Chapter 14Transmission lines

Tutorial concepts:

Please note that all page references are not really that important, but they help a lot if you have the book

Introduction (P456)o A transmission line is a pair of conductors that conduct electronic energy (P510)o Electrical model of a transmission line (P458)

There are a few factors in consideration when it comes to transmission lines

The resistance of the line1. Frequency2. Magnetic field3. Cross-sectional area

Conductance of the dielectric (at low frequencies, this can be neglected because it is quite small)

RLC (see: introduction-> Types of cables-> coaxial cables-> these cables are unbalanced lines(P459))

1. Resistance (exists within all conductors)2. Inductance (exists within all conductors)3. Capacitance (exists between any two conductors)

There are two types of models in transmission lines, the balanced line and the unbalanced line

o Types of cables

Coaxial cables (P510, P456)

These cables are unbalanced lines (P459)

Notice some things in this pictureG = dielectric conductance (it should be in Siemens but it’s represented as a resistance in parallel)C = the capacitance that exists between the two parallel lines separated by the dielectricR = internal cable resistanceL = cable inductance due to frequency

Solid dielectric (P456)

Air dielectric (P465)

Parallel-line cables (P457)

These cables are naturally balanced (P459)

Notice some things in this pictureG = dielectric conductance (it should be in Siemens but it’s represented as a resistance in parallel)C = the capacitance that exists between the two parallel lines separated by the dielectricR = internal cable resistanceL = cable inductance due to frequency

Television twin-lead

Open-wire line

Shielded-twin lead

Twisted pair cable

Characteristic Impedance

The ratio between voltage and current on an infinitely long transmission line (P510)

In my own words: it is the virtual resistance as if the transmission lines were infinitely long, this resistance plays a part as kind of standard that kind of changes according to certain factors as depicted in the following formula (formula 14.1)

All these properties are per unit length

Zo = characteristic impedanceR = conductor resistanceL = inductanceG = dielectric constant (Siemens)C = capacitanceω = operating frequency

If the line were to be lossless, then R and G are both 0 and then all we would have after manipulating the equation would be

In example 14.1:A coaxial cable has a capacitance of 90pF/m and a characteristic impedance of 50Ω. Find the inductance of 1m length: We have Zo and we have C, if we solve for C with some basic mathematics, we get L = 225 nH/m.

There are some methods to calculate characteristic impedances in different types of transmission lines

In an air dielectric parallel line (check parallel lines in P459 or just some parts earlier in this script)

D = distance between the centers of the two conductorsr = radius of each conductor

(P461)

Or in coaxial cables (check P456 or just some parts earlier in this script)

Єr = relative permittivity of the dielectric, compared with that of free space (dielectric constant)D = inside diameter of outer cable (say if the outer cable is a tube, this unit will take in consideration the diameter closest to the dielectric)d = diameter of the inner core cable

(P462)

Velocity factor(P463)o c = 300x106 m/s (speed of light!)o the propagation velocity (velocity of which the signal passes through the line) is

just a bit slower than that of the speed of light (P510)o vf is the velocity factor which is the ratio between the propagation velocity and

the speed of light. The book says “ratio of the speed of propagation on a line to that of light in free space” (P510)

Therefore(P463)

A bit earlier, we were introduced to Єr, which is a lines dielectric constant. Vf is almost entirely dependent on the dielectric used in the line, as in

It is almost like vf is indirectly proportional to the dielectric constant Reflections(P464)

o Sometimes in transmission lines, some of the signal voltage is reflected back from the load, this voltage reflects back and return to the source, and does not dissipate as heat as many think, it simply returns to the source and acts like a resisting force against the incident signal (initial signal).

o It is vital for us to calculate the time it takes for the signal to propagate through the line, so using the previous pointer on velocity factors (sometimes, velocity factors will be produced in the question paper and you will have to find the propagation velocity).

To find time, it’s a simple formula (P465)

T = time for signal to travel length LL = length of the lineVp = propagation velocity on the line

o The Reflection Coefficient is the ratio of the reflected voltage against the incident voltage. It is calculated as follows

(P466; Eq 14.9)

o On page 468,