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electrical_engineering_1:circuits_under_different_frequencies [2022/12/09 18:20]
mexleadmin 		electrical_engineering_1:circuits_under_different_frequencies [2023/09/19 23:37] (aktuell)
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Zeile 1: Zeile 1:
-====== 7. Networks at variable frequency ======+====== 7 Networks at variable frequency ======
  
 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]] 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]]
Zeile 5: Zeile 5:
 ==== Introduction ==== ==== 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: It applies to the capacitor:
  
 \begin{align*}  \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} +\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*} \end{align*}
  
Zeile 16: Zeile 17:
  
 \begin{align*}  \begin{align*} 
-\underline{U}_L = j\omega \cdot L \cdot \underline{I}_L \quad \rightarrow \quad \underline{Z}_L = j\omega \cdot L +\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*} \end{align*}
  
Zeile 37: Zeile 39:
 <WRAP> <imgcaption imageNo01 | Two-Terminal Network to Four-Terminal Network> </imgcaption> \\ {{drawio>ZweipolundVierpol.svg}} \\ </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 $U_I$ is applied, on the other side $U_O$ 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.+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 a four-terminal network, 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*}  \begin{align*} 
-\underline{A} & = {{\underline{U}_O^\phantom{O}}\over{\underline{U}_I^\phantom{O}}} \\  +\underline{A} & = {{\underline{U}_{\rm O}^\phantom{O}}\over{\underline{U}_{\rm I}^\phantom{O}}} \\  
-              & \text{with} \; \underline{U}_O = U_O \cdot e^{j \varphi_{uO}} \\  +              & \text{with} \; \underline{U}_{\rm O} = U_{\rm O} \cdot {\rm e}^{{\rm j} \varphi_{u\rm O}} \\  
-              & \text{and} \; \underline{U}_I = U_I \cdot e^{j \varphi_{uI}} \\ +              & \text{and}  \; \underline{U}_{\rm I} = U_{\rm I} \cdot {\rm e}^{{\rm j} \varphi_{u\rm I}} \\ 
               \\                \\ 
-\underline{A} & = \frac {\underline{U}_O^\phantom{O}}{\underline{U}_I^\phantom{O}} = \frac {U_O \cdot e^{j \varphi_{uO}}}{U_I\cdot e^{j \varphi_{uI}}} \\  +\underline{A} & = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}}  
-              & = \frac {U_O}{U_I}\cdot e^{j (\varphi_{uO}-\varphi_{uI})} \\ +                = \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*}  \end{align*} 
  
 \begin{align*}  \begin{align*} 
-\boxed{\underline{A} = \frac {\underline{U}_O^\phantom{O}}{\underline{U}_I^\phantom{O}} = \frac {U_O}{U_I}\cdot e^{j \Delta\varphi_{u}}} +\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*} \end{align*}
  
 <callout icon="fa fa-exclamation" color="red" title="Reminder:"> <callout icon="fa fa-exclamation" color="red" title="Reminder:">
  
-  * The complex-valued quotient ${\underline{U}_O}/{\underline{U}_I}$ is called the **transfer function**. +  * The complex-valued quotient ${\underline{U}_{\rm O}}/{\underline{U}_{\rm I}}$ is called the **transfer function**. 
-  * The frequency-dependent magnitude of the quotient $A(\omega)={U_O}/{U_I}$ is called **amplitude response**  and the angular difference $\Delta\varphi_{u}(\omega)$ is called **phase response**.+  * 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> </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~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
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 <WRAP> <imgcaption imageNo02 | RL-series> </imgcaption> \\ {{drawio>RLReihenschaltung.svg}} \\ <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 $\underline{U}_O$: +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*}  \begin{align*} 
-\underline{A} = \frac {\underline{U}_O^\phantom{O}}{\underline{U}_I^\phantom{O}} = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}\cdot e^{j\left(\frac{\pi}{2} - arctan \frac{\omega L}{R} \right)} +\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*} \end{align*}
  
Zeile 77: Zeile 82:
  
   * the amplitude response: $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}$ and   * the amplitude response: $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}$ and
-  * the phase response: $\Delta\varphi_{u} = arctan \frac{R}{\omega L} = \frac{\pi}{2} - arctan \frac{\omega L}{R}$+  * 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. The main focus should first be on the amplitude response. Its frequency response can be derived from the equation in various ways.
  
