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Evans in the late s and early s. For reference and review, the various forms, for representation of point P in Figure b are as follows: Obviously, not all the above data are necessary in every application. These polar units operate on alternating current through a full-wave rectier and provide very sensitive, high-speed operation on very low energy levels. If the current is limited to within rated values, the diode recovers its nonconducting characteristics when the reverse voltage falls below the zener value. Many other switching-type operations generate tran- frequency.

Figure Voltage plot for a solid phase a-to-ground fault on a solidly grounded system. Also, the relatively high tower footing resistance may appreciably limit the fault current. Arc resistance is discussed in more detail in Chapter The diagrams shown are for effectively grounded systems. In all cases, the dotted or uncollapsed voltage triangle exists in the source the generator and the maximum collapse occurs at the fault location. The voltage at other locations will be between these extremes, depending on the point of measurement.

Among the operating conditions to be considered are maximum and minimum generation, selected lines out, line-end faults with the adjacent breaker open, and so forth.

With this information, the relay engineer can select the proper relays and settings to protect all parts of the power system in a minimum amount of time.

Three-phase fault data are used for the application and setting of phase relays and single- phase-to-ground fault data for ground relays. The method of symmetrical components is the foundation for obtaining and understanding fault data on three-phase power systems.

Formulated by Dr. Fortescue in a classic AIEE paper in , the symmetrical components method was given its rst practical application to system fault analysis by C.

Wagner and R. Evans in the late s and early s. Lewis and E. Harder added measurably to its development in the s. Today, fault studies are commonly made with the digital computer and can be updated rapidly in response to system changes. Manual calculations are practical only for simple cases. A knowledge of symmetrical components is impor- tant in both making a study and understanding the data obtained.

It is also extremely valuable in analyzing faults and relay operations. A number of protective relays are based on symmetrical compo- nents, so the method must be understood in order to apply these relays successfully. In short, the method of symmetrical components is one of the relay engineers most powerful technical tools. Although the method and mathematics are quite simple, the practical value lies in the ability to think and visualize in symmetrical components.

This skill requires practice and experience. Figure Phasor diagrams for the various types of faults occurring on a typical power system. Phasors, Polarity, and Symmetrical Components 21 the positive, negative, and zero sequence components. This reduction can be performed in terms of current, voltage, impedance, and so on. The positive sequence components consist of three phasors equal in magnitude and out of phase Fig.

The negative sequence components are three phasors equal in magnitude, displaced with a phase sequence opposite to that of the positive sequence Fig. The zero sequence components consist of three phasors equal in magnitude and in phase Fig. Note all phasors rotate in a counterclockwise direction. In the following discussion, the subscript 1 will identify the positive sequence component, the subscript 2 the negative sequence component, and the subscript 0 the zero sequence component.

For example, V a1 is the positive sequence component of phase-a voltage, V b2 the negative sequence component of phase-b voltage, and V c0 the zero sequence component of phase-c voltage. All components are phasor quantities, rotating counterclockwise. Since the three phasors in any set are always equal in magnitude, the three sets can be expressed in terms of one phasor. For convenience, the phase-a phasor is used as a reference.

Note that the b and c components always exist, as indicated by Eq. Note that dropping the phase subscripts should be done with great care. Where any possibility of misunderstanding can occur, the additional effort of using the double subscripts will be rewarded. Equations to can be solved to yield the sequence components for a general set of three-phase phasors: If any sequence component exists by measure- ment or calculation in one phase, it exists in all three phases, as shown in Eq.

Neutral is established by connecting together the terminals of three equal resistances as shown with each of the other resistor terminals connected to one of the phases. Substituting Eq. V 0 V ng Neutral and ground are distinctly independent and differ in voltage by V 0. Grounding and its inuence on relaying are discussed in Chapters 7 and Interconnections of the three sequence networks allow any series or shunt disconti- nuity to be investigated.

For the rest of the power- system network, it is assumed that the impedances in the individual phases are equal and the generator phase voltages are equal in magnitude and displaced from one another. Based on this premise, in the symmetrical part of the system, positive sequence current ow produces only positive sequence voltage drops, negative sequence current ow produces only negative sequence voltage drops, and zero sequence current ow produces only zero sequence voltage drops.

For an unsymmetrical system, interaction occurs between components. For a particular series or shunt discontinuity being repre- Figure Power system neutral. Phasors, Polarity, and Symmetrical Components 23 sented, the interconnection of the networks produces the required interaction. Any circuit that is not continuously transposed will have impedances in the individual phases that differ. This fact is generally ignored in making calculations because of the immense simplication that results.

From a practical viewpoint, ignoring this effect, in general, has no appreciable inuence. Except in the area of a fault or general unbalance, each sequence impedance is con- sidered to be the same in all three phases of the symmetrical system.

A brief review of these quantities is given below for synchronous machinery, transfor- mers, and transmission lines. X 00 d indicates the subtransient reactance, X 0 d the transient reactance, and X d the synchronous reactance. These direct-axis values are necessary for calculating the short-circuit current value at different times after the short circuit occurs. Since the sub- transient reactance values give the highest initial current value, they are generally used in system short-circuit calculations for high-speed relay applica- tion.

The transient reactance value is used for stability consideration and slow-speed relay application. The unsaturated synchronous reactance is used for sustained fault-current calculation since the voltage is reduced below saturation during faults near the unit. The negative sequence reactance of a turbine generator is generally equal to the subtransient X 00 d reactance.

X 2 for a salient-pole generator is much higher. The ow of negative sequence current of opposite phase rotation through the machine stator winding produces a double frequency component in the rotor. As a result, the average of the subtransient direct-axis reactance and the subtransient quadrature- axis reactance gives a good approximation of negative sequence reactance. Since the machine is braced for only three-phase fault current magnitude, it is seldom possible or desirable to ground the neutral solidly.

The armature winding resistance is small enough to be neglected in calculating short-circuit currents. This resistance is, however, important in determining the dc time constant of an asymmetrical short-circuit current. Typical reactance values for synchronous machin- ery are available from the manufacturer or handbooks. However, actual design values should be used when available. Values are available from the nameplate. The zero sequence reactance is either equal to the other two sequence reactances or innite except for the three-phase, core-type transformers.

In effect, the magnetic circuit design of the latter units gives them the effect of an additional closed delta winding. The resistance of the windings is very small and neglected in short-circuit calculations. The sequence circuits for a number of transformer banks are shown in Figure The impedances indicated are the equivalent leakage impedances between the windings involved.

For two-winding transformers, the total leakage impedance Z LH is measured from the L winding, with the H winding short-circuited. Z HL is measured from the H winding with the L winding shorted. Except for a 1: For three-winding and autotransformer banks, there are three leakage impedances: In the rst convention, the windings are labeled H high , L low , M medium ; in the second H high , L low , and T tertiary.

Unfortu- nately, the L winding in the second convention is 24 Chapter 2 equivalent to M in the rst. The tertiary winding voltage is generally the lowest. On a common kVA base, the equivalent wye leakage impedances are obtained from the following equations: The wye is a mathematical equivalent valid for current and voltage calculations external to the transformer bank.

The junction point of the wye has no physical signicance. One equivalent branch, usually Z M Z L , Figure Equivalent positive, negative, and zero sequence circuits for some common and theoretical connections for two- and three-winding transformers.

Phasors, Polarity, and Symmetrical Components 25 may be negative. On some autotransformers, Z H is negative. The equivalent diagrams shown in Figure are satisfactory when calculations are to be made relative to one segment of a power system. However, a more complex representation is required when phase cur- rents and voltages are to be determined at points in the system having an intervening transformer between them and the point of discontinuity being examined.

For delta-wye transformers, a phase shift must be accommodated. For ANSI standard transformers, the high-voltage phase-to-ground voltage leads the low- voltage phase-to-ground voltage by , irrespective of which side the delta or wye is on. This phase shift may be included in the equivalent per unit diagram by showing a 1 The phase shift in the negative sequence network for the delta-wye transformer is the same amount, but in the opposite direction, to that in the positive sequence network.

