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**Revealing the Mystery of Negative Impedance Converters**

**Circuit idea:** *Inverting the impedance by inverting either the voltage or current*

Negative impedance converters (NICs) are ones of the most interesting, odd, "mystic" and still unexplained electronic circuits... a real nightmare for students... and their teachers:) It is hard to believe but still there are not "human-friendly" explanations of these legendary circuits (even the famous Mr. Horovitz has mentioned but not explained the NIC in his bestseller The Art of Electronics, page 251); instead, he has afforded this opportunity to his students. For us, "circuit thinkers", the understanding of this clever circuit (in its two versions) would be crucial for understanding the very phenomenon of negative impedance.

## What is negative impedance converter?Edit

It is just an op-amp implementation of a true negative resistor. But what is a true negative resistor? It is the opposite "element" of the ordinary "positive" resistor; it is a circuit adding (injecting) the same energy that the equivalent "positive" resistor would dissipate. So, it is nothing else than a source... but this is not the ordinary constant source; it is a "self-varying" (dynamic) source. And as there are two kinds of sources (in contrast with the only one kind of "positive" resistor), there are two kinds of true negative resistors (S-shaped and N-shaped), accordingly, there are two kinds of their op-implementations (VNIC and INIC) as well. First, a NIC can behave as a dynamic voltage source producing voltage that is proportional to the current passing through it (named voltage inversion NIC or VNIC) or as a dynamic current source producing current that is proportional to the voltage across it (named current inversion NIC or INIC). You can consider the VNIC as a 1-port current-to-voltage (to voltage, not to voltage drop!) converter and INIC - as a 1-port voltage-to-current converter.

## How to create negative impedance convertersEdit

True negative impedance elements are amazing and extremely useful electronic devices (circuits). Unfortunately, they do not exist in nature; there are only ordinary passive elements with "positive" impedance (resistors, capacitors and inductors). Then how do we create them?

The idea is simple but powerful - *we can make negative impedance by inverting some initial positive impedance*. Thus the original positive elements will serve as shaping elements for creating "mirror" negative elements. But how do we invert an electrical quantity? We can see the solution around us when we invert some (usually "bad") quantity by adding a bigger opposite ("good") quantity. So, we may convert the "bad" voltage drop across an initial "reference" positive resistor into a "good" voltage across a new negative resistor by adding a (two times) higher voltage (connecting in series a doubling variable voltage source)

But how do we invert the positive impedance? In the simplest case, how do we invert the positive resistance?

The answer is simple if only we know the Ohm's law:) It presents the resistance as a ratio between the voltage and the current (R = V/I); so when the two variables are positive, the resistance is positive as well. To make negative resistance, we have to invert one of them - the voltage or the current:

### Inverting the voltage polarityEdit

In the case of the S-shaped negative resistance R_{S}, we invert the voltage (R_{S} = -V/I = -R). This means that if we pass current through the S-shaped negative resistor, the input terminal becomes negative (instead positive as in the case of the ordinary "positive" resistor). That is why, circuits implementing this technique are named *voltage-inversion negative impedance converters (VNIC)*. Note the power is also inverted (P_{S} = -V.I = -P).

### Inverting the current directionEdit

In the case of the N-shaped negative resistance R_{N}, we invert the current (R_{N} = V/-I = -R). This means that if we apply positive voltage across the N-shaped negative resistor, the current goes out of the negative resistor and enters the positive terminal of the voltage source (instead to leave the positive terminal of the voltage source and to enter the negative resistor as in the case of the ordinary "positive" resistor). That is why, circuits implementing this technique are named *current-inversion negative impedance converters (INIC)*. Note the power is also inverted (P_{N} = V.-I = -P).

Now, we have only to answer the questions, "How do we invert the voltage?" and "How do we invert the current?" To do that, we need more than Ohm's law...

