# Circuit Idea/Negative Differential Resistance

<<< contents - NI - NR - NIC - VNIC - INIC - page stage >>>

Demystifying the Negative Differential Resistance Phenomenon

Circuit idea: Making constant-voltage and constant-current nonlinear resistors overact.

How many times have you tried to understand what negative differential resistance (NDR) is? After browsing great number of web resources you probably have already known that a negative differential resistior has a negative slope of its IV curve, the current is a decreasing function of the voltage, it can compensate resistive losses, it can amplify, etc. But you probably need more than these dry facts; it is most likely you want to know why and how this odd element does this magic... just because you are a reasoning human being, not a computer...

In this exciting circuit story, you will find answers to your questions. You will learn, once and for all, the simple truth about the negative differential resistance phenomenon and its amazing manifestations... you will see there is nothing mystic here... Something more, this story is not only for negative differential resistors; it is a story for all kinds of nonlinear resistors...

The negative differential resistance is closely related to the true negative resistance. To see the close connection between them, a lot of reciprocal links are placed inside the two stories.

An S-shaped current-controlled negative differential resistor (e.g., a neon lamp) exhibits negative resistance in the middle region (red) of its IV curve where an increase in the current results in a decreased voltage. The instant ohmic resistance is still positive.
An N-shaped voltage-controlled negative differential resistor (e.g., a tunnel diode) exhibits negative resistance in the middle region (red) of its IV curve where an increase in the voltage results in a decreased current. The instant ohmic resistance is still positive as above.

## What negative differential resistance is

### The so strange NDR behavior...

If we increase the voltage across an ordinary ohmic resistor, the current flowing through it increases proportionally according to Ohm's law (IOUT = VIN/R). Also, if we increase the current through the resistor, the voltage across it increases as well (VOUT = IIN.R). Thus the voltage and the current change in the same directions. But there are mysterious two-terminal electronic components (negative differential resistors, shortly NDR), having just the opposite behavior in the middle part of their IV curves - the voltage across and the current through them change in opposite directions. Some of them - neon lamps, thyristors, have an S-shaped IV curve while other - tunnel, Gunn and lambda diode, have an N-shaped IV curve (see the pictures on the top).

It is a common practice to show the IV curve of negative differential resistors in its final S or N shape and just to say that the middle part has a negative slope. But to deeply understand the phenomenon, we need to know how and why this part of the curve is inclined to the left, how negative differential resistors do this magic, what the use of such a strange behavior is ... simply, what the secret of the NDR phenomenon is...

### Revealing the secret of negative differential resistance

Although the negative differential resistance seems to be a mystic phenomenon, it is actually based on an extremely simple, clear and intuitive trick - the great dynamizing idea that is brought here to the utmost degree. A differential negative resistor is nothing more than a kind of "self-varying", dynamic resistor that changes extremely its instant (ohmic, chordal) resistance depending on the current passing through the resistor or on the voltage applied across it; thus a differential negative resistor is actually an over-dynamic resistor.[1].

Negative differential resistors are dynamic but still positive resistors. They have different kinds of resistance in the three parts of their IV curves (located in the 1st or the 3th quadrant) - "positive" in the end parts and negative in the middle part. The three parts form the whole IV curve that, depending on the NDR behavior in the middle part, can be S- or N-shaped. When the input quantity (no matter current or voltage) increases, the S-shaped NDR decreases while the N-shaped NDR increases its instant resistance.

Depending on the proportion between the negative differential resistance and the "positive" resistance of the input source, both the negative resistors can operate either in a linear or a bi-stable mode. When operating in a linear mode, they change gradually their instant resistance in the negative resistance region while in a bi-stable mode they do it sharply (jump over it).

Each of the two NDR can be driven both by voltage and current. To operate in a linear mode, an S-shaped NDR has to be driven by current while an N-shaped NDR - by voltage; v.v., to operate in a bi-stable mode, an S-shaped NDR has to be driven by voltage while an N-shaped NDR - by current. So, in contrast to the widespread but misleading viewpoint, there are no particular current controlled (CCNR) and voltage controlled (VCNR) negative resistors - there are only S-shaped and N-shaped NDR, and each of them can be both current and voltage controlled.

In the middle negative resistance region, negative differential resistors behave as two-terminal active elements (such as transistors). They cannot be used independently; they need an additional power supply to be connected. Thus the combination of the negative differential resistor and the power supply can be considered as another kind of a true negative resistor (the other kind of a true negative resistor is a combination of a constant "positive" resistor and a varying voltage source). From another viewpoint, this combination can be thought as of an electrical source with negative internal resistance. It is a usual practice to think of a negative differential resistor as of a true negative resistor implicitly assuming the existence of a power supply.

## How to create negative differential resistance (compare with NR)

### How to show the secret of NDR

We, circuit ideators:), know that the truth about circuit phenomena is hidden rather in the movement from simple to complex circuits (in the circuit evolution) than in the final perfect circuit solutions. So, the best way to reveal the NDR phenomenon is to show the metamorphosis of the negative differential resistor from the ordinary ohmic resistor to the sophisticated negative one, to show the evolution of the bare ohmic resistance... According to this idea, we can first reveal the secret of S-shaped negative resistors by following the succession of typical resistances in the order high ohmic > decreased > zeroed > S-shaped NDR. Then we can reveal the secret of the dual N-shaped negative resistors by a similar succession of other typical resistances - low ohmic > increased > infinite > N-shaped NDR. To visualize the resistor evolution, we will gradually draw the particular segments of the seesaw NDR IV curves and will explain how they are obtained. Note they are straight lines although the whole IV curves represent nonlinear resistances.

### An S-shaped negative differential resistor (compare with S-shaped NR)

(these elements are frequently named "current controlled negative resistors" - CCNR, although they can be controlled by voltage as well)

#### Looking for the basic S-shaped NDR idea

We can see many examples (analogies) of this arrangement in our routine, where we experience some opposition when implementing our purposes and someone unnoticely helps us. As a result, some kind of a "pressure" or a "flow" appears. A good example can be the plumbing analogy where you control the main water faucet and I control some sink faucet. Thus you control the water flow or pressure and I control the magnitude of the water impediment.

