Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
circuit_design:6_filter_circuits_ii [2021/12/05 19:28] slinn |
circuit_design:6_filter_circuits_ii [2023/09/19 22:17] mexleadmin |
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- | ====== 6. Filter Circuits II - Higher Order Filters ====== | + | ====== 6 Filter Circuits II - Higher Order Filters ====== |
===== 6.1 Bandpass filter ===== | ===== 6.1 Bandpass filter ===== | ||
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- | \\ {{drawio> | + | \\ {{drawio> |
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- | When analyzing different signals, only a part of the entire frequency spectrum is desired. In <imgref pic0>, the channels of the WLAN standard 802.11 are shown as an example; these are used alternately for data transmission. Another example arises with vibration spectra of a motor in a machine, which contains not only the vibrations (usable for diagnostics), but also interference from other machine parts. Other examples are cabled data transmission or [[https:// | + | When analyzing different signals, only a part of the entire frequency spectrum is desired. In <imgref pic0>, the channels of the WLAN standard 802.11 are shown as an example; these are used alternately for data transmission. Another example arises with the vibration spectra of a motor in a machine, which contains not only the vibrations (usable for diagnostics) but also interference from other machine parts. Other examples are cabled data transmission or [[https:// |
- | To separate the desired frequencies, | + | To separate the desired frequencies, |
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The range between the two frequencies is called the **passband**, | The range between the two frequencies is called the **passband**, | ||
- | Outside the passband, the gain drops off. A real filter | + | Outside the passband, the gain drops off. A real filter |
Also, there are various ideal filters where outside the passband, gain does not approach zero, but just falls below a threshold. | Also, there are various ideal filters where outside the passband, gain does not approach zero, but just falls below a threshold. | ||
Often the sloping region is called the **transition region** and the region below the threshold is called the **blocking region**. | Often the sloping region is called the **transition region** and the region below the threshold is called the **blocking region**. | ||
- | The threshold itself is called **blocking area**. In <imgref pic1b>, the ranges are drawn. However, the terms are not clearly defined; in various textbooks, the transition region is already called the blocking region. | + | The threshold itself is called |
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=== Assembling the bandpass filter === | === Assembling the bandpass filter === | ||
- | This filter can be composed | + | This filter can be composed |
- | $$\underline{A}_{BP}= {{\underline{U}_A}\over{\underline{U}_E}} = {{\underline{U}_A}\over{\underline{U}_1}} \cdot {{\underline{U}_1}\over{\underline{U}_E}} = \underline{A}_{TP} \cdot \underline{A}_{HP}$$ | + | $$\underline{A}_{\rm BP}= {{\underline{U}_{\rm O}}\over{\underline{U}_{\rm I}}} = {{\underline{U}_{\rm O}}\over{\underline{U}_1}} \cdot {{\underline{U}_1}\over{\underline{U}_{\rm I}}} = \underline{A}_{\rm LP} \cdot \underline{A}_{\rm HP}$$ |
