Chapter 44 Transmission lines

At the end of this chapter you should be able to: • appreciate the purpose of a • define the transmission line primary constants R, L, C and G • calculate phase delay, wavelength and velocity of propagation on a transmission line • appreciate current and voltage relationships on a transmission line

• define the transmission line secondary line constants Z0, γ , α and β • calculate and propagation coefficient in terms of the primary line constants • understand and calculate distortion on transmission lines • understand wave reflection and calculate reflection coefficient • understand standing waves and calculate standing wave ratio

are often used to couple f.m. and television receivers to 44.1 Introduction their antennas. At frequencies greater than 1000MHz, transmission A transmission lineis asystem of conductors connecting lines are usually in the form of a which may one point to another and along which electromagnetic be regarded as coaxial lines without the centre con- energy can be sent. Thus telephone lines and power ductor, the energy being launched into the guide or distribution lines are typical examples of transmission abstracted from it by probes or loops projecting into lines; in electronics, however, the term usually implies the guide. a line used for the transmission of radio-frequency (r.f.) energy such as that from a radio transmitter to the antenna. 44.2 Transmission line primary An important feature of a transmission line is that it constants should guide energy from a source at the sending end to a load at the receiving end without loss by radiation. Let an a.c. generator be connected to the input termi- One form of construction often used consists of two nals of a pair of parallel conductors of infinite length. similar conductors mounted close together at a constant A sinusoidal wave will move along the line and a finite separation. The two conductors form the two sides of current will flow into the line. The variation of voltage a balanced circuit and any radiation from one of them with distance along the line will resemble the variation is neutralized by that from the other. Such twin-wire of applied voltage with time. The moving wave, sinu- lines are used for carrying high r.f. power, for exam- soidal in this case, is called a voltage travelling wave. ple, at transmitters. The coaxial form of construction is As the wave moves along the line the of the commonly employed for low power use, one conduc- line is charged up and the moving charges cause mag- tor being in the form of a cylinder which surrounds the netic energy to be stored. Thus the propagation of such other at its centre, and thus acts as a screen. Such cables an electromagnetic wave constitutes a flow of energy.

