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100644 assets/images/2024/2024-03-02-ecg/invert_filter.png create mode 100644 assets/images/2024/2024-03-02-ecg/noise_opamp.png create mode 100644 assets/images/2024/2024-03-02-ecg/opamp-KVL.png create mode 100644 assets/images/2024/2024-03-02-ecg/opamp-opamp21.gif create mode 100644 assets/images/2024/2024-03-02-ecg/opamp-opamp21.png diff --git a/Gemfile b/Gemfile index dea2b9a..d84dd6e 100644 --- a/Gemfile +++ b/Gemfile @@ -8,4 +8,5 @@ group :jekyll_plugins do gem 'jekyll-timeago', '~> 0.13.1' gem 'jekyll-scholar', '~> 7.0.0' gem 'jekyll-feed', '~> 0.17.0' + gem 'jekyll-figure', '~> 0.2.0' end diff --git a/_bibliography/references.bib b/_bibliography/references.bib index c7ec9a7..221fe4f 100644 --- a/_bibliography/references.bib +++ b/_bibliography/references.bib @@ -15,3 +15,111 @@ @BOOK{book:Nahin1998-wi address = "Princeton, NJ", language = "en" } +@article{Venkatachalam2011, + title = {Signals and Signal Processing for the Electrophysiologist: Part I: Electrogram Acquisition}, + volume = {4}, + ISSN = {1941-3084}, + url = {http://dx.doi.org/10.1161/CIRCEP.111.964304}, + DOI = {10.1161/circep.111.964304}, + number = {6}, + journal = {Circulation: Arrhythmia and Electrophysiology}, + publisher = {Ovid Technologies (Wolters Kluwer Health)}, + author = {Venkatachalam, K.L. and Herbrandson, Joel E. and Asirvatham, Samuel J.}, + year = {2011}, + month = dec, + pages = {965–973} +}, +@inbook{inbook, + author = {Malmivuo, Jaakko and Plonsey, Robert}, + year = {1975}, + month = {01}, + pages = {277-289}, + title = {Bioelectromagnetism. 15. 12-Lead ECG System}, + isbn = {978-0195058239}, + url = {https://www.bem.fi/book/15/15.htm} +}, +@instamp{instamp, + title = {The Differential Amplifier}, + url = {https://www.electronics-tutorials.ws/opamp/opamp_5.html} +}, +@bandfilter{bandfilter, + title = {Active Band Pass Filter}, + url = {https://www.electronics-tutorials.ws/filter/filter_7.html} +}, +@noise{noise, + title = {Operational Amplifiers: Noise Calculations of Instrumentation Amplifier Circuits}, + url = {https://www.renesas.com/us/en/document/apn/r13an0011-noise-calculations-instrumentation-amplifier-circuits-rev100} +}, +@texas_noise{texas_noise, + title = {Noise Analysis in Operational Amplifier Circuits}, + url = {https://www.ti.com/lit/an/slva043b/slva043b.pdf?ts=1706030563174&ref_url=https%253A%252F%252Fsearch.brave.com%252F} +}, +@article{YDU2017, + title = {Design of an ECG Sensor Circuitry for Cardiovascular Disease Diagnosis}, + volume = {2}, + ISSN = {2573-2838}, + url = {http://dx.doi.org/10.15406/ijbsbe.2017.02.00032}, + DOI = {10.15406/ijbsbe.2017.02.00032}, + number = {4}, + journal = {International Journal of Biosensors & Bioelectronics}, + publisher = {MedCrave Group, LLC}, + author = {Y DU, Winncy }, + year = {2017}, + month = may +}, +@ARTICLE{air_pollution, + +AUTHOR={Zhang, Shugang and Lu, Weigang and Wei, Zhiqiang and Zhang, Henggui}, + +TITLE={Air Pollution and Cardiac Arrhythmias: From Epidemiological and Clinical Evidences to Cellular Electrophysiological Mechanisms}, + +JOURNAL={Frontiers in Cardiovascular Medicine}, + +VOLUME={8}, + +YEAR={2021}, + +URL={https://www.frontiersin.org/articles/10.3389/fcvm.2021.736151}, + +DOI={10.3389/fcvm.2021.736151}, + +ISSN={2297-055X}, + +ABSTRACT={Cardiovascular disease is the leading cause of death worldwide and kills over 17 million people per year. In the recent decade, growing epidemiological evidence links air pollution and cardiac arrhythmias, suggesting a detrimental influence of air pollution on cardiac electrophysiological functionality. However, the proarrhythmic mechanisms underlying the air pollution-induced cardiac arrhythmias are not fully understood. The purpose of this work is to provide recent advances in air pollution-induced arrhythmias with a comprehensive review of the literature on the common air pollutants