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- | <WRAP right>< | ||
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- | \\ {{drawio> | ||
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=== Amplitude response of the bandpass filter === | === Amplitude response of the bandpass filter === | ||
- | In <imgref pic2>, the amplitude response of the bandpass filter can be seen. Since in the amplitude response the transfer function is represented in $dB$ ($\underline{A}^{dB}$), | ||
- | Important: The cutoff frequency of the low-pass filter $f_{Gr,TP}$ must be larger than the cutoff frequency of the high-pass filter $f_{Gr,HP}$ (see <imgref pic2>). | + | In <imgref pic2>, the amplitude response of the bandpass filter can be seen. Since in the amplitude response, the transfer function is represented in $\rm dB$ ($\underline{A}^{\rm dB}$), multiplying the transfer functions of the low-pass and high-pass filters $\underline{A}_{\rm LP}$ and $\underline{A}_{\rm HP}$ results in an addition of $\underline{A}_{\rm LP}^{\rm dB}$ and $\underline{A}_{\rm HP}^{\rm dB}$. In the amplitude response, we can see that it results in a $20 ~\rm dB/dec$ change twice: once at $f_{\rm c, HP}$ and once at $f_{\rm c, LP}$. So the filter has an order of 2. |
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+ | Important: The cutoff frequency of the low-pass filter $f_{\rm c, LP}$ must be larger than the cutoff frequency of the high-pass filter $f_{\rm c, HP}$ (see <imgref pic2>). | ||
But what does the frequency response look like? This is to be derived in the following. | But what does the frequency response look like? This is to be derived in the following. | ||
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Zeile 75: | Zeile 76: | ||
The extremal value consideration yields: | The extremal value consideration yields: | ||
* for $ \boldsymbol{\omega \rightarrow 0} $:\\ The magnitude of the impedance of the capacitances becomes large \\ and thus $|\underline{X}_{C_1}| \gg R_1$ , as well as $|\underline{X}_{C_2}| \gg R_2$ \\ Thus $\underline{X}_{C_1}$ prevails at $\underline{Z}_1$ and $\underline{R}_2$ bei $\underline{Z}_2$. \\ $\rightarrow$ **A reverse differentiator results at low frequencies.** | * for $ \boldsymbol{\omega \rightarrow 0} $:\\ The magnitude of the impedance of the capacitances becomes large \\ and thus $|\underline{X}_{C_1}| \gg R_1$ , as well as $|\underline{X}_{C_2}| \gg R_2$ \\ Thus $\underline{X}_{C_1}$ prevails at $\underline{Z}_1$ and $\underline{R}_2$ bei $\underline{Z}_2$. \\ $\rightarrow$ **A reverse differentiator results at low frequencies.** | ||
- | * for $ \boldsymbol{\omega \rightarrow \infty} $:\\ The magnitude of the impedance of the capacitances becomes small and thus $|\underline{X}_{C_1}| \ll R_1$ , as well as $|\underline{X}_{C_2}| \ll R_2$ \\ Thus $\underline{R}_1$ predominates at $\underline{Z}_1$ and $\underline{X}_{C_2}$ predominates at $\underline{Z}_2$. \\ $\rightarrow$ **A reverse integrator results at high frequencies.** | + | * for $ \boldsymbol{\omega \rightarrow \infty} $:\\ The magnitude of the impedance of the capacitances becomes small and thus $|\underline{X}_{C_1}| \ll R_1$, as well as $|\underline{X}_{C_2}| \ll R_2$ \\ Thus $\underline{R}_1$ predominates at $\underline{Z}_1$ and $\underline{X}_{C_2}$ predominates at $\underline{Z}_2$. \\ $\rightarrow$ **A reverse integrator results at high frequencies.** |
