12.2.1 Substation

Raising or lowering line voltage is carried out in a substation. In addition, all switching, regulations, monitoring, and protection of the line take place in a substation. The main device for changing voltage is a transformer. The first transformer after a generator is the generator step up (GSU) transformer. Figure 12.1 shows a schematic of a small grid. This grid is fed only by one power plant. Larger grids have numerous feeding power stations. Any later installment to either feed this grid or take energy out of it must do so through a substation. In Figure 12.1, G represents a generator (power plant), GSU stands for generator step up, used for a stepping up transformer just after the generator to raise the generators voltage to that of the line it connects to, and CB denotes circuit breaker. A circuit breaker is an interrupter device as part of the protective means for part or all of a circuit. In case of malfunction or abnormal situations in a circuit, it cuts off electricity.

In addition to transformers a substation is equipped with other devices for various functions. Among the devices in a substation are lightning arresters and air break switches. An air break switch is a switch for high voltages in transmission lines. Because opening and closing of these switches is accompanied by electric arc flash, it is necessary to suppress the arc. In an air break switch, compressed air is used for this purpose. Points A, B, C, and D in Figure 12.1 signify switchgear substations, where no voltage change takes place, but different branches meet.

As can be seen, a grid can have many circuit breakers at various points. Circuit breakers are of various types and they act based on some criteria, the existence of which are sensed by the appropriate sensors somewhere in the circuit. The action of sensors, normally called relaying, is set to

Figure 12.1 General representation of a simple electric grid.

measure and detect overvoltage, undervoltage, overcurrent, overfrequency, underfrequency, overtemperature, open phase, reverse power flow in motors, and so on.

Figure 12.2 Substation. (From Hemami, A., Wind Turbine Technology, 1E ©2012 Delmar Learning, a part of Cengage Learning Inc. Reproduced with permission from www.cengage.com/permissions.)

Figure 12.2 shows a small substation.

Electricity to houses and small buildings is single phase. In North America the nominal voltage for electricity in the buildings is 110 V (but depending on region and municipalities it can be between 110 and 125 V). Nevertheless, for certain appliances that need more power and take more current, such as the kitchen stove (range) and dryer, to lower current, the required voltage is 220 V (nominal). Common practice in North America is to deliver to each house a center-tapped 220 V line. The center-tapped line is grounded and acts as the neutral for building wiring. In this way, while 220 V is available for those devices that need it, the voltage difference between the center line and each other line is 110 V. Wiring for various parts of a house is almost evenly distributed between the two half voltage sources, and the stove and dryer are connected to the full voltage. Figure 12.3 shows the schematics of this wiring. This method of wiring is not practiced in other countries with 220 V electricity.

At this point it is worth noting that the special wiring of the lights for staircases and corridors that can be turned on-off by two switches at the two ends of the pathway. For this purpose, two DPST (double-pole single-throw; see Chapter 5) switches are used. Figure 12.4 shows this wiring scheme. It is also possible to turn a light on and off from more than two points. In this case the middle switch(s) must have four connections; each time, two alternative terminals pairs are connected together.

12.3.1 Overhead Conductors

For transmission lines nowadays the conductors are made of aluminum. Compared with copper, aluminum has less conductivity and less strength. For this reason, for the same power transmission, aluminum conductors must be thicker. Also, for overhead lines their strength can be reinforced by steel.

Theoretically, conductors can be made out of rigid bars. However, for transportation this is impossible and they should be flexible. See Figure 12.5. To increase flexibility of thicker conductors, they are made of several strands. As the current carrying capacity requirement of cables increases, more strands are added, and accordingly more reinforcement is necessary. The customary number of strands in overhead cables is as follows:

Note that each layer has six strands more than the layer inside it. For example, 19 = 1 + 6 + (6 + 6) and 37 = 1 + 6 + 12 + (12 + 6), etc. See Figure 12.6.

There are several categories of aluminum conductors. These are some of the more common aluminum conductors: all aluminum conductor (AAC), all aluminum alloy conductor (AAAC), aluminum conductor alloy reinforced (ACAR), aluminum conductor steel reinforced (ACSR), aluminum conductor steel supported (ACSS), aluminum conductor carbon fiber reinforced (ACFR), and gap-type aluminum conductor steel reinforced (GTACSR).

Figure 12.5 For various reasons including transportation, conductors must be flexible. (From ©2014 Southwire Company, LLC. All rights reserved. PPI reproduction is pursuant to Southwire Company, LLC’s express permission.)

Figure 12.6 Typical bare wire conductor. (Courtesy of General Cable. ©General Cable Technologies Corporation, all rights reserved.)

There are other categories as well. The names are self-explanatory, to some extent, but can change from company to company. Not all conductors have strands with circular cross section. Figure 12.7 shows a cable with oval shape strands. Also, to increase the conductivity of cables for the same cross section, some cables have trapezoid shape strands that form circular layers, which resemble tubes of different diameters inside each other (see Figure 12.8). In this way, more use of space (thus, more conductivity) is made out of the same conductor diameter.

Figure 12.7 Conductor with oval cross section. (From ©2014 Southwire Company, LLC. All rights reserved. PPI reproduction is pursuant to Southwire Company, LLC’s express permission.)

Figure 12.8 Conductor with trapezoid strands. (Courtesy of General Cable. ©General Cable Technologies Corporation, all rights reserved.)

