f(Λ)

Summary on passive filters

Hi,

for my electronics workshop, I compiled a short summary on low- and highpass filters and how they function once they are combined to a bandpass.

Special care was taken in regards to how they are combined (Low + High vs High + Low).

At the end of this document, you will find the complete LaTeX\LaTeX code to compile the summary for yourself.

Chain Matrix Highpass

{% pdflatex %} \begin{circuitikz} \draw (1,2) to [C, l_=CC, -] (3,2); \draw (3,2) to[R=RR] (3,0); \draw (0,0) to[short, o-] (3,0) to[short, -o] (4,0); \draw (0,0) to [open, v^>=U1U_1] (0,2); \draw (0,2) to [short, o- ,i=I1I_1] (1,2); \draw (4,0) to [open, v>=U2U_2] (4,2); \draw (3,2) to [short, -o ,i=I2I_2] (4,2); \end{circuitikz} {% endpdflatex %} H=(zR+zCzRzC1zR1) H = \begin{pmatrix} \frac{z_R + z_C}{z_R}&z_C \\\\ \frac{1}{z_R}&1 \end{pmatrix}

Chain Matrix Lowpass

{% pdflatex %} \begin{circuitikz} \draw (1,2) to [R, l_=RR, -] (3,2); \draw (3,2) to[C=CC] (3,0); \draw (0,0) to[short, o-] (3,0) to[short, -o] (4,0); \draw (0,0) to [open, v^>=U1U_1] (0,2); \draw (0,2) to [short, o- ,i=I1I_1] (1,2); \draw (4,0) to [open, v>=U2U_2] (4,2); \draw (3,2) to [short, -o ,i=I2I_2] (4,2); \end{circuitikz} {% endpdflatex %} T=(zR+zCzCzR1zC1)T = \begin{pmatrix} \frac{z_R + z_C}{z_C} & z_R \\\\ \frac{1}{z_C} & 1 \end{pmatrix}

Bandpasses

Chain Matrix of Highpass followed by Lowpass

{% pdflatex %} \begin{circuitikz} \draw (0,0) to [short, o-] (3,0) to [short, -] (5,0) to [short, -o] (6,0); \draw (0,3) to [short, o-] (1,3) to [C, l_=CHC_H, -] (3,3) to [R, l_=RTR_T, -] (5,3) to [short] (5,3); \draw (3,3) to [R, l_=RHR_H] (3,0); \draw (5,3) to [C, l_=CTC_T] (5,0); \draw (0,0) to [open, v^>=U1U_1] (0,3); \draw (0,3) to [short, o- ,i=I1I_1] (1,3); \draw (6,0) to [open, v>=U2U_2] (6,3); \draw (5,3) to [short, -o ,i=I2I_2] (6,3); \end{circuitikz}

KHT=H⋅T=((zRT+zCT)⋅(zCH+zRH)zRH⋅zCT+zCHzCTzCH⋅zRTzRH+zRT+zCHzRH+zRT+zCTzCT⋅zRHzRTzRH+1)K_{HT} = H \cdot T = \begin{pmatrix} \frac{\left(z_{R_T} + z_{C_T} \right) \cdot \left(z_{C_H} + z_{R_H}\right)}{z_{R_H} \cdot z_{C_T}} + \frac{z_{C_H}}{z_{C_T}} & \frac{z_{C_H} \cdot z_{R_T}}{z_{R_H}} + z_{R_T} + z_{C_H}\\\\ \frac{z_{R_H} + z_{R_T} + z_{C_T}}{z_{C_T} \cdot z_{R_H}} & \frac{z_{R_T}}{z_{R_H}} + 1 \end{pmatrix}

Chain Matrix of Lowpass followed by Highpass

{% pdflatex %} \begin{circuitikz} \draw (0,0) to [short, o-] (3,0) to [short, -] (5,0) to [short, -o] (6,0); \draw (0,3) to [short, o-] (1,3) to [R, l_=RTR_T, -] (3,3) to [C, l_=CHC_H, -] (5,3) to [short] (5,3); \draw (3,3) to [C, l_=CTC_T] (3,0); \draw (5,3) to [R, l_=RHR_H] (5,0); \draw (0,0) to [open, v^>=U1U_1] (0,3); \draw (0,3) to [short, o- ,i=I1I_1] (1,3); \draw (6,0) to [open, v>=U2U_2] (6,3); \draw (5,3) to [short, -o ,i=I2I_2] (6,3); \end{circuitikz}