and arrhythmias. Six common air pollutants of widespread concern are discussed, namely particulate matter, carbon monoxide, hydrogen sulfide, sulfur dioxide, nitrogen dioxide, and ozone. The epidemiological and clinical reports in recent years are reviewed by pollutant type, and the recently identified mechanisms including both the general pathways and the direct influences of air pollutants on the cellular electrophysiology are summarized. Particularly, this review focuses on the impaired ion channel functionality underlying the air pollution-induced arrhythmias. Alterations of ionic currents directly by the air pollutants, as well as the alterations mediated by intracellular signaling or other more general pathways are reviewed in this work. Finally, areas for future research are suggested to address several remaining scientific questions.} +}, +@ARTICLE{pharmacokinetic, + +AUTHOR={Zou, Huixi and Banerjee, Parikshit and Leung, Sharon Shui Yee and Yan, Xiaoyu}, + +TITLE={Application of Pharmacokinetic-Pharmacodynamic Modeling in Drug Delivery: Development and Challenges}, + +JOURNAL={Frontiers in Pharmacology}, + +VOLUME={11}, + +YEAR={2020}, + +URL={https://www.frontiersin.org/articles/10.3389/fphar.2020.00997}, + +DOI={10.3389/fphar.2020.00997}, + +ISSN={1663-9812}, + +ABSTRACT={With the advancement of technology, drug delivery systems and molecules with more complex architecture are developed. As a result, the drug absorption and disposition processes after administration of these drug delivery systems and engineered molecules become exceedingly complex. As the pharmacokinetic and pharmacodynamic (PK-PD) modeling allows for the separation of the drug-, carrier- and pharmacological system-specific parameters, it has been widely used to improve understanding of the in vivo behavior of these complex delivery systems and help their development. In this review, we summarized the basic PK-PD modeling theory in drug delivery and demonstrated how it had been applied to help the development of new delivery systems and modified large molecules. The linkage between PK and PD was highlighted. In particular, we exemplified the application of PK-PD modeling in the development of extended-release formulations, liposomal drugs, modified proteins, and antibody-drug conjugates. Furthermore, the model-based simulation using primary PD models for direct and indirect PD responses was conducted to explain the assertion of hypothetical minimal effective concentration or threshold in the exposure-response relationship of many drugs and its misconception. The limitations and challenges of the mechanism-based PK-PD model were also discussed.} +}, + + +@Article{ideal_ecg, +AUTHOR = {Awal, Md. Abdul and Mostafa, Sheikh Shanawaz and Ahmad, Mohiuddin and Alahe, Mohammad Ashik and Rashid, Mohd Abdur and Kouzani, Abbas Z. and Mahmud, M. A. Parvez}, +TITLE = {Design and Optimization of ECG Modeling for Generating Different Cardiac Dysrhythmias}, +JOURNAL = {Sensors}, +VOLUME = {21}, +YEAR = {2021}, +NUMBER = {5}, +ARTICLE-NUMBER = {1638}, +URL = {https://www.mdpi.com/1424-8220/21/5/1638}, +PubMedID = {33652721}, +ISSN = {1424-8220}, +ABSTRACT = {The electrocardiogram (ECG) has significant clinical importance for analyzing most cardiovascular diseases. ECGs beat morphologies, beat durations, and amplitudes vary from subject to subject and diseases to diseases. Therefore, ECG morphology-based modeling has long-standing research interests. This work aims to develop a simplified ECG model based on a minimum number of parameters that could correctly represent ECG morphology in different cardiac dysrhythmias. A simple mathematical model based on the sum of two Gaussian functions is proposed. However, fitting more than one Gaussian function in a deterministic way has accuracy and localization problems. To solve these fitting problems, two hybrid optimization methods have been