Unterschiede

Hier werden die Unterschiede zwischen zwei Versionen angezeigt.

--- electrical_engineering_1:circuits_under_different_frequencies [2021/10/30 13:41]
slinn
+++ electrical_engineering_1:circuits_under_different_frequencies [2023/09/19 23:37] (aktuell)
mexleadmin
@@ Zeile 1: / Zeile 1: @@
-====== 7. Networks at variable frequency ======
+====== 7 Networks at variable frequency ======
-Further content can be found at {{https://www.elektroniktutor.de/analogtechnik/rei_swkr.html|elektroniktutor}}
+Further content can be found at this [[https://www.electronics-tutorials.ws/accircuits/series-circuit.html|Tutorial]] or that [[https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-natural-and-forced-response/a/ee-rlc-natural-response-intuition|Tutorial]]
 ==== Introduction ====
-At the previous chapters it was explained how the "influence of a sinusoidal current flow" of capacitor and inductors look like. To describe this, the impedance was introduced. This can be understood as a complex resistance for sinusoidal excitation.
+In the previous chapters, it was explained what the "influence of a sinusoidal current flow" of capacitors and inductors looks like. To describe this, the impedance was introduced. This can be understood as a complex resistance for sinusoidal excitation.
 It applies to the capacitor:
-\begin{align*} \underline{U}_C = \frac{1}{j\omega \cdot C} \cdot \underline{I}_C \quad \rightarrow \quad \underline{Z}_C = \frac{1}{j\omega \cdot C} \end{align*}
+\begin{align*}
+\underline{U}_C = \frac{1}{{\rm j}\omega \cdot C} \cdot \underline{I}_C \quad \rightarrow \quad
+\underline{Z}_C = \frac{1}{{\rm j}\omega \cdot C}
+\end{align*}
 and for the inductance
-\begin{align*} \underline{U}_L = j\omega \cdot L \cdot \underline{I}_L \quad \rightarrow \quad \underline{Z}_L = j\omega \cdot L \end{align*}
+\begin{align*}
+\underline{U}_L = {\rm j}\omega \cdot L \cdot \underline{I}_L \quad \rightarrow \quad
+\underline{Z}_L = {\rm j}\omega \cdot L
+\end{align*}
-Complex impedances can be dealt with in much the same way as ohmic resistances in Electrical Engineering 1 (see: [[:electrical_engineering_1:simple_circuits|simple_dc_circuits]], [[:electrical_engineering_1:non-ideal_sources_and_two_pole_networks|linear_sources_and_bipoles]], [[:electrical_engineering_1:network_analysis|analysis_of_dc_networks]]). In these transformations, the fraction $ j\omega \cdot$ is preserved. Circuits with impedances such as inductors and capacitors will show a frequency dependence accordingly.
+Complex impedances can be dealt with in much the same way as ohmic resistances in Electrical Engineering 1 (see: [[:electrical_engineering_1:simple_circuits|simple DC Circuits]], [[electrical_engineering_1:non-ideal_sources_and_two_terminal_networks|linear Sources and two-terminal network]], [[:electrical_engineering_1:network_analysis|Analysis of DC Networks]]). In these transformations, the fraction $ j\omega \cdot$ is preserved. Circuits with impedances such as inductors and capacitors will show a frequency dependence accordingly.
-===== 7.1 Frequency-dependent voltage divider =====
 <callout>
@@ Zeile 31: / Zeile 35: @@
 </callout>
-==== From two-pole to four-pole ====
+===== 7.1 From Two-Terminal Network to Four-Terminal Network =====
-<WRAP> <imgcaption imageNo01 | two-pole and four-pole> </imgcaption> \\ {{drawio>ZweipolundVierpol}} \\ </WRAP>
+<WRAP> <imgcaption imageNo01 | Two-Terminal Network to Four-Terminal Network> </imgcaption> \\ {{drawio>ZweipolundVierpol.svg}} \\ </WRAP>
-Until now, components such as resistors, capacitors and inductors have been understood as two-terminal. This is also obvious, since there are only two connections. In the following however circuits are considered, which behave similar to a voltage divider: On one side a voltage $U_E$ is applied, on the other side $U_A$ is formed with it. This results in 4 terminals. The circuit can and will be considered as a four-terminal circuit in the following. But the input and output values will be complex.
+Until now, components such as resistors, capacitors, and inductors have been understood as two-terminal. This is also obvious since there are only two connections. In the following however circuits are considered, which behave similarly to a voltage divider: On one side a voltage $U_\rm I$ is applied, and on the other side $U_\rm O$ is formed with it. This results in 4 terminals. The circuit can and will be considered as a four-terminal network in the following. However, the input and output values will be complex.
