In this article we will discuss about:- 1. Introduction to Travelling Waves 2. Surge Impedance and Velocity of Propagation 3. Specifications 4. Reflection and Refraction5. Typical Cases of Line Terminations 6. Equivalent Circuit 7. Bifurcated Line 8. Reactive Termination 9. Successive Reflections, Bewley Lattice or Zigzag Diagram.
Introduction to Travelling Waves:
A transmission line is a distributed parameter circuit and a distinguishing feature of such a circuit is its ability to support travelling waves of voltage and current. A circuit with distributed parameters has a finite velocity of electromagnetic field propagation. In such a circuit the changes in voltage and current, owing to switching and lightning, do not occur simultaneously in all parts of the circuit but spread out in the form of travelling waves or surges.
When a transmission line is suddenly connected to a voltage source by closing of a switch, the whole of the line is not energized all at once (the voltage does not appear instantaneously at the other end). This is due to presence of distributed constants (inductance and capacitance in a loss free line).
When switch S is closed, the inductance L1 acts as an open circuit and C1 as short circuit instantaneously. The same instant next section cannot be charged because the voltage across capacitor C1 is zero. So unless the capacitor C1 is charged to some value whatsoever, charging of the capacitor C2 through L2 is not possible which, of course, will take some finite time.
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The same argument applies to the third section, fourth section and so on. So we see that the voltage at the successive sections builds up gradually. This gradual build-up of voltage over the transmission line conductors can be regarded as though a voltage wave is travelling from one end to the other end and the gradual charging of the capacitances is due to associated current wave. The current wave, which is accompanied by a voltage wave sets up a magnetic field in the surrounding space.
At junctions and terminations these surges undergo reflections and refractions. In an extensive network with many lines and junctions, the number of travelling waves initiated by a single incident wave will mushroom at a considerable rate as the waves split and multiple reflections occur.
It is true that the total energy of the resultant waves cannot exceed the energy of the incident wave. However, it is possible for the voltage to build up at certain junctions due to reinforcing action of several waves. For a complete study of the phenomenon the use of Bewley lattice diagram or digital computer is necessary. In this article we will study some of the elementary aspects of wave propagation, reflections and refractions.
Surge Impedance and Velocity of Propagation:
The gradual establishment of line voltage can be considered as due to voltage wave travelling from the supply source end towards the far end, and the progressive charging of the line capacitances will account for the associated current wave.
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Assume that in a very small time δt the conditions of a current I and a voltage E are established along a length δx of the line (Fig. 8.2). The emf E is balanced by the back emf generated by the magnetic flux which is produced by the current in this length of the line. The inductance of the length δx is Lδx, (L is inductance of line per unit length) so that the flux built up is IL δx and the back emf is the rate of build-up viz. IL δx/δt.
So we have E = [IL (δx/δt)] = ILv …(8.1)
where v is the velocity of propagation of wave.
ADVERTISEMENTS:
The current I carries a charge I δt in the time δt, and this charge remains on the line to charge it up to the potential of E.
Since the capacitance of length δx of the line is C δx (C is the capacitance of the line per unit length), its charge is ECδx, so we have –
I δt = EC δx
or I = [EC (δx/δt)] = EC v …(8.2)
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The switching of an emf E on to the line results therefore in a wave of current I and velocity v where I and v are given by Eqs. (8.1) and (8.2).
Dividing Eq. (8.1) by Eq. (8.2), we have –
E/I = ILv/ECv = I/E . L/C
or E2/I2 = L/C
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or E/I = √L/C = Zn (say) …(8.3)
The expression is a ratio of voltage and current which has the dimensions of impedance and is therefore here designated as surge impedance of the line. It is also called the natural impedance because this impedance has nothing to do with the load impedance, but depends only on the line constants. The value of this impedance is 400 Ω to 600 Ω for an overhead line and 40 to 60 Ω for a cable.
