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electrical_engineering_1:introduction_in_alternating_current_technology [2023/12/20 09:53] mexleadmin |
electrical_engineering_1:introduction_in_alternating_current_technology [2023/12/20 09:55] mexleadmin |
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Zeile 476: | Zeile 476: | ||
Up to now, we used the following formula to represent alternating voltages: | Up to now, we used the following formula to represent alternating voltages: | ||
- | $$u(t)= \sqrt{2} | + | $$u(t)= \sqrt{2} U \cdot \sin (\varphi)$$ |
This is now interpreted as the instantaneous value of a complex vector $\underline{u}(t)$, | This is now interpreted as the instantaneous value of a complex vector $\underline{u}(t)$, | ||
Zeile 504: | Zeile 504: | ||
Generally, from now on not only the voltage will be considered as a phasor, but also the current $\underline{I}$ and derived quantities like the impedance $\underline{X}$. \\ | Generally, from now on not only the voltage will be considered as a phasor, but also the current $\underline{I}$ and derived quantities like the impedance $\underline{X}$. \\ | ||
Therefore, the known properties of complex numbers from Mathematics 101 can be applied: | Therefore, the known properties of complex numbers from Mathematics 101 can be applied: | ||
- | * A multiplication with $j$ equals a phase shift of $+90°$ | + | * A multiplication with $j\omega$ equals a phase shift of $+90°$ |
- | * A multiplication with $-j$ equals a phase shift of $-90°$ | + | * A multiplication with ${{1}\over{j\omega}}$ equals a phase shift of $-90°$ |
===== 6.5 Complex Impedance ===== | ===== 6.5 Complex Impedance ===== | ||
Zeile 556: | Zeile 556: | ||
\\ \\ | \\ \\ | ||
The relationship between ${\rm j}$ and integral calculus should be clear: | The relationship between ${\rm j}$ and integral calculus should be clear: | ||
- | - The derivative of a sinusoidal value - and therefore a phasor - can simply be written as " | + | - The derivative of a sinusoidal value - and therefore a phasor - can simply be written as " |
- | - The integral of a sinusoidal value - and therefore a phasor - can simply be written as " | + | - The integral of a sinusoidal value - and therefore a phasor - can simply be written as " |
\begin{align*} | \begin{align*} | ||
\int {\rm e}^{{\rm j}(\omega t + \varphi_x)} | \int {\rm e}^{{\rm j}(\omega t + \varphi_x)} | ||
- | = {{1}\over{\rm j}} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)} | + | = {{1}\over{\rm j\omega}} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)} |
- | = | + | = -{{\rm j}\over{\omega}} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)} |
\end{align*} | \end{align*} | ||
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