2.5.4 Attenuation on a Low-Loss Line Recall that, the propagation constant, is given by Y = √(R+JwL)(G+ 3wC). This can be written as and thus (2.141) With a low-loss line, R< wL and G < wC, and so, using a Taylor series approximation (see Equation (1.174)), 1/2 (1+) ≈1+- G JwC JwC = + ( R√√² + G√ =) + R G w√ LC ((1+ JBL) (1 + JCC). ἀπ y = jw√ N + 1/2 + GZo R JwL G Hence for low-loss lines (in Np/m if SI units are used), R Zo ≈1 + − +JwVLC. (2.140) (2.145) and B w√LC. (2.142) (2.143) g (2.144) (2.146)

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Engineering Electromagnetics: how did my textbook get this apporximation for gamma and alpha? I don't understand the derivation, it says to use the Taylor Series and I did.

 

2.5.4 Attenuation on a Low-Loss Line
Recall that y, the propagation constant, is given by
Y = √(R+JwL) (G+JwC).
This can be written as
and
thus
Y = jw√LC₁
With a low-loss line, R < wL and G < wC, and so, using a Taylor series
approximation (see Equation (1.174)),
1/2
α
22
R
JwL
1+
R
JwL
G
1+-
'JwC¹
L
• + (R√√= + 0 √ 7) +
с
R
(1 + JBL) (1 + C)
(1+)
7 = 1/2
1/2
1+
Hence for low-loss lines (in Np/m if SI units are used),
R
+(2+GZ₁) (2.145)
Zo
+JwVLC.
(2.140)
and w√LC.
(2.141)
(2.142)
(2.143)
g (2.144)
(2.146)
Transcribed Image Text:2.5.4 Attenuation on a Low-Loss Line Recall that y, the propagation constant, is given by Y = √(R+JwL) (G+JwC). This can be written as and thus Y = jw√LC₁ With a low-loss line, R < wL and G < wC, and so, using a Taylor series approximation (see Equation (1.174)), 1/2 α 22 R JwL 1+ R JwL G 1+- 'JwC¹ L • + (R√√= + 0 √ 7) + с R (1 + JBL) (1 + C) (1+) 7 = 1/2 1/2 1+ Hence for low-loss lines (in Np/m if SI units are used), R +(2+GZ₁) (2.145) Zo +JwVLC. (2.140) and w√LC. (2.141) (2.142) (2.143) g (2.144) (2.146)
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