# Circuit Idea/Negative Resistance

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The page is under a major reconstruction. Now use Negative impedance and Negative differential resistance.

Circuit idea: Making dynamic resistors and dynamic sources overact.

Negative resistance is a property of some elements and electric circuits, in which current through and voltage across them change in opposite directions (in contrast to a simple ohmic resistor, where current and voltage change in the same direction).

## Background

Creation. Negative resistance is created on the base of positive resistance by modifying the instant (static) resistance in a limited region of the operating range. In the negative resistance region, the IV curve folds up and shows a negative slope; in contrast to this, a resistor will have a positive slope along the whole IV curve.

Varieties. Depending on the way of modification, there are two kinds of negative resistors - absolute ("true") negative resistors (ANR) and negative differential resistors (NDR). In the negative resistance region, true negative resistors behave as dynamic sources while differential resistors behave as dynamic resistors.
Depending on the shape of the IV curve, there are two kinds of negative resistors - with S-shaped and with N-shaped IV curve.[nb 1] Both the curves consist of three sections where the middle part with a negative slope represents the negative resistance region and the end parts with positive slopes represent positive resistance.

Modes. To operate in the negative resistance region (linear mode), N-shaped negative resistors should be driven by electric sources with low internal resistance (e.g., voltage sources) while S-shaped negative resistors should be driven by electric sources with high internal resistance (e.g., current sources). Otherwise, they will act like bistable devices with positive feedback (Schmitt triggers). In the general case, elements and circuits with negative resistance are driven by real voltage sources with some positive internal resistance (it can include additional resistances). The proportion between the magnitudes of the positive and negative resistances determines the operating mode.

Properties. Whereas positive resistors consume energy from circuits, the equivalent negative resistors add the same energy to circuits. A true negative resistor does this by itself as it contains its own power source; a negative differential resistor does it by regulating the energy of an additional power supply. Thus the basic property of negative resistance is to neutralize an equivalent positive resistance.

## "Absolute" negative resistance

### Properties

Strictly speaking, there are not true negative resistors exactly as there are not true energy sources since they will violate thermodynamics laws; there are only energy converters. Whereas positive resistors consume energy from circuits, the equivalent true negative resistors add the same energy to circuits. For example, if the same current I flows through a positive resistor and through an S-shaped negative resistor with the same resistance R, the positive resistor subtracts a voltage drop V = R.I from while the negative resistor adds voltage V = R.I to the circuit. Thus the positive resistor acts as a two-terminal (1-port) current-to-voltage drop converter while the equivalent true negative resistor acts as a current-to-voltage converter.

In contrast to ordinary constant sources, true negative resistors are dynamic sources whose voltage depends linearly on the current flowing through them or whose current depends linearly on the voltage across them. They are auxuliary sources that cannot operate independently; they begin operating only after the main source begins operating.

### Basic circuit

Absolute negative resistors are composed devices (circuits) consisting of two connected in series components: an ohmic resistor and a dynamic voltage source. This arrangement is one of the possible Miller theorem realizations.

The two-terminal negative resistance circuit shown in Fig. 6 is an op-amp implementation of an "absolute" negative resistor with N-shape IV curve (shown below in Figure 7a). The two resistors R1 and the op amp constitute a non-inverting amplifier with gain A = 2 that serves as the dynamic voltage source needed. It amplifies the input voltage V across the two input circuit terminals and applies it through the resistance R back to the input. For a positive input voltage, the current I = V/R is reversed and "pushed" back into the input source instead to be drawn from it as in the case of positive resistance R. The circuit as though converts the "positive resistance" R into negative one by inverting the current direction; thus the name negative impedance converter with current inversion (INIC). There is also a dual circuit - negative impedance converter with voltage inversion (VNIC).

The input port of the circuit can be connected into another network as if it was a negative resistance component.

### Operation

Negative impedance converters are established to operate in a linear mode with the purpose to use their negative resistance regions. They can operate in a bistable mode as well but the simpler op-amp non-inverting and inverting Schmitt triggers (special cases of INIC and VNIC) are usually used in this case. To operate in a linear mode (one-valued function), an N-shaped "true" negative resistor should be driven by an input source with low internal resistance RG < RNE (e.g., a perfect voltage source with RG = 0) and an S-shaped negative resistor should be driven by an input source with high internal resistance RG > RNE (e.g., a perfect current source with infinite RG).

These circuits have constant ohmic resistance but different voltage in the three parts of their IV curves (when the input quantity varies from zero to maximum). In the middle part, the op-amp operates in active mode and manages to create negative resistance. The IV curve has a negative slope in the middle region and passes through the origin of the coordinate system; it can enter the 2nd and 4th quadrants as additional energy is supplied. This is to be compared with NDR devices such as the tunnel diode where the negative slope portion of the curve does not pass through the origin. In the end parts, the op-amp is saturated and the circuit behaves as a voltage source with internal resistance R.