The phase shift then, for an ANSI standard transformer, would be 1 The phase shift must be used in all the combinations of Figure where a wye and delta winding coexist.

This effect is extremely important when consideration is being given to the behavior of devices on both sides of such a transformer. As a rule of thumb, the Hz reactance is roughly 0: The zero sequence impedance is always different from the positive and negative sequence impedances.

Zero sequence impedance can vary from 2 to 6 times X 1 ; a rough average for overhead lines is 3 to 3. The resistance terms for the three sequences are usually neglected for overhead lines, except for lower- voltage lines and cables. In the latter cases, line angles of 30 to may exist, and resistance can be signicant. A good compromise is to use the impedance value rather than reactance and neglect the angular differ- ence in fault calculations.

This yields a lower current to assure that the relay will be set sensitively enough. This mutual impedance becomes an increasingly important factor as more lines are crowded into common rights of way. Thus, three network diagrams are required to separate the three sequence compo- nents for individual consideration: These sequence network diagrams consist of one phase to neutral of the power system, showing all the compo- nent parts relevant to the problem under considera- tion.

Typical diagrams are illustrated in Figures through The positive sequence network Fig. Balanced loads may be shown from any bus to the neutral bus. Generally, however, balanced loads are neglected. Compared to the system low-impedance high-angle quantities, they have a much higher impedance at a Figure Example system and positive sequence network.

In short, balanced loads complicate the calculations and generally do not affect the fault currents signicantly. With two exceptions, the negative sequence network Fig. For all practical calculations involving faults or discontinuities remote from the generating plant, however, X 1 is assumed to be equal to X 2. The zero sequence network Fig. First of all, it has no voltage: Rotating machinery does not produce zero sequence voltage. Also, the transformer connections require special consideration and grounding impe- dances must be included.

Figure shows the zero sequence circuits for many transformers. A three-line system diagram is usually not required to determine the zero sequence network, but if a question arises as to the ow of zero sequence currents, the three-line diagram can be useful. From this three- phase system diagram, the zero sequence network requirements can be resolved by determining whether or not equal and in-phase currents can exist in each of the three phases. If the zero sequence current component can ow, the zero sequence network must reect its path.

For simplicity, Figure shows the generators solidly grounded. In practice, however, solid ground- ing is used only in very special cases. Current reference direction in any circuit element must be the same in all three networks to avoid confusion. Current ow in one or more of the networks may reverse for some types of unbalances, particularly if the networks are complex. Reverse ow should be treated as a negative current to ensure that it will be properly subtracted when determining the phase currents.

Each sequence network is, of course, a per unit diagram representing one of the three phases of the symmetrical power system. Therefore, a resistor reactor, impedance connected between the system neutral and ground, as shown in Figure , must be multiplied by 3 as indicated. In the system, 3I 0 ows through R; in the zero sequence network, however, I 0 ows through 3R, producing an equivalent voltage drop.

Figure Negative sequence network for example system. Figure Zero sequence network for example system. Figure Example system generators shown solidly grounded for simplication. Phasors, Polarity, and Symmetrical Components 27 5. In such areas, negative and zero sequence voltages are generated, as previously described.

Sequence network connections for various types of common faults are shown in Figures through From the three-phase diagrams of the fault area, the sequence network connections representing the fault can be derived. These diagrams do not show fault impedance, and fault studies do not include this effect except in very rare cases.

The single-sequence impedance Z 1 ; Z 2 ; Z 0 practically equivalent to X 1 ; X 2 ; X 0 shown in the gures is the net impedance between the neutral bus and selected fault location. Based on zero load, all generated voltages V AN are equal and in phase. Since the three-phase fault is balanced, symmetrical components are not required for this calculation. However, since the positive sequence network repre- sents the system, the network can be connected as shown in Figure to represent the fault.

For a phase-a-to-ground fault, the three networks are connected in series Fig. Figure illustrates a phase-b-c-to-ground fault and its sequence network interconnection. The phase-b-to-phase-c fault and its sequence connections are shown in Figure Fault studies normally include only three-phase faults and single-phase-to-ground faults. Three-phase faults are the most severe phase faults, whereas single- phase-to-ground faults are the most common.

Studies of the latter faults provide useful information for ground relaying. A fundamental study of both series and shunt unbalances was made by E.

Harder in The shunt unbalances summarized in Figure are taken from Harders study. Note that all the faults shown in Figures through are also represented in Figure Figure Sequence network connections and voltages. Figure Three-phase fault and its network connection.

Figure Phase-to-ground fault and its sequence net- work connections. Inside the topmost box for each shunt condition is a four-line representation of the shunt to be connectedtothe systemat point x. The three lower boxes for each shunt condition are the positive, negative, and zero sequence representations of the shunt.

The sequence connections for the series unbalances, such as open phases and unbalanced series impe- dances, are shown in Figure As before, these diagrams are taken from E.

Harders study. Here again, the diagrams inside the topmost box for each series condition represent the area under study, from point x on the diagrams left to point y on the right. The power system represented by the box is open between x and y to insert the circuits shown inside the box. Points x and y can be any distance apart, as long as there is no tap or other system connection between them. The positive, negative, and zero sequence interconnections for the discontinuity shown in the top box are illustrated in the three boxes below it.

Simultaneous faults require two sets of interconnec- tions from either Figures or or both. As shown in Figure , ideal or perfect transformers can be used to isolate the two restrictions. It is sometimes necessary to use two transformers as shown in Figure f.

In this case, the rst transfor- mer ratios 1: These can be replaced by an equivalent transformer with ratios 1: Figure a, for example, represents an open phase-a conductor with a simultaneous fault to ground on the x side. The sequence networks are connected for the open conductor according to Figure j, with three 1: The manual calcula- tions required, which involve the solution of simultaneous equations, may be quite tedious.

To simplify this reduction, with negligible effect on the results, the following basic assumptions are some- times made: All generated voltages are equal and in phase. All resistance is neglected, or the reactance of machines and transformers is added directly with line impedances. All shunt reactances are neglected, including loads, charging, and magnetizing reactances. All mutual reactances are neglected, except on parallel lines. By using these assumptions, the positive sequence network can be drawn with a single-source voltage V an connected to the generator impedances by a bus.

If voltages are different, either the voltages must be retained in the network or Thevenin theorem and superposition must be used to reduce the network and calculate fault currents and voltages. Note that for the series unbalances of Figure , a difference in Figure Double phase-to-ground fault and its sequence network connections. Figure Phase-to-phase fault and its sequence network connections.

Phasors, Polarity, and Symmetrical Components 29 voltageeither magnitude, phase angle, or bothis required for current to ow. The single-sequence impedances Z 1 , Z 2 and Z 0 of Figures through will be different for each fault location because of the different network reduc- tions.

During the network reduction, the distribution of currents in the various branches should be calculated, both as a check and to determine the current ow through the relays involved in a fault. These distribu- tion factors are calculated with the assumption that 1 per unit current ows in these single-sequence impe- dances at the fault or point of discontinuity.

Network reduction calculations for the system of Figure are illustrated in Figures , , and 2- In these gures, X 1 , X 2 , and X 0 are the impedances between the neutral bus and the fault at bus G. I 1 , I 2 , and I 0 are all assumed to be equal to 1 per unit. Analog or digital studies should be tailored to produce outputs that allow each branch current in Figure Sequence network interconnections for shunt balanced and unbalanced conditions.

For single-phase-to- ground faults, 3I 0 is required for relays. When using the computer for sequence network reduction, the impedance data are input for the positive and zero sequence networks, along with bus and fault node points. The network is then solved for three-phase and single-phase-to-ground faults. Tabu- lated printed data are provided for phase-a fault current and three-phase fault voltages, along with the corresponding 3I 0 , 3V 0 values for the phase-to-ground fault.

