commit c795832288e2e42de47b6954abc5a719cd2a879d parent b0e20d9d9f9019cbe643cfdbbf97693b978022ec Author: Martin Kloeckner <mjkloeckner@gmail.com> Date: Mon, 13 Jul 2026 23:22:53 -0300 Agregar diseño de circuito de primer orden pasa bajos Diffstat:
16 files changed, 1159 insertions(+), 16 deletions(-) diff --git a/tp/informe/informe.pdf b/tp/informe/informe.pdf Binary files differ. diff --git a/tp/informe/informe.tex b/tp/informe/informe.tex @@ -98,7 +98,8 @@ citecolor=black } -\usepackage{subfig} +% \usepackage{subfig} +\usepackage{subcaption} \usepackage{fancyvrb} \fvset{xleftmargin=\mathindent} @@ -994,13 +995,148 @@ transferencia, es por esto que cada uno de los circutios o ``etapas'' se diseña por separado y luego se conecta la salida de uno a la entrada de otro, en ``cascada''. -\hypertarget{filtro-pasaaltos-de-primer-orden}{% -\subsection{Filtro pasa altos de primer orden}\label{filtro-pasaaltos-de-primer-orden}} +\hypertarget{filtro-pasa-altos-de-primer-orden}{% +\subsection{Filtro pasa altos de primer orden}\label{filtro-pasa-altos-de-primer-orden}} La transferencia $H_{1}(s)$ es de la forma de la ecuación +\eref{eq-trans-1gr-generica}, la cual se puede implementar mediante un divisor +resistivo con impedancias $\mathbb{Z}_{1}$ y $\mathbb{Z}_{2}$.\footnotemark[9] + +\begin{align} + T(s) = K\cdot \dfrac{s+z_{1}}{s+p_{1}} + \label{eq-trans-1gr-generica} +\end{align} + +Para un divisor resistivo con impedancias no inductivas, como el que se muestra +en la figura \ref{fig-impedance-divider}, la función de transferencia resulta +como se muestra en \eref{eq-trans-res-div}. En la misma ecuacion se muestra la +igualdad utilizando admitancias, que se definen como la inversa de la impedancia +$\mathbb{Y}=1/\mathbb{Z}$. Se utilizan admitancias porque simplifican las +expresiones de resistores y capactirores en paralelo. + +\begin{figure}[ht] + \centering + \begin{subfigure}{0.40\textwidth} + \centering + \includegraphics[width=\textwidth]{sch/complex_resistor_divider.png} + \caption{Valores genéricos} + \label{fig-impedance-divider} + \end{subfigure} + \hfill + \begin{subfigure}{0.55\textwidth} + \centering + \includegraphics[width=\textwidth]{sch/passive_highpass.png} + \caption{Valores calculados normalizados} + \label{fig-impedance-divider-norm-values} + \end{subfigure} + + \caption{Circuito pasa altos de primer orden con divisor resistivo} + \label{fig:circuito-considerado} +\end{figure} + +\begin{align} + \dfrac{V_{out}(s)}{V_{in}(s)} = + T(s) = + \dfrac{\mathbb{Z}_{2}}{\mathbb{Z}_{1}+\mathbb{Z}_{2}} = + \dfrac{\mathbb{Y}_{1}}{\mathbb{Y}_{1}+\mathbb{Y}_{2}}; + \quad \mathbb{Y}_{1} = s\ C_{1} + \dfrac{1}{R_{1}}; + \quad \mathbb{Y}_{2} = s\ C_{2} + \dfrac{1}{R_{2}}; + \label{eq-trans-res-div} +\end{align} + +\footnotetext[9]{Sección 3.2, ``Realization with Passive Elements'', \cite{schaumann2001}} + +Aplicando las expresiones de las admitancias de \eref{eq-trans-res-div} en la +ecuacion \eref{eq-trans-1gr-generica} de la transferencia generica, y con +$\mathbb{G}=1/R$ la conductancia, la transferencia del divisor resistivo resulta +como se muestra en \eref{eq-trans-res-div-comp}. Luego, sacando factor comun +$C_{1}$ en el numerador y $\left(C_{1} + C{2}\right)$ en el denominador se +obtiene una expresión similar a la expresión de la transferencia generica de +\eref{eq-trans-1gr-generica}. + +% Dado que no se utilizarán inductores y que se tienen resistores y capacitores en +% paralelo, la expresión de la transferencia en funcion de los resistores y +% capacitores se muestra a continuación, partiendo de la expresion de admitancia y +% con $\mathbb{G}=1/R$ la conductancia. + +\begin{align} + T(s) = + \dfrac{s\ C_{1} + G_{1}}{s\ \left(C_{1}+C_{2}\right)+G_{1}+G_{2}} = + \dfrac{C_{1}}{C_{1}+C_{2}} + \cdot\dfrac{s + + \dfrac{G_{1}}{C_{1}}}{s+\dfrac{G_{1}+G_{2}}{C_{1}+C_{2}}} + \label{eq-trans-res-div-comp} +\end{align} + +Igualando la ecuación \eref{eq-trans-res-div-comp} con la expresion generica de +\eref{eq-trans-1gr-generica} resulta como se muestra a continuacion en +\eref{eq-trans-res-div-comp-1}. + +\begin{align} + T(s) + = \dfrac{C_{1}}{C_{1}+C_{2}} + \cdot\dfrac{s + + \dfrac{G_{1}}{C_{1}}}{s+\dfrac{G_{1}+G_{2}}{C_{1}+C_{2}}} + = K\cdot \dfrac{s+z_{1}}{s+p_{1}} + \label{eq-trans-res-div-comp-1} +\end{align} + +De la expresion \eref{eq-trans-res-div-comp-1} se obtiene: + +\begin{align} + s &= z_{1} = \dfrac{G_{1}}{C_{1}} = \dfrac{1}{R_{1}\ C_{1}} + \label{eq-trans-res-div-z1}\\[6pt] + s &= p_{1} = \dfrac{G_{1}+G_{1}}{C_{1}+C_{2}} + = \dfrac{1}{\left(\dfrac{R_{1}\ R_{2}}{R_{1} + R_{2}}\right) + \left(C_{1}+C_{2}\right)} + \label{eq-trans-res-div-p1}\\[0pt] + K_{\text{LF}} &= T(0) = \dfrac{G_{1}}{G_{1}+G_{2}} + \label{eq-trans-res-div-dc-gain} \\[0pt] + K_{\text{HF}} &= T(\infty) = \dfrac{C_{1}}{C_{1}+C_{2}} \label{eq-trans-res-div-inf-gain} +\end{align} + +Para el caso de la transferencia $H_{1}(s)$ se tiene que $z_{1} = 250$, $p_{1} = +500$; la ganancia a frecuencias nulas $K=T(0)=0.5$ y para frecuencias muy altas +$K=T(\infty)=1$. Dada la ganancia $K$, entonces de +\eref{eq-trans-res-div-dc-gain} se obtiene $R_{1}=R_{2}$, y de +\eref{eq-trans-res-div-inf-gain} resulta que $C_{2}=0$. Imponiendo $C_{1} = +22nF$, entonces de \eref{eq-trans-res-div-z1}: $R_{1} = R_{2} \approx +181818\Omega$ y dado que $181818\Omega$ no es un valor comercial estandarizado +se toma el valor $180k\Omega$, esto último implica un error de aproximadamente +$1\%$. + +Reemplazando los valores en el circuito de la figura +\ref{fig-impedance-divider}, resulta en el circuito de la figura +\ref{fig-impedance-divider-norm-values}. El circuito con los valores +reemplazados se realizó una simulación en LTSpice de manera de comparar el +desempeño con la transferencia $H_{1}$. El resultado se muestra en el diagrama +de Bode de la figura \ref{fig-pasa-altos-primer-orden}, en la cual se observa que +la diferencia es imperceptible entre la respuesta en frecuencia del circuito +simulado y la transferencia real. + +\begin{figure}[!ht] + \centering + \includegraphics[width=\textwidth]{plot/bode_passive_highpass.png} + \caption{Diagrama de Bode de circuito simulado y teorico para $H_{1}(s)$} + \label{fig-pasa-altos-primer-orden} +\end{figure} + + +% circuito pasivo de solo con resistores y capascitores\footnotemark[9] tal como +% se muestra en la figura \eref{fig-pasa-bajos-primer-orden-inversor} del cual se +% tienen expresiones tabuladas para los componentes en funcion del polo ($p$), +% cero ($z$) y ganancia ($K$). Notese que si bien la transferencia es de primer +% orden debido a que asi lo es el polinomio del denominador $s+500$, se tiene un +% circuito con $2$ almacenadores de energÃa, $C_{1}$ y $C_{2}$, esto no implica +% que el circuito sea de segundo orden ya que $C_{1}$ y $C_{2}$ comparten el nodo, +% por lo que no son linealmente independientes. + + +\iffalse +La transferencia $H_{1}(s)$ es de la forma de la ecuación \eref{eq-transferencia-lineal}, la cual se puede implementar mediante un circuito amplificador inversor\footnotemark[9] como el que se muestra en la -figura \eref{fig-pasabajos-primer-orden-inversor} del cual se tienen expresiones +figura \eref{fig-pasa-bajos-primer-orden-inversor} del cual se tienen expresiones tabuladas para los componentes en funcion del polo ($p$), cero ($z$) y ganancia ($K$). Notese que si bien la transferencia es de primer orden debido a que asi lo es el polinomio del denominador $s+500$, se tiene