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| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
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electrical_engineering_1:simple_circuits [2021/09/23 09:26] tfischer |
electrical_engineering_1:simple_circuits [2023/10/18 01:21] (aktuell) mexleadmin |
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| - | ====== 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_terminal_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 |
| - | https:// | + | For the direction of the arrows different conventions are available. Here (and quite often in Germany) the [[https:// |
| + | This convention is | ||
| - | === Generator Reference Arrow System === | + | === Generator Reference Arrow System |
| <WRAP group>< | <WRAP group>< | ||
| Zeile 127: | 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 142: | Zeile 149: | ||
| <callout color=" | <callout color=" | ||
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| === Load Reference Arrow System === | === Load Reference Arrow System === | ||
| Zeile 151: | 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 159: | 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 teference | + | The reference |
| - | {{youtube> | + | We will instead use voltage arrows from plus to minus |
| + | {{youtube> | ||
| </ | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ===== 2.3 Nodes, Branches, and Loops ===== | ||
| - | ===== 2.3 Knots, branches | + | < |
| + | Explanation of the different network structures \\ | ||
| + | (Graphs | ||
| - | <WRAP right> | + | {{youtube> |
| - | Erklärung der verschiedenen Netzwerkstrukturen \\ (Graphen und Bäume werden erst in späteren Kapiteln benötigt) | + | |
| - | {{youtube> | + | |
| </ | </ | ||
| < | < | ||
| - | === Ziele === | + | === Learning Objectives |
| - | + | ||
| - | Nach dieser Lektion sollten Sie: | + | |
| - | | + | By the end of this section, you will be able to: |
| - | - eine Schaltung damit übersichtlicher darstellen können. | + | |
| + | - use them to reshape a circuit. | ||
| </ | </ | ||
| Zeile 206: | Zeile 220: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | <WRAP right> | ||
| - | < | ||
| - | </ | ||
| - | {{drawio> | ||
| - | < | + | |
| + | 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: | ||
| - | Elektrische Stromkreise haben typischerweise die Struktur von Netzen. Netze bestehen aus zwei elementaren Strukturelementen: | + | - <fc # |
| + | - <fc # | ||
| - | - <fc #cd5c5c>**Zweige/ | + | <WRAP> |
| - | | + | <imgcaption BildNr8 | nodes, branches and loops> |
| + | </imgcaption> \\ | ||
| + | {{drawio> | ||
| + | </WRAP> | ||
| - | Bei elektrischen Schaltkreisen ist zu beachten: | + | Sometimes there is a differentiation between " |
| - | | + | Branches in electrical networks are also called two-terminal networks. |
| - | - <fc # | + | Their behavior is described by current-voltage characteristics and explained in more detail in the chapter [[non-ideal_sources_and_two_terminal_networks]]. |
| - | Zweige in elektrischen Netzwerken bezeichnet man als Zweipole. | + | In addition, another term is to be explained: \\ |
| - | Ihr Verhalten wird durch Strom-Spannungs-Kennlinien beschrieben und im Kapitel [[non-ideal_sources_and_two_terminal_networks]] näher erklärt. | + | |
| - | Zudem soll noch ein weiterer Begriff erklärt werden: \\ | + | A **<fc #ffa500>loop</ |
| - | Eine **<fc #ffa500>Masche</ | + | |
| - | Da auch ein Voltmeter als Komponente zwischen zwei Knoten vorhanden sein kann, ist es auch möglich eine Masche über eine Angabe einer Spannung zu schließen | + | 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~~ | ||
