Transmission Line Transients

Transcription

1 8 5 Transmission Line Transients CHAPTER OBJECTIES After reading this chapter, you should be able to: Proide an analysis of traelling waes on transmission lines Derie a wae equation Understand the effect of traelling wae phenomena when line is terminated through resistance, inductance and capacitance Draw the Bewley Lattice Diagram 5. INTRODUCTION When a transmission line is connected to a oltage source, the whole of the line is not instantly energized. Some time elapses between the initial and the final steady states. This is due to the distributed parameters of the transmission lines. The process is similar to launching a oltage wae, which traels along the length of the line at a certain elocity. The traelling oltage wae also called surge, may be caused by switching or lightning. The oltage wae is always accompanied by a current wae. The surge reaches the terminal approach such as cable boxes, transformers and switch gears, and may damage them if they are not properly protected. As the waes trael along the line their wae shapes and magnitudes are also modified. This is called distortion. The study of traelling waes helps in knowing the oltages and currents at all points in a power system. It helps in the design of insulators, protectie equipment, and the insulation of the terminal equipment and oerall insulation coordination. Generally, a power system operates under a steady-state condition. Howeer, transients are initiated due to disturbances like switching, occurrence of short-circuit faults or lightning discharge which may result in current and oltages higher in magnitude as compared to those in steady-state conditions.

2 8 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION 5. TYPES OF SYSTEM TRANSIENTS Depending upon the speed of the transients, these can be classified as: Surge phenomena (extremely fast transients) Short circuit phenomena (medium fast transients) Transient stability (slow transients) Lightning and switching causes transient or surge phenomena. These transients (surges) trael along the transmission line with a elocity of light (3 8 m/s), i.e., in one millisecond, it traels 3 km along a transmission line. Thus, the transient phenomenon associated with these traelling waes occurs during the first few milliseconds after their initiation. The present-line losses cause fast attenuation of these waes, which die out after a few reflections. The reflection of surges at open line ends, or at transformers which presents high inductance, causes maximum oltage buildup which may eentually damage the insulation of high-oltage equipment with consequent short circuit. Lightning on 33/ k substation is shown in Fig TRAELLING WAES ON A TRANSMISSION LINE A transmission line is a circuit with distributed parameters. A typical characteristic of a circuit is to support traelling waes of oltage and current, in addition to which, it also has a finite elocity of electromagnetic field propagation. In such a circuit, the changes in oltage and current due to switching or lightning are spread out in all parts of the circuit in the form of traelling waes or surges. Connected to a source of electrical energy due to sudden lightening or switching, the transmission line encounters a traelling wae of oltage and current passing through it at a finite elocity. For explanation of the traelling wae phenomenon, consider a lossless transmission line which has series inductance (LΔx) of length Δx and shunt capacitance (CΔx) of length Δx as shown in Fig. 5.. After the switch S is closed, the applied oltage will not appear instantly at the load end because of S L x L x L x L x L x L x C x C x C x C x Fig. 5. Equialent circuit of a long transmission line L o a d

3 TRANSMISSION LINE TRANSIENTS 83 inductance and capacitance of the lossless transmission line. When the switch S is closed, the current passing through the first inductance is zero because it acts as an open circuit, and the oltage across the first capacitor is zero because it acts as a short circuit at the same time. At this instant, the next sections cannot be charged because the oltage across the first capacitor is zero. When the first capacitance is charged through first inductance, the capacitance of the next section starts charging and so on. It is therefore clear from the discussion that the oltage at the successie sections builds up gradually and finally the oltage wae reaches the other end of the line. 5.4 THE WAE EQUATION Assume a transmission line with distributed parameters as shown in Fig Let R Resistance of line per unit length L Inductance of line per unit length C Capacitance of line per unit length G Shunt conductance of line per unit length S x L o a d Fig. 5.3 Transmission line representation with distributed parameters For a small section of the line Δx, the resistance, inductance, capacitance, and conductance are RΔx, LΔx, CΔx and GΔx, respectiely. The oltage at distance (x Δx) from the sending-end is (x Δx). By Taylor theorem, ( ) ( )+ x+ Δx x Δx x (5.) The difference in oltages between the distances x and (x Δx) due to the resistance and inductance from Eq. (5.) is x ( ) ( x+ x) x ( ) x ( ) + x x Δ Δ i ( RΔx) i+ ( LΔx) t i Δx ( RΔx) i+ ( LΔx) x t Ri L i or x + t (5.)

4 84 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION Similarly, the difference in current can be written as: i ix ix x x G x C x ( ) ( ) + Δ ( Δ ) + ( Δ ) t i or G C (5.3) x + t When we consider a lossless transmission line, i.e., R, G, then the Eqs. (5.) and (5.3) become (5.4) L i x t and i (5.5) C x t Differentiating Eq. (5.4) partially with reference to distance x and Eq. (5.5) with reference to time t Also, From Eqs. (5.6) and (5.7) LC x t Similarly differentiating Eq. (5.4) with reference to t and Eq. (5.5) with reference to x and and i L x t t i C x x t From Eqs. (5.9) and (5.), we get i LC i x t i L x x t i C xt t i i t x x t (5.6) (5.7) (5.8) (5.9) (5.) (5.) The Eqs. (5.8) and (5.) are identical in form and gie similar solutions. They are called the traelling wae equations in a lossless transmission line. They represent the distribution of oltage and current along the line in terms of time and distance. Solutions of Eqs. (5.8) and (5.) represent the oltage and current waes that can trael in either direction, i.e., in the forward or backward direction without change in shape or magnitude with a elocity equal to. LC If the wae traelling in the forward or positie x direction can be expressed as a function of ( LCx t) then the function, f( LCx t).

