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Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
electrical_engineering_1:introduction_in_alternating_current_technology [2023/07/17 02:41] mexleadmin [6.5.2 Application on pure Loads] |
electrical_engineering_1:introduction_in_alternating_current_technology [2023/12/20 09:55] (aktuell) mexleadmin |
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Zeile 1: | Zeile 1: | ||
- | ====== 6. Introduction to Alternating Current Technology ====== | + | ====== 6 Introduction to Alternating Current Technology ====== |
Up to now, we had analyzed DC signals (chapters 1. - 4.) and abrupt voltage changes for (dis)charging capacitors (chapter 5.). In households, we use alternating voltage (AC) instead of a constant voltage (DC). This is due to at least three main facts | Up to now, we had analyzed DC signals (chapters 1. - 4.) and abrupt voltage changes for (dis)charging capacitors (chapter 5.). In households, we use alternating voltage (AC) instead of a constant voltage (DC). This is due to at least three main facts | ||
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*} | ||
</ | </ | ||
Zeile 817: | Zeile 817: | ||
in the following, some of the numbers are given. | in the following, some of the numbers are given. | ||
- | Calculate the RMS value of the missing | + | Calculate the RMS value of the missing |
- $I_R = 3~\rm A$, $I_L = 1 ~\rm A$, $I_C = 5 ~\rm A$, $I=?$ | - $I_R = 3~\rm A$, $I_L = 1 ~\rm A$, $I_C = 5 ~\rm A$, $I=?$ | ||
- $I_R = ?$, $I_L = 1.2~\rm A$, $I_C = 0.4~\rm A$, $I=1~\rm A$ | - $I_R = ?$, $I_L = 1.2~\rm A$, $I_C = 0.4~\rm A$, $I=1~\rm A$ | ||
Zeile 824: | Zeile 824: | ||
<panel type=" | <panel type=" | ||
- | The following two currents with similar frequencies, | + | The following two currents with similar frequencies, |
* $i_1(t) = \sqrt{2} \cdot 2 ~A \cdot \cos (\omega t + 20°)$ | * $i_1(t) = \sqrt{2} \cdot 2 ~A \cdot \cos (\omega t + 20°)$ | ||
* $i_2(t) = \sqrt{2} \cdot 5 ~A \cdot \cos (\omega t + 110°)$ | * $i_2(t) = \sqrt{2} \cdot 5 ~A \cdot \cos (\omega t + 110°)$ | ||
Zeile 832: | Zeile 832: | ||
<panel type=" | <panel type=" | ||
Two complex impedances $\underline{Z}_1$ and $\underline{Z}_2$ are investigated. | Two complex impedances $\underline{Z}_1$ and $\underline{Z}_2$ are investigated. | ||
- | The resulting impedance for a series circuit is $60~\Omega$. | + | The resulting impedance for a series circuit is |
- | The resulting impedance for a parallel circuit is $25~\Omega$. | + | The resulting impedance for a parallel circuit is $25~\Omega + \rm j \cdot 0 ~\Omega $. |
What are the values for $\underline{Z}_1$ and $\underline{Z}_2$? | What are the values for $\underline{Z}_1$ and $\underline{Z}_2$? | ||
+ | |||
+ | # | ||
+ | It's a good start to write down all definitions of the given values: | ||
+ | * the given values for the series circuit ($\square_\rm s$) and the parallel circuit ($\square_\rm p$) are: \begin{align*} R_\rm s = 60 ~\Omega , \quad X_\rm s = 0 ~\Omega \\ R_\rm p = 25 ~\Omega , \quad X_\rm p = 0 ~\Omega \\ \end{align*} | ||
+ | * the series circuit and the parallel circuit results into: \begin{align*} | ||
+ | * the unknown values of the two impedances are: \begin{align*} \underline{Z}_1 = R_1 + {\rm j}\cdot X_1 \tag{3} \\ \underline{Z}_2 = R_2 + {\rm j}\cdot X_2 \tag{4} \\ \end{align*} | ||
+ | |||
+ | Based on $(1)$,$(3)$ and $(4)$: | ||
+ | \begin{align*} | ||
+ | R_\rm s & | ||
+ | &= R_1 + {\rm j}\cdot X_1 &&+ R_2 + {\rm j}\cdot X_2 \\ | ||
+ | \rightarrow 0 & | ||
+ | \end{align*} | ||
+ | Real value and imaginary value must be zero: | ||
+ | \begin{align*} | ||
+ | R_1 &= R_{\rm s} - R_2 \tag{5} \\ | ||
+ | X_1 &= - X_2 \tag{6} | ||
+ | \end{align*} | ||
+ | |||
+ | Based on $(2)$ with $R_\rm s = \underline{Z}_1 + \underline{Z}_2$ | ||
+ | \begin{align*} | ||
+ | R_{\rm p} &= {{\underline{Z}_1 \cdot \underline{Z}_2}\over{\underline{Z}_1 + \underline{Z}_2}} \\ | ||
+ | & | ||
+ | R_{\rm p} \cdot R_{\rm s} & | ||
+ | & | ||
+ | & | ||
+ | \end{align*} | ||
+ | |||
+ | Substituting $R_1$ and $X_1$ based on $(5)$ and $(6)$: | ||
+ | \begin{align*} | ||
+ | R_{\rm p} \cdot R_{\rm s} & | ||
+ | \rightarrow 0 & | ||
+ | \end{align*} | ||
+ | |||
+ | Again real value and imaginary value must be zero: | ||
+ | \begin{align*} | ||
+ | 0 & | ||
+ | & | ||
+ | \rightarrow | ||
+ | |||
+ | 0 &= R_{\rm s} R_2 - R_2^2 + X_2^2 - R_{\rm p} \cdot R_{\rm s} \\ | ||
+ | &= R_{\rm s} ({{1}\over{2}} R_{\rm s}) - ({{1}\over{2}} R_{\rm s})^2 - X_2^2 - R_{\rm p} \cdot R_{\rm s} \\ | ||
+ | &= {{1}\over{4}} R_{\rm s}^2 + X_2^2 - R_{\rm p} \cdot R_{\rm s} \\ | ||
+ | \rightarrow | ||
+ | |||
+ | \end{align*} | ||
+ | |||
+ | The concluding result is: | ||
+ | \begin{align*} | ||
+ | (5)+(7): \quad R_1 &= {{1}\over{2}} R_{\rm s} \\ | ||
+ | (7): \quad R_2 &= {{1}\over{2}} R_{\rm s} \\ | ||
+ | (6)+(8) | ||
+ | (8): \quad X_2 &= \pm \sqrt{R_{\rm p} \cdot R_{\rm s} - {{1}\over{4}} R_{\rm s}^2 } | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | \begin{align*} | ||
+ | R_1 &= 30~\Omega \\ | ||
+ | R_2 &= 30~\Omega \\ | ||
+ | X_1 &= \mp \sqrt{600}~\Omega \approx \mp 24.5~\Omega \\ | ||
+ | X_2 &= \pm \sqrt{600}~\Omega \approx \pm 24.5~\Omega \\ | ||
+ | \end{align*} | ||
+ | # | ||
+ | |||
</ | </ | ||