继电保护原理与应用(第三版)3.1-3.5

3.1 Introduction介绍
Phasors and polarity are two important and useful tools in power system protection. They aid in understanding and analyzing of the connections, operation, and testing of relays and relay systems. In addition, these concepts are essential in understanding power system performance during both normal and abnormal operation. Thus, a sound theoretical and practical knowledge of phasors and polarity is a fundamental and valuable resource.
相量和极性是电力系统保护的两个重要工具。它们有助于理解和分析继电器和继电器系统的连接、操作和测试。此外，这些概念对于理解电力系统在正常和非正常运行期间的性能至关重要。因此，良好的相量和极性的理论和实践知识是一个基础和宝贵的资源。
3.2 Phasors相量
The IEEE Dictionary (IEEE 100) defines a phasor as ‘‘a complex number.’’ Unless otherwise specified, it is used only within the context of steady-state alternating linear systems. It continues: ‘‘the absolute value (modulus) of the complex number corresponds to either the peak amplitude or root-meansquare (rms) value of the quantity, and the phase (argument) to the phase angle at zero time. By extension, the term phasor can also be applied to impedance, and related complex quantities that are not time dependent.’’
IEEE字典（IEEE 100）将相量定义为“复数”。除非另有规定，否则仅在稳态交替线性系统的上下文中使用。它继续：“复数的绝对值（模数）对应于量的峰值振幅或均方根（rms）值，相位（参数）对应于零时间的相位角。推而广之，相量一词也可用于阻抗，以及与时间无关的相关复数
In this book, phasors will be used to document various AC voltages, currents, fluxes, impedances, and power. For many years phasors were referred to as ‘‘vectors,’’ but this use is discouraged to avoid confusion with space vectors. However, the former use lingers on, so occasionally a lapse to vectors may occur.
在这本书中，相量将被用来记录各种交流电压，电流，通量，阻抗和功率。多年来，相量被称为“矢量”，但不鼓励这种使用，以避免与空间矢量混淆。然而，前一种用法仍然存在，因此偶尔会出现向量的错误。
3.2.1 Phasor Representation相量表示
The common pictorial form for representing electrical and magnetic phasor quantities uses the Cartesian coordinates with x (the abscissa) as the axis of real quantities and y (the ordinate) as the axis of imaginary quantities. This is illustrated in Figure 3.1. Thus, a point c on the complex plane x–y can be represented as shown in this figure, and mathematically documented by the several alternative forms given in the following equation:
表示电和磁相量的常用图形形式使用笛卡尔坐标，其中x（横坐标）为实数轴，y（纵坐标）为虚数轴。如图3.1所示。因此，复平面x–y上的点c可表示为该图所示，并通过以下等式中给出的几种替代形式进行数学记录：

FIGURE 3.1 Reference axes for phasor quantities: (a) Cartesian xy coordinates. (b) Impedance phasor axes. (c) Power phasor axes. 图3.1相量参考轴：（a）笛卡尔xy坐标。（b） 阻抗相量轴。（c） 电源相量轴。
Sometimes the conjugate form is useful: 有时共轭形式很有用：

where c is the phasor; c* is its conjugate; x is the real value (alternate: Re c or c’ ); y is the imaginary value (alternate: Im c or c”); jcj is the modulus (magnitude or absolute value); and f is the phase angle (argument or amplitude) (alternate: arg c).
式中，c是相量；c*是其共轭；x是实值（备选值：Re c或c’）；y是虚值（备选值：Im c或c”）；jcj是模数（幅度或绝对值）；f是相角（参数或幅度）（备选值：arg c）。
The modulus (magnitude or absolute value) of the phasor is相量的模数（幅值或绝对值）为

From Equation 3.1 and Equation 3.3, 从方程式3.1和方程式3.3中，

3.2.2 Phasor Diagrams for Sinusoidal Quantities正弦量的相量图
In applying the preceding notation to sinusoidal (AC) voltages, currents, and fluxes, the axes are assumed as fixed, with the phasor quantities rotating at a constant angular velocity. The international standard is that phasors always rotate in the counterclockwise direction. However, as a convenience, on the diagrams the phasor is always shown as fixed for the given condition. The magnitude of the phasor c can be either the maximum peak value or the rms value of the corresponding sinusoidal quantity. In normal practice, it represents the rms maximum value of the positive half-cycle of the sinusoid unless otherwise specifically stated.