-For quadripoles, the relation of "what goes out" (e.g. $\underline{U}_O$ or $\underline{U}_2$) to "what goes in" (e.g. voltage $\underline{U}_I$ or $\underline{U}_1$) is important. Thus, the output and input variables ($\underline{U}_O$) and ($\underline{U}_I$) give the quotient:
+For a four-terminal network, the relation of "what goes out" (e.g. $\underline{U}_\rm O$ or $\underline{U}_2$) to "what goes in" (e.g. voltage $\underline{U}_\rm I$ or $\underline{U}_1$) is important. Thus, the output and input variables ($\underline{U}_\rm O$) and ($\underline{U}_\rm I$) give the quotient:
-\begin{align*} \underline{A} & = \frac {\underline{U}_O}{\underline{U}_I} \\ & \text{with} \; \underline{U}_O = U_O \cdot e^{j \varphi_{uO}} \\ & \text{and} \; \underline{U}_I = U_I \cdot e^{j \varphi_{uI}} \\ \\ \underline{A}& = \frac {\underline{U}_O}{\underline{U}_I} = \frac {U_O \cdot e^{j \varphi_{uO}}}{U_I\cdot e^{j \varphi_{uI}}} \\ & = \frac {U_O}{U_I}\cdot \cdot e^{j (\varphi_{uO}-\varphi_{uI})} \\ \end{align*} \begin{align*} \boxed{\underline{A} = \dfrac {\underline{U}_O}{\underline{U}_I} = \frac {U_O}{U_I}\cdot e^{j \Delta\varphi_{u}}} \end{align*}
+\begin{align*}
+\underline{A} & = {{\underline{U}_{\rm O}^\phantom{O}}\over{\underline{U}_{\rm I}^\phantom{O}}} \\
+              & \text{with} \; \underline{U}_{\rm O} = U_{\rm O} \cdot {\rm e}^{{\rm j} \varphi_{u\rm O}} \\
+              & \text{and}  \; \underline{U}_{\rm I} = U_{\rm I} \cdot {\rm e}^{{\rm j} \varphi_{u\rm I}} \\
+              \\
+\underline{A} & = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}}
+                = \frac {U_{\rm O}            \cdot {\rm e}^{{\rm j}  \varphi_{u\rm O}}}{U_{\rm I}\cdot {\rm e}^{{\rm j} \varphi_{u\rm I}}} \\
+              & = \frac {U_{\rm O}}{U_{\rm I}}\cdot {\rm e}^{{\rm j} (\varphi_{u\rm O}-\varphi_{u\rm I})} \\
+\end{align*}
-<callout icon="fa fa-exclamation" color="red" title="Note:">
+\begin{align*}
+\boxed{\underline{A} = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}} = \frac {U_\rm O}{U_\rm I}\cdot {\rm e}^{{\rm j} \Delta\varphi_{u}}}
+\end{align*}
-  * The complex-valued quotient ${\underline{U}_O}/{\underline{U}_I}$ is called the **transfer function**.
+<callout icon="fa fa-exclamation" color="red" title="Reminder:">
-  * The frequency-dependent magnitude of the quotient $A(\omega)={U_O}/{U_I}$ is called **amplitude response**  and the angular difference $\Delta\varphi_{u}(\omega)$ is called **phase response**.
+  * The complex-valued quotient ${\underline{U}_{\rm O}}/{\underline{U}_{\rm I}}$ is called the **transfer function**.
+  * The frequency-dependent magnitude of the quotient $A(\omega)={U_{\rm O}}/{U_{\rm I}}$ is called **amplitude response**  and the angular difference $\Delta\varphi_{u}(\omega)$ is called **phase response**.
 </callout>
-The frequency behaviour of the amplitude response and the frequency response is not only important in electrical engineering and electronics, but will also play a central role in control engineering.
+The frequency behavior of the amplitude response and the frequency response is not only important in electrical engineering and electronics but will also play a central role in control engineering.
 ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== RL series connection ====
+===== 7.2 RL Series Circuit =====
-<WRAP> <imgcaption imageNo02 | RL-series> </imgcaption> \\ {{drawio>RLReihenschaltung}} \\ <WRAP>
+<WRAP> <imgcaption imageNo02 | RL-series> </imgcaption> \\ {{drawio>RLReihenschaltung.svg}} \\ <WRAP>
-First, a series connection of a resistor $R$ and an inductor $L$ shall be considered (see <imgref imageNo02 >). This structure is also called RL-element. \\ Here, $\underline{U}_I= \underline{X_I} \cdot \underline{I}_I$ with $\underline{X}_I = R + j\omega \cdot L$ and corresponding for $\underline{U}_O$: \begin{align*} \underline{A} = \dfrac {\underline{U}_O}{\underline{U}_I} = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}\cdot e^{j\left(\frac{\pi}{2} - arctan \frac{\omega L}{R} \right)} \end{align*}
+First, a series connection of a resistor $R$ and an inductor $L$ shall be considered (see <imgref imageNo02 >). This structure is also called RL-element. \\
+Here, $\underline{U}_{\rm I}= \underline{X_\rm I} \cdot \underline{I}_{\rm I}$ with $\underline{X}_{\rm I} = R + {\rm j}\omega \cdot L$ and corresponding for $\underline{U}_{\rm O}$:
+\begin{align*}
+\underline{A} = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}}
+              = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan \frac{\omega L}{R} \right)}
+\end{align*}
 This results in the following for
   * the amplitude response: $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}$ and
-  * the phase response: $\Delta\varphi_{u} = \frac{\pi}{2} - arctan \frac{\omega L}{R}$
+  * the phase response: $\Delta\varphi_{u} = \arctan \frac{R}{\omega L} = \frac{\pi}{2} - \arctan \frac{\omega L}{R}$
 The main focus should first be on the amplitude response. Its frequency response can be derived from the equation in various ways.
