UNIT II TANSMISSION LINE THEOY Name of the subject: Transmission Lines and Networks Name of the Faculty: LAXMINAAYANAN.G Sub Code: EC T55 Yr/Sem/Sec:III/V/ TANSMISSION LINE EQUATION: Short section PQ of length dx at a distance x from the sending end A. By making dx very small, the current is considered constant for voltage calculation and voltage is constant for current calculation. At P let the voltage be V and current be I At Q 1 the voltage will be V + dv and current will be I + di The series impedance of small section dx will be ( + jw ) dx Shunt admittance be (G + jwc ) dx dx is very small, the voltage deop from P to Q may be considered due to the current I flowing through the series empedance ( + jw ) dx. The decrease in ct from P to Q may be considered to be due to the voltage V applied to the shunt admittance (G + jwc) dx. Potential difference between P and Q is due to current flowing through series impedance ( + jwl) dx. V ( v + dv ) I ( + jwl) dx. dv (+jwl) Idx (+jwl) I 1
Current different between P and Q is due to voltage applied to shunt admittance (G+jwC) dx I (I + di) V (G + jwc) dx -di (G+jwC) v dx - (G+jwC) V It is convenient to eliminate first I and then V so that each equation involves only one independent valuable. Differentiating each equation and sub for and P 2 V P 2 I The above two equation are referred to as differential equations of the transmission line, fundamental to circuit of distributed constant. These equations are standard linear differential equations with constant co-efficient whose solution are V a I c Where a and b are constants with dimension of voltage while c and d are constants with the dimensions of current. The exponential are replaced by hyperbolic function V a ( coshpx + sin h Px ) + b ( coshpx - sin h Px ) a coshpx + a sin h Px + b coshpx b sin h Px (a + b) coshpx + (a-b) sinhpx A coshpx + B sinhpx I c (coshpx + sinhpx) + d(coshpx sinhpx) (c + d) coshpx + (c d) sinhpx C coshpx + D sinhpx 2
The above I and V equation are a second very useful form for the voltage and current values at any point on a transmission line. Instead of four constant A, B, C, and D, above equation can be simplified to only two unknown constants, by substituting the values of V P Z 0 Which I also a complex constant for a given frequency V I Again these equations can also be expressed in exponential form I a Where a and b are old constants. The relation between the old and new constant are a + b A and a - b B a and b The constants A and B or a and b that remain may be expressed in terms of voltage and current values at either end of the line (i.e.,) sending end or receiving end. 3
Determination of Constants A and B sending end are known Let us suppose that constants at the sending end are known and let Is and Vs are the current and voltage respectively at the sending end. At the sending end x 0 and V Vs Vs A coshpx0 + BsinhPx0 Vs AX1 + BX0 A Similarly putting x 0 and I Is Is Is (B coshpx0 + A sinhpx0) B B - Is Z 0 Thus A and B are expressible in terms of the current and voltage at the sending end. V Vs coshpx Is Z 0 sinhpx I (-Is Z 0 coshpx + Vs sinhpx) I Is coshpx - sinhpx These equations are general line equations expressing respectively the current and voltage at a point distance x from the sending end in terms of sent current and voltage. PIMAY AND SECONDAY CONSTANT The primary and secondary constants of a transmission line can be computed from the knowledge of Zoc and Z sc. These quantities can be found by measurement of the input impedance of the line under two conditions: with the far end of the line shorted with the far end of the line open The input impedance of a line is normally determined by an A.C. bridge (WIEN) as shown in the following figure. The input end of the line PQ is connected in BD arm of the bridge. The opposite arm CD contains an adjustable resistor and variable capacitor C. The impedance Z across CD arm and consequently and C are so adjusted that no current flows (null-point) through galvanometer G Under the condition of null 4
1 2 Z IN Z where, Z IN is the input impedance of the line and Z is the impedance of arm CD which is the parallel combination of and C. If 1 and 2 are so chosen that have equal values, then Z Z IN 1 Or Z Z IN This shows that under null condition the impedance of the arm CD will be the input impedance of the line. Thus we have to determine the value of Z. 1 1 Z jc Or Z 1 jc Multiplying the right hand numerator and denominator by (1 jc ) in order to separate the real and imaginary parts, we get Therefore, X 2 2 2 1 C (1 jc) Z X jy, say 2 2 2 (1 C) and 2 C Y 1 C 2 2 2 Thus, Z X Y 2 2 1 C 2 2 2 1 Y 1 Also, tan tan ( C) X 5