Figure 12.9 Aluminum conductor with composite core. (From ©2014 Southwire Company, LLC. All rights reserved. PPI reproduction is pursuant to Southwire Company, LLC’s express permission.)

More recently, carbon fiber reinforcement cables have been introduced; instead of steel, these cables have strands of carbon fiber composite material in the middle. Carbon fiber composite cable (CFCC) offers desired properties such as less weight and smaller thermal expansion compared with steel. It has 1/5 of the weight and 1/12 of thermal expansion of those of steel, for a price. Figure 12.9 shows a picture of such a cable.

A gap-type conductor consists of a core of high strength steel surrounded by a small gap filled with temperature resistant grease. Stacked around this gap are the trapezoid shape stands of aluminum.

In the seven-strand conductor there are six aluminum strands around one steel cable. The number of steel strands depends on the specification of a particular conductor. For a 37-strand cable, there are 30 aluminum and 7 steel strands, but for the 61-strand cable the number of steel strands can be 7 or 19 and the rest are aluminum. There are always exceptions, and these depend on the manufacturer.

Electric poles and supporting structures come in different forms and sizes, mainly based on the voltage of the power they transmit. Some typical ones are shown in Figure 12.10. For the final distribution to consumers’ poles of approximately 12 m (40 ft) are used, and the height of larger structures varies between 18 and 42 m (60 and 140 ft).

12.3.2 Electrical Properties of Electric Wires and Cables

The principal electrical property of a piece of metal is the resistance R that it shows to the flow of electrical current. It is very important to know the resistance of electric conductors when they are used in electric circuits. Knowing R allows one to determine voltage drop and the energy transformed into heat in parts of an electric circuit, in motor windings, and so forth. In addition to this property, for wires and cables, there is another property that determines how much current is allowed to pass through a conductor. This property, called ampacity (made from the two words “ampere” and “capacity”), defines the current capacity of an inductor based on the heat that is generated owing to electrical current, the structure and material of the conductor, and ambient temperature. It is easy to understand, therefore, that whereas the resistance of a wire can be almost constant, its ampacity depends on the temperature and some other working conditions, and it cannot be a constant.

Resistance of a piece of any material (even an insulator) to the flow of electricity is proportional to its length and inversely proportional to its cross-section area. So, for example, if the length of a wire doubles, its resistance doubles, but if cross section doubles, resistance halves. R depends also on the material; for example, copper is a much better conductor than iron. This is why wires are made up of copper and not iron. The effect of the material is designated by the Greek letter ρ (rho, pronounced ro), which represents the resistance of a piece of the material with specific dimensions. The value ρ is called the specific resistance or resistivity of a substance.

Figure 12.10 Various transmission line supports.

On the basis of the aforementioned discussion, resistance R of a piece of wire can be found from

where ρ is the specific resistance, l is the length, and A is the cross-section area of the wire. Table 12.2 shows the specific resistance of some metals and nonmetals in metric system. The numbers shown are for the measurements made at 20°C. For other temperatures, values must be corrected by using the temperature coefficients in column 3 of Table 12.2.

Table 12.2 also shows the conductivity of materials. Conductivity is the inverse of resistivity. It is an indicator of how conductive a material is. Its unit in metric system is, thus, 1/ohm-meter. Also shown in the table is temperature coefficient, which represents how much the specific resistance of a metal changes with temperature. This implies that ρ does not have a constant value. Values given in tables are usually for the resistivity at room temperature (20°C). The relationship to calculate specific resistance of a metal at a different temperature than that known is

where t₂ is the temperature at which we need to know the specific resistance ρ₂, t₁ is the temperature at which the value is known (ρ₁), α₁ is the temperature coefficient at t₁.

Equation 12.2 can be readily substituted by

where R₁ is the known value of the resistance at t₁ and R₂ is the resistance at t₂.

The variation of the temperature coefficient α can be found from

α₀ is the temperature coefficient at zero degrees.

Specific Resistance

Specific resistance is electrical resistance of a specific size of a material, and it is shown by the Greek letter ρ. In the metric system, ρ is the resistance of a piece of metal (10 mm × 10 mm × 10 mm), that is 1 cm long and has a cross section of 1 cm² (see figure below). Because in the metric system the units for the length and area are meter and square meter, respectively, ρ is defined in ohm-meter (Ω.m) so that when inserted in the relationship for resistance $R=\rho \frac{l}{A}$ the resultant is Ω.

$$\Omega .\text{m}\frac{\text{m}}{{\text{m}}^{\text{2}}}\to \Omega$$

In the US customary system, ρ is defined by the resistance of 1 foot of metal with a cross section of 1 circular mil (CM). See Appendix D for the definition of circular mil. The unit of measurement for ρ, therefore, is ohm-circular mil per foot (Ω.CM/ft).

Circular mil: Unit for measuring the thickness (cross section) of wires. A CM is the area of a circle whose diameter is one mil (1/1000 of an inch).

The standard wire sizes are given in Table 12.4 for both aluminum and copper wires. Table 12.4 also indicates the electric resistance per 1000 m and 1000 ft of these wires. AWG stands for American wire gauge.