Multiplying Eqs. (8.1) and (8.2), we have –
EI =ILv x EC v = E I LC v2
or v2 = I/LC
or v = √1/LC …(8.4)
Substituting the values of L and C for overhead lines in above equation, we have –
Since the product of L and C is the same for all overhead lines, it follows that the velocity of propagation is also the same. This velocity is the same as the velocity of light, but as we have assumed a resistanceless line in the above analysis, the velocity in practice will be from 5 to 10 percent less than this. Normally a velocity of approximately 285 m/µs is assumed.
The velocity of propagation over the cables will be smaller than that over the overhead lines because in case of overhead lines ԑr = 1 while for cables ԑ r > 1, being the dielectric constant. The velocity of wave propagation in case of cables can be given as –
where ԑ r varies from 2.5 to 4 in case of cables.
Specifications of Travelling Waves:
A travelling wave is characterized by four specifications, as illustrated in Fig. 8.3.
1. Crest:
Crest of the wave is the maximum amplitude of the wave and is usually expressed in kV or kA.
2. Front:
Front of the wave is the portion of the wave before crest and is expressed in time from beginning of the wave to the crest value in ms or µs. However, for waves having a slow initial rate of rise, i.e., a long toe, it is better to consider virtual front as determined by straight lines between 30% and 90% points [Fig. 8.3(b)]. The virtual front is 1.667 × t1. The extension of this straight line to X-axis gives O1 i.e., the virtual zero.
3. Tail:
Tail of the wave is the portion beyond the crest. It is expressed in time (µs) from beginning of the wave to the point where the wave has reduced to 50% of its value at crest. For waves having a slow initial rate of rise, i.e., long toe, the tail time is measured from O1 to 50% value on tail [Fig. 8.3(b)],
4. Polarity:
Polarity is the polarity of crest voltage or current. A positive wave of 500 kV crest, 1 µs front and 25 µs tail will be represented as + 500/1.0/25.0.
A travelling wave can be represented mathematically in a number of ways. The simplest and most commonly used representation is the infinite rectangular or step wave illustrated in Fig. 8.4. Such a wave jumps suddenly from zero to full value and is maintained at that value there-after.
As this wave has front causing maximum gradients and sustained tail producing maximum oscillations in machine windings, it is most dangerous to apparatus/equipment. Hence, the analysis based on it is liable to err on the safer side.
Reflection and Refraction of Travelling Waves:
If a travelling wave arrives at a point where the impedance suddenly changes the wave is partly transmitted and partly reflected. Loading points, line-cable junctions and even faults constitute such discontinuities. Independent waves meeting along a line will combine in accordance with their polarity to provide different voltage and current levels at the meeting point.
It is convenient to adopt a standard sign convention, and in what follows, forward waves of current and voltage are given the same polarity. If the wave is being reflected the corresponding current and voltage waves are given opposite polarity. This may be illustrated by considering waves of current and voltage being transmitted along a line of characteristic impedance ZC terminated by an impedance Z (Fig. 8.5).
Let E and I represent the incident waves, ET and IT represent the transmitted (or refracted) waves and ER and IR the reflected waves. The state of affairs is illustrated in Fig. 8.6.
The following relations hold good for incident, transmitted and reflected voltage and current waves –
E = I ZC …(8.7a)
ET = IT Z …(8.7b)
ER = – IR ZC …(8.7c)
The negative sign in Eq. (8.7c) is because of the fact that ER and IR are travelling in the negative direction of x or backwards on the same line.
The transmitted voltage and current will be respectively the algebraic sum of incident and reflected voltage and current waves.
ET = E + ER …(8.8a)
IT = I + IR …(8.8b)
Substituting the values of I, IR and IT from Eqs. (8.7a, b, c) in Eq. (8.8 b), we have –
ET/Z = [(E/ZC) – (ER/ZC)] …(8.9)
From Eqs. (8.8 a) and (8.9), we have –
ET/Z = [(E/ZC) – (ET – E/ZC)]
or ET x ZC /Z + ET = 2 E
or ET [1 + (ZC/Z)] = 2 E
or ET = {E [2Z/(Z + ZC)]} …(8.10)
IT = (ET/Z) = [2 E/(Z + ZC)] = (2 ZC I/Z + ZC) …(8.11)
ER = ET – E = {[2 ZE/(Z + ZC)] – E} = [E (Z – ZC)/(Z + ZC)] …(8.12)
or IR = (– ER/ZC)= [(– E/ZC)x (Z – ZC/Z + ZC)] = [I (ZC Z/ZC + Z)] …(8.13)
The coefficients (2Z/Z + ZC)and (Z – ZC/Z + ZC)are called coefficients of refraction and reflection respectively.
