\rm S} \cdot {{ R_2 R_4 }\over{R_1  (R_2 + R_4)+ R_2 R_4 }} \\
+                 &= U_{\rm S} \cdot {{ R_2 R_4 }\over{R_1  R_2 + R_1 R_4 + R_2 R_4 }} \\
+\end{align*}
+</callout>
+<callout>
+=== Circuit 2 : only consider $U_{\rm I}$, ignore $U_{\rm S}$ ===
+{{drawio>electrical_engineering_1:exc541circ2.svg}}
+\begin{align*}
+U_{\rm O}^{(2)}  &= U_{\rm I} \cdot {{R_1||R_2}\over{R_4 + R_1||R_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}} }} \\
+                 &= U_{\rm I} \cdot {{ R_1 R_2 }\over{R_4 (R_1 + R_2)+ R_1 R_2 }} \\
+                 &= U_{\rm I} \cdot {{ R_1 R_2 }\over{R_4 R_1 + R_4 R_2+ R_1 R_2 }} \\
+\end{align*}
+</callout>
+<callout>
+=== Superposition: Let's sum it up! ===
+\\
+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, one can see that the denominator for both $U_{\rm O}^{(1)}$ and $U_{\rm O}^{(2)}$ is the same. \\
+We can also simplify further when looking at often-used sub-terms (here: $R_2$)
+\begin{align*}
+U_{\rm O}                                            &=  {{ 1 }\over{R_4 R_1 + R_4 R_2+ R_1 R_2 }} \cdot (U_{\rm S} \cdot R_2 R_4  + U_{\rm I} \cdot R_1 R_2 ) \\
+U_{\rm O} \cdot (R_4 R_1 + R_4 R_2+ R_1 R_2 )        &=   U_{\rm S} \cdot R_2 R_4  + U_{\rm I} \cdot R_1 R_2  \\ \\
+U_{\rm O} \cdot ({{R_1 R_4}\over{R_2}} + R_4 + R_1 ) &=   U_{\rm S} \cdot     R_4  + U_{\rm I} \cdot R_1      \tag 1 \\
+\end{align*}
+The formula $(1)$ is the general formula to calculate the output voltage $U_{\rm O}$ for a changing input voltage $U_{\rm I}$, where the supply voltage $U_{\rm S}" is constant. \\
+</callout>
+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 input voltage $U_{\rm I}= +15 ~\rm V$, the output voltage shall be $U_{\rm O} = 3.3 ~\rm V$
+This leads to two situations:
+<callout>
+=== 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*}
+                          &=   U_{\rm S} \cdot     R_4  + U_{\rm I,min} \cdot R_1     \\
+- U_{\rm I,min} \cdot R_1  &=   U_{\rm S} \cdot     R_4       \\
+ {{R_1}\over{R_4}}         &=-{{U_{\rm S}}\over {U_{\rm I,min}}} = k_{14} \tag 2 \\
+\end{align*}
+So, with formula $(2)$, we already have a relation between $R_1$ and $R_4$. Yeah 😀 \\
+The next step is situation 2
+</callout>
+<callout>
+=== 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 S} \cdot     R_4  + U_{\rm I,max} \cdot k_{14} R_4 \\
+U_{\rm O,max} \cdot (k_{14}{{ R_4  }\over{R_2}} + 1   + k_{14}     ) &=   U_{\rm S}                + U_{\rm I,max} \cdot k_{14}  \\
+                     k_{14}{{ R_4  }\over{R_2}}  + 1   + k_{14}      &= {{U_{\rm S}                + U_{\rm I,max} \cdot k_{14} }\over{        U_{\rm O,max} }} \\
+                     k_{14}{{ R_4  }\over{R_2}}                      &= {{U_{\rm S}                + U_{\rm I,max} \cdot k_{14} }\over{        U_{\rm O,max} }}        - (1   + k_{14})\\
+                           {{ R_4  }\over{R_2}}                      &= {{U_{\rm S}                + U_{\rm I,max} \cdot k_{14} }\over{ k_{14} U_{\rm O,max} }} - {{1   + k_{14} }\over{k_{14}}} \tag 3 \\
+\end{align*}
+So, another relation for $R_4$ and $R_2$.  😀 \\
+</callout>
+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}}  =-{{5 ~\rm V}\over {-15 ~\rm V }} = {{1}\over{3}} \\