12.3.3 Insulated Conductors and Ampacity

In the previous section we discussed the resistance of materials to electric current. We learned that almost all conductors are from copper or aluminum, and those from aluminum are reinforced by steel to give them mechanical strength when hanging from electric poles and towers. We also noticed that the sizes of wires are standardized and one must select a size among those that are commercially available. A different question that needs to be answered is how much current is allowed for a particular conductor. Ampacity (ampere capacity) is the answer to this question. Ampacity determines how many amperes of electric current a conductor can handle safely.

Ampacity is determined based on the amount of heat generated in a conductor owing to the current and the fact that this heat must be taken away, so that the conductor temperature does not increase anymore beyond a certain safe level. Otherwise, the buildup of heat can cause a problem. In this sense, the ambient temperature and the nature of conductor and its surroundings are the parameters that affect the ampacity rating of a conductor. In this respect, the same cable has more ampacity when in the air than when in a conduit.

Figure 12.11 illustrates a three-conductor underground cable. These cables come in different sizes with one, two, and three conductors and various insulation materials. Good electrical insulation is absolutely necessary for underground cables.

Overhead lines are normally in the air and cooled by streams of free air, whereas the underground cables are either in a conduit or buried underground. Also, their insulation decreases the rate of cooling compared to bare wires. The ambient temperature can be –30°C or +30°C, for instance, and it can vary significantly from winter to summer. Selection of cables must be based on the worst case scenario and the highest ambient temperature.

Figure 12.11 A three-conductor underground cable. (Courtesy of CME Wire and Cable, Inc.)

On the basis of the material and other conditions the ampacity of conductors are standardized. Tables 12.5 and 12.6 are examples of the ampacity of conductors according to National Electric Code (NEC) standard. These tables are only samples. Refer to NEC publications for a complete set of standards. Table 12.5 corresponds to ampacities of up to three conductors in a conduit, and Table 12.6 is for the same types of conductors when hanging in free air. In both cases the line voltage cannot exceed 2000 V.

As can be seen from these tables, ampacities are restricted for temperatures shown (30°C in this case) and must be adjusted for other temperatures based on the correction factors given in Table 12.7. The following examples show how these tables can be used. Letter combinations identified in the table are for the insulation types. Table 12.8 exemplifies the pertinent information. For full list of these types one must consult National Electric Code publications.

Solution

From Table 12.7 we find that for temperature range 36°C–40°C a correction factor of 0.82 must be considered. If the desired current of 40 A is divided by this number, we get 40 ÷ 0.82 ≅ 50 A. Now we must find the standard wire corresponding to the

Table 12.5 Allowable Ampacities of Insulated Conductors in Conduits

Table 12.6 Allowable Ampacities of Insulated Conductors in Free Air

Table 12.7 Temperature Correction Factors for Tables 12.5 and 12.6

Ambient Temperature (°C)	Temperature Rating of Conductor			Ambient Temperature (°F)
	60°C	75°C	90°C
10 or less	1.29	1.20	1.15	50 or less
11–15	1.22	1.15	1.12	51–59
16–20	1.15	1.11	1.08	60–68
21–25	1.08	1.05	1.04	69–77
26–30	1.00	1.00	1.00	78–86
31–35	0.91	0.94	0.96	87–95
36–40	0.82	0.88	0.91	96–104
41–45	0.71	0.82	0.87	105–113
46–50	0.58	0.75	0.82	114–122
51–55	0.41	0.67	0.76	123–131
56–60	–	0.58	0.71	132–140
61–65	–	0.47	0.65	141–149
66–70	–	0.33	0.58	150–158
71–75	–	–	0.50	159–167
76–80	–	–	0.41	168–176
81–85	–	–	0.29	177–185

Source: NFPA 70^®-2014, National Electrical Code^®, Copyright © 2013, National Fire Protection Association, Quincy, MA. Reprinted with permission. This reprinted material is not the complete and official position of the NFPA on the referenced subject, which is represented only by the standard in its entirety.

Note: For ambient temperatures other than 30°C (86°F), multiply the allowable ampacities specified in the ampacity tables by the appropriate correction factor shown above.

given conditions and with 50 A ampacity. We need to look in Table 12.5 under the column for the type of wire TW, but this ampacity of 50 A cannot be found in the table. Hence, we need to choose the gauge corresponding to the higher current of 55 A. The required wire is, thus, gauge 6.

12.4.1 Electric System Faults

In a three-phase power system, a fault, in general, can be a result of a short circuit or an inadvertent open circuit in one or two of the phases. A short

Table 12.8 Example of Definitions for Various Insulated Cables

circuit can happen owing to physical wear of insulation as a result of age, thermal stress and fatigue, high voltage, high current, harsh environment, abuse, and the like. An open circuit can occur owing to lightning, overload, load surge, malfunction of a device, loss of synchronization, and so on. In either case, there is a significant imbalance between the currents in the three phases.

Depending on the severity of a fault, currents of several times magnitude can flow through wires, transformers, and a generator on the power side. If the fault persists, it can cause other failures in a system by over-heating and damaging parts, wires insulations, and insulators. In a power system it can cause fire or in the best situation if the protective devices act right and cut off the electricity (clearing the fault), a lot of consumers will be left without electricity until the cause of the fault is found and removed. In a bolt short circuit, that is, when a current carrying line is solidly shorted without any impedance in between, currents of several hundred times are not uncommon.

