The basic problem of synthesizing a nonlinear resistor, inductor, or capacitor with a prescribed i-v, φ-i, or q-v curve is solved by introducing three new linear two-port network elements, namely the mutator, the reflector, and the scalor. The mutator has the property that a nonlinear resistor is transformed into a nonlinear inductor, or a nonlinear capacitor, upon connecting this resistor across port two of an appropriate mutator. The reflector has the property that a given i-v, φ-i, or q-v curve can be reflected about an arbitrary straight line through the origin. The scalor is characterized by the property that any i-v, φ-i, or q-v curve can be compressed or expanded along a horizontal direction, or along a vertical direction. Using these new elements as building blocks, it is shown that any prescribed single-valued (which need not be monotonic) i-v, φ-i, or q-v curve can be synthesized. Active circuit realizations for each of these new elements are given. Laboratory models of mutators, reflectors, and scalors have been built using discrete components. Oscilloscope tracings of typical mutated, reflected, and scaled i-v, φ-i, and q-v curves are given. The experimental results are in good agreement with theory at relatively low operating frequencies. The practical problems that remain to be solved are the stability and frequency limitation of the present circuits.

Synthesis of new nonlinear network elements

In conjunction with the capacitor, the differential voltage controlled current source (DVCCS) is shown to yield in a relatively simple manner and in integrated form all components needed for linear circuit construction. The effects of dissipative losses in the DVCCS are tabulated for several configurations. Finally, a new building block having two outputs is introduced.

Generation of all finite linear circuits using the integrated DVCCS

Theorem: An arbitrary symmetric N × N matrix of real rational functions in the complex-frequency variable (a) can be realized as the immittance matrix of an N-port network containing only resistors, capacitors, and N negative-RC impedances, and (b) cannot, in general, be realized as the immittance matrix of an N-port network containing resistors, capacitors, inductors, ideal transformers, and M negative-RC impedances if M < N. The necessary and sufficient conditions for the immittance-matrix realization of transformerless networks of capacitors, self-inductors, resistors, and negative resistors follow as a special case of the theorem. In addition, an earlier result is extended by presenting a procedure for the realization of an arbitrary N × N short-circuit admittance matrix as an unbalanced transformerless active RC network requiring no more than N controlled sources. The passive RC structure has the interesting property that it can always be realized as a (3 N + 1)-terminal network of two-terminal impedances with common reference node and no internal nodes. The active subnetwork can always be realized with N negative-impedance converters.

Synthesis of transformerless active N-port networks

Necessary and sufficient conditions exist, which show that only a subset of all rational matrices can be realized as the transfer or admittance matrix of a common-ground network of resistors, inductors, capacitors, and grounded unity-gain voltage amplifiers (VGUGAs). This paper shows that by eliminating the ground constraint on the amplifiers (VUGAs) one can realize any rational transfer or admittance matrix with a common-ground RC :VUGA network. The synthesis is based upon a simple circuit whose transfer function is the ratio of two independent admittances. This is used with one capacitor and one resistor to form a differentiator. The differentiators are cascaded to form an admittance polynomial whose coefficients are determined by RC products. Transfer functions are realized by forming ratios of these polynomial admittances. The synthesis procedure results in selecting component values by inspection and permits tuning the circuits with the resistors.

Synthesis of Rational Transfer and Admittance Matrices with Active RC Common-Ground Networks Containing Unity-Gain Voltage Amplifiers

A synthesis procedure–easily implemented as a digital computer program– is presented whereby the network function T (s) can be realized as an active RC multiport network with grounded ports. Based on V (s) = T (s) U (s), where T (s) is a q × p matrix of real rational functions of the complex variable s, the realization requires a minimum number of grounded capacitors–n = degree { T (s)}–and no more than 2 (p+n) inverting, grounded voltage amplifiers or p+n differential output, grounded voltage amplifiers. Note: These properties of the realization are desirable if the network is to be fabricated as an integrated circuit.

Synthesis of active RC multiport networks with grounded ports

The following basic theorem concerning active RC networks is proved: Theorem: An arbitrary N × N matrix of real rational functions in the complex-frequency variable (a) can be realized as the short-circuit admittance matrix of a transformerless active RC N-port network containing N real-coefficient controlled sources, and (b) cannot, in general, be realized as the short-circuit admittance matrix of an active RC network containing less than N controlled sources.

I. W. Sandberg

Papers

Synthesis of N-port active RC networks