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Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
electrical_engineering_1:introduction_in_alternating_current_technology [2023/03/27 09:08] mexleadmin |
electrical_engineering_1:introduction_in_alternating_current_technology [2023/12/20 09:55] (aktuell) mexleadmin |
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- | ====== 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 | ||
- | - Often the voltage given by the **power plant is AC**. This is true for example in all power plants which use electric generators. In these, the mechanical energy of a rotating system is transformed into electric energy | + | - Often the voltage given by the **power plant is AC**. This is true for example in all power plants which use electric generators. In these, the mechanical energy of a rotating system is transformed into electric energy |
- For long-range power transfer the power losses $P_{\rm loss}$ can be reduced by reducing the currents $I$ since $P_{\rm loss}=R\cdot I^2$. Therefore, for constant power transfer, the voltage has to be increased. This is much easier done with AC voltages: **AC enables the transformation of a lower voltage to a higher** by the use of alternating magnetic fields in a transformer. | - For long-range power transfer the power losses $P_{\rm loss}$ can be reduced by reducing the currents $I$ since $P_{\rm loss}=R\cdot I^2$. Therefore, for constant power transfer, the voltage has to be increased. This is much easier done with AC voltages: **AC enables the transformation of a lower voltage to a higher** by the use of alternating magnetic fields in a transformer. | ||
- AC signals have **at least one more value** which can be used for understanding the situation of the source or load. This simplifies the power and load management in a complex power network. | - AC signals have **at least one more value** which can be used for understanding the situation of the source or load. This simplifies the power and load management in a complex power network. | ||
Zeile 12: | Zeile 12: | ||
Besides the applications in power systems AC values are also important in communication engineering. Acoustic and visual signals like sound and images can often be considered as wavelike AC signals. Additionally, | Besides the applications in power systems AC values are also important in communication engineering. Acoustic and visual signals like sound and images can often be considered as wavelike AC signals. Additionally, | ||
- | In order to understand these systems a bit more, we will start this chapter with a first introduction to AC systems. | + | To understand these systems a bit more, we will start this chapter with a first introduction to AC systems. |
< | < | ||
Zeile 106: | Zeile 106: | ||
</ | </ | ||
- | In order to analyze AC signals more, often different types of averages are taken into account. The most important values are: | + | To analyze AC signals more, often different types of averages are taken into account. The most important values are: |
- the arithmetic mean $\overline{X}$ | - the arithmetic mean $\overline{X}$ | ||
- the rectified value $\overline{|X|}$ | - the rectified value $\overline{|X|}$ | ||
Zeile 147: | Zeile 147: | ||
Without limiting the generality, we use $\varphi=0$ and $t_0 = 0$ | Without limiting the generality, we use $\varphi=0$ and $t_0 = 0$ | ||
\begin{align*} | \begin{align*} | ||
- | \overline{|X|} &= {{1}\over{T}}\cdot \int_{t=0 | + | \overline{|X|} &= {{1}\over{T}}\cdot \int_{t=0 |
\end{align*} | \end{align*} | ||
Since $sin(\omega t)\geq0$ for $t\in [0,\pi]$, the integral can be changed and the absolute value bars can be excluded like the following | Since $sin(\omega t)\geq0$ for $t\in [0,\pi]$, the integral can be changed and the absolute value bars can be excluded like the following | ||
\begin{align*} | \begin{align*} | ||
- | \overline{|X|} | + | \overline{|X|} |
- | | + | &= 2 \cdot {{1}\over{T}}\cdot [-\hat{X}\cdot {{T}\over{2\pi}}\cdot |
- | | + | &= 2 \cdot {{1}\over{T}}\cdot {{T}\over{2\pi}}\cdot |
- | | + | &= {{1}\over{\pi}}\cdot \hat{X} \cdot [1+1] \\ |
- | \boxed{\overline{|X|} = {{2}\over{\pi}}\cdot \hat{X} \approx 0.6366 \cdot \hat{X}}\\ | + | \boxed{\overline{|X|} |
+ | = {{2}\over{\pi}}\cdot \hat{X} \approx 0.6366 \cdot \hat{X}}\\ | ||
\end{align*} | \end{align*} | ||
</ | </ | ||
Zeile 175: | Zeile 176: | ||
\begin{align*} | \begin{align*} | ||
- | | + | |
- | U_{DC} \cdot I_{\rm DC} &= {{1}\over{T}} \int_{0}^{T} u(t) \cdot i(t) {\rm d}t \\ | + | U_{DC} \cdot I_{\rm DC} |
- | | + | |
- | | + | |
- | \rightarrow I_{\rm DC} &= \sqrt{{{1}\over{T}} \int_{0}^{T} i^2(t) {\rm d}t} | + | \rightarrow |
\end{align*} | \end{align*} | ||
Zeile 204: | Zeile 205: | ||
\begin{align*} | \begin{align*} | ||
X & | X & | ||
- | & | + | & |
- | & | + | & |
& | & | ||
& | & | ||
Zeile 231: | Zeile 232: | ||
The following simulation shows the different values for averaging a rectangular, | The following simulation shows the different values for averaging a rectangular, | ||
- | Be aware that one has to wait for a full period | + | Be aware that one has to wait for a full period to see the resulting values on the right outputs of the average generating blocks. |
< | < | ||
Zeile 295: | Zeile 296: | ||