KTH=T⋅H=((zRT+zCT)⋅(zCH+zRH)zRH⋅zCT+zRTzRHzCH⋅zRTzCT+zRT+zCHzRH+zCH+zCTzCT⋅zRHzCHzCT+1)K_{TH} = T \cdot H = \begin{pmatrix} \frac{\left(z_{R_T} + z_{C_T} \right) \cdot \left(z_{C_H} + z_{R_H}\right)}{z_{R_H} \cdot z_{C_T}} + \frac{z_{R_T}}{z_{R_H}}& \frac{z_{C_H} \cdot z_{R_T}}{z_{C_T}} + z_{R_T} + z_{C_H}\\\\ \frac{z_{R_H} + z_{C_H} + z_{C_T}}{z_{C_T} \cdot z_{R_H}} & \frac{z_{C_H}}{z_{C_T}} + 1 \end{pmatrix}

Chain Matrix of frequency dependent voltage divider

{% pdflatex %} \begin{circuitikz} \draw (0,3) to [short, o- ,i=I1I_1] (1,3) to R, l_=RSR_S to C, l_=CSC_S, -* to short, -* to [short, -o, i=I2I_2] (8,3); \draw (0,0) to short, o-* to [short, -*] (7,0) to short, -o; \draw (7,3) to [R, l_=RPR_P] (7,0); \draw (5,3) to [C, l_=CPC_P] (5,0); \draw (0,0) to [open, v^>=U1U_1] (0,3); \draw (8,0) to [open, v>=U2U_2] (8,3); \end{circuitikz}

KST=((zRP+zCP)⋅(zCS+zRS)zCP⋅zRP+1zCS+zRSzCP+zRPzCP⋅zRP1)K_{ST} = \begin{pmatrix} \frac{\left(z_{R_P} + z_{C_P} \right) \cdot \left(z_{C_S} + z_{R_S}\right)}{z_{C_P} \cdot z_{R_P}} +1 & z_{C_S}+z_{R_S}\\\\ \frac{z_{C_P}+z_{R_P}}{z_{C_P} \cdot z_{R_P}} & 1 \end{pmatrix}

Note:

KTH≠KHT≠KSTK_{TH} \neq K_{HT} \neq K_{ST}

LaTeX\LaTeX

This includes pictures of the circuits based on circuitikz.

summary_filters.tex

\documentclass[
    a4paper,
    11pt,
    DIV=12,
    ]
{scrartcl}
\usepackage{tikz}
\usepackage[european resistors,american inductors, american voltages]{circuitikz}
\linespread{1.2}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\begin{document}