developed to select the optimal ECG model parameters. The first method is the combination of an approximation and global search technique (ApproxiGlo), and the second method is the combination of an approximation and multi-start search technique (ApproxiMul). The proposed model and optimization methods have been applied to real ECGs in different cardiac dysrhythmias, and the effectiveness of the model performance was measured in time, frequency, and the time-frequency domain. The model fit different types of ECG beats representing different cardiac dysrhythmias with high correlation coefficients (>0.98). Compared to the nonlinear fitting method, ApproxiGlo and ApproxiMul are 3.32 and 7.88 times better in terms of root mean square error (RMSE), respectively. Regarding optimization, the ApproxiMul performs better than the ApproxiGlo method in many metrics. Different uses of this model are possible, such as a syntactic ECG generator using a graphical user interface has been developed and tested. In addition, the model can be used as a lossy compression with a variable compression rate. A compression ratio of 20:1 can be achieved with 1 kHz sampling frequency and 75 beats per minute. These optimization methods can be used in different engineering fields where the sum of Gaussians is used.}, +DOI = {10.3390/s21051638} +} \ No newline at end of file diff --git a/_config.yml b/_config.yml index 435b1ed..3949fae 100644 --- a/_config.yml +++ b/_config.yml @@ -38,6 +38,7 @@ plugins: - jekyll-mermaid - jekyll-feed - jekyll/scholar + - jekyll-figure mermaid: src: https://cdn.jsdelivr.net/npm/mermaid/dist/mermaid.min.js @@ -46,6 +47,12 @@ mermaid: sass: style: compressed +scholar: + style: ieee + +jekyll-figure: + paragraphs: false + exclude: - "*.gemspec" - LICENSE.md diff --git a/_includes/head.html b/_includes/head.html index 636ecf6..ade0690 100644 --- a/_includes/head.html +++ b/_includes/head.html @@ -6,13 +6,14 @@ + - + diff --git a/_posts/2023-08-17-nernst-potential.md b/_posts/2023-08-17-nernst-potential.md index 02ac636..a50a082 100644 --- a/_posts/2023-08-17-nernst-potential.md +++ b/_posts/2023-08-17-nernst-potential.md @@ -113,4 +113,6 @@ $$\Delta V = \frac{1}{\beta q}\ln\left(\frac{n_1}{n_2}\right)$$ Which is the Nernst Potential with $\beta = \frac{1}{k_B T}$. -{% bibliography %} \ No newline at end of file +--- + +{% bibliography --cited %} \ No newline at end of file diff --git a/_posts/2024-03-02-ecg.md b/_posts/2024-03-02-ecg.md new file mode 100644 index 0000000..9a7989c --- /dev/null +++ b/_posts/2024-03-02-ecg.md @@ -0,0 +1,173 @@ +--- +layout: post +title: "The ECG Circuit" +description: "Theoretical analysis of the electrocardiogram circuit" +#image: assets/images/pic01.jpg +--- + +An Electrocardiogram (ECG) device is designed to detect and enhance faint electrical signals generated by the heart, allowing medical professionals to analyze them for diagnostic purposes. As the incoming signals are relatively weak, in the range of millivolts (0.001 to 100 mV according to {% cite YDU2017 %}), the ECG device needs to amplify them to a level that can be effectively processed by a microcontroller. Additionally, the device must employ adequate filtering techniques to remove unwanted noise from the signal; it is recommended to neglect frequencies exceeding a 0.05 to 100 Hz range. Another important consideration is the presence of a direct current (DC) component of approximately 500 mV due to contact between the electrodes and the patient's skin {% cite Venkatachalam2011 %}. + + +## 12 Lead ECG System + +The 12-lead ECG system is a method of measuring the electrical activity of the heart using four electrodes placed on each of the limbs and six on the chest. From the limb electrodes, an extra virtual electrode, called the Wilson's central terminal (VCT), is created by averaging the signals from the two electrodes on each arm and one on the left