#### A setup for emulating an S-shaped NDR

The best way to see how such weird electronic devices do these magics (when the voltage increases, the current decreases or when the current increases, the voltage decreases) is to place ourselves in their place and begin performing their functions, to emulate their behavior by an empathy. What does actually an NDR do? It just changes its instant ohmic resistance depending in some definite way on the input variable (voltage or current). Then how do we emulate it? Of course, simply by using a humble variable resistor (a rheostat) - fig. 1 and fig. 6. Let's do it as a funny game: you will control the input current source; I will control the rheostat (you are the source, I am the load:) Thus I and the rheostat combine to form a "man-controlled" negative differential resistor and you drive it.

Fig. 1. A setup for emulating an S-shaped negative differential resistor

To see the negative resistance regions in the IV curves of negative resistors, they have to operate in a linear mode (you will understand what this means later). For this purpose, we have to supply S-shaped negative resistors by a current source; you will understand why below. Otherwise, they will operate in the even more exotic bi-stable mode.

We can present graphically the circuit operation of the simple electrical circuit (a source connected to a load) shown in fig. 1 if we consider that the voltage across the two elements is the same and the current flowing through them is the same as well. For this purpose, we have to superimpose their IV curves on the same coordinate system: the IV curve of the variable resistor is a straight line beginning from the coordinate origin and with a slope depending on the instant resistance; the IV curve of the input current source is a horisontal line shifted from the X-axis. The intersection point A (alias operating point) represents the instant magnitudes of the current IA and the voltage VA.

Now connect a voltmeter across the resistor R and an ammeter in series with it to monitor the voltage and the current. Are you ready? Let's start the "game"!

#### High ohmic resistance

Fig. 2. High constant (ohmic) resistance along the section 0-1.

In routine, when applying some kind of "flow" and experiencing a constant high impediment, the obtained "pressure" is proportional to our efforts. For example, imagine I have slightly opened the sink faucet in our plumbing analogy; so, if you begin opening more and more the main faucet to increase the water flow, the water pressure begins increasing quickly and proportionally to the water flow.

According to this life situation, in the beginning, I set the rheostat's slider in the end position (highest resistance R) and you begin increasing continuously the input current IIN. As a result, voltage drop VOUT appears across the resistor R. According to Ohm's law, it is proportional to the current passing through the resistor - VOUT = R.IIN. Note the voltage depends only on the current and the Ohm's law equation is a function of one variable.

On the graphical representation (fig. 2), when you vary the current IIN of the input current source, its (your) IV curve moves vertically remaining parallel to itself. As a result, the operating point A slides over the IV curve of the ohmic resistor R from point 0 to point 1 that is a straight line. The slope of the R IV curve represents graphically the value of the ohmic resistance R. It is a "static", constant, steady resistance as it does not depend on the location of the operating point A.

#### Virtually decreased resistance

But there are many situations in our life where the contrary impediment changes when we implement our purposes. Imagine our antagonist has a unsteady behavior and decreases (inconspicuously for us) his/her initial high opposition. As a result, we have the illusion that the opposition has decreased; the "pressure" decreases as well. As an example, in the plumbing analogy, if I begin opening gradually the sink faucet when you open the main faucet to increase the water flow, the water pressure will increase lazier and you will have the feeling that the water resistance is decreased. But this is an illusion since it is not simply decreased; it is decreasing.

Fig. 3. Virtually decreasing the resistance along the section 1-2.

Let's now apply this clever dynamizing trick to decrease virtually the resistance R in such an exotic way. Imagine when you reach the point 1 (fig. 3), I decide to help you along the whole section 1-2. For this purpose, I begin moving slightly the slider to decrease gradually the rheostat's resistance. As a result, the voltage begins depending both on the current and the resistance. The Ohm's law equation becomes a function of two variables - VOUT = IIN1.RIN2.

This is the simple but clever dynamizing trick - if the input current IIN1 increases, the resistance RIN2 slightly decreases and their product VOUT increases slowly! You have the illusion that the resistance R has permanently decreased but actually, it is continuously decreasing and you see new, lower dynamic (differential) resistance dR1 < R. The "static" resistance R is converted into a smaller differential resistance dR1. "Bad" zeners and ordinary diodes have such property.

Now look at the picture (fig. 3) to see this trick in a more attractive way. When you increase the input current IIN from point 1 to point 2, its IV curve moves upward remaining parallel to itself. But as, at the same time, I decrease the resistance R, its IV curve rotates counterclockwise. As a result, the operating point A slides along a new more vertical IV curve, which represents the new dynamic resistance dR1 < R.

As a whole, this bent IV curve represents nonlinear resistance but note the segment 1-2 is a straight line and it resembles ordinary ohmic resistance. Looking only at this part of the curve, you may have the illusion that you investigate an ohmic resistor. But if you look at the whole curve, you will note that the line continuation does not pass through the coordinate origin; so this is not an ohmic resistor...

#### Virtually zeroed resistance

Now imagine that the dynamizing idea above is improved so that our antagonist decreases (again, inconspicuously for us) his/her opposition to such extent that we do not experience any impediment when implementing our purposes. The result is amazing: we have the illusion that the opposition has dissapeared, we realize our purposes without any efforts and the "pressure" stays constant! We can see this situation in our plumbing analogy. In this region, you continue opening the main faucet but now I am opening the sink faucet with the same rate of change. What a simple but clever idea - just change the flow and the impediment with the same rate but in different directions! The result is the same - the pressure stays constant and you have the feeling that there is no any impediment!

Fig. 4. Virtually zeroing the resistance along the section 2-3.

Then let's apply this trick to zero virtually the resistance R. Now imagine when you reach the point 2 (fig. 4), I begin moving vigorously enough the slider thus decreasing considerably the rheostat's resistance R so that, when in Ohm's law (VOUT = IIN1.RIN2) the input current IIN1 increases, the resistance RIN2 decreases with the same rate of change and their product VOUT stays unchangeable. You have the feeling that there is no resistance and the input current source is shorted. What a great idea! We have made a constant-voltage nonlinear resistor (a voltage stabilizer). All kinds of diodes (zener, LEDs, etc.), varistors and some circuits with parallel negative feedback (active diodes, active zeners, etc.) act in this way.

On the graphical representation (fig. 4), when you increase the input current IIN from point 2 to point 3, its IV curve moves upward as before. But I decrease vigorously enough the resistance R at the same time; so, its IV curve rotates rapidly enough contraclockwise. As a result, the operating point A slides upward from point 2 to point 3 along the vertical IV curve of the new dynamic resistance dR2 = 0.

