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Is there a device that has \$s\$ as its transfer function, that is, a physical system implementing a derivative?

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@Ariser already gave you a good answer in the context of electronics: a differentiator implemented with opamps.

From a strictly theoretical point of view (system theory) a system with a transfer function \$H(s)=s\$ is not realizable because it is not causal, i.e. it would have an impulse response which is not zero before the impulse is applied to the input.

See this reference for more information.

Note that the differentiator using an opamp is a true differentiator only if we assume the opamp is ideal. Real opamps have internal circuitry that contributes many higher order poles to the overall transfer function, so the aforementioned circuit will have poles which make its transfer function causal (and also limit its bandwidth, which otherwise would be infinite).

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  • \$\begingroup\$ Thanks Lorenzo. I was looking for an answer from a control or systems perspective, as I was already aware of the differentiator circuit mentioned in other answers. \$\endgroup\$ – RYS Dec 8 '15 at 23:54
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You can build such a device with limitations from an OP-amp, 1 resistor and 1 capacitor.

schematic

simulate this circuit – Schematic created using CircuitLab

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  • \$\begingroup\$ And what is the caveat? Derivative over limited bandwidth? \$\endgroup\$ – RYS Dec 8 '15 at 21:24
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    \$\begingroup\$ @RYS: Pretty unstable in practice. \$\endgroup\$ – Fizz Dec 8 '15 at 21:28
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    \$\begingroup\$ @Chu Same goes for an integrator, because no device can store infinite energy and generate infinite voltage. \$\endgroup\$ – Ariser Dec 8 '15 at 22:06
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    \$\begingroup\$ @Ariser, yes, infinite gain at zero frequency is equally impossible \$\endgroup\$ – Chu Dec 8 '15 at 22:07
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Another way to look at it is that a capacitor has current that is (ideally) proportional to the derivative of the voltage across it.

If you connect a capacitor to a transimpedance amplifier (current to voltage converter) you get a differentiator as others have discussed.

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