在将上述符号应用于正弦（AC）电压、电流和通量时，假设轴是固定的，相量以恒定的角速度旋转。国际标准是相量总是逆时针旋转。然而，为了方便起见，在图表上，对于给定的条件，相量总是显示为固定的。相量c的大小可以是相应正弦量的最大峰值或均方根值。在正常实践中，除非另有特别说明，否则它表示正弦波正半周期的rms最大值。
Thus, a phasor diagram shows the respective voltages, currents, fluxes, and so on, existing in the electric circuit. It should document only the magnitude and relative phase-angle relations between these various quantities. Therefore, all phasor diagrams require a scale or complete indications of the physical magnitudes of the quantities shown. The phase-angle reference is usually between the quantities shown, so that the zero (or reference angle) may vary with convenience. As an example, in fault calculations using reactance X only, it is convenient to use the voltage V reference at 908. Then and the j value cancels, so the fault current does not involve the j factor. On the other hand, in load calculations it is preferable to use the voltage V at 08 or along the x-axis so that the angle of the current I represents its actual lag or lead value
因此，相量图显示了电路中存在的各个电压、电流、通量等。它应该只记录这些不同量之间的大小和相对相位角关系。因此，所有相量图都需要显示物理量的刻度或完整指示。相位角参考通常在所示的量之间，因此零点（或参考角）可能随方便而变化。例如，在仅使用电抗X的故障计算中，使用908处的电压V基准是很方便的。然后，j值取消，因此故障电流不涉及j因子。另一方面，在负载计算中，最好使用08处或沿x轴的电压V，以便电流I的角度表示其实际滞后或超前值
Other reference axes that are in common use are shown in Figure 3.1b and Figure 3.1c. For plotting a graph of impedance, resistance, and reactance, the R–X axis of Figure 3.1b is used. Inductive reactance is X and capacitive reactance is X.
其他常用的参考轴如图3.1b和图3.1c所示。绘制阻抗、电阻和电抗图时，使用图3.1b的R–X轴。感性电抗为X，容性电抗为X。
For plotting power phasors, Figure 3.1c is used. P is the real power (W, kW, MW) and Q is the reactive power (var, kvar, Mvar). These impedance and power diagrams are discussed in latter chapters. Although represented as phasors, the impedance and power phasors do not rotate at system frequency.
对于绘制功率相量，使用图3.1c。P为实际功率（W、kW、MW），Q为无功功率（var、kvar、Mvar）。这些阻抗和功率图将在后面的章节中讨论。虽然表示为相量，但阻抗和功率相量不以系统频率旋转。
3.2.3 Combining Phasors
The various laws for combining phasors are presented for general reference:
Multiplication. The magnitudes are multiplied and the angles are added:、
组合相量的各种定律供一般参考：
乘法。大小相乘，角度相加：

Division. The magnitudes are divided and the angles are subtracted: 分部。震级被除以，角度被减去：

Powers. 权力。

3.2.4 Phasor Diagrams Require a Circuit Diagram相量图需要电路图
The phasor diagram, defined earlier, has an indeterminate or vague meaning unless it is accompanied by a circuit diagram. The circuit diagram identifies the specific circuit involved, with the location and assumed direction for the currents, and the location and assumed polarity for the voltages to be documented in the phasor diagram. The assumed directions and polarities are not critical, because the phasor diagram will confirm if the assumptions are correct, and provide the correct magnitudes and phase relations. These two complementary diagrams (circuit and phasor) are preferably kept separate to avoid confusion and errors in interpretation. This is discussed further in Section 3.3.
先前定义的相量图具有不确定或模糊的含义，除非附有电路图。电路图确定了所涉及的具体电路，包括电流的位置和假定方向，以及要记录在相量图中的电压的位置和假定极性。假设的方向和极性并不重要，因为相量图将确认假设是否正确，并提供正确的量级和相位关系。最好将这两个互补图（电路和相量）分开，以避免混淆和解释错误。这将在第3.3节中进一步讨论。
3.2.5 Nomenclature for Current and Voltage电流和电压术语
Unfortunately, there is no standard nomenclature for current and voltage, so confusion can exist among various authors and publications. The nomenclature used throughout this book has proved to be flexible and practical over many years of use, and it is compatible with the polarities of power system equipment.
不幸的是，没有电流和电压的标准命名法，因此在不同的作者和出版物中可能存在混淆。这本书中使用的术语经过多年的使用证明是灵活和实用的，它与电力系统设备的极性兼容。
3.2.5.1 Current and Flux电流和磁通
In the circuit diagrams, current or flux is shown by either (1) a letter designation, such as I or u, with an arrow indicator for the assumed direction of flow; or (2) a letter designation with double subscripts, the order of the subscripts indicating the assumed direction. The direction is thus assumed to be the flow during the positive half-cycle of the sine wave. This convention is illustrated in Figure 3.2a. Thus, in the positive half-cycle, the current in the circuit is assumed to be flowing from left to right, as indicated by the direction of the arrow used with Is, or denoted by subscripts, as with Iab, Ibc, and Icd. The single subscript, such as Is, is a convenience to designate currents in various parts of a circuit and has no directional indication, so an arrow for the direction must be associated with these. Arrows are not required with Iab, Ibc, or Icd, but are often used for added clarity and convenience. It is very important to appreciate that, in these circuit designations, the arrows do not indicate phasors. They are only assumed as directional and locational indicators.