\\ | \\ | ||
Zeile 83: | Zeile 84: | ||
The transfer function is again to be derived from a complex-valued inverting amplifier: | The transfer function is again to be derived from a complex-valued inverting amplifier: | ||
- | $\underline{A}_V = {{\underline{U}_A}\over{\underline{U}_E}} = - {{\underline{Z}_2}\over{\underline{Z}_1}} = - {\underline{Z}_2}\cdot {1\over{\underline{Z}_1}} = - \Large{{{R_2\cdot {1\over{j\omega C_2}}}\over{{R_2 + {1\over{j\omega C_2}}}}}}\cdot{1 \over{{R_1 + {1\over{j\omega C_1}}}}}= | + | $\underline{A}_{\rm V} = {{\underline{U}_{\rm O}}\over{\underline{U}_{\rm I}}} = - {{\underline{Z}_2}\over{\underline{Z}_1}} = - {\underline{Z}_2}\cdot {1\over{\underline{Z}_1}} = - \Large{{{R_2\cdot {1\over{{\rm j}\omega C_2}}}\over{{R_2 + {1\over{{\rm j}\omega C_2}}}}}}\cdot{1 \over{{R_1 + {1\over{{\rm j}\omega C_1}}}}}= |
- | $\boxed{\underline{A}_V = - \color{blue}{R_2 \over R_1 } \cdot \large\color{teal}{1 \over {1+ j\omega \cdot C_2 R_2}} \cdot \large\color{brown}{{j\omega \cdot C_1 R_1} \over {1+ j\omega \cdot C_1 R_1}}}$ | + | $\boxed{\underline{A}_{\rm V} = - \color{blue}{R_2 \over R_1 } \cdot \large\color{teal}{1 \over {1+ {\rm j}\omega \cdot C_2 R_2}} \cdot \large\color{brown}{{{\rm j}\omega \cdot C_1 R_1} \over {1+ {\rm j}\omega \cdot C_1 R_1}}}$ |
\\ | \\ | ||
- | Clever | + | Better |
- $- \color{blue}{R_2 \over R_1 }$: This corresponds to a [[3_opamp_basic_circuits_i# | - $- \color{blue}{R_2 \over R_1 }$: This corresponds to a [[3_opamp_basic_circuits_i# | ||
- | - $\large\color{teal}{1 \over {1+ j\omega \cdot C_2 R_2}}$: This corresponds to a [[5_filter circuits_i# | + | - $\large\color{teal}{1 \over {1+ {\rm j}\omega \cdot C_2 R_2}}$: This corresponds to a [[5_filter circuits_i# |
- | - $\large\color{brown}{{j\omega \cdot C_1 R_1} \over {1+ j\omega \cdot C_1 R_1}}$ This corresponds to a [[5_filter circuits_i# | + | - $\large\color{brown}{{{\rm j}\omega \cdot C_1 R_1} \over {1+ {\rm j}\omega \cdot C_1 R_1}}$ This corresponds to a [[5_filter circuits_i# |
\\ | \\ | ||
This results in a function via the extremal value consideration: | This results in a function via the extremal value consideration: | ||
- | * for $ \boldsymbol{\omega \rightarrow 0 } $:\\ $\underline{A}_V = - \Large{R_2 \over R_1 } \cdot \Large\color{teal}{1 \over {1+ \color{black}{\underbrace{\color{teal}{j\omega \cdot C_2 R_2}}_{\rightarrow 0}}}} \cdot \Large\color{brown}{{j\omega \cdot C_1 R_1} \over {1+ \color{black}{\underbrace{\color{brown}{j\omega \cdot C_1 R_1}}_{\rightarrow 0}}}} \rightarrow - {R_2 \over R_1 } \cdot \color{teal}{ 1 \over 1} \cdot \Large\color{brown}{{j\omega \cdot C_1 R_1} \over 1} \rightarrow - \color{brown}{\normalsize{j\omega \cdot C_1 \color{black}{R_2}}}$ \\ The equation is the same as that of a reverse differentiator \\ \\ | + | * for $ \boldsymbol{\omega \rightarrow 0 } $:\\ $\underline{A}_{\rm V} = - \Large{R_2 \over R_1 } \cdot \Large\color{teal}{1 \over {1+ \color{black}{\underbrace{\color{teal}{{\rm j}\omega \cdot C_2 R_2}}_{\rightarrow 0}}}} \cdot \Large\color{brown}{{{\rm j}\omega \cdot C_1 R_1} \over {1+ \color{black}{\underbrace{\color{brown}{{\rm j}\omega \cdot C_1 R_1}}_{\rightarrow 0}}}} \rightarrow - {R_2 \over R_1 } \cdot \color{teal}{ 1 \over 1} \cdot \Large\color{brown}{{{\rm j}\omega \cdot C_1 R_1} \over 1} \rightarrow - \color{brown}{\normalsize{{\rm j}\omega \cdot C_1 \color{black}{R_2}}}$ \\ The equation is the same as that of a reverse differentiator \\ \\ |