PHYSICAL SIGNIFICANCE OF EQUATION FO INFINITE LINE : The propagation of electric waves along any uniform line the reuse may be deduced in terms of lts for a hypothetical line of infinite length having electrical constants per unit length. A signal fed into a line of infinite length could not reach the far end in a finite time; the condition of the far end can have no effect at the input end. Z 0 When an A.C voltage is applied to the sending end of an infinite line, a finite current will flow due to the capacitance C and the leakage conductance G between the two wires of the line. Vsi and Isi sending end voltage and current of an infinite line Current at any point distance x from the sending end is I c C sending end of the infinite line C and D can determine by considering an infinite line. At the sending end of the infinite line X 0 and I I si I si c + d At the receiving end of the infinite line X and I 0 0 c x + d x 0 Either c 0 or 0 but cannot be equal to zero, therefore c 0, when c 0 Isi d I Isi This equation gives current at any point of an infinite line. Similarly voltage at any point of an infinite line can be V Vsi Where I 1 is the current at a unit distance from sending end. Then a distance x from sending end. 6
Where I is the current at distance x P I I s < - I I s < Similarly, V V S < - Infinite Line is equivalent to a finite line terminated in its Z 0. If a finite length of line is joined with a similar kind of infinite line. Their total input impedance is the same as that of infinite itself together. They make one infinite line. However, the infinite line alone presents an impedance Z 0 al AB, because the input impedance of an infinite line is Z 0. It must be conducted that a finite line has an input impedance Z 0 when it is terminated in Z 0 or a finite line terminated by its Z 0 behaves as an infinite line. 7
Consider a line of length l, terminated its characteristics impedance Z 0. Let the voltage and current at the termination be V and I. Thus x l, V and I in V cos Px - sin Ph Px I cos h Px - sin h Px cos h Pl - sin h Pl h Pl - sin h Pl Multiply and Divide by we get 1 But V S / I S is the input impedance of the line Z 0 Z in Thus the input impedance of a line terminated in its characteristic impedance Z 0 is the characteristics of the line. 8
ATTENUATION AND PHASE CONSTANTS: Propagation Constant P is a complex quantity P α + jβ α eal part called alternation constant Determines the reduction or alternation in voltage and current along the line and higher its value the quicker the reduction. Units in neper / Km 1 neper 8,686 db. β Imaginary part called phase constant or wave constant. Determined the variation in phase position of voltage and current along the line. Unit is radian / Km 1 rad 57.3 α and β both are function of frequency eferred as alternation function or phase function. Propagation Constant should have a positive angle when expression in its polar form hence α and β should be.+ve. Phase Constant when multiplied by the length of transmission line is termed as electrical length of a line. Similarly alternation Constant when multiplied by the length of line is termed as total attenuation or line attenuation. 9
P α + jβ Also p Adding the two equation G - + 2 G - + G - + Subtract the two equation 2 G - + These values of attenuation constant and phase co nstant in terms of primary constant, L, G SKIN EFFECT The electrical properties of a tx line are determined by the primary constant of the line. All are called by the above formula except at radio frequency where L and are controlled by skin effect. When an alternating current flows in a conductor the alternating magnetic flux within the conductor induces an e.m.f. This e.m.f. causes current density to decrease in the interior of the wire and to increase towards the outer surface. This is known as skin effect. (i.e) the.f current travels on the outer surface or the skin of a conducting material. The skin depth or nominal depth of penetration is given by 10
metres resistivity of the conductor in Ω / metre frequency in H z absolute magnetic permeability of the conductor in henry / metre. For copper Ω / metre henry / metre metres Thus as the frequency increases, the skin depth decreases, At a frequency of 60 H z At 10 KH z 10 KH z Due to skin effect the series impedence (Z + jwl) of a tx line change with increases of frequency. Under consideration of skin effect the series impedance is termed as internal impedance. The resistance of which is normally referred as effective rile the resistance (e) while the inductance is known as internal inductance (L i ) For cylindrical conductor e e L i To overcome the skin effects 1. By using tubular conductor. This might be practical for short indoor lengths such as bus but impractical for long outdoor exposed length. In such case, copper strip bent, rolled and welded to steel care is used. 2. Litz wire is used for intermediate frequencies and is quite effective upto 100 k H z 3. Silver is a better conductor than copper, and silver plating of higher frequency conductor is often used to reduce skin effect losses. Silver plated plastic rods and silver plated copper tubling are used frequently in UHF bands. 11
