TRANSMISSION LINE THEORY (TEM Line) A uniform transmission line is defined as the one whose dimensions and electrical properties are identical at all planes transverse to the direction of propaation. Circuit Representation of TL s A uniform TL may be modeled by the followin circuit representation:
R: Series resistance per unit lenth of line (for both conductors (ohm/m)). L: Series inductance per unit lenth of line (Henry/m). G: Shunt conductance per unit lenth of line (mho/m). C: Shunt capacitance per unit lenth of line (Farad/m). The line is pictured as a cascade of identical sections, each of z lon. Since z can always be chosen small compared to the operatin wavelenth, an individual section of line may be analyzed usin ordinary ac circuit theory. In the followin analysis, we let z, so the results are valid at all frequencies (hence for any physical time variation). Applyin the Kirchhoff s voltae law to the line section ives:
i( z, t) v( z, t) ( Rz) i( z, t) ( Lz) v( z z, t) t Rearranin yields: v( z z, t) v( z, t) i( z, t) R i( z, t) L z t Lettin z, we et, v( z, t) i( z, t) R i( z, t) L z t Now applyin Kirchhoff s current law to the line section ives: v( z z, t) i( z, t) Gzv z z, t Cz i( z z, t) t Rearranin yields: i( z z, t) i( z, t) v( z z, t) G v( z z, t) C z t Lettin z
i( z, t) v( z, t) G v( z, t) C z t Then the time domain TL or telerapher equations are: v( z, t) i( z, t) R i( z, t) L z t i( z, t) v( z, t) G v( z, t) C z t The solution of these equations, toether with the electrical properties of the enerator and load, allow us to determine the instantaneous voltae and current at any time t and any place z alon the uniform TL. Lossless Line: For the case of perfect conductors (R=) and insulators (G=), the telerapher equations reduce to the followin form:
v( z, t) i( z, t) L z t i( z, t) v( z, t) C z t v( z, t) i( z, t) L z z z t i( z, t) v L L C t z t t Or,
v( z, t) i( z, t) LC z t i( z, t) i( z, t) LC z t Wave equation s for voltae and current on a lossless TL. Althouh real lines are never lossless, lolessness approximation for practical TL s is very usefull. TRANSMISSION LINES WITH SINUSOIDAL EXCITATION We will only consider the sinusoidal steady-state solutions. Transmission-Line Equations: Under sinusoidal steady state conditions, the TL equations take the form:
d ( z) ( R jl) I( z) dz di( z) ( G jc) ( z) dz Where () zand I() zare voltae and current phasors. The real sinusoidal voltae and current waveforms are obtained from: jt v( z, t) Re ( z) e jt i( z, t) Re I( z) e Wave Propaation on a TL The second order differential equations for () zand I() zare: d z ( ) dz ( ) dz d I z ( z) I z ( )
where j R j LG j C 1/ complex propaation constant. attenuation constant (Np/m). phase constant (rad/m). The solution for () zis: () z e e z z ( z) ( z) ( z) Where and are constants independent of z, and ( z) ( z) e e z z Represent voltae waves travelin on the line in the positive and neative z directions respectively.
WAE PROPAGATION ON A TL The second order equations for (z) and I(z) are: d z ( ) dz ( ) dz d I z The solution for (z) is: ( z) I z z z () z e e ( ) Where is a constant of voltae waves travelin in the forward direction, is a constant of voltae waves travelin in the reverse direction, independent of z. Now consider the equation: d () z dz ( R jl) I( z)
If we substitude the solution for (z) into the above equation we et, z z e e ( R jl) I( z) or, z I() z e e ( R jl) ( R jl) z R jl( G jc) ( R jl) R jl ( G jc) ( R jl) R jl 1/ 1/
Define R jl 1/ ( G jc) the characteristic impedance of the TL. Then, 1 ( R jl) So, I() z e e I e I e Where, I z z z z I ( z) I ( z) I Constants independent of z.
I z z e I z z e Lossless Line: For the lossless line R=, G=. i) L C ii) j R j LG j C jl jc jlc LC Propaation constant (no loss)
iii) () z e e jz jz 1 jz jz I( z) ( e e ) j j e e then iv) If v) v( z, t) cos t z cos t z 1 i( z, t) cos t z cos t z 1 f LC f LC vi) u p u u p p 1 LC LC f v 1 p LC f So,
TERMINATED TRANSMISSION LINE Infinitely Lon TL For the infinite line, only forward travelin waves exist and therefore at z=,
I I in I Suppose now that the infinite line is broken at z. Since the line to the riht of z is still infinite, its input impedance is and therefore replacin it by a load impedance of the same value does not chane any of the conditions to the left of z. This means that a finite line terminated in its characteristic impedance is equivalent to an infinitely lon line. Like the infinite line, a finite lenth line, terminated in has no reflections. Also, its input impedance is equal to and independent of the line lenth. in I in in I
I in So, And I z jz () z e e I() z e e z jz The time-averaed power absorbed by the load is: * 1 1 P Re I Re e e e e * l jl l jl L L L 1 l l PL e e If (lossless line) 1 PL P in