For example, when the input voltage varies from minimum (negative) to maximum (positive), an N-shaped negative resistor produces constant negative voltage VSAT- in the first part (Fig. 7a); in the negative resistance region, its voltage increases from negative to positive, and in the last part, the voltage VSAT+ is constant positive.

Figure 7a: An N-shaped IV curve (black) of a true negative resistor (the INIC from Fig. 8) driven by a voltage source. The operating point C is represented by the intersection point of two superimposed IV curves: the red one of the input voltage source VIN and the green one of the true negative resistor (dynamic internal voltage source VOA and a constant ohmic resistor R connected in series). When the input voltage varies in the negative resistance region, the internal voltage source VOA varies and the operating point C moves and pictures the negative resistance part (2-4) of the curve.

Figure 7b: An S-shaped IV curve (black) of a true negative resistor (VNIC, unshown here) driven by a current source. The operating point C is represented by the intersection point of two superimposed IV curves: the green one of the input current source IIN and the red one of the true negative resistor (dynamic internal voltage source VOA and a constant ohmic resistor R connected in series). When the input current varies in the negative resistance region, the internal voltage source VOA varies and the operating point C moves and pictures the negative resistance part (2-4) of the curve.

### Generalization

True negative resistance circuits invert (by means of additional sources) the polarity of the voltage across or the direction of the current through an ohmic resistor to convert it into a negative resistor. This idea can be extended by replacing the "original" ohmic resistor with a nonlinear resistor (e.g., a diode), capacitor, inductor or other impedance to obtain a negative diode, negative capacitor, negative inductor or other element with negative impedance.

For example, the "positive" impedance of the stray capacitance CSTR (Fig. 10) consuming undesired current can be neutralized (made infinite) by connecting in parallel a negative capacitor with the same but negative capacitance. It is implemented by a "positive" capacitor with the same capacitance C = CSTR and a voltage source with redoubled voltage VH = 2VCstr. This negative capacitor eliminates the undesired current ICstr by producing the same current IC = ICstr. As a result, the stray capacitance does not consume any current from the input source as the negative capacitor does provide all the current needed to charge the stray capacitance. The op-amp implementation of this negative capacitor (Fig. 9) is just the negative impedance converter with current inversion (INIC) from Fig. 6 where the resistor R is replaced by the capacitor C.

w:Wien bridge oscillator is another example of using true negative impedance where the complex impedance of a series RC network is converted into negative impedance that neutralizes the impedance of a parallel RC network.

Although these circuits are named "negative impedance converters", actually they do not convert solely impedance into negative one. They can convert any passive element connected in this place into an active one just by copying the voltage drop across the passive element or the current through it and inserting the same voltage or current into the circuit.

### Similarities

Fig. 10: In an op-amp inverting integrator, the op-amp behaves as a negative capacitor cancelling the "positive" capacitance C.

Negative feedback. Op-amp inverting circuits with shunt negative feedback can mimick true negative impedance. Examples are w:transimpedance amplifier, diode logarithmic converter, capacitive integrator and inductive differentiator where the properly supplied op-amp behaves accordingly as a true negative resistor, diode, capacitor and inductor. It produces output voltage that is a mirror copy of the voltage drop across the impedance element connected in the feedback loop. This voltage compensates the voltage drop across the "positive" impedance and the resulting voltage is almost zero (w:virtual ground); thus the op-amp negative impedance cancels (zeros) the positive impedance. So the op-amp acts as a kind of a negative impedance converter with voltage inversion (VNIC).

For example, in the widespread circuit of an op-amp inverting integrator (Fig. 10), the op-amp serves as a negative capacitor neutralizing (zeroing) the impedance of the capacitor C. As a result, the impedance between the op-amp inputs is almost zero (the circuit represents one useful implementation of w:Miller effect).

These circuits are not exactly negative impedance circuits as they use an additional third wire to sense the virtual ground while the genuine negative impedance circuits are two-terminal. The op-amp inverting circuits only behave as negative impedance circuits.

Positive feedback. The electronic part of LC oscillators is usually implemented as an amplifier with positive feedback. It behaves as a circuit with true negative resistance that compensates the "positive" internal resistance of the passive LC circuit. As above, it is not a genuine two-terminal negative resistor; it only behaves as a negative resistor.

### Applications

#### Impedance cancellation

The basic property of negative resistance is to neutralize an equivalent positive resistance: connecting an S-shaped negative resistor in series with an equivalent ohmic resistor gives zero total resistance; connecting an N-shaped negative resistor in parallel to an equivalent ohmic resistor gives infinite total resistance. Because of this compensating property, true negative resistance can be used to cancel the effects of positive impedances, for example, by eliminating (zeroing) the internal resistance of a voltage source or making the internal resistance of a current source infinite. This property is used in telephony line repeaters and in circuits such as the Howland current source, Deboo integrator and load cancellers.

#### Oscillators

All feedback oscillators imply the presence of negative resistance. There are many such topologies, including the w:Dynatron oscillator, w:Colpitts oscillator, w:Hartley oscillator, w:Wien bridge oscillator, and some types of w:relaxation oscillators. If the feedback loop is broken and the input impedance examined it will be found to include negative resistance. Thus, in LC oscillators, electronic circuits behaving as negative impedance devices compensate the losses inside an LC tank.