It will be observed that the transmitted or refracted current and voltage always have positive polarity. The polarity of the reflected waves depends on the magnitude relationship between ZC and Z. If ZC > Z, the voltage wave is negative and the current wave positive, but vice-versa if Z > ZC.
Typical Cases of Line Terminations:
1. Short-Circuited Line:
If the line is short circuited at the receiving end, i.e., Z = 0, then the transmitted and reflected waves are given as –
ET = 0 …(8.14 a)
IT = 2 I …(8.14 b)
ER = – E …(8.14 c)
IR = I …(8.14 d)
The unique characteristic of the short circuit is that voltage across it is zero. When an incident voltage wave E arrives a short circuit, the reflected voltage wave must be – E to satisfy the condition that the voltage across the short circuit is zero.
The waves are shown in Fig. 8.7:
2. Open-Circuited Line:
If the line is open circuited at the receiving end, i.e., Z is infinite, the transmitted and reflected waves are given as –
ET = 2 E (doubling effect) …(8.15 a)
IT = 0 …(8.15b)
ER = E …(8.15c)
IR = – I …(8.15 d)
An open circuit at the end of a line demands that the current at that point is always zero. Thus when an incident current wave I arrives at the open circuit, a reflected wave equal to – I is at once initiated to satisfy the boundary condition. The waves are shown in Fig. 8.8.
3. Line Terminated by an Impedance Equal to Surge Impedance:
If the line is terminated with an impedance equal to surge impedance, i.e., Z = ZC, we have –
ET = E …(8.16 a)
IT = I …(8.16 b)
ER = 0 …(8.16 c)
IR = 0 …(8.16 d)
It means that the line is correctly terminated and there will be no reflection and ET and IT will be equal respectively to E and I.
4. Line Connected to a Cable:
A wave travelling over the line and entering the cable, as shown in Fig. 8.9, looks into a different impedance and, therefore, it suffers reflection and refraction at the junction.
The refracted or transmitted voltage is given as –
ET = 2 Z2 E/Z1 + Z2 …(8.17)
The other waves can be had by employing the relations of Eqs. (8.11), (8.12) and (8.13). The surge impedances of the overhead line and cable are approximately 500 Ω and 50 Ω respectively. With these values it can be seen that the voltage entering the cable will be –
ET = {E x [(2 x 50)/(50 + 500)]} = 2/11 E
or roughly 20% of the incident voltage.
This is the reason that an overhead line is terminated near a station by connecting the station equipment to the overhead line through a short length of underground cable. Besides the reduction in magnitude of the voltage wave, it also reduces the steepness of the wave. It is because of capacitance of the cable.
The reduction in steepness is very important because it is one of the factors for reducing the voltage distribution along the equipment windings. In connecting the overhead line to a station equipment through a cable the important point to be remembered is that the length of the cable should not be shorter than the expected length of the wave otherwise successive reflections at the junction may cause piling up of voltage and the voltage at the junction may attain the value of incident voltage.
Equivalent Circuit for Travelling Wave Studies:
An equivalent circuit for studies of travelling waves can be easily developed by making use of Thevenin’s theorem that provides a mathematical technique for replacing a two terminal network by a voltage source VT, and resistance RT connected in series.
The voltage source VT, called the Thevenin’s equivalent voltage, is the open-circuit voltage that appears across the load terminals when the load is removed or disconnected and resistance RT, called the Thevenin’s equivalent resistance, is equal to the resistance of the network looking back into the load terminals.
