Passive integrator circuit

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.

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,br
$Y=Xfrac\{Z\_C\}\{Z\_C+Z\_R\}=Xfrac\{frac\{1\}\{j\; omega\; C\{frac\{1\}\{j\; omega\; C\}+R\}=Xfrac\{1\}\{1+j\; omega\; RC\},$
brwhere $X$ and $Y$ are input and output signals' amplitudes respectively, and $Z\_R$ and $Z\_C$ are the resistor's and capacitor's impedances.brTherefore, the complex transfer function isbr
$K(j\; omega)=frac\{1\}\{1+j\; omega\; RC\}=frac\{1\}\{1+frac\{j\; omega\}\{omega\_0,$
brwhere
$omega\_0=frac\{1\}\{RC\}.$br

Amplitude transfer function
$H(omega)=|K(j\; omega)|=frac\{1\}\{sqrt\{1+left(frac\{omega\}\{omega\_0\}\; ight)^2.$
brPhase transfer function
$phi\; (omega)=arg\; K(j\; omega)=-arctan\; frac\{omega\}\{omega\_0\}.$
br

Transfer functions for the second circuit are the same (with $omega\_0=frac\{R\}\{L\}$).

Impulse response

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

brwhere $au=frac\{1\}\{omega\_0\}$ is a time constant.

Applications

A passive integrator circuit is often used in cheap digital audio systems (i.e. cheap soundcards) as a reconstruction filter.

It is also one of the basic electronic circuits, being widely used in circuit analysis based on the equivalent circuit method.

See also

Passive differentiator circuitbr
RC circuitbr
Electronic filter