A fault in a power system is in one of the following forms: one phase shorted to ground, two phase lines shorted together, two phase lines shorted to ground, all phases shorted to ground, one line open, and two lines open.

In the case of any of these faults, if a system is not provided with the proper protective means, the consequences are costly. The importance of a protective measure is realized when a fault occurs. Thus, to reduce the risk of losing a power system and its equipment, or the electrical equipment in a consumer site, each side (generation and consumption) needs to be protected by the proper means. At the very lowest level and very fundamental to all protective means is grounding of an electric circuit at the source and at the consumption sites.

12.4.2 Grounding

Grounding is the major feature of the protective means for a power system, an electric device, and personnel. The other feature is clearing a fault, which is done by devices that break a circuit and clear the fault by removing the faulty device or portion of the circuit containing a fault.

For equipment, grounding causes the fault currents find their way through ground, and not through the equipment parts and windings. For personnel, grounding causes the voltage of a faulty equipment frame/case to stay at the ground level, so that if a person touches the frame, there is no potential difference between the body part in contact with the equipment and the ground.

Grounding implies physically connecting a point of a device or circuit to the earth, which provides a common reference point for zero voltage and a path for return of a current that otherwise has no path to flow through, to the source. For a power system, grounding is connecting the neutral point of the system to the earth, and for equipment, grounding implies connecting the noncarrying conductive material(s) such as motor frames and enclosures to earth.

In normal operation of electric systems the current through ground is negligible. This current may be due to unbalanced loads in a three-phase system, but it is still relatively small, especially that the soil resistance comes into effect. It is only when a fault happens that the connections to ground and the earth play their role in the currents that seek a path to follow. In such a case a high current is flowing in the ground wire of a grounded system. Table 12.9 shows the specific resistance of various earth materials.

For most electrical equipment, grounding is performed by a solid or bolt connection to ground, whereas for a system, such as the electric network, generators (like in all wind turbines), and in larger and expensive equipment, grounding can be either solidly or through a current limiting device (see Figure 12.12).

Figure 12.12 Neutral Grounding Resistor.

The purpose of a current limiting device is to reduce the current that will flow through a device, and a system or its parts in case of a fault, which can cause damage. A common current limiting device for this purpose is a neutral grounding resistance (NGR). Such a resistor, nonetheless, must be carefully selected based on a number of criteria including the involved voltage and the allowed delay for the protecting device (e.g., a circuit breaker). A low resistance neutral grounding or a high resistance neutral grounding can be employed. In the former, the allowed current is higher. A current of 50 A or more (up to 400 A) is allowed for 10 sec. In the latter the maximum allowed current is 25 A for potentials of 600 V or less. A current of 5 A is more common. For higher voltages (13.8 kV and up), solid grounding is normally employed.

Figure 12.13 Using a single phase transformer in conjunction with a grounding resistor.

Instead of directly connecting a grounding resistor to the ground bus, a single-phase transformer can be employed, as shown in Figure 12.13. In this case a smaller resistor is used based on the transformer turns ratio.

Recall from Chapter 8 that, as its specific property, an inductor resists a change in the current through it. This property makes it a candidate for current limiting, so it is desirable for grounding. As another means for grounding, a reactor can be used. A reactor is a winding (inductor) with or without a core, dry or oil immersed, which is specifically designed for a number of purposes, including grounding. Among other reactor applications are inrush-current limiting (for capacitors and motors), harmonic filtering, reactive power compensation, reduction of ripple currents, damping of switching transients, flicker reduction, load balancing, and power conditioning.

12.4.2.2 Grounding in Power Transmission

We have seen in Chapter 10 that three-phase transformers can be used in four different configurations (see Section 10.6). These are delta-delta, delta-wye, wye-delta, and wye-wye. Each of these combinations has its advantages and disadvantages. A wye connection provides a suitable point for grounding. A delta connection provides a path for capturing third-order harmonics, which is quite important in power transmission and distribution.

Grounding transformer: Transformer used only for providing a reference point for connection to ground, in order for detection of faults in power generation and distribution.

Figure 12.14 Common arrangement for generators in traditional power plants.

It is more common to have one side delta connected and the other side wye connected to benefit from both advantages. However, it might be necessary to use a grounding transformer for the side with delta connection. The standard configuration by utilities for power transmission is “delta high – grounded wye low.” This implies that the higher voltage side is delta connected and the lower voltage side is star connected and it is grounded. This has the advantage that the low voltage bus has always a ground reference. Moreover, the common practice in traditional power plants (thermal and hydraulic plants) is to have the generator-step-up (GSU) transformer delta connected at the generator side (lower voltage) and the transformer secondary star-connected and grounded. A grounding transformer is then used for the generator to provide a solid ground for fault protection. This is depicted in Figure 12.14. A grounding transformer must have the correct size and rating for its purpose. Thus, its selection must be based on a careful study, and it is an additional cost to any facility.

In wind farms, however, because there are many wind turbines in a wind farm, the transformer at the generator side is star connected and grounded and the collector side (high voltage) is delta connected.

The standard configuration by utilities for power transmission is “Delta high – grounded wye low.”

A grounding transformer is a special wye-delta transformer that is designed for this purpose only. The secondary is not connected to any load and the primary is connected to the three-phase line at the location the grounding is desired. In the case of significant unbalance primary currents (a fault), very high currents circulate in the secondary windings. The transformer, therefore, must have been designed and rated for handling the high currents.