5 Similarly, the wae traelling in the backward or negatie x direction can be expressed as another function, φ( LCx+ t). It can be proed that f( LCx t) is a solution of Eq. (5.). To do this, let us write ( LCx t) s (5.) One solution is f (s) (5.3) Differentiating Eq. (5.3) with reference to x But from Eq. (5.), Taking the second deriatie with reference to x, we obtain (5.4) In a similar way, it can be shown that f (5.5) t s From Eqs. (5.4) and (5.5), we get which is same as Eq. (5.8). This equation satisfies the function f( LCx t) and also φ( LCx+ t). Hence, the complete solution can be written as: f ( LCx t)+ φ ( LCx + t) This may be written in the form: + Where f LCx t direction and f s x s x LC x f s LC f x s LC x t s x ( ) φ ( ) LCx+ t LC (5.6) (5.7) represents the incident wae, i.e., the wae traelling in x increasing represents the reflected wae, i.e., the wae traelling in x decreasing direction. Thus, the transient oltage wae is seen to hae two oltage wae components traelling in the forward direction (i.e., when x is increasing) and traelling in the backward direction (i.e., when x is decreasing). The wae traelling in the forward direction is called the Forward traelling wae and the wae traelling in the backward direction is called the Backward traelling wae. The solution for the current may be written similarly. From Eqs. (5.5) and (5.6), we get ( ) i C f LCx t x LC x ( ) φ LCx + t x TRANSMISSION LINE TRANSIENTS 85 (5.8)

6 86 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION After integrating Eq. (5.8), we get i f ( LCx t LCx t LC ) φ ( + ) (5.9) The quantity L/ C is defined as the characteristic impedance of the line and is denoted by symbol Z. The characteristic impedance of a lossless line is a real quantity. It has the characteristics of resistance and the dimensions of ohm. Therefore, it is also called characteristic resistance or surge resistance. The surge resistance is denoted by R. L Z R C Substituting Z from the aboe equation in Eq. (5.9), we get i f ( LCx t) φ ( LCx+ t) Z i Z i Z Z This may be written as, i i + i [ ] for a lossless line. (5.) (5.) where, and represents the incident current wae, i i and represents the reflected Z Z current wae. Consider a oltage wae which propagates in the Direction of Direction of Total at forward direction and a wae which propagates in propagation propagation instant the backward direction, as illustrated in Fig. 5.4(a). The backward wae is called the reflected wae. The positie direction of current is taken as the direction i i i of propagation of the wae itself. In case of a forward wae, the direction of current and oltage are the same. But for a backward wae, the direction of propagation of current is opposite to that of oltage, Fig. 5.4(a) Direction of wae propagation so it is taken as negatie. Figs. 5.4(b) and (c) represent the wae shapes for forward and reflected waes for oltage and current, and their resultant at any instant. The mathematical relation between them f i fz r i f rz r (iii) Total oltage is gien as: i Z i Z A function of the form Inductance of the line/km/phase Capacitance of the line/km/phase ( ) f LCx± t (i) Forward oltage wae (ii)reflected oltage wae Fig. 5.4(b) Wae shapes of oltage represents a traelling wae because, for any alue of t, a corresponding alue of x can be found such that LCx ± t has a constant alue and, therefore, defines a fixed point on ±. Corresponding alues ( ) f ( LCx t) ( ) ( LCx t) of x and t which define the same points on a wae are gien by LCx t and +.

7 TRANSMISSION LINE TRANSIENTS 87 (i) Forward current wae i f i (ii) Reflected current wae r i if + ir (iii) Total current Fig. 5.4(c) Wae shapes of current Test Yourself. Why is characteristic impedance also called characteristic resistance? 5.5 EALUATION OF SURGE IMPEDANCE In this section, we will be calculating the surge impedance for oerhead transmission lines and underground cables. (i) Oerhead transmission line where, D is the distance between the centres of the conductors and r is the radius of the conductor and D r. Z (ii) Cable 7 L ln ( D r) H/phase/m πε π 9 36π 9 C F/phase/m ln Dr ln Dr 8ln Dr 7 L ln ( R r) H/phase/m πε C F/phase/m ln ( Rr) 9 ε r 8ln ( Rr) where, R is the radius of the cable and r is the radius of the conductor. Assuming a dielectric haing a relatie dielectric constant of ε r Z ( ) ( ) ( ) ( L C) 6 L ln C 7 ln ( Dr) Dr 9 8 ( Dr) 6ln ( ) Ω ln ε r R r Ω ( )

8 88 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION A alue of 5 Ω is usually assumed for the surge impedance of an oerhead line while a alue of 5 Ω is assumed for the surge impedance of a cable. Test Yourself. Why is the surge impedance in oerhead lines more than in underground cables? 5.6 IMPORTANCE OF SURGE IMPEDANCE Knowledge of surge impedance is extremely useful as it enables the calculation of the transient oltages and currents which may occur in a circuit. For example, if a line carrying a current I has been interrupted suddenly, the maximum alue of the oscillating oltage produced and is equal to IZ, which may reach a dangerous alue if Z is high. Power transmission systems are always complex in character, i.e., they consist of sections or elements, such as generators, transformers, transmission lines, and loads with different electrical constants. In such a compound circuit, oscillations harmful in one port of the circuit may reach a dangerous state in another port, due to ariations in Z o. For example, consider the case of a long transmission line connected to an underground cable. An oscillation current in the cable of lower impedance of its own, may gie rise to dangerously high oscillating oltages when it enters into the oerhead part of the line owing to the far higher natural impedance of the latter. Similarly, if a transformer is connected across the end of an oerhead line, the natural impedance of such a transformer may be between Ω and 4 Ω, which is ery much higher than that of the line itself. Consequently, an oscillating current which only gies rise to moderate oscillating oltages in the line may produce destructie oltages in the winding of the transformer. Therefore, a consideration of Z of different apparatus gies considerable information regarding the relatie danger and the preentie action to be taken. 5.7 TRAELLING WAE A lightning discharge or sudden switching in or out results in the impression of an electric energy suddenly in a transmission line. This moes along the line at a speed of light (approximately) as a traelling wae or impulse until it has been dierted from the wae front and has, by piling up the oltage locally in the windings of reactie apparatus, had destructie consequences. The traelling wae is also called impulse wae. This is shown in Fig At point a the oltage of the wae is zero. As the wae moes along the line, the oltage at a will rise from zero to the peak alue at b and again fall to zero on moing further. The left portion of the peak alue oltage (k) % 5% a b T T Fig. 5.5 Traelling wae shape Time( μs)