- | * for $ \omega \rightarrow \infty $:\\ $\underline{A}_V = - \Large{R_2 \over R_1 } \cdot \Large\color{teal}{1 \over {1+ j\omega \cdot C_2 R_2}} \cdot \Large\color{brown}{{j\omega \cdot C_1 R_1} \over \color{brown}{1+ {j\omega \cdot C_1 R_1}}} \rightarrow - {R_2 \over R_1 } \cdot \color{teal}{ 1 \over {j\omega \cdot C_2 R_2}} \cdot \Large\color{brown}{1 \over 1} \rightarrow - \color{teal}{1 \over {j\omega \cdot C_2 \color{black}{R_1}}}$ \\ The equation is equivalent to that of an inverse integrator | + | * for $ \omega \rightarrow \infty $:\\ $\underline{A}_{\rm V} = - \Large{R_2 \over R_1 } \cdot \Large\color{teal}{1 \over {1+ {\rm j}\omega \cdot C_2 R_2}} \cdot \Large\color{brown}{{{\rm j}\omega \cdot C_1 R_1} \over \color{brown}{1+ {{\rm j}\omega \cdot C_1 R_1}}} \rightarrow - {R_2 \over R_1 } \cdot \color{teal}{ 1 \over {{\rm j}\omega \cdot C_2 R_2}} \cdot \Large\color{brown}{1 \over 1} \rightarrow - \color{teal}{1 \over {{\rm j}\omega \cdot C_2 \color{black}{R_1}}}$ \\ The equation is equivalent to that of an inverse integrator |
\\ | \\ | ||
== Determination of magnitude and phase from complex-valued observation == | == Determination of magnitude and phase from complex-valued observation == | ||
- | For the magnitude $|\underline{A}_V|$ of the transfer function, the following hint can be used: $|a\cdot b\cdot c| = |a| \cdot |b| \cdot |c| $. \\ | + | For the magnitude $|\underline{A}_{\rm V}|$ of the transfer function, the following hint can be used: $|a\cdot b\cdot c| = |a| \cdot |b| \cdot |c| $. \\ |
- | Thus, for the magnitude $|\underline{A}_V|$, we get: \\ | + | Thus, for the magnitude $|\underline{A}_{\rm V}|$, we get: \\ |
- | $ | + | $ |
- | $\xrightarrow{\color{teal}{\omega_{Gr, TP}}, \ \ \color{brown}{\omega_{Gr, HP}}}$ | + | $\xrightarrow{\color{teal}{\omega_{\rm c, LP}}, \ \ \color{brown}{\omega_{\rm c, HP}}}$ |
- | $\boxed{|\underline{A}_V| = {R_2 \over R_1 } \cdot \large{1 \over \sqrt{1+ \omega^2 / \color{teal}{\omega_{Gr, TP}^2}}} \cdot \large{{\omega / \color{brown}{\omega_{Gr, HP}}} \over \sqrt{1+ \omega^2 \color{brown} | + | $\boxed{|\underline{A}_{\rm V}| = {R_2 \over R_1 } \cdot \large{1 \over \sqrt{1+ \omega^2 / \color{teal}{\omega_{\rm c, LP}^2}}} \cdot \large{{\omega / \color{brown}{\omega_{\rm c, HP}}} \over \sqrt{1+ \omega^2 \color{brown}{\omega_{\rm c, HP}^2}}}}$ |
- | For the phase $\varphi$ must be conjugate complexly extended again. \\ | + | The phase $\varphi$ must be conjugate complexly extended again. \\ |
- | At first this produces an unwieldy equation - but a real-valued constant can be separated from it. | + | At first, this produces an unwieldy equation - but a real-valued constant can be separated from it. |
- | $\underline{A}_V = - \color{blue}\large{R_2 \over R_1 } $ | + | $\underline{A}_{\rm V} = - \color{blue}\large{R_2 \over R_1 } $ |
- | $\cdot \large\color{teal }{ 1 \over \color{lightgray}{\boxed{\color{teal }{\small{1+ j\omega \cdot C_2 R_2}}}}}$ | + | $\cdot \large\color{teal }{ 1 \over \color{lightgray}{\boxed{\color{teal }{\small{1+ |
