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circuit_design:6_filter_circuits_ii [2023/03/28 16:54]
mexleadmin 		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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 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://en.wikipedia.org/wiki/Electroencephalography|bands of brain waves]]. 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://en.wikipedia.org/wiki/Electroencephalography|bands of brain waves]].
  
-To separate the desired frequencies, a filter can be used that only passes a given band between two frequencies (//frequency band//). This is possible with a **bandpass filter**.+To separate the desired frequencies, a filter can be used that only passes a given band between two frequencies (frequency band). This is possible with a **bandpass filter**.
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
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 $       |\underline{A}_{\rm V}| = {R_2 \over R_1 } \cdot \large{1 \over \sqrt{1+ \omega^2 C_2^2 R_2^2}} \cdot \large{{\omega \cdot C_1 R_1} \over \sqrt{1+ \omega^2 C_1^2 R_1^2}} $ $       |\underline{A}_{\rm V}| = {R_2 \over R_1 } \cdot \large{1 \over \sqrt{1+ \omega^2 C_2^2 R_2^2}} \cdot \large{{\omega \cdot C_1 R_1} \over \sqrt{1+ \omega^2 C_1^2 R_1^2}} $
 $\xrightarrow{\color{teal}{\omega_{\rm c, LP}}, \ \ \color{brown}{\omega_{\rm c, HP}}}$  $\xrightarrow{\color{teal}{\omega_{\rm c, LP}}, \ \ \color{brown}{\omega_{\rm c, HP}}}$ 
-$\boxed{|\underline{A}_{\rm V}| = {R_2 \over R_1 } \cdot \large{1 \over \sqrt{1+ \omega^2 / \color{teal}{\omega_{\rm c, LP}^2}}} \cdot \large{{\omega / \color{brown}{\omega_{\rm c, HP}}} \over \sqrt{1+ \omega^2 \color{brown} / {\omega_{\rm c, HP}}^2}}}$+$\boxed{|\underline{A}_{\rm V}| = {R_2 \over R_1 } \cdot \large{1 \over \sqrt{1+ \omega^2 / \color{teal}{\omega_{\rm c, LP}^2}}} \cdot \large{{\omega / \color{brown}{\omega_{\rm c, HP}}} \over \sqrt{1+ \omega^2 \color{brown}{\omega_{\rm c, HP}^2}}}}$
  
  
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 <panel type="info" title="Exercise 6.1.1 Bandpass based on the inverting amplifier"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 6.1.1 Bandpass based on the inverting amplifier"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-  - Consider again the [[#uebertragungsfunktion1| transfer function]] and find the complex gain for $\omega_0 = \large\sqrt{\color{teal}{\omega_{\rm c, LP}} \color{brown}{\omega_{\rm c, HP}}}$. \\ Is this value positive (= no phase shift) or negative (= phase shift by $\pm 180°$)?+  - Consider again the [[#uebertragungsfunktion1| transfer function]] and find the complex gain for $\omega_0 = \large\sqrt{\color{teal}{\omega_{\rm c, LP}} \cdot  \color{brown}{\omega_{\rm c, HP}}}$. \\ Is this value positive (= no phase shift) or negative (= phase shift by $\pm 180°$)?
   - Consider the circuit in the simulation below at the following points:   - Consider the circuit in the simulation below at the following points:
     - Increase of $+20~\rm dB/Dec$ at low frequencies.     - Increase of $+20~\rm dB/Dec$ at low frequencies.