Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| electrical_engineering_1:simple_circuits [2021/09/24 08:13] – tfischer | electrical_engineering_1:simple_circuits [2024/10/24 08:13] (aktuell) – mexleadmin | ||
|---|---|---|---|
| Zeile 1: | Zeile 1: | ||
| - | ====== 2. Simple DC circuits ====== | + | ====== 2 Simple DC circuits ====== |
| - | < | + | So far, only simple circuits consisting of a source and a load connected by wires have been considered. \\ |
| + | In the following, more complicated circuit arrangements will be analyzed. These initially contain only one source, but several lines and many ohmic loads (cf. <imgref BildNr91> | ||
| + | |||
| + | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | |||
| - | So far, only simple circuits consisting of a source and a load connected by wires have been considered. \\ In the following, more complicated circuit arrangements will be analysed. These initially contain only one source, but several lines and many ohmic loads (cf. <imgref BildNr91> | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ===== 2.1 ideal components | + | ===== 2.1 Idealized Components |
| < | < | ||
| - | === goals === | + | === Learning Objectives |
| - | After this lesson | + | By the end of this section, |
| - Know the representation of ideal current and voltage sources in the U-I diagram. | - Know the representation of ideal current and voltage sources in the U-I diagram. | ||
| - Know the internal resistance of ideal current and voltage sources. | - Know the internal resistance of ideal current and voltage sources. | ||
| Zeile 26: | Zeile 27: | ||
| Every electrical circuit consists of three elements: | Every electrical circuit consists of three elements: | ||
| - | - **Consumers**: | + | - **Consumers** |
| - into electrostatic energy (capacitor) | - into electrostatic energy (capacitor) | ||
| - into magnetostatic energy (magnet) | - into magnetostatic energy (magnet) | ||
| Zeile 32: | Zeile 33: | ||
| - into mechanical energy (loudspeaker, | - into mechanical energy (loudspeaker, | ||
| - into chemical energy (charging an accumulator) | - into chemical energy (charging an accumulator) | ||
| - | - **sources (generators)**: sources convert energy from another form of energy into electrical energy. (e.g. generator, battery, photovoltaic). | + | - **Sources** also called **Generator** (in German: Quellen): sources convert energy from another form of energy into electrical energy. (e.g. generator, battery, photovoltaic). |
| - | - **wires (interconnections)**: the wires of interconnection lines link consumers to sources. | + | - **Wires** also called **Interconnections** (in German Leitungen or Verbindungen): The wires of interconnection lines link consumers to sources. |
| These elements will be considered in more detail below. | These elements will be considered in more detail below. | ||
| Zeile 39: | Zeile 40: | ||
| ==== Consumer ==== | ==== Consumer ==== | ||
| - | * The colloquial term ' | + | * The colloquial term ' |
| * A resistor is often also referred to as a consumer. In addition to pure ohmic consumers, however, there are also ohmic-inductive consumers (e.g. coils in a motor) or ohmic-capacitive consumers (e.g. various power supplies using capacitors at the output). Correspondingly the equation " | * A resistor is often also referred to as a consumer. In addition to pure ohmic consumers, however, there are also ohmic-inductive consumers (e.g. coils in a motor) or ohmic-capacitive consumers (e.g. various power supplies using capacitors at the output). Correspondingly the equation " | ||
| - | * Current-voltage characteristics (vgl. <imgref BildNr4> | + | * Current-voltage characteristics (see <imgref BildNr4> |
| - | * Current-voltage characteristics of a load always run through the origin, because without current there is no voltage and vice versa. | + | * Current-voltage characteristics of a load always run through the origin, because without current there is no voltage, and vice versa. |
| * Ohmic loads have a linear current-voltage characteristic which can be described by a single numerical value. \\ The slope in the $U$-$I$-characteristic is the conductance: | * Ohmic loads have a linear current-voltage characteristic which can be described by a single numerical value. \\ The slope in the $U$-$I$-characteristic is the conductance: | ||
| Zeile 48: | Zeile 49: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 55: | Zeile 56: | ||
| ==== Sources ==== | ==== Sources ==== | ||
| - | < | + | * Sources act as generators of electrical energy |
| + | * A distinction is made between ideal and real sources. \\ The real sources are described in the following chapter " | ||
| - | Ideal Sources | + | The **ideal voltage source** generates a defined constant output voltage $U_\rm s$ (in German often $U_\rm q$ for Quellenspannung). |
| - | {{youtube>IZDh_EUuhRs}} | + | In order to maintain this voltage, it can supply any current. |
| + | The current-voltage characteristic also represents this (see <imgref BildNr6> | ||
| + | The circuit symbol shows a circle with two terminals. In the circuit, the two terminals are short-circuited. \\ | ||
| + | Another circuit symbol shows the negative terminal of the voltage source as a "thick minus symbol", | ||
| + | |||
| + | The **ideal current source** produces a defined constant output current $I_\rm s$ (in German often $I_\rm q$ for Quellenstrom). | ||
| + | For this current to flow, any voltage can be applied to its terminals. | ||
| + | The current-voltage characteristic also represents this (see <imgref BildNr7> | ||
| + | The circuit symbol shows again a circle with two connections. This time the two connections are left open in the circle and a line is drawn perpendicular to them. | ||
| + | |||
| + | <WRAP> | ||
| - | \\ | ||
| <WRAP group>< | <WRAP group>< | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| <WRAP column 45%> | <WRAP column 45%> | ||
| Zeile 70: | Zeile 81: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | * Sources act as generators | + | Another Explanation |
| - | * A distinction is made between ideal and real sources. \\ The real sources are described in the following chapter ([[non-ideal_sources_and_two_pole_networks]]). | + | {{youtube> |
| - | The **ideal voltage source** generates a defined constant output voltage $U_s$ (in German often $U_q$ for Quellenspannung). | + | \\ |
| - | In order to maintain this voltage, it can supply any current. | + | |
| - | The current-voltage characteristic also represents this (see <imgref BildNr6> | + | |
| - | The circuit symbol shows a circle with two terminals. In the circuit, the two terminals are short-circuited. \\ | + | |
| - | Another circuit symbol shows the negative terminal of the voltage source as a "thick minus symbol", | + | |
| - | + | ||
| - | The **ideal current source** produces a defined constant output current $I_s$ (in German often $I_q$ for Quellenstrom). | + | |
| - | For this current to flow, any voltage can be applied to its terminals. | + | |
| - | The current-voltage characteristic also represents this (see <imgref BildNr7> | + | |
| - | The circuit symbol shows again a circle with two connections. This time the two connections are left open in the circle and a line is drawn perpendicular to them. | + | |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== wire connection | + | ==== Wire Connection |
