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electrical_engineering_and_electronics_1:block23 [2025/12/14 22:50] – [Worked examples] mexleadminelectrical_engineering_and_electronics_1:block23 [2025/12/15 00:12] (aktuell) – mexleadmin
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-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  $R_{\rm O}\rightarrow \infty$). Therefore, the currents through the resistors $R_1$ and $R_2$ are the same: $i_1 = i_2$ (given, that  $R_{\rm O}\rightarrow \infty$).
  
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 === Bang-Bang Control === === 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=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-ACDEzBi-DqxeYGwophb6ktxYgVD3YnwpmLIAASvfcLJX0RmIsb5GPes4MkwEJrMG4CWggfbDEwEB-G+dBfuKZCUucP6RDIpqoYI77FnQhz6vOyEgAAwjwH55MGISVLkJ7dlOUAYCQPAAEZBEe9CMLQdBHMeqy6IkPJBEIomyIoipCQGnZiGSBRjvICjSeKkBGGygYaQICogAAMgA9l4AAmVygQpQSEPKkhdGQmZCNmTYFkMxa1nmMDVhWjb1o2eYOeIlYkE2Uk8OI8ogCoEB9hkwhcJ+vBhV0EVReAMXMHFH6YKFyboJFLHTh0UVlvAC6legcWUUAA 1000,500 noborder}}
 </WRAP> </WRAP>
 \\ \\ \\ \\
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 When such a signal is fed directly into a comparator, small noise amplitudes around the threshold can cause rapid switching of the output (chatter). 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. As long as the input signal remains between these two values, the output state does not change.
  
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 ==== Exercises ==== ==== Exercises ====
- 
-===== Exercises ===== 
  
 <panel type="info" title="Task 23.1 Comparator Output States"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Task 23.1 Comparator Output States"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
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 </WRAP></WRAP></panel> </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>
 +
 +
  
 ===== Embedded resources ===== ===== Embedded resources =====