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Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
circuit_design:5_filter_circuits_i [2023/03/28 15:02] mexleadmin |
circuit_design:5_filter_circuits_i [2023/07/17 11:59] (aktuell) mexleadmin |
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Zeile 11: | Zeile 11: | ||
=== Introductory example === | === Introductory example === | ||
- | < | + | < |
Various applications work in harsh environments, | Various applications work in harsh environments, | ||
Zeile 54: | Zeile 54: | ||
The conversion to a level in $dB$ is defined for current or voltage ratios by the following equation: | The conversion to a level in $dB$ is defined for current or voltage ratios by the following equation: | ||
- | $\boxed{A_{\rm V}^{\rm dB}=20 ~{\rm dB} \cdot \log_{10}\left(\frac{U_2}{U_1}\right)=20 dB \cdot \log_{10} A_\rm V}$(nbsp)(nbsp)(nbsp)(nbsp) | + | $\boxed{A_{\rm V}^{\rm dB}=20 ~{\rm dB} \cdot \log_{10}\left(\frac{U_2}{U_1}\right)=20 |
\\ | \\ | ||
Zeile 93: | Zeile 93: | ||
- For $A_{\rm V}= \color{green}{1} $ we get < | - For $A_{\rm V}= \color{green}{1} $ we get < | ||
- | $ A_{\rm V}^{\rm dB}(\color{green}{1}) = 20 {\rm dB} \cdot \underbrace{log_{10}\left(\color{green}{1}\right)}_\color{blue}{x} \qquad \qquad \rightarrow \boldsymbol{A_{\rm V}^{\rm dB}(\color{green}{1})} \quad = 20 {\rm dB} \cdot 0 \quad \ \boldsymbol{= 0 {\rm dB}}$ \\ | + | $ A_{\rm V}^{\rm dB}(\color{green}{1}) = 20 {\rm dB} \cdot \underbrace{\log_{10}\left(\color{green}{1}\right)}_\color{blue}{x} \qquad \qquad \rightarrow \boldsymbol{A_{\rm V}^{\rm dB}(\color{green}{1})} \quad = 20 {\rm dB} \cdot 0 \quad \ \boldsymbol{= 0 {\rm dB}}$ \\ |
Here $\color{blue}{x}$ has the value that gives $10^\color{blue}{x} = \color{green}{1}$, | Here $\color{blue}{x}$ has the value that gives $10^\color{blue}{x} = \color{green}{1}$, | ||
- For $A_{\rm V}= \color{green}{0.01} $ we get < | - For $A_{\rm V}= \color{green}{0.01} $ we get < | ||
- | $ A_{\rm V}^{\rm dB}(\color{green}{0.01}) = 20 {\rm dB} \cdot \underbrace{log_{10}\left(\color{green}{0.01}\right)}_\color{blue}{x} \qquad \rightarrow \boldsymbol{A_{\rm V}^{\rm dB}(\color{green}{0.01})} = 20 {\rm dB} \cdot (-2) \boldsymbol{= -40 {\rm dB}}$ \\ | + | $ A_{\rm V}^{\rm dB}(\color{green}{0.01}) = 20 {\rm dB} \cdot \underbrace{\log_{10}\left(\color{green}{0.01}\right)}_\color{blue}{x} \qquad \rightarrow \boldsymbol{A_{\rm V}^{\rm dB}(\color{green}{0.01})} = 20 {\rm dB} \cdot (-2) \boldsymbol{= -40 {\rm dB}}$ \\ |
Here $\color{blue}{x}$ has the value, that gives $10^\color{blue}{x} = \color{green}{0.01}$, | Here $\color{blue}{x}$ has the value, that gives $10^\color{blue}{x} = \color{green}{0.01}$, | ||
- For $A_{\rm V}= \color{green}{2} $, we get < | - For $A_{\rm V}= \color{green}{2} $, we get < | ||
- | $ A_{\rm V}^{\rm dB}(\color{green}{2}) = 20 {\rm dB} \cdot \underbrace{log_{10}\left(\color{green}{2}\right)}_\color{blue}{x} \qquad \qquad \rightarrow \boldsymbol{A_{\rm V}^{\rm dB}(\color{green}{2})} \quad\approx 20 {\rm dB} \cdot 0.30103 \boldsymbol{\approx 6 {\rm dB}}$ \\ | + | $ A_{\rm V}^{\rm dB}(\color{green}{2}) = 20 {\rm dB} \cdot \underbrace{\log_{10}\left(\color{green}{2}\right)}_\color{blue}{x} \qquad \qquad \rightarrow \boldsymbol{A_{\rm V}^{\rm dB}(\color{green}{2})} \quad\approx 20 {\rm dB} \cdot 0.30103 \boldsymbol{\approx 6 {\rm dB}}$ \\ |
