Inductance of a Co-axial Line m.m.f. round any closed path = current enclosed

Inductance of a Co-axial Line And Flux density: For 1 m length, axially, the flux linkages 

Inductance of a Co-axial Line This expression for inductance is valid for the space between r and R. However, flux is also "linked" inside the inner and outer conductors. Internal flux linkages for radius x < r.

Inductance of a Co-axial Line since the flux links only ( ) of one turn  L2L2 Thus for h.f. applications the inductance of a coaxial line is taken as:

Capacitance of a Coaxial Line Assign a line charge of 1 C/m to inner and outer conductors. Electric flux density at x = Electric field intensity Hence the Capacitance is

Characteristic Impedance Coaxial Line neglecting internal linkages If  r =1 and  r = 1 If, for example. ( )

Derivation of General Transmission Line Equations Representation of a Uniform Transmission Line

Derivation of General Transmission Line Equations R = distributed resistance/metre G = distributed conductance/metre L = distributed inductance/metre C = distributed capacitance/metre

Derivation of General Transmission Line Equations Setting up Differential Equations Potential drop across  x is:- The decrease in current across  x is where i x + v x are functions of both x and time.

Derivation of General Transmission Line Equations Let and where I x + V x are phasor quantities and are functions of x alone.

Derivation of General Transmission Line Equations Hence differentiating with respect to x which becomes, on using or where

Derivation of General Transmission Line Equations  is termed the propagation coefficient. The general steady state solution of Equation is:- V x = Ae  x + Be -  x I = A z0z0 exex e-xe-x z0z0 B -

Lossless or High-frequency Lines Many transmission lines operate at relatively high frequencies. Under those conditions the lossy terms, R and G, pale into insignificance when compared with  L and  C.

Traveling Waves The basic equations for the transmission line 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 Waves and 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 Waves If the line is considered as lossless then neither the incident or reflective waves decay as they progress along the 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 Waves The value of the maximum voltage is A + |  |A. A + |  |A.

The condition for no standing waves is that |  | = 0, no reflection and the line is matched. Standing Waves

Using some imagination the current is seen as being at right angles to the voltage, i.e. space quadrature

Reflections on Unmatched Lines We have already seen that transmission lines that are not matched have both incident and reflected waves on them. We will now consider expressing the equations previously derived in terms of reflection coefficients.