#### "Decreasing" negative differential resistance

Finally, imagine that the dynamizing idea is enormously reinforced so that our antagonist goes too far decreasing many times more his/her opposition than needed. The result is quite unexpected and surprising - we increase our efforts but the "pressure" does not increase or stay constant; instead imagine it even decreases! In the plumbing analogy, you are opening the main faucet but I am opening the sink faucet with a higher rate of change. What a wonderful trick - change the impediment more vastly than the input water flow and you will see that the water pressure goes down instead to rise!

Fig. 5. Making the resistance negative (creating S-shaped negative resistance) along the section 3-4.

If we put this idea in practice, we will manage to convert the ordinary "positive" resistance R into negative differential resistance. Practically, this means to make the constant-voltage nonlinear resistor overact, to go too far when changing its resistance. Well, let's do it. When you reach the point 3 (fig. 5), I begin moving extremely vigorously the slider thus decreasing enormously the rheostat's resistance R. The situation with Ohm's law (VOUT = IIN1.RIN2) becomes now extremely interesting - the input current IIN1 increases but the resistance RIN2 decreases more quickly and their product VOUT decreases! What a magic! You increase the input variable (the current) but the output variable (the voltage) decreases?!?!

Figuratively speaking, the two input variables (the current and the resistance) "fight" each other striving to change (in opposite directions) the output variable (the voltage). There are three situations: the first variable dominates; the two variables compensate each other; the second variable dominates. First, if the rate of the resistance change is smaller than the rate of the current change, the voltage only slowers its rate of change but continues increasing. Then, if the two rates are equal, the voltage does not change at all. Now the rate of the resistance change is bigger than the rate of the current change; so, the voltage begins decreasing. In electronics, thyristors and neon bulbs act in this way (in certain parts of their IV curves).

If you are observant enough, you can see the connection with the previous step thinking of the negative differential resistor as of an "over-acting" constant-voltage nonlinear resistor.

In the graphical representation (fig. 5), when you increase the input current IIN from point 3 to point 4, its IV curve moves upward as usual. But now, as I decrease extremely vigorously the resistance R at the same time, its IV curve rotates enormously rapidly counterclockwise. As a result, the operating point A slides upwards from point 3 to point 4 over the IV curve of the new negative differential resistance dR3 < 0. It is inclined (folded up) to the left and has a negative slope.

#### After the S-shaped NDR region

Unfortunately, there are no unlimited things in this world:( So, at point 4 (fig. 5), we have already exhausted a great part of the initial resistance and begin slowing the rate of change. Thus we can create again zero, decreased and finally, ohmic resistance. The IV curve folds up clockwise and resembles the letter "S".

#### Final conclusions about the S-shaped NDR properties

• S-shaped negative differential resistor is an overacting constant-voltage nonlinear resistor.
• When the current through an S-shaped NDR increases, it decreases enormously its instant ohmic resistance so that the voltage across it decreases.
• An S-shaped NDR starts with high initial resistance (to decrease the resistance it has to have what to decrease:) and finishes with low final resistance.

### An N-shaped negative differential resistor

(these elements are frequently named "voltage controlled negative resistors" - VCNR, although they can be controlled by current as well)

Once we revealed the secret of the S-shaped NDR, let's unveil, in the similar way, the mystery of the dual N-shaped NDR.

#### A setup for emulating an N-shaped NDR

Fig. 6. A setup for emulating an N-shaped negative differential resistor.

To operate in a linear mode, we have to supply N-shaped negative resistors by a voltage source; otherwise, they will operate in the exotic bi-stable mode. So, to emulate an N-shaped negative differential resistor, we can use the same setup as in the fig. 1 only if we replace the current source with a voltage one - fig. 6. As above, the voltage across and the current through the two elements is the same; so we may superimpose their IV curves on the same coordinate system. The IV curve of the variable resistor is again a straight line beginning from the coordinate origin and with a slope depending on the instant resistance but the IV curve of the input voltage source is a vertical line shifted from the Y-axis. The operating point A represents the instant magnitudes of the current IA and the voltage VA.

#### Low ohmic resistance

In routine, when applying some kind of "pressure" and experiencing a constant low impediment, the obtained "flow" is proportional to our efforts (a life manifestation of Ohm's law). For example, imagine I have vastly opened the sink faucet in our plumbing analogy; so, if you begin opening more and more the main faucet to increase the water pressure, the water flow begins increasing quickly and proportionally to the pressure (a hydraulic manifestation of Ohm's law).

Fig. 7. Low constant (ohmic) resistance along the section 0-1.

According to this life situation, in the beginning, I set the rheostat's slider (fig. 6) in the end position according to the lowest resistance R and you begin increasing continuously the input voltage VIN. As a result, current IOUT appears across the resistor R. According to Ohm's law, it is proportional to the voltage applied across the resistor - IOUT = VIN/R.

On the graphical representation (fig. 7), when you vary the voltage VIN of the input voltage source, its (your) IV curve (red colored) moves horizontally. As a result, the operating point A slides over the IV curve of the ohmic resistor R from point 0 to point 1 that is a straight line. The slope of R IV curve represents graphically the value of the ohmic resistance R. It is "static", constant, steady resistance as it does not depend on the location of the operating point A.

#### Virtually increased resistance

Remember the situations in our life when the contrary impediment changes when we implement our purposes. Now imagine the "antagonist" begins increasing (inconspicuously for us) his/her initial low opposition. As a result, we have the illusion that the opposition has increased and the "flow" decreases. In the plumbing analogy, if I begin closing gradually the sink faucet when you open the main faucet to increase the water pressure, the water flow will increase lazier and you will have the feeling that the water resistance is increased.

Fig. 8. Virtually increasing the resistance along the section 1-2.

Let's apply the dynamizing trick now to increase virtually the resistance R. Imagine when you reach the point 1 (fig. 8), I begin moving slightly the slider to increase gradually the rheostat's resistance. As a result, the current begins depending both on the voltage and the resistance. The Ohm's law equation becomes a function of two variables - IOUT = VIN1/RIN2.

Now if the input voltage VIN1 increases, the resistance RIN2 slightly increases and their ratio VOUT/RIN2 increases slowly! You have the illusion that the resistance R has permanently increased but actually, it is continuously increasing and you see new, higher dynamic (differential) resistance dR1 > R. The "static" resistance R as though is converted into higher differential resistance dR. In this way we can explain what a nonlinear resistor is - we can think of it as of a dynamic linear (ohmic) resistor. For example, incandescent light bulbs have such property in certain part of their IV curves.