## How to implement conceptually the resistance inversionEdit

### V-inverted resistorEdit

When the input current I_{IN} flows through the positive resistor R, it creates a voltage drop V_{R} = I_{IN}.R (see the attached conceptual picture). We can use this voltage to drive an additional "helping" voltage source (VCVS, on the right in the picture) so that to produce the inverted voltage V_{OUT} = -I_{IN}.R. But as the voltage drop V_{R} will subtract from this voltage, it has to be two times higher (2V_{OUT}) so that the resulting voltage V_{OUT} = -I_{IN}.R across the whole "resistor" will be the same but inverted as the initial voltage across the resistor.

So the trick is to add two times higher negative voltage to the initial positive voltage drop with the purpose to create a negative voltage.

### I-inverted resistorEdit

Let's how do the same with the current...

Now we can invert (reverse) the current direction by connecting an additional "opposing" voltage source in series, which voltage is two times higher than the input voltage. As a result, the same but opposite current enters back in the input source.

So the trick is to add two times bigger reverse current to the initial direct current with the purpose to create a "negative" current.

## How to realize NICs by fixed gain amplifiersEdit

## How to realize NICs by op-ampsEdit

## Presenting the op-amp NIC as a bridgeEdit

### Voltage-inversion op-amp bridge NICEdit

#### Conceptual bridge VNICEdit

#### Op-amp bridge VNICEdit

#### Functional bridge VNICEdit

#### What is the element to be V-inverted?Edit

### Current-inversion op-amp bridge NICEdit

#### Conceptual bridge INICEdit

#### Op-amp bridge INICEdit

The op-amp keeps up the voltage drop V_{E} across R_{E} equal to the input voltage V_{IN} (the op-amp acts as a voltage-to-voltage converter or voltage follower) by passing a current I_{E} = V_{E}/R_{E} = V_{IN}/R_{E} through the right resistor R_{R} (so R_{E} acts as a voltage-to-current converter). The current I_{E} creates a voltage drop V_{RR} = I_{E}.R_{R} across R_{R} (so R_{E} acts as a current-to-voltage converter). The op-amp keeps up the voltage drop across the left resistor R_{L} equal to the voltage drop across the right resistor R_{R} (the op-amp acts as another voltage-to-voltage converter or voltage follower) by passing a current I_{OUT} = V_{RL}/R_{L} = V_{RR}/R_{L} = (I_{E}.R_{R})/R_{L} = ((V_{IN}/R_{E}).R_{R})/R_{L} through the input source. So, the input resistance is -R_{L}.R_{E}/R_{R}.

If R_{L} = R_{R} = R (the usual case), the circuit injects the same current I_{OUT} = -I_{E} that would be drawn by the resistor R_{E} if it was connected directly to the input source. So, it behaves as a "negative resistor" R_{E} having the same voltage as the positive R_{E} but with an inverted current; thus the name of the circuit - "negative impedance converter with current inversion" (INIC). The circuit "inverts" every positive/negative element (resistor, capacitor or inductor) connected in the place of the resistor R_{E} to the "opposite" negative/positive element with equivalent impedance; it is just a "current inverter" (actually, the very INIC consists of the two resistors R_{L} and R_{R}, and the op-amp). According to this explanation, the current I_{OUT} enters the input source when it produces a positive voltage.

#### Functional bridge INICEdit

#### What is the element to be I-inverted?Edit

## ReferencesEdit

## See alsoEdit

Investigating the linear mode of negative impedance converters with voltage inversion

Investigating the linear mode of negative impedance converters with current inversion

Negative impedance converter considers NIC with current inversion (INIC).

## External linksEdit

Theory of the negative impedance converter is a genuine source from 50's

Understanding negative impedance converters (VNIC) - reveals in three consecutive steps the basic idea behind negative impedance converters with voltage inversion (VNIC).

Negative Resistance Revived - condensed version of article originally published in *Amateur Radio*, November 1995.

Negative-resistance circuits - nice material from Answers.com.

Handbook of operational amplifier active RC networks - a formal but well-written electronic book.

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