| * The ideal connection line is resistance-free and transmits current and voltage instantaneously. | * The ideal connection line is resistance-free and transmits current and voltage instantaneously. | ||
| * Real existing influences (e.g. voltage drop) of connections are considered via separately drawn components (e.g. ohmic resistance). | * Real existing influences (e.g. voltage drop) of connections are considered via separately drawn components (e.g. ohmic resistance). | ||
| - | ===== 2.2 Reference-arrow Systems and first consideration | + | ===== 2.2 Reference-Arrow Systems, Sign Convention, |
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson | + | By the end of this section, |
| - | - Be able to apply and distinguish between the producer and consumer reference arrow systems. | + | - apply and distinguish between the producer and consumer reference arrow systems |
| + | - similarly use passive and active sign conventions. | ||
| </ | </ | ||
| - | In the chapter [[preparation_properties_proportions|1. Preparation and Proportions]] the conventional directional sense of currents | + | In the chapter [[preparation_properties_proportions|1. Preparation and Proportions]] the direction |
| - | In <imgref BildNr5> such a meshed net is shown. In this circuit a switch $S_1$ and a current $I_2$ are marked. Once the state of the switch is swapped, the direction of the current changes. | + | In <imgref BildNr5> such a meshed net is shown. In this circuit, a switch $S_1$ and a current $I_2$ are marked. Once the state of the switch is swapped, the direction of the current changes. |
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Generator | + | ==== Sign and Arrow-Systems |
| For the direction of the arrows different conventions are available. Here (and quite often in Germany) the [[https:// | For the direction of the arrows different conventions are available. Here (and quite often in Germany) the [[https:// | ||
| This convention is | This convention is | ||
| - | === Generator Reference Arrow System === | + | === Generator Reference Arrow System |
| <WRAP group>< | <WRAP group>< | ||
| Zeile 128: | Zeile 131: | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| With **sources** (or generators), | With **sources** (or generators), | ||
| - | For generators, the arrow__foot__ of the current is attached to the arrow__head__ of the voltage. Voltage and current arrows are antiparallel ($\uparrow \downarrow$). | + | For generators, the arrow__foot__ of the current is attached to the arrow__head__ of the voltage. Voltage and current arrows are antiparallel ($\uparrow \downarrow$). \\ |
| + | Similarly, the active sign convention for one component states: The current enters the component on the more negative terminal. Or vice versa: The current exits the component on the positive terminal. | ||
| + | |||
| + | Both expressions " | ||
| For generators holds: | For generators holds: | ||
| Zeile 143: | Zeile 149: | ||
| <callout color=" | <callout color=" | ||
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| === Load Reference Arrow System === | === Load Reference Arrow System === | ||
| Zeile 152: | Zeile 158: | ||
| In the case of **consumers**, | In the case of **consumers**, | ||
| For consumers, the arrow__foot__ or arrow__head__ of the current and voltage are related. Voltage and current arrows are parallel ($\uparrow \uparrow$). | For consumers, the arrow__foot__ or arrow__head__ of the current and voltage are related. Voltage and current arrows are parallel ($\uparrow \uparrow$). | ||
| + | Here we have to use the passive sign convention: The current enters the component on the more positive terminal. Or vise versa: The current exits the component on the negative terminal. | ||
| + | |||
| + | Both expressions again come to the same result, when drawing the arrows. | ||
| For consumers, the following holds: | For consumers, the following holds: | ||
| Zeile 160: | Zeile 169: | ||
| </ | </ | ||
| + | |||
| </ | </ | ||
| <callout icon=" | <callout icon=" | ||
| < | < | ||
| - | < | ||
| - | </ | ||
| - | {{drawio> | ||
| - | </ | ||
| * **Before the calculation, | * **Before the calculation, | ||
| - | * the generator arrow system | + | * the active sign convention/generator arrow system is used for all sources (e.g. all voltage and current sources): the current is antiparallel to the voltage arrow. |
| - | * the motor arrow system | + | * the passive sign convention/motor arrow system is used for all consumers (e.g. all passives like resistors, capacitors, inductors, diodes, etc.): the current is parallel to the voltage arrow. |
| - | * for loads, where the direction of the power is not known, the motor arrow system is recommented | + | * for loads, where the direction of the power is not known, the motor arrow system is recommended |
| * **After the calculation** means | * **After the calculation** means | ||
| * $I>0$: The reference arrow reflects the conventional directional sense of the current | * $I>0$: The reference arrow reflects the conventional directional sense of the current | ||
| * $I<0$: The reference arrow points in the opposite direction to the conventional directional sense of the current | * $I<0$: The reference arrow points in the opposite direction to the conventional directional sense of the current | ||
| * Reference arrows of the current are drawn **in** the wire if possible. | * Reference arrows of the current are drawn **in** the wire if possible. | ||
| + | |||
| + | |||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| </ | </ | ||
| - | < | + | < |
| - | The reference arrow system | + | The reference arrow system |
| - | {{youtube> | + | We will instead use voltage arrows from plus to minus |
| + | {{youtube> | ||
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | + | ===== 2.3 Nodes, Branches, and Loops ===== | |
| - | ===== 2.3 Nodes, Branches and Loops ===== | + | |
| < | < | ||
| Explanation of the different network structures \\ | Explanation of the different network structures \\ | ||
| (Graphs and trees are only needed in later chapters) | (Graphs and trees are only needed in later chapters) | ||
| - | {{youtube> | + | |
| + | {{youtube> | ||
| </ | </ | ||
| < | < | ||
| - | === Goals === | + | === Learning Objectives |
| - | After this lesson | + | By the end of this section, |
| - | - Be able to identify the nodes, branches and loops in a circuit. | + | - identify the nodes, branches, and loops in a circuit. |
| - | - Be able to use them to make a circuit | + | - use them to reshape |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | |||
| + | |||
| + | |||
| + | Electrical circuits typically have the structure of networks. Networks consist of two elementary structural elements: | ||
| + | - <fc # | ||
| + | - <fc # | ||
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| + | </ | ||
| + | Please note in the case of electrical circuits, we will use the following definition: | ||
| + | |||
| + | - <fc # | ||
| + | - <fc # | ||
| + | |||
| + | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Electrical circuits typically have the structure of networks. Networks consist of two elementary structural elements: | + | Sometimes there is a differentiation between " |
| - | - <fc # | + | |
| - | - <fc # | + | |
| - | + | ||
| - | Please note in the case of electrical circuits: | + | |
| - | + | ||
| - | - <fc # | + | |
| - | - <fc # | + | |
