Harmonics phase shifter for a three-phase system with voltage control by integral-cycle triggering mode of thyristors

American Journal of Applied Sciences, Nov, 2008 by I Badran, A.L. Mahmood, M.T. Lazim

INTRODUCTION

Load voltage control by means of switching a pair of inverse parallel connected thyristors or triac is well established. It is customary to use modes of thyristor triggering known as integral-cycle triggering whereby burst of complete cycles of current are followed by complete cycles of extinction (1-4). Integral-cycle triggering results in conduction patterns that contain subharmonics of the supply frequency and so constitute a form of step-down frequency changing that can be considered as a form of frequency changer. Also integral-cycle triggering results in a considerable reduction in the amplitudes of the higher order harmonics as compared with other triggering techniques and it is possible that Radio Frequency Interference (RFI) is negligible (2). The phase-control switching can produce higher order harmonics and heavy inrush current while switching on in a cold start (5), while integral-cycle control circuits have the advantage of low inrush current due to zero voltage switching ease in construction and low hardware cost. Therefore, integral-cycle control loads have been widely used in resistive loads, such as heaters, oven, furnaces and spot welders (6-8). Also it is used in speed control of single-phase induction motor (9) and dc series motor (10).

As a frequency changing scheme, integral-cycle triggering was found not feasible for applications in the three-phase systems exploiting this technique for ac motor speed control (11). This is because the amplitudes and phase displacement angles of the higher order harmonic and subharmonic components of the integral-cycle controlled waveform are determined by the conduction period N and the control period T and the order of the individual harmonic. The three-phase analysis of voltage and current waveforms and phase relationships of the generated harmonic and subharmonic components for different circuit configurations are described in (12), (13).

Consider a three-phase resistive load with line voltage control as shown in Fig. 1. The resulting three-load voltage waveforms are identical and so are the three-load current waveforms. The supply frequency components are found to be balanced, since they are 120[degree] apart in time-phase while the phase displacement angles of a particular subharmonic or higher order harmonic are unbalanced (11), (12). Due to the unbalanced phase relationships of the subharmonic components, these components represent a source of trouble when these voltage waveforms are used to feed ac machines for speed control purposes (11). In this research an attempt is made to study the phase unbalanced characteristics of the subharmonics as well as the higher order harmonics generated due to integral-cycle triggering and to find a new phase shifting technique that is capable of correcting the unbalanced phases based on microprocessor implementation (14)

[FIGURE 1 OMITTED]

THE PROPOSED PHASE SHIFTING TECHNIQUE

Figure 2 shows the waveforms of the load voltages for the case when using integral-cycle control with control period T = 2 and conduction period N = 1, for the circuit shown in Fig. 1 with R-L load. Let the notation 1, 2, 3 denote the three phases A, B and C respectively. Thus the load voltage ([v.sub.LJ]) for any [j.sup.th]) phase will have the general form (12).

[FIGURE 2 OMITTED]

[v.sub.Lj] = [square root of (2)] V sin ([omega]Tt - [[gamma].sub.i]) = [[gamma].sub.j]/T][less than or equal to][omega]t[less than or equal to][[[2[pi]N [phi] [[gamma].sub.j]]/T (1)

where

j = 1,2,3

[[gamma].sub.1] = 0, [[gamma].sub.2] = 2[pi]/3, = [[GAMMA].SUB.3]4[pi]/3

[phi] = [tan.sup.-1][omega]L]/R]

Fourier analysis of Eq. 1 results in the following mathematical expressions: for n [not equal to] T, the dc component is:

[a.sub.0] = [[square root of (2) V]/[[pi]T]][1 - cos[phi] (2)

The amplitude [c.sub.n] of nth order subharmonic or higher order harmonic as well as its phase displacement [[psi].sub.n] are found as follows:

[a.sub.nj] = [square root of (2)]V/[pi]([T.sup.2] - [n.sup.2])[T{cos[eta][[gamma].sub.j]/T - Cos[phi]Cosn/T(2[pi]N [phi] [gamma].sub.j)} - n sin[phi]sin n/T(2[pi] [phi] [[gamma].sub.j])] (3)

[b.sub.nj] = [square root of (2)V/[pi]([T.sup.2] - [n.sup.2])[T{sin[eta][[gamma].sub.j]/T - Cos[phi]sin n/T(2[pi]N [phi] [gamma].sub.j)} n sin[phi]cos n/T(2[pi] [phi] [[gamma].sub.j])] (4)

[c.sub.nj] = [square root of][[a.sub.nj.sup.2] [b.sub.nj.sup.2]]

[c.sub.nj] = [square root of 2] V/[pi]([T.sup.2] - [n.sup.2])[[T.sup.2]{cos n/T (2[pi]N [pi]) - cos[phi]}.sup.2]} [{T sin n/T(2[pi]N [phi]) - n sin[phi]}.sup.2] (5)

[chi].sub.nj] = [tan.sup.-1] [a.sub.ni]/[b.sub.nj]

>[MATHAMATICAL EXPRESSION NOT REPRODUABLE IN ASCII] (6)

For n = T, the supply frequency component, the Fourier coefficients are:

[a.sub.Tj] = [[square root of (2)]v/4[pi]T]][cos [[gamma].sub.j] - 2(2[pi]N [phi]) [[gamma].sub.j] - cos(2[phi] [[gamma].sub.j])] (7)

[b.sub.Tj] = [[square root of (2)]v/4[pi]T]][sin [[gamma].sub.j] 2(2[pi]N [phi])cos[[gamma].sub.j] - sin(2[phi] [[gamma].sub.j])] (8)