Transformer with active elements
FIELD OF THE INVENTION 
     This invention relates to an electric energy transformer which uses active elements. 
     BACKGROUND OF THE INVENTION 
     A conventional transformer comprises inductors consisting of electric wire wound onto a core forming a magnetic circuit. These heavy and bulky elements are particularly troublesome when it is desired to use them in miniaturized electronic circuits. Attempts have therefore been made to replace such elements by electronic circuits capable of performing the same functions. A voltage v 1  with a frequency F = (ω2π) applied to the primary of a transformer generates therein a current i 1  which causes a current i 2  with a voltage v 2  to appear in the secondary. The convention relationships between these currents and voltages are written as: ##EQU1## WHERE L 1 , L 2  are the values of the primary inductance and of the secondary inductance, respectively, and M is the mutual induction coefficient of the transformer. 
     Quadrupoles that satisfy expressions (1) and, therefore, provide a transformer function equivalent to that of a transformer have been developed in the past. 
     However, the prior art circuits are generally costly because they require an excessive number of operational amplifiers. 
     OBJECTS OF THE INVENTION 
     Accordingly, the object of the present invention is to provide an active quadrupole capable of performing the functions of a transformer, but requiring a reduced number of operational amplifiers. 
     SUMMARY OF THE INVENTION 
     These and other object, features and advantages of the invention are accomplished by the transformer with active elements disclosed herein. An active transformer is disclosed of the type which includes two primary terminals to which is applied a voltage v 1  with a current i 1 , and two secondary terminals supplying a voltage v 2  with a current i 2 . The transformer is characterized in that it includes a first circuit whose input, connected to the primary terminals, generates an input current of the form i 1  &#39; = jC 1  ωv 1   and whose output supplies a current of the form i 2  &#34; = -jNωv 1  at the secondary. The transformer also includes a second circuit whose input, connected to the secondary terminals, generates an input current of the form i 2  &#39; = jC 2  ωv 2 , and whose output supplies a current i 1  &#39; = jC 2  ωv 2 , and whose output supplies a current i 1  &#34; = -jNωv 2  at the primary. The terms C 1 , C 2  and N are respectively the dual equivalents of the terms L 1 , L 2  and M found in the relations associated with the conventional transformer. The currents flow in such a way that they satisfy the relations i 1  = i 1  &#39; - i 1  &#34; and i 2  = i 2  &#39; -  2  &#34;. 
    
    
     DESCRIPTION OF THE FIGURES 
     The foregoing and other objects, features and advantages of the invention will be apparent from the following more particular description of the preferred embodiment of the invention, as illustrated in the accompanying drawings. 
     FIG. 1 is a schematic representation of the transformer of the present invention. 
     FIG. 2 illustrates an embodiment of the circuits of FIG. 1. 
     FIG. 3 illustrates an embodiment of circuits 1b and 2b of FIG. 2. 
    
    
     DISCUSSION OF THE PREFERRED EMBODIMENT 
     The invention is based upon the use of the so-called dual principle which permits converting all current vs. voltage relations into voltage vs. current relations. 
     If we put ##EQU2## then expressions (1) can be written as ##EQU3## 
     In accordance with the duality principle, a capacitor is associated with each inductor such that ##EQU4## 
     Similarly, term N = (M/r 1  r 2 ) can be associated with M. 
     Expressions (2) become ##EQU5## 
     If v 1  &#39; = v 1  and v 2  &#39; = v 2 , then i 1  &#39; = i 1  and i 2  &#39; = i 2 . Hence: ##EQU6## 
     A quadrupole which satisfies expressions (4) is schematically illustrated in FIG. 1. A voltage v 1  with a current i 1  is applied to the input terminals A 1 , B 1 . Terminal B 1  is connected to one of the inputs of a circuit 1. Terminal A 1  is connected both to the second input of circuit 1 and to the output of a circuit 2. The output of circuit 1 is connected to output terminal B 2  and to an input terminal of circuit 2 the second input terminal of which is connected to output terminal A 2 . 
     If i 1  &#39; is the current applied to the input of circuit 1 connected to terminal A 1 , and if i 1  &#34; is the current from circuit 2, then 
     
         i.sub.1 = i.sub.1 &#39; - i.sub.1 &#34; 
    
     Similarly, as regards terminal B 2 , we can write 
     
         i.sub.2 = i.sub.2 &#39; - i.sub.2 &#34; 
    
     FIG. 2 illustrates an embodiment of the circuits of FIG. 1 when circuits 1 and 2 have been divided into two parts. Considering the reversibility properties of the transformer, identical circuits are used at 1a, 2a and at 1b, 2b. Circuit 1a includes an operational amplifier Aopl, a capacitor C 1  and a resistor R 1 , and converts voltage v 1  to ##EQU7## Since ##EQU8## 
     The admittance of circuit 1b being equal to (N/C 1  R 1 ), the current ##EQU9## 
     Because of the symmetries of the circuits, one may write 
     
         i.sub.1 &#34; = -jNωv.sub.2. 
    
     If it is assumed that operational amplifier Aopl is perfect, its input impedance being infinite, then all of the current i 1  &#39; flows through C 1  and R 1 , so that ##EQU10##  and, similarly, i 2  &#39; = jC 2  ωv 2 . 
     Thus, one finds again ##EQU11## 
     A voltage to current converter capable of generating i 1  &#39; and i 2  &#39; from voltages f(v 1 ) and f(v 2 ) can be implemented in different manners and, in particular, as shown in FIG. 3. The circuit input is at the point labeled IN. The circuit comprises four resistors R connected in series between the V +  and V -   poles of a pair of supplies which are symmetric relative to ground. Point P&#39; is connected to the base region of the PNP transistor T 1  the emitter region of which is connected to V 30   terminal through a resistor R 3 . The collector region T 1  is connected to the collector region of a NPN transistor T 2  the base region of which is connected to the point Q&#39; and the emitter region of which is connected to terminal V -   through a resistor R 3  &#39;. The circuit output is obtained at the point common to T 1  and T 2 . This point is also connected to the point common to resistors R 4  and R 4  &#39; which are connected between V +   and V - . R 4  and R 4  &#39;  are high resistance elements intended to balance the R CE  of T and T&#39; so as to maintain the reference level of the OUT terminal to ground. If one selects 
     
         R.sub.3 = R.sub.3 &#39; = (R.sub.1 /m) and R &lt;&lt; β R.sub.1, 
    
     where β is the gain of transistors T 1  and T 2 , one obtains an output current ##EQU12## 
     In order for the preceding expressions to be verified, one selects ##EQU13## Considering that, usually, ##EQU14## 
     Accordingly, we find again ##EQU15## 
     A similar circuit makes it possible to obtain ##EQU16## 
     The circuit is extremely simple and performs the transfer function of a transformer, but provides no galvanic insulation. The latter characteristic can however be provided in the circuit of the present invention, particularly by placing optoelectronic couplers 10 and 12 operating in linear mode at P and Q, respectively. 
     While the invention has been particularly shown and described with reference to a preferred embodiment thereof, it will be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope of the invention.