I 2 and V 2 values should also be obtained for negative sequence relays. These voltage and current values are needed for not only faults near the relay, but also those several buses or lines away. Among the operating conditions normally considered are maximum and minimum generation, selectedlines out of service, andline-end faults where the adjacent breaker is open. This information allows the correct relay types and settings to be selected in a minimal amount of time for the entire power system.

The following steps must be performed for calculat- ing fault currents and voltages: Obtain a complete single-line diagram for the entire system, including generators, transformers, and transmission lines, along with the positive, negative, and zero sequence impedances for each component. Prepare a single-line impedance diagram from the system diagram or establish the nodes in a digital study for the positive, negative, and zero sequence networks.

Reduce the impedance values of all network branches to a common base. Values may be expressed as per unit on a common kVA base or as ohms impedance on a common voltage base.

Figure Sequence network interconnections for series balanced and unbalanced conditions. Phasors, Polarity, and Symmetrical Components 31 Obtain, or have the computer obtain, the equivalent single impedance of each sequence network, current distribution factors, and equivalent source voltage for the positive-phase sequence network.

All quantities must be referred to the proper base.

Interconnect the networks or utilize the computer program to represent the fault type involved, and calculate the total fault current at the fault.

Determine the current distribution and voltages as required in the system. Total fault current is seldom of use as relays generally see a fraction of that current except for radial circuits. Alternatively, they Figure Representations for simultaneous unbalances. All the impedances have been reduced to a common base, as indicated in the diagram. The positive sequence network for this system is shown in Figure , the zero sequence network in Figure The negative sequence network is equal to Figure , except that V an is not present.

To perform this sample calculation of a phase-to- ground fault on the bus at station D, the networks must be reduced to a single reactance value between the neutral bus and fault point. Of the several delta loops, at least one must be converted to wye-equivalent in order to reduce the networks.

After one loop is chosen arbitrarily , the equivalent X, Y, Z branches for an equivalent wye are dotted in as shown in Figures and The X, Y, Z conversion from delta to wye- equivalent is a simple process: The X branch of the wye-equivalent is the product of the two delta reactances on either side divided by the sum of the three delta impedances. The same relation applies to Figure Network reduction for example system Figure fault at bus G. Figure Network reduction and current distribution.

Figure Final network reduction for fault at bus G in Figure Figure Single line diagram for a typical loop-type power system. Thus, in Figures and , the networks are reduced as follows: Positive and negative sequence networks Zero sequence network X 1 62 j Since the two upper branches of each network are in parallel, they can be reduced as follows: Positive and negative sequence networks Zero sequence network 0: Figure Zero sequence network reduction for the system of Figure The remaining branches are in parallel and can also be reduced: These factors are expressed as the ratio of each term in the numerator and denominator.

Determining these fac- tors provides a convenient check on the calculations, since the sum of the two fractions must be 1. Distribution factors can be determined by working back through the reduction.

The factors should be written on the diagrams as shown in Figure The distribution factors for the upper parallel branches of Figure c are determined as follows: Positive sequence network The delta current distribution factors are obtained from the X, Y, Z equivalents.

The conversion technique is straightforward: The voltage drop across two of the wye branches is equivalent to the drop across the delta branch. Calculating from Figure c, we obtain Positive sequence network 0: The three networks are connected in series for the phase-to-ground fault Fig. For convenience, the sequence currents are calculated in per unit values: Figure Per unit current distribution for AG fault at D. These currents may be expressed in either per unit or ampere values.

Currents in the fault are calculated for each phase as follows: I 1 I 0 0 For each branch, the per unit positive, negative, and zero sequence currents can then be used to determine the individual phase currents by using Eqs. These are recorded in Figure Next, the sequence and phase voltages at each bus are determinedas inFigure It is convenient tocalculate the voltages in per unit values. Note that the impedances listed in Figure appear in percent, rather than ohms, and may be converted easily to per unit.

In the following calculations, the values in parenth- eses are volts, converted from the per unit values for the kV system of Figure V line-to-neutral 1: V ag j0: All the distributed current and voltage values for the system are displayed in Figure In this example, only a kV system fault, with its currents and voltages, was involved. The effect of the phase shift through the transformer banks could not, however, have been neglected if currents and voltages were required for the opposite side of the power transformers.

If the transformer bank is wye-connected on the high-voltage side, as shown in Figure , the general equations for one phase are I A nI a I c V an nV An V Bn nV AB The lowercase subscripts represent high-side quantities and the capital letter subscripts low-side quantities. In Figure Current and voltage distribution for a single phase-to-ground fault at bus D of the system of Figure Phasors, Polarity, and Symmetrical Components 37 the balanced or symmetrical transformer bank, the sequences are independent.

Consequently, positive sequence only is rst applied to Eqs. If a power transformer bank is connected delta on the high-voltage side, as shown in Figure , the general equations for one phase are I a 1 n I A I B V A 1 n V a V c Applying only positive sequence quantities to Eqs.

In either case, the positive sequence quantities are shifted in one direction, while the negative sequence quantities are shifted in the opposite direction. Zero sequence quantities are not affected by phase shift. These either pass directly through the bank or, more commonly, are blocked by the connections. Thus, in a wye-delta bank, zero sequence current and voltage on one side cannot pass through the bank to the other side.

With a ground fault, current ows in not only the faulted phase a, but also the unfaulted b and c phases. The positive and zero sequence distribution factors on any loop system will be different. Conse- quently, the positive, negative, and zero sequence currents will not add up to zero in the unfaulted phases. On a radial system one with a source at one end only for both the positive and zero sequences , the three network distribution factors will all be equal to 1.

For a phase-a-to-ground fault on these circuits, I b equals I c , which equals 0. In practice, only 3I 0 and related 3V 0 ; V 2 , and I 2 values would be recorded for a phase-to-ground fault. The phase currents and voltages shown in Figure were provided for academic purposes. The reason for showing 3I 0 , rather than the faulted phase current, can be seen from Figure In most circuits, there is a signicant difference between the I a and 3I 0 currents in any loop network.

In a radial system, however, I a is equal to 3I 0 and ground relays operate on 3I 0. On phase-to-ground faults, the phase relays will receive current and may start to operate. Coordination between ground and phase relays is usually not necessary. The principal reason there are so few coordination problems is that phase relays must be set above load 5 A secondary , whereas ground relays are conventionally set at 0. Phasors, Polarity, and Symmetrical Components 39 not miscoordinate with the phase relays.

If higher ground settings are used, the likelihood of miscoordi- nation is increased. Under any fault condition, the total current owing into the ground must equal the total current owing up the neutrals. With an autotransformer, however, current can ow down the neutral. In this case, the fault current plus the autotransformer neutral current equals the current up the other transformer neutrals.

The convention that current ows up the neutral when current is owing down into the earth at the fault has given rise to the idea that the grounded wye-delta transformer bank is a ground source, a source of zero sequence current. This long-established idea is not, in fact, correct. The fault is the true source.

It is a converter of positive sequence into negative sequence and, for ground faults, into zero sequence current. This is illustrated by a voltage plot for various faults on a simple system Fig.

For simplicity, assume Z 1 equals Z 2 equals Z 0. During faults, the voltage inside the generators does not change unless the fault persists long enough for the internal ux to change. No appreciable voltage change should occur in high- or medium-speed relaying. For a solid three-phase fault, the voltage at the fault is zero. Therefore, high positive sequence-phase currents ow to produce the gradient shown in the plot of Figure For a phase-to-phase fault, negative sequence voltage is produced by the fault itself.

Negative sequence current, then, ows through- Figure Voltage gradient for various types of faults. The same general conditions also apply to phase-to-ground faults, except that since V a is zero, V 2 and V 0 are negative. In summary, the positive sequence voltage is always highest at the generators or sources and lowest at a fault. In contrast, negative and zero sequence voltages are always highest at the fault and lowest at the sources.

