Passive integrator circuit

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Circuit 1

Circuit 2

Passive integrator circuit is a simple quadripole consisting of two passive elements. It is also the simplest (first-order) low-pass filter.

We'll analyze only the first circuit, the second is absolutely similar.

1 Transfer function
2 Impulse response
3 Applications
4 See also

[edit] Transfer function

A transfer ratio is a gain factor for the sinusoidal input signal with given frequency.

A transfer function shows the dependence of the transfer ratio from the signal frequency, given that the input signal is sinusoidal.

According to Ohm's law,

$Y=X\frac{Z_C}{Z_C+Z_R}=X\frac{\frac{1}{j \omega C}}{\frac{1}{j \omega C}+R}=X\frac{1}{1+j \omega RC},$

where $X$ and $Y$ are input and output signals' amplitudes respectively, and $Z R$ and $Z C$ are the resistor's and capacitor's impedances.
Therefore, the complex transfer function is

$K(j \omega)=\frac{1}{1+j \omega RC}=\frac{1}{1+\frac{j \omega}{\omega_0}},$

where

$\omega_0=\frac{1}{RC}.$

Amplitude transfer function

$H(\omega)=|K(j \omega)|=\frac{1}{\sqrt{1+\left(\frac{\omega}{\omega_0}\right)^2}}.$

Phase transfer function

$\phi (\omega)=\arg K(j \omega)=-\arctan \frac{\omega}{\omega_0}.$

Amplitude and phase transfer functions for a passive integrator circuit

Transfer functions for the second circuit are the same (with $\omega_0=\frac{R}{L}$ ).

[edit] Impulse response

The circuit's Impulse response can be derived as an inverse Laplace transform of the complex transfer function:

$h(t)=\mathcal{L}^{-1} \left \{K(p) \right \}=\int_{\beta-j \infty}^{\beta+j \infty} K(p)e^{pt} \, dp=\omega_0 e^{-\omega_0 t}=\frac{1}{\tau} e^{-\frac{t}{\tau}},$

where $\tau=\frac{1}{\omega_0}$ is a time constant.