RC circuit: harmonic response

Control theory

This article refers to the analysis of a simple low pass filter. The circuit is made of a resistor and capacitor, connected in series. In order to keep things simple, the calculations will be carried out in the Laplace domain.

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Using the voltage divider formula, we can write:

V_u=V_i\frac{Z_C}{Z_R+Z_C}=
=V_i\frac{\frac{1}{sC}}{R+\frac{1}{sC}}=V_i\frac{\frac{1}{sC}}{\frac{RCs+1}{sC}}

If the formula is simplified and divided by V_i, we can write:

G(s)=\frac{V_u}{V_i}=\frac{1}{1+RCs}

This formula has to be compared with the more generic Bode form formula.

K\frac{(1+T_1s)+...(1+\frac{2\zeta_1s}{\rho_{n1}}+\frac{s^2}{\rho_{n1}^2})+...}{s^n(1+\tau_1s)+...(1+\frac{2\xi_1s}{\omega_{n1}}+\frac{s^2}{\omega_{n1}^2})+...}

The two formulas have to be made equivalent to one another. This can be achieved using the following parameter values.

Parametro Valore
K 1
\tau_1 Time constant in the denominator. There is just one: RC