# Negative resistance

Figure 1: A working mechanism of a resonant tunneling diode device and negative differential resistance in output characteristic. Notice the negative resistance characteristic after the first current peak due to reduction of first energy level below source fermi level with gate bias.

Negative resistance is a property of some electric circuits where an increase in the current entering a port results in a decreased voltage across the same port. This is in contrast to a simple ohmic resistor, which exhibits an increase in voltage under the same conditions. Negative resistors are theoretical and do not exist as a discrete component. However, some types of diodes (e.g., tunnel diodes) can be built that exhibit negative resistance in some part of their operating range. Such a differential negative resistance is illustrated in Figure 1 with a resonant-tunneling diode. Electric discharges through gases exhibit negative resistance, and some chalcogenide glasses,[1] organic semiconductors, and conductive polymers exhibit a similar region of negative resistance as a bulk property. In electronics, negative resistance devices are used to make bistable switching circuits and electronic oscillators, particularly at microwave frequencies.

## Properties

Figure 2: The IV curve of a theoretical negative resistor

Figure 2 shows a graph of a negative resistor, showing the negative slope. In contrast to this, a resistor will have a positive slope. Tunnel diodes and Gunn diodes[2] exhibit a negative resistance region in their IV (current – voltage) curve. They have two terminals like a resistor; but are not linear devices. Unijunction transistors also have negative resistance properties when a circuit is built using other components.

For negative resistance to be present there must be active components in the circuit providing a source of energy. This is because current through a negative resistance implies a source of energy just as current through positive resistance implies that energy is being dissipated. A resistor produces voltage that is proportional to the current through it according to Ohm's law. The IV curve of a true negative resistor has a negative slope and passes through the origin of the coordinate system (the curve can only enter the 2nd and 4th quadrants if energy is being supplied). This is to be compared with devices such as the tunnel diode where there is no source of energy within the device. The negative slope portion of the curve is entirely in the first quadrant and the curve passes through the origin into the third quadrant, never entering the second or fourth quadrants.[3]

## History

In early research it was noticed that arc discharge devices and some vacuum tube devices such as the dynatron exhibit negative differential resistance effects.[4] If the screen grid is at a higher potential than the plate electrode the secondary emission from the plate will result in large numbers of electrons being attracted to the screen grid. If the plate voltage is reduced further the plate current to the screen grid will increase. This effect is a negative resistance and when a screen grid tube is operated in this negative resistance region it is called a dynatron.[5] Small-signal transit-time effects in vacuum tube diodes can result in alternating positive and negative conductance. This occurs when the transit time of the electrons is slightly over one cycle of the AC signal.[6] Practical and economic devices only became available with solid state technology. The typical true negative impedance circuit—the negative impedance converter – is due to John G. Linvill (1953)[7] and the popular element with negative differential resistance—the tunnel diode – is due to Leo Esaki (1958).[8]

## Implementations

Figure 3: Amplifier exhibiting negative resistance through positive feedback

Any amplifier with sufficient positive feedback will present a negative resistance at its input. Referring to figure 3,

$i = \frac{v - Av}{R_1} + {v \over R_{\mathrm {in}}}$

and,

${1 \over R} \triangleq {i \over v} = \frac {1-A}{R_1} + {1 \over R_{in}}$

If Rin is large,

$R \simeq \frac {R_1}{1-A}$

### Operational amplifiers

Figure 4: Negative resistance circuit

The negative resistance circuit shown in Figure 4 is an opamp implementation of the negative impedance converter. The two resistors R1 and the op amp constitute a negative feedback non-inverting amplifier with gain A = 2. The input resistance (for an ideal opamp) is given by;

$R_\text{in} \triangleq {v \over i} = -R \,\!$

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

In the general case $Z$ can be selected to produce any value of negative impedance. Negative capacitances and negative inductances can both be simulated by these means.

### Components exhibiting negative differential resistance

Tunnel diodes are heavily doped[9] semiconductor junctions that have an "N" shaped transfer curve. A vacuum tube can also be made to exhibit negative resistance.[10] Other negative resistance diodes have been built that have an "S" shaped transfer curve.[11] When biased so that the operating point is in the negative resistance region, these devices can be used as an Amplifier. These devices can also be biased so that they will switch between two states very quickly, as the applied voltage changes.[9]

### Neuronal models

Some instances of neurons display regions of negative slope conductances (RNSC) in voltage-clamp experiments.[12] The negative resistance here is implied were one to consider the neuron a typical Hodgkin–Huxley style circuit model.

## Applications

### Oscillators

Many electronic oscillator circuits use one-port negative resistance devices, such as magnetron tubes, tunnel diodes and Gunn diodes. In these circuits, a resonator, such as an LC circuit, quartz crystal, or cavity resonator, is connected across the negative resistance device, and a DC bias voltage applied. The negative resistance of the active device can be thought of as cancelling the (positive) effective loss resistance of the resonator, creating sustained oscillations. These circuits are frequently used for oscillators at microwave frequencies. Oscillators have also been built using the negative resistance region of amplifying devices like vacuum tubes, as in the dynatron oscillator.

