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electrical_engineering_and_electronics_1:block05 [2025/09/28 23:30] – angelegt mexleadminelectrical_engineering_and_electronics_1:block05 [2026/01/10 13:23] (aktuell) – mexleadmin
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-====== Block 05 — Resistive networks ======+====== Block 05 — Resistive Networks ======
  
-===== Learning objectives =====+===== 5.0 Intro ===== 
 + 
 +==== 5.0.1 Learning Objectives ==== 
 +<callout>
 After this 90-minute block, you can After this 90-minute block, you can
   * reduce series/parallel resistor networks to an equivalent resistance $R_{\rm eq}$,   * reduce series/parallel resistor networks to an equivalent resistance $R_{\rm eq}$,
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   * recognize and analyze **bridge** circuits (Wheatstone; balance condition),   * recognize and analyze **bridge** circuits (Wheatstone; balance condition),
   * check results by units and by “sanity bounds” (e.g., $R_{\rm eq}$ in parallel is below the smallest branch).   * check results by units and by “sanity bounds” (e.g., $R_{\rm eq}$ in parallel is below the smallest branch).
 +</callout>
 +
 +==== 5.0.2 Preparation at Home ====
 +
 +And again: 
 +  * Please read through the following chapter.
 +  * Also here, there are some clips for more clarification under 'Embedded resources' (check the text above/below, sometimes only part of the clip is interesting). 
  
 +For checking your understanding please do the following exercises:
 +  * 2.5.3
 +  * 2.7.8
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-===== 90-minute plan  =====+==== 5.0.3 90-minute Plan  ====
   * 0–10 min — Recap KCL/KVL; sign conventions.   * 0–10 min — Recap KCL/KVL; sign conventions.
   * 10–30 min — Series & parallel; quick numeric checks.   * 10–30 min — Series & parallel; quick numeric checks.
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-===== Core Content ===== +===== 5.1 Core Content ===== 
-==== Recap (from Block 04) ==== +==== 5.1.1 Unloaded Voltage Divider ====
-Kirchhoff’s laws on **nodes** and **loops** plus reshaping circuits are the tools behind everything in this block. The node sign rule we use: arrows **toward** the node positive, **away** negative, $\sum I=0$.  +
- +
-<panel type="info" title="Dimensional check"> +
-For the **current divider** relation $\displaystyle \frac{I_1}{I_2}=\frac{G_1}{G_2}$ both sides are ratios → dimensionless. Since $[G]=1~{\rm S}$, the unit cancels. (We will re-derive it below.)  +
-</panel> +
- +
-~~PAGEBREAK~~ ~~CLEARFIX~~ +
- +
-==== Unloaded voltage divider ====+
  
 The series circuit of two resistors $R_1$ and $R_2$ shall be considered now. \\ The series circuit of two resistors $R_1$ and $R_2$ shall be considered now. \\
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-<panel type="info" title="Exercise 2.5.1 unloaded voltage divider"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>+<panel type="info" title="Exercise"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
 <WRAP> <WRAP>
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== The loaded Voltage Divider ====+==== 5.1.2 The loaded Voltage Divider ====
  
 If - in contrast to the abovementioned, unloaded voltage divider - a load $R_{\rm L}$ is connected to the output terminals (<imgref BildNr15>), this load influences the output voltage. If - in contrast to the abovementioned, unloaded voltage divider - a load $R_{\rm L}$ is connected to the output terminals (<imgref BildNr15>), this load influences the output voltage.
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 $ U_1 = \LARGE{{U} \over {1 + {{R_2}\over{R_L}} + {{R_2}\over{R_1}} }}$ $ U_1 = \LARGE{{U} \over {1 + {{R_2}\over{R_L}} + {{R_2}\over{R_1}} }}$
  
-or on a potentiometer with $k$ and the sum of resistors $R_{\rm s} = R_1 + R_2$:+An alternative representation of the formula sticks more to the application. \\  
 +It uses: 
 +  - the position on a potentiometer given as $k={{R_1}\over{R_1 + R_2}}$ and  
 +  - the sum of resistors $R_{\rm s} = R_1 + R_2$. 
 +Both are more often used in real setups.
  
-$ U_1 = \LARGE{{k \cdot U} \over { 1 + k \cdot (1-k) \cdot{{R_{\rm s}}\over{R_{\rm L}}} }}$+Mathematically, both parameter lead to $R_1 = k \cdot R_{\rm s}$ and $R_2 = (1 - k) \cdot R_{\rm s}$. \\ 
 +When these tyo relations are included in rhe the formula above, we get:  
 + 
 +$ U_1 = U \cdot k \cdot \LARGE{{1} \over { 1 + k \cdot (1-k) \cdot{{R_{\rm s}}\over{R_{\rm L}}} }}$
  
 <imgref BildNr65> shows the ratio of the output voltage $U_1$ to the input voltage $U$ (y-axis), in relation to the ratio $k={{R_1}\over{R_1 + R_2}}$.  <imgref BildNr65> shows the ratio of the output voltage $U_1$ to the input voltage $U$ (y-axis), in relation to the ratio $k={{R_1}\over{R_1 + R_2}}$. 
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== Bridge networks (Wheatstone) ====+==== 5.1.3 Bridge Networks (Wheatstone) ====
 A four-resistor bridge can be seen as **two voltage dividers in parallel**. The detector (bridge branch) sees the **difference** of the two divider node voltages. The **balance condition** (zero detector current) is A four-resistor bridge can be seen as **two voltage dividers in parallel**. The detector (bridge branch) sees the **difference** of the two divider node voltages. The **balance condition** (zero detector current) is
 \begin{align*} \begin{align*}
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== Strategy for network reduction ====+==== 5.1.4 Strategy for network reduction ====
   * Reshape (without changing node connections), then collapse **clear** series or parallel groups.   * Reshape (without changing node connections), then collapse **clear** series or parallel groups.
-  * If blocked by a three-terminal cluster, apply **Δ–Y** (or **Y–Δ**), then collapse again.+  * (If blocked by a three-terminal cluster, apply **Δ–Y** (or **Y–Δ**), then collapse again. {{wp>Y-Δ_transform|Y–Δ and Δ–Y}} ist not covered and necessary in this course)
   * Repeat until a simple ladder remains; finish with KCL/KVL if needed.    * Repeat until a simple ladder remains; finish with KCL/KVL if needed. 
  
-In this subchapter, a methodology is discussed, which should help to reshape circuits. In subchapter [[#2.6 Star-Delta-Circuit]] towards the end a network was already transformed in a way, that it does not contain triangular meshes anymore. Now, this procedure shall be systematized. +In this subchapter, a methodology is discussed, which should help to reshape circuits.  
-Starting points are tasks, where for a resistor network the total resistance, total current, or total voltage has to be calculated.+The following concept works at tasks, where the total resistance, total current, or total voltage has to be calculated for a resistor network.
  
 An example of such a circuit is given in <imgref imageNo89>. Here $I_0$ is wanted.  An example of such a circuit is given in <imgref imageNo89>. Here $I_0$ is wanted. 
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 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
-===== Exercises =====+===== 5.2 Exercises =====
  
 <panel type="info" title="Exercise 2.4.3 Three Resistors"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 2.4.3 Three Resistors"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
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 + 
 +{{page>electrical_engineering_and_electronics:task_cgeyprm6oboukcci_with_calculation&nofooter}} 
 +{{page>electrical_engineering_and_electronics:task_erlctd760zmvox0t_with_calculation&nofooter}} 
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