Harmonic distortion problems include equipment overheating, motor failures, capacitor failure and inaccurate power metering. The topic of. POWER SYSTEM HARMONICS. Second Edition. Jos Arrillaga and Neville R. Watson. University of Canterbury, Christchurch, New Zealand. Definitions and Standards Power system harmonics are defined as sinusoidal voltage and currents at frequencies that are integer multiples of the main.
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ABSTRACT. This paper is intended to give an overview of power system harmonics and is aimed at those who have some electrical background but little or no. Power system harmonics are not a new phenomenon. In fact, a text published by Steinmetz in. devotes considerable attention to the study of harmonics in. This document has been created to give general awareness of power system harmonics, their causes, effects and methods to control them especially when.
Accuracy of the normal PI model decreases as the frequency increases and this can be observed at the wavelength frequency. Similarly 3. For steady-state analysis at one particular have been calculated for frequency. In expanded form. Standing wave effects show the different accuracies provided by the two models.
This figure shows the per unit positive sequence voltages of a km. Computer derivation of the correction factors for conversion from the nominal PI to the equivalent PI model. The equivalent PI model avoids the problems of determining the number of sections needed and round-off error that accumulates in this situation.
The LR2 algorithm of Wilkinson and Reinsch is used for accurate calculations in the derivation of the eigenvalues and eigenvectors. Its primitive network. If a tertiary winding is also present, the primitive network consists of nine instead of six coupled coils and its mathematical model will be a 9 x 9 admittance matrix. The interphase coupling can usually be ignored e. The connection matrix [C] between the primitive network and the actual transformer buses is derived from the transformer connection.
By way of example consider the Wye G-Delta connection of Figure 3. The following connection matrix applies:. In general, any two-winding three-phase transformer may be represented by two coupled compound coils as shown in Figure 3.
If the parameters of the three phases are assumed balanced, all the common threephase connections can be modelled by three basic submatrices.
The submatrices [Ypp],[Yps], are given in Table 3. For transformers with neutrals connected through an impedance, an extra coil is added to the primitive network for each unearthed neutral and the primitive admittance increases in dimension. However, by noting that the injected current in the neutral is zero, these extra terms can be eliminated from the connected network admittance matrix. Once the admittance matrix has been formed for a particular connection it represents a simple subsystem composed of the two busbars interconnected by the transformer.
The data for these elements are usually given in terms of their rated megavolt-amps and rated kilovolts.
The equivalent phase admittance in per unit is calculated from these data. The coupled admittances to ground at bus k are formed into a 3 x 3 admittance matrix as shown in Figure 3. The admittance matrix is incorporated directly into the system admittance matrix, contributing only to the self-admittance of the particular bus.
While provision for off-diagonal terms exists, the admittance matrix for shunt elements is usually diagonal, as there is normally no coupling between the components of each phase. Consider, as an example, the three-phase capacitor bank shown in Figure 3.
A 3 x 3 matrix representation similar to that for a line section is illustrated. The element admittance matrix will be diagonal and proportional to frequency. The presence of any series inductance in the capacitor banks is ignored. In terms of ABCD parameters described in section 3. Series elements are connected directly between two buses and for modelling purposes they constitute a subsystem in the network subdivision. A three-phase coupled series admittance between two busbars i and k is shown in Figure 3.
The series capacitor, used for transmission line reactance compensation, is an example of an uncoupled series element; in this case the admittance matrix is diagonal. For a lumped series element, the ABCD parameter matrix equation is: The cross section of a cable, although extremely complex can be simplified to that of Figure 3.
The impedances of the insulation are given by. If the insulation is missing, e.
The earth return impedance can be calculated approximately with equation 3. This approach, also used by Bianchi and Luoni [IS] to find the sea return impedance is quite acceptable considering the fact that sea resistivity and other input parameters are not known accurately.
Because a good approximation for many cables having bonding between the sheath and the armour and the armour earthed to the sea is Vslrc. Similarly, for each cable the per unit length harmonic admittance is calculated, i. Therefore, when converted to core, sheath and armour quantities, 3. Therefore, for frequencies of interest, the cable per unit length harmonic impedance, Z', and admittance, I", are calculated with both the zero and positive.
Voltage kV , Based on 2. Resistance 0. In the absence of rigorous computer models, such as described above, power companies often use approximations to the skin effect by means of correction factors.
A three-dimensional graphic representation is used to provide simultaneous information of the harmonic levels along the line. At each harmonic up to the 25th harmonic , one per unit positive sequence current is injected at the Islington end of the line. The voltages caused by this current injection are, therefore, the same as the calculated impedance, i.
Figures 3. The difference in harmonic magnitudes along the line are due to standing wave effects and shifting of the resonant frequencies caused by line terminations.
The 25th harmonic clearly illustrates the standing wave effect, with voltage maxima and minima alternating at quarter of the wavelength intervals.
At any particular frequency, a peak voltage at a point in the line will indicate the presence of a peak current of the same frequency at a point about a quarter wavelength away. This is clearly seen in Figure 3. When the line is short-circuited at the extreme end, the harmonic current penetration is completely different, as shown in Figure 3.
The high current levels at the receiving end of the line are due to the short-circuit condition. However, this. The reason is that the points plotted correspond only to harmonic frequencies and resonances do not fall exactly on these frequencies; i. Coupling Between Harmonic Sequences It is the zero sequence penetration, rather than the positive sequence.