9 is called the wae front and the right portion is called the tail of the wae. Usually, the impulse wae is designated by time T taken to attain maximum alue and the time T taken for the tail to fall 5% of the peak alue. Thus, suppose T is s and T is 5 s, then the wae is designated as /5 wae. The equation of an impulse wae is of the form: ( ) TRANSMISSION LINE TRANSIENTS 89 αt βt e e (5.) where, represents a factor that depends on the peak alue. α and β are constants which control the wae front and wae tail times, respectiely. Their alues are α.436 and β.467 for./5 μs, impulse wae. 5.8 EALUATION OF ELOCITY OF WAE PROPAGATION Consider the Eq. (5.), where, Differentiating this expression with respect to time t, we get x t is the rate of change of distance, which is equialent to elocity. elocity of wae propagation, υ m/s (5.3) LC (i) Oerhead transmission line Where, D is the distance between the centres of the conductors and R is the radius of the conductor and D > R. (ii) Cable ( ) s LCx t LC x t x t LC L 7 ln ( D R) H/phase/m 9 πε π 36π 9 C ln DR ln DR 8ln DR ( ) ( ) ( ) υ ( ) LC ln DR 8ln DR 7 L ln ( R r) H/phase/m πε C F/phase/m ln ( Rr) 9 ε r 8ln Rr ( ) ( ) F/phase/m ( ) ( ) m/s

10 9 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION where, R is the radius of the cable and r is the radius of the conductor. Assuming a dielectric haing a relatie dielectric constant of ε r υ ( ) 7 9 LC ln ( Rr) ε 8ln ( Rr) 3 ε r 8 m/s r (5.4) Example 5. A cable has a conductor of radius.75 cm and a sheath of inner radius.5 cm. Find (i) the inductance per meter length (ii) capacitance per meter length (iii) surge impedance and (i) elocity of propagation, if the permittiity of insulation is 4. Solution: Radius of conductor, r.75 cm Inner radius of sheath, R.5 cm Permittiity of insulation, ε r 4 (i) The inductance per metre length of the cable is, 7 R L ln H/m r 7 5 ln H/m (ii) The capacitance per metre length of the cable is, πε ε C r F/m R ln r ln F/m (iii) Surge impedance of the cable is, Z C L C Ω 9 (i) The elocity of wae propagation is, υ LC m/s 7 9

11 TRANSMISSION LINE TRANSIENTS REFLECTION AND REFRACTION COEFFICIENT (LINE TERMINATED THROUGH A RESISTANCE) Z O Consider a lossless transmission line which has a surge impedance of Z terminated through a resistance R as shown in Fig When the wae traels along the line and absorbs any change (line end, change of series or shunt impedance), then it is partly or totally reflected. R The expression for reflected current is: Fig. 5.6 Line terminated through a resistance i Z where, and i are the reflected oltage and current waes, respectiely. Let and i be the transmitted oltage and current waes and and i be the incident waes. From Fig. 5.6, Incident current, i Z Reflected current, i Z and, transmitted current, i R Since i i + i and + R Z Z ( ) Z Z Z Z (5.5) Therefore, the transmitted oltage, R (5.6) Z + R and, transmitted current, i R Z + R Z Z Z + R Z i Z + R (5.7) Z From Eq. (5.7), the coefficient of transmitted or refraction current waes is (5.8) Z + R

12 9 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION R and, transmitted coefficient for oltage waes (5.9) Z + R Similarly, substituting for in terms of, the Eq. (5.5), becomes + R Z Z R Z R+ Z R Z and i Z Z R+ Z R Coefficient of reflection for current waes Z R+ Z R and, reflected coefficient for oltage waes + Z R+ Z (5.3) (5.3) Example 5. A rectangular wae traels along a 5 km line terminated with a resistance of Ω. The line has a resistance of.3 Ω/km and a surge impedance of 4 Ω. If the oltage at the termination point after two successie reflections is k, determine the amplitude of the incoming surge. Solution: Length of the line 5 km Terminated resistance, R Ω Line resistance.3 Ω/km Surge impedance, Z C 4 Ω Termination oltage k The line resistance for 5 km, Ω/km R The amplitude of the incoming surge, R+ Z C k 3 Example 5.3 A oltage haing a crest alue of 3 k is traelling on a 75 k line. The protectie leel is 7 k and the surge impedance of the line is 3 Ω. Calculate (i) the current in the line before reaching the arrester, (ii) current through the arrester, (iii) the alue of arrester resistant for this condition and (i) reflect oltage. erify the reflection and refraction coefficients. Solution: Z c 3 Ω, 3 k, a 7 k (i) i Z c A

13 TRANSMISSION LINE TRANSIENTS 93 (ii) The oltage equation is Z c i a a 3 3 3i a i a 4333 A 3 (iii) Resistance of arrester, R 3 a Ω ia 4333 (i) Reflected oltage, or 3 k 3 Reflection coefficients Refraction coefficients R Z Reflection coefficients R+ Z R 8. 6 Refraction coefficients. 567 R+ Z c c c ( ) ( ) LINE OPEN-CIRCUITED AT THE RECEIING END Consider Fig. 5.6, when the receiing end is open-circuited, i.e., R, the equialent circuit is shown in Fig R Consider the transmitted coefficient of oltage wae Z + R When R, the transmitted coefficient of oltage wae + Z R + Z (5.3) S dx + _ Fig. 5.7 Case of an open-circuit ended line

