HYPERCOMPLEX INSTANTANEOUS POWERS IN LINEAR AND PASSIVE ELECTRICAL NETWORKS IN PERIODICAL NON-HARMONIC STEADY-STATE

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1 BULEINUL INSIUULUI POLIEHNIC DIN IAŞI Pulicat de Universitatea ehnică Gheorghe Asachi din Iaşi omul LVII (LXI), Fasc., Secţia ELECROEHNICĂ. ENERGEICĂ. ELECRONICĂ HYPERCOMPLEX INSANANEOUS POWERS IN LINEAR AND PASSIVE ELECRICAL NEWORKS IN PERIODICAL NON-HARMONIC SEADY-SAE BY HUGO ROSMAN * Gheorghe Asachi echnical University of Iaşi Faculty of Electrical Engineering, Energetics and Applied Informatics Received: Septemer, Accepted for pulication: Decemer, Astract. he method proposed y V.N. Nedelcu to characterize the energy regime of a linear and passive electrical networ, excited y a periodical harmonic signal, utilizing the symolic method, ased on representation of harmonic signals of same frequency through complex images, is extended to the case when the networ in excited y a periodical, non-harmonic signal utilizing the symolic method ased on representation of such signals ( originals ), through hypercomplex images. Key words: instantaneous hypercomplex powers; periodical non-harmonic steady-state; symolic hypercomplex method.. Introduction Let e a linear and passive one-port (LPOP), woring in harmonic steady-state, excited y the voltage * adi_rotaru5@yahoo.com ( ω γ ) u = U cos t + ; () u

2 46 Hugo Rosman the current which flows through the LPOP eing ( ω γ ) i= Icos t +. () he instantaneous power exchanged y the LPOP with the outside is where i p= ui= UI cosϕ + UI cos ωt + γ + γ, (3) u i ϕ = γ γ. (4) It is well nown that in this case, the LPOP s energy regime is characterized, in main, y the active and reactive powers. Having in view that relation (3) may e written u i p= UIcosϕ + cos ωt + γi + UI sinϕsin ωt + γi, (5) the active power, P= UI cosϕ, represents the average value of the instantaneous power while the reactive power, Q= UI sin ϕ, may e defined as the amplitude of his oscillating component. With the view to eliminate this existent disparity in the definition manner of active and reactive power, V.N. Nedelcu (for instance, 963) has adopted an artifice which may e rendered evident if the symolic complex method to represent the harmonic signals is utilized. Namely to signals () and () are attached theirs complex instantaneous images or theirs complex effective images j( ωt+ γu ) j e ωt+ γ u = U, i= e I i, (6) jγu U = U e, I = I e ; (7) jγi to the instantaneous power exchanged y the LPOP with the outside it is possile to attach the complex instantaneous apparent power * * j ωt e ()e j σ () t s= u i+ i = UI + U I = st, (8) which is different from the expression of complex apparent power, S = u i = U I = P+ jq. (9) * *

3 Bul. Inst. Polit. Iaşi, t. LVII (LVXI), f., 47 In these conditions the instantaneous power (3) represents the real part of the complex instantaneous apparent power p=r es (). () V.N. Nedelcu (op. cit.) has proposed for the instantaneous power, as real part of the complex instantaneous apparent power, the denomination of instantaneous active power. Consequently the imaginary part of the complex instantaneous apparent power * ϕ ( ω γu γi) q=i ms () = s s = UIsin + U Isin t + + () represents the instantaneous reactive power. It results that the complex apparent power is the average value of the complex instantaneous power * jϕ d e j, () S = s = s t = u i = UI = P+ Q where the active power, P, and the reactive power, Q, are defined as average values of the instantaneous active power (3), P= p = p dt = UI cosϕ, (3) respectively the average value of the instantaneous reactive power Q= q = q dt = UI sinϕ, (4) = πωeing the period of the harmonic signals u (t), i(t). Relations (8) and () lead to expression where s= S + s f, (5) jωt jωt s f = ui= UI e = S fe = pf + jqf. (6)

4 48 Hugo Rosman with ( ω γ γ ) ( ω γ γ ) S = U I, p = U Icos t + +, q = UIsin t + +. (7) f f u i f u i Here S f represents the complex apparent fluctuating power, p f the active instantaneous fluctuating power, while q f the reactive instantaneous fluctuating power. he aim of this paper is to extend the powers study proceeding proposed y V.N. Nedelcu, to the more generally case of the periodical, nonharmonic steady-state. In this case is useful to utilize the hypercomplex symolic method of representation of periodic, non-harmonic signals through hypercomplex images, elaorated y B.A. Rozenfeld (949). In a previous paper (Rosman, ) an LPOP in periodical non-harmonicsteady-state was considered, supplied y the voltage ( ω γ ), (8) u () t = U cos t + u the current which flows through the LPOP eing with ( ω γi ), (9) it () = I cos t + γ γ = ϕ. () u i Utilizing the polar hypercomplex symolic representation of periodic non-harmonic signals through hypercomplex images, it is useful to attach to the signals (8) and (9) such images (Rosman, op. cit) namely respectively ( ω γ ) ( ω γ ), () uˆ = U cos t + + j U sin t + u u ( ω γi ) ( ω γi ). () iˆ= I cos t + + j I sin t + Here functions, j are orthonormalized. If in the vector space of images û, respectively î, the vectorial product is introduced, associative and