The presence of. However, the levels of zero sequence current are low notice the scale change between positive and zero sequence plots. Differences in Phase Voltages In conventional harmonic analysis using single-phase positive sequence models, a transmission line is assumed to have one resonant. However, the use of the three-phase algorithm to model the IslingtonKikiwa unbalanced transmission line shows that the resonant frequencies are different for each phase.
In this case, the spread of frequencies can be seen from Figure 3. The results clearly indicate that harmonics in the transmission system are unbalanced and three-phase in nature. Effect of Mutual Coupling in Double Circuits The unbalanced behaviour of double circuit lines is well documented at fundamental frequency [ 18, The three-phase harmonic penetration algorithm is used in this section to determine the importance of modelling mutual coupling at harmonic frequencies.
The line used is shown in Figure 3. The figure also displays the coupling between the positive sequence and the other sequence networks, i. Results for the case of a coupled line are illustrated in Figure 3. Moreover, the magnitude and resonant frequency of Z , is very different in the two cases. Robinson  reported that telephone interference caused by zero sequence currents did not coincide with high levels of power system harmonics.
Such a method. The series impedance and shunt equivalent matrices are combined into one admittance matrix that represents the transposed section. Applying a partial inversion algorithm to Equation 3.
The first relates to a harmonic voltage excited open-ended line. In fundamental frequency studies the effect of transpositions is generally accounted for by averaging the distributed parameters of the three transposed sections and using them in a single nominal or equivalent PI-circuit. VS and V R are vectors of a size determined by the number of coupled conductors. D are converted back into an admittance matrix which properly represents the effects of transpositions.
The nodal admittance matrix equation of the three-phase transmission line may be written as 3. D' parameters. Conductor type: Details of the test line: The test line. These two cases are illustrated by the simplified diagrams of Figures 3. Reproduced from I by permission of IEE. It is thus appropriate to consider the effectiveness of transpositions in the presence of harmonic as well as power frequency voltage sources.
Equal distances between transpositions and the natural impedance matrix. The second important case is the harmonic current excited short-circuited ended line. Effect of transpositions with Voltage Excitation Harmonic voltage sources are thought to be of little significance at the moment and are generally ignored when assessing harmonic distortion.
This case produces the highest voltage harmonic levels and must. In each case.. These figures indicate that in the absence of voltage compensation R The test line is fed from 1 p. It is realised that the presence of 1 p.. Open-ended Line The fundamental frequency behaviour of the open-ended line is illustrated in Figures Although such transmission distances are impractical without compensation.
For distances approaching the quarter wavelength. Such attenuation is caused by the series and shunt resistive components of the equivalent PI-model. Such behaviour is exemplified in Figure See Key for Figure 3. Thus the region of resonant distances has been expanded in Figure The resonant peaks of the three phases occur at very different distances. Reproduced from  by permission of IEE wavelength i. It is also interesting to note the dramatic voltage amplification which occurs for electrical distances equal to the first quarter wavelength.
Reproduced from  by permission of IEE. O LkO didonce. For Key see Figure 3. The improved symmetry of the phase voltages at the three quarter wave distance. Reproduced from  by permission of IEE The immediate effect of the transpositions is the compensation of geometrical line asymmetry..
Thus the deterioration of voltage balance current standing wave along the line. This can only result in electrical symmetry if the average currents in each of the transposed sections are similar. The level of imbalance of the untransposed line Figure 3.
Line Loaded If an ideal uncoupled and unattenuated line is loaded with its characteristic impedance. It must be noted that in a coupled multiconductor line such impedance is a matrix.
We cannot therefore expect to see the uniform 1 p. L 0 50 For this particular loading conditions the effectiveness of transpositions is limited to distances of about and km for the 3rd and 5th harmonics.
In contrast. Up to the first quarter wavelength the effect of natural fundamental frequency loading on the harmonic voltages is very similar to the fundamental frequency. By way of example. Subsequent harmonic peaks are seen to reduce rapidly with loading. Beyond those distances the transposed lines produce higher levels of imbalance.
The harmonic behaviour of a loaded transmission line without and with transpositions is illustrated in Figures 3. For longer lines. This figure displays the variation of 5th harmonic voltage at the receiving end of a km line with one per unit voltage injection at the sending end.
Results for the fundamental frequency. Reproduced from  by permission of IEE illustrates a dramatic increase in the voltage imbalance as the load reduces from the natural level 1 p. As the line load increases above the natural level. With harmonic current excitation the situation may be quite different. Considering the relatively insignificant levels of harmonic voltage excitation expected from a well-designed system. Because of their large dc smoothing inductance compared to the ac system impedance.
Reproduced from  by permission of IEE Figure 3. Effect of Transpositions with Current Excitation The main cause of power system harmonic distortion is the large static power converter. A qualitative justification for this behaviour has been made in Figure 3. As the harmonic order increases. The harmonic voltages at the point of current harmonic injection follow the same pattern as those of the open circuit line with harmonic voltage excitation.
This is clearly illustrated in Figure 3. Reproduced from  by permission of IEE Thus. Similarly to the voltage excited open line. Each line section is represented by its harmonic admittance matrix 3. With the assumption that the compensating inductances and capacitances are uncoupled and linearly dependent with frequency their corresponding harmonic matrix admittances are: An equivalent circuit of a typical long distance transmission line with conventional compensation elements is illustrated in Figure 3.
