There are several different tools available to model a filter’s frequency response. For the most part, filter response modeling is a very straight forward. We can develop a modeling equation base on the frequency response of a passive, RC low pass filter.
The passive RC low pass filter is a simple voltage divider circuit where the impedance of the capacitor changes based on frequency. Using the resistor and capacitor impedances we can create the filter transfer function. From this transfer function we can create our own filter modeling equation.
The voltage divider equation is given by:
The transfer function, H, is simply VOUT divided by the VIN.
Having the transfer function equation simplified and expressed in frequency (ω), we can find the transfer function magnitude to plot the frequency response.
A filter’s cutoff frequency is defined as the frequency where the filter output is 3dB lower than the input. For the passive RC low pass filter this is defined as:
Substituting the cutoff frequency equation the transfer function magnitude response becomes:
Note that the transfer function magnitude is only an expression of frequency that forms a generic frequency response equation. Normally a filter frequency response is expressed in dB, which uses the log function. Using several log identities we are able to create a simple frequency response equation.
Calculating 20*log of the magnitude transforms our result into dB.
From the simple passive RC low pass filter we have created a single equation that calculates the filter frequency response in dB that only requires the filter cutoff frequency. To model a high pass filter, f and fc are flipped.
As an example, the filter response model below uses the above equation in JavaScript and plots the filter frequency response using chart.js based on your inputs.