Also, a zigzag transformer can be used for grounding. The description of a zigzag transformer is given in Appendix C.

A grounding transformer is a special wye-delta transformer with its secondary not connected to any load. It is designed for the purpose of grounding only.

12.5.1 Transmission Lines Effects

In addition to the ohmic resistance in the wires and electric conductors, which we have discussed in this and Chapters 4–6, conductors carrying AC electricity have inductive and capacitive effects. Consider, for instance, a pair of transmission lines that extend for several miles and compare them to two plates forming a capacitor. You will see that there is a similarity in terms of the two metallic surfaces, being at a distance from each other, with a common cross-section area separated by a non-conductive material (air). Thus, the two conductors form a capacitor. The same thing is true for the magnetic effect of the current in each conductor on itself and on the other conductor. Thus, there exists inductance between the two conductors.

The amount of capacitive and inductive effects might be small and negligible for short wires, for example, in the household wiring having a low voltage, but they are not negligible for the very long transmission lines and the high voltages they carry.

Capacitive and inductive effects of the transmission lines, thus, need to be included in the calculation of their current and voltage drop. If these effects are added, the schematic representation of an electric network between load(s) and the source(s) is similar to Figure 12.15b, instead of Figure 12.15a. Resistances and inductances are in series with each other along the length of a conductor, but the capacitive effects are shown as parallel. These effects are indicated for one line only, but they correspond to both conductors. In practice, there are usually more than two conductors, particularly for three-phase lines. But, for three-phase systems the effects of two conductors do not add up as in the case of single-phase wires because there are no return lines carrying the exact same currents at the same moment.

Values for ohmic resistance can be readily defined for each conductor in terms of its length, cross-section area, the conductor material, and its specific resistance. These data are usually available for any cable in terms of the resistance for each meter of the conductor, or 1000 ft of it, as we have seen in the tables in Section 12.3.3.

Capacitor value and inductor value for a set of conductors, however, depend on the factors particular to their setup. Depending on conductors spacing, their radii, the way they are laid out, the number of conductors, and how many are in each bundle sharing the same current, a value can be determined for the inductance of a given length (e.g., 1 km or 1 mile) of a transmission line. Similarly, a value can be obtained for the capacitance. In this way, more precise values for the current, voltage drop, lost energy, and so on can be found for a transmission line of a given length.

Another phenomenon that causes loss in transmission lines is corona. Corona is the effect of potential difference (voltage) on the air surrounding the transmission lines, ionizing the air. This effect reflects in the form of a glow coming out of lines. Corona depends on the voltage and spacing between the wires. The amount of energy lost this way, however, is small and is often neglected.

Figure 12.15 Typical representation of a transmission line. (a) Two long transmission wires for AC. (b) Assumed capacitors and inductors to represent the capacitance and inductance effects of the nearby wires in AC transmission.

Details of calculation of the equivalent inductor and capacitor for various line arrangements and spacing conditions is lengthy and involves mathematics that are out of the scope of this book. We may only use the results accompanied by some examples. To comprehend the subject, never theless, a reader is expected to have acquaintance with logarithm of a number because all calculations involve natural (or Naperian) logarithm (see the boxed text in this chapter for logarithm).

Figure 12.16 illustrates the relative spacing of possible transmission lines arrangements (compare with Figure 12.10). Figure 12.16a shows a general case where the three lines of a three-phase system are at locations 1, 2, and 3. These lines, furthermore, are at general distances D₁₂, D₁₃, and D₂₃ from each other. In the discussions that follows it is assumed that the loads on the three phases are the same and we are dealing with balanced loads. Except in the case of Figure 12.16c where all lines are equidistance from each other, values for the inductance and capacitance of a length of the lines are not the same. As a result, voltage drops in the three lines are different. Consequently, at the load site the voltage at the three lines carrying three phases are not the same. Also, the phase relationships can be disturbed and the three phases will no longer be 120° from each other.

Figure 12.16 Conductor positions in some common three-phase transmission lines. (a) In three corners of a triangle. (b) Along one horizontal line. (c) At the three corners of an equilateral triangle. (d) Double-circuit three-phase, as on transmission structures.

It is very undesirable if the voltages arriving at a site from the same source are different and a proper three-phase system is not delivered at the line end. In many arrangements for transmission lines, it is not common, and sometimes impossible, to have the lines at the corners of an equilateral triangle (e.g., Figure 12.16b and d). To overcome this problem, it is common practice to rotate the lines, so that each of the phases occupies the same physical position equally along the length of the transmission line. In this way, the total inductance and capacitance of all three lines will be equalized. This rotation is called transposition, and a line treated this way is said to be transposed. This is illustrated in Figure 12.17, which illustrates that each of the three-phase lines A, B, and C are placed equally in the three physical positions denoted by 1, 2, and 3 along the length of the transmission line. When all the three phases of a line equally occupy the three positions, so that all the voltage differences cancel each other, the line is said to be completely transposed.

Figure 12.17 Transposition in transmission lines. (a) Physical position of the three-phase lines. (b) Interchanging the wire positions at some point along the line.