The magnitudes of the harmonic voltages for the loaded line are smaller than those of an open-ended line. Harmonic Current Excitation In this case. This in turn can be converted back to an equivalent admittance matrix. The effect of shunt inductive compensation in the harmonic behaviour of the unloaded line is shown in Figure 3. The results of combining series and shunt capacitive compensation for the case of a heavily-loaded line are shown in Figure 3.
For the positive sequence shunt admittance values of the test system. For this loading condition Figure 3. The addition of shunt inductive Compensation effectively increases the characteristic impedance and thus reduces the load that causes the optimum voltage profile. Harmonic Voltage Excitation The test line is a km of the same configuration as in section 3.
If the purpose is to observe harmonic voltages at the far end of the transmission system. It is noted that shunt capacitors tend to amplify harmonic distortion at the compensation points. The results. In the absence of compensation.
In particular. It is. For generality the test system. The distances of the main transmission components are: Under perfectly symmetrical ac supply and switching conditions. The New Zealand HVdc link. Towers in the coastal sections are identical to those in Figure 3. S1 inland line SI coastal line: Derivation of Parameters Considering the perfectly balanced self and mutual impedance of the line.
A cross-section of the submarine cable is shown in Figure 3. A relative permittivity of 3. With reference to the circuit diagram of Figure 3. For the circuit of Figure 3. The final form of the ABCD parameter matrix for a particular section is. For the situation under consideration. Y I and Y2 must be replaced with the appropriate matrices.
YII y41 y12 y32? V2 only: It may. Since the receiving end converter can be approximated by a voltage source. V I and This leads to the 2 x 2 admittance matrix in terms of II. Of note here. As can be readily seen. This resonant point may also be observed from the Benmore end.
Although the resonances are again shifted to the left.
These models. Method of symmetrical co-ordinates applied to the solution of polyphase networks.
The formulation of earth path and conductor impedances taking into account skin effect has been carried out and then used to derive simpler solutions based on tabulated coefficients.
Frequency-dependent models have also been derived for the transformers and VAR compensating equipment. Several practical examples of application of the models to ac and dc transmission have been included. A frequency-dependent equivalent PI model has been described suitable for computer implementation. Harmonic phenomena associated with the Benmore-Haywards HVdc transmission scheme.
Line constants of overhead lines and underground cables. E notes. G and Luoni. Peter Peregrinus Ltd. Circulating currents in parallel untransposed multicircuit lines. T J and Harker. J H and Reinsch. The complex ground return plane. Time domain modelling of frequency dependent threephase transmission line impedance.
Calculation of maximum harmonic currents and voltages on transmission lines. Induced currents and losses in single-core submarine cables. Proc IEE. Admittance matrix model of a synchronous machine for harmonic analysis. John Wiley. Arnold C P and Harker. H g Voltage Direct Current Transmission.
Electrical Transmission of Power and Signals. R A and Hickey. University of Canterbury.. The thyristor-controlled static phase-shifter. Wave propagation in overhead wires with ground return. I Numerical evaluations. Course E. W N and Wedepohl. University of British Columbia. Edison Electric Institute Ineffectiveness of transmission line transpositions at harmonic frequencies.
J F and Arrillaga. Modelling of power system transformers in the complex conjugate harmonic space. Linear Algebra. Harmonic instability between controlled static converters and ac networks.
A and Deri. H and Guth. IEEE Traiu. Power system modelling. Brown Boveri Rev. McGraw Hill. Vol Handbook for Automatic Computations. Three phase power system harmonic penetration. H W New Zealand Engineer. PWRS-2 4. W I and McNamee. J and Duke. Development of equivalent PI and T matrix circuits for long untransposed transmission lines. M S and Dillon. Bell System Tecltnical Journal. Electric Energy Systems Theory: An Introduction.
Zero sequence harmonic current generation in transmission lines connected to large converter plant. Thyristor-controlled quadrature boosting. E Calculation of electrical parameters for short and long polyphase transmission lines.
Derivation of harmonic impedances of the inter-island HVdc link. Line Constants Program Manual. Bonneville Power Administration. Nashville Indiana. Part of the EMTP package.
Ineffectiveness of transmission line VAR compensation at harmonic frequencies. Represent the individual network elements. The derivation of the harmonic voltages and currents will. Most power system non-linearities manifest themselves as harmonic current sources. This model is commonly used to derive the system harmonic impedances at the point of common coupling as required in filter design.
A comprehensive algorithm of general applicability should have the following capabilities: Provide graphical interfaces for the specification and display of the system to be analysed. A Fourier analysis is then applied to obtain the harmonic currents injected by each non-linear component into the linear system.
In such case. The simplest model involves a single harmonic source and performs a single phase harmonic analysis. Figure 4. The main difficulty is to determine which model best represents the various system components at the required frequency and obtain appropriate parameters for them. These effects give rise to unbalanced self and mutual admittances of the network elements.
The current injections. For the three-phase system. The three-phase nature of the power system always results in some load or transmission line asymmetry. With this information. The nodal admittance matrix of the network at a frequency f is of the form: A separate system admittance matrix is generated for each frequency of interest.
The matrix is triangulated using Gaussian elimination. The resulting matrix equation for an n-node system with n. In remain unchanged since the currents above these in the current vector are zero. The injected currents at most ac busbars will be zero. To calculate an admittance matrix for the reduced portion of a system comprising of just the injection busbars.