leg. This central terminal is used as a reference point for the six electrodes on the chest. Another three leads, called the augmented limb leads, measure the potential difference between the limb electrodes and the Wilson's central terminal. All the 12 leads can be seen in {% cite inbook %}. + +# Components + +## Instrumentation Amplifier + +The instrumentation amplifier is chosen as a safer alternative to measure the potential difference between electrodes to avoid causing discomfort to the patient. This is due to its ideally infinite input impedance, drawing a negligible amount of current from the patient's body. + +{: refdef: style="text-align: center;"} +{% figure caption:"Theoretical Schematic of an ECG circuit." label:fig_schematic%} +![Theoretical Schematic of the ECG circuit.]({{ site.baseurl }}/assets/images/2024/2024-03-02-ecg/ecg/ecg_simple.svg){: width="80%" } +{% endfigure %} +{: refdef} + +A schematic of the instrumentation amplifier {% cite instamp %} is presented in {% figref fig_schematic %}. The power supplies are ignored for simplicity. The use of the Op-Amps at the input terminals ensure that almost no current is extracted from the sample. The resulting signal is then fed into a standard differential amplifier. + +{: refdef: style="text-align: center;"} +{% figure caption:"Realistic Schematic of an ECG circuit." label:fig_bandfilter %} +![Realistic Schematic of the ECG circuit.]({{ site.baseurl }}/assets/images/2024/2024-03-02-ecg/ecg/ecg_with_numbers.svg){: width="80%" } +{% endfigure %} +{: refdef} + +### Gain Calculation + +For simplicity, we will assume that all resistances have the same value of $R$ except for $R_g$. As for ideal Op-Amps no current flows on the input ends, a KVL across path P1 highlighted in magenta on the {% figref fig_schematic %} yields: + +\begin{equation}\label{eq_r_g} +V_1 - R_g I - V_2 = 0 \implies R_g I = V_1 - V_2 +\end{equation} + +As the current along the path P2 highlighted in orange on the {% figref fig_schematic %} is the same since the bifurcations are met against the inputs of Op-Amps, we have that the KVL along this curve is: + +$$ V_\mathrm{out} + R I + R I + (2 R I + R_g I) - R I - R I = 0 \implies V_\mathrm{out} = -(2 R I + R_g I) $$ + +. + +Replacing $I$ with \eqref{eq_r_g} + +$$ V_\mathrm{out} = -(2R + R_g)(V_1 - V_2)/R_g = (2R/R_1 + 1)(V_2 - V_1)$$ + +. + +That is, the gain of this amplifier is entirely controlled by $R_g$ if all the resistances are kept the same. For this design, this gain factor can be chosen within the range of 51 to 101 to account for the different scales of the signals. + +## Band Pass Filter + +Once the signal is captured by the instrumental amplifiers, we still have to filter out the incoming noise and DC component due to the contact between the electrodes and the patient's skin. Both of these tasks can be accomplished with the use of a band pass filter, which allows only a certain range of frequencies to pass through. Typically, a Notch filter is also included to remove the 50/60 Hz noise caused by the power grid, but this model assumes that the input power is already stabilized a priori, as there are some situations where the signal will have meaningful components around this range {% cite Venkatachalam2011 %}. There are many ways to design a band pass filter, the one presented the {% figref fig_schematic %}, which was obtained on {% cite bandfilter %}, is called an Inverting Band Pass Filter. The reason for such a name becomes apparent when we calculate the transfer function of the system. + +### Transfer Function Calculation + +To start, the parallel RC combination at the top of the circuit can be easily calculated with the use of the equivalent impedance $Z_2$ of a parallel circuit: + +$$ Z_2 = 1/(1/R_2 + 1/Z_{C_2}) $$ + +In here, $Z_{C_2}$ is the impedance of the capacitor $C_2$, which is just $1/(s C_2)$. + +Doing a KVL path from the input $V_\mathrm{in}$ to the ground while passing through the input terminals of the Op-Amp, we have: + +\begin{equation}\label{eq:band_filter_current} + V_\mathrm{in} - R_1 I - Z_{C_1} I = 0 \implies I = V_\mathrm{in}/(R_1 + Z_{C_1}) +\end{equation} + +The reason I chose only a single symbol for the current $I$ is that the ideally infinite impedance of the inverting input of the Op-Amp ensures that the current won't split between the two branches. + +Another KVL also starting from $V_\mathrm{in}$ and passing through $R_1$, $C_1$, the parallel $R C$ until $V_\mathrm{out}$ yields: + +$$ V_\mathrm{in} - R_1 I - Z_{C_1} I - Z_2 I = V_\mathrm{out} \implies V_\mathrm{out} = - Z_2 I $$ + +Replacing the current with the result found in \eqref{eq:band_filter_current}, we have: + +$$ V_\mathrm{out} = - V_\mathrm{in}\frac{Z_2}{R_1 + Z_{C_1}} = - V_\mathrm{in}\frac{1}{(1/R_2 + 1/Z_{C_2})(R_1 + Z_{C_1})} $$ + +The minus signal at the output is an indication that the signal is inverted. Replacing the capacitor impedances with their respective expressions, the transfer function of the circuit $V_\mathrm{out}/V_\mathrm{in}:=H$ becomes: + +$$ H(s) = -\frac{s C_1 R_2}{(1 + s R_2 C_2)(1 + s C_1 R_1)} $$ + +The RC combinations in the transfer function stand for the inverse of the frequency thresholds of the circuit. A more comprehensive analysis can be done by reducing the number of variables by letting $R_1 C_1 = 1/\omega_1$, $R_2 C_2 = 1/\omega_2$, and $R_2 = r R_1$. + +\begin{equation}\label{eq:transfer_function_pretty} + H(s) = -\frac{r s \omega_2}{(\omega_2 - \omega_1)} \left(\frac{1}{\omega_1 + s} - \frac{1}{\omega_2 + s}\right) +\end{equation} + +An important edge case is when $s = 0$, corresponding to an wave of zero frequency, or a DC signal. For this case, the transfer function is zero, meaning that the filter excludes DC signals as a natural consequence of its design. + +# Input Referred-Noise + +Input referred noise models rely on the superposition principle, modelling the noisy output signal as a sum of the ideal output plus a noisy source caused by the components. With this idea, the output noise is calculated by adding various noise sources in the circuit, and assuming that no external input is present. To make calculations easier, each noise source is treated separately and then added together to obtain the total output noise. + +### Noise Model Instrumentation Amplifier + +In the case of an instrumentation amplifier, the noise can be modelled in function of each of the three Op-Amps as in the {% figref fig_in_amp_noise_simpler %} {% cite noise %}. + +{: refdef: style="text-align: center;"} +{% figure caption:"Simplified noise model for the instrumentation amplifier." label:fig_in_amp_noise_simpler %} +![Simplified noise model for the instrumentation amplifier.]({{ site.baseurl }}/assets/images/2024/2024-03-02-ecg/inamp_noise_simpler.png) +{% endfigure %} +{: refdef} + +The mean squared contribution of the noise caused by the Instrumentation Amplifier ($E_\mathrm{no}^2$) is {% cite noise %} + +\begin{equation}\label{eq_ENO} + \overline{E}_\mathrm{no}^2 = G^2 \Delta E (e_n^2 + 2 I_n^2 (R_S / 2)^2) +\end{equation}. + +In the equation \eqref{eq_ENO}, $G$ stands for the gain of the amplifier, $e_n$ is the voltage noise of the Op-Amp, $I_n$ is the current noise of the Op-Amp, and $R_S$ is the source resistance. The symbol $\Delta E$ is the noise equivalent bandwidth and is calculated as {% cite noise %} + +$$ \Delta E= 1.57 \times \text{"INA Bandwidth"}(G) $$ + +### Noise Model Band Pass Filter + +For the case of the band pass filter, we will make use of the Op-Amp noise model as presented in {% figref fig_opamp_noise %} {% cite texas_noise %}. + +{: refdef: style="text-align: center;"} +{% figure caption:"Noise model for the Op-Amp." label:fig_opamp_noise %} +![Noise model for the Op-Amp.]