Passive differentiator circuit

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Circuit 1

Circuit 2

Passive differentiator circuit is a simple quadripole consisting of two passive elements. It is also the simplest (first-order) high-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,

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

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

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

where

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

Amplitude transfer function

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

Phase transfer function

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

Amplitude and phase transfer functions for a passive differentiator 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:

$h(t)=\mathcal{L}^{-1} \left \{K(p) \right \}=\delta (t)-\omega_0 e^{-\omega_0 t}=\delta (t)-\frac{1}{\tau} e^{-\frac{t}{\tau}}$

where $\tau=\frac{1}{\omega_0}$ is a time constant, and $δ(t)$ is a Dirac delta function

An impulse response of a passive differentiator circuit

[edit] Applications

A passive differentiator circuit is one of the basic electronic circuits, being widely used in circuit analysis based on the equivalent circuit method.