-  - Limit value consideration of the RL arrangement (in the equation and in the system)
+  - Extreme frequency consideration of this RL circuit (in the equation and the system)
   - Plotting amplitude and frequency response
   - Determination of prominent frequencies
@@ Zeile 71: / Zeile 92: @@
 These three points are now to be gone through.
-=== Limit value consideration of the RL arrangement ===
+==== 7.2.1 RL High Pass ====
-For the limit consideration we look at what, happens when the frequency $\omega$ runs to the definition range limits, i.e. $\omega \rightarrow 0$ and $\omega \rightarrow \infty$:
+For the first step, we investigate the limit consideration: We look at what happens when the frequency $\omega$ runs to the definition range limits, i.e. $\omega \rightarrow 0$ and $\omega \rightarrow \infty$:
   * For $\omega \rightarrow 0$, $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}} \rightarrow 0$ as the numerator approaches zero and the denominator remains greater than zero.
-  * For $\omega \rightarrow \infty$, $A \rightarrow1$, because in the root in the denominator $(\omega L)^2$ becomes larger and larger in the ratio $R^2$ to . So the root tends to $\omega L$ and thus to the numerator.
+  * For $\omega \rightarrow \infty$, $A \rightarrow 1$, because in the root in the denominator $(\omega L)^2$ becomes larger and larger in the ratio $R^2$ to . So the root tends to $\omega L$ and thus to the numerator.
 It can thus be seen that:
   * at small frequencies there is no voltage $U_2$ at the output.
-  * at high frequencies $A = \frac {U_O}{U_I} = \rightarrow 1$, so the voltage at the output is equal to the voltage at the input.
+  * at high frequencies $A = \frac {U_{\rm O}}{U_{\rm I}} = \rightarrow 1$, so the voltage at the output is equal to the voltage at the input.
-The RL element shown here therefore only allows large frequencies to pass (= pass through) and small ones are filtered out. The circuit corresponds to a **high pass**. \\ This can also be derived from understanding the components: At small frequencies, the current in the coil and thus the magnetic field changes only slowly. So only a negligibly small reverse voltage is induced. The coil acts like a short circuit at low frequencies. At higher frequencies, the current generated by $U_I$ through the coil changes faster, the induced voltage $U_i = - dI / dt$ becomes large. As a result, the coil inhibits the current flow and a voltage drops across the coil. If the frequency becomes very high, only a negligible current flows through the coil - and hence through the resistor. The voltage drop at $R$ thus approaches zero and the output voltage $U_O$ tends towards $U_I$.
+Result: \\
+The RL element shown here therefore only allows large frequencies to pass (= pass through) and small ones are filtered out. \\
+The circuit corresponds to a **high pass**. \\ \\
-For further consideration, the equation of the transfer function $\underline{A} = \dfrac {\underline{U}_O}{\underline{U}_I}$ is to be rewritten so that it becomes independent of component values. This allows for a generalized representation. This representation is called **normalization**:
+This can also be derived from understanding the components:
+  * At small frequencies, the current in the coil and thus the magnetic field changes only slowly. So only a negligibly small reverse voltage is induced. The coil acts like a short circuit at low frequencies.
+  * At higher frequencies, the current generated by $U_I$ through the coil changes faster, the induced voltage $U_{\rm i} = - {\rm d}I / {\rm d}t$ becomes large. \\ As a result, the coil inhibits the current flow and a voltage drops across the coil.
+  * If the frequency becomes very high, only a negligible current flows through the coil - and hence through the resistor. The voltage drop at $R$ thus approaches zero and the output voltage $U_\rm O$ tends towards $U_\rm I$.
-\begin{align*} \underline{A} = \dfrac {\underline{U}_O}{\underline{U}_I} = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}\cdot e^{j\left(\frac{\pi}{2} - arctan \frac{\omega L}{R} \right)} \quad \xrightarrow{\text{normalization}} \quad \quad \underline{A}_{norm} = \frac {\omega L / R}{\sqrt{1 + (\omega L / R)^2}}\cdot e^{j\left(\frac{\pi}{2} - arctan \frac{\omega L}{R} \right)} = \frac {x}{\sqrt{1 + x^2}} \cdot e^{j\left(\frac{\pi}{2} - arctan x \right)} \end{align*}
+The transfer function can also be decomposed into amplitude response and frequency response. \\
+Often these plots are not given in with linear axis but:
+  * the amplitude response with a double logarithmic coordinate system and
+  * the phase response single logarithmic coordinate system.
-This equation behaves quite the same as the one considered so far.
+By this, the course from low to high frequencies is easier to see. The following simulation in <imgref imageNo5> shows the amplitude response and frequency response in the lower left corner.