WAVELENGTH, VELOCITY OF POPAGATION AND GOUP VELOCITY: Velocity of propagation is also called phase velocity.these three unit of a tx line can be easily calculated if the values of and frequency of operation are known. Wavelength The distance that a wave travels along the line in order that the total shift is 2 rad In case of.f. lines with air dielectric approximates the wavelength of a radio waves of the same frequency. In case of cable with solid dielectric having closely the free space wavelength dielectric constant K, the wavelength is very closely the free space wavelength divided by Velocity of Propagation: Velocity with which a signal of single frequency propagates along the line at a particular frequency f. Since the change of 2π in phase angle represents one cycle in time t and occurs in a distance of one wavelength 6 then Vp x f Vp x Vp f f Group Velocity In case of distortion less or loss line is not a constant multiple of W. As a result of the components in a complex waveform normally shift in phase relation during propagation. This is known as dispersion which results in distortion. When dispersion exists, the significant value of V P is often difficult to define in complex wave. When s max differmall dispersion, a significant velocity of propagation is group velocity. 12
Small dispersion takes place when the max difference in the frequencies of the components in a given signal is small. Thus group velocity is defined as the velocity of the envelope of a complex signal. W 1 & W 2 are two close angular frequencies β 1 & β 2 phase constants Vg Vg Vp Differentiate wrt 1 Vp 1-1 - Substitute the value of β WAVEFOM DISTOTION When the received signal is not the exact replica of the transmitted signal, then the signal is said to be distorted. There exists some kind of distortion in the signal. There are three types of distortions present in the transmitted wave along the transmission line: 13
Due to variation of characteristic impedance Z0 with frequency Frequency distortion due to the variation of attenuation constant α with frequency Phase distortion due to the variation of phase constant β with frequency Distortion due to Z0 Varying with frequency The characteristic impedance Z0 of the line varies with the frequency while the line is terminated in an impedance which does not vary with frequency in similar fashion as that of Z 0. This causes the distortion. The power is absorbed at certain frequencies while its get reflected for certain frequencies. So there exists the selective power absorption, due to this type of distortion. It is known that, Z 0 L 1 j jl G jc G C 1 j G If for the line, the condition LG C is satisfied then L C and hence G o L o Therefore, Z 0 0 0 G C L C 1 j 1 j G For such a line Z0 does not vary with frequency and it is purely resistive in nature. such a line can be easily and correctly terminated in an impedance which matches with Z0 at all the frequencies. For such a line, selective power absorption. Z L G C. This eliminates the distortion and hence Frequency Distortion The attenuation constant α, is a function of frequency. Hence the different frequencies transmitted along the line will be attenuated to the different extent. For example a voice signal consists of many frequencies. And all these frequencies will not be attenuated equally along the transmission line. Hence received signal will not be exact replica of the input signal at the sending end. Such a distortion is called a frequency distortion. 14
Such a distortion is very serious and important for audio signals but not much more important for video signals. Thus in high frequency radio broadcasting such frequency distortion is eliminated by the use of equalizers. The frequency and phase distortion of such equalizers are inverse to those of the line. Thus nullifying the distortion, making the overall frequency response, uniform in nature. Phase Distortion It is known that for a line, 1 2 2 2 2 2 2 2 2 L G C G LC The phase constant β also varies with frequency. Now the velocity v is given by, v Thus the velocity of propagation of waves also varies with frequency. Hence some waves will reach receiving end very fast while some waves will get delayed than others. Hence all frequencies will not have same transmission time. Thus the output wave at the receiving end will not be exact replica of the input wave at the sending end. This type of distortion is called