A line with a surge impedance ZC is shown in Fig. 8.10(a). A surge wave E travels on the line towards the terminals XX’. The open-circuit voltage across the terminals XX’ is 2 E and the impedance seen from the terminals XX’, with the sending end of the line short circuited is ZC. Thus the circuit given in Fig. 8.10(a) can be replaced by the equivalent circuit shown in Fig. 8.10(b).
From the equivalent circuit shown –
Current through Z, IT = [2 E/(Z + ZC)] …(8.18)
and voltage across Z, ET = [2 EZ/(Z + ZC)] …(8.19)
The above results are the same as Eqs. (8.11) and (8.10). The values of ER and IR can be determined from Eqs. 8.8(a) and (b). The equivalent circuit is very useful for the studies of travelling waves and can be employed for resistive as well as reactive terminations.
Bifurcated Line:
Let a line of natural impedance ZC bifurcate into two branches of natural impedances Z1 and Z2. As far as the voltage wave is concerned, the refracted (transmitted) portion will be the same for both branches as they are in parallel but the refracted currents will be different as Z1 ≠ Z2.
Let the incident wave be (E, I) travelling to the right, the reflected wave (ER, IR) travelling to the left and the transmitted waves (ET, IT1) and (ET, IT2) travelling towards the right, as shown in Fig. 8.11(a).
The analysis is exactly similar to that of a line terminated by impedance Z and the desired results can be had by using Eqs. (8.10), (8.11), (8.12), (8.13) and (8.8a) and (8.8b). Equivalent circuit is shown in Fig. 8.11 (b).
From equivalent circuit shown in Fig. 8.11 (b), we have –
Z = [Z1 Z2/(Z1 + Z2)] …(8.20)
ET = {[2 E/(Z1 Z2)/(Z1 + Z2)] + ZC x [Z1Z2/(Z1 + Z2)]} …(8.21)
IT1 = ET/Z1 …(8.22)
and IT2 = ET/Z2 …(8.23)
ER = ET – E …(8.24)
IR = IT1 + IT2 – I …(8.25)
The above procedure can be extended for the analysis of a line branching off into any number of lines.
Reactive Termination:
1. Reflection from Terminal Inductance:
Let the line be terminated with an inductance L, as illustrated in Fig. 8.12.
The voltage across the inductance is given as –
eT = eL = [L (diL/dt)] …(8.26)
Since from Eqs. (8.86), (8.7a) and (8.7c)
iL = iT = i + iR = [(e/ZC) – (eR/ZC)] …(8.27)
Substituting value of iL from Eq. (8.27) in Eq. (8.26), we have –
eL = [L (d/dt)] [(e/ZC) – (eR/ZC)] …(8.28)
and since from Eq. (8.8a)
eL = e + eR
So e + eR = [L (d/dt)] [(e/ZC) – (eR/ZC)]
or {– e + [(L/ZC)(de/dt)]}= {eR + [(L/ZC)(deR/dt)]} …(8.29)
Let the incident wave e be of constant value E, so that its time derivative is zero. So Eq. (8.29) becomes –
[(L/ZC)(deR/dt)] = – (E + eR)
or [deR/(E + eR)] = (– ZC/L)dt …(8.30)
Integrating both sides of above Eq. (8.30), we have –
loge (E + eR) = (– ZC/L)t + A …(8.31)
where A is a constant of integration which can be evaluated from initial condition i.e., when t = 0, iL = 0, because when the wave arrives at the inductance terminal the inductance L does not carry any current, the whole of it is reflected. It means that when t = 0, eR = E and therefore
loge (E + E) = 0 + A
or A = loge 2 E
Substituting A = loge 2 E in Eq. (8.31), we have –
loge (E + eR) = (– ZC/L)t + loge 2 E
or loge [(E + eR)/2 E] = (– ZC/L)t
From the above Eq. (8.32) it is observed that the reflected voltage varies exponentially from + E to – E which is explained by the fact that as the wave arrives the inductance terminal it does not carry any current initially, but with the passage of time, it carries more and more current, finally (theoretically with elapse of infinite time) it behaves as if the line is short circuited in which case the reflected voltage will be – E.