在电路图中，电流或磁通量的表示用（1）字母，如I或u，用箭头表示假定的流动方向；或（2）用双下标表示，用下标的顺序表示假定的流动方向。因此，假设该方向为正弦波正半周期内的流动方向。此惯例如图3.2a所示。因此，在正半周期内，电路中的电流假设为从左向右流动，如is所用箭头方向所示，或如Iab、Ibc和Icd所用下标所示。单个下标（如Is）便于指定电路各部分中的电流，并且没有方向指示，因此方向箭头必须与这些相关联。Iab、Ibc或Icd不需要箭头，但通常用于增加清晰度和方便性。重要的是要认识到，在这些电路名称中，箭头并不表示相量。它们仅被假定为方向和位置指示器。

FIGURE 3.2 Phasor diagram for the basic circuit elements: (a) Circuit diagram showing location and assumed directions of current and voltage drops. I and V are locational and directional indicators, not phasors. (b) Phasor diagrams showing current and voltage magnitudes, and phase relations.
图3.2基本电路元件的相量图：（a）显示电流和电压降位置和假设方向的电路图。I和V是位置和方向指示器，而不是相量。（b） 显示电流和电压大小以及相位关系的相量图。
3.2.5.2 Voltage电压
Voltages can be either drops or rises. Much confusion can result by not clearly indicating which is intended or by mixing the two practices in circuit diagrams. This can be avoided by standardizing to one practice. As voltage drops are far more common throughout the power system, all voltages are shown and are always considered to be drops from a higher voltage to a lower voltage during the positive half-cycle. This convention is independent of whether V, E, or U is used for voltage in many countries. In this book, V is used and as indicated, it is always a voltage drop.
电压可以是下降或上升。如果不能清楚地指出这两种做法的意图，或者在电路图中混用这两种做法，可能会造成许多混乱。这可以通过将标准化为一种做法来避免。由于电压降在整个电力系统中更为常见，因此所有电压都会显示出来，并且总是被认为是在正半周期内从较高电压降到较低电压。在许多国家，这个惯例与电压是V、E还是U无关。在这本书中，V是用来表示，它总是一个电压降。
The consistent adoption of only drops throughout does not need to cause difficulties. A generator or source voltage becomes a minus drop because current flows from a lower voltage to a higher voltage. This practice does not conflict with the polarity of equipment, such as transformers, and it is consistent with fault calculations using symmetrical components.
始终只采用drops并不需要造成困难。由于电流从较低的电压流向较高的电压，发电机或电源电压变成负电压降。这种做法与设备（如变压器）的极性不冲突，并且与使用对称组件的故障计算一致。
Voltages (always drops) are indicated by either (1) a letter designation with double subscripts; or (2) a small plus () indicator shown at the point assumed to be at a relatively high potential. Thus, during the positive halfcycle of the sine wave, the voltage drop is indicated by the order of the two subscripts used, or from the ‘‘’’ indicator to the opposite end of the potential difference. This is illustrated in Figure 3.2a, where both methods are shown. It is preferable to show arrows at both ends of the voltage-drop designations, to avoid possible confusion. Again, it is most important to recognize that both these designations in the circuit diagrams, especially if arrows are used, are only location and direction indicators, not phasors.
电压（总是下降）由（1）带有双下标的字母表示；或（2）在假定处于相对高电位的点处显示的小加号（）指示器表示。因此，在正弦波的正半周期内，电压降由使用的两个下标的顺序表示，或从“”指示器到电位差的另一端。如图3.2a所示，其中两种方法均显示。最好在电压降名称的两端显示箭头，以避免可能的混淆。同样，最重要的是要认识到，电路图中的这两个名称，特别是如果使用箭头，只是位置和方向指示灯，而不是相量。
It may be helpful to consider current as a ‘‘through’’ quantity and voltage as an ‘‘across’’ quantity. In this sense, in the representative Figure 3.2a, the same current flows through all the elements in series, so that Iab Ibc Icd IS. By contrast, voltage Vab applies only across nodes a and b, voltage Vbc across nodes b and c, and Vcd across nodes c and d.
将电流视为“通过”量，将电压视为“穿过”量可能会有所帮助。从这个意义上说，在代表性的图3.2a中，相同的电流流过串联的所有元件，因此IbaIbcIcd为零。相比之下，电压Vab仅适用于节点a和b，电压Vbc适用于节点b和c，而Vcd适用于节点c和d。
3.2.6 Phasor Diagram相量图
With the proper identification and assumed directions established in the circuit diagram, the corresponding phasor diagram can be drawn from the calculated or test data. For the circuit diagram in Figure 3.2a, two types of phasor diagrams are shown in Figure 3.2b. The top diagram is referred to as an open type, where all the phasors originate from a common origin. The bottom diagram is referred to as a closed type, where the voltage phasors are summed together from left to right for the same series circuit. Both types are useful, but the open type is preferred to avoid the confusion that may occur with the closed type. This is amplified in the following section.
