Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
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electrical_engineering_1:circuits_under_different_frequencies [2022/01/12 09:30] tfischer |
electrical_engineering_1:circuits_under_different_frequencies [2023/09/19 23:37] (aktuell) mexleadmin |
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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:// | Further content can be found at this [[https:// | ||
| Zeile 5: | Zeile 5: | ||
| ==== Introduction ==== | ==== Introduction ==== | ||
| - | At the previous chapters it was explained | + | In the previous chapters, it was explained |
| 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*} | ||
| - | Complex impedances can be dealt with in much the same way as ohmic resistances in Electrical Engineering 1 (see: [[: | + | Complex impedances can be dealt with in much the same way as ohmic resistances in Electrical Engineering 1 (see: [[: |
| < | < | ||
| Zeile 33: | Zeile 35: | ||
| </ | </ | ||
| - | ===== 7.1 From Two-pole to Four-pole ===== | + | ===== 7.1 From Two-Terminal Network |
| - | < | + | < |
| - | 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 | + | 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 |
| - | For four-poles, 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{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}}{U_{\rm I}}\cdot | ||
| \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 |
| \end{align*} | \end{align*} | ||
| <callout icon=" | <callout icon=" | ||
| - | * The complex-valued quotient ${\underline{U}_O}/ | + | * The complex-valued quotient ${\underline{U}_{\rm O}}/ |
| - | * The frequency-dependent magnitude of the quotient $A(\omega)={U_O}/{U_I}$ is called **amplitude response** | + | * The frequency-dependent magnitude of the quotient $A(\omega)={U_{\rm O}}/{U_{\rm I}}$ is called **amplitude response** |
| </ | </ | ||
| - | The frequency | + | The frequency |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 67: | Zeile 70: | ||
| ===== 7.2 RL Series Circuit ===== | ===== 7.2 RL Series Circuit ===== | ||
| - | < | + | < |
| - | 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}} |
| + | | ||
| \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 limit consideration | + | For the first step, we investigate |
| * 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. |
| - | 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, | + | 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 | + | 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 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. | ||
| + | |||
| + | By this, the course from low to high frequencies is easier to see. The following simulation in <imgref imageNo5> | ||
| + | |||
| + | <WRAP centeralign> | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| + | |||
| + | |||
| + | For further consideration, | ||
| + | This allows for a generalized representation. This representation is called **normalization**: | ||
| + | |||
| + | <WRAP centeralign> | ||
| \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)} \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)} | + | \large{\underline{A} |
| - | \end{align*} | + | |
| + | \quad \quad \vphantom{\HUGE{I \\ I}} \large{\xrightarrow{\text{normalization}}} \vphantom{\HUGE{I \\ I}} \quad \quad \quad | ||
| + | \large{\underline{A}_{norm} | ||
| + | | ||
| + | \large{ | ||
| + | \end{align*} | ||
| + | </ | ||
| This equation behaves quite the same as the one considered so far. | This equation behaves quite the same as the one considered so far. | ||
| Zeile 111: | Zeile 146: | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | The transfer function can also be decomposed into amplitude response and frequency response. This can be done by | + | <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. |
| - | * the amplitude response double logarithmic | + | < |
| - | * the phase response | + | |
| - | 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. | + | 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, |
| - | + | ||
| - | 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, | + | |
| 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 >> | + | So this point marks a limit. Far to the left, the ohmic resistance is significantly greater |
| - | The point $x=1$ just marks the cut-off frequency. It holds | + | The point $x=1$ just marks the cut-off frequency. |
| + | <WRAP group> | ||
| + | <WRAP quarter column > </ | ||
| + | <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}\quad \quad \left \{\begin{array}{l}x \ll 1 & \widehat{=} \omega L \ll R &: \quad\quad \frac{U_O}{U_I}=x &, | + | \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*} | \end{align*} | ||
| + | </ | ||
| + | <WRAP quarter column > | ||
| + | \begin{align*} | ||
| + | \left\{\begin{array}{l} | ||
| + | x \ll 1 & \widehat{=}& | ||
| + | x \gg 1 & \widehat{=}& | ||
| + | x = 1 & \widehat{=}& | ||
| + | \end{array} \right. | ||
| + | \end{align*} | ||
| + | </ | ||
| + | <WRAP quarter column > </ | ||
| + | </ | ||
| + | |||
| + | |||
| <callout icon=" | <callout icon=" | ||
| - | * 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 |
| These statements apply to single-stage passive filters, i.e. one RL or one RC element. Multistage filters are considered in circuit engineering. </ | These statements apply to single-stage passive filters, i.e. one RL or one RC element. Multistage filters are considered in circuit engineering. </ | ||
| - | 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_{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 ==== | ||
| - | < | + | < |
| - | 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. 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: | In this case, the normalized transfer function is given by: | ||
| \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 |
| \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> | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 171: | Zeile 229: | ||
| ==== 7.3.1 RC High Pass ==== | ==== 7.3.1 RC High Pass ==== | ||
| - | < | + | < |
| 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. | 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 178: | 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 |
| \end{align*} | \end{align*} | ||
| Zeile 184: | Zeile 242: | ||
| \begin{align*} | \begin{align*} | ||
| - | R &= \frac{1}{\omega_{c} C} \omega_{c} = \frac{1}{RC} \\ | + | R &= \frac{1}{\omega_{\rm c} C} \\ |
| - | 2 \pi f_{c} &= \frac{1}{RC} \quad \rightarrow \quad \boxed{f_{c} =\frac{1}{2 \pi\cdot RC} } | + | \omega_{\rm c} &= \frac{1}{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*} | ||
| + | |||
| + | <WRAP centeralign> | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| Zeile 192: | Zeile 258: | ||
| ==== 7.3.2 RC Low Pass ==== | ==== 7.3.2 RC Low Pass ==== | ||
| - | < | + | < |
| Again, the voltage at the impedance is to be used as the output voltage. This results in a low-pass filter. | Again, the voltage at the impedance is to be used as the output voltage. This results in a low-pass filter. | ||
| Zeile 199: | 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 |
| \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 $f_{\rm c} =\frac{1}{2 \pi\cdot RC}$ |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | <WRAP centeralign> | ||
| + | < | ||
| + | </ | ||
| + | {{url> | ||
| + | </ | ||