14 94 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION Z and transmitted coefficient of current wae + R (5.33) R and reflection coefficient of oltage wae + Z R+ Z (5.34) R Similarly, reflection coefficient of current wae Z Z R (5.35) R+ Z + Z R From Eq. (5.3), the transmitted coefficient is two, i.e., the oltage at the open-ended line is. This means that the oltage of the open-ended line is raised by due to reflection. Transmitted wae incident wae + reflected wae For an open-ended line, a traelling oltage wae is reflected back with a positie sign and the coefficient of reflection is unity [from Eq. (5.34)]. From Eq. (5.33), the transmission coefficient of current is zero, i.e., the current at the open-ended line is zero. This means a current wae of magnitude i traels back with a negatie sign and the coefficient of reflection is unity [from Eq. (5.35)]. The aboe cycle is repeated for oltage and current waes. This cycle occupies the time taken for a wae to trael four times the length of line and is explained through Fig oltage Current At At l t O + ' I I ' At l ' I ' At 3l ' -I ' At 4l ' I ' Fig. 5.8 ariation of oltage and current in an open-circuit ended line 5.9. LINE SHORT-CIRCUITED AT THE RECEIING END S dx Consider Fig when the receiing end is shortcircuited, i.e., R, the equialent circuit is shown in Fig. 5.9 (a). Consider the transmitted coefficient of oltage R wae Z + R When R, ± Fig. 5.9(a) Case of a short-circuit ended line

15 TRANSMISSION LINE TRANSIENTS 95 R the transmitted coefficient of oltage wae (5.36) Z + R Z and transmitted coefficient of current wae (5.37) Z + R R and reflection coefficient of oltage wae + Z (5.38) R+ Z R Similarly, reflection coefficient of current wae Z (5.39) R+ Z From Eq. (5.36), the transmitted coefficient is zero, i.e., the oltage at the short-circuit ended line is zero. This means that a oltage wae of magnitude traels back with a negatie sign and the coefficient of reflection is unity [from Eq. (5.38)]. Transmitted wae incident wae + reflected wae From Eq. (5.37), the transmission coefficient of current is two, i.e., the current at the short-circuit ended line is i as seen in Fig. 5.9(b). This means that the current of the short-circuit ended line is raised by i due to reflection. From this discussion, we can conclude that the line oltage is periodically reduced to zero but at each reflection of either end, the current is increased by the incident current, i. Theoretically, the Z current will become infinite for infinite reflections, but practically the current will be limited by the resistance of the line in an actual system and its final alue will be, i. R At At l t O + oltage Current I I At l I At 3l 3I At4l 4I Fig. 5.9(b) ariation of oltage and current in short-circuit ended line Test Yourself Is the reflection coefficient of current wae in an open-circuit condition? If yes, justify.

16 96 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION 5. LINE CONNECTED TO A CABLE When a wae traels towards the cable from the line (see Fig. 5.), because of the difference in impedances at the junction, part of the wae is reflected and the rest is transmitted. The transmitted oltage wae [from Eq. (5.6)] is gien by Z L Z C Fig. 5. Line connected to a cable Z c Z + Z and, the transmitted current wae [from Eq. (5.7)] is gien by Z i i c Z + Z Similarly, reflected oltage and current waes are Zc ZL Zc + ZL Zc Z and i i Z + Z L L c c c L L Example 5.4 An oerhead line with inductance and capacitance per km length of.3 mh and.9 F, respectiely is connected in series with an ungrounded cable (see Fig. 5.) haing inductance and capacitance of. mh/km and.3 F/km, respectiely. Calculate the alues of reflected and refracted (transmitted) waes of oltage and current at the junction due to a oltage surge of k traelling to the junction (i) along the line towards the cable and (ii) along the cable towards the line. Z L Z C Fig. 5. Circuit diagram for Example 5.4 Solution: The natural impedance of oerhead line, Z The natural impedance of cable, Z c 3 L.3 6 C.9.8 Ω 3 L. 6 C Ω (i) The oltage wae of magnitude k which is initiated in an oerhead line is partly reflected and partly transmitted on the cable at the junction of the line and the cable. Z c Therefore, transmitted (refracted) oltage, Zc + Z L k L

17 TRANSMISSION LINE TRANSIENTS 97 Zc ZL and, reflected oltage, Zc + ZL k ZL Transmitted current, i i Z + Z Z c L L ZL Zc + ZL ka and, reflected current, i ZL A. 88 (ii) The oltage wae of magnitude k which is initiated in the cable is partly reflected and partly transmitted on the oerhead transmission line at the junction of the cable and the line. Z L Therefore, transmitted oltage, ZL + Z c k ZL Zc and, reflected oltage, ZL + Zc k Zc Transmitted current, i i ZL + Zc Zc Zc ZL + Zc ka and, reflected current, i Zc ka 5. 8 Example 5.5 Two stations are connected together by an underground cable haing a surge impedance of 5 Ω joined to an oerhead line with a surge impedance of 4 Ω. If a surge haing a maximum alue of k traels along the cable towards the junction with the oerhead line, determine the alue of the reflected and the transmitted wae of oltage and current at the junction. Z L Z C Fig. 5. Circuit diagram for Example 5.5 Solution: Surge impedance of the cable, Z c 5 Ω