\section{Highpass}
\begin{circuitikz}
\draw (1,2) to [C, l_=$C$, -*] (3,2);
\draw (3,2) to[R=$R$] (3,0);
\draw (0,0) to[short, o-*] (3,0) to[short, -o] (4,0);
\draw (0,0) to [open, v^>=$U_1$] (0,2);
\draw (0,2) to [short, o- ,i=$I_1$] (1,2);
\draw (4,0) to [open, v>=$U_2$] (4,2);
\draw (3,2) to [short, -o ,i=$I_2$] (4,2);
\end{circuitikz}
\subsection{Chain Matrix}
$$H =
\begin{pmatrix}
\frac{z_R + z_C}{z_R}&z_C \\
\frac{1}{z_R}&1
\end{pmatrix}
$$
\section{Lowpass}
\begin{circuitikz}
\draw (1,2) to [R, l_=$R$, -*] (3,2);
\draw (3,2) to[C=$C$] (3,0);
\draw (0,0) to[short, o-*] (3,0) to[short, -o] (4,0);
\draw (0,0) to [open, v^>=$U_1$] (0,2);
\draw (0,2) to [short, o- ,i=$I_1$] (1,2);
\draw (4,0) to [open, v>=$U_2$] (4,2);
\draw (3,2) to [short, -o ,i=$I_2$] (4,2);
\end{circuitikz}
\subsection{Chain Matrix}
$$T =
\begin{pmatrix}
\frac{z_R + z_C}{z_C} & z_R \\
\frac{1}{z_C} & 1
\end{pmatrix}
$$
\section{Bandpass, second order}
\subsection{Highpass followed by Lowpass}
\begin{circuitikz}
\draw (0,0) to [short, o-*] (3,0) to [short, -*] (5,0) to [short, -o] (6,0);
\draw (0,3) to [short, o-] (1,3) to [C, l_=$C_H$, -*] (3,3) to [R, l_=$R_T$, -*] (5,3) to [short] (5,3); 
\draw (3,3) to [R, l_=$R_H$] (3,0);
\draw (5,3) to [C, l_=$C_T$] (5,0);
\draw (0,0) to [open, v^>=$U_1$] (0,3);
\draw (0,3) to [short, o- ,i=$I_1$] (1,3);
\draw (6,0) to [open, v>=$U_2$] (6,3);
\draw (5,3) to [short, -o ,i=$I_2$] (6,3);
\end{circuitikz}
\subsubsection{Chain Matrix}
$$K_{HT} = H \cdot T = 
\begin{pmatrix}
\frac{\left(z_{R_T} + z_{C_T} \right) \cdot \left(z_{C_H} + z_{R_H}\right)}{z_{R_H} \cdot z_{C_T}} +
\frac{z_{C_H}}{z_{C_T}} & \frac{z_{C_H} \cdot z_{R_T}}{z_{R_H}} + z_{R_T} + z_{C_H}\\
\frac{z_{R_H} + z_{R_T} + z_{C_T}}{z_{C_T} \cdot z_{R_H}} & \frac{z_{R_T}}{z_{R_H}} + 1
\end{pmatrix}
$$
\subsection{Lowpass followed by Highpass}
\begin{circuitikz}
\draw (0,0) to [short, o-*] (3,0) to [short, -*] (5,0) to [short, -o] (6,0);
\draw (0,3) to [short, o-] (1,3) to [R, l_=$R_T$, -*] (3,3) to [C, l_=$C_H$, -*] (5,3) to [short] (5,3); 
\draw (3,3) to [C, l_=$C_T$] (3,0);
\draw (5,3) to [R, l_=$R_H$] (5,0);
\draw (0,0) to [open, v^>=$U_1$] (0,3);
\draw (0,3) to [short, o- ,i=$I_1$] (1,3);
\draw (6,0) to [open, v>=$U_2$] (6,3);
\draw (5,3) to [short, -o ,i=$I_2$] (6,3);
\end{circuitikz}
\subsubsection{Chain Matrix}
$$K_{TH} = T \cdot H = 
\begin{pmatrix}
\frac{\left(z_{R_T} + z_{C_T} \right) \cdot \left(z_{C_H} + z_{R_H}\right)}{z_{R_H} \cdot z_{C_T}} +
\frac{z_{R_T}}{z_{R_H}}& \frac{z_{C_H} \cdot z_{R_T}}{z_{C_T}} + z_{R_T} + z_{C_H}\\
\frac{z_{R_H} + z_{C_H} + z_{C_T}}{z_{C_T} \cdot z_{R_H}} & \frac{z_{C_H}}{z_{C_T}} + 1
\end{pmatrix}
$$
\subsection{Frequency dependent voltage divider}
\begin{circuitikz}
\draw (0,3) to [short, o- ,i=$I_1$] (1,3) 
    to [R, l_=$R_S$](3,3) 
    to [C, l_=$C_S$, -*](5,3) 
    to [short, -*](7,3) 
    to [short, -o, i=$I_2$] (8,3); 
\draw (0,0) to [short, o-*](5,0) to [short, -*] (7,0) to [short, -o](8,0);
\draw (7,3) to [R, l_=$R_P$] (7,0);
\draw (5,3) to [C, l_=$C_P$] (5,0);
\draw (0,0) to [open, v^>=$U_1$] (0,3);
\draw (8,0) to [open, v>=$U_2$] (8,3);
\end{circuitikz}
\subsubsection{Chain Matrix}
$$K_{ST} =
\begin{pmatrix}
\frac{\left(z_{R_P} + z_{C_P} \right) \cdot \left(z_{C_S} +
z_{R_S}\right)}{z_{C_P} \cdot z_{R_P}} +1 & z_{C_S}+z_{R_S}\\ 
\frac{z_{C_P}+z_{R_P}}{z_{C_P} \cdot z_{R_P}} & 1
\end{pmatrix}
$$

\section{Note:}
$$K_{TH} \neq K_{HT} \neq K_{ST}$$
\end{document}

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