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electrical_engineering_and_electronics_1:block23 [2025/12/14 22:46] – mexleadminelectrical_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  $R_{\rm O}\rightarrow \infty$). 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 === === 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>
 \\ \\ \\ \\
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). 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.
  
Zeile 208: Zeile 228:
  
 ==== Conceptual checks ==== ==== Conceptual checks ====
-  1. Explain in one or two sentences why a comparator is normally operated without negative feedback. +  - Explain in one or two sentences why a comparator is normally operated without negative feedback. 
-  2. What information about the input signal does the comparator output represent when \(u_{\rm O}\) is in saturation? +  - What information about the input signal does the comparator output represent when \(u_{\rm O}\) is in saturation? 
-  3. Why is \(u_{\rm d}=0\) a special point for a comparator, even though it is not a stable operating point?+  - Why is \(u_{\rm d}=0\) a special point for a comparator, even though it is not a stable operating point? 
 + 
 +==== 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>
  
-==== Worked examples ==== 
  
-... 
  
 ===== Embedded resources ===== ===== Embedded resources =====