Transmission Lines Two wire open line a Strip line d w a Coaxial Cable h b

Transmission

The field distribution due to a pair of wires Two wire open line The field distribution due to a pair of wires

Two wire open line Line Behavior When the switch is closed how long does the lamp take before it lights? If we make it easy and let the length of the wires between battery and lamp be 3.108 m, then the time between the switch closing and lamp lighting will be approximately ______s.

Two wire open line Does this mean that electrons are traveling at the speed of light? Does it mean that the mechanism relies on electromagnetic energy being transported from start to finish of the structure? If so where is the energy stored during transit? If it is electromagnetic energy that is stored then it has to be stored in inductance for magnetic energy and capacitance for electric energy. This means that the transmission line has to have an inductance (per unit length) and a capacitance (per unit length).

Low Loss line at High Frequency Two wire open line Low Loss line at High Frequency Inductance and capacitance are uniformly and continuously distributed as L (Henrys/m) and C (Farads/m) respectively. When the switch is closed and a voltage V is applied to the line through a source impedance Zs, simple reasoning shows that the C's take a finite time to charge up through the L's; thus, the voltage propagates at a finite rate towards the load i.e. volts do not reach the load instantaneously.

Transmission in Capacitive Element Let L and C be distributed inductance and capacitance per unit length. In time t, electric flux q = (C ·x)v (from Q = CV) is produced in time x/u sec i(amps) = q/ t = Cx• v/ t (capacitive charging or "displacement" current) i = Cx• v /(x/ u ) = Cvu (I)

Transmission in Magnetic Element In time t, magnetic flux linkages =(L•x)i (from = LI) ; are produced in time x/u sec v(volts) = /t = L•x•i/t=Liu (ind Volts)…(II) Multiplying I and II:- vi =vi CLu2 Or u = 1/  LC ms-1velocity of propagation Dividing I by II:- i/v =vC/iL or v2/i2 =L/C v/I = L/C……Characteristic Impedance Z0

Energy Transmission ; i2 L = v2C ½ Li2 = ½ Cv2 Note:- the equality of Electric and Magnetic field energy in unit length of transmission line.

Movement of Energy Rate of energy input to line due to advance of wavefront, is vi watts. resistive load, connected directly to voltage source. Rate of energy input, due to thermal dissipation in R, is vi watts (also i2R).

Movement of Energy Rate of energy input = vi watts. Idea of a 'matched line': looks like an infinitely long line. Note: if Rg = Z0, maximum power if transferred from the generator to the line.

Movement of Energy As can be seen from the intuitive picture of a transmission line, wave propagation characteristics are dependent on the inductance and capacitance of the line. Thus we need to find expressions for the L and C of typical lines.

Inductance in Two wire line Assume r << S Due to conductor A, for 1 amp Linkages per metre, axially, L

Inductance in Two wire line Due to conductor B, an identical expression is Obtained:-

Mutual Inductance Between 2 Twin Lines External field due to 1 amp in line A, at radius

Mutual Inductance Between 2 Twin Lines Flux linkages per metre axially through circuit (between conductor 1 & 2) is :-  = B =

Capacitance two wire line Capacitance of a Twin Line (r << S) Assign a line charge of 1 C/m to both conductors. D at x from conductor A is;

Capacitance two wire line and Similarly for VB(=VA) 

R, L, C of Two wire line  Rdc= 2A2 0 d loge loge L =  a 0 C = per unit length  Rdc= ohm/m 2A2 0 d loge loge L = H/m  a 0 C = F/m loge (d/a)

Characteristic Impedence Z0Twin-wire line Characteristic Impedence For air, therefore Z0 = 120 loge(b/a), (approx ) 0 d loge L =  a 0 C = loge (d/a)