# How we approach RLC circult from RLGC model?

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

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?

• Your equation for the impedance of the RLC circuit is incorrect. The correct expression is $Z_{RLC} = R + i\left( \omega L - \frac{1}{\omega C} \right)$. Commented Dec 19, 2022 at 0:00