Now look at the picture (fig. 8) to see this trick in a more attractive way. When you increase the input voltage VIN1 from point 1 to point 2, the vertical IV curve of the input voltage source moves from left to right remaining parallel to itself. But as, at the same time, I increase the resistance R, its IV curve rotates clockwise. As a result, the operating point A slides along a new more inclined to the right IV curve, which represents the new dynamic resistance dR1 > R.

#### Virtual infinite resistance

Now imagine that the dynamizing idea above is enforced so that our antagonist increases (again, inconspicuously for us) his/her opposition to such extent that we experience infinite impediment when implementing our purposes. The result is amazing: we have the illusion that the opposition has become infinite, we cannot realize our purposes at all and the "flow" stays constant! Let's see this situation in the plumbing analogy. In this region, you continue opening the main faucet but now I am closing the sink faucet with the same rate of change. Another clever idea - just change the pressure and the impediment with the same rate and in the same directions! Then the flow stays constant (there is a flow but there is no change of the flow).

Fig. 9. Making infinite resistance along the section 2-3.

We can strenghten the dynamizing idea above to the extent to make the resistance R virtually infinite. For this purpose, when you reach the point 2 (fig. 9), I begin moving vigorously enough the slider to increase considerably the rheostat's resistance R so that, when in Ohm's law (IOUT = VIN1/RIN2) the input voltage VIN1 increases, the resistance RIN2 increases with the same rate of change and their ratio VIN1/RIN2 stays unchangeable. As a result, you have the feeling that the circuit is open and the input voltage source is unloaded. Another great idea! We have made a constant-current nonlinear resistor (a current stabilizer). The output parts (collector-emitter, drain-source) of all kinds of transistors (bipolar, FET, MOS, etc.), tubes and resettable fuses act in this way.

On the graphical representation (fig. 9), when you increase the input voltage VIN1 from point 2 to point 3, the vertical IV curve of the input voltage source continues moving from left to right as before. But I increase vigorously enough the resistance R at the same time; so, its IV curve rotates rapidly enough clockwise. As a result, the operating point A slides from point 2 to point 3 along the horizontal IV curve of the new dynamic resistance dR2 = ∞.

#### "Increasing" negative differential resistance

In the middle of our "excursion" along the IV curve, imagine that the dynamizing idea is enormously reinforced so that our antagonist goes too far increasing many times more his/her opposition than needed. The result is very, very surprising - we increase our efforts but the "flow" does not increase or stay constant; instead, it decreases as the pressure above! In the plumbing analogy, you are opening the main faucet but I am firmly closing the sink faucet with a higher rate of change. Another wonderful trick - increase the impediment more vastly than the input water pressure increases and you will see that the water flow goes down instead to rise!

Fig. 10. Making the resistance negative (creating N-shaped negative resistance) along the section 3-4.

Let's apply this idea in the electricity. If we make the constant-current nonlinear resistor overact, we will manage to convert in another way the ordinary "positive" resistance R into negative but N-shaped differential resistance. Let's do it. When you reach the point 3 (fig. 10), I begin moving extremely vigorously the slider to increase enormously the rheostat's resistance R. The situation with Ohm's law (IOUT = VIN1/RIN2) becomes again extremely interesting - the input voltage VIN1 increases but the resistance RIN2 increases more quickly and their ratio VIN1/RIN2 decreases! Another magic! You increase the input variable (the voltage) but the output variable (the current) decreases again! Let's scrutinize this situation as above.

The two input variables (the voltage and the resistance) "fight" each other striving to change the output variable (the current). There are three situations again. First, if the rate of the resistance change is smaller than the rate of the voltage change, the current only slowers its rate of change but continues increasing. Then, if the two rates are equal, the current stays constant. Finally (this case), the rate of the resistance change is bigger than the rate of the voltage change and the current begins decreasing. In electronics, tunel diode, Gunn diode and lambda diode act in this way (in certain parts of their IV curves).

If you are observant again, you can see a connection with the previous step thinking of this kind of a negative differential resistor as of an "over-acting" constant-current nonlinear resistor.

On the graphical representation (fig. 10), when you increase the input voltage VIN1 from point 3 to point 4, the vertical IV curve of the input voltage source continues moving from left to right as before. But as I increase extremely vigorously the resistance R at the same time, the resistor's IV curve rotates enormously rapidly clockwise. As a result, the operating point A slides downwards from point 3 to point 4 over the IV curve of the new negative differential resistance dR3 < 0. As in the case of the "decreasing" (S-shaped NDR), it has a negative slope but now it is inclined (folded up) to the right.

#### After the N-shaped NDR region

Remember again there are no unlimited things in our world:((( As above, at point 4 (fig. 8), we are already exhausted a great part of the initial high conductance (the other name of "low resistance":) and begin slowing the rate of change. Thus we can create infinite, decreased and finally, ohmic resistance. The IV curve folds up counterclockwise and resembles the letter "N".

#### Final conclusions about the N-shaped NDR properties

• N-shaped negative differential resistor is an overacting constant-current nonlinear resistor.
• When the voltage across an N-shaped NDR increases, it increases enormously its instant ohmic resistance so that the current through it decreases.
• An N-shaped NDR starts with low initial resistance (to increase the resistance it has to have what to increase:) and finishes with high final resistance.

### General considerations about creating the NDR

The ordinary "positive" resistance and the negative differential resistance are closely related - negative differential resistance is created on the base of positive resistance, simply by modifying it dynamically. Since there is no internal source, an NDR IV curve always starts from the coordinate origin (0,0). This means if the voltage applied across an N-shaped NDR is zero, the current through it is zero as well and v.v., if the current passed through an S-shaped NDR is zero, the voltage across it is zero again.

Note the NDR cannot occupy the whole IV curve; we can create a negative slope only in some limited middle part of the whole IV curve of a positive resistor (in the end parts, the NDR saturates and the resistance becomes positive; thus the odd "S" or "N" shape). For this purpose, we have somehow to change the monotonous motion of the operating point (that draws the IV curve) when the input quantity (voltage or current) changes within the negative resistance region; we have to change its direction back to front; we have to fold up the curve.

