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A frequency-dependent negative resistor (FDNR) is a circuit element that exhibits a purely real
If all the impedances (including the source and load impedances) of a passive
Wait gives the circuit shown to the right as suitable for a grounded FDNR.
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 ordina ...
−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 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 characteri ...
(GIC) or gyrator
A gyrator is a passivity (engineering), passive, Linear circuit, linear, lossless, two-port network, two-port electrical lumped-element model, network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after t ...
. The impedance of a FDNR is
:
or
:
when ''s'' = j''ω''.
The concept of frequency-dependent negative resistors was introduced by L. T. Bruton and is expounded upon within several chapters of his book.
They are also discussed in Temes & LaPatra, Chen and Wait, Huelsman & Korn.
Application
If all the impedances (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 tr ...
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 function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
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 an electric current flows through it. An inductor typically consists of an insulated wire wound into a ...
which are often problematic components. This technique is especially useful when all the capacitors are grounded. If the technique is applied to capacitors
In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...
that are not grounded, the resulting FDNRs are floating (neither end is grounded), which in practice, can be difficult to stabilize.
The resulting circuit requires practical considerations because FDNR lowpass filters require a DC path to ground. The DC transfer function of the analog prototype filter has a value of (R6)/(R1 + R6) whereas 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 leakage resistance in the capacitors so that the actual DC performance of the transformed ladder is unpredictable. Ra and Rb are added to the circuit to mitigate these problems. If Rb/(Ra + Rb + L3/k + L5/k) = (R6)/(R1 + R6) then the DC gain of the transformed circuit is the same as the predecessor circuit. Finally, if Ra and Rb 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