Frequency dependent negative resistance electronic symbol

A frequency-dependent negative resistor (FDNR) is a circuit element that exhibits a purely real

negative resistance In electronics, negative resistance (NR) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it. This is in contrast to an ordi ...

−1/(''ω''²''kC'') that decreases in magnitude at a rate of −40 dB per decade. The element is used in implementation of low-pass

active filter An active filter is a type of analog circuit implementing an electronic filter using active components, typically an amplifier. Amplifiers included in a filter design can be used to improve the cost, performance and predictability of a filter ...

s modeled from

ladder A ladder is a vertical or inclined set of rungs or steps used for climbing or descending. There are two types: rigid ladders that are self-supporting or that may be leaned against a vertical surface such as a wall, and rollable ladders, such ...

filters. The element is usually implemented from a

generalized impedance converter A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characte ...

(GIC) or

gyrator A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four con ...

. The impedance of a FDNR is :

Z = \frac

or :

Z = \frac,

when ''s'' = j''ω''. The definition and application of frequency-dependent negative resistors is discussed in Temes & LaPatra, Chen and Wait, Huelsman & Korn. The technique is attributed to L. T. Bruton.

Application

Low pass ladder filter FDNR implementation

If all the

impedances In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit. Quantitatively, the impedance of a two-terminal circuit element is the ratio of the compl ...

(including the source and load impedances) of a passive

ladder filter Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected. Filter design characterises filter circuits primarily by their ...

are divided by ''s''k, the

transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...

is not changed. The effect of this division is to transform resistors into capacitors, inductors into resistors and capacitors into FDNRs. The purpose of this transformation is to eliminate

inductors An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...

which are often problematic components. This technique is especially useful when all the capacitors are grounded. If the technique is applied to

capacitors A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a ...

that are not grounded, the resulting FDNRs are floating (neither end is grounded), which in practice, can be difficult to stabilize. The resulting

circuit Circuit may refer to: Science and technology Electrical engineering * Electrical circuit, a complete electrical network with a closed-loop giving a return path for current ** Analog circuit, uses continuous signal levels ** Balanced circu ...

has two problems. Practical FDNRs require a DC path to ground. The DC transfer function has a value of (R₆)/(R₁ + R₆). The transformed ladder filter realizes the DC transfer gain as the ratio of two capacitors. In the ideal case, this is valid, but in the practical case there is always some, usually unpredictable, finite resistance across the capacitors so that the DC performance of the transformed ladder is unpredictable. R_a and R_b are added to the circuit to mitigate these problems. If R_b/(R_a + R_b + L₃/k + L₅/k) = (R₆)/(R₁ + R₆) then the DC gain of the transformed circuit is the same as the predecessor circuit. Finally, if R_a and R_b are large with respect to the other resistors there is little effect on the filter's transition band and high frequency behavior.

Implementation

Wait gives the circuit shown to the right as suitable for a grounded FDNR.

References

{{Reflist Analog circuits Linear filters