The elements are the self. The reduced matrix equation is and the order of the admittance matrix is three times the number of injection busbars. Advantage is taken of the symmetry and sparsity of the admittance matrix . V 2 and V3. To compare measured and simulated impedances at a current injection busbar it is thus necessary to derive equivalent phase impedances from the 3 x 3 admittance matrix. This then allows the unknown busbar voltages and unknown harmonic currents to be found.
If V2 represents the known voltage sources then I2 are the unknown variables. Partitioning the matrix equation to separate the two types of nodes gives: Restricted measurements on the physical network limit the ability to compare a three-phase model with test results. The remaining busbars are represented as a harmonic current injection I. The data obtained from live three-phase systems only includes the phase voltages and currents of the coupled phases. Reducing a system to provide an equivalent admittance matrix.
Whenever required. XJ"is the generator subtransient reactance and X. Besides the transmission system. An example of a typical variation of the inductive coefficient of a transformer with frequency is shown in Figure 4. Such models are discussed in Sections 4. The modelling of multi-phase transformers as part of a transmission system is discussed in Chapter 3 Section 3.
When calculating harmonic flows throughout the network each node has to be explicitly represented in the analysis. The transmission system has the greatest influence on the parameters of the matrix and special attention has been given to their modelling in Chapter 3.
A frequency dependent multiplying factor can be added to the reactance terms to account for skin effect. Often the purpose of the studies is to determine the voltage harmonic distortion levels at points of common coupling PCC between a distorting component and other consumers. The magnetising admittance is usually ignored since under normal operating conditions its contribution is not significant.
For short lines. A typical simplified dominant configuration of a distribution feeder is shown in Figure 4. Utilities should be encouraged to develop data base of their electrical regions. The aggregate nature of the load makes it difficult to establish models based purely on theoretical analysis.
The following guidelines are recommended for the derivation of distribution feeder equivalents . Attempts to deduce models from measurements have but been made [3. All elements should be uncoupled three-phase branches. Simpler equivalents for the transmission and distribution systems should be used only for remote points.. The weighting coefficient Jlt.
The representation should be more detailed nearer the points of interest. Other elements. The equivalent resistance is estimated from the active power at fundamental frequencies. There are basically three types of loads. In each case the derivation of equivalent conductance and susceptance harmonic bandwidths from specified P active and Q reactive power flows will need extra information on the actual composition of the load. Some early measurements  showed that maximum plant conditions can result in reduced impedances at lower frequencies and increased impedances at higher frequencies.
Power distribution companies will have a reasonable idea of the proportion of each type in their system depending on the time of day and should provide such information. Power factor correction PFC capacitance should be estimated as accurately as possible and allocated at the corresponding voltage level.
X impedance i. Simulation studies  have also demonstrated that the addition of detailed load representation can result in either an increase or decrease in harmonic flow. Alternative models for load representation should be used according to their composition and characteristics.
The equivalent inductance represents the relatively small motor content when known. A detailed analysis of the induction motor response to harmonic frequencies. In this situation the load model A suggested by reference  can be used. To illustrate the importance of the loading level on the harmonic impedances. The motor impedance at any frequency can be expressed as: The A model is a parallel connection of inductive reactance and resistance whose values are: C configuration and their non-linear characteristics cannot fit within the linear harmonic equivalent model.
The presence of system non-linearities has been discussed earlier. Figures 4. Modelling the power electronic loads is a more difficult problem because. In studies concerning mainly the transmission network the loads are usually equivalent parts of the distribution network.
When studying the transmission network it is strongly recommended to model at least part of the next lower voltage level and place the load equivalents there. In the absence of detailed information the power electronic loads are often left open-circuited when calculating harmonic impedances.
QS At harmonic frequencies: EO Any local plant components such as synchronous compensators. Considering the large number of studies involved in filter design.
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Santoso, S. Power Delivery, 11 2 , — Ringrose, M. Power Quality Assessment, Amsterdam. Haykin, S. A Comprehensive Foundation. Macmillan, pp. Zadeh, L. Control, 8 3 , — Tanaka, K. The increasing use of power electronic devices for the control of power apparatus and systems has been the reason for the greater concern about waveform distortion in recent times.
A power electronic converter can be viewed as a matrix of static switches that provides a flexible interconnection between input and output nodes of an electrical power system. Through these switches power can be transferred between input and output systems operating at the same or different frequencies one or both of which can be d.
The most common power electronic aid is the single-phase rectifier, used to power most modern office and domestic appliances. Although the individual ratings are always small, their combined effect can be an important source of waveform distortion. Because of their considerable power ratings, three-phase static power converters are the main contributors to the harmonic problem.
The terms rectification and inversion are used for power transfers from a. According to the relative position of the firing instant of the switches from one cycle to the next on the steady state, four basically different power electronic control principles are in common use: Inverter fed a. The primary current, however, will not be purely sinusoidal, because the flux is not linearly proportional to the magnetising current, as explained in the next section.
In Figure 3. When the hysteresis effect is included, as in the case of Figure 3. In this case the current corresponding to any point on the flux density wave of Figure 3. The distortion illustrated in Figures 3. With three-limb transformers the triplen harmonic m. Thus, flux density and e. However, the elimination of triplen harmonics in the delta-connected windings is only fully effective when the voltages are perfectly balanced.
The magnetising current harmonics often rise to their maximum levels in the early hours of the morning, i. The problem of overvoltage saturation is particularly onerous in the case of transformers connected to large rectifier plant following load rejection. It has been shown  that the voltage at the converter terminals can reach a level of 1.