-~~PAGEBREAK~~ ~~CLEARFIX~~
+<WRAP centeralign>
-<code>
+<imgcaption imageNo5 | RL high pass filter>
+</imgcaption>\\
+{{url>https://www.falstad.com/afilter/circuitjs.html?cct=$+1+0.000005+5+50+5+50%0A%25+4+1630997.1347384118%0Ar+64+80+224+80+0+35%0AO+224+80+336+80+0%0Ag+224+208+224+240+0%0A170+64+80+32+80+3+20+1000+5+0.1%0Al+224+80+224+208+0+0.001+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0Ao+1+16+0+34+2.5+0.00009765625+1+-1+out%0A 500,500 noborder}}
+</WRAP>
-<wrap hide>
-\\  === Plotting amplitude and frequency response ===
-</wrap>
-</code>
+For further consideration, the equation of the transfer function $\underline{A} = \dfrac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}}$ is to be rewritten so that it becomes independent of component values $R$ and $L$.\\
+This allows for a generalized representation. This representation is called **normalization**:
-The transfer function can also be decomposed into amplitude response and frequency response. This can be done by
+<WRAP centeralign>
+\begin{align*}
+\large{\underline{A}  = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}}
+                      = \frac {\omega L}    {\sqrt{R^2 +    (\omega L)^2}}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan \frac{\omega L}{R} \right)}}
+ \quad  \quad \vphantom{\HUGE{I \\ I}} \large{\xrightarrow{\text{normalization}}} \vphantom{\HUGE{I \\ I}} \quad \quad \quad
+\large{\underline{A}_{norm}
+                      = \frac {\omega L / R}{\sqrt{1  + (\omega L / R)^2}}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan \frac{\omega L}{R} \right)} }
+\large{               = \frac {x}           {\sqrt{1  + x^2             }}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan x \right)} }
+\end{align*}
+</WRAP>
-  * the amplitude response double logarithmic and
-  * the phase response single logarithmic
-logarithmically. <imgref imageNo03 > shows the two plots. On the x-axis, $x = \omega L / R$ has been plotted as the normalization variable. This represents a weighted frequency.
+This equation behaves quite the same as the one considered so far.
+~~PAGEBREAK~~ ~~CLEARFIX~~
+<imgref imageNo03 > shows the two plots. On the x-axis, $x = \omega L / R$ has been plotted as the normalization variable. This represents a weighted frequency.
-<WRAP> <imgcaption imageNo03 | Amplitude and phase response of the RL high-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRLHochpass}} </WRAP>
+<WRAP> <imgcaption imageNo03 | Amplitude and phase response of the RL high-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRLHochpass.svg}} </WRAP>
-Here, too, the behavior determined in the limit value observation can be seen: at small frequencies $\omega$ (corresponds to small $x$), the amplitude response tends toward zero. At high frequencies, the ratio $U_A / U_E = 1 $ is established.
+Here, too, the behavior determined in the limit value observation can be seen:
+  * at small frequencies $\omega$ (corresponds to small $x$), the amplitude response tends toward zero.
+  * At high frequencies, the ratio $U_{\rm O} / U_{\rm I} = 1 $ is established.
 Interesting in the phase response is the point $x = 1$.
-  * Further to the left of this point (i.e. at smaller frequencies) a tenfold increase of the frequency $\omega$ produces a tenfold increase of $U_A / U_E$.
+  * Further to the left of this point (i.e. at smaller frequencies) a tenfold increase of the frequency $\omega$ produces a tenfold increase of $U_{\rm O} / U_{\rm I}$.
-  * Further to the right of this point (i.e. at higher frequencies) $U_A / U_E = 1$ remains.
+  * Further to the right of this point (i.e. at higher frequencies) $U_{\rm O} / U_{\rm I} = 1$ remains.
-So this point marks a limit. Far to the left, the ohmic resistance is significantly greater the amount of impedance of the coil: $R >> \omega L$. far to the right is just the opposite.
+So this point marks a limit. Far to the left, the ohmic resistance is significantly greater than the amount of impedance of the coil: $R \gg \omega L$. far to the right is just the opposite.
+The point $x=1$ just marks the cut-off frequency. \\ It holds
+<WRAP group>
+<WRAP quarter column > </WRAP>
+<WRAP quarter column >
+\begin{align*}
+\vphantom{\HUGE{I }} \\
+\underline{A}_{\rm norm} = \frac{x}{\sqrt{1 + x^2}}    \cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - arctan x \right)}
+                         = \frac{U_{\rm O}}{U_{\rm I}} \cdot {\rm e}^{{\rm j}\varphi}
+\end{align*}
+</WRAP>
+<WRAP quarter column >
+\begin{align*}
+\left\{\begin{array}{l}
+x \ll 1 & \widehat{=}& \omega L \ll R \, : \quad & \frac{U_{\rm O}}{U_{\rm I}}=x                  &, \varphi = \frac{\pi}{2}  \, \widehat{=} \, 90° \\
+x \gg 1 & \widehat{=}& \omega L \gg R \, : \quad & \frac{U_{\rm O}}{U_{\rm I}}=1                  &, \varphi = 0              \; \widehat{=} \, 0° \\
+x   = 1 & \widehat{=}& \omega L =   R  , : \quad & \frac{U_{\rm O}}{U_{\rm I}}=\frac{1}{\sqrt{2}} &, \varphi = \frac{\pi}{4}  \, \widehat{=} \, 45°
+\end{array} \right.
