【正文】 m(t) = V Sin ωt (2) Iim(t) = I Sin ( ωt – Ф) Comparing Eqs (1) and (2), it is obvious that the imaginary or fictitious phase of the voltage or current in a single phase power supply can be created in the time domain by shifting the real ponent on the time axis to the right by an equivalent phase shift of л/2. According to Eqs(1) and (2), the αβ orthogonal coordinate systems for both of the voltage and current are defined as follows: vα = vRe(t) and vβ = vim(t) (3) iα = iRe(t) and iβ = iim(t) According to reference [1], each of the α and β ponents of the voltage and current are bined to form a vector, x(t). This vector can be represented by the following equation: This vector is represented in the Gaussian plex domain as a four sided symmetrical trajectory, . Because of the symmetry of the x(t) trajectory shown in , it is evident that the voltage and current investigation for the plex power system (including both of real and imaginary voltage and current ponents), could be carried out within quarter of the periodic time of the voltage and current waveforms (T/4). Thus, Fourier transforms applied for the harmonic analysis of nonsinusoidal waveforms could be carried out during this time interval only as it will be shown later. shows the arrangement of the real and fictitious/imaginary circuits of the plex single phase power system under investigation. As it is shown on this Figure, the real and fictitious circuits should be synchronised by the so called “SYNC” signal. This implies that the x(0) is a priori zero. 2 pqr INSTANTANEOUS REACTIVE POWER In this section the use of the pqr instantaneous reactive power method, described in references [2], [3] and [5], for pensating of the reactive power and harmonic filtering is explained. Consider a single phase power system with a cosinusoidal voltage supplying a solid state controlled rectifier, thus yielding a nonsinusoidal supply current waveform. The supply current is assumed to have a square waveform. Thus, the supply voltage and the fundamental ponent of the supply current could be written as: vRe(t) = V Cos ωt (5) i1Re(t) = I1 Cos (ωt –л/3) The supply voltage and current waveforms as well as the fundamental current waveform are depicted in . The voltage and fundamental current ponents are given as: vim(t) = V Sin ωt (6) iim(t) = I1 Sin (ωt – л/3) The instantaneous active and reactive power equations for the plex power system under consideration are given in the αβ domain, as described in references [1], [3] and [5], as follows: depicts the time variation of p and q for the plex single phase power system under consideration. In this figure PAV and QAV respectively are the average values of the active and reactive power. The instantaneous power factor, Ф, is defined as: It is important to point out that the values of p, q and Ф in Eqs (7) and (8) are instantaneous values. The pqr theory is introduced in reference [3], where the current, voltage and power equations are projected in pqr rotating frame of reference. shows the voltage ponents in both of the fixed αβ and rotating pq frame of reference for a single phase power system. The raxis is considered to be identical to the zero axis, hence the voltage transformation equation from the fixed frame of reference αβ to the rotating frame of reference pqr, can be written as: The currents in the rotating frame of reference, ip, iq and ir are related to the currents in the stationary frame of reference, iα and iβ by similar equations as the voltage equations in . Moreover, the following relations can be derived in the pqr rotating frame of reference: 3 DERIVATION OF REFERENCE CURRENT EXPRESSIONS FOR THE ACTIVE FILTER In this section instantaneous expressions for the reference currents for an active power filter to pensate for the harmonic distortion or reactive power or both in the single phase power system under investigation are derived. Because of the symmetry of the plex voltage and current vectors trajectories, , the average value of the active and reactive powers for both of the real and imaginary/fictitious phases can be evaluated from as follows: According to reference [1], the instantaneous expressions for the active and reactive power in the real phase of the single phase power system under analysis are given as: The real phase average value, fundamental and ripple ponents of the active and reactive power are extracted from Eqs (12) and (13) and are depicted in Fig .7 and respectively. The real phase current, iα, can be derived from as follows: In , p~ and q~ respectively are the ripple active and reactive power ponents. Reference current for the active filter of the single phase system under consideration can assume different expressions depending on the special requirements of pensating for the reactive power or filtering the distortion harmonics. Three special cases are listed below: i) Reference current for distortion harmonic filtering and reactive power pensation ii) Reference current for average reactive power pensation iii) Reference current harmonic distortion pensation 4 EXPERIMENTAL RESULTS A test rig was set up to verify the theoretical derivations above. An active power filter is implemented with the curre