In all that follows we assume that the lines are solid, circular, and all have the same radius r. The distances are denoted by D with one or two indexes when necessary (like in D₁₂). The numbers calculated for inductance and capacitance are for a given length, and, thus, for the whole length of a transmission line they must be multiplied by the line length. Results of the calculations are per phase for three-phase lines. In single-phase systems the numbers must be doubled for inductance and halved for capacitance because there are two lines and one is the return of the other line.

Before going ahead we need to define the two following terms. In fact, these are the parameters that change, from one line arrangement to another. Geometrical mean distance (GMD) is a number corresponding to the equivalent distance between two or more lines (cables). Geometrical mean radius (GMR) is a number corresponding to the effective radius of one or more lines.

12.5.2 Line Inductance

The following general formula can be used for finding the inductance of each line in a transmission system. If three phase, the line is assumed to be transposed and the load(s) balanced.

GMD and GMR are as defined above, and they vary on a case-by-case basis. The coefficient (2 × 10^–7) is for metric system, and the answer is in henries per meter. For the imperial system a conversion is necessary, as

will be seen later in some examples. We consider GMD and GMR for the following three cases:

1. Case of single phase (two lines):

   GMD = D = the distance between the lines

   $$\text{GMR}=r^{\prime }=\frac{r}{e^{1/4}}=0.7788r$$

   where r is the wire radius. Because in this case the return line has the same characteristic and current as the main line, the value obtained for L must be doubled to represent the closed loop.

• 2. Case of three-phase single circuit line (Figure 12.16a, b, and c):

  For all of the above cases

  $$\text{GMR}=\sqrt[3]{D_{12}\times D_{13}\times D_{23}}$$

  (12.6)

  $$\text{GMR}=\frac{r}{e^{1/4}}=0.7788r$$

  (12.7)

  (D₁₂, D₁₃, and D₂₃ are the distances between lines as shown in Figure 12.16, r is the radius of the cables.)

• 3. Case of double circuit three-phase lines (Figure 12.16d):

  Note the way numbers and letters are assigned to the three-phase lines, and the definition of various distances.

  $$\text{GMD}=\sqrt[3]{D_{{\text{AB}}_{\text{eq}}}\times D_{{\text{AC}}_{\text{eq}}}\times D_{{\text{BC}}_{\text{eq}}}}$$

  (12.8)

  where

  $$\begin{array}{l}{D}_{{\text{AB}}_{\text{eq}}}=\sqrt[4]{D_{12}\times D_{12^{\prime }}\times D_{1^{\prime }2}\times D_{1^{\prime }{2}^{\prime }}}\hfill \\ D_{{\text{AB}}_{\text{eq}}}=\sqrt[4]{D_{13}\times D_{13^{\prime }}\times D_{1^{\prime }3}\times D_{1^{\prime }{3}^{\prime }}}\hfill \\ \begin{array}{l}{D}_{{\text{AB}}_{\text{eq}}}=\sqrt[4]{D_{23}\times D_{23^{\prime }}\times D_{2^{\prime }3}\times D_{2^{\prime }{3}^{\prime }}}\\ GMR=\sqrt[3]{{\text{GMR}}_{\text{A}}\times {\text{GMR}}_{\text{B}}\times {\text{GMR}}_{\text{C}}}\end{array}\hfill \end{array}$$

  (12.9)

  where

  $$\begin{array}{l}{\text{GMR}}_{\text{A}}=\sqrt{\frac{r}{e^{\frac{1}{4}}}(D_{11^{\prime }})}\hfill \\ {\text{GMR}}_{\text{B}}=\sqrt{\frac{r}{e^{\frac{1}{4}}}(D_{22^{\prime }})}\hfill \\ {\text{GMR}}_{\text{C}}=\sqrt{\frac{r}{e^{\frac{1}{4}}}(D_{33^{\prime }})}\hfill \end{array}$$

  (12.10)

The values obtained from the preceding relationships determine the inductance of each line of a three-phase system. The values are in henries per meter.

Note that $\frac{1}{e^{\frac{1}{4}}}=\frac{1}{e^{0.25}}=\frac{1}{1.2840}=0.7788$ Thus, in all the above equations, this value can be directly used (e = 2.7183).

After L is determined, the inductive reactance of the line is determined from the familiar relationship X_L = 2πfL.

(For all distance definition between lines, see Figure 12.16d; all lines are assumed to be the same size with a radius r.)

Figure 12.18 Wire position arrangement for Example 12.11.

12.5.3 Line Capacitance

Line capacitance of electric conductors can be found in the same way that their inductances were formulated for various line arrangements. The only difference is in the value of geometric mean radius, as far as the formulas are concerned. In addition, the effect of the ground comes into picture for capacitance. However, this effect is only a small fraction of the values of capacitance due to the wires themselves and can be ignored unless the cables are very near to the ground.

The value of GMR in Equation 12.7 for the case of capacitance is the actual radius of the conductor. In this respect, for all the calculations of capacitance the term $\frac{1}{e^{\frac{1}{4}}}=\frac{1}{e^{0.25}}=\frac{1}{1.2840}=0.7788$ (when it appears in the equations for inductance) must be replaced by 1. The general formula for capacitance of conductors in the metric system is

and the answer found this way is in F/m (farads per meter). For any length of conductors this value must be multiplied by the length.

For the simple case of two wires, GMR is simply the radius of the wires and GMD is their distance. The following examples determine the capacitance for some of the same conductors as previously seen in this section.

