Transmission lines:
The transmission line is system of material boundaries that forms continuous path which can direct transmission of electromagnetic energy from power station to other stations or for transmitting energy from one point to another. Transmission line is uniform if there is no change in cross sectional geometry. Wavelengths on transmission line are compatible with size at gigahertz frequencies and capacitances and inductances are very small.
Change in potential difference per unit length i.e.
ΔV/ΔL = -LdI/dt = -j(wL + R)I
Or in limit of ΔL becoming dl
dV/dl = -(jwL + R)I
Where L is self inductance because of magnetic field around conductor V and R, is resistance of conductor. Change in current flowing in line is provided by:
dI/Dl = cdV/dt - (jwc + G)V
Where C is capacitance per unit length (formed due to finite distance between conductors of line) and G is conductance per unit length (whose existence is because of dielectric losses of dielectric medium in between conductors). If we suppose that conductors have zero resistance and that they are divided by perfect insulator (in which case transmission line becomes lossless), equations which are basic equations of transmission line becomes,
dV/dl = -LdI/dt
dI/dl = -CdV/dt
Differentiating equations with l and t respectively we have
d2V/dl2 = -L(d2I/dldt)
d2I/dtdx = -Cd2V/dt2
(1/LC)(d2V/dl2) = d2V/dt2
(1/LC)(d2I/dl2) = d2I/dt2
Equations are familiar wave equations of voltage and current respectively mean that both voltage and current propagate as waves along transmission lines. Velocity of propagation is V = 1/√LC. It can be illustrated that characteristic impedance of the transmission line z0 = √LC.
Parallel wire and coaxial cable transmission line:
Two major examples of transmission lines are (i) parallel wire and (ii) coaxial cable.
To get propagation velocity, V and characteristics impedance of any transmission line inductance per unit length, L and capacitance per unit length should be determined.
a) Parallel wire transmission lines: Recall that electric field E of conductor is given by Gauss law for electric as,
E = λ/(πrε0εr = q/(2πrε0εrl))
λ(linear charge density) = q/l, for conductor in the medium of relative permittivity (dielectric constant), εr. For two wires, each of radius x and separated by distance y.
E = E1 + E2 = q/2πrε0εrl + q/2πrε0εrl = q/πrε0εrl
Capacitance of pair is obtained from
C = q/V = q/x∫2yEdr = [q/(q/(πε0εrlx∫2y(dr/k)))]
Giving
C/l = πε0εr/ln(2y/x) i.e. capacitance per unit length. Inductance per unit length is attained as follows: magnetic flux of two conductors (whose shape are approximately cylindrical),
ΦB = 2lx∫2y-xμ0μrIdr/2πr ≈ [μ0μrIln(2y/x)/π]
But self inductance, L = ΦB/i = [μ0μrIln(2y/x)/π]
Therefore velocity of propagation, V = 1/√LC = 1/√ε0εrμ0μr as expected. Characteristics impedance,
z0 = √L/C = (μ0μr/π2ε0εr)1/2ln(2y/x)
b) Coaxial cable transmission line: the capacitance for coaxial cable is obtained following procedure utilized for the parallel wire except that ∫Edr is from a to x and b to y, a being radius of central conductor and b, distance between centres of both conductors. Capacitance per unit length is therefore
C/l = 2πε0εr/ln(y/x)
Inductance of coaxial cable is attained as follows: flux of magnetic field, ΦB, through closed circuit formed by joining conductors at the end of section of cable of length, l, is lx∫y(μ0μrI/2πr)dr = (μ0μr/2π)Illn(y/x)
Self inductance, L = ΦB/I = L = (μ0μr/2π)ln(y/x)
Velocity of propagation, V = 1/√LC
Substituting V = 1/√ε0εrμ0μr, and characteristics impedance,
z0 = √L/C(μ0μr/4π2ε0εr)1/2 lny/x
Equivalent circuit:
In Figure 1 above is illustrated two infinitely long lines carrying equal and opposite current. Part between point X and Y is subsection which comprises of impedance z =R+ jωL and admittance y= G+ jωC. Resistance, R and inductance, L are due to length and diameter of conductors while capacitance exists by virtue of close separation of conductors. Conductors being separated by the imperfect insulator or dielectric necessitate leakage of current. This signifies shunt conductance that together with shunt capacitance provides admittance. 2 conductors can therefore be represented in which case resistance, R and conductance, G are negligible. Each subsection is the equivalent circuit and infinitely loss line is considered as cascade of infinite number of such circuits.
Tutorsglobe: A way to secure high grade in your curriculum (Online Tutoring)
Expand your confidence, grow study skills and improve your grades.
Since 2009, Tutorsglobe has proactively helped millions of students to get better grades in school, college or university and score well in competitive tests with live, one-on-one online tutoring.
Using an advanced developed tutoring system providing little or no wait time, the students are connected on-demand with a tutor at www.tutorsglobe.com. Students work one-on-one, in real-time with a tutor, communicating and studying using a virtual whiteboard technology. Scientific and mathematical notation, symbols, geometric figures, graphing and freehand drawing can be rendered quickly and easily in the advanced whiteboard.
Free to know our price and packages for online physics tutoring. Chat with us or submit request at [email protected]
avail algebra assignment help service by qualified and skilled tutors to get authentic papers at fair prices and to boost up your grades.
Fraunhofer Diffraction tutorial all along with the key concepts of Diffraction at a Single Slit, Observed Pattern, Diffraction by a circular aperture and Diffraction by a rectangular aperture
Plant-Water Relations tutorial all along with the key concepts of Cellular water relations, Stomatal mechanism, Transpiration process and Water translocation
www.tutorsglobe.com offers stereoisomers homework help, stereoisomers assignment help, online tutoring assistance, organic chemistry solutions by online qualified tutor's help.
tutorsglobe.com chemical properties of halogen family assignment help-homework help by online halogen family tutors
Modern Examples of Evolution tutorial all along with the key concepts of Sickle-cell Anaemia, Industrial Melanism, Chloroquine - Resistance Malaria Parasite, Pesticide-Resistance Insects, HIV, Phylogenetic categorization of Organisms
Negative Feedback tutorial all along with the key concepts of Benefits of Negative Feedback, Gain Stability, Decreased Distortion, Feedback over Several Stages, Forms of Negative Feedback, Shunt-Derived Series-Fed Voltage Feedback, Current-Series Feedback Amplifier
tutorsglobe.com frenkel defects assignment help-homework help by online imperfections in solids tutors
www.tutorsglobe.com offers Fifth Generation Language homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
Systematic Classification of Protozoa tutorial all along with the key concepts of The Flagellates, The Zooflagellates, The Amoebas, Nutrition of Amoeba, The Sporozoa, The Ciliates and Other Ciliated Protozoa
tutorsglobe.com flagellar functions assignment help-homework help by online flagellation in bacteria tutors
tutorsglobe.com rainwater harvest through bore wells assignment help-homework help by online collection of rainwater from terrace tutors
tutorsglobe.com types of ligands assignment help-homework help by online terminologies tutors
Thermodynamic Potentials tutorial all along with the key concepts of Internal Energy, Enthalpy, Helmholtz Free Energy, Gibbs free energy, isothermal process
www.tutorsglobe.com offers Layering homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
1936227
Questions Asked
3689
Tutors
1477017
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!