Unterschiede

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--- electrical_engineering_and_electronics_1:block23 [2025/12/14 22:48] – [Conceptual checks] mexleadmin
+++ electrical_engineering_and_electronics_1:block23 [2025/12/15 00:12] (aktuell) – mexleadmin
@@ Zeile 104: / Zeile 104: @@
-The **golden rules** ($R_{\rm I}=0$, $R_{\rm O}\rightarrow \infty$, $A_{\rm D}\rightarrow \infty$) also apply here. \\ \\
+The **golden rules** ($R_{\rm I}\rightarrow \infty$, $R_{\rm O}=0$, $A_{\rm D}\rightarrow \infty$) also apply here. \\ \\
 Therefore, the currents through the resistors $R_1$ and $R_2$ are the same: $i_1 = i_2$ (given, that  $R_{\rm O}\rightarrow \infty$).
@@ Zeile 151: / Zeile 151: @@
 === Bang-Bang Control ===
-See {{wp>Bang–bang_control}}
+In the shown simulation, **{{wp>Bang–bang_control}}** is realized with a comparator including hysteresis. and a simple first-order plant (RC network).
-<WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l5BOJyWoVaAmTkDsewAOMAVgDZIz8AWMkEk8Se5kgUwFowwAoHXEJlyZw1aiADMOUePEBXAPoB5HgA9BCAZjKNqlQWUIhxREErkAXAA6WAOgGcAygEsA5gDsAhgBseAd0FIIzAxQKMEOkgeACcwkAjwTHDIwPgYxOCkkDITLOYcNICwLITizKMozwyZOJDxdEYORjB4eEloTVIECRI8HIkpI2poXrhtajAEEIZcE1aovQl4uiSjbHEJQyh6BWoFSAVVJJawMgUW-dg4MAUuBUwFQnv9hRIFCUvWzDeDuQcAET4eGqpWKIEIzHEAGleNRMEskhtMOhkegpOgFvDJJAkWjJpJkVAeHCllJcZIJGjCZjSaF0ZISFSMcSsZtyRJcEyiSTJIR2UgCcyeXpyRMuTTBHzsSsBRJ9BK5Rt9BJ8XLIizSZglXQBmj5RrJHTlYzpdzWTlTRy9eqeRIpWrJLL9cKcaaxaaouoWgidOBCCitiYjP9nAAzUPQBwuDw+NTGITYgTUO2SFrGcB0ACqcbEzBVkLIIhVS1yIHsAAsADRyHMTCkQagmgayRJlqu+dRwoyDYyU+vp04gbOLSVrQhDQebCpQXYvI6EOWnT7XZeQG53G5PB6-BwqWI4YKhIQiOrbFpwHgAY0Ewhqx7i4hg8y1lLQaEYEg6uC6PT6yZ7jBRK4N4iAeIGCFq2xRPw1SnMwpQUOmigqF6ZAQFQ4C3vgA50AASnGYCzCA2FEMwuAlq2mAEZASwtHM6AtEmra8OoCDMCcTAQHROEgPh6jws0mAQFqdDFC2g58aO4AIGsAo4A2GYgAAwjwuHZLkwSiXk2x6OmzD6SMPApuIABiEDsakcAgFwvG8CmSxmWelnMDZuFUfZzCOQZ8zmVc8BvgF6A2SpQA 1000,500 noborder}}
+The circuit can be interpreted as follows:
+  * The comparator with positive feedback (via $R_1$ and $R_2$) forms a **Schmitt trigger** with an upper threshold $U_{\rm sh,u}$ and a lower threshold $U_{\rm sh,l}$.
+  * The output of the comparator switches only between its two saturation values ($U_{\rm sat,max}$ and $U_{\rm sat,min}$), which is characteristic of bang-bang behavior.
+  * The resistor–capacitor combination ($R$, $C$) represents a **controlled system** (plant) with inertia: the capacitor voltage changes only gradually.
+The operating principle is:
+  * If the output voltage $u_{\rm O}$ is high, the capacitor is charged through $R$, causing the feedback signal to increase.
+  * As soon as the capacitor voltage reaches the **upper threshold** $U_{m sh,u}$, the comparator switches abruptly to its lower saturation level.
+  * The capacitor now discharges (or charges in the opposite direction), until the voltage reaches the **lower threshold** $U_{\rm sh,l}$.