| - | Im Gegensatz zu den anderen Ursache-Wirkungs-Beziehungen ändert sich bei den vernetzten Stromkreisen fast immer das gesamte Verhalten, wenn in einem Zweig / an einem Knoten eine Änderung auftritt. \\ Dies ist vergleichbar mit anderen Änderungen | + | Please keep in mind, that usually the entire behavior of networked circuits almost always changes when a change occurs |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Vereinfachungen | + | ==== 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. | ||
| + | |||
| + | < | ||
| + | < | ||
| {{elektrotechnik_1: | {{elektrotechnik_1: | ||
| </ | </ | ||
| </ | </ | ||
| - | Mit der Kenntnis von Knoten, Zweigen und Maschen lassen sich Schaltungen vereinfachen. | + | For practical tasks, repeated trial and error can be useful. |
| - | Schaltungen lassen sich beliebig umformen, solange nach der Umformung alle Zweige an den gleichen Knoten bleiben | + | It is important to check afterward that the same components are connected to each node as before the transformation. |
| - | Die <imgref BildNr9> zeigt wie eine solche Umformung möglich ist. | + | |
| - | Bei praktischen Aufgaben kann ein wiederholtes Ausprobieren sinnvoll sein. | + | Further examples can be found in the following video |
| - | Wichtig dabei ist eine nachträgliche Kontrolle, dass an jedem Knoten die selben Komponenten wie vor der Umwandlung angeschlossen sind. | + | {{youtube> |
| - | + | ||
| - | Weitere Beispiele sind in folgendem Video zu finden | + | |
| - | {{youtube> | + | |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | <panel type=" | + | <panel type=" |
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Geben Sie für die Markierungen | + | For the markings |
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | < | + | |
| - | < | + | {{youtube> |
| + | </ | ||
| + | |||
| + | |||
| + | <panel type=" | ||
| + | < | ||
| + | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Vereinfachen Sie die Schaltungen | + | Reshape the circuits |
| </ | </ | ||
| - | ===== 2.4 Kirchhoffsche Gleichungen ===== | ||
| - | + | ===== 2.4 Kirchhoff' | |
| - | <WRAP right> | + | |
| - | Darstellung und Anwendung der Kirchhoffschen Gesetze | + | |
| - | {{youtube> | + | |
| - | </ | + | |
| < | < | ||
| - | === Ziele === | + | === Learning Objectives |
| - | Nach dieser Lektion sollten Sie: | + | By the end of this section, you will be able to: |
| - | - die Kirchhoffschen Gleichungen bzw. Knoten- und Maschensatz kennen und anwenden können. | + | Know and apply Kirchhoff' |
| </ | </ | ||
| + | |||
| + | < | ||
| + | {{wp> | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Der Knotensatz (1. Kirchhoffsche Gleichung) | + | ==== Kirchhoff' |
| - | Der Knotensatz formuliert in der Sprache der Mathematik die Erfahrung, dass sich in elektrischen Leitern keine Ladungs"anhäufungen" | + | Kirchhoff' |
| - | Dies ist von besonderer Relevanz an einem Netzknoten | + | This is of particular relevance at a network node (<imgref BildNr10> |
| - | Zur Formulierung der Gleichung werden bei diesem Netzknoten die Bezugspfeile der Ströme alle in gleicher Weise festgelegt. | + | To formulate the equation at this node, the reference arrows of the currents are all set in the same way. |
| - | Das heißt: alle zeigen vom Knoten weg oder auf ihn zu. | + | That means: all point away from or towards the node. |
| - | <callout icon=" | + | <callout icon=" |
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Die Summe aller Ströme, welche aus den Knoten zulaufen, muss Null sein. | + | The sum of all currents flowing from the nodes must be zero. |
| $\boxed{I_1 + I_2 + I_3 + ... + I_n = \sum_{x=1}^{n} I_x=0}$ | $\boxed{I_1 + I_2 + I_3 + ... + I_n = \sum_{x=1}^{n} I_x=0}$ | ||
| - | Es gilt von nun an folgende Festlegung: | + | From now on, the following definition applies: |
