# Negative impedance converter

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The negative impedance converter (NIC) is a one-port op-amp circuit acting as a negative load which injects energy into circuits in contrast to an ordinary load that consumes energy from them. This is achieved by adding or subtracting excessive varying voltage in series to the voltage drop across an equivalent positive impedance. This reverses the voltage polarity or the current direction of the port and introduces a phase shift of 180° (inversion) between the voltage and the current for any signal generator. The two versions obtained are accordingly a negative impedance converter with voltage inversion (VNIC) and a negative impedance converter with current inversion (INIC). The basic circuit of an INIC and its analysis is shown below.

An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op-amp produces an output potential that is typically hundreds of thousands of times larger than the potential difference between its input terminals. Operational amplifiers had their origins in analog computers, where they were used to perform mathematical operations in many linear, non-linear, and frequency-dependent circuits.

In electronics, negative resistance (NR) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it.

In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside circuit, that meets the port condition; the currents flowing into the two nodes must be equal and opposite.

## Basic circuit and analysis

INIC is a non-inverting amplifier (the op-amp and the voltage divider ${\displaystyle R_{1}}$, ${\displaystyle R_{2}}$ on the figure) with a resistor (${\displaystyle R_{3}}$) connected between its output and input. The op-amp output voltage is

${\displaystyle V_{\text{opamp}}=V_{\text{S}}\left(1+{\frac {R_{2}}{R_{1}}}\right).}$

The current going from the operational amplifier output through resistor ${\displaystyle R_{3}}$toward the source ${\displaystyle V_{\text{S}}}$ is ${\displaystyle -I_{\text{S}}}$, and

${\displaystyle -I_{\text{S}}={\frac {V_{\text{opamp}}-V_{\text{S}}}{R_{3}}}=V_{\text{S}}{\frac {~{\frac {R_{2}}{R_{1}}}~}{R_{3}}}.}$

So the input ${\displaystyle V_{\text{S}}}$ experiences an opposing current ${\displaystyle -I_{\text{S}}}$ that is proportional to ${\displaystyle V_{\text{S}}}$, and the circuit acts like a resistor with negative resistance

${\displaystyle R_{\text{in}}\triangleq {\frac {V_{\text{S}}}{I_{\text{S}}}}=-R_{3}{\frac {R_{1}}{R_{2}}}.}$

In general, elements ${\displaystyle R_{1}}$, ${\displaystyle R_{2}}$, and ${\displaystyle R_{3}}$ need not be pure resistances (i.e., they may be capacitors, inductors, or impedance networks).

A capacitor is a passive two-terminal electronic component that stores electrical energy in an electric field. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. The capacitor was originally known as a condenser or condensator. The original name is still widely used in many languages, but not commonly in English.

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil around a core.

## Application

By using an NIC as a negative resistor, it is possible to let a real generator behave (almost) like an ideal generator, (i.e., the magnitude of the current or of the voltage generated does not depend on the load).

An example for a current source is shown in the figure on the right. The current generator and the resistor within the dotted line is the Norton representation of a circuit comprising a real generator and ${\displaystyle R_{s}}$ is its internal resistance. If an INIC is placed in parallel to that internal resistance, and the INIC has the same magnitude but inverted resistance value, there will be ${\displaystyle R_{s}}$ and ${\displaystyle -R_{s}}$ in parallel. Hence, the equivalent resistance is

A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.

Norton's theorem holds, to illustrate in DC circuit theory terms :

${\displaystyle \lim \limits _{R_{\text{NIC}}\to R_{s}+}R_{s}\|(-R_{\text{INIC}})\triangleq \lim \limits _{R_{\text{INIC}}\to R_{s}+}{\frac {-R_{s}R_{\text{INIC}}}{R_{s}+-R_{\text{INIC}}}}=\infty .}$

That is, the combination of the real generator and the INIC will now behave like a composed ideal current source; its output current will be the same for any load ${\displaystyle Z_{L}}$. In particular, any current that is shunted away from the load into the Norton equivalent resistance ${\displaystyle R_{s}}$ will be supplied by the INIC instead.

The ideal behavior in this application depends upon the Norton resistance ${\displaystyle R_{s}}$ and the INIC resistance ${\displaystyle R_{\text{NIC}}}$ being matched perfectly. As long as ${\displaystyle R_{\text{INIC}}>R_{s}}$, the equivalent resistance of the combination will be greater than ${\displaystyle R_{s}}$; however, if ${\displaystyle R_{\text{INIC}}\gg R_{s}}$, then the effect of the INIC will be negligible. However, when

${\displaystyle {\frac {1}{R_{\text{INIC}}}}>{\frac {1}{R_{s}}}+{\frac {1}{R_{L}}},\quad {\text{(i.e., when}}\,R_{\text{INIC}}

the circuit is unstable (e.g., when ${\displaystyle R_{\text{INIC}} in an unloaded system). In particular, the surplus current from the INIC generates positive feedback that causes the voltage driving the load to reach its power supply limits. By reducing the impedance of the load (i.e., by causing the load to draw more current), the generatorNIC system can be rendered stable again.

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

Positive feedback is a process that occurs in a feedback loop in which the effects of a small disturbance on a system include an increase in the magnitude of the perturbation. That is, A produces more of B which in turn produces more of A. In contrast, a system in which the results of a change act to reduce or counteract it has negative feedback. Both concepts play an important role in science and engineering, including biology, chemistry, and cybernetics.

In principle, if the Norton equivalent current source was replaced with a Norton equivalent voltage source, a VNIC of equivalent magnitude could be placed in series with the voltage source's series resistance. Any voltage drop across the series resistance would then be added back to the circuit by the VNIC. However, a VNIC implemented as above with an operational amplifier must terminate on an electrical ground, and so this use is not practical. Because any voltage source with nonzero series resistance can be represented as an equivalent current source with finite parallel resistance, an INIC will typically be placed in parallel with a source when used to improve the impedance of the source.

### Negative impedance circuits

The negative of any impedance can be produced by a negative impedance converter (INIC in the examples below), including negative capacitance and negative inductance. [1] NIC can further be used to design floating impedances - like a floating negative inductor. [2] [3]

 Negative impedance circuit${\displaystyle Z_{\text{in}}={v \over i}=-Z}$ Negative resistance circuit${\displaystyle R_{\text{in}}={v \over i}=-R}$ Negative capacitance circuit${\displaystyle Z_{\text{in}}={v \over i}={j \over {\omega C}}}$ Negative inductance circuit${\displaystyle Z_{\text{in}}={v \over i}=-j\omega CR_{1}^{2}}$

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## References

1. Chen, W.-K. (2003). The Circuits and Filters Handbook. CRC Press. pp. 396–397. ISBN   0-8493-0912-3.
2. Mehrotra, S. R. (2005). "The Synthetic floating negative inductor using only two op-amps". Electronics World. 111 (1827): 47.
3. USpatent 3493901,Deboo, G. J.,"Gyrator type circuit",issued 1970-02-03, assigned to NASA