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In the text, it introduces a practical model to investigate a transmission line (like BNC cable), it considers the transmission line has resistive $R$, inductance $L$, conductance $G$ and capacitance $C$. The model is illustrated as follow

enter image description here

It is easy to derive the (telegraph) equations and figure out the impedance Z to be

$$Z = \sqrt{\frac{R+iX_L}{G+i/X_C}}$$

where $i$ is the imaginary unit, $\omega$ is the angular frequency, $X_L$ is the inductive reactance and $X_C$ is the capacitive reactance.

And in other section, it introduce a RC circuit and RLC circuit, in which, the impedance are $$Z_{RC} = \sqrt{R^2 + X_C^2}, \qquad Z_{RLC}=\sqrt{R^2 + (X_L - X_C)^2}$$

It is pretty confusing because from RLGC model, if we make the electrical conductance $G$ to zero and consider no inductance ($L=0$), so the circuit becomes RC circuit, but from the first equation for the impedance given by the RLGC model, the impedance should be

$$Z = \sqrt{-iRX_C}$$

Why are they not the same? How to approach RC and RLC case from RLGC model?

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They are different because in a transmission line we have distributed resistance, capacitance, conductance and inductance (meaning that each tiny segment of transmission line has its own tiny resistance, capacitance, conductance and inductance) while in RLC circuits we have lumped resistance, inductance and capacitance. Also RLGC doesn't model a transmission line with the circuit you've shown above but with infinite number of them in series.

We know well how to deal with lumped elements and circuits containing them, but dealing with distributed elements and circuits (e.g transmission lines) is often much harder and we have to resort to solving Maxwell equations directly. So I think not only there's no point in approaching RLC circuits from RLGC model but also it's impractical.

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But what happen if the transmission light is so uniform and short so we only have one section, then the RLGC model is just a model of RLC in series and with one G connected to C in parallel. Also, RLGC is just a mathematical model, why it fail to reduce to RLC circuit? – user1285419 May 3 '13 at 0:00
What I mean is if I only show the equivalent circuit as in the question but didn't tell that it is the simplified model for a transmission line. So how can you tell the difference? Note that to get the impedance for RLGC model, the only math we use is the loop and junction law, so the result of RLGC model should be able to reduced RLC circuit if we remove the G term. – user1285419 May 3 '13 at 4:07
Note that to get the impedance for RLGC model, the only math we use is the loop and junction law That's not true. The formula you're given for characteristic impedance is derived from telegrapher equation assuming infinite number of infinitesimal components (integration). – Azad May 3 '13 at 7:42
Ok. I think I get the point now. But I still have one question. If we separate the transmission line into infinite number of sections, why we assume in each section, the R, L, G, C are constant and same for each section? – user1285419 May 3 '13 at 14:00

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