The symmetrical magnetising current associated with a single transformer core saturation contains all the odd harmonics. If the fundamental component is ignored, and if it is assumed that all triplen harmonics are absorbed in delta windings, then the harmonics being generated are of orders 5, 7, 11, 13, 17, In conventional six-pulse rectifier schemes it is usual to filter these harmonics from the a.
In this case the fifth- and seventh-order harmonics produced by a saturated converter transformer are not filtered and have to be absorbed by the a. The composition of the magnetising current versus the exciting voltage is typically as shown in Figure 3.
When the transformer is re-energised the flux density illustrated in Figure 3. For a normally designed transformer this can create peak flux densities of about 3.
When this is compared to the saturation flux density levels of around 2. This effect gives rise to magnetising currents of up to 5—10 per unit of the rating as compared to the normal values of a few percentage points. Such an inrush current is shown in Figure 3. The decrement of the inrush current with time is mainly a function of the primary winding resistance. For the larger transformers this inrush can go on for many seconds because of their relatively low resistance.
By way of illustration, the Fourier series of the waveshape of Figure 3. The harmonic content, shown as a percentage of the rated transformer current, varies with time, and each harmonic has peaks and nulls.
Magnetisation It has been shown in previous sections that a transformer excited by sinusoidal voltage produces a symmetrical excitation current that contains only odd harmonics. Under magnetic imbalance, the shape of the magnetising characteristic and the excitation current are different from those under no-load conditions.
If the flux is unbalanced, as shown in Figure 3. The existence of an average flux implies that a direct component of excitation current is present in Figure 3. Under such unbalanced conditions, the transformer excitation current contains both odd and even harmonic components. The asymmetry can be caused by any load connected to the secondary of the transformer, leading to a direct component of current, in addition to the sinusoidal terms. The direct current may be a feature of the design, as in a transformer feeding a half-wave rectifier, or may result from the unbalanced operation of some particular piece of equipment, such as a three-phase converter with unbalanced firing.
A similar effect can occur as a result of geomagnetically induced currents GIC. These are very low frequency currents typically 0. They enter the transformer windings by way of earthed star connections and produce asymmetrical flux, causing half-cycle saturation.
The linearity is better for the lower-order harmonics. Moreover, the harmonics generated by the transformer under d. This independence is most noticeable at low levels of the direct current and for the lower harmonic orders. Distribution of A. Windings Figure 3. Under such idealised conditions the air-gap m. The frequency-domain representation of the rectangular m. However, in practice the windings are distributed along the surface with g slots per pole per phase and the m.
Sum of m. Hence the m. The corresponding m. However the flux is never exactly distributed in this way, particularly in salient pole machines. A non-sinusoidal field distribution can be expressed as a harmonic series: The winding e. The triplen harmonics in a three-phase machine are generally eliminated by phase connection, and it is usual to select the coil span to reduce fifth and seventh harmonics.
Standby generators with neutral require special consideration in this respect, as illustrated by the following example. A standby generator had to be designed to supply a kVA load consisting mainly of fluorescent lighting appliances.
A neutral current of 40 A was considered sufficient for the generator design. However, in practice the machine generated A of thirdharmonic zero sequence and had to be rewound with a two-thirds pitch i. The fundamental rotor m. Relative to the rotor, however, these two waves have different velocities. In any closed rotor circuit each of these will generate currents of frequency 2mgf by considering the ratio of speed to wavelength and these superimpose a time-varying m.
The resultant stator e. Slot harmonics can be minimised by skewing the stator core, displacing the centre line of damper bars in successive pole faces, offsetting the pole shoes in successive pairs of poles, shaping the pole shoes, and by the use of composite steel—bronze wedges for the slots of turboalternators.
It can be shown that the distribution factor for slot harmonics is the same as for the fundamental e. Fractional instead of integral slotting should be used. If the stator current is of positive sequence, the field produced by this current in the rotor is stationary, and only causes armature reaction at the fundamental frequency. On the other hand, the flux produced by a negative sequence stator current can be divided into two components rotating in opposite senses, which therefore induce two e.
The latter, using the same reasoning, will produce fifth-harmonic voltage and this in turn will create some seventh harmonic, etc. This mechanism is illustrated in Figure 3. The above reasoning can be extended to the presence of harmonic currents in the stator.
This effect is illustrated in Figure 3. In practice, as the machine poles are not completely salient and the transmission line is approximately symmetrical, the effect discussed above is not significant. Consequently, when carrying out harmonic penetration studies, the salient pole effect is normally neglected and the synchronous machine represented as a linear impedance. Time harmonics are produced by induction motors as a result of the harmonic content of the m. A harmonic of order n in the rotor m.
This harmonic induces an e. Harmonics can also occur as a result of electrical asymmetry. Slip frequency e. The frequencies of stator e. Interaction of harmonic and mains frequency currents results in beats at this low frequency 2sf being registered on connected meters. Following arc ignition the voltage decreases due to the short-circuit current, the value of which is only limited by the power system impedance.
The main harmonic sources in this category are the electric arc furnace, dischargetype lighting with magnetic ballasts, and to a lesser extent arc welders. The current levels, limited mainly by the furnace cables and leads and transformer, can reach values of over 60 kA. Those impedances have a buffering effect on the supply voltage and thus the arcing load appears as a relatively stable current harmonic source. However, the stochastic voltage changes due to the sudden alterations of the arc length produce a spread of frequencies, predominantly in the range 0.