Also when the potential difference across the inductance is zero, the current passing though it will be twice that of incident current (refer to Eq. 8.14b)
Again eL = e + eR [refer to Eq. (8.8a)]
= E + E (2 e(– ZC/L)t – 1) = 2 Ee (– ZC/L)t …(8.33)
Current through inductance, iL = [(E/ZC)– (eR/ZC)]
From the above (8.34) it is observed that the current through inductance also varies exponentially. eL, iL and eR as functions of time are shown in Fig. 8.13(a). The disposition of voltage waves at different instants is shown in Figs. 8.13 (b, c and d).
If the line is terminated with an inductance in parallel with resistance R, it can be shown that –
2. Reflection from Terminal Capacitance:
When the line is terminated with a capacitor, a charging current flows through it and at the moment of arrival of the incident wave (of value E volts) the capacitor acts as a short circuit. At this instant the reflected voltage wave is -E (because initially the charge across the capacitor is zero) and the current of the incident wave is momentarily doubled.
As the capacitor gets charged the terminal voltage rises. Finally the capacitor becomes fully charged; it then behaves as an open-circuit for the terminal current is zero and the voltage of the incident voltage is doubled.
The reflected voltage varies exponentially and its mathematical expression can be arrived in the same way as in case of terminal inductance.
The expressions are given as –
eT, iT and eR as functions of time are shown in Fig. 8.15(a). The disposition of voltage wave at different instants is shown in Figs. 8.15 (b, c, d). In travelling wave studies, time t is usually expressed is µs, so L and C should be expressed in µH and µF respectively.
The capacitance termination is of more practical importance owing to the paradox that for lightning waves a transformer acts as a capacitance rather than as an inductance. The lightning surge arrives at a transformer as a travelling wave, and the front of the wave is so steep and rises so suddenly to a maximum, that there is practically no time for current to start to flow through the large inductance of the transformer winding.
However, there is some small capacitance in the transformer (between turns and between winding and the core), and the transformer’s reaction to the lightning surge is governed largely by this capacitance, rather than by inductance.
3. Reflection from a Line Terminated with a Parallel Combination of Capacitance and Resistance:
When a travelling wave meets a termination composed of parallel combination of a capacitance C and resistance R, as illustrated in Fig. 8.16, the problem is the same as that illustrated in Fig. 8.15, except that Z in Eqs. (8.10) and (8.11) must be replaced by
The voltage at the termination is thus
It must be remembered that p = d/dt and E is a voltage which is zero until t = 0 and E after t = 0,
ET may be determined as follows:
This is a linear differential equation for ET of which the solution is –
where A is an arbitrary constant and is determined by the fact that ET can rise at a finite rate from its zero value.
This gives –
where ET0 is the voltage at the end when there is no capacitance. Fig. 8.17 shows the curve for voltage at the termination, ET. The effect of capacitance is to cause the voltage at the end to rise to the full value gradually instead of abruptly, i.e., it flattens the wave front.
It is usual to specify the condition of the wave front by stating the time the wave takes to increase from 10 to 90 percent of its value and multiplying by 1.25. If the wave attains x of its value in time t.
The specifying time in this case is therefore
In the case of a capacitance at a point of a line which stretches in both directions away from it, ZC = R and the time is 1.375 CZC second.
Thus a 0.01 µF capacitance in a line of surge impedance ZC = 500 Ω flattens the wave so that the time of the wave front becomes 1.375 × 10-8 × 500 = 6.875 micro-second.
Flattening the wave front has a very beneficial effect, as it reduces the stress on the line-end windings of a transformer connected to the line.
Successive Reflections, Bewley Lattice or Zigzag Diagram:
In an extensive network having many junctions and terminations, the number of transmitted and reflected waves, initiated by a single incident wave, increases as the wave meets different junctions.
Generally it becomes difficult to keep track of the transmitted and reflected waves, but with the use of Bewley’s lattice diagram one can know at a glance the position and direction of motion of every incident, reflected and transmitted wave on the system at every instant of time.
