# How to design filters with transfer function?

I want to design filters base on transfer function and Bode plots. For example, lets say I want a low pass filter, the limit is 36hz. How should I design a transfer function base on 36 hz limit? (pls feel free to make up other requirements, for example Phase margin, Gain margin and gain crossover. Its just an example problem to help me understand the transfer function)

I know I could use a resistor and capacitator to make a first order low pass filter, or an op amp to make a second order low pass filter, and change their parts' values to get this 36 hz low pass filter, but I want to start from the transfer function and verifying it in bode plot. How should I start?

• I don't understand what you mean by "start from the transfer function." Shouldn't 'how you start' depend on what you have to start with?
– jonk
Dec 4, 2022 at 5:33
• Filtering is a huge topic. Filter synthesis, the process of making a filter after starting with pole and zero locations, can fill a text book. But for a single pole low-pass, the cutoff frequency (omega, in radians per second) is just 1/Tau where Tau is R x C. The cutoff frequency (f, in Hz) is just omega/(2 Pi). Dec 4, 2022 at 5:39

The classical sequence for designing a lowpass filter is as follows:

• Specification of the requirements: Passband edge and damping requirements (minimum attenuation for the most critical disturbing frequency)
• Based on these requirements - selection of the desired lowpass approximation (Butterworth, Chebyshev,...) and determination of the (smallest) filter order which can meet these requirements;
• Selection of a suitable filter topology (out of many alternative structures). This is a typical trade-off activity because many aspects have to be considered: Power, weight, cost, passive/active sensitivity to non-idealities and tolerances,....
• Derivation of the corresponding transfer function for the selected topology (can be found also in relevant books).
• Comparison with the general form of the transfer function - expressed using relevant pole data (pole frequency wp, pole-quality factor Qp and DC gain Ao).
• Determination of Qp and wp for the actual application (can be found in filter tables for the different approximations);
• Derivation of usable formulas from the above mentioned comparison and calculation of the corresponding parts values (some values can/must be chosen).

Look at the AC response for a low pass filter and the pole (for an ideal filter, real filters already decrease -3db). The pole $$\w_0\$$ is where the transfer function starts to decrease. You can stack more than one pole at one location. Each pole gets the transfer function.

$$\frac{1}{1+s/w_0}$$

• The pole is at w=wo (and NOT at 1/wo). Furthermore, at w=wo the magnitude of the transfer function is already down by 3dB, Hence, it does not "start to decrease at the pole. This decrease starts already at w=0 (DC).
– LvW
Dec 4, 2022 at 14:46
• I was thinking ideal low pass, I have corrected the answer Dec 4, 2022 at 17:46