-  - Extreme frequency consideration of this RL circuit (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   - Plotting amplitude and frequency response
   - Determination of prominent frequencies   - Determination of prominent frequencies
Zeile 89: Zeile 94:
 ==== 7.2.1 RL High Pass ==== ==== 7.2.1 RL High Pass ====
  
-For the first step we investigate the limit consideration: Wwe 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 $\omega \rightarrow 0$, $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}} \rightarrow 0$ 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: It can thus be seen that:
  
   * at small frequencies there is no voltage $U_2$ at the output.   * 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.
  
-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**. \\ \\+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**. \\ \\
  
 This can also be derived from understanding the components:  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 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.  +  * 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 $U_O$ tends towards $U_I$.+  * 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 $U_\rm I$.
  
 The transfer function can also be decomposed into amplitude response and frequency response. \\  The transfer function can also be decomposed into amplitude response and frequency response. \\ 
-Often this plots are not given in with linear axis but:+Often these plots are not given in with linear axis but:
   * the amplitude response with a double logarithmic coordinate system and   * the amplitude response with a double logarithmic coordinate system and
   * the phase response single logarithmic coordinate system.    * the phase response single logarithmic coordinate system. 
  
-By this, the course from low to high frequencies are easier to see. The following simulation in <imgref imageNo5> shows the amplitude response and frequency response in the lower left corner.+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.
  
 <WRAP centeralign>  <WRAP centeralign> 
Zeile 120: Zeile 127:
  
  
-For further consideration, the equation of the transfer function $\underline{A} = \dfrac {\underline{U}_O^\phantom{O}}{\underline{U}_I^\phantom{O}}$ is to be rewritten so that it becomes independent of component values $R$ and $L$.\\ +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**: This allows for a generalized representation. This representation is called **normalization**:
  
 <WRAP centeralign> <WRAP centeralign>
-$\large{\underline{A} = \frac {\underline{U}_O^\phantom{O}}{\underline{U}_I^\phantom{O}} = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}\cdot e^{j\left(\frac{\pi}{2} - arctan \frac{\omega L}{R} \right)}}$ +\begin{align*}  
-$ \quad  \quad \vphantom{\HUGE{I \\ I}} \large{\xrightarrow{\text{normalization}}} \vphantom{\HUGE{I \\ I}} \quad \quad \quad $ +\large{\underline{A}  = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}}  
-$\large{\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 {\omega L}    {\sqrt{R^2 +    (\omega L)^2}}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan \frac{\omega L}{R} \right)}} 
-$\large{= \frac {x}{\sqrt{1 + x^2}} \cdot e^{j\left(\frac{\pi}{2} - arctan x \right)} }$ + \quad  \quad \vphantom{\HUGE{I \\ I}} \large{\xrightarrow{\text{normalization}}} \vphantom{\HUGE{I \\ I}} \quad \quad \quad  
-<WRAP>+\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>
  
  
Zeile 141: Zeile 152:
 Here, too, the behavior determined in the limit value observation can be seen:  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 small frequencies $\omega$ (corresponds to small $x$), the amplitude response tends toward zero. 
-  * At high frequencies, the ratio $U_O / U_I = 1 $ is established.+  * At high frequencies, the ratio $U_{\rm O} / U_{\rm I} = 1 $ is established.
  
 Interesting in the phase response is the point $x = 1$. 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 $U_O / U_I$. +  * 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) $U_O / U_I = 1$ 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: $R \gg \omega L$. 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 The point $x=1$ just marks the cut-off frequency. \\ It holds
  
 +<WRAP group>
 +<WRAP quarter column > </WRAP>
 +<WRAP quarter column > 
 \begin{align*}  \begin{align*} 
-\underline{A}_{norm} = \frac{x}{\sqrt{1 + x^2}} \cdot e^{j\left(\frac{\pi}{2} - arctan x \right)} = \frac{U_O}{U_I} \cdot e^{j\varphi} +\vphantom{\HUGE{I }} \\ 
-\quad \quad \left\{\begin{array}{l} +\underline{A}_{\rm norm} = \frac{x}{\sqrt{1 + x^2}}    \cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - arctan x \right)}  
-x \ll 1 & \widehat{=}& \omega L \ll R &: \quad\quad \frac{U_O}{U_I}=x                  &, \varphi = \frac{\pi}{2}  \, \widehat{=} \, 90° \\ +                         = \frac{U_{\rm O}}{U_{\rm I}} \cdot {\rm e}^{{\rm j}\varphi}  
-x \gg 1 & \widehat{=}& \omega L \gg R &: \quad\quad \frac{U_O}{U_I}=1                  &, \varphi = 0              \: \widehat{=} \, 0° \\ +\end{align*} 
-x   = 1 & \widehat{=}& \omega L = R   &: \quad\quad \frac{U_O}{U_I}=\frac{1}{\sqrt{2}} &, \varphi = \frac{\pi}{4}  \, \widehat{=} \, 45°+</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{array} \right.
 \end{align*} \end{align*}
 +</WRAP>
 +<WRAP quarter column > </WRAP>
 +</WRAP>
 +
 +
  