To picture this situation in a more colorful and empathic way, let's put ourselves in the place of the operating point and begin travelling from the coordinate origin:) When reaching the first key point, we decide to turn: in the case of an S-shaped NDR, we have to turn to the left (countercklockwise); in the case of an N-shaped NDR, we have to turn to the right (clockwise). But how do we "turn" back? How do we fold up the curve?

The trick is simple but clever: in the one-variable function IOUT = VIN/R or VOUT = IIN.R (Ohm's law) we begin changing the resistance as a second variable simultaneously with the first one. Thus, in this region, we actually have a function of two variables (IOUT = VIN1/RIN2 or VOUT = IIN1.RIN2) but we continue considering it as a function of one variable (we see only the one variable - the input voltage or current). Usually, the resistance is a function of the input quantity - R = R(VIN) or R = R(IIN); so we have a compound function IOUT = VIN/R(VIN) or VOUT = IIN.R(IIN). The whole trick is that we are misled - seeing only the straight line of the negative resistance region, we have the feeling that we see an ordinary but only changed (decreased/increased, zeroed/infinite or negative) ohmic resistance. But this is an illusion since we actually investigate not constant but varying ohmic resistor; we see dynamic but we think we continue seeing constant resistance! So, if we look at the whole curve, we will see that the continuation of the line does not pass through the coordinate origin. Thus we have artificially changed the resistance in this region; we have made dynamic resistance. As a result, the curve changes its slope up to negative but only within the borders in the negative resistance region where we have what (resistance) to change.

Remember the same clever trick - introducing a second variable or making it dependent on the first one, was used in creating true negative resistors and was generalized in Miller theorem. Only there we insert additional voltage or current (instead to vary the resistance) as a second variable. Thus, in the one-variable function IOUT = VIN/R or VOUT = IIN.R (Ohm's law) we begin changing the additional variable simultaneously with the first one. For example, we can change the voltage VIN2 of an additional voltage source connected in series with the resistor. Thus, in this region, we actually have a function of two variables (IOUT = (VIN1 + VIN2)/R) but we consider it as a function of one variable (we see only the one variable - the input voltage). In this way, we have artificially changed the resistance in this region; we have made another kind of dynamic resistance...

Let's now return to the negative differential resistance. After the end of the region, the reserve of high resistance (S-shaped curve) or high conductance (N-shaped curve) is depleted and the resistance reaches its minimum (S-shaped curve) or maximum (N-shaped curve). The magic of negative resistance ceases and the ordinary positive resistance establishes in the final part of the IV curve (the negative resistor is saturated). More colorfully, at this second key point, we decide to turn back: now, in the case of an S-shaped NDR, we have to turn to the right (cklockwise); in the case of an N-shaped NDR, we have to turn to the left (counterclockwise). For this purpose, we slow the rate of the resistance change and finally stop changing it at all; our "trip" is finished.

Similar considerations can be written for the first part with positive resistance where the initial resistance is maximum (S-shaped curve) or minimum (N-shaped curve) and the negative resistor is saturated... Let's finally generalize all this wisdom in one sentence:

Negative differential resistance is created in some limited region of the resistor IV curve by vigorously changing its instant resistance.

## Operation

Above we have driven the negative differential resistor with resistance RNE by a perfect current source with infinite internal resistance (RG = ∞) or by a perfect voltage source with zero internal resistance (RG = 0). In the general case, the NDR is driven by a real voltage source with some internal resistance RG. It is interesting fact that the negative differential resistor has a different behavior depending on the proportion between the magnitudes of the input positive resistance RG and the negative resistance RNE; i.e., it can operate in two different modes - linear and bistable.

### Linear mode

In this mode, there is only one intersection point of the two superimposed IV curves (of the input source and of the NDR). The IV characteristic is a single-valued function and the output quantity is proportional to the input one.

#### Current controlled S-shaped NDR

Fig. 11a: An S-shaped IV curve (blue) of a current-driven NDR.

To operate in a linear mode, an S-shaped NDR should be driven by a high-resistive enough input source with internal resistance RG > RNE; in the extreme case, this is a perfect current source with infinite RG. In Fig. 11a, the operating point A is represented by the intersection point of two superimposed IV curves: the green one of the input current source IIN and the orange one of the NDR (dynamic positive resistor R).

When the input current varies from zero to maximum, an S-shaped NDR keeps up high ohmic resistance in the first part. In the negative resistance region, it behaves as a dynamic resistor that enormously decreases its ohmic (chordal) resistance from high to low; so, its IV curve rotates counterclockwise and the operating point C moves up and pictures the negative resistance part of the curve. In the last part, it has low ohmic resistance. And v.v., when the input current returns from maximum to zero, an S-shaped NDR begins with low ohmic resistance in the last part; in the negative resistance region, it behaves as a dynamic resistor that enormously increases its ohmic resistance from low to high, and in the first part, it has high ohmic resistance again.

#### Voltage controlled N-shaped NDR

Fig. 11b: An N-shaped IV curve (blue) of a voltage-driven NDR.

To operate in a linear mode, an N-shaped NDR should be driven by a low-resistive enough input source with internal resistance RG < RNE; in the extreme case, this is a perfect voltage source with RG = 0. In Fig. 11b, the operating point A is represented by the intersection point of two superimposed IV curves: the red one of the input voltage source VIN and the orange one of the NDR (dynamic positive resistor R).

When the input voltage varies from zero to maximum, an N-shaped NDR keeps up low positive resistance in the first part. In the negative resistance region, it behaves as a dynamic resistor that enormously increases its ohmic (chordal) resistance from low to high; so, its IV curve rotates clockwise and the operating point C moves down and pictures the negative resistance part of the curve. In the last part, it has high positive resistance. V.v., when the input voltage returns from maximum to zero, an N-shaped NDR keeps up high positive resistance in the last part; in the negative resistance region, it behaves as a dynamic resistor that enormously increases its ohmic resistance from high to low, and in the first part, it has low positive resistance again.

### Bistable mode

In this mode, there is in total three intersection points of the two superimposed IV curves: the middle point is unstable; only the end points are stable. The IV characteristic is a multivalued function and the output quantity can take only two end stable values (i.e., the negative resistor "jumps" between the two states). But how? To get to know, we will do something very interesting and unique - we will peek in the very transition between the two states; we will see what this changeable element does during the transistion. But why? Just because we are curious human beings:)

#### How the negative resistor "jumps"

Note the negative differential resistor is an exotic 2-terminal amplifying element since its output and input are inherently connected to each other; they are just the same! So, there is an intrinsic feedback in this device. Let's see why.