This effect is more evident during the melting phase, caused by continuous motion of the melting scrap and the interaction of electromagnetic forces between the arcs.
During the refining part of the process the arc is better behaved, but there is still some modulation of the arc length by waves on the surface of the molten metal. Typical time-averaged frequency spectra of the melting and refining periods are shown in Figure 3. However, the levels of harmonic currents vary markedly with time, and are better displayed in the form of probabilistic plots, such as that shown in Figure 3.
Three sets of averaged harmonic current levels obtained by different investigators are listed in Table 3. This effect is clearly illustrated in Figure 3. This effect is particularly important in the case of fluorescent lamps, given the large concentration of this type of lighting.
Additional magnetic ballasts are needed to limit the current to within the capability of the fluorescent tube and stabilise the arc. In a three-phase, four-wire load the triplen harmonics are basically additive in the neutral, the third being the most dominant.
With reference to the basic fluorescent circuit of Figure 3. The voltage across the tube itself Figure 3. The waveform in Figure 3. The latter consists almost exclusively of third harmonic.
Lighting circuits often involve long distances and have very little load diversity. With individual power factor correction capacitors, the complex LC circuit can approach a condition of resonance at third harmonic. This effect has been illustrated by laboratory results in a balanced three-phase fluorescent lamp installation . The results of greatest interest refer to the effects of increasing the neutral reactance and isolating the capacitor star point see Figure 3.
In the graph of Figure 3. FL, fluorescent lamp; B, ballast; C, power factor correction capacitor; L, variable inductor; S, switch to isolate capacitor star point a b c Figure 3. The nominal Hz voltage is the calculated product of the Hz lamp current per phase and the circuit , Switch S closed star point connected to neutral ; zero-sequence impedance Hz.
It can be seen that with the capacitor star point connected to the neutral, the third harmonic neutral current can by far exceed the nominal value calculated by the conventional method based on three times the nominal lamp current. With the star point disconnected the neutral current is less than the nominal value. The results demonstrated in the laboratory test were in fact a confirmation of actual field test results taken on a kVA fluorescent installation.
Whenever possible, the design procedure recommended to avoid resonance is to try to avoid individual lamp compensation, providing, instead, capacitor banks adjacent to distribution boards connected either in star with floating neutral or in delta. Power Supplies Many commercial and domestic appliances require direct current for their operation. The single-phase diode bridge rectifier Figure 3.
The circuit of Figure 3. The Fourier series of the current pulse of Figure 3. Earlier technology used a transformation stage to control the required low voltage levels and the transformer leakage inductance had a smoothing effect that resulted in low harmonic current levels. Instead, modern appliances use the switch-mode power supply concept, whereby the input rectifier is directly connected to the a. This process provides a very compact design and efficient operation, tolerating large variations in input voltage.
Personal computers and most office and domestic appliances, as well as the electronic ballast of modern fluorescent lighting systems, are now of this type. However, the lack of a. Particularly troublesome is the third harmonic, which adds arithmetically in the neutral of the three-phase network. The current harmonics calculated from equation 3. Typical examples of single-phase distorting appliances are TV receivers, personal computers and microwave ovens.
TV Receivers Figure 3. The main harmonics are in order of magnitude the third, fifth, seventh and ninth. PC and Printer Figure 3. To illustrate the cumulative effect of this type of load, imagine the case of a high office block, which may have up to personal computers.
From the spectrum of Figure 3. Microwave Oven Figure 3. A typical configuration of the locomotive power supply, shown in Figure 3.
At the start, the back e. Therefore during the initial accelerating period, with maximum d. To alleviate the situation at low speeds, one of the bridges is often bypassed and phase control exercised on the other. When the speed builds up, and the second bridge operates on minimum delay, phase control is exercised on the first bridge. The relevant waveforms are illustrated in Figure 3. V 7 11 11 3 3 3 3 7 7 7 11 11 11 11 Variation in harmonic currents with locomotive operation An indication  of the wide variation in harmonic current magnitudes corresponding to the waveform of Figure 3.
This is achieved by means of a series smoothing reactor on the d. As the a. Under these conditions, the d. The switches must block voltages of both polarities, but are only required to conduct current in one direction. Thus, large converters have traditionally been of the current-source type because of the availability of efficient highly rated thyristors.
Under perfectly symmetrical a. If in the analysis of the waveform of Figure 3. The relevant Fourier coefficients, with reference to a 1 per unit-d. Its Fourier series is obtained by combining equations 3. For the square wave of Figure 3. The time-domain waveforms can also be synthesised from the combination of the time-domain representation of the individual harmonics. Figure 3. For clarity only the fundamental, third and fifth harmonics have been shown and the complex waveform produced is therefore not complete.
Some useful observations can now be made from equation 3. This statement may not be obvious. Let us clarify it with reference to the fundamental, third although not present in the symmetrical case and fifth harmonic current components.
The r. The Fourier series of the waveform shown in Figure 3. Moreover, to maintain pulse operation the two six-pulse groups must operate with the same control angle and therefore the fundamental frequency currents on the a. The time-domain representation of the pulse waveform is shown in Figure 3. The addition of further appropriately shifted transformers in parallel provides the basis for increasing pulse configurations.
Although theoretically possible, pulse numbers above 48 are rarely justified due to the practical levels of distortion found in the supply voltage waveforms, which can have as much influence on the voltage crossings as the theoretical phase-shifts.