({{ site.baseurl }}/assets/images/2024/2024-03-02-ecg/noise_opamp.png) +{% endfigure %} +{: refdef} + +The topology of the circuit is similar to the band pass filter presented in the {% figref fig_schematic %}, with the exception that some of the resistances will have to be replaced by impedances. A conversion table between the two is presented in {% figref tab_impedance %}. + +{% figure caption: "Conversion table between the resistances of the band pass filter and their respective impedances." label:tab_impedance %} +|*Noise Model*|*Band Pass Filter*| +|---|---| +|$R_1$|$Z_1 = R_1 + \frac{1}{s C_1}$| +|$R_2$|$Z_2 = \left(\frac{1}{R_2} + s C_2\right)^{-1}$| +|$R_3$|0| +{% endfigure %} + +Making the appropriate variable changes in the total mean square noise voltage of the Op-Amp ($E_\mathrm{filter}^2$) as presented in {% cite texas_noise %}, we have: + +\begin{equation}\label{eq_noise_filter} +\overline{E}^2_\mathrm{filter} = \frac{1}{2\pi} \mathrm{abs}\left(\int \mathrm{d} \omega \left( \overline{E}^2_\mathrm{no} \left( \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{e_2}^2 + \overline{e_n}^2 \left(1 + \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{I}_\mathrm{nn}(Z_2^2(i \omega)) \right)\right) +\end{equation} + +. + +Where $Z_1$ and $Z_2$ are the respective impedances presented in {% figref tab_impedance %}. A little bit of algebraic manipulation reveals that the ratio $Z_2/Z_1$ is simply equation \eqref{eq:transfer_function_pretty} without the minus sign. So, we can further simplify \eqref{eq_noise_filter} as: + +$${\overline{E}_\mathrm{filter}}^2 = 1/(2\pi)\left(\int \mathrm{d} \omega ({\overline{E}_\mathrm{no}}^2 H^2(i \omega) + \overline{e_2}^2 + \overline{e_n}^2 (1-H(i \omega))^2 + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))\right)$$ + +Rearranging: + +$$ {\overline{E}_\mathrm{filter}}^2 = 1/(2\pi) \mathrm{abs}(\int \mathrm{d} \omega (({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2) H^2(i \omega) + (\overline{e_2}^2 + \overline{e_n}^2) \-2\overline{e_n}^2H(i \omega) + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))) $$ + +In the worst case scenario, this integral is bounded by the triangle inequality: + +$$ {\overline{E}_\mathrm{filter}}^2 \leq 1/(2\pi) \int \mathrm{d} \omega (({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2) \mathrm{abs}(H(i \omega))^2 + (\overline{e_2}^2 + \overline{e_n}^2) + 2( \overline{e_n}^2) \mathrm{abs}(H(i \omega)) + {\overline{I}_\mathrm{nn}} \mathrm{abs}(Z_2(i \omega))^2) $$ + +In which I have assumed that all the voltage and current noise sources aren't complex. Now, we will abuse the boundedness of the $\mathrm{abs}(H(i \omega))$ and $\mathrm{abs}(Z_2(i \omega))$ functions to use the ML inequality: + +$$ {\overline{E}_\mathrm{filter}}^2 \leq (\Delta \omega)/(2\pi) (({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2)(2r \omega_2/(\omega_2 \- \omega_1))^2 \ + (\overline{e_2}^2 + \overline{e_n}^2) \ + 2( \overline{e_n}^2) (2r \omega_2/(\omega_2 \-\omega_1)) + R_2 {\overline{I}_\mathrm{nn}} ) $$ + +. + +The term $\Delta \omega$, not to be confused with $\omega_2 \- \omega_1$, stands for the noise bandwidth of the Op-Amp in rad/s. Equation \eqref{eq:noise_upper_bound} with the definition of ${\overline{E}_\mathrm{no}}$ {% cite eq:ENO} provide a maximum upper bound to the output noise caused by the conjuction of the instrumentation amplifier and the band pass filter. + +Since all the $\overline{e_j}$ factors have a $k_B T$ term for noise due to thermal agitation, stabilizing the circuit's temperature with coolers will help with noise reduction. + +--- + +{% bibliography --cited %} \ No newline at end of file diff --git a/assets/images/2024/2024-03-02-ecg/bode_ang.svg b/assets/images/2024/2024-03-02-ecg/bode_ang.svg new file mode 100644 index 0000000..8229cf9 --- /dev/null +++ b/assets/images/2024/2024-03-02-ecg/bode_ang.svg @@ -0,0 +1,1457 @@ + + + + + + + + 2024-01-27T00:29:53.360303 + image/svg+xml + + + Matplotlib v3.7.