| - | * Ströme, deren Strompfeile auf den Knoten hin zeigen, werden in der Rechnung addiert. | + | * Currents whose current arrows point towards the node are added to the calculation. |
| - | * Ströme, deren Strompfeile vom Knoten weg zeigen, werden | + | * Currents whose current arrows point away from the node are subtracted |
| </ | </ | ||
| - | < | + | === Parallel circuit of resistors === |
| - | < | + | |
| + | From Kirchhoff' | ||
| + | |||
| + | < | ||
| + | < | ||
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | === Parallelschaltung von Widerständen === | + | Since the same voltage $U_{ab}$ is dropped across all resistors, using Kirchhoff' |
| - | Aus dem Knotensatz lässt sich der Gesamtwiderstand für parallel geschaltete Widerstände herleiten (<imgref BildNr11> | + | $\large{{U_{ab}}\over{R_1}}+ {{U_{ab}}\over{R_2}}+ ... + {{U_{\rm ab}}\over{R_n}}= {{U_{\rm ab}}\over{R_{\rm eq}}}$ |
| - | Da an allen Widerständen die gleiche Spannung | + | $\rightarrow \large{{{1}\over{R_1}}+ {{1}\over{R_2}}+ ... + {{1}\over{R_n}}= {{1}\over{R_{\rm eq}}} = \sum_{x=1}^{n} {{1}\over{R_x}}}$ |
| - | $\large{{U_{ab}}\over{R_1}}+ {{U_{ab}}\over{R_2}}+ ... + {{U_{ab}}\over{R_n}}= {{U_{ab}}\over{R_{ersatz}}}$ | + | Thus, for resistors connected in parallel, the equivalent conductance |
| - | $\rightarrow \large{{{1}\over{R_1}}+ {{1}\over{R_2}}+ | + | __In general__: the equivalent resistance of a parallel circuit is always smaller than the smallest resistance. |
| - | Bei parallel | + | Especially for two parallel |
| - | __Allgemein gilt__: Der Ersatzwiderstand einer Parallelschaltung ist stets kleiner als der kleinste Widerstand. | + | === Current divider === |
| - | Speziell für zwei parallele Widerstände $R_1$ und $R_2$ gilt: $R_{ersatz}= \large{{R_1 \cdot R_2}\over{R_1 + R_2}}$ | + | < |
| - | + | Derivation of the current divider with examples | |
| - | === Stromteiler === | + | {{youtube> |
| - | + | ||
| - | < | + | |
| - | Herleitung des Stromteilers mit weiteren Betrachtungen | + | |
| - | {{youtube> | + | |
| </ | </ | ||
| - | + | \\ \\ | |
| - | Aus dem Knotensatz lässt sich auch die Stromteiler-Regel herleiten. | + | The current divider rule shows in which way an incoming current on a node will be divided into two outgoing branches. |
| - | Diese besagt, dass sich bei parallel geschalteten Widerständen | + | The rule states that the currents |
| $\large{{I_1}\over{I_g}} = {{G_1}\over{G_g}}$ | $\large{{I_1}\over{I_g}} = {{G_1}\over{G_g}}$ | ||
| $\large{{I_1}\over{I_2}} = {{G_1}\over{G_2}}$ | $\large{{I_1}\over{I_2}} = {{G_1}\over{G_2}}$ | ||
| + | |||
| + | 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~~ | ||
| - | <panel type=" | + | <wrap anchor # |
| + | <panel type=" | ||
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| - | In der Simulation | + | In the simulation |
| - | - Welche Ströme erwarten Sie in den einzelnen Zweigen, wenn die Eingangsspannung von $5V$ auf $3,3V$ gesenkt würde? __Nachdem__ Sie Ihr Ergebnis überlegt hatten, können Sie durch Bewegen des Sliders | + | - What currents would you expect |
| - | - Überlegen Sie sich was passiert wenn Sie den Schalter umlegen __würden__, | + | - 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 |
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | Zwei Widerstände von $18\Omega$ | + | Two resistors of $18~\Omega$ |
| + | Calculate the total resistance and how the currents are split to the branches. | ||
| + | |||
| + | <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~~ | ||
| - | ==== Der Maschensatz (2. Kirchhoffsche Gleichung) | + | ==== Kirchhoff' |
| - | Auch der Maschensatz beschreibt in in der mathematischen Sprache eine praktischer Erfahrung: | + | Also, Kirchhoff' |