Similarly to the case of the pulse connection, the alternative phase-shifts involved in higher pulse configurations require the use of appropriate factors in the parallel transformer ratios to achieve common fundamental frequency voltages on their primary and secondary sides.
Generally harmonics above the 49th can be neglected as their amplitude is too small. As we have seen in previous sections, high-pulse configurations are combinations of three-pulse groups, i. The current waveform has now lost the even symmetry with respect to the centre of the idealised rectangular pulse.
Using as a reference the corresponding commutating voltage i. These can now be expressed in terms of the delay firing , and overlap angles and their magnitudes, related to the fundamental components, are illustrated in Figures 3.
In summary, the existence of system impedance is seen to reduce the harmonic content of the current waveform, the effect being much more pronounced in the case of uncontrolled rectification. With large firing angles the current pulses are practically unaffected by a. To illustrate the use of these graphs let us consider the case of a six-pulse rectifier connected via a 50 MVA Y—Y transformer of unity turns ratio to the kV system.
If the rated d. From Equation 3. The corresponding d. From equations 3. These curves and equations show some interesting facts. Equation 3. Voltage Smoothing Considering the limited inductance of the motor armature winding and the larger variation of firing angle, the constant d. The d. Under nominal loading the firing delay is kept low, but during motor start or light load conditions the delay increases substantially and the current may even be discontinuous.
This extreme operating condition is illustrated in Figure 3. Each phase consists of two positive and two negative current pulses, which are derived from the general expression 3. When d. An approximate method described by Dobinson  derives the harmonic components of the a.
The method ignores the effect of the commutation reactance, which at large delay angles is negligible. With reference to Figure 3. The information included in Figures 3. When operating at full load i. However, under operating conditions requiring firing delays, the half-wave symmetry of the current waveform is lost, as shown in Figure 3. At low loads these controllers not only have a very poor power factor but introduce severe waveform distortion, particularly at even harmonics.
More often than not the controllers and motors initially installed are larger than required to cope with future expansion, and operation is then at a fraction of the full load. Under these conditions the second harmonic component often reaches levels close to the fundamental current. In general each of the main three parts of the system is always in error to a lesser or greater extent: C supply Id Vd d.
Trace ii shows the d. Trace iii shows the supply voltage and current of phase A 2 3 The d. The firing angle control systems often given rise to substantial errors in their implementation. As a result the large static converters often produce harmonic orders and magnitudes not predicted by the Fourier series of the idealised waveforms. Filters are not normally provided for uncharacteristic harmonics and as a result their presence often causes more problems than the characteristic harmonics.
By way of example, Table 3. All the harmonic voltages are unbalanced, particularly the third and ninth. The table also illustrates the presence of all current harmonic orders, odd and even, with the uncharacteristic orders causing higher voltage distortion than the characteristic ones.
A realistic quantitative analysis of the uncharacteristic harmonic components can only be achieved by a complete three-phase computer model of the system behaviour with detailed representation of the converter controls. Imperfect a.
Source Deviations from the perfectly balanced sinusoidal supply can be caused by i presence of negative sequence fundamental frequency in the commutating voltage; ii harmonic voltage distortion of positive or negative sequence; and iii asymmetries in the commutation reactances. In general an imperfect a. The first problem can be eliminated by using equidistant firing control but the second problem still remains. This effect, illustrated in Figure 3. Under normal operating conditions the expected levels of asymmetry and distortion are small and their effects can be approximated with reasonable accuracy.
The a. In practice, of course, the result of unbalance is a combination of each effect, which may either increase or decrease the harmonic content of the individual cases. Five cases were considered, as follows: The effect of each of these cases on the first ten harmonic currents generated by the converter are shown in Table 3.
These results show that the effects of phase reactance unbalance and firing angle unbalance are different for the two types of converter transformer. These are more severe for the star—star transformer i. With reference to supply distortion, if a small positive- or negative-sequence signal Vn per unit of the normal fundamental voltage is added to the otherwise ideal threephase supply of a pulse converter configuration, the order and maximum level Vk of uncharacteristic harmonic voltage at the rectified output come under one of the following categories : Case 1: There are also higher-order harmonics in each case, not shown, of much lower amplitude.
The tabulated Vk is expressed per unit of the maximum average d. The values given above are in practice applicable also to converters of higher pulse number. Thus, while characteristic harmonics can be reduced by using high pulse number, the uncharacteristic harmonics due to a. It should be noted that the above is an approximation due to neglect of commutation reactance but is generally sufficiently valid at low harmonic orders below the fifth.
The cause of the imperfection may also be some asymmetry in the commutation reactances, i. In this case the maximum level of uncharacteristic a. Table 3. Unequal commutation reactances also cause uncharacteristic voltages on the d. Only even harmonics occur, given by Vn max. Current Modulation  If we now assume a perfect three-phase supply and equidistant firing, the addition of a small current harmonic component Ik of order k on the d.
It is again approximate, valid only at low frequencies, because the commutation reactance has been neglected. Control System Imperfections No general rules can be given in this case. By way of example, Ainsworth  describes the effect of modulating the harmonic content of the voltage applied to the oscillator of the d.
Assuming constant d. The magnitude in per unit of the maximum average rectified voltage of the d. The total a. Firing Asymmetry A. Kimbark  describes the effect of late and early firings with reference to a six-pulse bridge converter.
This produces triplen harmonic currents. The voltage and current relationships across the converter can be expressed as follows: This process is illustrated in Figure 3. In the absence of commutation overlap, the transfer functions are rectangular, as shown in Figure 3.