+\end{align*}
+</WRAP>
+<WRAP quarter column > </WRAP>
+</WRAP>
-The point $x=1$ just marks the cutoff frequency. It holds
-\begin{align*} \underline{A}_{norm} = \frac {x}{\sqrt{1 + x^2}} \cdot e^{j\left(\frac{\pi}{2} - arctan x \right)} = \frac {U_A}{U_E} \cdot e^{j\varphi}\quad \quad \\left{ \begin{array}{l} x \ll 1 & \widehat{=} \omega L \ll R &: \quad\quad \frac{U_A}{U_E}=x &, \varphi = \frac{\pi}{2} & \widehat{=} 90° x \gg 1 & \widehat{=} \omega L \gg R &: \quad\quad \frac{U_A}{U_E}=1 &, \varphi = 0 & \widehat{=} 0° x = 1 & \widehat{=} \omega L = R &: \quad\quad \frac{U_A}{U_E}=\frac{1}{\sqrt{2}} &, \varphi = \frac{\pi}{4} &, \widehat{=} 45{°} \end{array} \right. \end{align*}
 <callout icon="fa fa-exclamation" color="red" title="Reminder:">
-  * The **cutoff frequency**  for high-pass and low-pass filters is the frequency at which the ohmic resistance just equals the value of the impedance.
+  * The **cut-off frequency** $f_\rm c$ for high-pass and low-pass filters is the frequency at which the ohmic resistance just equals the value of the impedance.
-  * The cutoff frequency separates a range in which the filter allows signals through from one in which they are suppressed (=blocked).
+  * The cut-off frequency separates a range in which the filter allows signals through from one in which they are suppressed (=blocked).
-  * At the cutoff frequency, the phase $\varphi = 45°$ and the amplitude $A = \frac{1}{\sqrt{2}}$.
+  * At the cut-off frequency, the phase $\varphi = 45°$ and the amplitude $A = \frac{1}{\sqrt{2}}$.
+  * In German the cut-off Frequency is called //Grenzfrequenz// $f_{\rm Gr}$
 These statements apply to single-stage passive filters, i.e. one RL or one RC element. Multistage filters are considered in circuit engineering. </callout>
-The cut-off frequency in this case is given by:
+The cut-off frequency, in this case, is given by:
-\begin{align*} R &= \omega L \omega_{Gr} &= \frac{R}{L} 2 \pi f_{Gr} &= \frac{R}{L} \quad \rightarrow \quad \boxed{f_{Gr} = \frac{R}{2 \pi \cdot L}} \end{align*}
-== SYNC, CORRECTED BY ELDERMAN ==
+\begin{align*}
+R               &= \omega L \\
+\omega _{\rm c} &= \frac{R}{L} \\
+\pi f_{\rm c} &= \frac{R}{L} \quad \rightarrow \quad \boxed{f_{\rm c} = \frac{R}{2 \pi \cdot L}} \end{align*}
-=== low pass ===
+==== 7.2.2 RL Low Pass ====
-<WRAP> <imgcaption imageNo04 | Circuit, pointer diagram, and amplitude and phase response of RL low-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRLTiefpass}} </WRAP>
+<WRAP> <imgcaption imageNo04 | Circuit, pointer diagram, and amplitude and phase response of RL low-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRLTiefpass.svg}} </WRAP>
-So far, only one variant of the RL element has been considered, namely the one where the output voltage $\underline{U}_A$ is tapped at the inductance. Here we will briefly discuss what happens when the two components are swapped.
+So far, only one variant of the RL element has been considered, namely the one where the output voltage $\underline{U}_{\rm O}$ is tapped at the inductance. \\
+ Here we will briefly discuss what happens when the two components are swapped.
 In this case, the normalized transfer function is given by:
-\begin{align*} \underline{A}_{norm} = \frac {1}{\sqrt{1 + (\omega L / R)^2}}\cdot e^{-j arctan \frac{\omega L}{R} } \end{align*}
+\begin{align*}
+\underline{A}_{\rm norm} = \frac {1}{\sqrt{1 + (\omega L / R)^2}}\cdot {\rm e}^{-{\rm j} \; \arctan \frac{\omega L}{R} }
+\end{align*}
-The cutoff frequency is again given by $f_{Gr} = \frac{R}{2 \pi \cdot L}$.
+The cut-off frequency is again given by $f_{\rm c} = \frac{R}{2 \pi \cdot L}$.
+<WRAP centeralign>
+<imgcaption imageNo6 | RL low pass filter>
+</imgcaption>\\
+{{url>https://www.falstad.com/afilter/circuitjs.html?cct=$+1+0.000005+5+50+5+50%0A%25+4+1630997.1347384118%0Ar+224+208+224+80+0+35%0AO+224+80+336+80+0%0Ag+224+208+224+240+0%0A170+64+80+32+80+3+20+1000+5+0.1%0Al+64+80+224+80+0+0.001+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0Ao+1+16+0+34+2.5+0.00009765625+1+-1+out%0A 500,500 noborder}}
+</WRAP>
 ~~PAGEBREAK~~ ~~CLEARFIX~~
-==== RC Series ====
+===== 7.3 RC Series Circuit =====
-=== RC high pass ===
+==== 7.3.1 RC High Pass ====
-<WRAP> <imgcaption imageNo05 | Circuit, pointer diagram, and amplitude and phase response of the RC high-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRCHochpass}} </WRAP>
+<WRAP> <imgcaption imageNo05 | Circuit, pointer diagram, and amplitude and phase response of the RC high-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRCHochpass.svg}} </WRAP>
 Now a voltage divider is to be constructed by a resistor $R$ and a capacity $C$. Quite similar to the previous chapters, the transfer function can also be determined here.