| - | Zwischen zwei Punkten | + | Between two points |
| - | Die Potentialdifferenz ist damit insbesondere unabhängig davon auf welchem Weg ein Netzwerk zwischen den zwei Punkten | + | Thus the potential difference is in particular independent of the way a network is traversed between the two points |
| - | Dies lässt sich durch die Betrachtung von Maschen beschreiben. | + | This can be described by considering the meshes. |
| - | <callout icon=" | + | <callout icon=" |
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | In jeder beliebigen Masche eines elektrischen Netzwerks ist die Summe aller Spannungen gleich null (<imgref BildNr12> | + | In any mesh of an electrical circuit, the sum of all voltages is always zero (<imgref BildNr12> |
| $\boxed{U_{1} + U_{2} + ... + U_{n} = \sum_{x=1}^{n} U_x = 0}$ | $\boxed{U_{1} + U_{2} + ... + U_{n} = \sum_{x=1}^{n} U_x = 0}$ | ||
| - | Zur Berechnung muss ein Umlaufsinn festgelegt werden. Diese kann zunächst beliebig gewählt werden. | + | To calculate this, a convention for the loop direction must be specified. Theoretically, |
| - | Es gilt dann aber folgende Festlegung: | + | Independently, |
| - | * Spannungen, deren Spannungspfeile im Umlaufsinn zeigen, werden | + | For example: |
| - | * Spannungen, deren Spannungspfeile gegen Umlaufsinn zeigen, werden | + | * Voltages, whose voltage arrows point __in__ the direction of circulation are __added__ |
| + | * Voltages, whose voltage arrows point __against__ the direction of rotation are __subtracted__ | ||
| </ | </ | ||
| - | === Beweis des Maschensatzes | + | === Proof of Kirchhoff' |
| - | + | ||
| - | Drückt man die Spannungen 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 414: | Zeile 470: | ||
| $U_{41}= \varphi_4 - \varphi_1 $ | $U_{41}= \varphi_4 - \varphi_1 $ | ||
| - | Werden diese Spannungen in die Maschengleichung eingesetzt, so wird | + | If these voltages are added, this leads to Kirchhoff' |
| $U_{12}+U_{23}+U_{34}+U_{41} = 0$ \\ \\ | $U_{12}+U_{23}+U_{34}+U_{41} = 0$ \\ \\ | ||
| - | === Reihenschaltung von Widerständen | + | === Series circuit of resistors |
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Über den Maschensatz lässt sich der Gesamtwiderstand einer Reihenschaltung | + | 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 $ |
| - | Da bei der Reihenschaltung der Strom durch alle Widerstände gleich sein muss - also $I_1 = I_2 = ... = I$ - | + | Since in a series circuit, the current through all resistors must be the same - i.e. $I_1 = I_2 = ... = I$ - it follows that: |
| - | $R_1 + R_2 + ... + R_n = R_{ersatz} = \sum_{x=1}^{n} R_x $ | + | $R_1 + R_2 + ... + R_n = R_{\rm eq} = \sum_{x=1}^{n} R_x $ |
| - | __Allgemein gilt__: Der Ersatzwiderstand einer Reihenschaltung ist stets größer als der größte Widerstand. | + | __In general__: The equivalent resistance of a series circuit is always greater than the greatest resistance. |
| - | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | ==== Application ==== |
| - | <panel type=" | + | === Kelvin-Sensing === |
| - | Gegeben sind drei gleiche Widerstände mit je $20k\Omega$. \\ | + | Often resistors are used to measure a current |
| - | Welche Werte sind durch beliebige Verschaltung von einem bis drei Widerstände realisierbar? | + | 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~~ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | Three equal resistors of $20~k\Omega$ each are given. \\ | ||
| + | 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 unbelasteter und belasteter Spannungsteiler | + | ===== 2.5 Voltage Divider |
| - | ==== Der unbelastete Spannungsteiler ==== | + | < |
| - | < | + | 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 dividers..) |
| - | Herleitung des unbelasteten Spannungsteilers | + | {{youtube> |