### Amplifiers

Figure 5: Negative resistance microwave amplifier using a circulator
Figure 6: 8–12 GHz tunnel diode amplifier, circa 1970

A device exhibiting negative resistance can be used to amplify a signal and this is an especially useful technique at microwave frequencies. Such devices do not present as pure negative resistance at these frequencies (in the case of the tunnel diode a large parallel capacitance is also present) and a matching filter is usually required. The reactive components of the device's equivalent circuit can be absorbed into the filter design so the circuit can be represented as a pure resistance followed by a bandpass filter. The output of this arrangement is fed into one port of a three-port circulator. The other two ports constitute the input and output of the amplifier with the direction of circulation as shown in the diagram. Treating R0 as being positive, the reflection coefficients at the two ends of the filter are given by;

$\Gamma_1 = \frac{Z_1 - R_0}{Z_1 + R_0}$ and, $\Gamma_2 = \frac{Z_2 - R_1}{Z_2 + R_1}$

Since the filter has no resistive elements, there is no dissipation and the magnitudes of the two reflection coefficients must be equal,

$\left| \Gamma_1 \right| = \left| \Gamma_2 \right|$

The input power entering the circulator is directed at the matching filter, is reflected at both the input and output of the filter and a portion finally arrives at the load. This portion is given by;

$\frac{P_\mbox{out}}{P_\mbox{in}} = \left| \Gamma_1 \right|^2$

For a well matched filter, the reflection coefficients will be very small in the passband and very little power will reach the load. On the other hand if R0 is replaced by a negative resistance such that,

$R_0' = - R_0 \,\!$ then,
$\Gamma_1' = \frac{Z_1 + R_0}{Z_1 - R_0}$ and,
$\left| \Gamma_1' \right| = \left| \Gamma_2' \right| = \frac{1}{\left| \Gamma_1 \right|}$

Now the reflection coefficients are very large and more power is reaching the load than was injected in the input port. The net result of terminating one port in a negative resistance is amplification between the remaining two ports.[13]

### Mixers and frequency converters

The highly non-linear characteristics of tunnel diodes makes them useful as frequency mixers. The conversion gain of a tunnel diode mixer can be as high as 20 dB if it is biased to operate in the negative resistance region.[14]

### Antenna design

Another concept of negative resistance exists in the domain of radio frequency antenna design. This is also known as negative impedance. It is not uncommon for an antenna containing multiple driven elements to exhibit apparent negative impedance in one or more of the driven elements.[15][16] This situation is different from the meaning of negative resistance in this article: the ratio of V/I is indeed negative but the slope of the IV curve remains positive at all points. The same is true of any source of electric energy. For instance a battery has a negative V/I ratio (positive I defined as going into the positive terminal) but a positive slope equal to its internal resistance.

### Impedance cancellation

Negative impedances can be used to cancel the effects of positive impedances, for example, by eliminating 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[17] and in circuits such as the Howland current source,[18] Deboo integrator[19] and load cancellers.[20][21]

## References

1. ^ Abdel-All, A.; Elshafieb, A.; Elhawaryb, M.M. (2000). "DC electric-field effect in bulk and thin-film Ge5As38Te57 chalcogenide glass". Vacuum 59 (4): 845–853. doi:10.1016/S0042-207X(00)00378-X.
2. ^ W. Alan Davis, Microwave Semiconductor Circuit Design, p. 329, Van Nostrand Reinhold ISBN 0-442-27211-1
3. ^ N. Balkan, B. K. Ridley, A. J. Vickers, Negative Differential Resistance and Instabilities in 2-D Semiconductors, p. 2, Springer, 1993 ISBN 0-306-44490-9.
4. ^ For instance G Crisson, "Negative Impedances and the Twin 21-Type Repeater", The Bell System Technical Journal, p. 492, January 1931.
5. ^ Electronics and Radio Engineering, pp 196-197, Frederick E. Terman, 1955
6. ^ Electronics and Radio Engineering, pp 212-215, Frederick E. Terman, 1955
7. ^ Linvill, J.G. (1953). "Transistor Negative-Impedance Converters". Proceedings of the IRE 41 (6): 725–729. doi:10.1109/JRPROC.1953.274251.
8. ^ Belevitch, V (1962). "Summary of the history of circuit theory". Proceedings of the IRE 50 (5): 853. doi:10.1109/JRPROC.1962.288301.
9. ^ a b RCA Tunnel Diode Manual
10. ^ J. Groszkowski (1936). "A New Electron Tube Having Negative Resistance". Proceedings of the IRE 24 (7): 1041. doi:10.1109/JRPROC.1936.228352.
11. ^ Nyle Steiner Zinc Negative Resistance Oscillator 22 March 2001
12. ^ MacLean and Schmidt, "Voltage-sensitivity of motoneuron NMDA receptor channels is modulated by serotonin in the neonatal rat spinal cord", Journal of Neurophysiology, vol. 86, no. 3, 1 September 2001.
13. ^ Matthaei, Young, Jones Microwave Filters, Impedance-Matching Networks, and Coupling Structures, pp. 4–9, McGraw-Hill 1964.
14. ^ Solid-state microwave devices. tpub.com
15. ^ Roger L. Freeman, Radio System Design for Telecommunication, page 805, John Wiley & Sons, 2006 ISBN 0470050438.
16. ^ J. Belrose, "VLF, LF and MF antennas", in: Alan W. Rudge (ed), The Handbook of Antenna Design, Volume 2, page 612, IEE, 1983 ISBN 0906048877.
17. ^ Neil J. Boucher, The Paging Technology Handbook, p. 143, John Wiley and Sons, 1995 ISBN 0-930633-17-2
18. ^ Impedance and admittance transformations using operational amplifiers
19. ^ Consider The "Deboo" Integrator For Unipolar Noninverting Designs
20. ^ Wang, W. et al., "A Comprehensive Study on Current Source Circuits", IFMBE Proceedings, Vol 17, pp. 213–216, Springer, 2007 ISBN 3-540-73840-1 doi:10.1007/978-3-540-73841-1_57.