The phasor diagrams of Figure illustrate the same phenomena, from a different viewpoint. In a three-phase fault, the voltages collapse symmetrically, except inside the generator. The three currents have a large symmetrical increase and lagging shift of angle. Other phase faults shown in Figure are characterized by the relative collapse of two of the phase-to-neutral voltages, compared to the relatively normal third phase-to-neutral voltage. Two of the phase currents have a large lagging increase.

For a single-phase-to-ground fault, on the other hand, one phase-to-neutral voltage is collapsed relative to the other two phases. Similarly, one phase current has a large value and lags the line-to-ground voltage.

With wye-delta transformers between the fault and measurement point, the positive sequence quantities shift in one direction, and the negative sequence quantities shift in the opposite direction.

As a result, a phase-to-ground fault on the wye side of a bank has the appearance of a phase-to-phase fault on the delta side. Figures and offer a nal look at sequence currents and voltages for faults. Note that the positive sequence currents and voltages, shown in the left-hand columns, have approximately the same phase relations Figure Sequence currents for various faults.

Figure Sequence voltages for various faults. Phasors, Polarity, and Symmetrical Components 41 for all types of faults. At the fault are various nonsymmetrical currents and voltages, as shown in the far right-hand column.

The negative and, some- times, the zero sequence quantities provide the transi- tion between the symmetrical left-hand column and nonsymmetrical right-hand column.

These quantities rotate and change to produce the nonsymmetrical, or unbalanced, quantity when added to the positive sequence. These phasors can be constructed easily by remembering which fault quantity should be mini- mum or maximum. In a phase c-a fault, for example, phase-b current will be small. Thus, I b2 will tend to be opposite I b1. Since phase-b voltage will be relatively uncollapsed, V b1 and V b2 will tend to be in phase.

After one sequence phasor is established, the others can be derived from Eq. A number of other protective relays use combinations of the sequence quantities, as summar- ized in Table A zero sequence 3I 0 current lter is obtained by connecting three current transformers in parallel. A zero sequence 3V 0 voltage lter is provided by the wye-grounded-broken-delta connection for a voltage transformer or an auxiliary. Positive and negative sequence current and voltage lters are described in Chapter 3.

Because a number of these fault-detecting or decision units are used in a variety of relays, they are called basic units. Basic units fall into several categories: Combinations of units are then used to form basic logic circuits applicable to protective relays. When the current or voltage applied to the coil exceeds the pickup value, the plunger moves upward to operate a set of contacts. The force F which moves the plunger is proportional to the square of the current in the coil. The plunger units operating characteristics are largely determined by the plunger shape, internal core, magnetic structure, coil design, and magnetic shunts.

Plunger units are instantaneous in that no delay is purposely introduced. Typical operating times are 5 to 50 msec, with the longer times occurring near the threshold values of pickup.

The unit shown in Figure a is used as a high- dropout instantaneous overcurrent unit. The steel plunger oats in an air gap provided by a nonmag- netic ring in the center of the magnetic core. When the coil is energized, the plunger assembly moves upward, carrying a silver disk that bridges three stationary contacts only two are shown. A helical spring absorbs the ac plunger vibrations, producing good contact action.

The pickup range can be varied from a two-to-one to a four-to-one range by the adjusting core screw. The more complex plunger unit shown in Figure b is used as an instantaneous overcurrent or voltage unit.

An adjustable ux shunt permits more precise settings over the nominal four-to-one pickup range. This unit is relatively independent of frequency, operating on dc, Hz, or nominal Hz frequency. It is available in high- and low-dropout versions. The armature is hinged at one side and spring-restrained at the other. When the associated electrical coil is energized, the armature moves toward the magnetic core, opening or closing a set of contacts with a torque proportional to the square of the coil current.

The pickup and dropout values of clapper units are less accurate than those of plunger units. Four clapper units are shown in Figure Those illustrated in Figures a and b have the same general design, but the rst is for dc service and the second for ac operation. In both units, upward movement of the armature releases a target, which drops to provide a visual indication of operation the target must be reset manually.

The dc ICS unit Fig. The ac IIT unit Fig. It is equipped with a lag-loop to smooth the force variations due to the alternating current input.

Its adjustable core provides pickup adjustment over a nominal four-to-one range. The SG Fig. The SG has provisions for four contacts two make and two break , and the MG will accept six. The AR clapper unit Fig. Figure Plunger-type units.

Figure Four clapper units. A permanent magnet across the structure polarizes the armature-gap poles, as shown. The nonmagnetic spacers, located at the rear of the magnetic frame, are bridged by two adjustable magnetic shunts.

This arrangement enables the magnetic ux paths to be adjusted for pickup and contact action. With balanced air gaps Fig. With the gaps unbalanced Fig. The resulting polarization holds the armature against one pole with the coil deener- gized. The coil is arranged so that its magnetic axis is in line with the armature and at a right angle to the permanent magnet axis. Current in the coil magnetizes the armature either north or south, increasing or decreasing any prior polarization of the armature.

If, as shown in Figure b, the magnetic shunt adjust- ment normally makes the armature a north pole, it will move to the right. Direct current in the operating coil, which tends to make the contact end a south pole, will overcome this tendency and the contact will move to the left.

Depending on design and adjustments, this polarizing action can be gradual or quick. The left-gap adjustment Fig. Some units use both an operating and a restraining coil on the armature. The polarity of the restraint coil tends to maintain the contacts in their initial position. Current of sufcient magnitude applied to the operating coil will provide a force to overcome the restraint, causing the contacts to change position.

A combination of normally open or normally closed contacts is available. These polar units operate on alternating current through a full-wave rectier and provide very sensitive, high-speed operation on very low energy levels. The operating equation of the polar unit is K 1 I op K 2 I r K 3 f where K 1 and K 2 are adjusted by the magnetic shunts; K 3 is a design constant; f is the permanent magnetic ux; I op is the operating current; and I r is the restraint current in milliamperes.

The induction disc unit Fig. The cylinder unit see Fig. Modern units, however, although using the same operating principles are quite different. All operate by torque derived from the interaction of uxes produced by an electromagnet with those from induced currents in the plane of a rotatable aluminum disc.

The E unit in Figure a has three poles on one side of the disc and a common magnetic member or keeper on the opposite side. The main coil is on the center leg. Current I in the main coil produces ux, which passes through the air gap and disc to the keeper. A small portion of the ux is shunted off through the side air gap.

A short- circuited lagging coil on the left leg causes f L to lag both f R and f T , producing a split-phase motor action.

The phasors are shown in Figure Flux f T is the total ux produced by main coil current I. The three uxes cross the disc air gap and induce eddy currents in the disc. These eddy currents react with the pole uxes and produce the torque that rotates the disc. With the same reference direction for the three uxes as shown in Figure b, the ux shifts from left to right and rotates the disc clockwise, as viewed from the top.

There are many alternative versions of the induction disc unit. The unit shown in Figure , for example, may have a single current or voltage input. The disc always moves in the same direction, regardless of the direction of the input. If the lag coil is open, no torque will exist. Other units can thus control torque in the induction disc unit.

Most commonly, a directional unit is connected in the lag coil circuit. When the directional units contact is closed, the induction disc unit has torque; when the contact is open, the unit has no torque. Induction disc units are used in power or directional applications by substituting an additional input coil for the lag coil in the E unit. The phase relation between the two inputs determines the direction of the operating torque. A spiral spring on the disc shaft conducts current to the moving contact.

This spring, together with the shape of the disc an Archimedes spiral and design of the electromagnet, provides a constant minimum operating current over the contact travel range. A permanent magnet with adjustable keeper shunt Figure Induction disc unit. Figure Phasors and operations of the E unit induction disc. The spring tension, damping magnet, and magnetic plugs allow separate and relatively independent adjustment of the units inverse-time current characteristics.

Shown in Figure , the basic unit used for relays has an inner steel core at the center of the square electromagnet, with a thin-walled aluminum cylinder rotating in the air gap. Cylinder travel is limited to a few degrees by the moving contact attached to the top of the cylinder and the stationary contacts. A spiral spring provides reset torque. Operating torque is a function of the product of the two operating quantities applied to the coils wound on the four poles of the electromagnet and the sine of the angle between them.

