082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.6 Constant coefficient equations 577
For many transmission lines of interest the signal that is being carried varies sinusoidally
with time. Therefore the voltage and current depend on both position along
theline,z,andtime,t.However,itiscommontoseparatethetimedependencefrom
the voltage and current expressions in order to simplify the analysis. It must be remembered
that any voltage and current variation with position has superimposed
uponitasinusoidalvariationwithtime.Therefore,ignoringthetime-dependentelement
we write the voltage as v and the current asi, knowing that they are functions
ofz.
y
i
i + di
di
Cdz
y L
Ldz
Gdz
y R
Rdz
y + dy
i C
iG
z
dz
Figure19.9
A section ofatransmissionline.
ConsiderthecircuitofFigure19.9whichrepresentsasectionofthetransmission
line oflength δz. Applying Kirchhoff’s voltage law tothe circuityields
v+δv−v+v L
+v R
=0
δv=−v L
−v R
where v L
is the voltage across the inductor and v R
is the voltage across the resistor.
Using the individual component laws for the inductor and resistorgives
δv = −ijωLδz −iRδz
= −i(R +jωL)δz
Notethat δihasbeenignoredbecauseitissmallcomparedtoi.Nowconsidertheparallelcombinationofthecapacitorandresistor(withunitsofconductance).Applying
Kirchhoff’s current law tothiscombination yields
δi=i C
+i G
wherei C
isthecurrentthroughthecapacitorandi G
isthecurrentthroughtheresistor.
Using the individual component laws for the capacitor and resistor gives
δi = −vjωCδz − vGδz
= −v(G +jωC)δz
Dividing these two circuitequations by δzyields
δv
δz
= −i(R +jωL)
δi
δz = −v(G+jωC)
➔