- | $\cdot \large\color{teal }{{1- j\omega \cdot C_2 R_2} \over \color{lightgray}{\boxed{\color{teal }{\small{1- j\omega \cdot C_2 R_2}}}}}$ | + | $\cdot \large\color{teal }{{1- {\rm j}\omega \cdot C_2 R_2} \over \color{lightgray}{\boxed{\color{teal }{\small{1- |
- | $\cdot \large\color{brown}{{ j\omega \cdot C_1 R_1} \over \color{ pink }{\boxed{\color{brown}{\small{1+ j\omega \cdot C_1 R_1}}}}}$ | + | $\cdot \large\color{brown}{{ |
- | $\cdot \large\color{brown}{{1- j\omega \cdot C_1 R_1} \over \color{ pink }{\boxed{\color{brown}{\small{1- j\omega \cdot C_1 R_1}}}}}$ | + | $\cdot \large\color{brown}{{1- |
- | $\underline{A}_V = \quad \quad \mathcal{C} \quad \quad \quad \quad$ | + | $\underline{A}_{\rm V} = \quad \quad \mathcal{C} \quad \quad \quad \quad$ |
- | $ \cdot \color{teal }{(1- j\omega \cdot C_2 R_2)}$ | + | $ \cdot \color{teal }{(1- {\rm j}\omega \cdot C_2 R_2)}$ |
- | $\ \cdot \color{brown}{ j\omega \cdot C_1 R_1 }$ | + | $\ \cdot \color{brown}{ |
- | $\ \cdot \ \color{brown}{(1- j\omega \cdot C_1 R_1)}$ | + | $\ \cdot \ \color{brown}{(1- |
- | $\underline{A}_V = \quad \quad \mathcal{C} \quad \quad \quad$ | + | $\underline{A}_{\rm V} = \quad \quad \mathcal{C} \quad \quad \quad$ |
- | $ \cdot (j + \omega R_2 C_2 + \omega R_1 C_1 - j \omega R_1 C_1 \omega R_2 C_2)$ | + | $ \cdot ({\rm j} + \omega R_2 C_2 + \omega R_1 C_1 - {\rm j} \omega R_1 C_1 \omega R_2 C_2)$ |
- | From this equation it is easy to read the proportions for real part $\Re(\underline{A}_V)$ and imaginary part $\Im(\underline{A}_V)$. \\ | + | From this equation, it is easy to read the proportions for real part $\Re(\underline{A}_{\rm V})$ and imaginary part $\Im(\underline{A}_{\rm V})$. \\ |
This gives for the phase $\varphi$ : | This gives for the phase $\varphi$ : | ||
- | $ | + | $ |
- | $\xrightarrow{\color{teal}{\omega_{Gr, TP}}, \ \ \color{brown}{\omega_{Gr, HP}}}$ | + | $\xrightarrow{\color{teal}{\omega_{\rm c, LP}}, \ \ \color{brown}{\omega_{\rm c, HP}}}$ |
- | $\boxed{\varphi = arctan \left( \frac{\color{teal}{\omega_{Gr, TP}} \color{brown}{\omega_{Gr, HP}} - \omega^2 }{\omega (\color{teal}{\omega_{Gr, TP}}+\color{brown}{\omega_{Gr, HP}})} \right)}$ | + | $\boxed{\varphi = \arctan \left( \frac{\color{teal}{\omega_{\rm c, LP}} \color{brown}{\omega_{\rm c, HP}} - \omega^2 }{\omega (\color{teal}{\omega_{\rm c, LP}}+\color{brown}{\omega_{\rm c, HP}})} \right)}$ |
Zeile 137: | Zeile 138: | ||
The formal for the phase $\varphi$ says | The formal for the phase $\varphi$ says | ||
The extremal consideration can now be carried out for some salient frequencies: | The extremal consideration can now be carried out for some salient frequencies: | ||
- | * for $ \boldsymbol{\omega \rightarrow 0} $:\\ $\varphi(0) = arctan \left( \frac{\mathcal{C}_1 - \omega^2}{\omega \mathcal{C}_2} \right) \rightarrow arctan \left( \frac{\mathcal{C}_1 - " | + | * for $ \boldsymbol{\omega \rightarrow 0} $:\\ $\varphi(0) = \arctan \left( \frac{\mathcal{C}_1 - \omega^2}{\omega \mathcal{C}_2} \right) \rightarrow |
- | * for $ \boldsymbol{\omega \rightarrow \infty} $: \\ $\varphi(\infty) = arctan \left( \frac{\mathcal{C}_1 - \omega^2}{\omega \mathcal{C}_2} \right) \rightarrow arctan \left( \frac{\mathcal{C}_1 - " | + | * for $ \boldsymbol{\omega \rightarrow \infty} $: \\ $\varphi(\infty) = \arctan \left( \frac{\mathcal{C}_1 - \omega^2}{\omega \mathcal{C}_2} \right) \rightarrow |