| - | Branches in electrical networks are also called two-pole. | + | Branches in electrical networks are also called two-terminal networks. |
| - | Their behaviour | + | Their behavior |
| In addition, another term is to be explained: \\ | In addition, another term is to be explained: \\ | ||
| - | A **<fc # | + | A **<fc # |
| Since a voltmeter can also be present as a component between two nodes, it is also possible to close a loop by a drawn voltage arrow (cf. $U_1$ in <imgref BildNr8> | Since a voltmeter can also be present as a component between two nodes, it is also possible to close a loop by a drawn voltage arrow (cf. $U_1$ in <imgref BildNr8> | ||
| + | |||
| + | A loop that does not contain other (smaller) loops is called a mesh. | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | Please keep in mind, that usually the entire | + | Please keep in mind, that usually the entire |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Simplifications | + | ==== Reshaping Circuits |
| + | |||
| + | With the knowledge of nodes, branches, and meshes, circuits can be simplified. | ||
| + | Circuits can be reshaped arbitrarily as long as all branches remain at the same nodes after reshaping | ||
| + | The <imgref BildNr9> shows how such a transformation is possible. | ||
| < | < | ||
| Zeile 247: | Zeile 272: | ||
| </ | </ | ||
| </ | </ | ||
| - | |||
| - | With the knowledge of nodes, branches and meshes, circuits can be simplified. | ||
| - | Circuits can be reshaped arbitrarily as long as all branches remain at the same nodes after reshaping | ||
| - | The <imgref BildNr9> shows how such a transformation is possible. | ||
| For practical tasks, repeated trial and error can be useful. | For practical tasks, repeated trial and error can be useful. | ||
| - | It is important to check afterwards | + | It is important to check afterward |
| Further examples can be found in the following video | Further examples can be found in the following video | ||
| - | {{youtube> | + | {{youtube> |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| <panel type=" | <panel type=" | ||
| - | < | + | < |
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 271: | Zeile 292: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | < | + | |
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| + | |||
| + | <panel type=" | ||
| + | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Simplify | + | Reshape |
| </ | </ | ||
| - | ===== 2.4 Kirchhoff' | ||
| - | + | ===== 2.4 Kirchhoff' | |
| - | < | + | |
| - | Representation and application of Kirchhoff' | + | |
| - | {{youtube> | + | |
| - | </ | + | |
| < | < | ||
| - | === Goals == | + | === Learning Objectives === |
| - | After this lesson | + | By the end of this section, |
| - | Know and be able to apply Kirchhof's circuit laws (Kirchhoff' | + | Know and apply Kirchhoff's circuit laws (Kirchhoff' |
| </ | </ | ||
| + | |||
| + | < | ||
| + | {{wp> | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Kirchhoff' | + | ==== Kirchhoff' |
| - | The Kirchhoff' | + | Kirchhoff' |
| This is of particular relevance at a network node (<imgref BildNr10> | This is of particular relevance at a network node (<imgref BildNr10> | ||
| To formulate the equation at this node, the reference arrows of the currents are all set in the same way. | To formulate the equation at this node, the reference arrows of the currents are all set in the same way. | ||
| Zeile 305: | Zeile 333: | ||
| <callout icon=" | <callout icon=" | ||
| - | < | + | < |
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 316: | Zeile 344: | ||
| From now on, the following definition applies: | From now on, the following definition applies: | ||
| - | * Currents whose current arrows point towards the node are added in the calculation. | + | * Currents whose current arrows point towards the node are added to the calculation. |
| * Currents whose current arrows point away from the node are subtracted in the calculation. | * Currents whose current arrows point away from the node are subtracted in the calculation. | ||
| </ | </ | ||
| + | |||
| + | === Parallel circuit of resistors === | ||
| + | |||
| + | From Kirchhoff' | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | === Parallel connection of resistors | + | Since the same voltage $U_{ab}$ is dropped across all resistors, using Kirchhoff' |
| - | From the Kirchhoff' | + | $\large{{U_{ab}}\over{R_1}}+ {{U_{ab}}\over{R_2}}+ ... + {{U_{\rm ab}}\over{R_n}}= {{U_{\rm ab}}\over{R_{\rm eq}}}$ |
| - | Since the same voltage | + | $\rightarrow \large{{{1}\over{R_1}}+ {{1}\over{R_2}}+ ... + {{1}\over{R_n}}= {{1}\over{R_{\rm eq}}} = \sum_{x=1}^{n} {{1}\over{R_x}}}$ |
| - | $\large{{U_{ab}}\over{R_1}}+ {{U_{ab}}\over{R_2}}+ ... + {{U_{ab}}\over{R_n}}= {{U_{ab}}\over{R_{substitute}}$ | + | Thus, for resistors connected in parallel, the equivalent conductance $G_{\rm eq}$ (German: |
| - | + | ||
| - | $\rightarrow \large{{{1}\over{R_1}}+ {{1}\over{R_2}}+ ... + {{1}\over{R_n}}= {{1}\over{R_{substitute}} = \sum_{x=1}^{n} {{1}\over{R_x}}$ | + | |
| - | + | ||
| - | Thus, for resistors connected in parallel, the equivalent conductance $G_{eq}$ (German: Ersatzleitwert) is the sum of the individual conductances: | + | |
| __In general__: the equivalent resistance of a parallel circuit is always smaller than the smallest resistance. | __In general__: the equivalent resistance of a parallel circuit is always smaller than the smallest resistance. | ||
| - | Especially for two parallel resistors $R_1$ and $R_2$ applies: $R_{eq}= \large{{R_1 \cdot R_2}\over{R_1 + R_2}}$ | + | Especially for two parallel resistors $R_1$ and $R_2$ applies: $R_{\rm eq}= \large{{R_1 \cdot R_2}\over{R_1 + R_2}}$ |
| === Current divider === | === Current divider === | ||
| < | < | ||
| - | Derivation of the current divider with further considerations | + | Derivation of the current divider with examples |
| - | {{youtube> | + | {{youtube> |
| </ | </ | ||
| + | \\ \\ | ||
| + | The current divider rule shows in which way an incoming current on a node will be divided into two outgoing branches. | ||
| + | The rule states that the currents $I_1, ... I_n$ on parallel resistors $R_1, ... R_n$ behave just like their conductances $G_1, ... G_n$ through which the current flows. \\ | ||
| - | The current divider rule can also be derived from the Kirchhoff' | + | $\large{{I_1}\over{I_{\rm res}} = {{G_1}\over{G_{\rm res}}}$ |
| - | This states that, for resistors $R_1, ... R_n$ their currents $I_1, ... I_n$ behave just like the conductances $G_1, ... G_n$ through which they flow. | + | |
| - | + | ||
| - | $\large{{I_1}\over{I_g}} = {{G_1}\over{G_g}}$ | + | |
| $\large{{I_1}\over{I_2}} = {{G_1}\over{G_2}}$ | $\large{{I_1}\over{I_2}} = {{G_1}\over{G_2}}$ | ||
| + | |||
| + | The rule also be derived from Kirchhoff' | ||
| + | - The voltage drop $U$ on parallel resistors $R_1, ... R_n$ is the same. | ||
| + | - When $U_1 = U_2 = ... = U$, then the following equation is also true: $R_1 \cdot I_1 = R_2 \cdot I_2 = ... = R_{\rm eq} \cdot I_{\rm res}$. \\ | ||
| + | - Therefore, we get with the conductance: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | <wrap anchor # | ||
| <panel type=" | <panel type=" | ||
| Zeile 364: | Zeile 398: | ||
| < | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| In the simulation in <imgref BildNr85> | In the simulation in <imgref BildNr85> | ||