Imagine you have assembled a dynamic voltage divider by two resistors - positive and negative (no matter S- or N-shaped), connected in series and you have supplied it by a voltage source (like the tunnel diode amplifier shown in fig. 15). If you change the input voltage, this urges the current through and the voltage across the negative resistor to change as well. But the negative resistor reacts to your "intervention" by changing its instant resistance and in this way it makes the same current and voltage change again. In other words, the negative resistor changes on itself its "own" current and voltage. In electronics (and in this world at all), this mechanism is referred to as feedback. In the bistable mode, the feedback is positive and the loop gain (between the input and the output) is bigger than one. As a result, the negative resistor switches between the two states in an avalanche-like manner accelerated by this self-reinforcing feedback. But how does the negative resistor do this magic? As in the emulating setups above, let's put ourselves in its place (empathy) to see what it "thinks":) and what it does during the transistion...

Beginning from the one end value (one of the ends of the negative resistance region) the negative resistor "is looking for" the equilibrium point (it has to be an intersection point between the two IV curves). For this purpose, it changes vigorously its instant (chordal) resistance looking for the equilibrium point but... it does not find it somewhere on the negative resistance part. Instead, it recedes further and further from the equilibrium point in an avalanche-like manner and finally, when the negative resistance region finishes, it reaches the equilibrium point somewhere on the other positive resistance part of the curve. Thus the NDR momentarily "jumps" from the one to the other positive part running quckly the negative area (the S-shaped NDR "jumps" vertically; the N-shaped NDR "jumps" horizontally).

To show in detail the mechanism of operation, two separate graphs are presented for the cases of increasing (Fig. 12a and 13a) and decreasing (Fig. 12b and 13b) input quantity. If you superimpose the two partial curves, they will represent the whole hysteresis curves for S-shaped (Fig. 12) and N-shaped (Fig. 13) negative resistors.

#### Voltage controlled S-shaped NDR

To operate in a bistable mode, an S-shaped NDR should be driven by a low-resistive enough input source with internal resistance RG < RNE; in the extreme case, this is a perfect voltage source with RG = 0.

Fig. 12a: An IV curve (blue) of an S-shaped NDR driven by increasing voltage (red).

Increasing voltage. When the input voltage varies from zero to maximum (Fig. 12a), the S-shaped NDR keeps up high ohmic resistance in the first part. In the middle part, when increasing voltage reaches VH, it decreases momentarily its instant (chordal) resistance from high to low. Its IV curve (orange) rotates counterclockwise while the voltage source IV curve stays immovable... then what is the trajectory of the operating point A? Let's try to guess - it moves up ("jumps up") along the voltage source IV curve and pictures this vertical part of its IV curve. Thus during the jump, the current increases instantly (jumps up) but the voltage stays constant (see more about the jump below). In the last part, the NDR has low ohmic resistance.

Fig. 12b: An IV curve (blue) of an S-shaped NDR driven by decreasing voltage (red).

Decreasing voltage. When the input voltage varies back from maximum to zero (Fig. 11b), the S-shaped NDR keeps up low positive resistance in the last part. In the middle part, when decreasing voltage reaches VL, it increases momentarily its ohmic (chordal) resistance from low to high. Its IV curve rotates clockwise; the operating point A moves down along the voltage source IV curve and pictures this vertical part of the curve. During the jump, the current decreases instantly (jumps down) but the voltage stays constant. In the last part, the NDR has high positive resistance.

#### Current controlled N-shaped NDR

To operate in a bistable mode, an N-shaped NDR should be driven by a high-resistive enough input source with internal resistance RG > RNE; in the extreme case, this is a perfect current source with infinite RG.

Fig. 13a: An IV curve (blue) of an N-shaped NDR driven by increasing current (green).

Increasing current. When the input current varies from zero to maximum (Fig. 13a), an N-shaped NDR behaves as follows. In the first part, it keeps up low positive resistance. In the middle part, when increasing current reaches IH, it increases momentarily its ohmic (chordal) resistance from low to high. Its IV curve rotates clockwise; the operating point A moves from left to right along the current source IV curve and pictures this horizontal part of the curve. During the jump, the voltage decreases instantly (jumps up) but the current stays constant (see more about the jump below). In the last part, it has high positive resistance.

Fig. 13b: An IV curve (blue) of an N-shaped NDR driven by decreasing current (green).

Decreasing current. When the input current varies back from maximum to zero (Fig. 13b), the N-shaped NDR keeps up high positive resistance in the last part. In the middle part, when decreasing current reaches IL, it decreases momentarily its ohmic (chordal) resistance from high to low. Its IV curve rotates counterclockwise; the operating point A moves from right to left along the current source IV curve and pictures this horizontal part of the curve. During the jump, the voltage decreases instantly (jumps down) but the current stays constant. In the first part, it has low positive resistance.

#### Where is the negative resistance during the "jump"?

You probably already are looking for the negative resistance region (section 3-4) on the NDR IV curve and do not find it. Where is it? As though it has dissapeared and the operating point A slides along the source's IV curve; it cannot move along the negative resistance section. To see why, let's consider a voltage controlled S-shaped NDR (fig. 12a). In this case, the operating point cannot move back along the section 3-4 (the voltage cannot go down) since the voltage source keeps up a constant voltage across the resistor during the "jump". But the negative resistor has somehow to get to the other stable point; it has to find a path to this point... and it passes over the voltage source IV curve (a vertical line). Similarly, in the case of a current controlled N-shaped NDR (fig. 13a), the operating point cannot move back along the section 3-4 (the current cannot go down) since the current source passes now a constant current through the resistor during the "jump". To get to the other stable point, the NDR has to find a path to this point... and it passes over the horizontal current source's IV curve.

Another surprising fact is that during the "jump", the current through and the voltage across the negative differential resistor change in the same direction! This contradicts the NDR definitions - "the current is a decreasing function of the voltage" or "the voltage is a decreasing function of the current". Then? It sounds surprising but there is no other way, we should conclude:

During the "jump", the negative differential resistor is not a negative resistor!

Maybe, it is a kind of a noncontrollable extremely rapidly changing resistor? Or something else? If you know, discuss on the talk page...

### A feedback viewpoint at the NDR operation

It is a powerful idea to see the great feedback phenomenon in the operation of negative differential resistors. Looking from this well-known viewpoint at these mysterious elements, we can understand better their odd behavior. Let's do it!

