3. Circuit Theory#
In electrical engineering, circuit theory provides the foundation for understanding how resistors, currents, and voltages interact within networks.
By applying Ohm’s and Kirchhoff’s laws, we can analyze series, parallel, and mixed resistor circuits.
3.1 Series Connection of Resistors#
In a series connection, resistors are connected one after another, so that the same current flows through all components.
Properties of a Series Circuit#
Current: identical through all resistors
$$ I = I_1 = I_2 = I_3 = \text{constant} $$Voltage: the total voltage equals the sum of all partial voltages
$$ U = U_1 + U_2 + U_3 $$
Since \( U = I \times R \):
- Equivalent Resistance:
$$ R_{eq} = R_1 + R_2 + R_3 $$
Voltage Divider Rule#
In a series connection, each resistor has a voltage proportional to its resistance.
$$ U_1 : U_2 : U_3 = R_1 : R_2 : R_3 $$and for any resistor:
$$ U_i = U_{total} \times \frac{R_i}{R_{eq}} $$3.2 Parallel Connection of Resistors#
In a parallel connection, all resistors are connected to the same two nodes.
Properties of a Parallel Circuit#
Voltage: identical across all branches
$$ U = U_1 = U_2 = U_3 = \text{constant} $$Current: the total current equals the sum of the branch currents
$$ I = I_1 + I_2 + I_3 $$
Since \( I = U / R \):
Equivalent Conductance:
$$ G_{eq} = G_1 + G_2 + G_3 $$Equivalent Resistance:
$$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} $$
For two resistors:
🔸 Note: The equivalent resistance of a parallel circuit is always smaller than the smallest branch resistance.
Current Divider Rule#
The branch currents are inversely proportional to the resistances:
$$ I_1 : I_2 : I_3 = \frac{1}{R_1} : \frac{1}{R_2} : \frac{1}{R_3} $$or in direct formula form:
$$ I_i = I_{total} \times \frac{R_{eq}}{R_i} $$3.3 Mixed Circuits#
In mixed circuits, resistors are both in series and in parallel.
To find the total resistance, the circuit is simplified step by step.
Example#
Given:
\( U = 12\,V \), \( R_1 = 470\,\Omega \), \( R_2 = 1.5\,k\Omega \), \( R_3 = 820\,\Omega \)
Solution:
Combine \( R_2 \) and \( R_3 \) in parallel:
$$ R_{23} = \frac{R_2 R_3}{R_2 + R_3} = 530.6\,\Omega $$Add \( R_1 \) in series:
$$ R_{eq} = R_1 + R_{23} = 1000.6\,\Omega $$Calculate total current:
$$ I = \frac{U}{R_{eq}} = 12 / 1000.6 = 12\,mA $$Find voltage drops:
$$ U_1 = I R_1 = 5.64\,V, \quad U_{23} = 6.36\,V $$Branch currents:
$$ I_2 = U_{23}/R_2 = 4.24\,mA, \quad I_3 = U_{23}/R_3 = 7.76\,mA $$
3.4 Voltage Divider#
A voltage divider is a series circuit used to generate a smaller voltage from a larger one.
Unloaded Voltage Divider#
$$ U_2 = U \times \frac{R_2}{R_1 + R_2} $$Example:
\( U = 24\,V, R_1 = 470\,\Omega, R_2 = 120\,\Omega \)
Loaded Voltage Divider#
When a load \( R_V \) is connected parallel to \( R_2 \), the effective resistance changes:
$$ R_{2V} = \frac{R_2 R_V}{R_2 + R_V} $$and the new output voltage becomes:
$$ U_{2V} = U \times \frac{R_{2V}}{R_1 + R_{2V}} $$Example:
\( U = 12\,V, R_1 = 120\,\Omega, R_2 = 120\,\Omega, R_V = 120\,\Omega \)
3.5 Measuring Resistance Using Current and Voltage#
To determine an unknown resistance, both current and voltage are measured simultaneously.
Voltage-Accurate Method#
The ammeter measures the total current, including the current through the voltmeter.
Example:
\( U_V = 3V, I_A = 10.3mA, R_V = 10k\Omega \)
Current-Accurate Method#
Here, the voltmeter measures the voltage across both the resistor and the ammeter.
Example:
\( U_V = 10V, I_A = 50mA, R_A = 1\Omega \)
3.6 Source with Internal Resistance#
Real voltage sources exhibit an internal resistance (Rᵢ) that causes voltage drop when current flows.
Key Relationships#
Open Circuit (no load):
$$ U_{open} = U_q $$Loaded Condition:
$$ U = U_q - I R_i $$Short Circuit:
$$ I_K = \frac{U_q}{R_i}, \quad U = 0 $$
Example:
A battery has \( U_q = 4.5V \) and \( U = 4.2V \) when \( I = 0.15A \).
Find \( R_i \):
💡 Summary:
Circuit theory provides the essential tools for analyzing all electrical systems.
Understanding how series and parallel resistors behave allows us to calculate voltage drops, currents, and losses precisely — both in theory and practice.