phase distortion or delay distortion. It is not much more important for the audio signals due to the characteristics of the human ear. But such a distortion is very serious in case of picture and video transmission. The remedy for this is use to co-axial cables for the picture transmission of television and video signals. A line in which the distortion are eliminated by satisfying certain conditions is called distortionless line. DISTOTION LESS LINE LOADING OF LINE TYPES OF LOADING: Signal tx-ed over lines are normally complex and consists of many frequency components. For ideal tx, the wave form at the receiving end line must be same as the waveform of the original input signal. The condition requires that all frequencies have the same attenuation and the same delay caused by a finite phase velocity or velocity of propagation. When these condition are not satisfied distortion exist. (i) Frequency distortion (ii) Delay distortion Frequency distortion: Various frequency components of the signal undergoes different attenuation. This is when the attenuation constant is not a function of the frequency. 15
Delay Distortions: The time required to transmit the various frequency components over the line and the consequent delay is not a constant. This is when the phase velocity V P is independent of frequency, delay distortion does not exist on lines. Since V P w / β it will be independent of frequency only when is equal to a constant multiplied. This type of distortion is also known as phase distortion. A transmission like will neither have delay distortion nor frequency distortion only if independent of w and β is a constant multiplied by w. is From, α β The above equation show that the α is not independent of w and therefore, the line will introduce frequency distortion, β is not a constant multiplied by w hence, the line will also introduce delay distortion. Distortion less Line However we have to know the condition on the line parameters that allows propagation without distortion. The line having parameters satisfy this condition is termed as distortion less line. Distortion less condition can help in designing new lines or modifying old ones to minimize distortion. This condition will be investigated by the expression for the propagation constant P. P α + jβ α β w The real part of the propagation constant is independent frequency and imaginary part β is w is a constant multiplied by w. This is the requested condition for distortion less transmission. From, P 16
( ) ( ) When P ( ) ( ) P α + j β ( ) ( ) DISTOTION LESS LINE, LOADING OF LINE, TYPES OF LOADING: Equating the real and imaginary parts α ) or ) or G β w Vp Vp The characteristic impedance Z 0 Z 0 Since / L G / C is at line distortion less line, Z 0 or Z 0 This shows that the characteristic impedance of distortion less line is purely real. 17
In a distortion less line, all frequency components have the same attenuation and phase velocity. The received and transmission waveforms have the same shape, but the received wave is reduced in amplitude because of attenuation. This assumes that line parameters are independent of frequency. For a lossless line α 0 And resistive component which absorb power and G are equal to zero. Subs 0 & G 0 in P α 0 β w Since is independent of w and β is a constant multiplied by ω, this also satisfies distortion less condition. Hence loss line is distortion less. INDUCTANCE LOADING OF TELEPHONE CABLES: Loading of Lines It is necessary to increase L/C ratio to achieve distortion less condition in the tx line. This can be done by increasing the inductance of a transmission line. Increasing inductance in series with line is termed as loading and such lines are called loaded line. Types of loading i) Lumped loading ii) Continuous loading iii) Patch loading Lumped loading The inductance of a tx line can be increased by the introduction of loading coil at uniform intervals, are called lumped loading. It acts as a low phase felter. So it is applicable only for a limited range of frequency. The loading coil have an internal resistance thus, increasing the total effective inductance increases. Further hysteris and eddy current losses which occur in the loading coils resulting in increase in. 18