| - | - What currents would you expect in each branch if the input voltage were lowered from $5V$ to $3.3V$? __After__ thinking about your result, you can adjust the '' | + | - What currents would you expect in each branch if the input voltage were lowered from $5~\rm V$ to $3.3V~\rm $? __After__ thinking about your result, you can adjust the '' |
| - Think about what would happen if you flipped the switch __before__ you flipped the switch. \\ After you flip the switch, how can you explain the current in the branch? | - Think about what would happen if you flipped the switch __before__ you flipped the switch. \\ After you flip the switch, how can you explain the current in the branch? | ||
| Zeile 376: | Zeile 410: | ||
| <panel type=" | <panel type=" | ||
| - | Two resistors of $18\Omega$ and $2 \Omega$ are connected in parallel. The total current of the resistors is $3A$. \\ | + | Two resistors of $18~\Omega$ and $2~\Omega$ are connected in parallel. The total current of the resistors is $3~\rm A$. \\ |
| - | Calculate the total resistance and how the currents | + | Calculate the total resistance and how the currents |
| + | |||
| + | <button size=" | ||
| + | The substitute resistor can be calculated to | ||
| + | \begin{equation*} | ||
| + | R_{eq} = \frac{R_1R_2}{R_1+R_2} = \frac{18~\Omega \cdot 2~\Omega}{18~\Omega+2~\Omega} | ||
| + | \end{equation*} | ||
| + | The current through resistor $R_1$ is | ||
| + | \begin{equation*} | ||
| + | I_1 = \frac{R_{eq}}{R_1} I =\frac{1.8~\Omega}{18~\Omega} \cdot 3~\rm A | ||
| + | \end{equation*} | ||
| + | The current through resistor $R_2$ is | ||
| + | \begin{equation*} | ||
| + | I_2 = \frac{R_{eq}}{R_2}I = \frac{1.8~\Omega}{2~\Omega} \cdot 3~\rm A | ||
| + | \end{equation*} | ||
| + | </ | ||
| + | <button size=" | ||
| + | The values of the substitute resistor and the currents in the branches are | ||
| + | \begin{equation*} | ||
| + | R_{eq} = 1.8~\Omega \qquad I_1 = 0.3~{\rm A} \qquad I_2 = 2.7~\rm A | ||
| + | \end{equation*} | ||
| + | </ | ||
| </ | </ | ||
| \\ | \\ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Kirchhoff' | + | ==== Kirchhoff' |
| - | Also the Kirchhoff' | + | Also, Kirchhoff' |
| - | Between two points $1$ and $2$ of a network there is only one potential difference. | + | Between two points $1$ and $2$ of a network, there is only one potential difference. |
| Thus the potential difference is in particular independent of the way a network is traversed between the two points $1$ and $2$. | Thus the potential difference is in particular independent of the way a network is traversed between the two points $1$ and $2$. | ||
| This can be described by considering the meshes. | This can be described by considering the meshes. | ||
| <callout icon=" | <callout icon=" | ||
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| Zeile 400: | Zeile 455: | ||
| $\boxed{U_{1} + U_{2} + ... + U_{n} = \sum_{x=1}^{n} U_x = 0}$ | $\boxed{U_{1} + U_{2} + ... + U_{n} = \sum_{x=1}^{n} U_x = 0}$ | ||
| - | To calculate this, a convention for the loop direction must be specified. Theoretically this can be chosen arbitrarily. In practice, often a specific direction (e.g. [[https:// | + | To calculate this, a convention for the loop direction must be specified. Theoretically, this can be chosen arbitrarily. In practice, often a specific direction (e.g. [[https:// |
| - | Independently, | + | Independently, |
| For example: | For example: | ||
| * Voltages, whose voltage arrows point __in__ the direction of circulation are __added__ in the calculation. | * Voltages, whose voltage arrows point __in__ the direction of circulation are __added__ in the calculation. | ||
| Zeile 409: | Zeile 464: | ||
| === Proof of Kirchhoff' | === Proof of Kirchhoff' | ||
| - | If one expresses the voltage in <imgref BildNr12> | + | If one expresses the voltage in <imgref BildNr12> |
| $U_{12}= \varphi_1 - \varphi_2 $ \\ | $U_{12}= \varphi_1 - \varphi_2 $ \\ | ||
| $U_{23}= \varphi_2 - \varphi_3 $ \\ | $U_{23}= \varphi_2 - \varphi_3 $ \\ | ||
| Zeile 419: | Zeile 474: | ||
| $U_{12}+U_{23}+U_{34}+U_{41} = 0$ \\ \\ | $U_{12}+U_{23}+U_{34}+U_{41} = 0$ \\ \\ | ||
| - | === Series | + | === Series |
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Using Kirchhoff' | + | Using Kirchhoff' |
| - | $U_1 + U_2 + ... + U_n = U_g$ | + | $U_1 + U_2 + ... + U_n = U_{\rm res}$ |
| - | $R_1 \cdot I_1 + R_2 \cdot I_2 + ... + R_n \cdot I_n = R_{ersatz} \cdot I $ | + | $R_1 \cdot I_1 + R_2 \cdot I_2 + ... + R_n \cdot I_n = R_{\rm eq} \cdot I $ |
| - | Since in series | + | Since in a series |
| - | $R_1 + R_2 + ... + R_n = R_{eq} = \sum_{x=1}^{n} R_x $ | + | $R_1 + R_2 + ... + R_n = R_{\rm eq} = \sum_{x=1}^{n} R_x $ |
| - | __In general__: The equivalent resistance of a series circuit is always greater than the greatest resistance.. | + | __In general__: The equivalent resistance of a series circuit is always greater than the greatest resistance. |
| + | |||
| + | ==== Application ==== | ||
| + | |||
| + | === Kelvin-Sensing === | ||
| + | |||
| + | Often resistors are used to measure a current $I$ via the voltage drop on the resistor $U = R \cdot I$. Applications include the measurement of motor currents in the range of $0.1 ... 500 ~\rm A$. \\ | ||
| + | Those resistors are called //shunt resistors// and are commonly in the range of some $\rm m\Omega$. | ||
| + | This measurement can be interfered by the resistor of the supply lines. | ||
| + | |||
| + | To get an accurate measurement often Kelvin sensing, also known as {{wp> | ||
| + | This is a method of measuring electrical resistance avoiding errors caused by wire resistances. \\ | ||
| + | The simulation in <imgref BildNr005> | ||
| + | |||
| + | Four-terminal sensing involves using: | ||
| + | * a pair of //current leads// or //force leads// (with the resistances $R_{\rm cl1}$ and $R_{\rm cl2}$) to supply current to the circuit and | ||
| + | * a pair of //voltage leads// or //sense leads// (with the resistances $R_{\rm vl1}$ and $R_{\rm vl2}$) to measure the voltage drop across the impedance to be measured. | ||
| + | The sense connections via the voltage leads are made immediately adjacent to the target impedance $R_{\rm s}$ at the device under test $\rm DUT$. | ||
| + | By this, they do not include the voltage drop in the force leads or contacts. \\ | ||
| + | Since almost no current flows to the measuring instrument, the voltage drop in the sense leads is negligible. | ||
| + | This method can be a practical tool for finding poor connections or unexpected resistance in an electrical circuit. | ||
| + | |||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 443: | Zeile 524: | ||
| <panel type=" | <panel type=" | ||
| - | Three equal resistors of $20k\Omega$ each are given. \\ | + | Three equal resistors of $20~k\Omega$ each are given. \\ |
| - | Which values are realizable by arbitrary interconnection of one to three resistors? | + | Which values are realizable by the arbitrary interconnection of one to three resistors?\\ |
| + | <button size=" | ||
| + | The resistors can be connected in series: | ||
| + | \begin{equation*} | ||
| + | R_{\rm series} = 3\cdot R = 3\cdot20~k\Omega | ||
| + | \end{equation*} | ||
| + | The resistors can also be connected in parallel: | ||
| + | \begin{equation*} | ||
| + | R_{\rm parallel} = \frac{R}{3} = \frac{20~k\Omega}{3} | ||
| + | \end{equation*} | ||
| + | On the other hand, they can also be connected in a way that two of them are in parallel and those are in series to the third one: | ||
| + | \begin{equation*} | ||