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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zyZ(F1g3`(LNr$#Sac4^B-$ZXB>vK0~Pl0*s22j~AwsNWsg{t$i)W>DIW2RZ0tXf!* zKcHt{=r?e%P0PP*7Iovta_?s(4u0rEAbh>z+N=?;nPH!!GQ{kBmA828eeA# z1j5&dP4m_4^N+m&e>!ikMNvFzyhkP4%Edyu;8!r4YZ-TJou8-7wgdPhv<++_#^^4q z>uzD{ZYg5!Y6%>bIP`xU;NWCoYvuDV Z2k`tpfSYLI3pn7?6Im6RPf{kY{tqjU_vQcq literal 0 HcmV?d00001 From a022acb56b668ba46c35b431356a7ebba8c71957 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=C3=8Dcaro?= Date: Thu, 21 Mar 2024 00:06:20 -0300 Subject: [PATCH 2/2] Edit ECG article --- _posts/2024-03-02-ecg.md | 28 +++++++++++++++++----------- 1 file changed, 17 insertions(+), 11 deletions(-) diff --git a/_posts/2024-03-02-ecg.md b/_posts/2024-03-02-ecg.md index 9a7989c..91fb081 100644 --- a/_posts/2024-03-02-ecg.md +++ b/_posts/2024-03-02-ecg.md @@ -46,7 +46,7 @@ $$ V_\mathrm{out} + R I + R I + (2 R I + R_g I) - R I - R I = 0 \implies V_\math . -Replacing $I$ with \eqref{eq_r_g} +Replacing $I$ with equation \eqref{eq_r_g} $$ V_\mathrm{out} = -(2R + R_g)(V_1 - V_2)/R_g = (2R/R_1 + 1)(V_2 - V_1)$$ @@ -78,7 +78,7 @@ Another KVL also starting from $V_\mathrm{in}$ and passing through $R_1$, $C_1$, $$ V_\mathrm{in} - R_1 I - Z_{C_1} I - Z_2 I = V_\mathrm{out} \implies V_\mathrm{out} = - Z_2 I $$ -Replacing the current with the result found in \eqref{eq:band_filter_current}, we have: +Replacing the current with the result found in equation \eqref{eq:band_filter_current}, we have: $$ V_\mathrm{out} = - V_\mathrm{in}\frac{Z_2}{R_1 + Z_{C_1}} = - V_\mathrm{in}\frac{1}{(1/R_2 + 1/Z_{C_2})(R_1 + Z_{C_1})} $$ @@ -128,43 +128,49 @@ For the case of the band pass filter, we will make use of the Op-Amp noise model {% endfigure %} {: refdef} -The topology of the circuit is similar to the band pass filter presented in the {% figref fig_schematic %}, with the exception that some of the resistances will have to be replaced by impedances. A conversion table between the two is presented in {% figref tab_impedance %}. +The topology of the circuit is similar to the band pass filter presented in the {% figref fig_schematic %}, with the exception that some of the resistances will have to be replaced by impedances. A conversion table between the two is presented in the table below ({% figref tab_impedance %}). {% figure caption: "Conversion table between the resistances of the band pass filter and their respective impedances." label:tab_impedance %} + |*Noise Model*|*Band Pass Filter*| |---|---| |$R_1$|$Z_1 = R_1 + \frac{1}{s C_1}$| |$R_2$|$Z_2 = \left(\frac{1}{R_2} + s C_2\right)^{-1}$| |$R_3$|0| + {% endfigure %} Making the appropriate variable changes in the total mean square noise voltage of the Op-Amp ($E_\mathrm{filter}^2$) as presented in {% cite texas_noise %}, we have: -\begin{equation}\label{eq_noise_filter} -\overline{E}^2_\mathrm{filter} = \frac{1}{2\pi} \mathrm{abs}\left(\int \mathrm{d} \omega \left( \overline{E}^2_\mathrm{no} \left( \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{e_2}^2 + \overline{e_n}^2 \left(1 + \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{I}_\mathrm{nn}(Z_2^2(i \omega)) \right)\right) +\begin{equation} +\label{eq_noise_filter} \end{equation} +$$ \overline{E}^2_\mathrm{filter} = \frac{1}{2\pi} \mathrm{abs}\left(\int \mathrm{d} \omega \left( \overline{E}^2_\mathrm{no} \left( \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{e_2}^2 + \overline{e_n}^2 \left(1 + \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{I}_\mathrm{nn}(Z_2^2(i \omega)) \right)\right) $$ + . -Where $Z_1$ and $Z_2$ are the respective impedances presented in {% figref tab_impedance %}. A little bit of algebraic manipulation reveals that the ratio $Z_2/Z_1$ is simply equation \eqref{eq:transfer_function_pretty} without the minus sign. So, we can further simplify \eqref{eq_noise_filter} as: +Where $Z_1$ and $Z_2$ are the respective impedances presented in {% figref