There is practical limited on the value of inductance than can be increased for the reduction of attenuation. Thus loading coil should be carefully designed so that it will not introduce any destination Continuous Loading A type of iron or some other magnetic material is wound on the transmission line to increase the permeability of the surrounding medium and thereby increase in inductance. The advantages of continuous loading over lumped loading is that attention factor uniformly with increase in frequency. increases Patch Loading It employs sections of continuously loaded cable separated by sections of unloaded cable. The length for the section is normally a quarter kilometer. In this method the advantages of continuous loading is obtained and the cost is reduced considerably. The four line parameters,l,c and are called primary constants of the transmission line. esistance ----> Loop resistance ----> Sum of resistance of both the wires for unit line length ----> Unit is ohm / km Inductance L ----> Loop inductance ----> Sum of inductance of both wires for unit line length ----> Unit is henry / km Conductance G ----> Shunt conductance between the two wires per unit length 19
-----> Its unit is mhos per km Capacitance C ------> Shunt capacitance between the two wires per unit line length ------> Its unit is farad / km All are referred to as constant but all will vary with frequency. For the purpose of transmission line theory, they will be assumed to be independent of frequency. The series Impedence Shut admittance Z + jwl Y G + jwc Loop Inductance It depends on the nature of tx line and its dimensions. (i) Open wire Line Then the self inductance of two wire taken together ie., 10 ( 10-7 henrys / metre relative magnetic permeability of the conductor material and unity for nonmagnetic material First term is called internal inductance of the line due to the internal flux leakage in the conductor. Second term is external inductance and is due to flux leakages with the flux external to the wire. (ii) (iii) Cable Inductance negligible Co-axial cable a radius of inner solid conductor b inner surface radius c outer surface radius The self inductance of coaxial cable is hens / meter µd - Permeability of the dielectric material. µc - Permeability of the Conductor. The first term is due to flux linkage between the conductor. Second term flux linkage within the conductor 20
At high frequency the flux linkage inside the conductor become zero., since the entire current tends to flow through the outer surface of the inner conductor and through the inner surface of the conductor. At higher frequencies, the self inductance is henry / metre x 2.3 henrys/ metre Shunt Capacitance is the relative permeability permeability (i) Open wire line A parallel line is constructed, so that the spacing d between wire is large with respect to the radius a of a conductor. Under this condition charge can be considered to be uniformly distributed around the periphery of each conductor. Farad / metre µ µf / metre dielectric constant of the dielectric material relative dielectric constant of the dielectric material (ii) Coaxial cable a radius of inner solid conductor b rad ius of inner surface of the outer conductor Farad / metre Loop esistance (i) Open wire Line 21
Conductivity of the material Frequency in H z (ii) Coaxial cable dc ohms / loop metre ac + ohms / loop metre OPEN AND SHOT CICUITED LINE - EFLECTED AND INCIDENT WAVES, STANDING WAVES, INPUT IMPEDANCE : At some points in the line, the two waves will always be in phase and will add, while at the other points the two waves will always be out of phase and will cancel. The places where the two waves add will be the points of maximum voltage and is termed antinodes and the points of cancellation will have minimum voltage and is termed nodes. (i) Open Circuited Lines A voltage difference can exist between two wires but no current can flow in open circuit. Thus at the open end termination of this line there exist a maximum voltage and minimum (nearly zero) current. Therefore, impedance at the open end will be finite. A quarter wavelengths from the open end, the incident wave will be 90 o earlier and the reflected wave 90 o later than they are at the end and thus will be 180 o out of phase. At this point the voltage will be zero. The current and voltage distribution along the open circuited line. In a high frequency lossless line the values of the different maximum are equal, however in a lossy line, these go on decreasing due to attenuation of the line. (ii) Short Circuited lines: Across the short circuited end between the two transmission line wires there can be no voltage difference, but there will be a maximum current flow. Therefore, at the short circuited termination, the current is maximum, the voltage is zero and the impedance will also be zero. The standing waves thus has a node or minimum at the short circuited end and at every half wave length from the end. 22