| + | R_{\rm res} = R + \frac{R\cdot R}{R+R} = \frac{3}{2}R = \frac{3}{2} \cdot 20~k\Omega | ||
| + | \end{equation*} | ||
| + | </ | ||
| + | <button size=" | ||
| + | \begin{equation*} | ||
| + | R_{series} = 60~k\Omega\qquad R_{\rm parallel} = 6.7~k\Omega\qquad R_{\rm res} = 30~k\Omega | ||
| + | \end{equation*} | ||
| + | </ | ||
| </ | </ | ||
| - | ===== 2.5 | + | ===== 2.5 Voltage Divider |
| - | ==== The unloaded voltage divider ==== | ||
| < | < | ||
| - | Derivation | + | Why are voltage dividers important? (a cutout from 0:00 to 10:56 from a full video of EEVblog, starting from 17:00 there is also a nice example for troubles with voltage |
| - | {{youtube> | + | {{youtube> |
| </ | </ | ||
| + | |||
| + | ==== The unloaded Voltage Divider ==== | ||
| < | < | ||
| + | === Learning Objectives === | ||
| - | === Goals === | + | By the end of this section, |
| - | + | ||
| - | After this lesson | + | |
| - to distinguish between the loaded and unloaded voltage divider. | - to distinguish between the loaded and unloaded voltage divider. | ||
| - to describe the differences between loaded and unloaded voltage dividers. | - to describe the differences between loaded and unloaded voltage dividers. | ||
| </ | </ | ||
| + | |||
| + | |||
| + | Especially the series circuit of two resistors $R_1$ and $R_2$ shall be considered now. | ||
| + | This situation occurs in many practical applications e.g. in a {{wp> | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Especially the series connection of two resistors $R_1$ and $R_2$ shall be considered now. | ||
| - | This situation occurs in many practical applications (e.g. {{wp> | ||
| In <imgref BildNr14> | In <imgref BildNr14> | ||
| - | Via the Kirchhoff' | + | < |
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | Via Kirchhoff' | ||
| - | $\boxed{ {{U_1}\over{U}} = {{R_1}\over{R_1 + R_2}} }$ | + | $\boxed{ {{U_1}\over{U}} = {{R_1}\over{R_1 + R_2}} \rightarrow U_1 = k \cdot U}$ |
| The ratio $k={{R_1}\over{R_1 + R_2}}$ also corresponds to the position on a potentiometer. | The ratio $k={{R_1}\over{R_1 + R_2}}$ also corresponds to the position on a potentiometer. | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | <panel type=" | + | <panel type=" |
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| - | In the simulation in <imgref BildNr81> | + | In the simulation in <imgref BildNr81> |
| - | - What voltage $U_out$ would you expect if the switch were closed? After thinking about your result, you can check it by closing the switch. | + | - What voltage $U_{\rm O}$ would you expect if the switch were closed? After thinking about your result, you can check it by closing the switch. |
| - | - First think about what would happen if you would change the distribution of the resistors by moving the wiper (" | + | - First, think about what would happen if you would change the distribution of the resistors by moving the wiper (" |
| - | - At which position do you get a $U_out = 3.5V$? | + | - At which position do you get a $U_{\rm O} = 3.5~\rm V$? |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== The loaded | + | ==== The loaded |
| + | |||
| + | If - in contrast to the abovementioned, | ||
| < | < | ||
| < | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | |||
| - | If - in contrast to the above-mendtioned, | ||
| A circuit analysis yields: | A circuit analysis yields: | ||
| - | $ 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 $R_s = R_1 + R_2$: | + | or on a potentiometer with $k$ and the sum of resistors $R_{\rm s} = R_1 + R_2$: |
| - | $ U_1 = \LARGE{{k \cdot U} \over { 1 + k \cdot (1-k) \cdot{{R_s}\over{R_L}} }} }$ | + | $ U_1 = \LARGE{{k \cdot U} \over { 1 + k \cdot (1-k) \cdot{{R_{\rm s}}\over{R_{\rm L}}} }}$ |
| + | |||
| + | <imgref BildNr65> | ||
| + | In principle, this is similar to <imgref BildNr14>, | ||
| < | < | ||
| Zeile 522: | Zeile 632: | ||
| </ | </ | ||
| - | <imgref BildNr65> | + | What does this diagram tell us? This shall be investigated by an example. First, assume an unloaded voltage divider with $R_2 = 4.0 ~\rm k\Omega$ and $R_1 = 6.0 ~\rm k\Omega$, and an input voltage |
| - | In principle, this is similar to <imgref BildNr14>, | + | Now this voltage |
| - | What does this diagram tell us now? This will be shown by an example. First, assume an unloaded | + | What is the practical use of the (loaded) |
| - | Now this voltage | + | * Voltage dividers are in use for controlling the output |
| - | + | * Another " | |
| - | + | ||
| - | zeigt in welchem Verhältnis die ausgegebene Spannung $U_1$ zur eingehenden Spannung $U$ steht (y-Achse), in Bezug zum Verhältnis $k={{R_1}\over{R_1 + R_2}}$. Prinzipiell gleicht dies der <imgref BildNr14> | + | |
| - | + | ||
| - | Was sagt dieses Diagramm nun aus? Dies soll an einem Beispiel gezeigt werden. Zunächst wird angenommen, dass ein __unbelasteter Spannungsteiler__ mit $R_2 = 4 k\Omega$ und $R_1 = 6 k\Omega$, sowie eine Eingangsspannung von $10V$ vorliegt. Damit ist $k = 0,6$, $R_s = 10k\Omega$ und $U_1 = 6V$. \\ Nun wird dieser Spannungsteiler mit einem Lastwiderstand belastet. Liegt dieser bei $R_L = R_1 = 10 k\Omega$, so reduziert sich $k$ auf etwa $0,48$ und $U_1$ auf $4, | + | |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | <panel type=" | + | <panel type=" |
| - | Ermitteln Sie aus der Schaltung | + | Determine from the circuit |
| + | <button size=" | ||
| + | According to the voltage division rule, the loaded voltage is | ||
| + | \begin{align*} | ||
| + | U_1 & | ||
| + | & | ||
| + | & | ||
| + | & | ||
| + | \end{align*} | ||
| + | The divided resistor $R_1$ and $R_2$ are put together to form $R_{\rm s}=R_1 + R_2$. | ||
| + | \begin{equation*} | ||
| + | U_1=\frac{R_1 R_{\rm L}}{R_1 R_2 + R_{\rm s} R_{\rm L}} U | ||
| + | \end{equation*} | ||
| + | With the equations given there is also $R_1=k(R_1+R_2)=k R_{\rm s}$ and $R_2 = R_{\rm s} - R_1 = R_{\rm s} - k R_{\rm s} = (1-k) R_{\rm s}$. | ||
| + | \begin{equation*} | ||
| + | U_1=\frac{k R_{\rm s} R_{\rm L}}{k R_{\rm s} (1-k) R_{\rm s} + R_{\rm s} R_{\rm L}}U | ||
| + | \end{equation*} | ||
| + | Dividing the numerator and denominator by $R_{\rm s} R_{\rm L}$ yields to | ||
| + | \begin{equation*} | ||
| + | U_1=\frac{k}{k(1-k)\frac{R_{\rm s}}{R_{\rm L}}+1}U | ||
| + | \end{equation*} | ||
| + | </ | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | <WRAP right> | + | In the simulation in <imgref BildNr82> |
| - | < | + | - What voltage '' |
| + | - At which position of the wiper do you get $3.50~\rm V$ as an output? Determine the result first by means of a calculation. \\ Then check it by moving the slider at the bottom | ||
| + | |||
| + | <WRAP> | ||
| + | < | ||
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| - | |||
| - | |||
| - | In der Simulation in <imgref BildNr82> | ||
| - | |||
| - | - Welche Spannung '' | ||
| - | - Bei welcher Aufteilung erhalten Sie $3,5V$. Ermitteln Sie das Ergebnis zunächst zur eine Rechnung.\\ Überprüfen sie es anschließend durch Verschieben des Slider unten rechts neben der Simulation. | ||
| - | |||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Sie wollten einen Kleinstmotor für einen kleinen Roboter testen. Anhand des Maximalstroms und des Innenwiderstands | + | You wanted to test a micromotor for a small robot. Using the maximum current and the internal resistance |
| - | + | - First, calculate the maximum current | |
