Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
electrical_engineering_1:circuits_under_different_frequencies [2022/01/10 02:02] 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: [[: |
- | + | ||
- | ===== 7.1 Frequency-dependent voltage divider ===== | + | |
< | < | ||
Zeile 35: | Zeile 35: | ||
</ | </ | ||
- | ==== 7.1.1 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~~ | ||
- | ==== 7.1.2 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 79: | 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 92: | ||
These three points are now to be gone through. | These three points are now to be gone through. | ||
- | === 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 113: | 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*} |
- | === 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~~ | ||
- | ==== 7.1.3 RC Series Circuit ==== | + | ===== 7.3 RC Series Circuit |
- | === 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 180: | 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 186: | Zeile 242: | ||
\begin{align*} | \begin{align*} | ||
- | R &= \frac{1}{\omega_{c} C} \omega_{c} &= \frac{1}{RC} 2 \pi f_{c} &= \frac{1}{RC} \quad \rightarrow \quad \boxed{f_{c} =\frac{1}{2 \pi\cdot RC} } | + | R &= \frac{1}{\omega_{\rm c} C} \\ |
+ | \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~~ | ||
- | === 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 200: | 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~~ | ||
- | ===== 7.2 Resonance phenomena ===== | + | < |
- | + | < | |
- | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | </ |
- | + | {{url> | |
- | ==== RLC - Series Resonant Circuit ==== | + | </ |
- | + | ||
- | < | + | |
- | + | ||
- | 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 $U_I$) 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} | + | |
- | \underline{U} | + | |
- | \underline{Z}_{ges} & | + | |
- | \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 | + | |
- | 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 or Q-factor** | + | |
- | + | ||
- | \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, | + | |
- | * 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** | + | |
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- | \begin{align*} | + | |
- | \boxed{ d_S = \frac{1}{Q_S} = R \sqrt{\frac{C}{L}} } | + | |
- | \end{align*} | + | |
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- | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | |
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- | < | + | |
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- | < | + | |
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- | ~~PAGEBREAK~~ ~~CLEARFIX~~ | + | |
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- | ====== Examples ====== | + | |
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- | === Decoupling capacitor on the microcontroller === | + | |
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- | [[http:// | + | |
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- | Further details can be found [[https:// | + | |
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- | === Crystal as Resonant Circuit === | + | |
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- | [[http:// | + | |
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- | [[https:// | + | |
- | [[https:// |