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electrical_engineering_and_electronics_1:block23 [2025/12/14 22:50] – [Worked examples] mexleadminelectrical_engineering_and_electronics_1:block23 [2026/01/10 10:08] (aktuell) – mexleadmin
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 ====== Block 23 — Comparator Circuits ====== ====== Block 23 — Comparator Circuits ======
  
-===== Learning objectives =====+===== 23.0 Intro ===== 
 + 
 +==== 23.0.1 Learning objectives ====
 <callout> <callout>
 After this 90-minute block, you will be able to After this 90-minute block, you will be able to
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 </callout> </callout>
  
-===== Preparation at Home =====+==== 23.0.2 Preparation at Home ====
  
 Well, again  Well, again 
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   * ...   * ...
  
-===== 90-minute plan =====+==== 23.0.3 90-minute plan ====
   - Warm-up (5–10 min):   - Warm-up (5–10 min):
     - Recall: op-amp with negative feedback vs. no feedback.     - Recall: op-amp with negative feedback vs. no feedback.
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     - Typical mistakes and outlook to further applications     - Typical mistakes and outlook to further applications
  
-===== Conceptual overview =====+==== 23.0.4 Conceptual overview ====
 <callout icon="fa fa-lightbulb-o" color="blue"> <callout icon="fa fa-lightbulb-o" color="blue">
   - A **comparator** is the “switching cousin” of the op-amp: it does not try to keep \(u_{\rm d}\approx 0\) with negative feedback. \\ Instead, it reports the **sign** of \(u_{\rm d}=u_{\rm p}-u_{\rm m}\) by saturating its output to one of two extreme levels.   - A **comparator** is the “switching cousin” of the op-amp: it does not try to keep \(u_{\rm d}\approx 0\) with negative feedback. \\ Instead, it reports the **sign** of \(u_{\rm d}=u_{\rm p}-u_{\rm m}\) by saturating its output to one of two extreme levels.
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 </callout> </callout>
  
-===== Core content =====+===== 23.1 Core content =====
  
-==== Comparator ====+==== 23.1.1 Comparator ====
  
 Up to now we focussed on operational amplifier, which is only usable in a closed-loop setup.  Up to now we focussed on operational amplifier, which is only usable in a closed-loop setup. 
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 </WRAP> </WRAP>
  
-==== Non-inverting Schmitt Trigger ====+==== 23.1.2 Non-inverting Schmitt Trigger ====
  
 Based on the comparator, we can try to setup a "op-amp like" circuitry.  Based on the comparator, we can try to setup a "op-amp like" circuitry. 
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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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 {{drawio>electrical_engineering_and_electronics_1:HysteresisV01.svg}} {{drawio>electrical_engineering_and_electronics_1:HysteresisV01.svg}}
  
-==== Applications ====+===== 23.2 Applications =====
  
-=== Bang-Bang Control ===+==== 23.2.1 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>
 \\ \\ \\ \\
-=== De-Noise ===+==== 23.2.2 De-Noise ==== 
  
 Real analog signals are often corrupted by noise.\\ Real analog signals are often corrupted by noise.\\
 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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 </WRAP> </WRAP>
  
-===== Common pitfalls =====+===== 23.3 Common pitfalls =====
   * **Treating a comparator like a linear op-amp**: assuming the output follows a linear gain law \(u_{\rm O}=A_{\rm D}\,u_{\rm d}\). In reality, the output almost always saturates at \(U_{\rm sat,min}\) or \(U_{\rm sat,max}\).   * **Treating a comparator like a linear op-amp**: assuming the output follows a linear gain law \(u_{\rm O}=A_{\rm D}\,u_{\rm d}\). In reality, the output almost always saturates at \(U_{\rm sat,min}\) or \(U_{\rm sat,max}\).
   * **Using negative-feedback intuition**: expecting the circuit to automatically enforce \(u_{\rm d}=0\). Without negative feedback, \(u_{\rm d}=0\) is only the *switching boundary*, not an operating point.   * **Using negative-feedback intuition**: expecting the circuit to automatically enforce \(u_{\rm d}=0\). Without negative feedback, \(u_{\rm d}=0\) is only the *switching boundary*, not an operating point.
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-===== Exercises ===== 
  
-==== Conceptual checks ====+===== 23.4 Learning Questions =====
   - 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.
   - 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?
   - 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 ==== +===== 23.5 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 =====