- | * for a **(circular) frequency** $\boldsymbol{\omega= \omega_0}$ **for which the argument of the** $\boldsymbol{arctan}$** function becomes zero**. \\ Thus the phase: \\ $\varphi(\omega_0) = arctan \left( 0 \right)$. \\ The corresponding frequency is given by: \\ $\large\frac{\color{teal}{\omega_{Gr, TP}} \color{brown}{\omega_{Gr, HP}} - \omega^2 }{\omega (\color{teal}{\omega_{Gr, TP}}+\color{brown}{\omega_{Gr, HP}})} = 0 \quad\rightarrow\quad \omega_0^2 = \color{teal}{\omega_{Gr, TP}} \color{brown}{\omega_{Gr, HP}} \quad\rightarrow\quad \omega_0 = \large\sqrt{\color{teal}{\omega_{Gr, TP}} \color{brown}{\omega_{Gr, HP}}}$ \\ \\ | + | * for a **(circular) frequency** $\boldsymbol{\omega= \omega_0}$ **for which the argument of the** $\boldsymbol{\arctan}$** function becomes zero**. \\ Thus the phase: \\ $\varphi(\omega_0) = \arctan \left( 0 \right)$. \\ The corresponding frequency is given by: \\ $\large\frac{\color{teal}{\omega_{\rm c, LP}} \color{brown}{\omega_{\rm c, HP}} - \omega^2 }{\omega (\color{teal}{\omega_{\rm c, LP}}+\color{brown}{\omega_{\rm c, HP}})} = 0 \quad\rightarrow\quad \omega_0^2 = \color{teal}{\omega_{\rm c, LP}} \color{brown}{\omega_{\rm c, HP}} \quad\rightarrow\quad \omega_0 = \large\sqrt{\color{teal}{\omega_{\rm c, LP}} \color{brown}{\omega_{\rm c, HP}}}$ \\ \\ |
- | * for the cutoff frequency of the high pass filter $\boldsymbol{\omega = \color{brown}{\omega_{Gr, HP} = {1 \over {R_1 C_1}}}}$. \\ For this, if the passband is sufficiently large, $\color{brown}{\omega_{Gr, HP}} \ll \color{teal}\omega_{Gr, TP}$ can be assumed. \\ Thus we get: \\ $\varphi(\color{brown}{\omega_{Gr, HP}}) = arctan \left( \large\frac{\color{teal}{\omega_{Gr, TP}} \color{brown}{\omega_{Gr, HP}} - \color{brown}{\omega_{Gr, HP}}^2 }{\color{brown}{\omega_{Gr, HP}} (\color{teal}{\omega_{Gr, TP}}+\color{brown}{\omega_{Gr, HP}})} \right) | + | * for the cutoff frequency of the high pass filter $\boldsymbol{\omega = \color{brown}{\omega_{\rm c, HP} = {1 \over {R_1 C_1}}}}$. \\ For this, if the passband is sufficiently large, $\color{brown}{\omega_{\rm c, HP}} \ll \color{teal}\omega_{\rm c, LP}$ can be assumed. \\ Thus we get: \\ $\varphi(\color{brown}{\omega_{\rm c, HP}}) = \arctan \left( \large\frac{\color{teal}{\omega_{\rm c, LP}} \color{brown}{\omega_{\rm c, HP}} - \color{brown}{\omega_{\rm c, HP}}^2 }{\color{brown}{\omega_{\rm c, HP}} (\color{teal}{\omega_{\rm c, LP}}+\color{brown}{\omega_{\rm c, HP}})} \right) |
- | * for the cutoff frequency of the lowpass filter $\boldsymbol{\omega = \color{teal}{\omega_{Gr, TP} = {1 \over {R_2 C_2}}}}$. \\ For this, if the passband is sufficiently large, $\color{brown}{\omega_{Gr, HP}} \gg \color{teal}{\omega_{Gr, TP}}$ can be assumed. \\ Thus we have: \\ $\varphi(\color{teal}{\omega_{Gr, TP}}) = arctan \left( \large\frac{\color{teal}{\omega_{Gr, TP}} \color{brown}{\omega_{Gr, HP}} - \color{teal}{\omega_{Gr, TP}}^2 }{\color{teal}{\omega_{Gr, TP}} (\color{teal}{\omega_{Gr, TP}}+\color{brown}{\omega_{Gr, HP}})} \right) | + | * for the cutoff frequency of the lowpass filter $\boldsymbol{\omega = \color{teal}{\omega_{\rm c, LP} = {1 \over {R_2 C_2}}}}$. \\ For this, if the passband is sufficiently large, $\color{brown}{\omega_{\rm c, HP}} \gg \color{teal}{\omega_{\rm c, LP}}$ can be assumed. \\ Thus we have: \\ $\varphi(\color{teal}{\omega_{\rm c, LP}}) = \arctan \left( \large\frac{\color{teal}{\omega_{\rm c, LP}} \color{brown}{\omega_{\rm c, HP}} - \color{teal}{\omega_{\rm c, LP}}^2 }{\color{teal}{\omega_{\rm c, LP}} (\color{teal}{\omega_{\rm c, LP}}+\color{brown}{\omega_{\rm c, HP}})} \right) |