通过在电路图中建立正确的标识和假定的方向，可以从计算或测试数据中得出相应的相量图。对于图3.2a中的电路图，图3.2b中显示了两种类型的相量图。顶部的图被称为开放式，其中所有的相量都来自一个共同的原点。下图被称为闭合型，其中电压相量从左到右为同一串联电路求和。这两种类型都很有用，但最好使用开放类型，以避免与封闭类型混淆。这将在下一节中详细说明。
3.3 Circuit and Phasor Diagrams for a Balanced Three-Phase Power System平衡三相电力系统的电路和相量图
A typical section of a three-phase power system is shown in Figure 3.3a. Optional grounding impedances ( ZGN ) and (ZHN) are omitted with solid grounding. This topic is covered in Chapter 7. (RSG) and (RSSG) represent the ground-mat resistance in the station or substation. Ground g or G represents the potential of the true earth, remote ground plane, and so on. The system neutrals n‘ , n or N, and n‘’ are not necessarily the same unless a fourth wire is used, as in a four-wire three-phase system. Upper- or lowercase N and n are used interchangeably as convenient for the neutral designation.
三相电力系统的典型截面如图3.3a所示。实心接地省略了可选接地阻抗( ZGN ) and (ZHN)
本主题在第7章中介绍。(RSG) and (RSSG) 表示车站或变电站的接地网电阻。接地g或g代表真实地球、远地平面等的电位。系统中性点n"，n或n，和n""不一定相同，除非使用第四根导线，如在四线三相系统中。大写或小写N和N可互换使用，以方便中性名称。
The various line currents are assumed to flow through this series section as shown, and the voltages are indicated for a specific point on the line section. These follow the nomenclature that was discussed previously. To simplify the discussion at this point, symmetrical or balanced operation of the three-phase power system is assumed. Therefore, no current can flow in the neutrals of the two transformer banks, so that with this simplification there is no difference of voltage between n’ , n or N, n‘’, and the ground plane g or G. As a result, Van = Vag; Vbn = Vbg; and Vcn = Vcg. Again, this is true only for a balanced or symmetrical system. With this, the respective currents and voltages are equal in magnitude and 1208 apart in phase, as shown in the phasor diagram (see Figure 3.3b), in both open and closed types. The phasors for various unbalanced and fault conditions are discussed in Chapter 4.
如图所示，假定各种线路电流流过该串联段，并为线路段上的特定点指示电压。它们遵循前面讨论过的命名法。为了简化这一点的讨论，假设三相电力系统对称或平衡运行。因此，两个变压器组的中性点中没有电流流动，因此通过这种简化，n"，n或n，n""和接地层g或g之间没有电压差。因此，Van=Vag；Vbn=Vbg；和Vcn=Vcg。同样，这只适用于平衡或对称系统。这样，在开路和闭合类型中，各电流和电压的幅值相等，相距1208，如相量图（见图3.3b）所示。第四章讨论了各种不平衡和故障情况下的相量。
The open-type phasor diagram permits easy documentation of all possible currents and voltages, some of which are not convenient in the closed-type phasor diagram. The delta voltage Vab, representing the voltage (drop) from phase a to phase b, is the same as Van - Vbn. Similarly, Vbc = Vbn - Vcn and Vca = Vcn - Van.
开放式相量图允许轻松记录所有可能的电流和电压，其中一些在封闭式相量图中不方便。表示从a相到b相的电压（压降）的电压增量Vab与Van-Vbn相同。类似地，Vbc=Vbn-Vcn和Vca=Vcn-Van。
As indicated, the closed-type phasor diagram can lead to difficulties. As seen in Figure 3.3b, its shape lends itself mentally to an assumption that the three vertices of the triangle represent a, b, and c phases of the power system, and that the origin 0 represents n = g. Questions arise with this closed-type phasor diagram about why Van = Vag has its phasor arrow as shown, because the voltage drop is from phase a to neutral; similarly for the other two phases. Also why Vab, Vbc, and Vca are pointing as shown, for they are drops from phase a to phase b, phase b to phase c, and phase c to phase a, respectively. It would appear that they should be pointing in the opposite direction.
如图所示，封闭式相量图可能会导致困难。如图3.3b所示，它的形状在心理上有助于一个假设，即三角形的三个顶点代表电力系统的a、b和c相，原点0代表n=g。这个封闭型相量图产生了问题，即为什么Van=Vag有如图所示的相量箭头，因为电压降是从a相到中性点的；其他两相也是如此。还有为什么Vab、Vbc和Vca如图所示指向，因为它们分别是从a相到b相、从b相到c相、从c相到a相的滴。看起来他们应该指向相反的方向。
The phasors shown on this closed phasor diagram (see Figure 3.3b) are absolutely correct and must not be changed. The difficulty is in combining the circuit diagram with the phasor diagram by the mental association of a, b, and c with the closed triangle. The open type avoids this difficulty. This also emphasizes the desirability of having two separate diagrams: a circuit diagram and a phasor diagram. Each serves particular, but quite different, functions.