18 98 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION Surge impedance of the oerhead line, Z L 4 Ω The oltage wae (magnitude of k) initiated in cable is partly reflected and partly transmitted on the oerhead transmission line at the junction of cable and line. Z L Therefore, transmitted oltage, ZL + Z c k ZL Zc and, reflected oltage, ZL + Zc k Zc Transmitted current, i i Z + Z Z L c c Zc ZL + Zc A and, reflected current, i Z c A 5 Example 5.6 The ends of two long transmission lines, A and C are connected by a cable B of length km. The surge impedances of A, B and C are 4, 5 and 5 Ω, respectiely. A rectangular oltage wae of 5 k magnitude and of infinite length is initiated in A and traels to C. Determine the first and second oltages impressed on C. Solution: Referring to Fig. 5.3, the oltage wae of magnitude 5 k is initiated in line A and is partly reflected and partly refracted onto cable B when reaching the junction J. The transmitted wae, Z B ZA + ZB k 4+5 This transmitted wae, when reaching the junction J, again obseres that a part of it is reflected and another refracted onto line C. This transmitted oltage wae thus is calculated as: 4 Zc ZB + Zc k 5+5 Transmission Transmission line A Cable B line C J J k (Refl.wae) 3 4 (Trans. wae) (Refl. wae) (Trans. wae) First impressed oltage on C 5 6 Second impressed oltage on C Fig. 5.3 Circuit diagram for Example 5.6

19 TRANSMISSION LINE TRANSIENTS k is the first oltage impressed on C. The reflected wae 3 at junction J is, 3 is transmitted and has reached junction J. From here, it is partially reflected and partially transmitted onto A. Let 5 be the reflected wae at junction J. Then, Howeer, 5 on reaching the junction J, gets partially transmitted onto line C. Let this be 6. Then, Zc ZB Zc + ZB k ZA ZB 3 ZA + ZB k Z c 5 ZB + Zc k Then, second impressed oltage k 5. REFLECTION AND REFRACTION AT A T-JUNCTION A oltage wae is traelling oer the lossless transmission line with natural impedance Z towards the junction as shown in Fig At the junction, due to the change of impedance, part of the wae is reflected back and the other is transmitted oer the parallel lines which hae natural impedances Z and Z 3, respectiely. Let, i and 3, i 3 be the oltages and currents in parallel branches. Z As far as the oltage wae is concerned, the reflected portion will be the same for both branches, Z i.e., 3, since they are parallel to each other. The following relations hold good at the transition point. + Z 3 i Z Fig. 5.4 T-Junction i Z i, i3 Z Z and, i i 3 i i 3 (5.37)

20 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION Substituting the alues of currents in Eq. (5.37) Z Substituting for + Z Z3 Z Z + + Z Z Z 3 Z The transmitted oltage is + Z Z Z 3 Z + + Z Z Z 3 and, the reflected oltage is Z Z Z3 + + Z Z Z 3 (5.38) (5.39) Example 5.7 A k surge traels on a line of 4 Ω surge impedance and reaches a junction where two branch lines of surge impedances 55 Ω and 35 Ω, respectiely are connected with the transmission line (see Fig. 5.5). Find the surge oltage and current transmitted into each branch line. Also find the reflected oltage and current. Solution: Z Z Surge oltage, Z Z Z + Z + Z c Z + Z k Z 55Ω Z C 4 Ω Z 35Ω Fig. 5.5 Circuit diagram for Example 5.7

21 TRANSMISSION LINE TRANSIENTS i A z 55 i A z 35 Reflected oltage, k 3 Reflected current, i i+ i i A 4 Example 5.8 A surge of k traels on a line of surge impedance 5 Ω and reaches the junction of the line with two branch lines as in Fig The surge impedances of the branch lines are 45 Ω and 5 Ω, respectiely. Find the transmitted oltage and currents. Also find the reflected oltage and current. Z 5 Ω Z 5 Ω Z 3 5 Ω Fig. 5.6 Circuit diagram for Example 5.8 Solution: The arious impedances are Z 5 Ω, Z 45 Ω, and Z 3 5 Ω The surge oltage (magnitude), k The surge reaches the junction and experiences reflection due to change in impedance and here the two lines (Z and Z 3 ) are parallel. Therefore, the transmitted oltage will hae the same magnitude and is gien by, Z + + Z Z Z k The transmitted current in branch line, i A Z 45

22 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION The transmitted current in branch line, i A Z 5 3 The reflected oltage, Z Z Z3 + + Z Z Z k The reflected current, i A Z 5 Example 5.9 An oerhead line has a surge impedance of 45 Ω. A surge oltage 5(e.5t e t ) k, where t is in s, traels along the line. The termination of the line is connected to two parallel oerhead line transformer feeders. The surge impedance of the feeder is 35 Ω. These two transformers are protected by surge dierters each of surge impedance being 4 Ω. Determine the maximum oltage which would initially appear across the feeder-end windings of each transformer due to the surge. Assume the transformer to hae infinite surge impedance. Solution: Figure 5.7 shows the circuit. Since AB and AC are parallel to each other, the oltage transmitted in them will be the same. The transmitted oltage in AB or AC is gien by z z z z B 45 Ω ' A 35 Ω 35 Ω 4 Ω C 4 Ω Fig. 5.7 Circuit diagram for Example 5.9

23 TRANSMISSION LINE TRANSIENTS 3 The conditions at junctions B and C are identical. The oltage transmitted at B or C z4 z + z 4 where, z 4 surge impedance of a dierter 4 Ω where, 5 (e.5t e t ).33 5 (e.5t e t ) d For maximum oltage to appear is zero, i.e., dt d 5. t t ( e e ) dt 5. t t 5. e + e t. 5t e 5. e 95t e. t μs Substituting the alue of t in the expression for gies the maximum transmitted oltage. Therefore, the maximum oltage appearing across the feeder end winding of each transformer is ( ) e e ( ) k 5. REACTANCE TERMINATION In this section, we consider the line terminated through capacitie and inductie reactance. 5.. LINE TERMINATED THROUGH CAPACITANCE Assume that a line is terminated through a capacitor C as shown in Fig. 5.8(a). When the wae is traelling along the line with natural impedance Z and terminated through C, then the transmitted oltage is determined from Eq. (5.6). S Z + _ C Fig. 5.8(a) Line terminated through a capacitor