The value obtained from Equation 12.11 is for the capacitance between each phase line and the neutral in a three-phase balanced system. For single phase the calculated value must be divided by 2 to find the capacitance between the hot line and neutral. In the above equation the effect of earth is assumed small and is not included.

In practice, for medium length lines (typically 80–250 km, 50–156 mi) after the values of the resistance, inductive reactance, and capacitive reactance are found, the line is modeled as a π circuit or a T circuit, as shown in Figure 12.19. In the π circuit model, the capacitance of each capacitor is half of the total calculated value. Likewise, in T circuit the values of impedances are half of the calculated value.

Figure 12.19 Modeling of transmission lines. (a) π model. (b) T model.

12.6.1 Kirchhoff’s Voltage Law

Problems similar to those in Figure 12.20a and b can be solved by employing Kirchhoff’s laws. We need to find some or all the currents in each of the seven resistances and/or finding the potential difference between various points, such as V_AB, V_AC, V_BC and so on.

Kirchhoff’s voltage law states that the algebraic sum of all voltages across the components in a loop is equal to the algebraic sum of the source voltages (there might be more than once source) in the loop. Attention must be paid to the polarity and the sign of a voltage. This is the reason for stating algebraic sum because some of the values can be negative. The voltage sign in a component depends on the direction of the current. For clarity, this is shown in Figure 12.22a. For a source the voltage sign is negative for the direction toward the positive terminal.

For those loops that contain no source the sum of the voltages across the components is zero because there is no source, and, consequently, their sum is zero. In Figure 12.23, three loops out of six possible loops are shown. Application of Kirchhoff’s voltage law to these loops reveals that

Figure 12.22 Voltage and current sign designation. (a) Voltage difference signs for a loop. (b) Current signs for a node.

Figure 12.23 Loops and currents in a network.

An alternative way of expressing Kirchhoff’s law is “The algebraic sum of all the voltages in a loop is equal to zero.” This statement involves the sign assignment for all components including the sources.

12.6.2 Kirchhoff’s Current Law

Kirchhoff’s current law states that the algebraic sum of all the currents in a node is zero. In other words, the sum of all currents coming to a node is equal to the sum of all currents leaving it. The currents coming toward a point are considered positive and the currents leaving the node are considered negative, as depicted in Figure 12.22b.

Figure 12.24 Circuit, for example, of Kirchhoff’s laws application.

In the circuit of Figure 12.23, there are five resistors and each one has a different current. Applying the Kirchhoff’s current law to nodes A, B, C, and D reveals that

The Kirchhoff’s current law defines the relationships between various currents in a circuit because these currents are not independent of each other. In this sense, in the circuit of Figure 12.23 one can assume only three unknown currents I₁, I₂, and I₃ because I₄, I₅, and I may be defined in terms of I₁, I₂, and I₃, as shown in Figure 12.24. The following example shows the application of Kirchhoff’s voltage and current laws.

12.6.3 Thevenin’s Theorem

Thevenin’s theorem is particularly helpful when multiple power supplies exist in a circuit. In Figure 12.25, for example, there are two power supplies, and we need to find the current through the resistor R. This could be part of a larger circuit, which provides 12 and 6 V across the points shown in the figure. Refer to Figure 12.25 for understanding the statement of Thevenin’s theorem.

Thevenin’s theorem states that the current through a resistor R connected across any two points A and B in an active network is obtained by dividing the potential difference between A and B (with R disconnected) by R + r, where r is the resistance of the circuit between A and B (with R disconnected) and the sources of the circuit replaced by their internal resistances.

Figure 12.25 Application of Thevenin’s theorem.

Figure 12.26 Equivalent circuit by Thevenin’s theorem.

Figure 12.27 Steps in implementing Thevenin’s theorem. (a) Finding equivalent r_Th between points A and B. (b) Finding the voltage difference V_Th, between points A and B.

Active network implies that the circuit considered is not dead and has EMF sources within it (EMF implies a voltage source). The internal resistance of sources corresponds to resistances that may exist in the power sources but can be zero (see Appendix A).

In other words, Thevenin’s theorem implies that to find the current through R (or the voltage difference between A and B), replace this circuit by an equivalent circuit as shown in Figure 12.26. In this circuit, there are two unknown values of which need to be determined: the resistance r and the voltage V.

To find r,

1. Disconnect R (see Figure 12.27a).
2. Replace each voltage source by its internal resistance.
3. Remove any current source.
4. Determine the resistance between A and B.

To find V,

1. Disconnect R (see Figure 12.27b).
2. Find the voltage between A and B.

Figure 12.28 Circuit of Example 12.16.

Solution

By Thevenin’s theorem this circuit can be replaced by a voltage V and a resistor r, as shown in Figure 12.29a.

The resistor r can be found by following the three above mentioned steps. Because the source internal resistances are zero, the two branches in parallel with the 32 Ω resistor contain 80 and 120 Ω. Thus the resistor r is calculated as

$$\frac{1}{r}=\frac{1}{80}+\frac{1}{120}=\frac{2.5}{120}=\frac{1}{48}$$

or

r = 48 Ω.

Figure 12.29 Thevenin’s circuit for Example 12.16. (a) Thevenin’s equivalent circuit. (b) Values for Thevenin’s model.

Figure 12.30 Finding voltage at A Thevenin’s equivalent circuit.