tab_impedance %}. A little bit of algebraic manipulation reveals that the ratio $Z_2/Z_1$ is simply equation \eqref{eq:transfer_function_pretty} without the minus sign. So, we can further simplify equation \eqref{eq_noise_filter} as: -$${\overline{E}_\mathrm{filter}}^2 = 1/(2\pi)\left(\int \mathrm{d} \omega ({\overline{E}_\mathrm{no}}^2 H^2(i \omega) + \overline{e_2}^2 + \overline{e_n}^2 (1-H(i \omega))^2 + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))\right)$$ +$${\overline{E}_\mathrm{filter}}^2 = \frac{1}{2\pi}\left(\int \mathrm{d} \omega ({\overline{E}_\mathrm{no}}^2 H^2(i \omega) + \overline{e_2}^2 + \overline{e_n}^2 (1-H(i \omega))^2 + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))\right)$$ Rearranging: -$$ {\overline{E}_\mathrm{filter}}^2 = 1/(2\pi) \mathrm{abs}(\int \mathrm{d} \omega (({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2) H^2(i \omega) + (\overline{e_2}^2 + \overline{e_n}^2) \-2\overline{e_n}^2H(i \omega) + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))) $$ +$$ {\overline{E}_\mathrm{filter}}^2 = \frac{1}{2\pi} \mathrm{abs}\left(\int \mathrm{d} \omega (({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2) H^2(i \omega) + (\overline{e_2}^2 + \overline{e_n}^2) - 2\overline{e_n}^2H(i \omega) + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))\right) $$ In the worst case scenario, this integral is bounded by the triangle inequality: -$$ {\overline{E}_\mathrm{filter}}^2 \leq 1/(2\pi) \int \mathrm{d} \omega (({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2) \mathrm{abs}(H(i \omega))^2 + (\overline{e_2}^2 + \overline{e_n}^2) + 2( \overline{e_n}^2) \mathrm{abs}(H(i \omega)) + {\overline{I}_\mathrm{nn}} \mathrm{abs}(Z_2(i \omega))^2) $$ +$$ {\overline{E}_\mathrm{filter}}^2 \leq \frac{1}{2\pi} \int \mathrm{d} \omega \left(({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2) \mathrm{abs}(H(i \omega))^2 + (\overline{e_2}^2 + \overline{e_n}^2) + 2\overline{e_n}^2 \mathrm{abs}(H(i \omega)) + {\overline{I}_\mathrm{nn}} \mathrm{abs}(Z_2(i \omega))^2 \right) $$ In which I have assumed that all the voltage and current noise sources aren't complex. Now, we will abuse the boundedness of the $\mathrm{abs}(H(i \omega))$ and $\mathrm{abs}(Z_2(i \omega))$ functions to use the ML inequality: -$$ {\overline{E}_\mathrm{filter}}^2 \leq (\Delta \omega)/(2\pi) (({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2)(2r \omega_2/(\omega_2 \- \omega_1))^2 \ + (\overline{e_2}^2 + \overline{e_n}^2) \ + 2( \overline{e_n}^2) (2r \omega_2/(\omega_2 \-\omega_1)) + R_2 {\overline{I}_\mathrm{nn}} ) $$ +\begin{equation}\label{eq_noise_upper_bound}\end{equation} + +$$ {\overline{E}_\mathrm{filter}}^2 \leq \frac{\Delta \omega}{2\pi} \left(({\overline{E}_\mathrm{no}}^2 + \overline{e_n}^2)\left(2r \frac{\omega_2}{\omega_2 - \omega_1}\right)^2 \ + (\overline{e_2}^2 + \overline{e_n}^2) \ + 2( \overline{e_n}^2) \left(2r \frac{\omega_2}{\omega_2 -\omega_1}\right) + R_2 {\overline{I}_\mathrm{nn}} \right) $$ . -The term $\Delta \omega$, not to be confused with $\omega_2 \- \omega_1$, stands for the noise bandwidth of the Op-Amp in rad/s. Equation \eqref{eq:noise_upper_bound} with the definition of ${\overline{E}_\mathrm{no}}$ {% cite eq:ENO} provide a maximum upper bound to the output noise caused by the conjuction of the instrumentation amplifier and the band pass filter. +The term $\Delta \omega$, not to be confused with $\omega_2 \- \omega_1$, stands for the noise bandwidth of the Op-Amp in rad/s. Equation \eqref{eq_noise_upper_bound} with the definition of ${\overline{E}_\mathrm{no}}$ equation \eqref{eq_ENO} provide a maximum upper bound to the output noise caused by the conjuction of the instrumentation amplifier and the band pass filter. Since all the $\overline{e_j}$ factors have a $k_B T$ term for noise due to thermal agitation, stabilizing the circuit's temperature with coolers will help with noise reduction.