Transfer functions and Bode plots for Lagged derivative systems

　・In Japanese
Prerequisites
　・Lagged derivative
　・RC Circuit
　・Scilab

■Lagged derivative Transfer Function

Calculate the transfer function of an lagged derivative system. As a concrete example of lagged derivative, consider the characteristics of Vr(t) versus V(t) in the following RC circuit.

The Laplace transform of the above differential equation is as follows.



The transfer function is as follows, which is called an lagged derivative system or a high-pass filter. It has the effect of advancing the phase while suppressing high frequency noise.

■How to draw the Bode plot of lagged derivative

For simplicity, let us set RC=1 in the above transfer function.


The usual way to solve this is to set s = jω and derive the absolute value and phase of G(jω), but here I will explain some tips to make it easier to draw.

Equation (1) contains a differential system and a first-order lag system, and each can be considered as an addition. This is because absolute values ​​use logarithms, and logarithms take advantage of the property that multiplication can be considered as addition. (Reference: Here)



＜Bode plot results＞
as below.

■simulation result

The black line is the input and the green line is the output, ω=0.5[rad/s]. You can see that the phase is advanced. However, the gain is small.

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