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 $\LaTeX$ code to compile the summary for yourself.

## Chain Matrix Highpass

$$H = \begin{pmatrix} \frac{z_R + z_C}{z_R}&z_C \\ \frac{1}{z_R}&1 \end{pmatrix}$$

## Chain Matrix Lowpass

$$T = \begin{pmatrix} \frac{z_R + z_C}{z_C} & z_R \\ \frac{1}{z_C} & 1 \end{pmatrix}$$

## Bandpasses

### Chain Matrix of Highpass followed by Lowpass

$$K_{HT} = H \cdot T = \begin{pmatrix} \frac{\left(z_{R_T} + z_{C_T} \right) \cdot \left(z_{C_H} + z_{R_H}\right)}{z_{R_H} \cdot z_{C_T}} + \frac{z_{C_H}}{z_{C_T}} & \frac{z_{C_H} \cdot z_{R_T}}{z_{R_H}} + z_{R_T} + z_{C_H}\\ \frac{z_{R_H} + z_{R_T} + z_{C_T}}{z_{C_T} \cdot z_{R_H}} & \frac{z_{R_T}}{z_{R_H}} + 1 \end{pmatrix}$$

### Chain Matrix of Lowpass followed by Highpass

$$K_{TH} = T \cdot H = \begin{pmatrix} \frac{\left(z_{R_T} + z_{C_T} \right) \cdot \left(z_{C_H} + z_{R_H}\right)}{z_{R_H} \cdot z_{C_T}} + \frac{z_{R_T}}{z_{R_H}}& \frac{z_{C_H} \cdot z_{R_T}}{z_{C_T}} + z_{R_T} + z_{C_H}\\ \frac{z_{R_H} + z_{C_H} + z_{C_T}}{z_{C_T} \cdot z_{R_H}} & \frac{z_{C_H}}{z_{C_T}} + 1 \end{pmatrix}$$

### Chain Matrix of frequency dependent voltage divider

$$K_{ST} = \begin{pmatrix} \frac{\left(z_{R_P} + z_{C_P} \right) \cdot \left(z_{C_S} + z_{R_S}\right)}{z_{C_P} \cdot z_{R_P}} +1 & z_{C_S}+z_{R_S}\\ \frac{z_{C_P}+z_{R_P}}{z_{C_P} \cdot z_{R_P}} & 1 \end{pmatrix}$$

### Note:

$$K_{TH} \neq K_{HT} \neq K_{ST}$$

## $\LaTeX$

This includes pictures of the circuits based on circuitikz.