The simplified modulation process, explained above, can be extended to the more realistic case where the commutation process is included. This extension, explained further in Chapter 8, is necessary to derive accurate quantitative information from the generalised table to be described in the following section. Link The frequency transfer relationships in the cross-modulation process of a line-commutated pulse a. The link can be an HVd. The resulting harmonic orders in system 2 are related to the frequency of system 2.
The DC column refers to the d. When the d. These can be divided into two groups: The back-to-back frequency conversion schemes represent the worst condition for non-integer harmonic frequencies. In this case, with small smoothing reactors the d. In six-pulse operation such schemes can produce considerable subharmonic content even under perfect a.
However, pulse converters do not produce subharmonic content under symmetrical and undistorted a. These will produce inter-harmonic currents as defined by equation 3. When the link interconnects two isolated a. This is a flickerproducing frequency. This same frequency will be referred to generator rotor shaft torque at 20 Hz, which may excite mechanical resonances.
Again, this type of cross-modulation effect is most likely to happen in back-toback schemes due to the stronger coupling between the two converters, although it is also possible with any HVd. Now consider two a. This frequency is generally too low to produce flicker but may induce mechanical oscillations.
However, the subharmonic levels expected from this second-order effect will normally be too small to be of consequence. Under these conditions the d. The simplest VSC configuration is the six-pulse diode bridge with a large capacitor across the output terminals. In this circuit the capacitor is charged every half-cycle of the supply frequency by two short current pulses, typically as shown in Figure 3.
However, unlike the single-phase power supply rectifier, the absence of neutral connection in the three-phase case eliminates the triplen harmonics. The addition of an a. Moreover, when bi-directional power flow is designed for, the switches must block a unidirectional voltage but be capable of conducting current in either direction. This type of converter suits the a. Advances in the ratings of GTO and IGBT devices is extending the application of the voltage-sourced conversion concept to very large motor drives and even to light HVd.
However the switching transients caused by the higher voltages impose additional stresses on the motor windings. The multi-level solution has been developed to generate high voltage waveforms using relatively low voltage switching devices. A multi-level voltage-source converter can switch its output between multiple voltage levels within each cycle, thus creating a better voltage waveform for a particular switching frequency, when compared to a conventional two-level inverter.
Theoretical phase output voltages for three- and five-level configurations are illustrated in Figure 3. Newton, M. The figure also shows the voltage THD calculated for two-, three-, five- and seven-level output waveforms. The following d. In this configuration the harmonic cancellation is achieved through the phase displacement of the voltage waveforms of phase-shifted transformer secondary windings.
This is a variation of the previous case, its main difference being the elimination of the phase-shifting transformers, i. Each phase consists of series connected single-phase full bridges, each bridge requiring an isolated d.
This alternative achieves the multilevel waveforms by the series or parallel connection of switches within the converter bridge itself. The multi-level flying capacitor converter . In previous configurations each phase leg consisted of a switch pair in parallel with a bus capacitor, and must be always connected to either the positive or negative node of the capacitor.
The chain circuit converter . This configuration consists of individually controlled units, which can then be assembled to form the three-phase converter. It provides modularity and ease of expansion. Unlike the previous multi-level configurations, where all the switches form part of the main conversion process, the pulse increase is now achieved by separate switching circuitry at reduced current levels. Inverter-Fed A. Drives Although the thyristor-controlled d.
The basic three-phase inverter bridge commonly used for a.
The VSI requires a constant d. CSI drives have better speed characteristics but require a motor with leading power factor either synchronous or induction-type with capacitors ; however, the use of turn-off switching devices removes this restriction. Motor Phase Voltage In the circuit of Figure 3. The effect of triplen harmonic elimination is shown in Figure 3.
Some high-power inverter-fed a. In this case the inverter output voltage waveforms are always square waves, as shown in Figure 3.
An alternative to independent d. The operating principle consists of chopping the basic inverter square wave output voltage of Figure 3. In its simplest form a saw-tooth wave  is used to modulate the chops, as shown in Figure 3. The saw-tooth has a frequency which is a multiple of three times the sine wave frequency, allowing symmetrical three-phase voltages to be generated from a three-phase sine wave set and one saw-tooth waveform.
This method controls lineto-line voltage from zero to full voltage by increasing the magnitude of the saw-tooth or a sine-wave signal, with little regard to the harmonics generated.
The most significant areas of voltage in the spectrum, apart from the fundamental, occur at the carrier frequency saw-tooth frequency and its two sidebands, and to a significant extent at each multiple of these frequencies in the spectrum. When the carrier frequency is six times the fundamental, the triplen harmonics cancel in the system; however, the phase waveforms of Figure 3.
If the carrier frequency is a large multiple of the fundamental, the first large harmonics encountered are high in the harmonic spectrum.
Single-phase bridge inverters can use either bipolar or unipolar PWM switching schemes. For odd values of k, harmonics only exist for even values of n and vice versa. With unipolar switching, each leg of the inverter is controlled independently so that there are periods when both sides of the load are connected to the same d. This has the effect of doubling the switching frequency as now the harmonics present are given by: The spectrum for the unipolar switching scheme is shown in Figure 3.
By way of example, Figure 3. In the method of reference  the period is divided into six regions. If the second and fifth regions of each phase waveform are filled with a train of pulses, or chops, only these pulses appear in the line-to-line voltage. Harmonic voltages occur at multiples of the carrier frequency i.