@@ Zeile 160: / Zeile 235: @@
 Here results as normalized transfer function:
-\begin{align*} \underline{A}_{norm} = \frac {\omega RC}{\sqrt{1 + (\omega RC)^2}}\cdot e^{\frac{\pi}{2}-j arctan \omega RC } \end{align*}
+\begin{align*}
+\underline{A}_{\rm norm} = \frac {\omega RC}{\sqrt{1 + (\omega RC)^2}}\cdot {\rm e}^{\frac{\pi}{2}-{\rm j} \; \arctan (\omega RC) }
+\end{align*}
-In this case, the normalization variable $x = \omega RC$. Again, the cutoff frequency is determined by equating $R$ and the magnitude of the impedance of the capacitance:
+In this case, the normalization variable $x = \omega RC$. Again, the cut-off frequency is determined by equating $R$ and the magnitude of the impedance of the capacitance:
-\begin{align*} R &= \frac{1}{\omega_{Gr} C} \omega_{Gr} &= \frac{1}{RC} 2 \pi f_{Gr} &= \frac{1}{RC} \quad \rightarrow \quad \boxed{f_{Gr} =\frac{1}{2 \pi\cdot RC} } \end{align*}}
+\begin{align*}
+               R &= \frac{1}{\omega_{\rm c} C} \\
+ \omega_{\rm c}  &= \frac{1}{RC} \\
+\pi  f_{\rm c} &= \frac{1}{RC} \quad \rightarrow \quad
+\boxed{f_{\rm c}  = \frac{1}{2 \pi\cdot RC} }
+\end{align*}
+<WRAP centeralign>
+<imgcaption imageNo7 | RC high pass filter>
+</imgcaption>\\
+{{url>https://www.falstad.com/afilter/circuitjs.html?cct=$+1+0.000005+5+50+5+50%0A%25+4+1630997.1347384118%0Ac+64+80+224+80+0+0.000001+0%0Ar+224+80+224+208+0+35%0AO+224+80+336+80+0%0Ag+224+208+224+240+0%0A170+64+80+32+80+3+20+1000+5+0.1%0Ao+4+16+0+34+5+0.00009765625+0+-1+in%0Ao+2+16+0+34+2.5+0.00009765625+1+-1+out%0A 500,500 noborder}}
+</WRAP>
 ~~PAGEBREAK~~ ~~CLEARFIX~~
-=== rc low pass ===
+==== 7.3.2 RC Low Pass ====
-<WRAP> <imgcaption imageNo06 | Circuit, pointer diagram, and amplitude and phase response of RC low-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRCTiefpass}} </WRAP>
+<WRAP> <imgcaption imageNo06 | Circuit, pointer diagram, and amplitude and phase response of RC low-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRCTiefpass.svg}} </WRAP>
 Again, the voltage at the impedance is to be used as the output voltage. This results in a low-pass filter.
@@ Zeile 176: / Zeile 264: @@
 Here results as normalized transfer function:
-\begin{align*} \underline{A}_{norm} = \frac {1}{\sqrt{1 + (\omega RC)^2}}\cdot e^{-j arctan \omega RC } \end{align*}
+\begin{align*}
+\underline{A}_{\rm norm} = \frac {1}{\sqrt{1 + (\omega RC)^2}}\cdot {\rm e}^{-{\rm j} \; \arctan (\omega RC) }
+\end{align*}
-Also, the cutoff frequency is given by $f_{Gr} =\frac{1}{2 \pi\cdot RC}$
+Also, the cut-off frequency is given by $f_{\rm c} =\frac{1}{2 \pi\cdot RC}$
 ~~PAGEBREAK~~ ~~CLEARFIX~~
-===== 7.2 Resonance phenomena =====
+<WRAP centeralign>
+<imgcaption imageNo8 | RC low pass filter>
-~~PAGEBREAK~~ ~~CLEARFIX~~
+</imgcaption>\\
+{{url>https://www.falstad.com/afilter/circuitjs.html?cct=$+1+0.000005+5+50+5+50%0A%25+4+1630997.1347384118%0Ac+224+208+224+80+0+0.000001+0%0Ar+64+80+224+80+0+35%0AO+224+80+336+80+0%0Ag+224+208+224+240+0%0A170+64+80+32+80+3+20+1000+5+0.1%0Ao+4+16+0+34+5+0.00009765625+0+-1+in%0Ao+2+16+0+34+2.5+0.00009765625+1+-1+out%0A 500,500 noborder}}
-==== RLC - Series Resonant Circuit ====
+</WRAP>
-<WRAP> <imgcaption imageNo06 | circuit of the series resonant circuit> </imgcaption> {{drawio>SchaltungdesSerienschwingkreises}} </WRAP>
-If a resistor $R$, a capacitor $C$ and an inductance $L$ are connected in series, the result is a series resonant circuit. In this case the output voltage is not clearly defined. It must be considered in the following how the voltages behave across the individual components. The total voltage (= input voltage $U_E$) results to:
-\begin{align*} \underline{U} = \underline{U}_R + \underline{U}_L + \underline{U}_C \end{align*}
-Since the current in the circuit must be constant, the total impedance can be determined here in a simple way:
-\begin{align*} \underline{U} &= R \cdot \underline{I} + j \omega L \cdot \underline{I} + \frac {1}{j\omega C } \cdot \underline{I} \underline{U} &= \left( R + j \omega L - j \cdot \frac {1}{\omega C } \right) \cdot \underline{I} \underline{Z}_{ges} &= R + j \omega L - j \cdot \frac {1}{\omega C } \end{align*}
-As the magnitude of the (input) voltage $U$ or the (input or total) impedance $Z$ and the phase result to:
-\begin{align*} U &= \sqrt{U_R^2 + (U_Z)^2} = \sqrt{U_R^2 + (U_L - U_C)^2} \end{align*}
-\begin{align*} Z &= \sqrt{R^2 + (Z)^2} = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2} \end{align*}
-\begin{align*} \varphi_u = \varphi_Z &= arctan \frac{\omega L - \frac{1}{\omega C}}{R} \end{align*}
-There are now 3 different situations to distinguish:
-  * If $U_L > U_C$ the whole setup behaves like an ohmic-inductive load. This is the case at high frequencies.
