Gegeben sind die beiden gleichfrequenten Wechselspannungen \(u_{1}(t)\) und \(u_{2}(t)\). Bestimmen Sie die durch Superposition entstehende resultierende Wechselspannung mit Hilfe der komplexen Rechnung \(\left(\omega=314 \mathrm{~s}^{-1}\right.\) ): a) \(u_{1}=100 \mathrm{~V} \cdot \sin (\omega t)\) b) \(u_{1}=-50 \mathrm{~V} \cdot \sin (\omega t)\) $$ u_{2}=150 \mathrm{~V} \cdot \cos \left(\omega t-\frac{\pi}{4}\right) \quad u_{2}=200 \mathrm{~V} \cdot \sin \left(\omega t+\frac{\pi}{3}\right) $$

The complex exponential form is a powerful way to represent sinusoidal functions using complex numbers. It leverages the inherent properties of exponential functions to simplify calculations.
The complex exponential form of a sinusoidal voltage can be represented as follows:
\ u(t) = U \times e^{j\theta} \ Here, \( U \) is the amplitude and \( \theta \) is the phase angle.
This form allows us to easily perform operations like differentiation and integration. Additionally, it helps in combining multiple sinusoidal signals efficiently.
For example, a cosine function \(u(t) = A \times \text{cos}(\theta)\) can be translated into its complex exponential form using Euler's formula:
\(u(t) = A \times \text{Re}( e^{j(\theta)} )\).

The superposition principle in electrical engineering states that the total response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.
This principle is crucial when dealing with linear systems, such as circuits with resistors, inductors, and capacitors.
In this problem, we have multiple voltage sources: \( u_{1}(t) \) and \( u_{2}(t) \).
To find the resultant voltage, we first transform each source into its complex form, perform the necessary mathematical operations, and then sum the results.
Afterward, we can revert to the time domain by taking the real part of the inverse complex form, providing us the actual time-domain signal.

Euler's formula is a fundamental bridge between trigonometric functions and complex exponentials.
It is represented as: \ e^{j\theta} = \text{cos}(\theta) + j\text{sin}(\theta) \ This relation simplifies the handling of sinusoidal functions in complex form.
This formula allows us to interchange between the exponential and trigonometric representations, making it highly versatile for electrical engineering applications.
For instance, if we have \( \text{cos}(\theta) \), we can write it as: \( \text{cos}(\theta) = \text{Re}( e^{j\theta} )\).
Similarly, \( \text{sin}(\theta)= \text{Im}( e^{j\theta} )\).
In our exercise, utilizing Euler’s formula helps convert sine and cosine terms into exponential expressions, which simplifies the algebra involved in combining voltages.

Phasor representation is a simplified way to analyze and manipulate sinusoidal signals.
A phasor is a complex number representing the amplitude and phase of a sinusoid.
For example, a sinusoidal voltage \(u(t) = U \times \text{cos}(\theta)\) can be represented as a phasor \( U \times e^{j\theta} \).
The phasors can be plotted on a complex plane with real and imaginary axes. This makes it much easier to perform calculations like addition, subtraction, multiplication, and division.
Once the operations are performed in the phasor domain, the result can be converted back to the time domain.
In our exercise, after converting the voltages to phasors, we can easily sum them to find the resultant voltage. Converting the result back to a time-domain signal provides the solution to the original problem.