## Negative differential resistance

Negative differential resistance is a kind of nonlinear resistance. So, negative differential resistors are dynamic but still positive resistors. They have different resistance in the three parts of their IV curves (located in the 1st or the 3th quadrant) that depends on the kind of the NDR.

Negative differential resistors are only two-terminal active elements (such as transistors) that cannot be used independently; they need an additional power supply. Thus the combination of the negative differential resistor and the power supply can be considered as a true negative resistor. 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.

### Operation

Depending on the proportion between the magnitudes of the input positive resistance RG and the negative resistance RNE, there are two modes of operation - linear and bistable.

#### Linear mode

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

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. When the input voltage varies from zero to maximum (Fig. 2a), 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, and in the last part, it has high positive resistance.

Figure 2a: An N-shaped IV curve (blue) of a negative differential resistor driven by a voltage source. 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 increases in the negative resistance region, the resistance increases and its IV curve rotates clockwise; the operating point C moves down and pictures the negative resistance part of the curve.

Figure 2b: An S-shaped IV curve (blue) of a negative differential resistor driven by a current source. 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). If the current increases in the negative resistance region, the resistance decreases and its IV curve rotates counterclockwise; the operating point C moves up and pictures the negative resistance part of the curve.

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. When the input current varies from zero to maximum (Fig. 2b), 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, and in the last part, it has low ohmic resistance.

#### 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 the end stable values. The switching between the two states is an avalanche-like process accelerated by the intrinsic positive feedback. Beginning from the one end value and "looking for" the equilibrium state, the negative resistor changes vigorously but in the "wrong" direction its instant (chordal) resistance. Thus it recedes further and further from the equilibrium point in an avalanche-like manner and finally reaches the other end value. To show in detail the mechanism of operation, two separate graphs are presented for the cases of increasing (Fig. 3) and decreasing (Fig. 4) input quantity. When superimposed, the two partial curves compose the whole hysteresis curves for N-shaped (3a + 4a) and S-shaped (3b + 4b) negative resistors.

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. When the input current varies from zero to maximum (Fig. 3a), an N-shaped NDR behaves as follows: in the first part, it keeps up low positive resistance; in the middle part, it increases momentarily its ohmic (chordal) resistance from low to high, and in the last part, it has high positive resistance. When the input current varies back from maximum to zero (Fig. 4a), the N-shaped NDR keeps up high positive resistance in the last part; in the middle part, it decreases momentarily its ohmic (chordal) resistance from high to low, and in the first part, it has low positive resistance.

Figure 3a: An IV curve (blue) of an N-shaped NDR driven by a current source (green). When increasing current reaches IH, the instant resistance increases momentarily. Its IV curve (orange) rotates clockwise; the operating point A moves ("jumps") from left to right along the current source IV curve and pictures this horizontal part of the curve. Thus during the jump, the voltage increases instantly (jumps up) but the current stays constant.

Figure 3b: An IV curve (blue) of an S-shaped NDR driven by a voltage source (red). When increasing voltage reaches VH, the instant resistance decreases momentarily. Its IV curve (orange) rotates counterclockwise; the operating point A moves up ("jumps up") along the voltage source IV curve and pictures this vertical part of the curve. Thus during the jump, the current increases instantly (jumps up) but the voltage stays constant.

Figure 4a: When decreasing current reaches IL, the instant resistance decreases momentarily. 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.

Figure 4b: When decreasing voltage reaches VL, the instant resistance increases momentarily. 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.

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. When the input voltage varies from zero to maximum (Fig. 3b), the S-shaped NDR keeps up high ohmic resistance in the first part; in the middle part, it decreases momentarily its ohmic (chordal) resistance from high to low, and in the last part, it has low ohmic resistance. When the input voltage varies back from maximum to zero (Fig. 4b), the S-shaped NDR keeps up low positive resistance in the last part; in the middle part, it increases momentarily its ohmic (chordal) resistance from low to high; in the first part, it has high positive resistance.

### Examples

w:Tunnel diodes are heavily doped semiconductor junctions that have an "N" shaped transfer curve. Gunn diodes exhibit a negative resistance region in their IV curve. w: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; w:neon lamps 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.

### Applications

#### Amplifiers

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 a active element and the power supply constitutes a true amplifier.

##### Tunnel diode amplifier

Figure 5: 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 two almost parallel IV curves. When the input voltage wiggles slightly, the intersection (operating) point moves vigorously.

When biased so that the operating point is in the negative resistance region, these devices can be used as an amplifier. To build such a one-port amplifier, four components are connected in series (Fig. 5): a constant-voltage power supply V, an input voltage source VIN, a positive resistor R and a negative differential resistor NDR (a tunnel diode). Actually, the two resistors constitute a dynamic w:voltage divider supplied by a varying composed voltage source (V + VIN). 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 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, the ratio R/RNDR has to be close to but less than unity.