Linear mode. What does a negative resistor do when operating in a linear mode (RG > RNE for an S-shaped NDR and RG < RNE for an N-shaped NDR)? It just keeps up the operating point on the negative resistance part of its IV curve. To see how it does this magic, look again at fig. 11 and imagine what happens when the input source changes its quantity. To deep in the NDR operation, assume that the negative resistor is too slow and lazy (as a human being:) and cannnot react instantly to the change.

When the input quantity sharply increases, the source IV curve moves and the operating point slides along the immovable IV curve of the chordal (instant, ohmic) NDR resistance since the negative resistor does not react at all in the first moment. Thus the operating point finally settles on the intersection point between the two IV curves that is apart from the NDR IV curve. To compensate the input intervention, the negative resistor begins changing its instant resistance in the according direction (S-shaped NDR - decreases, N-shaped NDR - increases) so that to send back the operating point on the negative resistance part. For this purpose, it rotates its chordal IV curve in the according direction (S-shaped NDR - counterclockwise, N-shaped NDR - clockwise) so that the operating point slides along the source IV curve and finally reaches the equilibrium on the negative resistance part. The the negative resistor operating in a linear mode, can be thought as an element with a positive feedback with a loop gain < 1.

Bi-stable mode. Now let's see what the negative resistor does when operating in a bistable mode (RG < RNE for an S-shaped NDR and RG > RNE for an N-shaped NDR). Look at fig. 14 as an N-shaped example to see. The negative resistor does just the same - trying to find the desired equilibrium point, it changes more and more its instant resistance but... in the wrong direction! It is misled:) and recedes further and further from the desired equilibrium point in an avalanche-like manner. Thus the negative resistor operating in a bistable mode, can be thought as an element with a positive feedback with a loop gain > 1.

## Examples

Fig. 14. A neon lamp is a typical example of an S-shaped negative differential resistor.

### Electronic elements and circuits

Tunnel diodes and Gunn diodes exhibit negative resistance region in their IV curve and have an "N" shaped transfer curve. Unijunction transistors also have negative resistance properties when a circuit is built using other components. Other negative resistance diodes have been built that have an "S" shaped transfer curve;[2] neon lamps (fig. 14) also have S-shaped IV curves. There are transistor circuits with positive feedback (a set of interconnected bipolar transistors, one PNP and the other NPN) exhibiting negative differential resistance.[3]

### Non-electrical devices

There are many mechanical systems that exhibit ranges of negative differential resistance. A general characteristic of negative resistance systems is that by driving them "firmly" it is possible to traverse the negative resistance region continuously (linear applications), but bistable switching action occurs if the system is driven "loosely" (bi-stable applications). More frequently, these mechanical NDR devices are used when operating in a bistable mode.

## Applications

### Linear applications

#### Compensating resistive losses...

The amazing property of negative resistors (both true and differential) is to compensate the equivalent "positive" resistance. They do this magic by adding the same energy into circuits (as much as it loses in the positive resistance). This technique is widely used in oscillators.

##### ...by an S-shaped NDR

For example, imagine a circuit consisting of series connected input voltage source, load and some undesired positive resistance (e.g., line resistance). When the input voltage increases, the voltage drop across the positive resistance increases as well. To compensate it, we can break the circuit and connect an equivalent S-shaped true negative resistor. It is a varying voltage source that produces the same voltage and adds it to the input voltage. As a result, the undesired resistance (the voltage drop) is compensated.

But negative differential negative resistors are just resistors. Then how can they compensate resistive losses? The trick is in some way paradoxical. We break the circuit and connect an additional voltage source (a power supply) and an S-shaped negative differential resistor with the same resistance. Voltage drop appears across the NDR and subtracts from the input voltage. When the input voltage increases, the voltage drop across the positive resistance increases as above. But now, the NDR decreases its resistance so that the voltage drop across it decreases as much as the undesired voltage drop across the positive resistance. This means the effective input voltage is increased with this value and the undesired voltage drop is compensated. So, an NDR first introduces additional voltage drop and then decreases it to compensate the undesired voltage drop across the positive resistance. BTW, this trick is widely used in our routine. For example, imagine you want to look good. For this purpose, you first become bad and then decrease your wickedness; as a result, you look good:)

More figuratively, you can think of an S-shaped NDR as of a "faithful" rheostat connected in series with the input voltage source. When the input voltage source tries to increase the current through the circuit, the rheostat moves its slider to decrease its resistanse and to increase the current; thus it "helps" the input source.

##### ...by an N-shaped NDR

A parallel connected N-shaped NDR can compensate positive resistance in a similar way. Initially it shunts the positive resistance. When the input voltage increases, the NDR increases its resistance and shunts less the positive resistance. In this way, it "helps" the input source to increase the voltage drop across the positive resistance.

As above, you can think of an N-shaped NDR as of a "faithful" rheostat connected in parallel with the load. When the input current source tries to increase the voltage across the load, the rheostat moves its slider to increase its resistanse and accordingly, to increase the voltage drop; thus it "helps" the input source.

#### Amplification

They say a negative differential resistor can act as an amplifier. If this is true, is there any difference between a negative resistor and an amplifier? If yes, what is the difference? Let's clarify the topic.

##### What does amplification actually mean?

Strictly speaking, there is no amplification; it is just impossible. We cannot amplify energy (power); we can only control, regulate it. Then what does an amplification mean? How do we amplify? How do we make an amplifier?

It sounds bluntly, but it is true that in (analog) electronics we use the possibly silliest idea for this purpose. In order to amplify some small input power (in electronics, usually presented by input voltage), we get many times bigger (at least, VPS = K.VINmax) power source and then, imagine, we throw out the excessive power! An example of this absurd (in energetics, they never do that): in order to "amplify" 10 times 1V input voltage by a 24V supplied amplifier, we throw out (as a heat) the power according to the rest 14V. Doing that, we actually dissipate, attenuate power. As a result, there is not amplification; there is only attenuation!

##### An ordinary 2-port amplifier

We realize this absurd idea by elements acting as electrically controlled resistors (in the past - carbon microphones, tubes; now - transistors, etc.) They change their resistance to resist the current (to dissipate power), proportionally to the magnitude of the input voltage. Thus we can assemble amplifiers by using only two components: a power supply and a regulating element.