| - | - Berechnen Sie zunächst den Maximalstrom | + | - Draw the corresponding electrical circuit with the motor connected as an ohmic resistor. |
| - | - Zeichnen Sie die entsprechende elektrische Schaltung mit angeschlossenem Motor als ohmschen Widerstand. | + | - At the maximum current, the motor should be able to deliver a torque of $M_{\rm max}=M(I_{\rm M, max})= 100~\rm mNm$. What torque would the motor deliver if you implement the setup like this? (Assumption: The torque of the motor increases proportionally to the motor current). |
| - | - Beim Maximalstrom soll der Motor ein Drehmoment von $M= 100mNm$ abgeben können. Welches Drehmoment würde der Motor abgeben, wenn Sie den Aufbau so umsetzen? (Annahme: Das Drehmoment des Motors steigt proportional zum Motorstrom). | + | - What might a setup with a potentiometer look like that would actually allow you to set a voltage between |
| - | - Wie könnte ein Aufbau mit Potentiometer aussehen, mit dem man tatsächlich eine Spannung zwischen | + | - Build and test your circuit |
| - | - Bauen Sie Ihre Schaltung | + | - Routing connections can be activated via the menu: '' |
| - | - Das Verlegen von Verbindungen lässt sich über das Menü '' | + | - Note that connections can only ever be connected at nodes. A red-marked node (e.g. at the $5 ~\Omega$ |
| - | - Beachten Sie, dass Verbindungen immer nur an Verbindungspunkten angeschlossen werden können. Der rot markierte Knoten am $5 \Omega$-Widerstand zeigt an, dass dieser nicht verbunden ist. Dieser könnte im ein Rasterschritt nach links verschoben werden, da dort ein Verbindungspunkt liegt. | + | - Pressing the ''< |
| - | - Mit Druck auf die ''< | + | - With a right click on a component it can be copied or values like the resistor can be changed via '' |
| - | - Mit Rechtsklick auf eine Komponente lässt sich diese kopieren oder Werte wie der Widerstand über '' | + | |
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | <WRAP group>< | + | Exercise on the voltage divider |
| - | Spannungsteiler, | + | |
| - | {{youtube> | + | |
| - | + | ||
| - | </ | + | |
| - | Übung zum Spannungsteiler | + | |
| {{youtube> | {{youtube> | ||
| - | </ | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
| Zeile 606: | Zeile 723: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | + | ===== 2.6 Circuits with three Connections | |
| - | ===== 2.6 Stern-Dreieck-Schaltung | + | |
| - | + | ||
| - | < | + | |
| - | + | ||
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | + | ||
| - | < | + | |
| - | </ | + | |
| - | {{url> | + | |
| - | + | ||
| - | </ | + | |
| < | < | ||
| - | === Ziele === | + | === Learning Objectives |
| - | Nach dieser Lektion sollten Sie: | + | By the end of this section, you will be able to: |
| - | + | - convert triangular loops into a star shape (and vice versa) | |
| - | - dreieckige Maschen in eine Sternform | + | |
| </ | </ | ||
| - | Zu Beginn des Kapitels wurde ein Beispiel eines Netzwerks gezeigt | + | At the beginning of the chapter, an example of a network was shown (<imgref BildNr91> |
| + | It is visible, that there are many $\Delta$-shaped (or triangle-shaped) loops resp. $\rm Y$-shaped | ||
| + | A method to calculate these will be discussed in more detail now. | ||
| - | Dazu zunächst ein Resume aus den bisherigen Erkenntnissen. Über den Knoten- und Maschensatz wurde klar, dass sowohl aus einer Reihen-, als auch aus einer Parallelschaltung ein Ersatzwiderstand ermittelt werden kann. Betrachtet man den Ersatzwiderstand als eine Blackbox - d.h. der innere Ausbau ist unbekannt - so könnte dieser also durch beide Schaltungsarten interpretiert werden (<imgref BildNr17>). | + | < |
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </WRAP> | ||
| - | Wie hilft uns das nun im Falle einer dreieckförmigen Masche? | + | First of all a summary of the previous findings: Using the node and loop rule it became clear that an equivalent resistance can be determined from a series as well as from a parallel circuit. If one considers the equivalent resistance as a black box - i.e. the internals are unknown - it could be interpreted by both types of a circuit (<imgref BildNr17> |
| - | Auch in für diesen Fall kann man eine Blackbox bereitstellen. Diese müsste sich aber immer gleich verhalten, wie die dreieckförmige Masche, also beliebige, angelegte Spannungen sollten gleiche Ströme erzeugen. | + | < |
| - | Anders gesagt: Die zwischen zwei Klemmen messbaren Widerständen müssen für beide Schaltungen identisch sein. | + | < |
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| - | Dazu sollen nun die verschiedenen Widerstände zwischen den einzelnen Knoten $a$, $b$ und $c$ betrachtet werden, siehe <imgref BildNr18> | + | Now how does this help us in the case of a $\Delta$-load (= triangular loop)? |
| + | Also in this case one can provide a black box. However, this should always behave in the same way as the $\Delta$-load, | ||
| + | In other words: The resistances measured between two terminals must be identical for the black box and for the known circuit. | ||
| + | For this purpose, the different resistances between the individual nodes $\rm a$, $\rm b$, and $\rm c$ are now to be considered, see <imgref BildNr18> | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{url> | + | {{url> |
| \\ \\ | \\ \\ | ||
| - | Berechung der Umformungsformeln: Sternschaltung | + | Calculation of the transformation formulae: Star connection |
| - | {{youtube> | + | {{youtube> |
| </ | </ | ||
| - | ==== Dreieckschaltung | + | ==== Delta Circuit |
| - | Bei der Dreieckschaltung sind die 3 Widerstände | + | In the delta circuit, the 3 resistors |
| + | The labeling with a superscript $\square^1$ refers to the three resistors | ||
| - | Für die Widerstände zwischen den zwei Anschlüssen | + | For the measurable resistance between two terminals |
| - | $R_{ab} = R_{ab}^1 || (R_{ca}^1 + R_{bc}^1) $ \\ | + | < |
| - | $R_{ab} = {{R_{ab}^1 \cdot (R_{ca}^1 + R_{bc}^1)}\over{R_{ab}^1 + (R_{ca}^1 + R_{bc}^1)}} = {{R_{ab}^1 \cdot (R_{ca}^1 + R_{bc}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} $ \\ | + | </ |
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | $R_{\rm ab} = R_{\rm ab}^1 || (R_{\rm ca}^1 + R_{\rm bc}^1) $ \\ | ||
| + | $R_{\rm ab} = {{R_{\rm ab}^1 \cdot (R_{\rm ca}^1 + R_{\rm bc}^1)}\over{R_{\rm ab}^1 + (R_{\rm ca}^1 + R_{\rm bc}^1)}} = {{R_{\rm ab}^1 \cdot (R_{\rm ca}^1 + R_{\rm bc}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} $ \\ | ||
| - | Gleiches gilt für die anderen Anschlüssen. Damit ergibt sich: | + | The same applies to the other connections. This results in: |
| \begin{align*} | \begin{align*} | ||
| - | R_{ab} = {{R_{ab}^1 \cdot (R_{ca}^1 + R_{bc}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} | + | R_{\rm ab} = {{R_{\rm ab}^1 \cdot (R_{\rm ca}^1 + R_{\rm bc}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} |
| - | R_{bc} = {{R_{bc}^1 \cdot (R_{ab}^1 + R_{ca}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} | + | R_{\rm bc} = {{R_{\rm bc}^1 \cdot (R_{\rm ab}^1 + R_{\rm ca}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} |
| - | R_{ca} = {{R_{ca}^1 \cdot (R_{bc}^1 + R_{ab}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} \tag{2.6.1} | + | R_{\rm ca} = {{R_{\rm ca}^1 \cdot (R_{\rm bc}^1 + R_{\rm ab}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} \tag{2.6.1} |
| - | ==== Sternschaltung | + | ==== Star Circuit |