The torque equation is T KI 1 I 2 sin f 12 K s where K is a design constant; I 1 and I 2 are the currents through the two coils; f 12 is the angle between I 1 and I 2 ; and K s is the restraining spring torque.

Different combinations of input quantities can be used for different applications, system voltages or currents, or network voltages. DArsonval Units In the DArsonval unit, shown in Figure , a magnetic structure and an inner permanent magnet form a two-pole cylindrical core.

A moving coil loop in the air gap is energized by direct current, which reacts with the air gap ux to create rotational torque. The DArsonval unit operates on very low energy input, such as that available from dc shunts, bridge networks, or rectied ac. The unit can also be used as a dc contact-making milliammeter or millivoltmeter. As the temperature changes, the different coefcients of thermal expansion of the two metals cause the free end of the coil or strip to move. A contact attached to the free end will then operate based on temperature change.

These networks, also known as sequence lters, are widely used. Basic Relay Units 47 I b , and I c inputs. Similarly, the secondaries of three- phase voltage transformers, connected in series with the primary in grounded wye, provide 3V 0. By using Thevenins theorem, these three-phase networks can be reduced to a simple equivalent circuit, as shown in Figure b. V F is the open circuit voltage at the output, and Z the impedance looking back into the three-phase network.

Z s is the self-impedance of the three-winding reactors secondary with mutual impedance X m. The open circuit voltage Fig. In some applications, the currents I b and I c are interchanged, changing Figure DArsonval-type unit. Figure Composite sequence current network. With switch r closed and switch s open, the zero sequence response of Eq.

The switches r and s are used in Figure as a convenience for description only. Several typical sequence network combinations are given in Table A network in common use is shown in Figure Since this network is connected phase to phase, there is no zero sequence voltage effect.

The network is best explained through the phasor diagram Fig. By design, the phase angle of Z R is lagging. For convenience, consider switches s to be closed and switches r open. Impedance Z R is thus connected across voltage V ab , and the autotrans- former across voltage V bc. With only positive sequence voltages Fig. The drop V by1 across the auto- transformer to the tap is in phase with voltage V bc across the entire transformer.

The tap is chosen so that jV xb1 j jV by1 j. The lter output V xy V F is the phasor sum of these two voltages. With only negative voltages applied Fig. Figure Sequence voltage network. Figure Phasor diagrams for the sequence voltage network of Figure with s closed and r open.

Thus, this is a positive sequence net- work. A negative sequence network can be made by reversing the b and c leads or, in Figure , by opening s and closing r. Then Figure a conditions apply to a negative sequence, giving an output V F ; Figure b conditions apply to a positive sequence with V F 0. This interchange of b and c leads to either the current or voltage networks offers a very convenient technique for checking the networks.

For example, the negative sequence current network should have no output on a balanced power-system load but by interchanging the b and c leads it should produce full output on test.

These components have been designed into logic units used in many relays. Before these logic units are described in detail, the semiconductor components and their characteristics will be reviewed Fig.

Relays use silicon-type components almost exclusively because of their stability over a wide temperature range. The device manifests a voltage drop for conduction in the forward direction of approximately 0. The limit of voltage to be applied in the reverse direction is dened by the rating of the diode. Failure of the diode is expected if a voltage in excess of the rating is applied in the reverse direction.

These devices are used in dc circuits to block interaction between circuits, for ac test circuits to generate a half-wave rectied current wave shape, or as a protective device around a coil to minimize the voltage associated with coil current interruption.

If the current is limited to within rated values, the diode recovers its nonconducting characteristics when the reverse voltage falls below the zener value. They are used for surge protection, voltage-regulating functions, and other applications in which a distinct conduction level is desired. Where conduction is desired in both directions with a threshold at a level at which conduction occurs, the back-to-back zener Fig. The characteristics of these devices are essentially the same in both the forward and reverse direction.

It has a voltage-dependent nonlinear charac- teristic. The thermistor depicted in Figure e is a nonlinear device whose resistance varies with tempera- ture. For this function, it is shifted from a nonconducting to conducting state by the base current I b. The transistor Figure Semiconductor components and their charac- teristics. The emitter current I e is the sum of I b and I c.

Very small values of I b are able to control much larger values of I c and I e Fig. The thyristor is also known as a silicon-controlled rectier SCR. With forward vol- tage applied, the thyristor will not conduct until gate current I g is applied to trigger conduction.

The higher the gate current, the lower the anode-to-cathode voltage V F required to start anode conduction. After conduction is established and the gate current is removed, the anode current I F continues to ow.

The minimum anode current required to sustain conduc- tion is called the holding current I H. When V e reaches the peak value V p , the device conducts and passes current I e. Current will continue to ow as long as V e does not fall below the minimum value V v. The unijunction transistor is used for oscillator and timing circuits. Figure The transistor and equivalent electrical sym- bols.

Figure Typical characteristic curves of transistor. Figure The thyristor and its characteristics. Basic Relay Units 51 4. A logic unit has only two states: Two logic conventions are used to indicate the voltages associated with the 0 and 1 states.

In normal logic, 0 is equivalent to zero voltage and 1 to normal voltage. In reverse logic, the corresponding voltage equivalents are reversed; 0 is equivalent to normal voltage and 1 to zero voltage. In positive logic, inputs and outputs are positive; in negative logic, both inputs and outputs are negative. Relay systems normally use positive logic, although some elements may use negative signal inputs and outputs.

Logic units are shown diagrammatically in their quiescent state, that is, the normal or at-rest state. The quiescent state corresponds to the normally deenergized representation in electromechanical relay circuitry. Two sets of symbols are in common use in the United States. The European practice is similar to this. Convention dictates that inputs are shown on the left-hand side and outputs on the right-hand side.

When a logic function has only two inputs, its output is usually simple to determine. For three or more inputs, particularly with combination logic functions, a logic or truth table offers a convenient method of determining the output. A logic table for a function with three inputs and one output is shown in Figure The table lists all possible combinations of zeros and ones for the inputs.

Each output could be 0 or 1, depending on the function. Detailed circuit descriptions will be kept to a minimum. For simplicity, the diagrams will show only two inputs per function and include electromechanical contact equivalents. The simplest type consists of forward-biased diodes and resistors Fig. The symbolic representation and electromechanical equivalents for this unit are given in Figure b, the logic table in Figure c.

The forward-biased diodes shunt the output terminal, and Figure Examples of logic symbols. Figure Example of logic table.

Figure The unijunction transistor and its character- istics. Since inputs are either 0 or 1, there is no in- between state that would allow partial output voltage. Thus, the output is either 0 or 1, as shown in the logic table. Three variations of the AND element are provided in Figure Again, the simplest type of unit consists of resistors and diodes. The symbolic representation and electromechanical equivalents for the unit are illustrated in Figure b, the logic table in Figure c.

Since the diodes are not biased, an input voltage applied to any input will produce an output voltage at X. The simplest relay system. Incorrect operation shown in Figure Generators ing a clear. The cause transformers supplying the relay or relay system of incorrect operation may be 1 poor application. Buses dividends—particularly when assistance from others is. A typical power system and its zones of protection are 2 no conclusion. Since operation includes current transformers.

Incorrect tripping of circuit breakers not associated Protection in each zone is overlapped to avoid the with the trouble area is often as disastrous as a failure possibility of unprotected areas. The purpose incorrect settings. Transmission and distribution circuits 3. Both types have high reliability records. As previously indicated.

The location of the current may be either failure to trip or false tripping. Although develop- 1. Equipment that can cause an incorrect protection within the guidelines outlined above. Figure The principle of overlapping protection around a circuit breaker. Introduction and General Philosophies 5 Transformer connections are particularly impor- tant.

Information on the following associated or To determine the degree of protection required. As new relay systems will often be required to operate with or utilize parts of the existing relaying.