-  * If $U_L$ equals $U_C$, the total input voltage $U$ is applied to the resistor. In this case, the total resistance $Z$ is minimal and only ohmic. \\Thus, the current $I$ is then maximal. If the current is maximum, then the responses of the capacitance and inductance - their voltages - are also maximum. This situation is the **resonance case**.
-  * If $U_L < U_C$ then the whole setup behaves like a resistive-capacitive load. This is the case at low frequencies.
-Again, there seems to be an excellent frequency, namely when $U_L = U_C$ or $Z_C = Z_L$ holds:
-\begin{align*} \frac{1}{\omega_0 C} & = \omega L \omega_0 & = \frac{1}{\sqrt{LC}} 2\pi f_0 & = \frac{1}{\sqrt{LC}} \rightarrow \boxed{ f_0 = \frac{1}{2\pi \sqrt{LC}} } \end{align*}
-The frequency $f_0$ is called **resonance frequency**.
-^ ^$\quad$^$f \rightarrow 0$^$\quad$^$f = f_0$^$\quad$^$f \rightarrow \infty$|
-|voltage $U_R$ \\ at the resistor| |$\boldsymbol{0}$| |$\boldsymbol{U}$ \\ since the impedances just cancel| |$ \boldsymbol{0}$|
-|voltage $U_L$ \\ at the inductor| |$\boldsymbol{0}$ \\ because $\omega L$ becomes very small| |$\boldsymbol{\omega_0 L \cdot I = \omega_0 L \cdot \frac{U}{R} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}\cdot U}}$| |$\boldsymbol{U}$ \\ since $\omega L$ becomes very large|
-|$\boldsymbol{U}$ \voltage $U_C$ \\ at the capacitor| |$\boldsymbol{U}$ \\ because $\frac{1}{\omega C}$ becomes very large| |$\boldsymbol{\frac{1}{\omega_0 C} \cdot I = \frac{1}{\omega_0 C} \cdot \frac{U}{R} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}\cdot U}}$| |$\boldsymbol{0}$ \because $\frac{1}{\omega C}$ becomes very small|
-The calculation in the table shows that in the resonance case, the voltage across the capacitor or inductor deviates from the input voltage by a factor $\color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}}$. This quantity is called **quality**  $Q_S$:
-\begin{align*} \boxed{ Q_S = \frac{U_C}{U} |_{\omega = \omega_0} = \frac{U_L}{U} |_{\omega = \omega_0} = \color{blue}{\frac{1}{R}\sqrt{\frac{L}{C}}} } \end{align*}
-The quality can be greater than, less than or equal to 1.
-  * If the quality is very high, the overshoot of the voltages at the impedances becomes very large in the resonance case. This is useful and necessary in various applications, e.g. in an RLC element as an antenna.
-  * If the Q is very small, overshoot is no longer seen. Depending on the impedance at which the output voltage is measured, a high-pass or low-pass is formed similar to the RC or RL element. However, this has a steeper slope in the blocking range. This means that the filter effect is better.
-The reciprocal of the Q is called **attenuation**  $d_S$. This is specified when using the circuit as a non-overshooting filter.
