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Revisitation of Reciprocity of Linear Resistive K-Ports

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Revisitation of Reciprocity of Linear Resistive K-Ports ECCTD’01 - European Conference on Circuit Theory and Design, August 28-31, 2001, Espoo, Finland RevisitationofReciprocityofLinearResistivek-Ports Giancarlo Storti-Gajani and Amedeo Premoli Dipartimento di Elettronica e Informazione, Campus Leonardo, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy. Abstract — This paper revisits reciprocity of lin- Now consider two different operating situations ear and resistive k-ports from an educational but v 0, i 0 and v 00, i 00 , each satisfying the branch re- rigorous point of view. We introduce the compan- {lations} of a {-port element.} ion concepts of antireciprocity and null-reciprocity, k extending the related theorems. In addition, rela- Definition 2 A cross pair of virtual powers is de- tionships between reciprocity and other properties of two-ports, such as symmetry, directionality and fined by interchanging the voltages and currents of power behavior, are discussed, by evidencing that different operating situations v 0, i 0 and v 00, i 00 : there is a symmetrical but nonreciprocal two-port. { } { } T T p0 = [v 00] i 0 and p00 = [v 0] i 00. 1 Introduction and preliminaries When some relationships between p0 and p00 exist, Many textbooks on basic circuit theory give a defi- we can introduce : nition of reciprocity based on the properties of ma- trices R and/or G of k-ports. The aim of this paper Definition 3 A resistive and linear k-port element is to revisit and extend reciprocity from an educa- is said to be : tional, but rigorous, point of view, by using the 1. reciprocal if p0 p00 = 0. − more general definition based on “virtual powers” 2. antireciprocal if p0 + p00 = 0 [3]. [1], [2], [3]. We consider only linear and resistive k- 3. null-reciprocal if p = 0 and p = 0. ports, disregarding the nonlinear case treated in [4]. 0 00 More exactly, we introduce the companion concepts v 0, i 0 and v 00, i 00 . of antireciprocity and null-reciprocity extending the ∀ { } { } known theorems on reciprocity. The implications of these properties on the representation matrices Def. (3) implies that any null-reciprocal k-port ele- ment is both reciprocal and antireciprocal. Note of k-ports are examined in detail. Moreover, for two-ports, relationships intercurring between reci- that antireciprocity, when null-reciprocity is ex- procity, symmetry, directionality and power behav- cluded, is a particular case of non-reciprocity. ior, are discussed by evidencing that, in a particular Example 1 Resistors are reciprocal, because but interesting case, a symmetrical two-port can be p0 = p00 = r i0 i00, and short and open circuits are nonreciprocal. null-reciprocal, because p0 = p00 = 0. By denoting with v and i the port voltages and Note that only homogeneous k-ports (i.e., de- currents of a k-port, linked by branch relations, the scribed by homogeneous linear representations) are scalar product v T i defines the effective power ab- here considered : non-homogeneous elements (i.e., sorbed by the k-port itself. described by affine representations) are always non- Definition 1 If branch relations between v and i T reciprocal when Def. (3) is adopted. However, it are disregarded, the scalar product p = v i defines is convenient to define reciprocity, antireciprocity the virtual power absorbed by the k-port. and null-reciprocity of non-homogeneous elements The main interest in Def. (1) is due to : by considering the associate homogeneous elements, Theorem 1 (Tellegen) The sum of the virtual obtained by cancelling all constant terms appearing powers absorbed by all elements of a circuit is null. in the corresponding affine representations. Proof The port voltages and currents of all ele- Example 2 Any voltage or current source is null- ments satisfy, respectively, KVL and KCL. reciprocal, since its associate homogeneous element We consider a “composite element”, i.e. an element is, respectively, a short or open circuit. embedding an arbitrary number of other (simple or, in turn, composite) elements. By applying Theo- Antireciprocity is related to effective power of rem (1) to Def. (1), we obtain : k-ports : Corollary 1 Any virtual power absorbed by a com- Property 1 Any k-port element is non-energic posite element is equal to the sum of the correspond- (i.e., the effective power is zero in any operating ing virtual powers of all embedded elements. situation) if and only if it is antireciprocal. III-129 aa Proof (“if”) Rewriting Def. (3) p0 + p00 = 0 for 3. null-reciprocal iff H = 0, a generic pair of coincident operating situations [H ab]T = H ba and H bb = 0. v 0, i 0 v 00, i 00 , we obtain that p0 = p00 = 0 is − Proof Set to zero both difference and sum of p0 just{ the} ≡ effective { } power. and p00 : (“only if”) is omitted for the sake of brevity. a T aa T aa a p0 p00 = [(i )00] [[H ] H ](i )0 + We extend the well-known reciprocity theorem ∓ ∓ z }| { to antireciprocity and null-reciprocity : b T ab T ba a + [(v )00] [[H ] + H ](i )0 ∓ Theorem 2 (Reciprocity) A composite k-port z }| { b T ab T ba a [(v )0] [[H ] + H ](i )00 + element embedding exclusively reciprocal, antirecip- ∓ rocal, or null-reciprocal elements is reciprocal, an- z }| { b T bb bb T b + [(v )00] [H [H ] ](v )0 = 0 (3) tireciprocal, or null-reciprocal, respectively. ∓ where the upper sign,z the lo}|wer sign,{ and both signs Proof Virtual powers pc0 and pc00, absorbed by a composite element, are equal to the sum of the re- hold, respectively, for reciprocity, antireciprocity, spective virtual powers of the embedded elements and null-reciprocity. Conditions (3) are verified for a b a b any value of (i )0,(v )0,(i )00 and (v )00, iff all the p10 , p20 , p30 , ... and p100 , p200 , p300, ... (Corol. 1); moreover, the two virtual powers of any cross pair overbraced matrices are zero. are related by p10 p100 = 0, p20 p200 = 0, p30 p300 = 0, Table 1 reports property (2) in terms of the entries ... according to∓ Def. 3. ∓ ∓ of generic hybrid matrices : 2 Representation matrices entries diagonal off-diagonal (pair) −→ Hybrid matrices We investigate the con- physical resistan. transresistan. voltage ratios ditions on the entries of the hybrid ma- meaning −→ conduct. transconduct. current ratios trices representing reciprocal, antireciprocal reciprocal: any h , h = h , h = h or null-reciprocal linear resistive k-ports [5]. µµ µν νµ µν − νµ antireciprocal: h = 0, h = h , h = h By ordering appropriately the k ports of an µµ µν − νµ µν − νµ element, any hybrid representation assumes null-reciprocal: h = 0, h = h = 0, h = h µµ µν νµ µν − νµ the partitioned form : Table 1. Constraints between the entries v a H aa H ab i a of hybrid matrices of k-ports. = (1) Table 1 comprehends, as particular cases, the sym- i b H ba H bb v b metry, skewsymmetry and the nullity of the resis- H tance and conductance matrices R and G. Non-hybrid matrices The application of Def. (3) Virtual powers p and| p assume{z the} expressions : 0 00 to k-ports, with k > 2, leads to the formulation of complex, and mostly useless, conditions on the T a T aa T a p0 = [i 0] v 00 = [(i )00] [H ] (i )0 + entries of these matrices. However, the following b T ab T ba a + [(v )00] [[H ] + H ](i )0 + property is reported by omitting the proof : b T bb b + [(v )00] H (v )0 Property 3 All k-ports, for any k, admitting only non-hybrid representations are neither reciprocal T a T aa T a nor antireciprocal. p00 = [i 00] v 0 = [(i )0] [H ] (i )00 + For two-ports, now we consider the non-hybrid rep- + [(v b) ]T [H ab]T + H ba (i a) + 0 00 resentations of two-ports : transmission matrices T b T bb b £ + [(v )0] ¤H (v )00 (2) and T 0. We express p0 and p00, by means of matrix T , versus i2 and v2 : Property 2 A k-port is, respectively, reciprocal, p0 = v100i10 v200i20 = t11 t21 v20 v200 + t12 t21 v20 i200 + antireciprocal or null-reciprocal iff the submatrices − + [t t 1] i0 v00 + t t i0 i00 of H in Eq. (1) satisfy the conditions : 11 22 − 2 2 12 22 2 2 aa aa T 1. reciprocal iff H = [H ] , p00 = v0 i00 v0 i00 = t t v0 v00 +t t v00 i0 + 1 1 − 2 2 11 21 2 2 12 21 2 2 ab T ba bb bb T [H ] = H and H = [H ] . + [t11 t22 1] i00 v0 + t12 t22 i0 i00 (4) − − 2 2 2 2 2. antireciprocal iff H aa = [H aa]T, − By investigating conditions p0 p00 = 0 like in prop- [H ab]T = H ba and H bb = [H bb]T. erty (2), we obtain that : ∓ − − III-130 Property 4 The two-port is : Case 2 : when h110 = 0 and h220 = 0, Eq. (5) are 1. reciprocal iff T = t11 t22 t12 t21 = 1. verified iff : h120 = h210 = 1. These conditions cor- | | − respond to a symmetrical± but non-reciprocal two- 2. antireciprocal iff any one of the two sets : port, since h120 = h210 = 0. first set : t11 = 0 , t22 = 0 , T = 1. 6 From h0 = h0 = 0 and h0 = h0 = 1 (see | | − 11 22 12 21 ± second set : t12 = 0 , t21 = 0 , T = 1. Case 2) we derive the branch relations | | 3. null-reciprocal iff t = 0 , t = 0 , T = 1. 12 21 | | v1 = v2 , i1 = i2 (6) For the class of two-ports admitting matrix T, Property (4) implies that : having adopted the associated reference directions Only the ideal transformer is null-reciprocal.
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