<WRAP right>< | <WRAP right>< | ||
< | < | ||
</ | </ | ||
- | \\ {{drawio> | + | \\ {{drawio> |
</ | </ | ||
Zeile 152: | Zeile 153: | ||
This results in the following for individual points: | This results in the following for individual points: | ||
- | ^ $\boldsymbol{\omega}\quad$ | | + | ^ $\boldsymbol{\omega}\quad$ | |
- | ^ $\boldsymbol{\varphi}$ | | + | ^ $\boldsymbol{\varphi}$ | |
^ | $+90°$ | ^ | $+90°$ | ||
- | The results also seem plausible with the course of the arc tangent (<fc # | + | The results also seem plausible with the course of the arc tangent (<fc # |
- | <fs x-large>ABER:</ | + | <fs x-large>BUT:</ |
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
Zeile 164: | Zeile 165: | ||
<panel type=" | <panel type=" | ||
- | - Consider again the [[# | + | - Consider again the [[# |
- Consider the circuit in the simulation below at the following points: | - Consider the circuit in the simulation below at the following points: | ||
- | - Increase of +20dB/dec at low frequencies. | + | - Increase of $+20~\rm dB/Dec$ at low frequencies. |
- Middle of the passband (" | - Middle of the passband (" | ||
- | - Drop of -20dB/dec at high frequencies \\ <WRAP outdent> <WRAP outdent> Which capacitor behaves like a short circuit at each point? \\ Knowing the behavior of the capacitors: What equivalent circuit describes the system in the forward region? </ | + | - Drop of $-20 ~\rm dB/Dec$ at high frequencies \\ <WRAP outdent> <WRAP outdent> Which capacitor behaves like a short circuit at each point? \\ Knowing the behavior of the capacitors: What equivalent circuit describes the system in the forward region? </ |
</ | </ | ||
- | < | + | < |
</ | </ | ||
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~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
==== 6.1.2 Multi-Feedback Bandpass ==== | ==== 6.1.2 Multi-Feedback Bandpass ==== | ||
- | < | + | < |
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</ | </ | ||
- | \\ {{drawio> | + | \\ {{drawio> |
</ | </ | ||
- | < | + | < |
< | < | ||
</ | </ | ||
- | \\ {{drawio> | + | \\ {{drawio> |
</ | </ | ||
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
- | ====== 6.2 Tape lock ====== | + | ====== 6.2 Band-Reject Filter |
- | Electrical Engineering | + | In Electrical Engineering |
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
<WRAP onlyprint> | <WRAP onlyprint> | ||
- | {{url> | + | {{url> |
- | From page [[https:// | + | From the page [[https:// |
</ | </ | ||
Zeile 210: | Zeile 211: | ||
Example: Evaluation of an infrared sensor: | Example: Evaluation of an infrared sensor: | ||
* Nodes are missing in the circuit from the manufacturer --> correct circuit is to be drawn. | * Nodes are missing in the circuit from the manufacturer --> correct circuit is to be drawn. | ||
- | * to which basic circuits do OPV 1 and 2 correspond? What filtenn | + | * to which basic circuits do OPV 1 and 2 correspond? What filter |
{{elektronische_schaltungstechnik: | {{elektronische_schaltungstechnik: | ||