+\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{ 3.3 ~\rm V \cdot {{1}\over{3}} }} - {{1   + {{1}\over{3}} }\over{ {{1}\over{3}} }} \\
+                     &= {{10}\over{1.1}} - 4 \\
+k_{42}               &\approx 5.09
+\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, a boundary for $R_{\rm S}$. The maximum voltage and the maximum current are given for the sensor. By this, we can calculate $R_{\rm S}$:
+\begin{align*}
+R_{\rm S}   &= {{ U_{\rm OC} }\over{ I_{\rm SC} }} = {{ U_{\rm S,max} }\over{ I_{\rm S,max} }} = {{ 15 ~\rm V }\over{ 1 ~\rm mA }} \\
+            &= 15 ~\rm k\Omega
+\end{align*}
+Therefore, $R_4 = R_{\rm S} + R_3$ must be larger than this. \\
+#@HiddenEnd_HTML~1,Solution ~@#
+#@HiddenBegin_HTML~Result1,Result~@#
+The sensor resistance is
+\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_3 + 15 {~\rm k\Omega}) = 20   {~\rm k\Omega} \\
+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_3 + 15 {~\rm k\Omega}) = 20 {~\rm k\Omega} \\
+R_2^0 &= {{1}\over{5.09}}(R_3 + 15 {~\rm k\Omega}) = 12 {~\rm k\Omega}
+\end{align*}
+#@HiddenEnd_HTML~Result1,Result~@#
+. Find the relationship between $R_1$, $R_2$, and $R_3$ by investigating Kirchhoff's nodal rule for the node where $R_1$, $R_2$, and $R_3$ are interconnected.
+#@HiddenBegin_HTML~Solution2,Solution~@#
+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  $U_{\rm I}$ to the nodal voltage $U_{\rm O}$:  $I_4 = {{U_{\rm I} - U_{\rm O}}\over{R_4}}$
+This led to the formula based on the Kirchhoff's nodal rule:
+\begin{align*}
+\Sigma I = 0 &= I_1 + I_2 + I_3 \\
+&= {{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 S}}\over{R_1}}  + {{U_{\rm I}}\over{R_4}}  \\
+      U_{\rm O}\cdot \left( {{1}\over{R_1}} + {{1}\over{R_2}} + {{1}\over{R_4}} \right) &=  {{U_{\rm S}}\over{R_1}}  + {{U_{\rm I}}\over{R_4}}  \\
+\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) &=  R_1 R_2 R_4 \cdot {{U_{\rm S}}\over{R_1}}  + R_1 R_2 R_4 \cdot {{U_{\rm I}}\over{R_4}}  \\
+      U_{\rm O}\cdot \left( R_2 R_4 + R_1 R_4 + R_1 R_2 \right) &=  R_2 R_4 \cdot U_{\rm S}  + R_1 R_2 \cdot U_{\rm I} \\
+      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.
+#@HiddenEnd_HTML~Solution2,Solution ~@#
+. What is the input resistance $R_{\rm in}(R_1, R_2, R_3)$ of the circuit (viewed from the sensor)?
+#@HiddenBegin_HTML~Solution3,Solution~@#
+\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*}
+#@HiddenEnd_HTML~Solution3,Solution~@#
+. What is the minimum allowed input resistance ($R_{\rm in, min}(R_1, R_2, R_3)$) for the sensor to still deliver current?
+#@HiddenBegin_HTML~Solution4,Solution~@#
+\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*}
+#@HiddenEnd_HTML~Solution4,Solution~@#
+#@TaskEnd_HTML@#
-For conditioning, the input signal is to be fed via the series resistor $R_3$ to the center potential of a voltage divider $R_1 - R_2$ with $R_1$ against $U_{\rm uC, max}$ (similar circuit see in simulation on the right).
-  - 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://de.wikipedia.org/wiki/E-Reihe|E24 series]] can be used to do this?
-</WRAP></WRAP></panel>
 {{page>aufgabe_4.5.2_mit_rechnung&nofooter}}
 {{page>aufgabe_4.5.3&nofooter}}