@inbook {IOPORT.00916409, author = {Takahashi, Tohru and Takahashi, Shin-ichi}, title = {RC active circuit synthesis via state variable method.}, year = {1990}, booktitle = {Linear circuits, systems, and signal processing: Advanced theory and applications}, isbn = {0-8247-8185-6}, pages = {29-57}, publisher = {New York, NY: Marcel Dekker}, abstract = {This is the paper number 2 of the volume Linear circuits, systems and signal processing'', edited by Nobuo Nagai, aiming to present significant contributions in this area in Japan. It is well known that any network function of an RC circuit can be synthesized by the use of passive elements (resistors, capacitors and multiwinding ideal transformers). The presence of ideal transformers makes these methods of little practicle interest. Therefore, many researchers tempted to obtain synthesis procedures which do not require transformers but could use active elements. The subject of this paper is how to synthesize RC active circuits by the use of state equations when some specifical network functions are prescribed. The starting point is the set of state space equations $$\text{(a)}\quad\dot x=A.x+b.u,\qquad\text{(b)}\quad y=c.x,$$ where $u$ is the input vector, $x$ the state vector and $y$ the output vector. The feature of the system is the fact that $u$ and $y$ have each only a nonzero element. The underlying idea of the presented procedure is that the equation (a) provides the characteristic polynomial $f(s)=s^n+a_1s^{n-1}+\dots+a_{n-1}s+a_n$ whereas (b) provides the numerator of the (prescribed) network function $T(s)$. It is known that for a given $f(s)$, one can write a particular state space equation (the so-called companion form) $x_c=A_c.z_c$ so that the entries of $A_c$ are simply related to the set of coefficients $\{a_k\}$. It is clear that the companion form gives no information about the topology of the network. The treatment chooses as typical networks the ladder ones. For such networks, the matrix A has a tridiagonal form (Schwarz matrix). First, it is shown that for an LC ladder network (which a single resistor at the input port) one can obtain the network elements $R,L,C$ in terms of the set $a_k$ via some adequate transforms including Routh parameters. This synthesis is not practical since one must sum up the states (inductor currents and capacitor voltages). Next, it is shown how to synthesize a $T_{\text{RC}}(s)$ transfer function using $R$ and $C$ element and in addition a summer (the active element). The last solution is restricted to the case when the characteristic polynomial has all roots with zero imaginary parts (the so-called aperiodic damping polynomials). To synthesize general transfer functions several procedures are presented. They use two summers (one for the states, the other for state feedback). The text is very clear and constitutes a reference material for RC active synthesis.}, reviewer = {D.Stanomir (Bucure\c{s}ti)}, identifier = {00916409}, }