For ground-relaying. For ground relaying. These data will provide answers to the Existing operating procedures and practices and following types of questions. Three-phase faults. Is pilot. This diagram should show in may operate before another. Is simultaneous Degree of protection required tripping of all breakers of a transmission line required? Fault study Is instantaneous reclosing needed? Are generator Maximum load and current transformer locations neutral-to-ground faults to be detected?

Whenever possible. In fact. The circuit shown is for the CR directional for switchboard mounting. Line and transformer impedances. A typical switchboard relay is shown in Figure The relay chassis. In the Flexitest case. In the following discussion. Transformer Data. It is desirable. The CR problem is essentially solved when the available directional time overcurrent relay in the Flexitest case.

While the example shown is an electromechanical Figure Typical internal schematic for a switchboard- relay. The important designations in the ac schematic for the relay.

Maximum loads should be consistent with the fault data and based on the same system conditions. In any event. Figure Typical ac schematic for a switchboard- mounted relay. Since these relays involve more relay. CRC ground directional time overcurrent relay of Figure Lower-voltage batteries are not recommended for tripping service when long trip leads are involved. The connections are for the CR phase and the breaker.

Line voltage cannot be used directly since. In complex and sophisticated circuitry. Detailed logic diagrams Three phase relays and one ground relay are shown plus ac and dc schematics are also required for a protecting this circuit. The connections are for three phase type CR and one CRC ground directional time overcurrent relays of Figure applied to trip a circuit breaker. In other applications. This device is simply a capacitor charged. An example of this arrangement is presented in Figure Any one could trip the complete view of the action to be expected from these associated circuit breaker to isolate the trouble or relays.

A station battery. A block diagram provides understanding of the deenergized. When the relay contacts close. Figure is a block diagram for the A typical control circuit is shown in Figure Relay polarity and terminal numbers 5. Introduction and General Philosophies 7 Figure Typical dc schematic for a switchboard- mounted relay. MDAR microprocessor relay. Such breakers have control circuits similar to those shown in Figure In this diagram.

The scheme shown includes red light supervision of Figure A typical rack type relay. The SBFU static the trip coil. A typical circuit diagram is shown in Figure Table compiles a few of the differences. C RST breaker showing the tripping and closing circuits. These distinctions are important. These technical tools are used 2. The system outlined below is In addition to a general knowledge of electrical power standard with most relay manufacturers. A vector is stationary positive half-cycle of the ac wave.

Confusion generally results when the circuit diagram is omitted or Figure Reference circuit diagram illustrating single the two diagrams are combined. If not. Phasor diagrams must be accompanied by a circuit diagram. The phasor diagram shows only the magni- tude and relative phase angle of the currents and voltages. Originally called vectors. In relaying. Figure Reference circuit diagram illustrating double The most common reference frame consists of the axis subscript notation.

Phasors must be referred to some reference frame. In much of the world. During the negative half-cycle of the ac wave. Figure b Reference axis and nomenclature for phasors. Vab or Eab is negative. If double subscript notation is used.

The convention for positive rotation is counterclockwise. In this sense. By arbitrary convention. In either case. At a chosen time in this instance at the time at which the phasor has advanced to The phasor diagram therefore represents the various phasors at any given common instant of time.

If locating arrows are used for voltage in the circuit Figure a Phasor generation of sinusoid. Current arrows not required but are of real quantities x and the axis of imaginary quantities usually shown in practice.

Voltage Vab or Eab is positive if terminal a is at a higher potential than terminal b when the ac wave is in the positive half- cycle. For simplicity. The absolute value of the phasor is jcj: The phasor diagram The division law is the inverse of multiplication: E jEjejy1 jEj For reference and review.

I2 equals jIj e: This method is preferred. Expo- gular Complex nential Polar Phasor 2. With this type of diagram. They are not the same.

The voltage from The resulting confusion is apparent when one notes n to g is zero when no zero sequence voltage exists. The in Figure Notation for three-phase systems varies consider- ably in the United States. The letter designations are preferred and used here to Figure Designation of the voltages and currents in a avoid possible confusion with symmetrical components three-phase power system.

In other countries. A typical three-phase system. Figure Alternative closed-type phasor diagram for the Figure Alternative closed system phasor diagram for basic circuit of Figure The alternative closed-system phasor diagram is shown Similarly. The The voltage drop from the polarity-marked to the connection can be changed from one rotation to the non-polarity-marked terminal on the primary other by completely interchanging all b and c connec.

Two such marks are necessary. Other protective relays. Phasor Rotation Phase rotation. Typical polarity markings for a directional unit are shown in Figure In this example. Phasor rotation is. This will be considered in later chapters. This applies irrespec- tive of the maximum sensitivity angle of the relay. Polarity is always associated with directional-type relay units. Of course. The primary function of the directional units is to limit 3. In current and voltage are in phase.

As current Ipq lags or leads this maximum torque position. It has maximum sensitivity when relay the practical application of phasors and polarity.

The char- acteristics discussed below are among the most common. The characteristic of the watt-type unit is as shown in Directional units can conveniently serve to illustrate Figure Directional units are often used to supervise the action of fault responsive devices such as overcurrent units. These highly sensitive units operate on load in the tripping direction. For all Phase Power Systems faults in the operating zone of the relay.

To ensure correct Four types of directional element connections Fig. The proper Figure Directional element connections. Faults on three-phase power position by 20 to One exception 90— connection is one in which the unity power- is in the lower-voltage areas.

Conse- the relationship of the system quantities and to identify quently. All will provide incorrect 4. The nature of the relay referred to is such that Simultaneous faults in two parts of the system are the maximum sensitivity occurs when the system generally impossible to relay properly under all current lags its unity power phase position by With this connection. The angle is that between the unity Phase-to-ground a-g. For example. Since this is both internal and external simultaneously.

The major types and causes connection is the one best suited to most power of failure are listed in Table Unless preceded by or caused by a fault. The Improper manufacturing Improper installation phasor diagrams are shown in Figure a for the Aging insulation phase relays and Figure for a commonly used Contamination ground relay. These abnormal conditions provide the discriminating For phase directional measurements. These condi- tions are. When faults appear drop polarity to nonpolarity by A study of these connections reveals that none is perfect.

The connection is one standard for Two-phase-to-ground a-b-g. Modern power systems. To ensure adequate protection. In Snow or ice addition to insulation failure. Vbg and Vcg become approximately 3 The power factor. There are three classes of grounding: In addition to load.

Several studies indicate 23 to 69 kV: In a symmetrical system. In dances predominate. The current will have an angle of 80 to lag for a phase fault at or near generator units. The angle will be less out in the system. If phase a is 4. Arc resistance is seldom an important factor in At these voltage levels. An ungrounded system is connected to ground through the natural shunt capacitance. If the transformer and generator impe. In contrast. Formulated by Dr.

A number of protective relays are based on symmetrical compo- nents. Fortescue in a classic AIEE paper in The method of symmetrical components is the foundation for obtaining and understanding fault data on three-phase power systems. The voltage at relay engineer can select the proper relays and settings to protect all parts of the power system in a minimum amount of time.

A knowledge of symmetrical components is impor- tant in both making a study and understanding the data obtained.

Although the method and mathematics are quite simple. Wagner and R. It is also extremely valuable in analyzing faults and relay operations.

This skill requires practice and experience. The conditions. Lewis and E. Manual calculations are practical only for simple cases. With this information. Harder added measurably to its development in the s. Three-phase fault data are used for the application and setting of phase relays and single- phase-to-ground fault data for ground relays. Evans in the late s and early s. Relay application requires a knowledge of system 4.

Among the operating conditions to be diagrams shown are for effectively grounded systems. In short. Note all phasors rotate in a From these equations. Vb2 or the negative sequence component of phase-b voltage. For convenience. This reduction can be performed in terms of current. All components are phasor quantities. Va1 is the positive sequence component of phase-a voltage. Since the three phasors in any set are always equal or in magnitude.