##### An NDR 1-port amplifier
Fig. 15: This funny picture reveals the basic idea behind a tunnel diode amplifier (for simplicity, biasing circuits are omitted and the input voltage source is floating). The operation is illustrated graphically inside the circuit diagram by superimposing the two almost parallel IV curves. When the input voltage wiggles slightly, the intersection (operating) point moves vigorously.

In the negative resistance region of their IV curve, NDR do the same - they change their resistance accordingly to the current through them or the voltage across them. So, they are electrically controlled resistors; they can act as needed "amplifying" elements. The only difference with the ordinary 3-terminal "amplifying" elements is that NDR are odd 2-terminal elements whose input and output part are the same. So, we can build an odd negative resistance amplifier by connecting in series four components (fig. 15): a constant-voltage power supply V, an input voltage source VIN, a positive resistor R and a negative differential resistor NDR (e.g., an N-shaped type implemented by a tunnel diode). Actually, the two resistors constitute a sort of dynamic voltage divider driven by a varying composed voltage source (V + VIN); its transfer ratio depends vigorously on the input voltage.

When the input voltage varies slightly, the negative differential resistor changes considerably its resistance according to the input voltage, which makes the voltage divider change noticeably its transfer ratio. As a result, the voltage drops across the positive and negative resistors vary considerably; so, some of them may be used as an output voltage. To obtain maximum gain but to stay still in the linear mode, the ratio R/RNDR has to be close to but less than unity.

This requirements is illustrated graphically in fig. 15 where the two IV curves are superimposed on the common IV coordinate. The IV curve of the negative resistor is the usual N-shaped curve that starts from the coordinate origin; in the negative resistance region, it is a straight line. The other one is a compound curve consisting of two subcurves: an IV curve of the voltage source and an IV curve of the "positive" resistor (the latter is frequently named "load line" although it is not always correct). The two main curves are almost parallel; so, when the input voltage wiggles slightly, the intersection (operating) point moves vigorously along the NDR IV curve.

In this arrangement, the differential negative resistor is not an amplifier; it is just a part of an amplifier (a 2-terminal active element). The combination of the differential negative resistor acting as an active element and the power supply constitutes a true amplifier.

The same effect can be achieved by varying the positive resistance instead the voltage. In this case, the dynamic voltage divider is assembled by two varying resistors - positive and negative differential. When the positive resistor changes slightly its resistance, the negative resistor changes vigorously its resistance as a respond to the "intervention". If we take the voltage drop across the negative resistor as an output, we will obtain a sensitive resistance-to-voltage converter.

With the same success we can make an amplifier by an S-shaped negative differential resistor NDR. To obtain maximum gain but to stay still in the linear mode, the ratio R/RNDR now has to be close to but bigger than unity. It seems we can make an amplifier with a humble neon lamp? Try it and describe it here:)

### Bi-stable applications

#### Electrical devices

Schmitt trigger. A negative differential resistor operating in a bi-stable mode actually acts as a kind of an exotic 2-terminal Schmitt trigger. It can be used in various switching applications, e.g. to make various relaxation oscillators (with neon lamps and other NDR).

Latch (flip-flop). A negative differential resistor operating in a bi-stable mode exhibits a hysteresis. If it is properly biased so that the quiescent point to be in the middle of the hysteresis cycle, it can act as a latch (flip-flop). So it can be used to memorize one bit of data. An example: a neon lamp acting as a flip-flop with light indication.

#### Mechanical devices

An NDR operating in a bi-stable mode is a common design element in mechanical systems that are designed to have "detents" or a "positive action" or a "click." A popular example is the well-known pen clicker. Good examples are also the keys on a computer keyboard and on a computer mouse, taking the key position and upward force to be analogous to voltage and current, respectively. As a key is pressed downward, it initially presents a firm and increasing upward force. Beyond a critical point, a zone is entered in which the upward force decreases, which feels like a "sudden" yielding. This is often referred to as a "collapse action" mechanism. There are several keyboard technology that give such collapse action, such as buckling spring switches.

Like electrical examples above, all these NDR devices operating in a bi-stable mode can act as mechanical Schmitt triggers (clicking push-buttons) or mechanical flip-flops (toggle switches).

## Making comparison

### Ohmic versus differential negative resistors

• Both they are positive resistors that absorb energy from circuits.
• Ohmic resistors have constant (steady) resistance while negative differential resistors have varying (dynamic) instant resistance.
• There is only one kind of ohmic resistors but there are two kinds of negative differential resistors (S- and N-shaped).
• Both they have IV curves beginning from the coordinate origin.

### Differential versus true negative resistors

• Negative differential resistors can be elements or electronic circuits while true negative resistors are only circuits.
• Both the negative resistors are dynamic electronic elements (circuits).
• Negative differential resistors are dynamic resistors while true negative resistors are dynamic electrical sources.
• NDR IV curve always starts from the coordinate origin since there is no internal source (thus if the applied voltage is zero, the current is zero as well and v.v.)
• NDR IV curve can occupy the first or third quadrant of the coordinate system.
• True negative resistors inject their own energy while differential negative resistors control external energy.
• Negative differential resistors cannot be used independently; they may be used in combination with electrical sources to build true negative resistors.
• True negative resistors created as negative impedance converters (NICs) consist of a constant ohmic resistor and a varying voltage source while these created on the base of NDR consist of a dynamic resistor and a constant voltage source.
• When connected in series, an S-shaped NDR compensates the undesired voltage drop across the equivalent positive resistance by decreasing the undesired voltage drop across itself while the true negative resistor does the same by adding an equivalent voltage.
• When connected in parallel, an N-shaped NDR compensates the undesired current consumed by the equivalent positive resistance by decreasing the (initially) consumed current while the true negative resistor does the same by adding an equivalent current.

### S-shaped versus N-shaped NDR

• Both they have negative resistance region represented by a part of straight line in their IV curves that is inclined to the left.
• Both they have IV curves biginning from the coordinate origin.
• S-shaped NDR create negative resistance by decreasing their instant resistance when the input voltage increases; N-shaped NDR create negative resistance by increasing their instant resistance when the input current increases.
• When connected in series, the S-shaped NDR "helps" the input voltage source to pass the desired current through the load by decreasing the undesired voltage drop across itself while the N-shaped NDR "opposes" it by increasing the voltage drop.
• When connected in parallel, the S-shaped NDR "opposes" the input current source to pass the desired current through the load by diverting a larger part of the current while the N-shaped NDR "helps" it by diverting a smaller part of the current.