 <callout icon="fa fa-exclamation" color="red" title="Reminder:"> <callout icon="fa fa-exclamation" color="red" title="Reminder:">
  
-  * The **cut-off frequency** $f_c$ 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 cut-off 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 cut-off 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 $f_{Gr}$+  * 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> 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*}  \begin{align*} 
-R            &= \omega L \\ +R               &= \omega L \\ 
-\omega_{c}  &= \frac{R}{L} \\ +\omega _{\rm c} &= \frac{R}{L} \\ 
-2 \pi f_{c} &= \frac{R}{L} \quad \rightarrow \quad \boxed{f_{c} = \frac{R}{2 \pi \cdot L}} \end{align*}+2 \pi f_{\rm c} &= \frac{R}{L} \quad \rightarrow \quad \boxed{f_{\rm c} = \frac{R}{2 \pi \cdot L}} \end{align*}
  
 ==== 7.2.2 RL Low Pass ==== ==== 7.2.2 RL Low Pass ====
Zeile 181: Zeile 206:
 <WRAP> <imgcaption imageNo04 | Circuit, pointer diagram, and amplitude and phase response of RL low-pass filter> </imgcaption> {{drawio>AmplitudenPhasengangRLTiefpass.svg}} </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 $\underline{U}_O$ is tapped at the inductance. \\+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.  Here we will briefly discuss what happens when the two components are swapped.
  
Zeile 187: Zeile 212:
  
 \begin{align*}  \begin{align*} 
-\underline{A}_{norm} = \frac {1}{\sqrt{1 + (\omega L / R)^2}}\cdot e^{-j \; arctan \frac{\omega L}{R} } +\underline{A}_{\rm norm} = \frac {1}{\sqrt{1 + (\omega L / R)^2}}\cdot {\rm e}^{-{\rm j} \; \arctan \frac{\omega L}{R} } 
 \end{align*} \end{align*}
  
-The cut-off frequency is again given by $f_{c} = \frac{R}{2 \pi \cdot L}$.+The cut-off frequency is again given by $f_{\rm c} = \frac{R}{2 \pi \cdot L}$.
  
 <WRAP centeralign>  <WRAP centeralign> 
Zeile 211: Zeile 236:
  
 \begin{align*}  \begin{align*} 
-\underline{A}_{norm} = \frac {\omega RC}{\sqrt{1 + (\omega RC)^2}}\cdot e^{\frac{\pi}{2}-j \; arctan (\omega RC) } +\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*} \end{align*}
  
Zeile 217: Zeile 242:
  
 \begin{align*}  \begin{align*} 
-R &= \frac{1}{\omega_{c} C} \\ +               R &= \frac{1}{\omega_{\rm c} C} \\ 
- \omega_{c} &= \frac{1}{RC} \\  + \omega_{\rm c}  &= \frac{1}{RC} \\  
-2 \pi f_{c} &= \frac{1}{RC} \quad \rightarrow \quad \boxed{f_{c} =\frac{1}{2 \pi\cdot RC} } +2 \pi  f_{\rm c} &= \frac{1}{RC} \quad \rightarrow \quad  
 +\boxed{f_{\rm c}  = \frac{1}{2 \pi\cdot RC} } 
 \end{align*} \end{align*}
  
Zeile 239: Zeile 265:
  
 \begin{align*}  \begin{align*} 
-\underline{A}_{norm} = \frac {1}{\sqrt{1 + (\omega RC)^2}}\cdot e^{-j \; arctan (\omega RC) } +\underline{A}_{\rm norm} = \frac {1}{\sqrt{1 + (\omega RC)^2}}\cdot {\rm e}^{-{\rm j} \; \arctan (\omega RC) } 
 \end{align*} \end{align*}
  
-Also, the cut-off frequency is given by $f_{c} =\frac{1}{2 \pi\cdot RC}$+Also, the cut-off frequency is given by $f_{\rm c} =\frac{1}{2 \pi\cdot RC}$
  
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