24 4 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION The transmitted oltage is, s Cs () Z + s ZCs s+ Cs ZC s s + ZC Taking the inerse Laplace transform on both sides of the aboe equation t ZC t () e (5.4) The shape of the oltage wae is as shown in Fig. 5.8(b). There is more practical importance for a case with terminal capacitance. oltage (t) ( e C ) Z t Time Fig. 5.8(b) ariation of oltage across the capacitor The final alue of the oltage at its terminals is. The effect of capacitance is to cause the oltage at the terminal to rise to the full alue gradually, instead of abruptly, so it flattens the wae front. Flattening the wae front is beneficial because it reduces the stress on the line-end windings of a transformer connected to the line. Test Yourself What is the benefit achieed by flattening the wae front of the incident wae in a transformer? Example 5. A k,.5 s rectangular surge traels on a line of surge impedance of 4 Ω. The line is terminated in a capacitance of 5 p.f. Find the oltage across the capacitance.

25 TRANSMISSION LINE TRANSIENTS 5 Solution: oltage wae magnitude k Time duration, t.5 μs Surge impedence, Z 4 Ω Terminated capacitance, C 5 p.f. Z c 4 Ω S ± C 5 pf Fig. 5.9 Circuit diagram for Example 5. t/cz t () ( e ) e k Example 5. A 5 k,.5 s duration rectangular surge passes through a line haing surge impedance of 4 Ω and approaches a station at which the concentrated earth capacitance is 3 3 p.f. Calculate the maximum alue of surge transmitted to the second line. Solution: oltage wae magnitude, 5 k Time duration, t.5 μs Surge impedance, Z 4 Ω Earth capacitance, C 3 3 p.f. The maximum alue of surge transmitted to the second line is gien by, tzc / t () ( e ) 5 e k ( ) Z Z Fig. 5. Capacitor connected at T C

26 6 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION Capacitor Connection at T From Fig. 5., the transmitted oltage across the capacitor of the circuit is calculated from Eq. (5.6). The transmitted oltage is Let, Then ( s) ZC ( s) s s + α Z Z + Z ZZC ( s) s Z+ Z s + α α Z ZZC s Z + Z s + α Z Z + Z s s + α Taking the inerse Laplace transform on both sides of the aboe equation, t () Zs + + Cs Z Z sz + C Z Z s + ZZC Z+ Z α ZZC Z Z + Z e Z Z + ZZ C t Z s ZZC Z+ Z + ZZC s 5.. LINE TERMINATED THROUGH INDUCTANCE From Fig. 5., the transmitted oltage across the inductor of the circuit is calculated from Eq. (5.6). Ls The transmitted oltage, ( s) Z + Ls s ( s+ Z L) Take inerse Laplace transform on both sides of the aboe equation, t () e Z L t (5.4)

27 TRANSMISSION LINE TRANSIENTS 7 Z i L L Fig. 5. Transmission line terminated by inductance Z L t Reflected oltage, t ( e ) (5.43) () Example 5. A step wae of k traels through a line haing a surge impedance of 35 Ω. The line is terminated by an inductance of 5 H. Find the oltage across the inductance and reflected oltage wae. Solution: oltage wae magnitude, k Surge impedance, Z 35 Ω Inductance connected, L 5 H Let, time duration t s Then, the oltage across the inductance is, t () e Z L t e e 7. t k 35 6 t 6 5 Z Reflected oltage, () t e L t e 35 t ( ).7t e k Example 5.3 A rectangular surge of.5 μs duration and magnitude k traels along a line of surge impedance 4 Ω. The latter is connected to another line of equal impedance through an inductor of 5 H. Calculate the maximum alue of the surge transmitted to the second line. Solution: oltage wae magnitude, k

28 8 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION Time duration, t.5 μs Surge impedance, Z 4 Ω Inductance connected, L 5 H The maximum alue of surge transmitted to the second line is gien by, t () e Z L t e 4 e k 5.3 BEWLEY S LATTICE DIAGRAM This is a graphical representation of the time-space relation, which shows the position and direction of motion at any instant of incident, reflected and transmitted current or oltage surges. In a Lattice diagram, the horizontal axes represent the distance traelled along the system and ertical axes represent the time taken to trael. At each instant of change in impedance, the reflected and transmitted alues (current or oltages) can be calculated by multiplying incident wae alues with reflected and transmitted coefficients. Case-: Receiing End is Open-circuited Consider a line connected to a source of constant oltage at one end and open circuited at the other, as shown in Fig. 5.(a). Let α s and α L be the reflection coefficients at the sending end and the load end respectiely, and t, the time taken by the wae to trael from one end to the other end. When time t s, the oltage is connected to the source end (s) and starts traelling along the line reaching the load end in time t s with the same magnitude. Since the load end is an open circuit, the wae reflected back with a magnitude of α L (because α L for an open-ended line) at time t + s, reaches source end in time t with a magnitude of. The reflected wae is reflected back once again with a magnitude of s from the source end after reaching the source end at time t + and this process is continued indefinitely. The same procedure can be implemented for current waes also. This procedure is illustrated in the Lattice diagram shown in Fig. 5.(b). Z R S Fig. 5.(a) Circuit diagram

29 TRANSMISSION LINE TRANSIENTS 9 Source end Load end t t t α S 3t α S 4t α S 5t Fig. 5.(b) Lattice diagram Case-: Receiing End is Connected with Resistance R Consider a line connected to a source of constant at one end and the other end is connected by a resistance R as shown in Fig. 5.3(a). Let α s and α L be the reflection coefficients at the sending-end and the load-end respectiely, and the time taken by the wae to trael from one end to the other in it. Z R s R Fig. 5.3(a) Circuit diagram