5 Bul. Inst. Polit. Iaşi, t. LVII (LVXI), f., 49 distriutive with respect to addition, the so otained vector space is a Hilert one. he so defined algera is commutative representing a real sum, of real numers field (generated y ) and the numerale set of complex numer fields (generated y the pair of elements, j ). he unity element of this algera is Also =. (3) = =, j =, j = j = j, p q = pjq = q p = j q p =, ( p q). (4) he symolic relation d m dt m m ( j mω ), m, (5) = is valid too, rendering evident the advantage of this symolic method to algerize the differential operations with respect the time.. he Hypercomplex Instantaneous Apparent Power Utilizing relation (8) as model which is valid in periodical, harmonic steady-state, it is possile to define a hypercomplex instantaneous apparent power, in periodical, non-harmonic steady-state namely Having in view that * * sˆ= uˆ î + î. (6) ( ω γi ) ( ω γi ) (7) î = I t + j I sin t + and sustituting in (6) expressions (), () and (7) it results ( ω γ ) ( ω γ ) sˆ= U cos t + j sin u + U t + u ( ω γ ) I cos t +. i (8)

6 5 Hugo Rosman aing into account relations (4) expression (8) ecomes finally sˆ = U I cos cos ϕ ωt γu γ i + + ju I sin sin. ϕ ωt γu γ i (9) It is reasonale to consider that the average value ˆ S = sˆ = sˆ dt = U I cosϕ + j U I sinϕ (3) represents the hypercomplex apparent power, which may e written as with, (3) Sˆ = P + j Q P = U I cos ϕ, Q = U I sinϕ, (3) representing the active, respectively the reactive power corresponding to the harmonic of order. It is also reasonale to consider that a) U I cos cos( ϕ + ωt + γu + γ i ) represents the hyper- complex active instantaneous power and ) ju I sin sin( ϕ + ωt + γu + γ i ) represents the hyper- complex reactive instantaneous power. he average values of these powers are with Pˆ = U I cos ϕ, Qˆ = j U I sinϕ, (33), (34) Pˆ = P, Qˆ = j Q

7 Bul. Inst. Polit. Iaşi, t. LVII (LVXI), f., 5 so that relation (3) ecomes Sˆ = Pˆ + Qˆ. (35) heirs moduli,, (36) P= U I cos ϕ, Q= U I sinϕ represent the well-nown expressions of active, respectively reactive power in periodical, non-harmonic steady-state. where 3. he Hypercomplex Instantaneous Apparent Fluctuating Power Expressions (9) and (3) lead to relation sˆ= Sˆ + s, (37) ˆf ( ω γ γ ) ( ω γ γ ) (38) sˆ = U I cos t j U I sin t+ + f u i u i represents the hypercomplex instantaneous apparent fluctuating power. It is easy to oserve that sˆ f = uî ˆ, (39) where relations () and () were taen into account. In relation (33) U I cos ωt + γu + γi represents the hypercomplex a) ( ) instantaneous fluctuating active power and ju I sin ωt + γu + γi represents the hypercomplex ) ( ) instantaneous fluctuating reactive power. he moduli of these powers may e considered as representing ( ω γ γ ) p = U I cos t+ + (4) f u i

8 5 Hugo Rosman the instantaneous fluctuating active power and ( ω γ γ ) q = U I sin t+ + (4) f u i the instantaneous fluctuating reactive power. 4. Conclusions Using a variant of the hypercomplex symolic representation of periodical, non-harmonic signals, proposed y B.A. Rozenfeld, the concepts of: hypercomplex instantaneous apparent power, hypercomplex apparent power, hypercomplex instantaneous active power, hypercomplex instantaneous reactive power, hypercomplex instantaneous apparent fluctuating power, hypercomplex instantaneous active fluctuating power, hypercomplex instantaneous reactive fluctuating power, which characterize the energy regime of a linear and passive one-port woring in periodical, non-harmonic steady-state, are introduced. hese powers represent a generalization of those proposed y V.N. Nedelcu, in harmonic steady-state, utilizing, with this view, the hypercomplex symolic method of representation of periodical, non-harmonic signals. REFERENCES Nedelcu V.N., Die einheitliche Leistungstheorie der unsymmetrischen und mehriwelligen Mehrphasensystem. EZ-A, 84, 5, (963). Rozenfeld B.A., Symolic Method and Vectorial Diagrams for Non-Sinusoidal Currents (in Russian). r. sem. vet. i tenz. anal., 7, (949). Rosman H., Aout a Symolic Representation Method of Periodical Non-Harmonic Signals. Proc. of the 6th Internat. Conf. On Electr. a. Power Engng. EPE, Oct. 8-3, Gh. Asachi echn. Univ., Jassy. Vol. I,, 7-9. PUERILE INSANANEE HIPERCOMPLEXE ÎN CIRCUIE ELECRICE LINIARE ŞI PASIVE, ÎN REGIM PERMANEN PERIODIC NEARMONIC (Rezumat) Se extinde propunerea lui V.N. Nedelcu de a caracteriza regimul energetic al unui uniport liniar şi pasiv, excitat de un semnal armonic, la cazul în care uniportul este excitat de un semnal periodic nearmonic. Se utilizează, în acest scop, metoda simolică de reprezentare a semnalelor periodice nearmonice prin imagini hipercomplexe.