Abstract-This paper proposes a new framework for linear active circuits that can encompass both circuit analysis and synthesis. The framework is based on a definition of port equivalence for admittance matrices. This is extended to cover circuits with ideal active elements through the introduction of a special type of limit-variable called the infinity-variable ( -variable). A theorem is developed for matrices containing -variables that may be utilized in both circuit analysis and synthesis. The notation developed in this framework can describe nonideal elements as well as ideal elements and therefore the framework encompasses systematic circuit modeling.

In this paper, we show how some basic building blocks for active-RC circuit design, such as amplifiers, impedance converters and simulated inductance circuits, may be synthesised in a systematic way by expansion of their port admittance matrices. The circuit topology emerges from the synthesis procedure, allowing all possible implementations to be identified and explored. Nullors representing ideal op-amps and transistors are represented within the nodal admittance matrix of a synthesised circuit by linked infinity parameters. In nodal admittance matrices describing ideal circuits synthesised, the replacement of linked infinity parameters by finite parameters provides a seamless transition to non-ideal analysis and practical circuit design.

A different approach to the synthesis of dipeptides is described based on the formation of the NHCHR 1 CONH À CHR 2 CO bond by carbenoid N À H insertion, rather than the formation of the peptide bond itself. Thus decomposition of triethyl diazophosphonoacetate catalysed by rhodium(ii) acetate in the presence of N-protected amino acid amides 8 gives the phosphonates 9. Subsequent Wadsworth ± Emmons reaction of 9 with aldehydes in the presence of DBU gives dehydro dipeptides 10. The reaction has been extended to a simple two-step procedure, without the isolation of the intermediate phosphonate, for conversion of a range of amino acid amides 11 into dehydro dipeptides 12 and to an Nmethylamide 11 h, and for conversion of a dipeptide to tripeptide (13 3 14). Direct conversion, by using methyl diazophenylacetate, of amino acid amides to phenylglycine-containing dipeptides 19 proceeds in good chemical yield, but with poor diastereoselectivity.

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A framework for symbolic analysis and synthesis of linear active circuits has previously been proposed which is based on the use of admittance matrices and infinity-variables. The notation has the important advantage that it can describe both ideal circuit elements, for which an infinite limit is implied, and nonideal circuit elements for which matrix elements are considered finite. The nullor is a very important circuit element because it can represent the ideal operational amplifier and the ideal transistor. For the nonideal case, the use of finite matrix elements implies that the operational amplifier and transistor are both modelled as a voltagecontrolled current source, which is fine if the transistor is a field effect transistor or if the operational amplifier is of the transconductance type, but not otherwise. The purpose of this paper is to apply the -variable framework in order to derive alternative models for the nullor that can be used to model voltage, current and transresistance operational amplifiers and bipolar junction transistors. We also show that the -variable description of an ideal transistor can include a factor to represent transistor geometry.