30 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION Source end Load end t α L Z t αα S L αα S L 3t α S α L 5t Fig. 5.3(b) Lattice diagram When time t s, the oltage is connected to the source end (s) and starts traelling along the line, and reaching the load end in time t s with the same magnitude. After reaching the junction, the wae is split into two parts, one part of the wae is transmitted and the other is reflected back with a magnitude of α L at time t + it traels towards source and reaches the source end in time t with a magnitude of α L. The wae, which has reached the source end, splits into two parts once again. One part is transmitted and the other is reflected back with a magnitude of α s α L from source end at time t +. This process is continued indefinitely. The same procedure can be implemented for current waes also. This procedure is illustrated in the Lattice diagram shown in Fig. 5.3(b). Example 5.4 Construct a Bewley lattice diagram when a pulse source of magnitude olts with a resistance of 5 Ω, is applied across a loss-free line with surge impedance of 4 Ω terminated with a resistance of Ω (see Fig. 5.4). Assume the line to be of km length. Solution: Zs Z 5 4 αs.4545 Zs + Z R Z 4 αl.33 R+ Z + 4 Z s 5r Z 4r R r Fig. 5.4 Circuit diagram for Example 5.4

31 TRANSMISSION LINE TRANSIENTS S L α.33 α S.4545 t 8/.4 t t.636 5t Fig. 5.5 Lattice diagram for Example 5.4 At the input of the line, the impressed oltage on line Z 4 8 Z + Z s The reflected oltage at load end. 8 8 α L This.4 backward pulse reaches the source after t s and gets reflection. The reflection oltage at source s (.4 ) This reflected oltage.98 reaches the load-end and is again reflected back. This processes is continued and is shown in Fig ATTENUATION OF TRAELLING WAES So far we hae studied the lossless oerhead transmission lines, so there is no attenuation. It is not true for practical systems. The analysis is more difficult due to the presence of losses. Howeer, these losses are ery much attactie because the energy of waes is dissipated through these losses. These losses are due to the presence of resistance R and conductance G of oerhead lines. Consider r, L, C and g as the parameters per unit length of an oerhead transmission line and and I as the oltage and current waes at x as shown in Fig The aim is to determine the oltage () and current (I) waes after traeling a distance x with time t s.

32 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION I I x x x Fig. 5.6 Wae traelling on a lossy line I r x L x g x C x x Fig. 5.7 Equialent circuit of differential element of oerhead line Let us consider a small distance dx traeled by the wae in time dt. The differential length, Δx, of the oerhead line is shown in Fig. 5.7 Power loss, P I r + g (5.44) Z r+ g On differentiation of Eq. (5.44) with respect to x, we get dp rdx rdx Z + (5.45) Power at a distance x, P I (5.46) Z Negatie sign indicates there is reduction in power as the wae traels with time. Differentiation of Eq. (5.46) with respect to is dp (5.47) Z d From Eqs. (5.45) and (5.47) + Z d rdx rdx Z d + dx Z r gz + d r gz dx Z ( )

33 TRANSMISSION LINE TRANSIENTS 3 d r gz + dx Z r gz + ln x+ K Z At x, K ln r gz ln + x+ ln Z r gz ln + x Z e r+ gz Z x where r α + gz Z α x e e α x (5.48) Similarly we can derie the expression for current, I I e αx (5.49) From eqs. (5.48) and (5.49) the oltage and current waes are attenuated exponentially as they trael oer the transmission line and the magnitude of attenuation depends upon the oerhead line parameters. From the empirical formula of Foust and Menger, oltage and current at any point of the oerhad line after traeling x distance can be calculated as k + Kx where K attenuation constant.37 for chopped waes.9 for short waes. for long waes CHAPTER AT A GLANCE. A lightning discharge or a sudden switching in or out in a power system suddenly impresses electrical energy in a transmission line, which moes along the line at nearly the speed of light and is called a traelling wae.. Types of system transients: Depending upon the speed of the transients, these can be classified as: surge phenomena, short circuit phenomena and transient stability. 3. Wae equations: LC i or x t x LC i t

34 4 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION 4. Incident wae: f LCx t or i ( / Z ) 5. Reflected wae: φ LCx+ t or i ( / Z ) 6. Ealuation of surge resistance: Oerhead transmission line, Z 6ln(D/R) Ω Underground cables, Z ( ) 7. Importance of surge impedance: Knowledge of surge impedance is extremely useful as it enables the calculation of the transient oltages and currents which may occur in a circuit. 8. Ealuation of elocity of wae propagation Oerhead transmission line, 3 8 m/s Underground cables, υ 3 8 ε ( ) r R r 6ln ( ) ε r m/s Ω SHORT ANSWER QUESTIONS. What is a traelling wae?. What are the factors that cause a traelling wae? 3. What are the alues of characteristic impedance for transmission lines, cables and transformers? 4. What is the elocity of propagation of a surge in oerhead lines and cables? 5. Write the equation of an impulse wae explaining the significance of each term. 6. What is the effect when the reflected wae meets (i) a short-circuited line (ii) an open-circuited line, and (iii) a resistance equal to characteristic impedance of the line? 7. What are the expressions for the oltage and current when a line is terminated by an (i) inductance (ii) a capacitance? 8. Why is the elocity of propagation same for all oerhead lines? 9. What is meant by crest of a wae?. What is meant by wae front?. A transmission line of surge impedance Z is terminated through a resistance R. Gie the coefficients of refraction and reflection.. What is the effect of shunt capacitance at the terminal of a transmission line? 3. What is the application of Bewley s diagram?