Voltage and current distributions here differs from the distributions of the open-circuited case only in the voltage and current are interchanged. That is, with short circuited load, the voltage on the line goes through minima at distances from the load that are even multiples of quarter wavelength and through maxima at a distance that are odd multiple of quarter wave length. The voltage and current distributions along short-circuited line are shown in fig. In a lossless line the distribution is as shown in where that of a lossy line. In a lossy line the voltage and current gets attenuated as they travel towards load. eflected and Incident waves When the load i.e., the termination is equal to the characteristic impedance of a transmission line, it is equivalent to an infinite line. Therefore all the energy sent down the line is completely absorbed by the load and no reflection takes place. However in dealing with open and short circuited lines, all the energy sent down the line is reflected back because there is no resistance at the terminating end to absorb it. It is therefore necessary to examine the result of reflected energy, in order to analyse the voltage and current distribution along the open and short circuited lines. Voltage and current at any point of a transmission line is given as follows px px V be ae 1 ( px px I be ae ) Z o The voltage and current existing in a transmission line as given by the above equations can be conveniently expressed as the sum of the voltage and current of the two waves viz. px ae. The first term px be px be in each of the above equations represents a voltage or current component of a travelling wave decreasing exponentially due to factor towards the load. Thus the term to the load. px be px e and in the direction of x increasing, represents the net sum of all individual waves that travel px Similarly, the second term ae, in each of the above equations represents a wave similar to the first but travelling in opposite direction. This wave is called the reflected wave and is generated 23
px at the load by the reflection of the incident wave. Thus the term ae represents the sum of all waves that travel away from the load, because the distance x will be negative as it is now measured in the direction of x decreasing. These two waves incident and reflected as shown in Fig are incident in nature except for consequences arising from their different direction of travel. It can, therefore, be concluded that the voltage and current at any point of a line may be interpreted as the superposition of two waves travelling in opposite direction. If the line is infinte x, reflected component, ae px p ae 1 1 a 0 e This shows that there is no reflected wave in an infinite line which confirm to earlier statements of no reflection in an infinite line or a finite line terminated in its characteristic impedance. STANDING WAVES: As shown in figure there will be two waves travelling in opposite direction between the input end and the load end. At some points in the line, the two waves will always be in phase and will add, while at the other points the two waves will always be out of phase and will cancel. The places where the two waves add will be the points of maximum voltage and is termed antinodes and the points of cancellation will have minimum voltage and is termed nodes. 24
INPUT IMPEDANCE Input Impedance of Open and Short Circuited Lines Input impedance of an open-circuited line is the impedance measured at the input of a finite length of line when its far end is open. It is normally denoted by Z. Similarly input impedance of a short circuited lines is the impedance measured at the input end of the finite length of line when its far end is shorted. It is usually denoted by Z. Consider a length of line l, having far end voltage and current Therefore, when xl, equations V V and I I sc oc V and I respectively.. Then substituting these values in the following V V cosh Px I Z sinh Px Vs I Is cosh Px sinh Px Z V V cosh Pl I sinh Pl...(a) s s Vs I Iscosh Pl sinh Pl...(b) Z In an open-circuited line, I 0, the equation (b) will become Vs 0 Is cosh Pl sinh Pl Z V Z s 0 cosh Pl Z0 Z0coth Pl sinh Pl s 0 0 0 0 Since V I s s is the input impedance Zoc 25
Z coth oc Z0 Pl... (c) Similarly, in a short-circuited line, V 0, and equation (a) will become as 0 V cosh Pl I Z sinh Pl V I s s s sinh Pl Z0 Z0tanh Pl cosh Pl s 0 But here, V I s s is the input impedance Zsc of the short circuited line, therefore Z tanh sc Z0 Pl...(d) Therefore for an infinite line of length l as a result of which tanh Pl and coth Pl both will become 1. Therefore it is proved that the input impedance of an infinite line is its characteristic impedance. OPEN AND SHOT CICUITED LINE - PIMAY AND SECONDAY CONSTANTS, TANSMISSION LINES AS CICUIT ELEMENTS, EACTIVE TEMINATION: Determination of Primary and Secondary Constants The primary and secondary constants of a transmission line can be computed from the knowledge of Zoc and Z sc. These quantities can be found by measurement of the input impedance of the line under two conditions: with the far end of the line shorted with the far end of the line open The input impedance of a line is normally determined by an A.C. bridge (WIEN) as shown in the following figure. The input end of the line PQ is connected in BD arm of the bridge. The opposite arm CD contains an adjustable resistor and variable capacitor C. The impedance Z across CD arm and consequently and C are so adjusted that no current flows (null-point) through galvanometer G Under the condition of null 1 2 Z IN Z where, Z IN is the input impedance of the line and Z is the impedance of arm CD which is the parallel combination of and C. If 1 and 2 are so chosen that have equal values, then 26