| - | {{youtube> | + | |
| </ | </ | ||
| + | |||
| + | ==== The unloaded Voltage Divider ==== | ||
| < | < | ||
| + | === Learning Objectives === | ||
| - | === Ziele === | + | By the end of this section, you will be able to: |
| + | - to distinguish between the loaded and unloaded voltage divider. | ||
| + | - to describe the differences between loaded and unloaded voltage dividers. | ||
| - | Nach dieser Lektion sollten Sie: | + | </ |
| - | - den belasteten und unbelasteten Spannungsteiler auseinanderhalten können. | ||
| - | - die Unterschiede zwischen belasteten und unbelasteten Spannungsteiler beschreiben können. | ||
| - | - | ||
| - | </ | + | 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>potentiometer}}. | ||
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Speziell die Hintereinanderschaltung von zwei Widerständen $R_1$ und $R_2$ soll nun näher betrachtet werden. | + | In <imgref BildNr14> |
| - | Diese Situation tritt in vielen praktischen Anwendungen auf (z.B. Potentiometer). | + | |
| - | In <imgref BildNr14> | + | |
| - | Über die Maschengleichung ergibt sich | + | < |
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| - | $\boxed{ {{U_1}\over{U}} = {{R_1}\over{R_1 + R_2}} }$ | + | Via Kirchhoff' |
| - | Das Verhältnis | + | $\boxed{ {{U_1}\over{U}} = {{R_1}\over{R_1 + R_2}} \rightarrow U_1 = k \cdot U}$ |
| + | |||
| + | The ratio $k={{R_1}\over{R_1 + R_2}}$ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | <panel type=" | + | <panel type=" |
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{url> | + | {{url> |
| </ | </ | ||
| - | In der Simulation | + | In the simulation |
| - | + | - 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. | |
| - | - Welche Spannung '' | + | - First, think about what would happen if you would change the distribution of the resistors by moving the wiper ("intermediate terminal"). \\ You can check your assumption by using the slider at the bottom right of the simulation. |
| - | - Überlegen Sie sich zunächst was passiert wenn Sie durch Verschieben des Schleifers | + | - At which position do you get a $U_{\rm O} = 3.5~\rm V$? |
| - | - Bei welcher Stellung erhalten Sie ein '' | + | |
| </ | </ | ||
| - | |||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Der belastete Spannungsteiler | + | ==== The loaded Voltage Divider |
| + | If - in contrast to the abovementioned, | ||
| - | < | + | < |
| - | < | + | < |
| </ | </ | ||
| - | {{drawio> | + | {{drawio> |
| </ | </ | ||
| - | Wird - im Gegensatz zum obigen, unbelasteten Spannungsteiler - an den Ausgangsklemmen eine Last $R_L$ angeschlossen (<imgref BildNr15> | + | A circuit analysis yields: |
| - | Durch eine Schaltungsanalyse ergibt sich: | + | $ 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 $R_{\rm s} = R_1 + R_2$: |
| - | bzw. an einem Potentiometer mit $k$ und $R_s = R_1 + R_2$: | + | $ U_1 = \LARGE{{k \cdot U} \over { 1 + k \cdot (1-k) \cdot{{R_{\rm s}}\over{R_{\rm L}}} }}$ |
| - | $ U_1 = \LARGE{{{k \cdot U} \over { 1 + k \cdot (1-k) \cdot{{R_s}\over{R_L}} | + | <imgref BildNr65> |
| + | In principle, this is similar to <imgref BildNr14>, | ||
| - | + | < | |
| - | < | + | < |
| - | < | + | |
| </ | </ | ||
| {{drawio> | {{drawio> | ||
| </ | </ | ||
| - | <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 of $10~\rm V$. Thus $k = 0.60$, $R_s = 10~\rm k\Omega$ and $U_1 = 6.0~\rm V$. |
| + | Now this voltage divider is loaded with a load resistor. If this is at $R_{\rm L} = R_1 = 10 ~\rm k\Omega$, $k$ reduces to about $0.48$ and $U_1$ reduces to $4.8~\rm V$ - so the output voltage drops. For $R_{\rm L} = 4.0~\rm k\Omega$, $k$ becomes even smaller to $k=0.375$ and $U_1 = 3.75~\rm V$. If the load $R_{\rm L}$ is only one-tenth of the resistor $R_{\rm s}=R_1 + R_2$, the result is $k = 0.18$ and $U_1 = 1.8~\rm V$. The output voltage of the unloaded voltage divider ($6.0~\rm V$) thus became less than one-third. | ||