此闭合相量图（见图3.3b）上显示的相量绝对正确，不得更改。难点在于通过a、b、c与闭合三角形的心理联系，将电路图与相量图结合起来。开放式避免了这个困难。这也强调了有两个单独的图的可取性：一个电路图和一个相量图。每一种服务都有特定的功能，但有很大的不同。

FIGURE 3.3 Phasor diagram for a typical three-phase circuit operating with balanced or symmetrical quantities. (a) Circuit diagram showing location and assumed directions of current and voltage drops. I and V are locational and directional indicators, not phasors. (b) Phasor diagrams showing current and voltage magnitudes and phase relations.
图3.3以平衡或对称量运行的典型三相电路的相量图。（a） 显示电流和电压降位置和假定方向的电路图。I和V是位置和方向指示器，而不是相量。（b） 显示电流和电压大小及相位关系的相量图。
3.4 Phasor and Phase Rotation相量和相位旋转
Phasor and phase rotation are two entirely different terms, although they almost look alike. The AC phasors always rotate counterclockwise at the system frequency. The fixed diagrams, plotted such as in Figure 3.3b, represent what would be seen if a stroboscopic light of system frequency were imposed on the system phasors. The phasors would appear fixed in space, as plotted.
相量和相旋转是两个完全不同的术语，尽管它们看起来几乎一样。交流相量始终以系统频率逆时针旋转。固定图（如图3.3b所示）表示，如果系统频率的频闪灯施加在系统相量上，将会看到什么。如图所示，相量在空间中看起来是固定的。
In contrast, phase rotation or phase sequence refers to the order in which the phasors occur as they rotate counterclockwise. The standard sequence today is: a, b, c; A, B, C; 1, 2, 3; or in some areas r, s, t. In Figure 3.3b, the sequence is a, b, c, as indicated. The IEEE dictionary (IEEE 100) defines only phase sequence; hence, this is preferred. However, phase rotation has been used over many years and is still in practice.
相反，相位旋转或相序是指逆时针旋转时相量出现的顺序。今天的标准顺序是：a，b，c；a，b，c；1，2，3；或者在某些区域r，s，t。在图3.3b中，顺序是a，b，c，如图所示。IEEE字典（IEEE 100）仅定义相序；因此，这是首选。然而，相位旋转已经使用了很多年，并且仍在实践中。
Not all power systems operate with phase sequence a, b, c, or its equivalent. There are several large electric utilities in the United States that operate with a, c, b phase sequence. Occasionally, this sequence is used throughout the system; for others, one voltage level may be a, b, c, and another voltage level, a, c, b. The specific phase sequence is only a name designation that was established arbitrarily early in the history of a company, and it is difficult to change after many years of operation.
并非所有的电力系统都以相序a、b、c或其等效物运行。美国有几家大型电力公司采用a、c、b相序运行。有时，该序列在整个系统中使用；对于其他系统，一个电压等级可以是a、b、c，另一个电压等级可以是a、c、b。特定的相序只是在公司历史早期任意建立的名称名称，在运行多年后很难改变。
Knowledge of the existing phase sequence is very important in threephase connections of relays and other equipment; therefore, it should be clearly indicated on the drawings and information documents. This is especially true if it is not a, b, c. The connections from a, b, c to a, c, b or vice versa can generally be made by completely interchanging phases b and c for both the equipment and the connections.
在继电器和其他设备的三相连接中，了解现有相序非常重要；因此，应在图纸和信息文件中明确说明。如果不是a、b、c，尤其如此。从a、b、c到a、c、b或相反的连接通常可以通过完全互换设备和连接的b和c相来实现。
3.5 Polarity极性
Polarity is important in transformers and in protection equipment. A clear understanding of polarity is useful and essential for the chapters that follow.
极性在变压器和保护设备中非常重要。对极性的清楚理解对于后面的章节是非常有用和必要的。
3.5.1 Transformer Polarity变压器极性
The polarity indications for transformers are well established by standards that apply to all types of transformers. There are two varieties of polarity: subtractive and additive. Both follow the same rules. Power and instrument transformers are subtractive, whereas some distribution transformers are additive. The polarity marking can be a dot, a square, or an X, or it can be indicated by the standardized transformer terminal markings, the practices varying over the years. Polarity designated by an X in this book.