un circuito con $2$ @@ -1012,8 +1148,8 @@ linealmente independientes. \begin{align} T(s) = K\cdot \dfrac{s+z_{1}}{s+p_{1}};\quad - z=\dfrac{1}{C_{1}\cdot R_{1}};\quad - p=\dfrac{1}{C_{2}\cdot R_{2}};\quad + z_{1}=\dfrac{1}{C_{1}\cdot R_{1}};\quad + p_{1}=\dfrac{1}{C_{2}\cdot R_{2}};\quad K=\dfrac{C_{1}}{C_{2}} \label{eq-transferencia-lineal} \end{align} @@ -1021,16 +1157,16 @@ linealmente independientes. \begin{figure}[!ht] \centering \includegraphics[width=0.65\textwidth]{sch/pasa-bajos_activo_primer_orden_cero_no_nulo_inversor.png} - \caption{Circuito pasa bajos primer orden correspondiente a transferencia $H_{1}(s)$} - \label{fig-pasabajos-primer-orden-inversor} + \caption{Circuito pasa altos primer orden correspondiente a transferencia $H_{1}(s)$} + \label{fig-pasa-bajos-primer-orden-inversor} \end{figure} En este caso la transferencia $H_{1}(s)$ tiene constantes $z=250$, $p=500$ y ganancia unitaria $K=1$, por lo que se pueden obtener los valores de los componentes $C_{i}$ y $R_{i}$ de las expresiones de -\ref{eq-transferencia-lineal}. Por ganancia unitaria los capacitores deben tener -igual valor, reescribiendo las relaciones de \ref{eq-transferencia-lineal} en -funcion de esto, resulta como en \ref{eq-transferencia-lineal-2}. +\eref{eq-transferencia-lineal}. Por ganancia unitaria los capacitores deben tener +igual valor, reescribiendo las relaciones de \eref{eq-transferencia-lineal} en +funcion de esto, resulta como en \eref{eq-transferencia-lineal-2}. \begin{align} K=1\Rightarrow C_{1}=C_{2};\quad @@ -1047,17 +1183,22 @@ resistores normalizados de tolerancia E24 (esto es $5\%$ de error) y capacitores de tolerancia E6 ($20\%$ de error) es $C_{1} = C_{2} = 22nF$, $R_{1}=180k\Omega$ y $R_{2}=91k\Omega$, de esta forma se tiene un error respecto al valor teórico de $1.00\%$ para $R_{1}$ y $0.10\%$ para $R_{2}$. +\fi -\hypertarget{filtro-pasabajos-de-segundo-orden}{% -\subsection{Filtro pasa bajos de segundo orden}\label{filtro-pasabajos-de-segundo-orden}} +\pagebreak +\hypertarget{filtro-pasa-bajos-de-segundo-orden}{% +\subsection{Filtro pasa bajos de segundo orden}\label{filtro-pasa-bajos-de-segundo-orden}} -La transferencia $H_{2}$ corresponde con un pasa bajos de segundo orden +La transferencia $H_{2}$ corresponde con un circuito Sallen-Key pasa bajos de +segundo orden. Dado que para $H_{1}$ se utilizó un filtro pasivo con ganancia +unitaria, entonces $H_{2}$ debe tener ganancia 12 para que asi sea igual a la +ganancia de $H$. \begin{figure}[!ht] \centering \includegraphics[width=0.80\textwidth]{sch/pasa-bajos_activo_segundo_orden_sallen-key.png} - \caption{Circuito pasa bajos segundo orden correspondiente a transferencia $H_{2}(s)$} - \label{fig-pasabajos-primer-orden-inversor} + \caption{Pasa bajos segundo orden Sallen-Key para transferencia $H_{2}(s)$} + \label{fig-pasa-bajos-primer-orden-inversor} \end{figure} \hypertarget{diagrama}{% diff --git a/tp/informe/plot/bode.png b/tp/informe/plot/bode.png Binary files differ. diff --git a/tp/informe/plot/bode_passive_highpass.png b/tp/informe/plot/bode_passive_highpass.png Binary files differ. diff --git a/tp/informe/sch/TL074.sub b/tp/informe/sch/TL074.sub @@ -0,0 +1,42 @@ +* TL074 OPERATIONAL AMPLIFIER "MACROMODEL" SUBCIRCUIT +* CREATED USING PARTS RELEASE 4.01 ON 06/16/89 AT 13:08 +* (REV N/A) SUPPLY 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Value 22n +SYMBOL res 336 144 R0 +WINDOW 0 41 41 Left 2 +WINDOW 3 39 67 Left 2 +SYMATTR InstName R2 +SYMATTR Value 180k +TEXT 272 280 Left 2 !.ac dec 100 .1 100k diff --git a/tp/informe/sch/passive_highpass_normalized.txt b/tp/informe/sch/passive_highpass_normalized.txt @@ -0,0 +1,602 @@ +Freq. 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