Negative impedance converter

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The negative impedance converter (NIC) is a configuration of an operational amplifier with the aim of creating a negative load. This is achieved by introducing a shift of 180° between the voltage and the current for any signal generator. The basic circuit implementing the NIC is the following:

Circuit analysis

If the operational amplifier is ideal, the inverting and non-inverting inputs have the same voltage, so the current [I_2] is simply given by:

[I_2 = \frac]

Then, considering the voltage from one ground to the other one, it is possible to write:

[(R_1 + R_2) I_2 + R_3 \cdot I_s - V_s = 0]

Replacing the previous relationship and rearranging we get:

[V_s = - I_s \cdot R_3 \frac]

from which we get the input resistance:

[\frac = R_ = - R_3 \frac]

Application

Using the NIC configuration, thus using a negative resistor, it is possible to let a real generator behave (almost) like an ideal one, i.e. the magnitude of the current or of the voltage they generate does not depend on the load.

An example is shown in the following picture:

The current generator and the resistor within the dotted line can be seen like the Norton representation of any circuit: it is a real generator and [R_s] is its internal resistance. If we put in parallel to that another resistor with the same magnitude creating it with a NIC configuration, we will have [R_s] and [-R_s] in parallel, thus the equivalent resistance is:

[R||(-R) = \frac \to \infty]

so the real generator will behave like an ideal one, and the whole current will go to the load [Z_L], whatever is the value of [Z_L] and [R_s].

References

[What Is the Idea behind Howland Current Source?] - a famous circuit using NIC to obtain a constant current.
[Keeping a constant current by adding an additional current] - using NIC to make a perfect Deboo integrator.
[How do we make "inverting" negative resistance?] - reveals the basic idea behind the dual inverting NIC.

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