The voltage V may be calculated as the voltage at A (assuming V_B = 0) based on a voltage drop in R₁ (the 80 Ω resistor) in the circuit of Figure 12.30 (32 Ω resistor removed). The current in this circuit is

$$\frac{12-6}{80+120}=\frac{6}{200}=0.03\text{ A}$$

and the voltage drop in R₁ is, accordingly, 80 × 0.03 = 2.4 V, which implies that the voltage at A is 12 − 2.4 = 9.6 V. The resulting circuit is indicated in Figure 12.29b. On the basis of this voltage and the resistor r (Thevenin’s equivalent circuit) the current through the 32 Ω resistor is 9.6 ÷ (48 + 32) = 0.12 A.

Thevenin’s theorem is not only for circuits with multiple sources. It can be employed for a circuit similar to that shown in Figure 12.23 (and solved by using Kirchhoff’s laws) when we are more interested to see the current in a resistor’s value is subject to change. If for various values of a resistor we assume we have to repeat all the calculation, it is tedious. By using Thevenin’s theorem we can readily find the current for each value of the resistor.

12.6.4 Principle of Superposition

The principle of superposition is another useful method that can be used in many problems. It has a broad application in many topics. In the context of this discussion it can be employed for solving problems similar to those tackled by Thevenin’s theorem when multiple sources exist. The principle is that a problem is broken into a number of overlapping but simpler cases. At the end, the solutions of the various cases are combined together. In our case of two sources in the same circuit, we break down into two overlapping problems, each one dealing with the same circuit, but only one source. We solve the simpler problems, and at the end, we add up the separately found currents for each component. As an example, we consider the same problem in Example 12.16.

Solution

We first consider the circuit without the 6 V source. The resulting circuit is shown in Figure 12.32b. The corresponding current I₁ in the circuit can be found as

$$I_1=\frac{12}{80+\frac{1}{\frac{1}{32}+\frac{1}{120}}}=0.114\text{ A}$$

Figure 12.32 Separating the sources in the circuit of Figure 12.27. (a) Original circuit. (b) The 6 V battery is removed. (c) The 12 V battery is removed.

This is the current in the 80 Ω resistor due to the 12 V battery. Only part of this current (I_1,32) flows through the 32 Ω resistor, which can be determined from the resistors’ ratio:

$$I_{1,32}=(0.114)\frac{120}{120+32}=0.090\text{ A}$$

Next, we consider the circuit without the 12 V source and follow the same process (Figure 12.32c).

$$I_2=\frac{6}{120+\frac{1}{\frac{1}{32}+\frac{1}{80}}}=0.042\text{ A}$$

This is the current through the 120 Ω resistor due to the 6 V battery. The part I_2,32 passing through the 32 Ω resistor is determined as

$$I_{2,32}=(0.042)\left(\frac{80}{80+32}\right)=0.030\text{ A}$$

The total current through the 32 Ω resistor is obtained by adding the two partial currents. Therefore,

I₃₂ = 0.090 + 0.030 = 0.120 A

This is the same answer as obtained by Thevenin’s theorem.

Thevenin’s theorem has the advantage that in case this resistor is subject to variation, we do not need to repeat all the calculation every time. We just simply enter the changed value of the resistor in the equivalent circuit.

Note that the directions of the two currents to be superimposed for the 32 Ω resistor are the same. For the other resistors this is not the case, and if we were to find the current in one of those (e.g., in the 80 Ω resistor) the two currents had to be subtracted from each other, according to the arrows shown in Figure 12.32b and c.

12.6.5 Norton’s Theorem

Norton’s theorem is a method similar to the Thevenin’s theorem for network analysis by reducing a circuit as seen by a load (i.e., all other components in the circuit around that load) to only one source and one resistor. The difference is that in Thevenin’s theorem a circuit is reduced to a voltage source and a resistor that is in series with the load. In Norton’s theorem the circuit is reduced to a current source and a resistor in parallel with the load. This is shown in Figure 12.33, together with the Thevenin’s equivalent circuit. The subscript Th has been added to the Thevenin’s parameters V and r, for clarity. The Norton’s parameters carry a subscript N.

The procedure to find the Norton’s equivalent circuit is the same as described for finding the Thevenin’s equivalent circuit. The difference is for the current source, which provides a (constant) current instead of a voltage to the circuit. This is described below. Suppose that R_L is the resistance of the load for which we want to find the Norton’s equivalent circuit as seen by the load. We need to find the resistance R_N and the current I_N.

Figure 12.33 Comparison of Thevenin’s and Norton’s equivalent circuits. (a) Original circuit. (b) Thevenin’s circuit. (c) Norton’s circuit.

Figure 12.34 Norton equivalent of a circuit as seen from a load between points A and B. (a) Original circuit. (b) Norton’s equivalent circuit.

To find R,

1. Disconnect R_L (see Figure 12.34a).
2. Replace each voltage source by its internal resistance.
3. Remove any current source if it exists.
4. Determine the resistance between A and B.

To find I_N,

1. Short R_L (short circuit between A and B).
2. Find the current flowing from A to B.

In fact, there is a correspondence between the values obtained for r_Th, R_N, and between V_TH and I_N. These are

Figure 12.28 Circuit of Example 12.19.

Figure 12.35 Norton’s Equivalent for the circuit in Figure 12.28.