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 RS, we invert the voltage (RS = -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 (PS = -V.I = -P).
Inverting the current directionEdit
In the case of the N-shaped negative resistance RN, we invert the current (RN = 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 (PN = 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
When the input current IIN flows through the positive resistor R, it creates a voltage drop VR = IIN.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 VOUT = -IIN.R. But as the voltage drop VR will subtract from this voltage, it has to be two times higher (2VOUT) so that the resulting voltage VOUT = -IIN.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.
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 VE across RE equal to the input voltage VIN (the op-amp acts as a voltage-to-voltage converter or voltage follower) by passing a current IE = VE/RE = VIN/RE through the right resistor RR (so RE acts as a voltage-to-current converter). The current IE creates a voltage drop VRR = IE.RR across RR (so RE acts as a current-to-voltage converter). The op-amp keeps up the voltage drop across the left resistor RL equal to the voltage drop across the right resistor RR (the op-amp acts as another voltage-to-voltage converter or voltage follower) by passing a current IOUT = VRL/RL = VRR/RL = (IE.RR)/RL = ((VIN/RE).RR)/RL through the input source. So, the input resistance is -RL.RE/RR.
If RL = RR = R (the usual case), the circuit injects the same current IOUT = -IE that would be drawn by the resistor RE if it was connected directly to the input source. So, it behaves as a "negative resistor" RE having the same voltage as the positive RE 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 RE to the "opposite" negative/positive element with equivalent impedance; it is just a "current inverter" (actually, the very INIC consists of the two resistors RL and RR, and the op-amp). According to this explanation, the current IOUT enters the input source when it produces a positive voltage.
Functional bridge INICEdit
What is the element to be I-inverted?Edit
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).
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.