Here $\color{blue}{x}$ has the value that gives $10^\color{blue}{x} = \color{green}{2}$. Thus, $\color{blue}{x}\approx 0.30103$. </ | Here $\color{blue}{x}$ has the value that gives $10^\color{blue}{x} = \color{green}{2}$. Thus, $\color{blue}{x}\approx 0.30103$. </ | ||
Zeile 149: | Zeile 149: | ||
- < | - < | ||
$A_{\rm V}^{dB}=58~{\rm dB} = 40~{\rm dB} + 18~{\rm dB} = \color{blue}{2}\cdot 20~{\rm dB} + \color{magenta}{3}\cdot 6~{\rm dB}$ \\ | $A_{\rm V}^{dB}=58~{\rm dB} = 40~{\rm dB} + 18~{\rm dB} = \color{blue}{2}\cdot 20~{\rm dB} + \color{magenta}{3}\cdot 6~{\rm dB}$ \\ | ||
- | This becomes linear | + | This becomes linear |
- < | - < | ||
$A_{\rm V}^{dB}=56~{\rm dB} = 80~{\rm dB} - 24~{\rm dB} = \color{blue}{4}\cdot 20~{\rm dB} + \color{magenta}{-4}\cdot 6~{\rm dB}$ \\ | $A_{\rm V}^{dB}=56~{\rm dB} = 80~{\rm dB} - 24~{\rm dB} = \color{blue}{4}\cdot 20~{\rm dB} + \color{magenta}{-4}\cdot 6~{\rm dB}$ \\ | ||
- | This becomes linear $ \qquad \qquad \qquad \qquad \qquad \qquad \ A_{\rm V} = 10^\color{blue}{4} \qquad \cdot \qquad | + | This becomes linear $ \qquad \qquad \qquad \qquad \qquad \qquad |
- < | - < | ||
$A_{\rm V}^{dB}=56~{\rm dB} = 40~{\rm dB} + 18~{\rm dB} - 3~{\rm dB} = \color{blue}{2}\cdot 20~{\rm dB} + \color{magenta}{3}\cdot 6~{\rm dB} + \color{teal}{-\frac{1}{2}}\cdot 6~{\rm dB}$ \\ | $A_{\rm V}^{dB}=56~{\rm dB} = 40~{\rm dB} + 18~{\rm dB} - 3~{\rm dB} = \color{blue}{2}\cdot 20~{\rm dB} + \color{magenta}{3}\cdot 6~{\rm dB} + \color{teal}{-\frac{1}{2}}\cdot 6~{\rm dB}$ \\ | ||
- | This becomes linear $ \qquad \qquad \qquad \qquad \qquad \qquad \qquad | + | This becomes linear $ \qquad |
\\ | \\ | ||
Zeile 223: | Zeile 223: | ||
- Phase response: linear phase over logarithmic frequency (i.e. single logarithmic representation) | - Phase response: linear phase over logarithmic frequency (i.e. single logarithmic representation) | ||
- | This gives a straight line in the amplitude response for functions of the form $A(\omega) \sim \omega^n$. | + | This gives a straight line in the amplitude response for functions of the form $A(\omega) \sim \omega^n$. |
+ | In particular, this is true for $A(\omega) \sim \omega$, i.e., a slope of $+ ~\rm 20dB/decade$, and for $A(\omega) \sim \frac{1}{\omega}$, | ||
+ | </ | ||
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
Zeile 365: | Zeile 367: | ||
$U_R=R\cdot I \qquad \qquad \qquad \qquad \qquad \rightarrow \underline{U}_R = R \cdot \underline{I}$ \\ | $U_R=R\cdot I \qquad \qquad \qquad \qquad \qquad \rightarrow \underline{U}_R = R \cdot \underline{I}$ \\ | ||
- | $U_C={ 1 \over C }\cdot(\int_{t_0}^{t_1} I_C \ {\rm d}t+ Q_0(t_0)) \qquad | + | $U_C={ 1 \over C }\cdot(\int_{t_0}^{t_1} I_C \ {\rm d}t+ Q_0(t_0)) \qquad |
However, this consideration can only be implemented under certain boundary conditions: | However, this consideration can only be implemented under certain boundary conditions: | ||
- | - **sinusoidal quantities**: | + | - **sinusoidal quantities**: |
- **Steady state**: The systems of equations consider only sinusoidal oscillations that have already existed for infinite time. This corresponds to a long time since the switch-on. This takes out disturbances that are generated by switching on. | - **Steady state**: The systems of equations consider only sinusoidal oscillations that have already existed for infinite time. This corresponds to a long time since the switch-on. This takes out disturbances that are generated by switching on. | ||
Zeile 475: | Zeile 477: | ||
< | < | ||