| - | 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 $4,8V$ - die Ausgangsspannung bricht | + | What is the practical use of the (loaded) voltage divider? \\ Here are some examples: |
| + | * Voltage dividers are in use for controlling the output of power supply ICs (see [[https:// | ||
| + | * Another " | ||
| ~~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 |
| + | | ||
| + | - Draw the corresponding electrical circuit with the motor connected as an ohmic resistor. | ||
| + | - 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: | ||
| + | - What might a setup with a potentiometer look like that would actually allow you to set a voltage between $0.5~\rm V$ to $4~\rm V$ on the motor? What resistance value should the potentiometer have? | ||
| + | - Build and test your circuit in the simulation below. For an introduction to online simulation, see: [[circuit_design: | ||
| + | - Routing connections can be activated via the menu: '' | ||
| + | - Note that connections can only ever be connected at nodes. A red-marked node (e.g. at the $5 ~\Omega$ resistor) indicates that it is not connected. This could be moved one grid step to the left, as there is a node point there. | ||
| + | - Pressing the ''< | ||
| + | - With a right click on a component it can be copied or values like the resistor can be changed via '' | ||
| - | - Berechnen Sie zunächst den Maximalstrom $I_{M,max}$ des Motors. | + | < |
| - | - Zeichnen Sie die entsprechende elektrische Schaltung mit angeschlossenem Motor als ohmschen Widerstand. | + | < |
| - | - 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). | + | |
| - | - Wie könnte ein Aufbau mit Potentiometer aussehen, mit dem man tatsächlich eine Spannung zwischen $0,5V$ bis $4V$ am Motor einstellen kann? Welchen Widerstandswert muss das Potentiometer haben? | + | |
| - | - Bauen Sie Ihre Schaltung in untenstehender Simulation auf und testen Sie diese. Eine Einführung zur Online-Simulation finden Sie unter: [[elektronische_schaltungstechnik: | + | |
| - | - Das Verlegen von Verbindungen lässt sich über das Menü '' | + | |
| - | - 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. | + | |
| - | - Mit Druck auf die ''< | + | |
| - | - 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 605: | Zeile 723: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | + | ===== 2.6 Circuits with three Connections | |
| - | ===== 2.6 Stern-Dreieck-Schaltung | + | |
| - | + | ||
| - | <WRAP right> | + | |
| - | + | ||
| - | < | + | |
| - | </ | + | |
| - | {{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 |
| + | |||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| - | $R_{ab} = R_{ab}^1 || (R_{ca}^1 + R_{bc}^1) $ \\ | + | $R_{\rm ab} = R_{\rm ab}^1 || (R_{\rm ca}^1 + R_{\rm 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}} $ \\ | + | $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> | ||
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| </ | </ | ||
| - | ===== 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_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 |
| \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> | ||
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| </ | </ | ||
| - | <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> | ||
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| </ | </ | ||
| - | {{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:// | ||
| </ | </ | ||
| + | <panel type=" | ||
| + | |||
| + | More German exercises can be found online on the pages of [[https:// | ||
| + | </ | ||