Z Z IN 1 Or Z Z IN This shows that under null condition the impedance of the arm CD will be the input impedance of the line. Thus we have to determine the value of Z. 1 1 Z jc Or Z 1 jc Multiplying the right hand numerator and denominator by (1 jc ) in order to separate the real and imaginary parts, we get Therefore, X 2 2 2 1 C (1 jc) Z X jy, say 2 2 2 (1 C) and 2 C Y 1 C 2 2 2 Thus, Z X Y 2 2 1 C 2 2 2 1 Y 1 Also, tan tan ( C) X 27
TANSMISSION LINES AS CICUIT ELEMENTS The four line parameters,l,c and are called primary constants of the transmission line. esistance ----> Loop resistance ----> Sum of resistance of both the wires for unit line length ----> Unit is ohm / km Inductance L ----> Loop inductance ----> Sum of inductance of both wires for unit line length ----> Unit is henry / km Conductance G ----> Shunt conductance between the two wires per unit length -----> Its unit is mhos per km Capacitance C ------> Shunt capacitance between the two wires per unit line length ------> Its unit is farad / km All are referred to as constant but all will vary with frequency. For the purpose of transmission line theory, they will be assumed to be independent of frequency. The series Impedence Shut admittance Z + jwl Y G + jwc 28
Loop Inductance It depends on the nature of tx line and its dimensions. (iv) Open wire Line Then the self inductance of two wire taken together ie., 10 ( 10-7 henrys / metre relative magnetic permeability of the conductor material and unity for non-magnetic material First term is called internal inductance of the line due to the internal flux leakage in the conductor. Second term is external inductance and is due to flux leakages with the flux external to the wire. (v) (vi) Cable Inductance negligible Co-axial cable a radius of inner solid conductor b inner surface radius c outer surface radius The self inductance of coaxial cable is hens / meter µd Permeability of the dielectric material. µc Permeability of the Conductor. The first term is due to flux linkage between the conductor. Second term flux linkage within the conductor At high frequency the flux linkage inside the conductor become zero., since the entire current tends to flow through the outer surface of the inner conductor and through the inner surface of the conductor. At higher frequencies, the self inductance is henry / metre x 2.3 henrys/ metre is the relative permeability 29
permeability Shunt Capacitance (iii) Open wire line A parallel line is constructed, so that the spacing d between wire is large with respect to the radius a of a conductor. Under this condition charge can be considered to be uniformly distributed around the pesiphery of each conductor. Farad / metre µf / metre dielectric constant of the dielectric material relative dielectric constant of the dielectric material (iv) Coaxial cable a radius of inner solid conductor b rad ius of inner surface of the outer conductor Farad / metre Loop esistance (iii) Open wire Line Conductivity of the material Frequency in H z 30
(iv) Coaxial cable dc ohms / loop metre ac + ohms / loop metre EACTIVE TEMINATION: When the transmission line is terminated by a pure reactance, complete reflection will take place as no energy can be absorbed. The starting wave pattern and the impedance at various points can then be found for the line, by the extension of the line by fictitious open and short circuited line as shown in the following Figure. For example, the line with Z 0 is terminated by X as shown in the Fig. A short-circuited line with the same characteristic impedance of length will have the same impedance, therefore jx 2 l 1 jx 0 tan X l1 tan 2 X0 Where Z0 is pure resistance Z0 at high frequencies. Hence l 1 is calculated. When the line is terminated in added, therefore jx as shown in figure, an open circuited of length l 2 can be 2 l 2 jx Z0 cot 2 l2 X 0 tan X 31
When X l2 tan 0 2 X X is inductive, it is better to replace it with a short circuited line, so that l 1 may be smaller than 4.Similarly when X is capacitive it is replaced by an open-circuited line for the same reason. 32