Komplexe Zahlen Wiederholung

1. 1. Complex Number Representations

A complex number  can be expressed in two main forms:

Rectangular Form

$z = x + iy$ where:

• is the real part.
• is the imaginary part.

Polar Form

$z = r, e^{i\varphi}$ with:

• (the amplitude or modulus),
• (the phase angle).

Conversion:

• Rectangular to Polar:

$r = \sqrt{x^2+y^2}, \quad \varphi = \operatorname{atan2}(y,x)$

• Polar to Rectangular:

$x = r\cos\varphi, \quad y = r\sin\varphi$

2. 2. Euler’s Formula

Euler’s formula provides the bridge between exponential and trigonometric representations:

$e^{i\varphi} = \cos\varphi + i\sin\varphi$

This is particularly useful when expressing sinusoidal functions as complex exponentials.

3. 3. Operations with Complex Numbers

Multiplication

For two complex numbers  and :

$z_1 z_2 = r_1 r_2, e^{i(\varphi_1+\varphi_2)}$

• Modulus: Multiply the amplitudes: .
• Phase: Add the phase angles: .

Division

For division, if :

$\frac{z_1}{z_2} = \frac{r_1}{r_2}, e^{i(\varphi_1-\varphi_2)}$

• Modulus: Divide the amplitudes: .
• Phase: Subtract the phase angles: .

Addition and Subtraction

These operations are performed in rectangular form:

$z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)$

It’s often easier to convert to rectangular form when summing or subtracting complex numbers.

4. 4. Application to AC Circuit Analysis

Phasors

A sinusoidal signal, for example,

$v(t) = \hat{V} \cos(\omega t + \varphi)$

can be represented as a phasor:

$\tilde{V} = \hat{V}, e^{i\varphi}$

• Amplitude:  (the peak voltage).
• Phase:  (the phase shift).

Phasor Operations

• Addition: When adding sinusoids of the same frequency, convert them to phasors and add:

$\tilde{V}_{\text{total}} = \tilde{V}_1 + \tilde{V}_2$

• Multiplication & Division: Useful for impedance calculations, where impedances are represented as complex numbers.

Determining Amplitudes and Phases

Once a result is expressed in polar form:

• Amplitude (Magnitude):

$r = |z| = \sqrt{x^2+y^2}$

• Phase (Angle):

$\varphi = \operatorname{atan2}(y,x)$

Note: Use the (\operatorname{atan2}) function to correctly account for the quadrant in which the complex number lies.