+  * At this point, the comparator switches back to the high saturation level.
+As a result, the system continuously oscillates between the two thresholds. The comparator output is a two-level (on/off) signal, while the capacitor voltage varies smoothly between $U_{\rm sh,l}$ and $U_{\rm sh,u}$.
+This example illustrates key properties of bang-bang control:
+  * the actuator (comparator output) has only two states,
+  * the controlled variable is kept within a **band** defined by the hysteresis,
+  * the switching frequency depends on the system dynamics (here the $RC$ time constant) and the hysteresis width.
+Such control principles appear in thermostats, relaxation oscillators, power electronics, and simple closed-loop controllers where simplicity and robustness are more important than exact regulation.
+<WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l5BOJyWoVaAmTkDsewAOMAVgDZIz8AWMkEk8Se5kgUwFowwAoHXENTjgwmQQGYx3MdRABXAPoA7AGIAhHgA96AzGUakxewoPAmAkkoAOcgC4AdAM4BlAJYBzJQEMANjwDuIDgmYNSywSAIdJA8AE5BkCZR4JhJ0QnwcSkhqSBksqImzDiZgYWRdOXlMV7Z4GEJIQ3ojBwG8PAg4tC4NOIk2MSQYIjo1NAk8OL5veSY1LjiTB18eHXJ0iCEzLIA0rzUmEsNki3op1A8h0v5XZjouOf3l9dbshdId+gxryPvz9wnt8rkcUv90GBxECXqDQuDwCRoT9YbcLmBHl8Ycd3pR6Odccibji6BjxASQUtCMTIvjohSmNTAV1yb95szKlD2Vj6ozEVzCeBUbj0bTLtpxNwgvpBDgpSYCiYAPJ2GwOFwebx+bREARkgRgBAmCU7QUgACqWgyXRGQWw1uOKRAjgAFgAaOSWnAQcQSoIGrriWQFMQu11asxG3IG86+gp0C1CY4mVIhXB0aZFKAKagKSAKTSpMlgMi52BwMCl+AVrgKCuEBSYXMKRSqDTxH2VUSCYSbZgjOA8ADG3b7XaEfbTphgHUgULIaAXXR6fQGqRGI0+jBi7hH4En4-ACDEzBi-DqxeYGwophb6ktxYgVD3YnwpmLIAASvfcLJX0RmIsb5GPes4MkwEJrMG4CWggfbDEwEB-G+dBfuKZCUucP6RDIpqoYI77FnQhz6vOyEgAAwjwH55MGISVLkJ7dlOUAYCQPAAEZBEe9CMLQdBHMeqy6IkPJBEIomyIoipCQGnZiGSBRjvICjSeKkBGGygYaQICogAAMgA9l4AAmVygQpQSEPKkhdGQmZCNmTYFkMxa1nmMDVhWjb1o2eYOeIlYkE2Uk8OI8ogCoEB9hkwhcJ+vBhV0EVReAMXMHFH6YKFyboJFLHTh0UVlvAC6legcWUUAA 1000,500 noborder}}
 </WRAP>
 \\ \\
@@ Zeile 161: / Zeile 181: @@
 When such a signal is fed directly into a comparator, small noise amplitudes around the threshold can cause rapid switching of the output (chatter).
-The Schmitt triggersolves this problem by its two distinct thresholds \(U_{\rm sh,u}\) and \(U_{\rm sh,l}\). \\
+The Schmitt trigger solves this problem by its two distinct thresholds \(U_{\rm sh,u}\) and \(U_{\rm sh,l}\). \\
 As long as the input signal remains between these two values, the output state does not change.
@@ Zeile 212: / Zeile 232: @@
   - Why is \(u_{\rm d}=0\) a special point for a comparator, even though it is not a stable operating point?
-==== Worked examples ====
+==== Exercises ====
+<panel type="info" title="Task 23.1 Comparator Output States"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A push-pull comparator is supplied with $0~{\rm V}$ and $5~{\rm V}$.
+The input voltages are given as:
+\[
+  u_{\rm p}=3.0~{\rm V}, \qquad u_{\rm m}=2.0~{\rm V}
+\]
+  - Determine the differential input voltage $u_{\rm d}$.
+  - State the resulting output voltage $u_{\rm O}$.