Where any possibility of misunderstanding can occur. For the rest of the power- 5. Figure Power system neutral. Note that dropping the phase subscripts should be done with great care. Equations to can be solved to yield the sequence components for a general set of three-phase phasors: From Eq. Note that the b and c components always exist. We positive sequence voltage drops. Neutral is from one another. Interconnections of the three sequence networks allow any series or shunt disconti- nuity to be investigated.

If any sequence component exists by measure- ment or calculation in one phase. These direct-axis values are necessary for The sequence circuits for a number of transformer calculating the short-circuit current value at different banks are shown in Figure The resistance of the windings is very small the transient reactance.

Both winding conventions shown above are in axis reactance gives a good approximation of negative common use. For two-winding current value. The transient reactance value is used for stability short-circuited. Unfortu- others. M medium. Since this generator reactance is invariably greater there are three leakage impedances: This general. The armature winding resistance is small enough to From a practical viewpoint.

Winding The negative sequence reactance of a turbine measured Shorted Open generator is generally equal to the subtransient X00d Impedance from winding winding reactance. Values are available from mers. Typical reactance values for synchronous machin- 5. Except in the area of a fault or general unbalance.

L low. ZHL is measured from the H winding consideration and slow-speed relay application. Since the Any circuit that is not continuously transposed will machine is braced for only three-phase fault current have impedances in the individual phases that differ.

Except for a 1: On a per unit basis. X00d indicates the subtransient reactance. A brief review of these quantities The positive and negative sequence reactances of all is given below for synchronous machinery.

For three-winding and autotransformer banks. X0d winding. In effect. The impedances times after the short circuit occurs. X2 for a salient-pole generator is much higher. Since the sub. One equivalent branch. The tertiary winding or voltage is generally the lowest. This phase shift may sequence self-impedance. In the latter cases. The phase shift in the negative sequence network for the delta-wye transformer is the same amount.

Zero negative. It Generally. This yields a lower current to accommodated. These sequence network diagrams consist of one phase to neutral of the power system. For ANSI standard transformers. Balanced loads The zero sequence impedance is always different may be shown from any bus to the neutral bus. Typical diagrams are illustrated in Figures sequence reactances are the same.

On some autotransformers. As a rule of thumb. The phase shift then. ZH is impedance for a positive and negative sequence. Zero sequence mutual impedance resulting from voltage phase-to-ground voltage by X1 is assumed to Each sequence network is.

Figure shows the generators solidly grounded. In the system. I0 dances must be included. Figure Zero sequence network for example system. Current reference Fig. For all practical ensure that it will be properly subtracted when calculations involving faults or discontinuities remote determining the phase currents. Rotating machinery does not produce zero neutral and ground. First of all. With two exceptions. From this three- phase system diagram.

The phase-b-to-phase-c fault and its sequence connections are shown in Figure From the three-phase diagrams of the fault area. X0 shown in the work connections.

Studies of the latter faults provide useful information for ground relaying. Three-phase faults are the most severe phase faults. Fault studies normally include only three-phase faults and single-phase-to-ground faults.

Figure illustrates a phase-b-c-to-ground fault and its sequence network interconnection. A fundamental study of both series and shunt unbalances was made by E. Note that all the faults shown in Figure Sequence network connections and voltages. Sequence network connections for various types of common faults are shown in Figures through Harder in Z0 Figure Phase-to-ground fault and its sequence net- practically equivalent to X1.

These diagrams do not show fault impedance. For a phase-a-to-ground fault. Based on zero load. In such areas. Since the three-phase fault is balanced. Figures through are also represented in Figure The single-sequence impedance Z1.

The All resistance is neglected. As before. The sequence connections for the series unbalances. As shown in Figure The three lower boxes simultaneous equations. All mutual reactances are neglected. Points times made: Here When manual calculations are performed. Simultaneous faults require two sets of interconnec- tions from either Figures or or both. All shunt reactances are neglected. Note that for the connections. Inside the topmost box for each shunt tions required by Figure f.

The To simplify this reduction. The manual calcula- condition is a four-line representation of the shunt to be tions required. It is sometimes necessary to use two transformers as shown in Figure f.

Figure Double phase-to-ground fault and its sequence Figure a. The sequence networks are connected for In Figure In this case. If voltages are different. By using these assumptions. The single-sequence impedances Z1. Analog or digital studies should be tailored to tion factors are calculated with the assumption that 1 produce outputs that allow each branch current in. Z2 and Z0 of Network reduction calculations for the system of Figures through will be different for each Figure are illustrated in Figures I0L are the per unit distribution factors.

These distribu. During the network reduction. The network is then solved for Obtain a complete single-line diagram for the entire three-phase and single-phase-to-ground faults. Values may be selected lines out of service. When using the computer for sequence network The following steps must be performed for calculat- reduction.

For single-phase-to. I2 and V2 values should also be obtained for Prepare a single-line impedance diagram from the negative sequence relays. This information allows the as ohms impedance on a common voltage base. Among the operating conditions normally Reduce the impedance values of all network considered are maximum and minimum generation.

For the typical loop system shown in Figure Total fault current is combinations of several machines. All quantities must be referred to the proper base.

All the impedances have been reduced to a common base. The positive sequence network for this system is shown in Figure Z conversion from delta to wye- equivalent is a simple process: The X branch of the wye-equivalent is the product of the two delta reactances on either side divided by the sum of the three delta impedances. The same relation applies to Figure Network reduction and current distribution.

The negative sequence network is equal to Figure Z branches for an equivalent wye are dotted in as shown in Figures and To perform this sample calculation of a phase-to- ground fault on the bus at station D.

Figure Final network reduction for fault at bus G in Figure The X. After one loop is chosen arbitrarily. Of the several delta power system. Since the two upper branches of the networks are reduced as follows: Figure Positive sequence network reduction for the system of Figure The networks now reduce to the simpler forms shown the Y and Z branches. Calculating from Figure c. These factors are expressed as the ratio of each term in the numerator and denominator.

Distribution factors can be determined by working back through the reduction. The conversion Figure shows the complete per unit distribution technique is straightforward: The voltage drop across for the original network of Figure The remaining branches are in parallel and can also be reduced: Positive and negative sequence networks Zero sequence network 0: Z equivalents.

Determining these fac- tors provides a convenient check on the calculations. Positive sequence network Positive sequence network The distribution factors for the upper parallel Figure Per unit current distribution for AG fault at D.

The factors should be written on the diagrams as shown in Figure It is convenient to calculate the voltages in per unit values. These are recorded in Figure The effect of the phase shift through the transformer banks could not. All the distributed current and voltage values for the The lowercase subscripts represent high-side quantities system are displayed in Figure Applying only positive sequence quantities to Eqs.

In most standards. On phase-to-ground faults. These either pass directly necessary. Ib 3Va2 equals Ic. The principal reason there are so few through the bank or. The reason for showing 3I0.

Coordination summarized in Table On a radial system one with a source at one n end only for both the positive and zero sequences. Zero sequence quantities are between ground and phase relays is usually not not affected by phase shift. Since through the bank to the other side. The positive and zero sequence distribution factors on any loop system will be different.

In a radial In either case. Ia is equal to 3I0 and ground relays shifted in one direction. If higher converter of positive sequence into negative sequence ground settings are used, the likelihood of miscoordi- and, for ground faults, into zero sequence current. During faults, the voltage the neutrals. No fault current plus the autotransformer neutral current appreciable voltage change should occur in high- or equals the current up the other transformer neutrals.

For a phase-to-phase fault, sequence current. This long-established idea is not, in negative sequence voltage is produced by the fault fact, correct. The fault is the true source. It is a itself. Phasors, Polarity, and Symmetrical Components The same general conditions also apply normal third phase-to-neutral voltage.

