05 Albach Kapitel 3.1 - 3.5

Material: HS25_NuS1_Kapitel_3_Einfache_elektrische_Netzwerke_Woche_5.pdf

Transclude of Belasteter-Spannungsteiler_2_sweep_R.asc

Transclude of Belasteter-Spannungsteiler_sweep_R.asc

Transclude of Belasteter-Spannungsteiler.asc

Transclude of Unbelasteter-Spannungsteiler.asc

3.1 Counting Arrows (Zählpfeile)

To analyze networks, we must define the direction for current and voltage. Counting arrows are used for this purpose2.

Current ( $I$ ): The arrow indicates the direction of positive current flow. This is a scalar quantity33333333.
Voltage ( $U$ ): The arrow indicates the direction of potential drop, pointing from higher potential to lower potential 4. $U_{12} = φ_{e} (P_{1}) - φ_{e} (P_{2})$ 5.

3.2 Voltage and Current Sources (Spannungs- und Stromquellen)

Sources are the active elements in a network.

Ideal Voltage Source: Maintains a constant voltage $U_{q}$ regardless of the current $I$ flowing through it. An ideal voltage source cannot be short-circuited, as this would imply infinite current666666666666666.
Ideal Current Source: Maintains a constant current $I_{q}$ regardless of the voltage $U$ across it. An ideal current source cannot be left in an open circuit, as this would imply infinite voltage777777777.

3.3 Counting Arrow Systems (Zählpfeilsysteme)

The relationship between the $U$ and $I$ arrows defines how power is calculated:

Consumer System (Verbraucherzählpfeilsystem): $U$ and $I$ arrows point in the same direction. This is used for passive components like resistors. Power $P = U \cdot I$ is consumed8.
Generator System (Erzeugerzählpfeilsystem): $U$ and $I$ arrows point in opposite directions. This is used for active components (sources) that deliver power9.

3.4 Kirchhoff’s Laws (Die Kirchhoff’schen Gleichungen)

These two fundamental laws govern all DC networks and are derived from the properties of stationary fields101010101010101010.

1. Node Rule (Knotenregel) - KCL

This law is based on the conservation of charge ( $\oint J \cdot d A = 0$ ) 1111111111.

Concept: At any node (junction point), the sum of all currents flowing in must equal the sum of all currents flowing _out_12121212.
Formula: $\sum_{v} I_{v} = 0$ 13.
Application: In a simple series circuit, the current is the same through every component14141414.

2. Mesh Rule (Maschenregel) - KVL

This law is based on the conservative nature of the $E$ -field ( $\oint E \cdot d s = 0$ )1515151515.

Concept: The sum of all voltage drops around any closed loop (mesh) must be zero 161616.
Formula: $\sum_{v} U_{v} = 0$ 17.
Sign Convention: When tracing a loop, voltages are added if their arrow matches the loop direction and subtracted if they oppose it181818181818181818.

3.5 Simple Resistor Networks (Einfache Widerstandsnetzwerke)

Series and Parallel Combinations

Series (Reihenschaltung): Resistors are connected end-to-end.
- Current $I$ is the same through all resistors19.
- Total resistance is the sum: $R_{g es} = R_{1} + R_{2} + ... + R_{n}$ 20.
Parallel (Parallelschaltung): Resistors are connected across the same two nodes.
- Voltage $U$ is the same across all resistors 21212121.
- The total conductance $G_{g es}$ (where $G = 1/ R$ ) is the sum: $G_{g es} = G_{1} + G_{2} + ... + G_{n}$ 22222222.
- For resistance: $1/ R_{g es} = 1/ R_{1} + 1/ R_{2} + ... + 1/ R_{n}$ 23232323.
- For two resistors: $R_{g es} = (R_{1} \cdot R_{2}) / (R_{1} + R_{2})$ 24242424.
- For $n$ identical resistors: $R_{g es} = R / n$ 25252525.

3.5.1 Voltage Divider (Spannungsteiler)

This is a series circuit used to “tap” a fraction of the total voltage262626262626262626.

Rule: The voltage drops across series resistors are proportional to their resistance.
Formula: For two resistors $R_{1}$ and $R_{2}$ in series, the voltage across $R_{2}$ is:

$U_{2} = U_{g es} \cdot \frac{R _{2}}{R _{1} + R _{2}} [cite: 6606]$

3.5.2 Loaded Voltage Divider (Belasteter Spannungsteiler)

When a load (like a voltmeter or another circuit) $R_{V}$ is connected to the output of a voltage divider, this load is in parallel with $R_{2}$ 27272727.

Effect: The load $R_{V}$ draws current, changing the total resistance of the lower part to $R_{p a r} = (R_{2} \cdot R_{V}) / (R_{2} + R_{V})$ 28.
Result: The output voltage drops to $U_{o u t} = U_{g es} \cdot R_{p a r} / (R_{1} + R_{p a r})$ 29. The loading effect is minimal if the load resistance is very high ( $R_{V} ≫ R_{2}$ )30.

3.5.3 Voltmeter Range Extension (Messbereichserweiterung)

To measure a voltage $U$ that is higher than a voltmeter’s maximum $U_{ma x}$ , a series resistor (Vorwiderstand) $R_{S}$ is added31313131. The meter and $R_{S}$ form a voltage divider, with $R_{S}$ chosen to absorb the excess voltage 32323232.

3.5.4 Current Divider (Stromteiler)

This is a parallel circuit where the total current $I_{g es}$ splits between the branches33.

Rule: The current splits inversely proportional to the branch resistances (or proportionally to the conductances).
Formula: For two resistors $R_{1}$ and $R_{2}$ in parallel, the current through $R_{2}$ is:

$I_{2} = I_{g es} \cdot \frac{R _{1}}{R _{1} + R _{2}} [cite: 6611]$

3.5.6 Resistance Measurement (Widerstandsmessung)

Measuring resistance $R$ with a voltmeter (internal resistance $R_{V}$ ) and an ammeter (internal resistance $R_{A}$ ) involves two main setups, each with a potential error source34343434.

“Correct Voltage” Setup (Spannungsrichtige Schaltung): The voltmeter is directly across $R$ . The ammeter measures $I_{A} = I_{R} + I_{V}$ (current for $R$ + current for $V$ ).
- Calculation: $R = U_{V} / I_{R} = U_{V} / (I_{A} - I_{V})$ 3535.
- Error: The ammeter reads high. This setup is accurate for small $R$ (when $R ≪ R_{V}$ , so $I_{V}$ is negligible) 36363636.
“Correct Current” Setup (Stromrichtige Schaltung): The ammeter is directly in series with $R$ . The voltmeter measures $U_{V} = U_{R} + U_{A}$ (voltage for $R$ + voltage for $A$ ).
- Calculation: $R = U_{R} / I_{A} = (U_{V} - U_{A}) / I_{A}$ 3737.
- Error: The voltmeter reads high. This setup is accurate for large $R$ (when $R ≫ R_{A}$ , so $U_{A}$ is negligible) 38383838.

~/Dash

Explorer