-\begin{align*} \boxed{ d_S = \frac{1}{Q_S} = R \sqrt{\frac{C}{L}} } \end{align*}
-~~PAGEBREAK~~ ~~CLEARFIX~~
-<WRAP>{{url>https://falstad.com/afilter/circuitjs.html?running=false&cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+256+80+304+80+0+0.000047+0%0Ar+192+80+256+80+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+304+160+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}} </WRAP>
-<WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l3BWcsDMYBM6EA4GTClgOwBsJIeFIALChQKYC0YYAUAErjrYgCc51HvyjgQ2QlUgiYCVgCcuQ8ukhKRKVgGMQKtdzVSY8SNSIhm0FAiK9sp6r3S8UvV9Six8rADY7se-3BSaU8IMGgEMGwSFkgiSBdeSARySFYAcz8eMGD9HVVpVhUzPOErdD4BEAB7AFcAF1ZqkQgpakhbcENoVp1JVhQeKTp2qTB4EAAxOXoAR1r6ADtNAE9WIA 500,400 noborder}} </WRAP> ~~PAGEBREAK~~ ~~CLEARFIX~~
-<WRAP>{{url>https://falstad.com/afilter/circuitjs.html?cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+304+160+304+80+0+0.000047+0%0Ar+192+80+256+80+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+256+80+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}} </WRAP> ~~PAGEBREAK~~ ~~CLEARFIX~~
-<WRAP>{{url>https://falstad.com/afilter/circuitjs.html?cct=$+1+0.000005+5+50+5+50%0A%25+4+1959.030510288011%0Ac+192+80+256+80+0+0.000047+0%0Ar+304+80+304+160+0+3%0Ag+304+160+304+192+0%0A170+192+80+160+80+3+20+1000+5+0.1%0Al+256+80+304+80+0+0.01+0.46265716582988115%0AO+304+80+352+80+0%0Ao+3+16+0+34+5+0.00009765625+0+-1+in%0A 500,400 noborder}} </WRAP>
-~~PAGEBREAK~~ ~~CLEARFIX~~
-===== Decoupling capacitor on the microcontroller =====
-[[http://www.falstad.com/circuit/circuitjs.html?cct=$+1+3.125e-9+0.7389056098930651+50+5+50%0AR+-256+144+-256+80+0+0+40000+0+5+0+0.5%0Ar+624+160+624+192+0+0.1%0Aw+624+144+624+160+0%0Aw+624+144+672+144+0%0Ag+672+208+672+224+0%0Ap+672+144+672+208+1+0%0Ap+-256+144+-256+208+1+0%0Ag+-256+208+-256+224+0%0Ar+176+144+224+144+0+0.02%0Al+112+144+176+144+0+3.0000000000000004e-7+0.0012646053584079516%0Aw+224+144+288+144+0%0Aw+-256+144+-208+144+0%0Al+-160+144+-96+144+0+0.000003+0.001714943880265035%0Ar+-96+144+-48+144+0+0.2%0Af+576+224+624+224+33+1.5+0.02%0Aw+624+192+624+208+0%0Ag+624+240+624+272+0%0AR+576+224+560+224+5+5+10000000+2.5+2.5+0+0.5%0Ar+288+272+288+208+0+0.01%0Al+288+208+288+144+0+1e-8+-3.95516952522712e-14%0Aw+336+144+288+144+0%0Aw+-160+144+-208+144+0%0Aw+624+144+560+144+0%0Ab+-192+96+-35+301+0%0Ax+-180+65+-54+68+4+18+Ersatzschaltbild%0Ac+-96+144+-96+208+0+1e-10+5.6099395909659755%0Ag+-96+208+-96+224+0%0Ax+-188+87+-34+90+4+18+5..50cm%5CsLeiterbahn%0Ax+83+87+242+90+4+18+0.5..5cm%5CsLeiterbahn%0Ax+91+65+217+68+4+18+Ersatzschaltbild%0Ag+176+208+176+224+0%0Ac+176+144+176+208+0+1e-10+5.828890665512928%0Ab+80+96+237+301+0%0Aw+80+144+112+144+0%0Ab+248+96+341+413+0%0Ax+232+444+358+447+4+18+Ersatzschaltbild%0Ax+231+470+337+473+4+18+100nF%5CsKerKo%0Aw+560+144+496+144+0%0Ax+555+87+673+90+4+18+Mikrocontroller%0Ax+552+65+678+68+4+18+Ersatzschaltbild%0Ab+521+96+726+301+0%0Aw+352+144+384+144+0%0Ab+352+96+509+301+0%0Ac+448+144+448+208+0+1e-11+5.8885188471740335%0Ag+448+208+448+224+0%0Ax+341+87+525+90+4+18+0.05..0.5cm%5CsLeiterbahn%0Al+384+144+448+144+0+3.0000000000000004e-8+0.0009675223894950857%0Ar+448+144+496+144+0+0.002%0Ax+-41+470+65+473+4+18+100nF%5CsKerKo%0Ax+-40+444+86+447+4+18+Ersatzschaltbild%0Ab+-24+96+69+413+0%0Aw+64+144+16+144+0%0Al+16+208+16+144+0+1e-8+1.5154544286133387e-13%0Ar+16+272+16+208+0+0.01%0Aw+-48+144+16+144+0%0Aw+64+144+80+144+0%0Ax+366+65+492+68+4+18+Ersatzschaltbild%0Aw+336+144+352+144+0%0Ax+186+352+225+355+4+32+S2%0As+288+320+288+352+0+1+false%0Ag+288+352+288+384+0%0Ac+288+272+288+320+2+1.0000000000000001e-7+5.013963438724142%0Ac+16+272+16+320+2+1.0000000000000001e-7+4.9684040165331345%0Ag+16+352+16+384+0%0As+16+320+16+352+0+1+false%0Ax+-79+352+-40+355+4+32+S1%0Ax+201+493+406+496+4+18+%22nahe%5Csam%5CsMikrocontroller%22%0Ao+6+8+0+4106+24.87604116742552+0.0001+0+2+5+0%0A|Simulation in Falstad]]\. Note: The simulation gives a highly simplified picture. The response of the microcontroller is shown reduced to a triangular signal, since the slope of the voltages cannot be represented. A real simulation requires a powerful SPICE program in which the [[https://en.wikipedia.org/wiki/Transmission_line|conduction theory ]]can be represented.
-Further details can be found [[https://en.wikipedia.org/wiki/Decoupling_capacitor|here (practice)]] or [[http://www.lothar-miller.de/s9y/categories/14-Entkopplung|here (board layout)]].