inductance of a co-axial line m.m.f. round any closed path = current enclosed

Inductance of a Co-axial Inductance of a Co-axial Line Line

m.m.f. round any closed path = current enclosed

Inductance of a Co-axial Inductance of a Co-axial LineLine

And Flux density:And Flux density:

For 1 m length, axially, the flux linkages


This expression for inductance is validThis expression for inductance is valid

for the space between r and R. However,for the space between r and R. However,

flux is also "linked" inside the inner andflux is also "linked" inside the inner and

outer conductors. Internal flux linkagesouter conductors. Internal flux linkages

for radius x < r. for radius x < r.


since the flux links only ( ) of one turn since the flux links only ( ) of one turn

L2

Thus for h.f. applications the inductance of a coaxial line is taken as:

Capacitance of a Coaxial Capacitance of a Coaxial Line Line

Assign a line charge of 1 C/m to inner and Assign a line charge of 1 C/m to inner and outerouter

conductors.conductors.

Electric flux density at x =Electric flux density at x =

Electric field intensity

Hence the Capacitance is

Characteristic Impedance Characteristic Impedance Coaxial LineCoaxial Line

neglecting internal linkages

If r =1 and r = 1

If , for example. ( )

Derivation of General Derivation of General Transmission Line Transmission Line

Equations Equations Representation of a Uniform Representation of a Uniform

Transmission Line Transmission Line


Equations Equations R = distributed resistance/metreR = distributed resistance/metre

G = distributed conductance/metreG = distributed conductance/metre

L = distributed inductance/metreL = distributed inductance/metre

C = distributed capacitance/metreC = distributed capacitance/metre


EquationsEquationsSetting up Differential EquationsSetting up Differential Equations

Potential drop across x is:-

The decrease in current across x is

where ix + vx are functions of both x and time.


EquationsEquations

LetLetand

where Ix + V x are phasor quantities and are functions of x alone.


EquationsEquationsHence differentiating with respect to xHence differentiating with respect to x

which becomes, on using

or

where


EquationsEquations is termed the propagation coefficient.is termed the propagation coefficient.

The general steady state solution of Equation is:-

Vx = Aex + Be-x

I =

A

z0ex e-x

z0

B-

Lossless or High-frequency Lossless or High-frequency Lines Lines

Many transmission lines operate at Many transmission lines operate at relativelyrelatively

high frequencies. Under those high frequencies. Under those conditions theconditions the

lossy terms, R and G, pale intolossy terms, R and G, pale into

insignificance when compared with insignificance when compared with L L and and

C. C.

Traveling Waves Traveling Waves The basic equations for the transmission The basic equations for the transmission

lineline

are: are:

Having solutions of the form

or knowing as we now do that we have 2 waves, incident, V+ and reflected, V- then we can rewrite the equations as:-

Traveling WavesTraveling Wavesandand

If z = 0 and we define this as the receiving end, then:-

If the voltage reflection coefficient, , is defined as the ratio of the reflected wave to the incident wave then, = V-/V+ . Hence:-

&

Traveling WavesTraveling WavesIf the line is considered as lossless then neither the If the line is considered as lossless then neither the

incidentincident

or reflective waves decay as they progress along the or reflective waves decay as they progress along the line.line. Considering the currents;

then

And at the load, z = 0 then

Hence the Current reflection Coefficient = - Voltage reflection Coefficient

Transmission Coefficients are 1 + reflection coefficients; = 1+

Traveling WavesTraveling Waves

The value of the maximum voltage isThe value of the maximum voltage is

A + |A + ||A.|A.

The condition for no standing waves is The condition for no standing waves is thatthat

||| = 0, no reflection and the line is | = 0, no reflection and the line is matched.matched.

Standing WavesStanding Waves

Standing WavesStanding WavesUsing some imagination the current is Using some imagination the current is

seenseen

as being at right angles to the voltage, as being at right angles to the voltage, i.e.i.e.

space quadrature space quadrature

Reflections on Unmatched Reflections on Unmatched Lines Lines

We have already seen that transmissionWe have already seen that transmission

lines that are not matched have bothlines that are not matched have both

incidentincident and and reflected wavesreflected waves on them. on them. WeWe

will now consider expressing the will now consider expressing the equationsequations

previously derived in terms of previously derived in terms of reflectionreflection

coefficients.coefficients.

inductance of a co-axial line m.m.f. round any closed path = current enclosed

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