06 Albach Kapitel 3.6 - 3.10

Material: HS25_NuS1_Kapitel_3_Einfache_elektrische_Netzwerke_Teil_2von2.pdf

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Schlüsselkonzepte:

Reale Spannungs- und Stromquellen und deren Ersatzschaltbilder (Kap. 3.6)
Leistungsanpassung (Kap. 3.7.2)
Stern-Dreieck-Umwandlung (Kap. 3.8.2)
Überlagerungsprinzip (Kap 3.9)
Analyse von umfangreichen Netzwerken (Vollständiger Baum, Auftrennen der Maschen) (Kap. 3.10, S. 156-164)

Lernziele:

Spannungs- und Stromquellen ineinander umrechnen
die Verbraucherleistung bei vorgegebener Quelle maximieren
Wirkungsgradberechnungen durchführen sowie
umfangreiche Gleichstromnetzwerke mit unterschiedlichen Methoden analysieren.

3.6 Real Voltage and Current Sources (Reale Spannungs- und Stromquellen)

Ideal sources maintain their value regardless of load. Real sources, however, have internal limitations, which are modeled using an internal resistance ( $R_{i}$ ).

Real Voltage Source: Modeled as an ideal voltage source $U_{0}$ in series with an internal resistance $R_{i}$ 1. $U_{0}$ is the open-circuit voltage (Leerlaufspannung), $U_{L}$ . The voltage at the terminals drops as the current $I$ increases: $U = U_{0} - I \cdot R_{i}$ .
Real Current Source: Modeled as an ideal current source $I_{0}$ in parallel with an internal resistance $R_{i}$ 2. $I_{0}$ is the short-circuit current (Kurzschlussstrom), $I_{K}$ . The current delivered to the terminals decreases as the voltage $U$ increases: $I = I_{0} - U / R_{i}$ .
Source Transformation: A real voltage source can be converted into an equivalent real current source (and vice-versa) if they have the same internal resistance ( $R_{i}$ ) and follow the relation $U_{0} = I_{0} \cdot R_{i}$ 3.

3.7 Interactions Between Source and Consumer (Wechselwirkungen zwischen Quelle und Verbraucher)

3.7.2 Power Matching (Leistungsanpassung)

This concept defines the condition for transferring the maximum possible power from a real source to a load.

Condition: Maximum power is delivered to a load $R_{L}$ when the load resistance is equal to the internal resistance of the source: ** $R_{L} = R_{i}$ **4.
Maximum Power: The maximum power transferred is $P_{ma x} = U_{0}^{2} / (4 R_{i})$ 5.

3.7.3 Efficiency (Wirkungsgrad)

Efficiency ( $η$ ) measures how much of the total power generated by the source ( $P_{g es}$ ) is successfully delivered to the load ( $P_{L}$ ), as opposed to being lost as heat in the internal resistance ( $P_{i}$ ).

Formula: $η = P_{L} / P_{g es} = R_{L} / (R_{i} + R_{L})$ .
At Power Matching: When $R_{L} = R_{i}$ (for maximum power transfer), the efficiency is $η = 50%$ . Half the power is delivered to the load, and the other half is lost in the source itself.
High Efficiency: High efficiency ( $η \to 100%$ ) is achieved when the load resistance is much larger than the internal resistance ( $R_{L} ≫ R_{i}$ ), but this comes at the cost of transferring very little total power.

3.8 Network Transformations (Netzwerkumwandlungen)

These are techniques to simplify complex parts of a circuit into a more basic equivalent.

3.8.1 Equivalent Two-Terminal Networks (Ersatzzweipole)

Any linear two-terminal network can be simplified into one of two equivalent forms:

Thévenin’s Theorem: Replaces the network with an equivalent voltage source, consisting of:
1. An ideal voltage source $U_{t h}$ (the open-circuit voltage $U_{L}$ measured at the terminals).
2. A series resistor $R_{t h}$ (the internal resistance $R_{i}$ seen from the terminals with all sources turned off)6.
Norton’s Theorem: Replaces the network with an equivalent current source, consisting of:
1. An ideal current source $I_{N}$ (the short-circuit current $I_{K}$ measured at the terminals).
2. A parallel resistor $R_{N}$ (which is identical to $R_{t h}$ )7.
Transformation: The two forms are equivalent, following the standard source transformation rule: $U_{t h} = I_{N} \cdot R_{t h}$ 8.

3.8.2 Star-Delta Transformation (Stern-Dreieck-Umwandlung)

This allows conversion between “Star” (T-network) and “Delta” ( $Π$ -network) resistor configurations, which cannot be simplified by simple series/parallel rules 9.

Delta-to-Star: $R_{1} = \frac{R _{12} R _{31}}{R _{12} + R _{23} + R _{31}}$ (and similarly for $R_{2}, R_{3}$ )10.
Star-to-Delta: $R_{12} = \frac{R _{1} R _{2} + R _{2} R _{3} + R _{3} R _{1}}{R _{3}}$ (and similarly for $R_{23}, R_{31}$ )11.

3.9 The Superposition Principle (Das Überlagerungsprinzip)

For any linear network with multiple independent sources, the total current or voltage in any branch can be found by calculating the effect of each source one at a time and then summing the results algebraically12.

Procedure:
1. Select one source to keep “on.”
2. Turn “off” all other independent sources:
  - Voltage sources are set to 0V, becoming a short circuit13.
  - Current sources are set to 0A, becoming an open circuit14.
3. Calculate the desired current or voltage from that one source.
4. Repeat for all other sources.
5. Sum the individual contributions (paying attention to signs) to find the total.

3.10 Analysis of Large Networks (Analyse umfangreicher Netzwerke)

These are systematic methods for solving any network using Kirchhoff’s laws.

3.10.1 Mesh Current Method (Maschenstromverfahren)

Basis: Kirchhoff’s Voltage Law (KVL).
Procedure:
1. Identify all independent meshes (loops) in the circuit.
2. Assign a fictitious “mesh current” (e.g., $I_{A}, I_{B}, ...$ ) circulating in each mesh 15.
3. Write a KVL equation for each mesh. The voltage drop across a resistor is $R \cdot I_{m es h}$ . Resistors shared by meshes will have voltage drops from multiple mesh currents (e.g., $R \cdot (I_{A} - I_{B})$ ).
4. Solve the resulting system of linear equations for the unknown mesh currents.
5. The actual current in any branch is the algebraic sum of the mesh currents flowing through it.

3.10.2 Node-Voltage Method (Knotenpotentialverfahren)

Basis: Kirchhoff’s Current Law (KCL).
Procedure:
1. Choose one node as the reference (ground, $φ = 0$ ).
2. Assign an unknown node voltage (e.g., $φ_{1}, φ_{2}, ...$ ) to all other essential nodes16.
3. Write a KCL equation for each non-reference node.
4. Express each current leaving the node in terms of the node voltages using Ohm’s Law (e.g., $I_{12} = (φ_{1} - φ_{2}) / R_{12}$ )17.
5. Solve the resulting system of linear equations for the unknown node voltages.

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