| - | Die Widerstände zwischen den Anschlüssen müssen nun denen bei der Sternschaltung gleichen. Auch bei der Sternschaltung sind 3 Widerstände verschalten, diese aber in Sternform. Die Sternwiderstände sind also alle mit einem weiteren Knoten | + | Given the idea, that the star circuit shall behave equally to the delta circuit, the resistance measured between the terminals must be similar. |
| + | Also in the star circuit, | ||
| + | $R_{\rm a0}^1$, $R_{\rm b0}^1$ | ||
| - | Auch hier wird vorgegangen wie bei der Dreieckschaltung: der Widerstand zwischen zwei Anschlüssen | + | Again, the procedure is the same as for the delta connection: the resistance between two terminals |
| + | The resistance of the further terminal | ||
| \begin{align*} | \begin{align*} | ||
| - | R_{ab} = R_{a0}^1 + R_{b0}^1 | + | R_{\rm ab} = R_{\rm a0}^1 + R_{\rm b0}^1 \\ |
| - | R_{bc} = R_{b0}^1 + R_{c0}^1 | + | R_{\rm bc} = R_{\rm b0}^1 + R_{\rm c0}^1 \\ |
| - | R_{ca} = R_{c0}^1 + R_{a0}^1 | + | R_{\rm ca} = R_{\rm c0}^1 + R_{\rm a0}^1 \tag{2.6.2} |
| \end{align*} | \end{align*} | ||
| - | Aus den Gleichungen | + | From equations |
| - | \begin{align} R_{ab} = {{R_{ab}^1 \cdot (R_{ca}^1 + R_{bc}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} = R_{a0}^1 + R_{b0}^1 \tag{2.6.3} \end{align} | + | \begin{align} R_{\rm ab} = {{R_{\rm ab}^1 \cdot (R_{\rm ca}^1 + R_{\rm bc}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} = R_{\rm a0}^1 + R_{\rm b0}^1 \tag{2.6.3} \end{align} |
| - | \begin{align} R_{bc} = {{R_{bc}^1 \cdot (R_{ab}^1 + R_{ca}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} = R_{b0}^1 + R_{c0}^1 \tag{2.6.4} \end{align} | + | \begin{align} R_{\rm bc} = {{R_{\rm bc}^1 \cdot (R_{\rm ab}^1 + R_{\rm ca}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} = R_{\rm b0}^1 + R_{\rm c0}^1 \tag{2.6.4} \end{align} |
| - | \begin{align} R_{ca} = {{R_{ca}^1 \cdot (R_{bc}^1 + R_{ab}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} = R_{c0}^1 + R_{a0}^1 \tag{2.6.5} \end{align} | + | \begin{align} R_{\rm ca} = {{R_{\rm ca}^1 \cdot (R_{\rm bc}^1 + R_{\rm ab}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} = R_{\rm c0}^1 + R_{\rm a0}^1 \tag{2.6.5} \end{align} |
| - | Die Gleichungen | + | Equations |
| - | Eine Variante ist die Formeln als ${{1}\over{2}} \cdot \left( (2.6.3) + (2.6.4) - (2.6.5) \right)$ | + | A variation is to write the formulas as ${{1}\over{2}} \cdot \left( (2.6.3) + (2.6.4) - (2.6.5) \right)$ |
| \begin{align*} | \begin{align*} | ||
| - | {{1}\over{2}} \cdot \left( {{R_{ab}^1 \cdot (R_{ca}^1 + R_{bc}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} + {{R_{bc}^1 \cdot (R_{ab}^1 + R_{ca}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} - {{R_{ca}^1 \cdot (R_{bc}^1 + R_{ab}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} \right) &= {{1}\over{2}} \cdot \left( R_{a0}^1 + R_{b0}^1 + R_{b0}^1 + R_{c0}^1 - R_{c0}^1 - R_{a0}^1 \right) \\ | + | {{1}\over{2}} \cdot \left( {{R_{\rm ab}^1 \cdot (R_{\rm ca}^1 + R_{\rm bc}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} |
| + | + {{R_{\rm bc}^1 \cdot (R_{\rm ab}^1 + R_{\rm ca}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} | ||
| + | - {{R_{\rm ca}^1 \cdot (R_{\rm bc}^1 + R_{\rm ab}^1)}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} \right) & | ||
| + | {{1}\over{2}} \cdot \left( | ||
| - | {{1}\over{2}} \cdot \left( {{R_{ab}^1 \cdot (R_{ca}^1 + R_{bc}^1)} + {R_{bc}^1 \cdot (R_{ab}^1 + R_{ca}^1)} - {R_{ca}^1 \cdot (R_{bc}^1 + R_{ab}^1)}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} \right) &= {{1}\over{2}} \cdot \left( 2 \cdot R_{b0}^1 | + | {{1}\over{2}} \cdot \left( {{R_{\rm ab}^1 \cdot (R_{\rm ca}^1 + R_{\rm bc}^1)} + {R_{\rm bc}^1 \cdot (R_{\rm ab}^1 + R_{\rm ca}^1)} |
| + | | ||
| + | | ||
| - | {{1}\over{2}} \cdot \left( {{R_{ab}^1 R_{ca}^1 + R_{ab}^1 R_{bc}^1 + R_{bc}^1 R_{ab}^1 + R_{bc}^1 R_{ca}^1 - R_{ca}^1 R_{bc}^1 - R_{ca}^1 R_{ab}^1}\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} \right) & | + | {{1}\over{2}} \cdot \left( {{R_{\rm ab}^1 R_{\rm ca}^1 + R_{\rm ab}^1 R_{\rm bc}^1 + R_{\rm bc}^1 R_{\rm ab}^1 + R_{\rm bc}^1 R_{\rm ca}^1 - R_{\rm ca}^1 R_{\rm bc}^1 - R_{\rm ca}^1 R_{\rm ab}^1}\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} \right) & |
| - | {{1}\over{2}} \cdot \left( {{ 2 \cdot R_{ab}^1 R_{bc}^1 }\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} \right) & | + | {{1}\over{2}} \cdot \left( {{ 2 \cdot R_{\rm ab}^1 R_{\rm bc}^1 }\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} \right) & |
| - | {{ R_{ab}^1 R_{bc}^1 }\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} & | + | {{ R_{\rm ab}^1 R_{\rm bc}^1 }\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} & |
| \end{align*} | \end{align*} | ||
| - | Auf ähnlichem Weg kann man nach $R_{a0}^1$ | + | Similarly, one can resolve to $R_{\rm a0}^1$ |
| - | ==== Stern-Dreieck-Transformation ==== | + | ==== Y-Δ-Transformation |
| - | <callout icon=" | + | <callout icon=" |
| - | < | + | < |
| - | Soll von einer **Dreieckschaltung in eine Sternschaltung** umgewandelt werden, so sind die Sternwiderstände ermittelbar über: | + | If a **delta circuit is to be converted into a star circuit**, the star resistors can be determined via: |
| \begin{align*} | \begin{align*} | ||
| - | \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{Sternwiderstand} \\ \text{an Anschluss | + | \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{star resistance} \\ \text{at the terminal |
| - | {{ \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{Produkt der} \\ \text{am Anschluss x liegenden} \\ \text{Dreieckwiderstände} \end{array} }}} } \over | + | {{ \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{product of} \\ \text{the delta resistances} \\ \text{connected with x} \end{array} }}} } \over |
| - | { \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{Summe aller} \\ \text{Dreieckwiderstände} \end{array} }}}}} \\ | + | { \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{sum of all} \\ \text{delta resistances} \end{array} }}}}} \\ |
| \\ | \\ | ||
| - | \text{also: | + | \text{therefore: |
| - | R_{a0}^1 &= {{ R_{ca}^1 \cdot R_{ab}^1 }\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} \\ | + | R_{\rm a0}^1 &= {{ R_{\rm ca}^1 \cdot R_{\rm ab}^1 }\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} \\ |
| - | R_{b0}^1 &= {{ R_{ab}^1 \cdot R_{bc}^1 }\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} \\ | + | R_{\rm b0}^1 &= {{ R_{\rm ab}^1 \cdot R_{\rm bc}^1 }\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} \\ |
| - | R_{c0}^1 &= {{ R_{bc}^1 \cdot R_{ca}^1 }\over{R_{ab}^1 + R_{ca}^1 + R_{bc}^1}} | + | R_{\rm c0}^1 &= {{ R_{\rm bc}^1 \cdot R_{\rm ca}^1 }\over{R_{\rm ab}^1 + R_{\rm ca}^1 + R_{\rm bc}^1}} |
| \end{align*} | \end{align*} | ||
| - | </ | + | </ |
| - | Soll von einer **Sternschaltung in eine Dreieckschaltung** umgewandelt werden, so sind die Dreieckwiderstände ermittelbar über: | + | If a **star circuit is to be converted into a delta circuit**, the star resistors can be determined via: |
| \begin{align*} | \begin{align*} | ||
| - | \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{Dreieckwiderstand} \\ \text{zwischen den} \\ \text{Anschlüssen | + | \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{delta resistance} \\ \text{between the} \\ \text{terminals |
| - | {{ \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{Summe aller Produkte} \\ \text{zwischen zwei} \\ \text{unterschiedlichen Sternwiderständen} \end{array} }}} } \over | + | {{ \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{sum of all} \\ \text{products between} \\ \text{varying star resistances} \end{array} }}} } \over |
