CR circuit: harmonic response

In this article, we will deal with a simple high pass filter, made using a capacitor and a resistor, connected in series. We already saw something similar in the article about a RC circuit, but this time the output voltage will be measured on the resistor instead of the capacitor. Obviously the circuit will behave in a completely different way. For the sake of simplicity, the calculations will be carried out in the Laplace domain.

Using the voltage divider formula, we can write:

V_u=V_i\frac{Z_R}{Z_R+Z_C}=

=V_i\frac{R}{R+\frac{1}{sC}}=V_i\frac{R}{\frac{RCs+1}{sC}}

Simplifying the fraction and dividing by $V_i$ , we can write:

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

We can now identify the parameters in the transfer function. From the formula above we can easily see that there is a binomial factor in the denominator. In the numerator we can identify a zero in the origin and two constants, R and C that will be part of the static gain K. As usual these values refer to the Bode form formula, reported below.

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

As a result we will choose the folowing parameters.

Parameter	Value
K	$RC$
n	-1
$\tau_1$	Time constant in the denumerator. There is just one: $RC$