35 TRANSMISSION LINE TRANSIENTS 5 MULTIPLE CHOICE QUESTIONS. For lossless line terminated by its surge impedance, the natural reactie power loading is a. /X b. /Z c. /Z c d. /R. A lossless line terminated with its surge impedance has a. flat oltage profile b. transmission line angle greater than actual length of line c. transmission line angle less than actual length of line d. a and b 3. An oerhead transmission line haing a surge impedance of 4 Ω is connected in series with an underground cable haing a surge impedance of Ω. If a source of 5 k traels from the line end towards the line-cable junctions, the alue of the transmitted-oltage wae at the junction is a. 3 k b. k c. 8 k d. 3 k 4. When a transmission line is energized through, propagate on it. a. oltage wae b. current wae c. both oltage and current d. power wae 5. The coefficient of reflection for current at an open-ended line is a.. b..5 c.. d. zero 6. The reflection between traelling oltage and current waes is gien as L L a. i b. c. i LC d. C i C i LC 7. Traelling oltage and current waes hae the same waeforms and trael together along the transmission line at a a. elocity of sound b. elocity of light c. slightly lesser than light d. more than the light 8. For an open-circuited line, the resulting current will be a. zero b. infinity c. equal to the incident oltage d. twice the incident oltage 9. For a short-circuited line, the resulting oltage will be a. infinity b. zero c. equal to the incident oltage d. twice the incident oltage. Steepness of the traelling wae is attenuated by a. line resistance b. line inductance c. line capacitance d. both b and c

36 6 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION. The steepness of the wae front can be reduced by connecting a. an inductor in series with the line b. a capacitor between line and earth c. both a and b d. an inductor between line and earth or a capacitor in series with the line. The reflection coefficient of the oltage wae in oerhead lines is gien by R Rl Rl Ro Rl + Ro a. b. c. d. R R R R R R R R l l l l 3. The reflection coefficient of a short circuit line is a. b. c..5 d. 4. The propagation constant of a transmission line is (.5 3 j.5 3 ) and the wae length of the traelling wae is a π b. π c d. π 5. 3 π The reflection coefficient at the load end of a short-circuited line is a. b. c. 9 d The reflection coefficient of the wae when load connected to a transmission line of surge impedance equals the load surge impedance is a. b. c. d. infinity 7. A surge oltage of k is applied to an oerhead line with its receiing end open. If the surge impedance of the line is 5 Ω, then the total surge power in the line is a. MW b. 5 MW c. MW d..5 MW 8. A surge of 6 k traelling in a line of neutral impedance of 5 Ω arries at the junction with two lines of neutral impedances of 5 Ω and 5 Ω, respectiely. The oltage transmitted in the branch line is a. 4 k b. 6 k c. 8 k d. 4 k 9. The coefficient of reflection for current for an open-ended line is a.. b.. c..5 d.. The real part of the propagation constant of a transmission line is a. attenuation constant b. phase constant c. reliability factor d. line constant. Two transmission lines each haing an impedance of Ω is separated by a cable. For zero reflection the impedance of the cable should be a. Ω b. Ω c. 4 Ω d. 6 Ω

37 TRANSMISSION LINE TRANSIENTS 7. An oerhead line with surge impedance of 4 Ω is terminated through a resistance R. A surge traelling oer the line will not suffer any reflection at the junction, if the alue of R is a. Ω b. 4 Ω c. Ω d. 6 Ω Answers:. d,. a, 3. b, 4. c, 5. c, 6. b, 7. c, 8. a, 9. b,. a,. c,. c, 3. a, 4. b, 5. d, 6. a, 7. a, 8. d, 9. b,. a,. b,. c REIEW QUESTIONS. Deelop an equialent circuit at the transition points of transmission lines for analyzing the behaiour of traelling waes.. Discuss the phenomena of wae reflection and refraction. Derie an expression for the reflection and refraction coefficients. 3. Describe the construction and the working principle of a zinc oxide gapless arrester with a neat sketch. 4. Starting from the first principles, show that surges behae as traelling waes. Derie expressions for surge impedance and wae elocity. 5. Explain Bewley s Lattice diagram and gie its uses. 6. Define surge impedance of a line. Obtain the expressions for oltage and current waes at a junction or at a transition point. PROBLEMS. A oltage haing a crest alue of k is traelling on a 4 k line. The protectie leel is k. The surge impedance of the line is Ω. Calculate (a) the current in the line before reaching the arrester, (b) the current through the arrester and (c) the alue of arrester resistance for this condition (d) the reflected oltage. erify the reflection and refraction coefficients.. A 5 k surge traels on an oerhead line of surge impedance 4 Ω towards its junction with a cable which has a surge impedance of 4 Ω. Find (a) transmitted oltage and current (b) reflected oltage and current. 3. A k surge traels on a transmission line of 4 Ω surge impedance and reaches a junction where two branch lines of surge impedances of 5 Ω and 3 Ω respectiely are connected with the transmission line. Find the surge oltage and current transmitted into each branch line. Also, find the reflected oltage and current. 4. A transmission line has an inductance of.93 H/km and a capacitance of.78 F/km. This oerhead line is connected to an underground cable haing an inductance of.55 mh/km and a capacitance of.87 F/km. If a surge of crest k traels in the cable towards its junction with the line, find the surge transmitted along the line.

38 8 ELECTRIC POWER TRANSMISSION AND DISTRIBUTION 5. A k, 3 s, rectangular surge traels on a line of surge impedance of 4 Ω. The line is terminated in a capacitance of 3 p.f. Find an expression for oltage across the capacitance. 6. An inductance of 7 H connects two sections of a transmission line each haing a surge impedance of 35 Ω. A 4 k, s rectangular surge traels along the line towards the inductance. Find the maximum alue of the transmitted wae.