+<button size="xs" type="link" collapse="Loesung_23_1_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_23_1_Tipps" collapsed="true">
+  * Recall that $u_{\rm d}=u_{\rm p}-u_{\rm m}$.
+  * For a push-pull comparator, the output directly saturates depending on the sign of $u_{\rm d}$.
+</collapse>
+<button size="xs" type="link" collapse="Loesung_23_1_Result">{{icon>eye}} Result</button><collapse id="Loesung_23_1_Result" collapsed="true">
+  * $u_{\rm d}=+1.0~{\rm V}$
+  * $u_{\rm O}=U_{\rm sat,max}=5~{\rm V}$
+</collapse>
+</WRAP></WRAP></panel>
+<panel type="info" title="Task 23.2 Schmitt Trigger Thresholds"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A non-inverting Schmitt trigger is built with the resistors
+\[
+  R_1=10~{\rm k\Omega}, \qquad R_2=100~{\rm k\Omega}
+\]
+The comparator saturates symmetrically at
+\[
+  U_{\rm sat,max}=+12~{\rm V}, \qquad U_{\rm sat,min}=-12~{\rm V}
+\]
+  - Calculate the upper threshold $U_{\rm sh,u}$.
+  - Calculate the lower threshold $U_{\rm sh,l}$.
+  - Sketch qualitatively the hysteresis characteristic $u_{\rm O}(u_{\rm I})$.
+<button size="xs" type="link" collapse="Loesung_23_3_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_23_3_Tipps" collapsed="true">
+  * Use the relations
+    \[
+      U_{\rm sh,u}=+\frac{R_1}{R_2}u_{\rm O}, \qquad
+      U_{\rm sh,l}=-\frac{R_1}{R_2}u_{\rm O}
+    \]
+</collapse>
+<button size="xs" type="link" collapse="Loesung_23_3_Result">{{icon>eye}} Result</button><collapse id="Loesung_23_3_Result" collapsed="true">
+  * $U_{\rm sh,u}=+1.2~{\rm V}$
+  * $U_{\rm sh,l}=-1.2~{\rm V}$
+</collapse>
+</WRAP></WRAP></panel>
+<panel type="info" title="Task 23.3 Application: De-Noising"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A noisy sensor signal fluctuates around $2.5~{\rm V}$ with a noise amplitude of $\pm 50~{\rm mV}$.
+A comparator without hysteresis is used to detect whether the signal is above or below $2.5~{\rm V}$.
+  - Explain why the output may switch rapidly.
+  - Explain qualitatively how a Schmitt trigger improves the situation.
+<button size="xs" type="link" collapse="Loesung_23_4_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_23_4_Tipps" collapsed="true">
+  * Consider the behavior of the comparator near $u_{\rm d}=0$.
+</collapse>
+<button size="xs" type="link" collapse="Loesung_23_4_Result">{{icon>eye}} Result</button><collapse id="Loesung_23_4_Result" collapsed="true">
+  * Without hysteresis: output chatter due to noise crossings.
+  * With hysteresis: two thresholds prevent switching for small fluctuations.
+</collapse>
+</WRAP></WRAP></panel>
+<panel type="info" title="Task 23.4 Thresholds from Resistor Ratio"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
+A Schmitt trigger uses resistors $R_1$ and $R_2$ for positive feedback. The output saturates at $\pm 8~{\rm V}$.
+  - Write expressions for $U_{\rm sh,u}$ and $U_{\rm sh,l}$.
+  - Explain how the ratio $R_1/R_2$ influences the control band of the bang-bang controller.
+<button size="xs" type="link" collapse="Loesung_23_7_Tipps">{{icon>eye}} Tips for the solution</button><collapse id="Loesung_23_7_Tipps" collapsed="true">
+  * Recall that the thresholds are proportional to the output saturation voltage.
+</collapse>
+<button size="xs" type="link" collapse="Loesung_23_7_Result">{{icon>eye}} Result</button><collapse id="Loesung_23_7_Result" collapsed="true">
+  * $U_{\rm sh,u}=+\dfrac{R_1}{R_2}\,8~{\rm V}$, $U_{\rm sh,l}=-\dfrac{R_1}{R_2}\,8~{\rm V}$.
+  * A larger ratio $R_1/R_2$ widens the control band.
+</collapse>
+</WRAP></WRAP></panel>
-...
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