For example, let us take the case of an open-circuited line having series resistance as well as inductance, and shunt leakance as well as capacitance. With such a line there is attenuation of both voltage and current waves as they travel along the line.
In the Heaviside distortionless line the voltage and current waves remain similar inspite of attenuation, and the condition for such a distortionless line is –
RC = GL …(8.43)
In such a line it can be shown that if a wave is of amplitude A1 at any point of the line, the amplitude A2 at some point distant x is –
A2 = A1 e-βx …(8.44)
where β is a constant, known as the attenuation constant. For the distortion-less line the value of this constant is √RG. With such a line any wave travelling the whole length of the line will suffer definite percentage attenuation, and, knowing this attenuation, it is possible to determine the amplitude of any wave at any instant. Let us assume the following data for an open-circuited line.
R = 0.3 Ω per km, G = 6.5 × 10-7 S per km; l= 640 km
So attenuation constant, β = √RG = √0.3 x 6.5 x 10-7 = 0.0004416
and e-βx = e-0.0004416 × 640 = e-0.2826 = 0.754 ... x = l = 640 km
If this quantity is denoted by a and the initial value of the voltage at the generating end by unity, then following sequence of events, as far as the reflected wave is concerned, will occur. The time taken to make one tour or line, t’ = (640 x 1,000)/(2.975 x 108) = 0.00215 s.
At zero time a wave of amplitude 1 start from generating or sending end G. At time t’ a wave of amplitude α= 0.754 strikes the open end and a reflected wave of + α = 0.754 commences the return journey. At time 2t’ this reflected wave is attenuated to α2 = 0.5685 and has reached G.
Here it is reflected to – α2 = – 0.5685 and after a time 3 t’ it reaches the open end attenuated to – α3 = – 0.4287. It is then reflected without change of sign and reaches G after a time 4t’ with an amplitude of – α4 = –0.3232. It is then reflected with a change of sign, thus starting off with an amplitude of + α4 = + 0.3232, and so on.
The Bewley lattice diagram is a space-time diagram with the space measured horizontally and time vertically, and the lattice for the above example is shown in Fig. 8.18:
Now, the increment of the voltage at the receiving end due to any reflection is twice the amplitude of the incident wave, because of the reflection without change of sign. Also, the final voltage at this end is the sum to infinity of all such increments.
Thus, in the above example, it is –
2 (0.754 – 0.4287 + 0.2437 – …).
It is simpler to express the series generally in term of α, thus,
Thus, even when open circuited, such a line gives a far-end voltage less than the sending-end voltage, the reason being that the shunt leakance causes a drop along the series resistance.
As far as the sending end is concerned, the voltage at the moment of incidence of a wave sent from the far end is unity plus the amplitude of this wave. The lattice shows the amplitudes of these incident waves to be + α2, – α4, and so on, successive sending-end voltages thus being 1, 1 + α 2, 1, 1 – α4, 1 and so on. Thus the voltage oscillates about unity, the amplitude of the oscillation rapidly diminishing so that eventually the voltage becomes the generator voltage of unity, as would be expected.
Now consider the case of the ideal resistanceless and leakanceless line joined at the far end by a resistance R. There is no attenuation as a wave travels along the line, but at the receiving end there is a reflection R –Zc/R + ZC which is less than unity. We can thus use this in place of α in the lattice diagram at the receiving end, and α = – 1 at the sending end. Suppose that R – ZC/R + ZC = 0.4, then the lattice diagram will be as illustrated in Fig. 8.19.
At the receiving end the increment of voltage is the sum of the incident and reflected waves at each reflection, so that the ultimate voltage at this point is the sum to infinity of the series.
Thus the voltage at the receiving end finally settles down to that at the sending end, and consequently the current settles down to the simple Ohm’s law value of E/R. The increments of current are obviously proportional to the increments of voltage at the receiving end, and therefore voltage-time and current-time curves for this end for α = 0.4 are as illustrated in Fig. 8.20. The tabulated values are as shown, and it will be seen that both voltage and current oscillate about the value unity and finally settles down to this value.