变压器的极性指示由适用于所有类型变压器的标准确定。极性有两种：减极性和加极性。两者遵循相同的规则。电力变压器和仪表变压器是减法的，而有些配电变压器是加法的。极性标记可以是一个点、一个正方形或一个X，也可以通过标准化的变压器端子标记来表示，这些做法多年来有所不同。在这本书中，极性是由X表示的。

FIGURE 3.4 Polarity definitions for transformers: (a) Subtractive polarity. (b) Additive polarity. 图3.4变压器的极性定义：（a）相减极性。（b） 加性极性。
The two fundamental rules of transformer polarity, illustrated in Figure 3.4 appling to both varieties are the following:
1. Current flowing in at the polarity mark of one winding flows out of the polarity mark of the other winding. Both currents are substantially in-phase.
2. The voltage drop from polarity to nonpolarity across one winding is essentially in phase with the voltage drop from polarity to nonpolarity across the other winding(s).
图3.4所示适用于两种类型的变压器极性的两个基本规则如下：
1在一个绕组的极性标记处流入的电流从另一个绕组的极性标记处流出。两个电流基本上是同相的。
2一个绕组从极性到非极性的电压降基本上与另一个绕组从极性到非极性的电压降同相。
The currents through and the voltages across the transformers are substantially in-phase because the magnetizing current and the impedance drop through the transformers are very small and can be considered negligible. This is normal and practical for these definitions.
通过变压器的电流和电压基本上是同相的，因为通过变压器的磁化电流和阻抗降非常小，可以忽略不计。这对于这些定义来说是正常的和实际的。
The current transformer (CT) polarity markings are shown in Figure 3.5. Note that the direction of the secondary current is the same, independent of whether the polarity marks are together on one side or on the other.
电流互感器（CT）极性标记如图3.5所示。请注意，二次电流的方向是相同的，与极性标记是在一侧还是另一侧在一起无关。

FIGURE 3.5 Polarity markings for CTs. 图3.5电流互感器的极性标记。
For CTs associated with circuit breakers and transformer banks, it is a common practice for the polarity marks to be located on the side away from the associated equipment.
对于与断路器和变压器组相关的CT，极性标记通常位于远离相关设备的一侧。
The voltage-drop rule is often omitted in the definition of transformer polarity, but it is an extremely useful tool to check the phase relations through wye–delta transformer banks, or in connecting up a transformer bank for a specific phase shift required by the power system. The ANSI=IEEE standard for transformers states that the high voltage should lead the low voltage by 308 with wye–delta or delta–wye banks. Thus, different connections are required if the high side is wye than if the high side is delta. The connections for these two cases are shown in Figure 3.6. The diagrams below the threephase transformer connection illustrate the use of the voltage-drop rule to provide or check the connections. Arrows on these voltage drops have been omitted (preferably not used), for they are not necessary and can cause confusion.
在变压器极性的定义中，电压降规则通常被忽略，但它是通过Y形-三角形变压器组检查相位关系的一个非常有用的工具，或者在连接变压器组以获得电力系统所需的特定相移时，它是一个非常有用的工具。ANSI=IEEE变压器标准规定，采用Y形三角形或三角形Y形组时，高压应领先低压308。因此，如果高压侧为Y形，则需要不同的连接，如果高压侧为三角形，则需要不同的连接。这两种情况的连接如图3.6所示。三相变压器连接的下图说明了使用压降规则来提供或检查连接。这些电压降上的箭头已被省略（最好不使用），因为它们不是必需的，并且可能导致混淆。
In Figure 3.6a, the check is made by noting that a to n from polarity to nonpolarity on the left-side winding is in phase with A to B from polarity to nonpolarity on the right-side winding. Similarly, b to n (polarity to nonpolarity) is in phase with B to C (polarity to nonpolarity) across the middle transformer, and c to n (polarity to nonpolarity) is in phase with C to A (polarity to nonpolarity) across the lower transformer. From this, by comparing the line-to-neutral voltages on the two sides, it is observed that phase-a-to-n voltage leads phase-A-to-neutral voltage. Accordingly, the wye side would be the high-voltage side if this is an ANSI=IEEE standard transformer.
在图3.6a中，通过注意左侧绕组从极性到非极性的a到n与右侧绕组从极性到非极性的a到B同相进行检查。类似地，b到n（极性到非极性）与中间变压器上的b到C（极性到非极性）同相，C到n（极性到非极性）与下部变压器上的C到A（极性到非极性）同相。由此，通过比较两侧的线对中性点电压，可以看出a-n相电压领先于a-n相电压。因此，如果这是ANSI=IEEE标准变压器，则Y形端将是高压侧。
This same technique of applying voltage drops to Figure 3.6b shows that for this three-phase bank connection the voltage-drop polarity to nonpolarity or phase a to n is in phase with the voltage-drop polarity to nonpolarity or phase A to phase C. Similarly, voltage-drop across phase b to n is in phase with voltage-drop phase B to phase A, and voltage-drop phase c to n is in phase with voltage-drop across phase C to phase B. By comparing similar voltages on the two sides of the transformer, phase-A-to-neutral voltage drop leads the phase-a-to-n voltage drop by 308, so the delta winding would be the highvoltage side if this is an ANSI=IEEE standard transformer bank. This technique is very useful to make the proper three-phase transformer connections from a desired or known voltage diagram or phase-shift requirement. It is a very powerful tool, which is simple and straightforward.