Now, we insert the functions representing the instantaneous signals and calculate the derivative: | Now, we insert the functions representing the instantaneous signals and calculate the derivative: | ||
\begin{align*} | \begin{align*} | ||
- | | + | |
- | & | + | & |
- | {I}\cdot \sin(\omega t + \varphi_i) | + | {I}\cdot \sin(\omega t + \varphi_i) |
\end{align*} | \end{align*} | ||
Zeile 309: | Zeile 310: | ||
\omega t + \varphi_i &= \omega t + \varphi_u + {{1}\over{2}}\pi \\ | \omega t + \varphi_i &= \omega t + \varphi_u + {{1}\over{2}}\pi \\ | ||
| | ||
- | \varphi_u -\varphi_i & | + | \varphi_u -\varphi_i & |
\end{align*} | \end{align*} | ||
Zeile 353: | Zeile 354: | ||
\begin{align*} | \begin{align*} | ||
| | ||
- | & | + | & |
- | | + | |
\end{align*} | \end{align*} | ||
Zeile 366: | Zeile 367: | ||
\omega t + \varphi_u &= \omega t + \varphi_i + {{1}\over{2}}\pi \\ | \omega t + \varphi_u &= \omega t + \varphi_i + {{1}\over{2}}\pi \\ | ||
| | ||
- | \boxed{\varphi = \varphi_u -\varphi_i = + {{1}\over{2}}\pi } | + | \boxed{\varphi = \varphi_u -\varphi_i = + {{1}\over{2}}\pi } |
\end{align*} | \end{align*} | ||
Zeile 399: | Zeile 400: | ||
</ | </ | ||
- | For the concept of AC two-terminal networks, we are also able to use the DC methods of network analysis | + | For the concept of AC two-terminal networks, we are also able to use the DC methods of network analysis to solve AC networks. |
===== 6.4 Complex Values in Electrical Engineering ===== | ===== 6.4 Complex Values in Electrical Engineering ===== | ||
Zeile 475: | 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 482: | Zeile 483: | ||
< | < | ||
</ | </ | ||
- | \\ {{drawio> | + | \\ {{drawio> |
</ | </ | ||
Zeile 494: | Zeile 495: | ||
\begin{align*} | \begin{align*} | ||
\underline{u}(t) & | \underline{u}(t) & | ||
- | & | + | & |
+ | \cdot {\rm e}^{{\rm j} \omega t} \\ | ||
& | & | ||
\end{align*} | \end{align*} | ||
Zeile 502: | 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 540: | Zeile 542: | ||
* $X = Z \sin \varphi$ | * $X = Z \sin \varphi$ | ||
- | ==== 6.5.2 Application on pure Loads ==== | + | value - and therefore a phasor - can simply |
With the complex impedance in mind, the <tabref tab01> can be expanded to: | With the complex impedance in mind, the <tabref tab01> can be expanded to: | ||
Zeile 547: | Zeile 549: | ||
^ Load $\phantom{U\over I}$ ^ ^ integral representation $\phantom{U\over I}$ ^ complex impedance $\underline{Z}={{\underline{U}}\over{\underline{I}}}$ | ^ Load $\phantom{U\over I}$ ^ ^ integral representation $\phantom{U\over I}$ ^ complex impedance $\underline{Z}={{\underline{U}}\over{\underline{I}}}$ | ||
- | | Resistance | + | | Resistance |
- | | Capacitance | + | | Capacitance |
- | | Inductance | + | | Inductance |
</ | </ | ||
\\ \\ | \\ \\ | ||
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*} | ||
+ | \int {\rm e}^{{\rm j}(\omega t + \varphi_x)} | ||
+ | | ||
+ | | ||
+ | \end{align*} | ||
+ | </ | ||
Once a fixed input voltage is given, the voltage phasor $\underline{U}$, | Once a fixed input voltage is given, the voltage phasor $\underline{U}$, | ||
Zeile 662: | Zeile 670: | ||
<WRAP indent>< | <WRAP indent>< | ||
The phasor diagram looks roughly like this: | The phasor diagram looks roughly like this: | ||
- | {{drawio> | + | {{drawio> |
</ | </ | ||
Zeile 697: | Zeile 705: | ||
The phasor diagram looks roughly like this. \\ | The phasor diagram looks roughly like this. \\ | ||
But have a look at the solution for question 5! | But have a look at the solution for question 5! | ||
- | {{drawio> | + | {{drawio> |
</ | </ | ||
Zeile 717: | Zeile 725: | ||
The calculated (positive) horizontal and (negative) vertical dimension for the voltage indicates a phasor in the fourth quadrant. Does it seem right? \\ | The calculated (positive) horizontal and (negative) vertical dimension for the voltage indicates a phasor in the fourth quadrant. Does it seem right? \\ | ||
The phasor diagram which was shown in answer 4. cannot be correct. \\ | The phasor diagram which was shown in answer 4. cannot be correct. \\ | ||
- | With the correct lengths and angles, the real phasor diagram | + | With the correct lengths and angles, the real phasor diagram looks like this: |
- | {{drawio> | + | {{drawio> |
Here the phasor is in the fourth quadrant with a negative angle. \\ | Here the phasor is in the fourth quadrant with a negative angle. \\ | ||
Zeile 757: | Zeile 765: | ||
<WRAP indent> | <WRAP indent> | ||
<button size=" | <button size=" | ||
- | The drawing of the voltage pointers is as follows: {{drawio> | + | The drawing of the voltage pointers is as follows: |
The voltage U is determined by the law of Pythagoras | The voltage U is determined by the law of Pythagoras | ||
\begin{align*} | \begin{align*} | ||
Zeile 780: | Zeile 788: | ||
<button size=" | <button size=" | ||
- | The drawing of the voltage pointers is as follows: {{drawio> | + | The drawing of the voltage pointers is as follows: {{drawio> |
The voltage $U_R$ is determined by the law of Pythagoras | The voltage $U_R$ is determined by the law of Pythagoras | ||
\begin{align*} | \begin{align*} | ||
Zeile 809: | 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 816: | 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 824: | 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*} | ||
+ | # | ||
+ | |||
</ | </ | ||