DOI: 10.1016/B978-1-85617-770-2.00044-6 Part 3 662 iue44.1 Figure o hrfr pert h eeao sa pncir- open an as load generator definite a the presents but does to cuit line appear The line. therefore the along not point any at measured be rtra ol ige‘upd meac fvalue of impedance ‘lumped’ single a would gen- the as to manner erator similar a in behaves line the and line otg is voltage inlns hs en eitne nutne capaci- , conductance. and resistance, tance being these lines, sion Z Z (ii) (i) 0 0 fe ufiin ietemgiueo h aemay wave the of magnitude the time sufficient After hr are There = once ietyars h eeao terminals. generator the across directly connected lcrclCrutTer n Technology and Theory Circuit Electrical V nhny e opmtetksit consideration into takes metre loop per henrys in eto 07 rmeuto 2) ae54 the in 574, considered page inductance is (23), equation line From twin 40.7. Section isolated induc- an The of them. through tance line flows transmission current a a when of conductors the rounding otpatcllines practical most and conductor where nutrand conductor each of area cross-sectional material, conductor the of resistivity the Inductance particular a line. in of conductors length two are the there consideration that into fact takes it since little specific a more is metre loop A per and conductor. in line stated the resistance a of of imperfection length the metre represents per ohms in stated is theline).Resistance of system, two-wire twicethelength represents a (for conductor the of length Resistance S / I S V hsalo h nryi bobdb the by absorbed is energy the of all Thus . orparameters four S L D n h edn-n urn is current sending-end the and = stedsac ewe ete fthe of centres between distance the is μ L R L 0 π sgvnby given is μ a sgvnby given is sdet h antcfil sur- field magnetic the to due is r nutr In conductor. each of radius the is  4 1 μ + r = ln soitdwt transmis- with associated .A nutnestated inductance An 1. D a Z R 0  ftesending-end the If . = henry/metre ρ l / A ,where A I l S sthe is sthe is then ρ is l (iii) ewe htflwn hog h ekg conductance leakage the through flowing that between ahavr hr egh( length short very T-sections cascaded a of each number large very a of network emntdi t hrceitcimpedance characteristic its in terminated line. the along distributed ae h nu meac ftentokwl lobe also will network the to equal of cas- impedance in input connected the are T-sections cade, identical of number a if meac ftentoki loeulto equal also is network the of impedance redsrbtdntr fteln.We h genera- the When line. the of tor nature the of distributed approaches it number true nearer the the sections, larger parameter the the lumped of line; approximation an distributed is uniformly This 44.1. Figure in shown aho h ortasiso ieconstants, line transmission and four the of Each G (iv) hc sls,adta hc rgesvl charges progressively which that and lost, is which , rnmsinln a ecniee ocnito a of consist to considered be can line transmission A rmCatr4,we ymtia -ewr is T-network symmetrical a when 41, Chapter From V G S nw sthe as known , nms rcia lines practical most In sgvnby given conductorsis two the between capacitance the 567, page con- (14), equation is From 40.3. line Section in twin sidered isolated an of line. capacitance transmission The a of conductors between field Capacitance partic- line. a of in length conductors two ular are there that fact the o odcac sleakance. is conductance represents for and name Another insulation. line the of of imperfection the length metre in per measured is Conductance other. the con- one to from ductor leak to current some allowing line Conductance scnetd current a connected, is Z 0 . C = C rmr constants primary G ln xssa euto h electric the of result a as exists πε sdet h nuaino the of insulation the to due is ( D 0 δ ε / l a r ε ftasiso ie as line, transmission of ) ) r /metre = I S 1 oswihdivides which flows r uniformly are , Z Z 0 0 Similarly, . h input the , R , L , C oiievlejs tteisatwe h iei con- is current line A the maximum in it. when a to instant to nected the shown rising at be line just voltage the value as the positive to let connected and be network shown generator as a Let filter 44.2. repetitive Figure a low-pass to reduces line T-section infinite the and simplifies and neglected, are iue44.2 Figure in delay, shown phase is the that net- it ladder 631, filters, page a T-section (27), is equation low-pass 44.2 Figure of in work shown line the Since delay Phase capacitor into simply is 44.1 losses Figure possessing filter in low-pass shown a that of section Each conductance the or both by loss caused is The line the line. in attenuation transmission the along moving wave capacitor each rtriptvlaeadtevlaea n on on point any gen- at the voltage between the line. difference the and phase voltage a input thus erator mov- and or time, zero be a therefore at will There value. be minimum its may towards ing input generator the voltage, l,b h iecapacitor time the by ple, each on depend and together other. current line the and along voltage travel manner waves this In capacitor. next to along the passed progressively as is charge and polarity, input opposite the before in turn falls, in voltage charged maximum is generator the capacitor the to When up voltage. charge turn input in will capacitors all eeosars t h otg ed urn through current a inductance sends voltage The it. across develops o hre n h otg eeoe cosi sends it across through developed current voltage a the and charges tor G 43Paedly aeeghand wavelength delay, Phase 44.3 h rcs ulndaoetkstm;frexam- for time; takes above outlined process The n h eisresistance series the and eoiyo propagation of velocity L 1 C C and 1 n hc esu thevoltagetravelling up sets which and h aaio hre n voltage a and charges capacitor The . L R 2 L 2 and and I nocapacitor into S C ostruhinductance through flows L G R 3 3 . a ece t maximum its reached has into r eoe,tecircuit the removed, are C R 3 n oo.Thus on. so and , and C 2 h capaci- The . G flosses If . β each ,is L 1 er fteline. the of metre where by: given h eoiyo propagation, of velocity The propagation of Velocity 2 is metre 2π of change phase leading point 2π initial by which point the latter at phase, the line same the the along is voltage point the next the and point given wavelength The Wavelength aito nfe pc sgvnby wavelength given The is space space. free free less in always in radiation is velocity line a the along than energy electrical of velocity sta flgt ..apoiaey300 approximately i.e. light, of that as where or h eoiyo ih.Snetevlct ln ieis line a along velocity than less the always Since light. of velocity the ol ei respace. free in it be than line would the on shorter always is frequency particular a rmeuto 2,wavelength (2), equation From (a) rbe 1. Problem nteln,ad()tesedo rnmsino a of transmission of speed signal. the (b) and line, the wavelengthon the (a) Determine phase 0.05rad/km. a of shift has 1910Hz at operating line transmission h eoiyo rpgto ffe pc stesame the is space free of propagation of velocity The L f stefeunyand frequency the is and π wavelength, / λ u β ec,paecag e metre, per change phase Hence, . C = = aallwr air-spaced parallel-wire A r h nutneadcpctneper capacitance and inductance the are c f λ h aeeghcrepnigt any to corresponding wavelength the , ω λ naln stedsac ewe a between distance the is line a on √ ain cus h hs hneper change phase the occurs, radians = ( ain ic noewvlnt a wavelength one in Since radian. LC f (2 λ radians/metre ) π = / β ) 2 λ β = π u h aeegh Hence wavelength. the ,isgivenby metres 2 rnmsinlines Transmission β π λ f λ = = = = = c × β ω 2 2. km 125.7 2 / 10 π π f / /0 where β 6 β . /.The m/s. 05 u = = 2 λ π c f (1) (2) (3) / of λ λ is 663 ,

Part 3