图3.6b中施加电压降的相同技术表明，对于这种三相组连接，电压降极性到非极性或相位a到n与电压降极性到非极性或相位a到相位C同相。同样，相位b到n的电压降与电压降相位b到相位a同相，相c到n的电压降与相c到相B的电压降同相。通过比较变压器两侧的相似电压，相A到中性点的电压降导致相A到n的电压降308，因此如果这是ANSI=IEEE标准变压器组，三角形绕组将是高压侧。这种技术对于根据所需或已知的电压图或相移要求进行适当的三相变压器连接非常有用。它是一个非常强大的工具，简单明了。

FIGURE 3.6 Voltage-drop polarity rule useful in checking or connecting wye–delta transformer banks: (a) Wye-connected side leads, delta-connected side 30°. (b) Deltaconnected side leads, wye-connected side 30°.
图3.6用于检查或连接Y形-三角形变压器组的电压降极性规则：（a）Y形连接侧引线，三角形连接侧30°。（b） 三角形连接侧导线，Y形连接侧30°。
Because the ANSI=IEEE standards have been in existence for several years, most transformer banks in service today follow this standard, except where it is not possible because of preexisting system conditions. Many years ago, in the absence of a standard, many different connections were used. Some of the older references and textbooks reflect this.
由于ANSI=IEEE标准已经存在了好几年，目前大多数运行的变压器组都遵循这一标准，除非由于先前存在的系统条件而不可能。许多年前，在没有标准的情况下，使用了许多不同的连接。一些旧的参考文献和教科书反映了这一点。
3.5.2 Relay Polarity继电器极性
Relays involving interaction between two input quantities from the power system may have the polarity marking that is necessary for their correct operation. There are no standards in this area, so if the polarity of the relay connections is important, the relay manufacturer must both specify the polarity markings and clearly document their meaning. Relays that sense the direction of current (or power) flow at a specific location and, thereby, indicate the direction of the fault, provide a good practical example of relay polarity. Directional units are usually not applied alone, but rather, in combinations with other units, such as fault sensors or detectors. A common practice is to use the output of the directional-sensing unit to control the operation of the fault sensors, which often is an instantaneous or an inverse-time–overcurrent unit, or both units together. Thus, if the current flow is in the desired operating direction (trip direction) and its magnitude is greater than the fault sensor’s minimumoperating current (pickup), the relay can operate. If the current is in the opposite direction (nontrip or nonoperate direction or zone), no operation can occur even though the magnitude of the current is higher than the pickup threshold current.
涉及电力系统两个输入量之间相互作用的继电器可能具有正确操作所需的极性标记。这方面没有标准，因此如果继电器连接的极性很重要，继电器制造商必须指定极性标记并清楚地记录其含义。在特定位置感测电流（或功率）流动方向，从而指示故障方向的继电器，为继电器极性提供了一个很好的实例。定向单元通常不是单独应用的，而是与其他单元（如故障传感器或检测器）结合使用。通常的做法是使用方向感测单元的输出来控制故障传感器的操作，这通常是瞬时或反时限过电流单元，或者两个单元一起。因此，如果电流在所需的工作方向（跳闸方向）并且其大小大于故障传感器的最小工作电流（拾取），则继电器可以工作。如果电流方向相反（非跳闸或非操作方向或区域），则即使电流的大小高于拾取阈值电流，也不会发生操作。
A directional-sensing unit requires a reference quantity that is reasonably constant, against which the current in the protected circuit can be compared. For relays intended to provide operation for phase-type faults, one of the system voltages in Figure 3.3b can be used as a reference. For all practical purposes, most system voltages do not change their phase positions significantly during a fault. In contrast, line currents can shift around 1808 (essentially reverse their direction or flow) for faults on one side of the circuit CTs relative to a fault on the other side of the CTs.
方向感测单元需要一个合理恒定的参考量，与之比较受保护电路中的电流。对于为相位型故障提供操作的继电器，图3.3b中的一个系统电压可用作参考。出于所有实际目的，大多数系统电压在故障期间不会显著改变其相位位置。相反，相对于电流互感器另一侧的故障而言，电路电流互感器一侧的故障线路电流可在1808左右移动（基本上反向或流动）。
Typical polarity indications for three commonly used directional-sensing units are shown in Figure 3.7. This uses the custom of showing several loops for voltage coils and a single loop for current coils, of placing the reference circuit or voltage circuit above the current circuit, and of placing the polarity markings diagonally, all as shown on the relay schematics in Figure 3.7.