Two of the to phase-to-ground faults, except that since Va is zero, phase currents have a large lagging increase. V2 and V0 are negative. For a single-phase-to-ground fault, on the other In summary, the positive sequence voltage is always hand, one phase-to-neutral voltage is collapsed relative highest at the generators or sources and lowest at a to the other two phases.

Similarly, one phase current fault. In contrast, negative and zero sequence voltages has a large value and lags the line-to-ground voltage. In a quantities shift in the opposite direction. As a three-phase fault, the voltages collapse symmetrically, result, a phase-to-ground fault on the wye side of a except inside the generator.

The three currents have bank has the appearance of a phase-to-phase fault on a large symmetrical increase and lagging shift of the delta side.

Note that the positive characterized by the relative collapse of two of the sequence currents and voltages, shown in the left-hand phase-to-neutral voltages, compared to the relatively columns, have approximately the same phase relations.

Figure Sequence currents for various faults. Assumes Figure Sequence voltages for various faults. A the far right-hand column. Positive and negative nonsymmetrical right-hand column. These phasors can be constructed easily by remembering which fault quantity should be mini- mum or maximum. In a phase c-a fault, for example, phase-b current will be small. Since phase-b voltage will be Component Quantities for Their Operation relatively uncollapsed, Vb1 and Vb2 will tend to be Sequence quantities in phase.

After one sequence phasor is established, Device no. Application used the others can be derived from Eq. A number of other protective relays use 46 Phase unbalance voltage V2 46 Phase unbalance current I2 combinations of the sequence quantities, as summar- Blown fuse detection V0 and not I0 ized in Table The force F which moves the plunger is or more fault-detecting or decision units, along with proportional to the square of the current in the coil.

Basic units fall into several categories: Plunger units are instantaneous in that no electromechanical units, sequence networks, solid-state delay is purposely introduced. Typical operating times units, integrated circuits, and microprocessor architec- are 5 to 50 msec, with the longer times occurring near ture. Combinations of units are then used to form the threshold values of pickup. The unit shown in Figure a is used as a high- dropout instantaneous overcurrent unit.

When the coil is energized, the plunger assembly moves upward, Four types of electromechanical units are widely used: A helical spring and thermal units. The air gap provides a ratio of 2. The pickup range can be varied Three types of magnetic attraction units are in from a two-to-one to a four-to-one range by the common use: When the It is available in high- and low-dropout versions.

The armature is hinged at one side and spring-restrained at the other. When the associated electrical coil is energized, the armature moves toward the magnetic core, opening or closing a set of contacts with a torque proportional to the square of the coil current.

The pickup and dropout values of clapper units are less accurate than those of plunger units. Four clapper units are shown in Figure In both units, upward movement of the armature releases a target, which drops to provide a visual indication of operation the target must be reset manually. The dc ICS unit Fig. The ac IIT unit Fig. It is equipped with a lag-loop to smooth the force variations due to the alternating current input.

Its adjustable core provides pickup adjustment over a nominal four-to-one range. The SG Fig. The SG has provisions for four contacts two make and two break , and the MG will accept six. The AR clapper unit Fig. Basic Relay Units Depending on design and adjustments, this polarizing action can be gradual or quick. The left-gap adjustment Fig. Some units use both an operating and a restraining coil on the armature.

The polarity of the restraint coil tends to maintain the contacts in their initial position. A combination of normally open or normally closed contacts is available. Figure Polar-type unit 2. There are two general types of magnetic induction 2.

The induction disc unit Fig. The cylinder unit applied to a coil wound around the hinged armature in see Fig. The nonmagnetic spacers, located at the rear of the magnetic frame, are bridged by two Originally, induction disc units were based on the adjustable magnetic shunts. This arrangement enables watthour meter design. With balanced air gaps Fig. With the gaps with those from induced currents in the plane of a unbalanced Fig.

The E unit in Figure a has through the armature. The coil is arranged so that its magnetic axis is in The main coil is on the center leg.

Current in the coil magnetizes and disc to the keeper. A short- ment normally makes the armature a north pole, it will circuited lagging coil on the left leg causes fL to lag move to the right. Direct current in the operating coil, both fR and fT , producing a split-phase motor action. There are many alternative versions of the induction disc unit. The unit shown in Figure , for example, may have a single current or voltage input.

The disc always moves in the same direction, regardless of the direction of the input. If the lag coil is open, no torque will exist. Other units can thus control torque in the induction disc unit.

Most commonly, a directional unit is connected in the lag coil circuit. Induction disc units are used in power or directional applications by substituting an additional input coil for the lag coil in the E unit.

The phase relation between the two inputs determines the direction of the operating torque. A spiral spring on the disc shaft conducts current to Figure Induction disc unit. A essentially in phase, in the shorted lag coil. Flux fT is permanent magnet with adjustable keeper shunt. The spring contacts. A spiral spring provides reset torque.

The torque equation is. The operation of a cylinder unit is similar to that of an where K is a design constant; I1 and I2 are the currents induction motor with salient poles for the stator through the two coils; f12 is the angle between I1 and windings. Shown in Figure , the basic unit used I2 ; and Ks is the restraining spring torque.

Different for relays has an inner steel core at the center of the combinations of input quantities can be used for square electromagnet, with a thin-walled aluminum different applications, system voltages or currents, or cylinder rotating in the air gap. Cylinder travel is network voltages. The unit can also be used as a dc contact-making milliammeter or millivoltmeter. A contact attached to the free end will then operate based on temperature change.

Static networks with three-phase current or voltage inputs can provide a single-phase output proportional to positive, negative, or zero sequence quantities. In zero sequence networks, three current transformer Figure Cylinder unit. The open circuit voltage Fig. VF is the open circuit voltage at the output. In some applications. The switches r and s are used in Figure as a convenience for description only.

With switch r closed and switch s open. By design. Several typical sequence network combinations are given in Table The drop Vby1 across the auto- transformer to the tap is in phase with voltage Vbc across the entire transformer. Figure Phasor diagrams for the sequence voltage Vxb2 is equal and opposite to Vby2. A network in common use is shown in Figure Since this network is connected phase to phase.

The network is best explained through the phasor diagram Fig. With only positive sequence voltages Fig. With only negative voltages applied Fig. The characteristics of the varistor are shown in Fig. Failure of the diode is expected if a voltage in excess of the rating is applied in the reverse direction. Before these logic conducting state by the base current Ib. These components have been designed For this function. Thermistor interchanging the b and c leads it should produce full output on test.

Then Figure a conditions level is desired. Relays use silicon-type components almost exclusively because of their stability over a wide temperature range. The transistor units are described in detail. These devices are used in dc circuits to block interaction between circuits. If Figure Semiconductor components and their charac- the current is limited to within rated values.

The device manifests a voltage drop for conduction in the forward direction of approximately 0. The characteristics of current or voltage networks offers a very convenient these devices are essentially the same in both the technique for checking the networks.

Where conduction is desired in both directions with Figure b conditions apply to a positive sequence a threshold at a level at which conduction occurs. It has a voltage-dependent nonlinear charac- teristic. They are A negative sequence network can be made by used for surge protection. The emitter below the minimum value Vv. Figure The thyristor and its characteristics. The higher the gate current. Very small values of transistor is used for oscillator and timing circuits.

With forward vol- tage applied. Ib are able to control much larger values of Ic and Ie Fig. Figure The transistor and equivalent electrical sym- bols. After conduction is established and the gate current is removed.

The minimum anode current required to sustain conduc- tion is called the holding current IH. Basic Relay Units 51 Figure Typical characteristic curves of transistor. When Ve reaches the peak is nonconducting until Ib is increased to a value at value Vp. The unijunction current Ie is the sum of Ib and Ic. A logic table for a function with three inputs and one output is shown in Solid-state logic units are combinations of solid-state Figure Figure b.

In solid- state relaying. Detailed circuit descriptions will be kept to a equivalent to normal voltage and 1 to zero voltage. The forward-biased diodes shunt the output terminal. The Logic units are shown diagrammatically in their simplest type consists of forward-biased diodes and quiescent state.