Source-Sink-Node (SSN)-Network Transformation

A new network transformation, called the source-sink-node (SSN)-network transformation, is presented which transforms an arbitrary network into an equivalent active network containing only source and sink nodes (a so-called SSN-type network). The only active elements in the transformed network are inverting operational amplifiers. Although the transformation is very general, it is primarily useful for the analysis and design of switched-capacitor (SC) networks. Thus, for example, it permits the synthesis of stray-insensitive SC networks which Manuscript received February 22, 1989. This letter was recommended by Associate Editor T. R. Viswanathan. A. Dabrowski is with the Institute of Electronics and Telecommunications Department of Electrical Engineering, Technical University of Poznan, PL-60 965 Poznan, Poland. G. S. Moschytz is with the Institute of Signal and Information Processing, Department of Electrical Engineering, Swiss Federal Institute of Technology, CH-8092 Ziirich, Switzerland. IEEE Log Number 8928235. 009%4094/90/0500-0663$01.00 01990 IEEE 664 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 5, MAY 1990 Fig. 1. Nodes in SSN-type network: a source node S, and a sink node sk. are SSN-transformed versions of stray-sensitive SC networks. The transformation procedure is illustrated by some examples. Among different an&aches to the analysis and design of switched-capacitor (SC) networks, signal-flow graph (SFG);echniques; [l]-[3] are of great interest, especially for small and medium-size SC networks. Althobgh, so far, numerous approaches for the derivation of SFG’s for SC networks have been proposed, e.g., [i]-[6], direct and by-inspection methods [l], [2] have so far been restricted to the so called source-sink-node (SSNhype SC networks, i.e., networks containing only three types of nodes (except for the grounded reference node): (i) source nodes, i.e., nodes driven by ideal voltage sources (e:g., op, amp output node); a source node can be represented by a grounded norator (Fig. 1); (ii) sink nodes, i.e., nodes at virtual ground; a sink node can be represented by a grounded nullator (Fig. 1); (iii) nodes connecting capacitors with switches; these nodes are either connected to source or to sink nodes or are disconnected from the network (when corresponding switches are open). Note that this definition can be applied to any active network if it contains only source and sink nodes. Although the class of SSN-type SC networks is slightly broader than that of stray-insensitive ones, it,is still quite restrictive, e.g., SC networks with unity-gain buffer amplifiers (instead of inverting op amps), which are conceivabie with ,advancing new technologies (such as GaAs [7]), do not belong to this class. A method has therefore been developed that permits the derivation of SFG’s for general SC networks [3]. It is composed of two steps: First, a given arbitrary SC network is transformed into the equivalent SSN-type network; second, known rules [l], [2] are applied to derive the SFG by inspection. In this letter, we present the source-sink node or SSN-network transformation. It is not restricted to SC networks and can be applied to any passive or active network: In addition to the elements contained in an SSN-type network (see above), a general network can contain the following: (i) general nodes, i.e., nodes which are neither source nor sink nodes; (ii) active elements other than inverting operational amplifiers (e.g., differential-input op amps, finite-gain amplifiers, unity-gain buffers, etc.). Consider the simple network shown in Fig. 2(a). It consists of an impedance 2, connected between a source node S, (with the node voltage Vk) and a general node G, (with the node voltage V[>. Our objective is to transform this network such that the general node G, is broken into two nodes: a source node S, with the same node voltage V, and a sink node sI supplying the same current Z/ to the transformed network (Fig. 2(e)). The transformed (SSN-type) network is equivalent to the original, even though the current leaving the source node S,, i.e., Zk = Z, in Fig. 2(a) and I; in Fig. 2(e) may be different for the two I = 5 i (sink;ode) Fig. 2. &N-network transformation. (a) Original circuit. (b) E&alent circuit with dummy unity-gain amplifier. (c) Complementary transformed circuii. (d) Nullator-norator model of the final t&formed circuit. (e) Final transformed circuit. networks (since Ik and I; are supplied from source nodes, they need not be the same). To transform the network of Fig. 2(a) into an SSN-type network we connect a series nullator-norator between the general node G, and ground (Fig. 2(b). This does not affect the original circuit because a nullator and norator in series corresponds to an open circuit [9]. Nevertheless, it can be interpreted as the insertion of a unity-gain amplifier (gain /? = l), as shown in Fig. 2(b). In order to obtain a source and sink node at G, we require a grounded norator and nullator, as pointed out earlier. We can achieve this by applying the so-called complementary transformation [8] to the network of Fig. 2(b), as shown in Fig. 2(c). However, according to the complementary transform, we now have the new gain p= p/(/3 1) =Q, instead of the unity gain we started out with. There is a second condition for the general node, however, that governs the current Z,, and that must also be satisfied, namely,