- | In the simulation above, the circuit from <imgref pic7_1> is shown again. In addition, two switches $S1$ and $S2$ are built into the circuit by which the various feedback paths can be disabled: | + | In the simulation above, the circuit from <imgref pic7_1> is shown again. In addition, two switches $\rm S1$ and $\rm S2$ are built into the circuit by which the various feedback paths can be disabled: |
- If only switch $\rm S1$ is closed, the circuit is an inverting amplifier. | - If only switch $\rm S1$ is closed, the circuit is an inverting amplifier. | ||
Zeile 673: | Zeile 675: | ||
</ | </ | ||
- | < | + | < |
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
Zeile 722: | Zeile 724: | ||
===== 5.4 High Pass Filter ===== | ===== 5.4 High Pass Filter ===== | ||
- | A high pass filter can be created from the inverting differentiator, | + | A high pass filter can be created from the inverting differentiator, |
< | < | ||
Zeile 761: | Zeile 763: | ||
~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
- | ===== 5.5 Overview high pass filter | + | ===== 5.5 Overview high pass Filter |
< | < | ||
- | < | + | < |
</ | </ | ||
\\ {{drawio> | \\ {{drawio> | ||
Zeile 806: | Zeile 808: | ||
The amplification factors are: | The amplification factors are: | ||
- | * $A_{V1} =80$ | + | * $A_{\rm V1} =80$ |
- | * $A_{V2} =0.0125$ | + | * $A_{\rm V2} =0.0125$ |
- | * $A_{V3} =250' | + | * $A_{\rm V3} =250' |
Calculate manually the resulting total amplification $A_{\rm V}$ as a linear factor and in $\rm dB$ ($A_{\rm V}^{\rm dB}$). | Calculate manually the resulting total amplification $A_{\rm V}$ as a linear factor and in $\rm dB$ ($A_{\rm V}^{\rm dB}$). | ||
Zeile 823: | Zeile 825: | ||
< | < | ||
^ Amp. Name $\rightarrow$ ^ lin. Factor $\rightarrow$^ re-arrange $\rightarrow$ ^ as exponents of $2$ and $10$ $\rightarrow$^ | ^ Amp. Name $\rightarrow$ ^ lin. Factor $\rightarrow$^ re-arrange $\rightarrow$ ^ as exponents of $2$ and $10$ $\rightarrow$^ | ||
- | | $A_{V1}$ | + | | $A_{\rm V1}$ | $80$ | $8 \cdot 10$ | $2^3 \cdot 10^1$ | $ 3 \cdot 6~{\rm dB} + |
- | | $A_{V2}$ | + | | $A_{\rm V2}$ | $0.0125$ |
- | | $A_{V3}$ | + | | $A_{\rm V3}$ | $250' |
</ | </ | ||
Zeile 849: | Zeile 851: | ||
- Briefly describe the differences in the amplitude response of the gain $A_{\rm V}$. | - Briefly describe the differences in the amplitude response of the gain $A_{\rm V}$. | ||
- What happens if instead of $R= 10 ~\rm k\Omega$, $C = 1.6 ~\rm µF$ the same time constant is implemented with $R= 10 ~\rm M\Omega$, $C = 1.6 ~\rm nF$? Verify this by comparing the Bode plot of the LM318 operational amplifier. | - What happens if instead of $R= 10 ~\rm k\Omega$, $C = 1.6 ~\rm µF$ the same time constant is implemented with $R= 10 ~\rm M\Omega$, $C = 1.6 ~\rm nF$? Verify this by comparing the Bode plot of the LM318 operational amplifier. | ||
- | - Simulate an inverting amplifier in Tina TI. To do this, replace the capacitor in the circuit with a resistor $R_2 = 10 \rm k\Omega$ and use the LM318 op-amp. | + | - Simulate an inverting amplifier in Tina TI. To do this, replace the capacitor in the circuit with a resistor $R_2 = 10 ~\rm k\Omega$ and use the LM318 op-amp. |
- Attach the Bode diagram. | - Attach the Bode diagram. | ||
- What should the Bode diagram (amplitude and phase response) look like for an ideal inverting amplifier? What deviation do you notice in the real setup? | - What should the Bode diagram (amplitude and phase response) look like for an ideal inverting amplifier? What deviation do you notice in the real setup? |