| - | { \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{Sternwiderstand} \\ \text{gegenüber von x und y} \end{array} }}}}} \\ | + | { \color{lightgray}{\boxed{ \color{black}{\begin{array}{} \text{star resistance} \\ \text{opposite |
| \\ | \\ | ||
| - | \text{also: | + | \text{therefore: |
| - | R_{ab}^1 &= {{ R_{a0}^1 \cdot R_{b0}^1 +R_{b0}^1 \cdot R_{c0}^1 +R_{c0}^1 \cdot R_{a0}^1 }\over{ R_{c0}^1}} \\ | + | R_{\rm ab}^1 &= {{ R_{\rm a0}^1 \cdot R_{\rm b0}^1 +R_{\rm b0}^1 \cdot R_{\rm c0}^1 +R_{\rm c0}^1 \cdot R_{\rm a0}^1 }\over{ R_{\rm c0}^1}} \\ |
| - | R_{bc}^1 &= {{ R_{a0}^1 \cdot R_{b0}^1 +R_{b0}^1 \cdot R_{c0}^1 +R_{c0}^1 \cdot R_{a0}^1 }\over{ R_{a0}^1}} \\ | + | R_{\rm bc}^1 &= {{ R_{\rm a0}^1 \cdot R_{\rm b0}^1 +R_{\rm b0}^1 \cdot R_{\rm c0}^1 +R_{\rm c0}^1 \cdot R_{\rm a0}^1 }\over{ R_{\rm a0}^1}} \\ |
| - | R_{ca}^1 &= {{ R_{a0}^1 \cdot R_{b0}^1 +R_{b0}^1 \cdot R_{c0}^1 +R_{c0}^1 \cdot R_{a0}^1 }\over{ R_{b0}^1}} | + | R_{\rm ca}^1 &= {{ R_{\rm a0}^1 \cdot R_{\rm b0}^1 +R_{\rm b0}^1 \cdot R_{\rm c0}^1 +R_{\rm c0}^1 \cdot R_{\rm a0}^1 }\over{ R_{\rm b0}^1}} |
| \end{align*} | \end{align*} | ||
| - | </ | + | </ |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | <panel type=" | + | <panel type=" |
| - | + | ||
| - | {{youtube> | + | |
| + | {{youtube> | ||
| + | {{youtube> | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
| Zeile 758: | Zeile 887: | ||
| </ | </ | ||
| - | ===== 2.7 Gruppenschaltung von Widerständen | + | ===== 2.7 Circuits with multiple Resistors |
| < | < | ||
| - | === Ziele === | + | === Learning Objectives |
| - | Nach dieser Lektion sollten Sie: | + | By the end of this section, you will be able to: |
| - | + | - simplify circuits consisting only of resistors. | |
| - | - Schaltungen, | + | - calculate the voltages and currents |
| - | - die Spannungen und Ströme | + | - simplify symmetrical circuits. |
| - | - symmetrische Schaltungen vereinfachen können. | + | |
| </ | </ | ||
| - | In diesem Unterkapitel wird auf eine Methodik eingegangen, welche beim Umformen von Schaltungen helfen soll. In Unterkapitel | + | In this subchapter, a methodology is discussed, which should help to reshape circuits. In subchapter |
| - | Ausgangspunkt sind Aufgaben, bei denen für ein Widerstandsnetzwerk der Gesamtwiderstand, Gesamtstrom oder die Gesamtspannung berechnet werden muss. | + | Starting points are tasks, where for a resistor network the total resistance, total current, or total voltage has to be calculated. |
| - | ==== einfaches Beispiel | + | ==== Simple Example |
| + | |||
| + | An example of such a circuit is given in <imgref imageNo89> | ||
| + | This current can be found by the (given) voltage $U_0$ and the total resistance between the terminals $\rm a$ and $\rm b$. So we are looking for $R_{\rm ab}$. | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| + | As already described in the previous subchapters, | ||
| + | It is important to note that these partial circuits for conversion into equivalent resistors may only ever have two connections (= two nodes to the " | ||
| - | Ein Beispiel für eine solche Schaltung ist in <imgref BildNr89> | ||
| - | |||
| - | Wie bereits in den vorherigen Unterkapitel beschrieben, | ||
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | < | + | < |
| + | As a result of the equivalent resistance one gets: | ||
| \begin{align*} | \begin{align*} | ||
| - | R_g = R_{12345} &= R_{12}||R_{345} = R_{12}||(R_3+R_{45}) = (R_1||R_2)||(R_3+R_4||R_5) \\ | + | R_{\rm |
| &= {{ {{R_1 \cdot R_2}\over{R_1 + R_2}} \cdot (R_3 + {{R_4 \cdot R_5}\over{R_4 + R_5}}) }\over{ {{R_1 \cdot R_2}\over{R_1 + R_2}} +R_3 + {{R_4 \cdot R_5}\over{R_4 + R_5}} }} \quad \quad \quad \quad \quad \quad \bigg\rvert \cdot{{(R_1 + R_2) \cdot (R_4 + R_5)}\over{(R_1 + R_2) \cdot (R_4 + R_5)}} \\ | &= {{ {{R_1 \cdot R_2}\over{R_1 + R_2}} \cdot (R_3 + {{R_4 \cdot R_5}\over{R_4 + R_5}}) }\over{ {{R_1 \cdot R_2}\over{R_1 + R_2}} +R_3 + {{R_4 \cdot R_5}\over{R_4 + R_5}} }} \quad \quad \quad \quad \quad \quad \bigg\rvert \cdot{{(R_1 + R_2) \cdot (R_4 + R_5)}\over{(R_1 + R_2) \cdot (R_4 + R_5)}} \\ | ||
| Zeile 805: | Zeile 936: | ||
| \end{align*} | \end{align*} | ||
| - | ==== Beispiel mit Dreieck-Stern-Transformation ==== | + | ==== Example of Δ-Y-Transformation ==== |
| - | Mit der Dreieck-Stern-Transformation lässt sich nun auch das anfängliche Beispiel umwandeln. Bei komplizierteren Schaltungen ist die wiederholte Dreieck-Stern-Transformation mit anschließendem Zusammenfassen der Widerstände sinnvoll, solange bis die entstandene Schaltung leicht mit Knoten- und Maschensatz berechenbar wird (< | + | With the Δ-Y-transformation now also the initial example can be transformed. For more complicated circuits, the repeated Δ-Y-transformation with a subsequent combining of the resistors is useful, until the resulting circuit is easily calculable with node and mesh theorem |
| < | < | ||
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| - | </WRAP> | + | < |
| - | ==== Beispiel mit Symmetrien | + | ==== Example with Symmetries |
| - | Ein gewisser Sonderfall betrifft mögliche Symmetrien | + | A certain special case concerns possible symmetries |
| < | < | ||
| - | < | + | < |
| - | </ | + | </ |
| - | {{url> | + | {{url> |
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | < | + | < |
| - | Über die Schalter kann nachgeprüft werden, ob ein Strom fließt, falls die jeweiligen Knoten verbunden werden. In der Simulation ist zu sehen, dass dies nicht der Fall ist. Im symmetrischen Aufbau sind diese Knoten jeweils auf dem gleichen Potential. | + | The switches can be used to check whether a current flows if the respective nodes are connected. In the simulation, it can be seen that this is not the case. In the symmetrical setup, these nodes are each at the same potential. |
| - | Damit lässt sich die Schaltung auch in die Form bringen, wie sie in < | + | This also allows the circuit to take the form shown in < |
| \begin{align*} | \begin{align*} | ||
| - | R_g = R || R + R || R || R || R + R || R || R || R + R || R = {{1}\over{2}}\cdot R + {{1}\over{4}}\cdot R + {{1}\over{4}}\cdot R + {{1}\over{2}}\cdot R = 1,5\cdot R | + | R_{\rm eq} = R || R + R || R || R || R + R || R || R || R + R || R = {{1}\over{2}}\cdot R + {{1}\over{4}}\cdot R + {{1}\over{4}}\cdot R + {{1}\over{2}}\cdot R = 1.5\cdot R |
| \end{align*} | \end{align*} | ||
| Zeile 840: | Zeile 971: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
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| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
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| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
| Zeile 859: | Zeile 990: | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
| Zeile 866: | Zeile 997: | ||
| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
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| - | <panel type=" | + | <panel type=" |
| {{youtube> | {{youtube> | ||
| Zeile 879: | Zeile 1010: | ||
| </ | </ | ||
| - | {{page> | + | {{page> |
| - | {{page> | + | {{page> |
| {{page> | {{page> | ||
| {{page> | {{page> | ||
| - | <panel type=" | + | <panel type=" |
| + | |||
| + | well explained example of a simplification due to symmetry: | ||
| + | |||
| + | {{youtube> | ||
| - | Weitere Anfgaben sind Online auf den Seiten von [[https:// | ||
| </ | </ | ||
| + | |||