三种常用方向感测装置的典型极性指示如图3.7所示。这使用了显示电压线圈的多个回路和电流线圈的单个回路的习惯，将参考电路或电压电路置于电流电路上方，并将极性标记对角放置，所有这些都如图3.7中继电器示意图所示。

FIGURE 3.7 Typical directional relay characteristics.
图3.7典型方向继电器特性。
The reference quantity is commonly called the ‘‘polarizing’’ quantity, especially for ground-fault relaying, where either current polarizing or voltage polarizing is used, or both. The polarity marks (Figure 3.7) are small plus symbols () placed, as illustrated, above one end of each coil, diagonally as shown, or on the opposite diagonal. As shown in Figure 3.5, relay operation is not affected whether the polarity marks are on one diagonal or the other.
参考量通常称为“极化”量，特别是对于接地故障继电器，其中使用电流极化或电压极化，或两者兼有。极性标记（图3.7）为小加号（），如图所示，位于每个线圈一端上方，如图所示为对角线，或位于相反的对角线上。如图3.5所示，无论极性标记在一条对角线上还是在另一条对角线上，继电器的运行都不受影响。
The meaning of the polarity for a specific relay must be stated clearly in words or by a diagram, such as the one shown in Figure 3.7. These show the basic design characteristics of an individual relay, independent of any connection or association with the power system. The terms maximum-torque line and zero-torque line come from the electromechanical designs long used and still common in the industry. With solid-state designs, these would be the operating lines or thresholds, but the well-established terminology no doubt will continue for many years for all types of designs.
必须用文字或图表（如图3.7所示）明确说明特定继电器极性的含义。这些显示了独立于电力系统的任何连接或关联的单个继电器的基本设计特征。最大转矩线和零转矩线这两个术语来源于机电设计，这两个设计在工业中已经使用了很长时间，并且仍然很常见。对于固态设计，这些将是操作线或阈值，但对于所有类型的设计，公认的术语无疑将持续多年。
The interpretation ofrelay polarity isillustrated in Figure 3.7 for three typical electromechanical units. Solid-state units can have adjustments for (1) the maximum-torque angle and (2) the angle limits of the operate zone, but the application and operation is the same for both types. In Figure 3.7a, the maximum-operating torque or energy occurs when the current flows from polarity to nonpolarity (Ipq) and leads by 308 the voltage drop from polarity to nonpolarity (Vrs). The minimum pickup of the directional unit is specified as the maximum-torque or operating condition. As seen, the unit will operate for currents from almost 608 lagging the reference voltage Vrs to almost 1208 leading. The operate (trip, contact close) zone or area is represented by the half plane, bordered on one side by the zero-torque (nonoperating) line and extending in the direction that contains both the reference (polarizing) and operating quantities. Higher-current values will be required when Ipq deviates from the maximum-torque line. The solid-state relays can adjust this torque line for increased sensitivity by adjusting it to the fault line. The operating torque at any angle is a function of the cosine of the angle between the current (Ipq) and the maximum-torque line, as well as the magnitudes of the operating quantities.
图3.7显示了三个典型机电装置的继电器极性解释。固态装置可以调整（1）最大扭矩角和（2）操作区的角度限制，但两种类型的应用和操作是相同的。在图3.7a中，当电流从极性流向非极性（Ipq）并使电压从极性流向非极性（Vrs）下降308时，出现最大工作转矩或能量。定向装置的最小吸合被指定为最大扭矩或工作条件。如图所示，该装置将运行的电流从约608滞后参考电压Vrs到约1208领先。操作（跳闸、触点闭合）区域或区域由半平面表示，一侧以零扭矩（非操作）线为边界，并在包含参考（极化）和操作量的方向上延伸。当Ipq偏离最大扭矩线时，需要更高的电流值。固态继电器可以通过将其调整到故障线路来调整此转矩线路以提高灵敏度。任何角度下的工作转矩是电流（Ipq）和最大转矩线之间夹角的余弦函数，以及工作量的大小。
For ground-fault protection, the 608 unit in Figure 3.7b is used with a 3 V0 reference (see Figure 3.9) and the zero (watt) unit of Figure 3.7c with a 3 I0 current reference (see Figure 3.10). The unit in Figure 3.7c is also used for power or var applications. A typical application isreverse power protection for a generator.
对于接地故障保护，图3.7b中的608单元与3 V0参考（见图3.9）一起使用，图3.7c中的零（瓦）单元与3 I0电流参考（见图3.10）一起使用。图3.7c中的装置也用于功率或无功应用。典型的应用是发电机的反向功率保护。
A similar type of electromechanical directional unit, as in Figure 3.7a, has its maximum-torque angle at 458 leading, instead of 308 leading. Both units are in wide use for phase-fault protection. Solid-state units with an adjustable angle feature can provide a range of angles.
类似类型的机电定向装置，如图3.7a所示，其最大扭矩角为458超前，而不是308超前。这两种装置都广泛用于相位故障保护。具有可调角度功能的固态单元可以提供一系列角度。