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

578 Chapter 19 Ordinary differential equations I
Inordertomodelacontinuoustransmissionlinewithevenlydistributedparameters,
δz isallowed totend tozero. Inthe limitthe two circuitequations become
dv
= −i(R +jωL) (19.19)
dz
di
= −v(G+jωC) (19.20)
dz
Differentiating Equation (19.19) yields
d 2 v
= −(R +jωL)di
dz2 dz
Substituting for di fromEquation (19.20) yields
dz
d 2 v
= (R +jωL)(G +jωC)v
dz2 This isusually writtenas
d 2 v
dz 2 = γ2 v where γ 2 = (R +jωL)(G +jωC) (19.21)
This is the differential equation that describes the variation of the voltage, v, with
position,z,alongthetransmissionline.Thegeneralsolutionofthisequationiseasily
shown tobe
v=v 1
e −γz +v 2
e γz (19.22)
where v 1
and v 2
areconstantsthatdependontheinitialconditionsforthetransmission
line. It is useful to write γ = α +jβ thus separating the real and imaginary parts of
γ. Equation (19.22) can then be writtenas
v = v 1
e −αz e −jβz + v 2
e αz e jβz (19.23)
The quantity v 1
e −αz e −jβz represents the forward wave on the transmission line. It
consists of a decaying exponential multiplied by a sinusoidal term. The decaying
exponentialrepresentsagradualattenuationofthewavecausedbylossesasittravels
along the transmission line. The quantity v 2
e αz e jβz represents the backward wave
produced by reflection. Reflection occurs if the transmission line is not matched
withitsload.Asthewaveistravellingintheoppositedirectiontotheforwardwave,
e αz stillrepresents an attenuation butinthiscase an attenuation aszdecreases.
Alosslesslineisoneinwhichtheattenuationisnegligible.Thiscasecorresponds
toα=0,andsoγ =jβ.Ifγ =jβthenγ 2 = −β 2 so that, from Equation (19.21),
(R + jωL)(G + jωC) must be real and negative. We see that this is the case when
R = 0 andG = 0. This agrees with what would be expected in practice as it is the
resistive and conductive termsthatlead toenergy dissipation.
Engineeringapplication19.10
Voltagereflectioncoefficient
Consider a lossless transmission line of the type already described in Engineering
application19.9.Weknowingeneralthattheforwardwaveatpositionzisgivenby
v 1
e −αz e −jβz . For the lossless line this simplifies to v 1
e −jβz . The reverse term, by the
samereasoning,is v 2
e jβz .