Electronic – the definition of characteristic impedance of a lattice network

This seems the simplest mathematical way to derive characteristic impedance. Consider a "lump" of transmission line connected to the continuation of that transmission line (\$Z_0\$): -

• R is series resistance of cable for a given length
• L is series inductance of cable for a given length
• G is parallel conductance of cable for a given length
• C is parallel capacitance of cable for a given length
• \$Z_0\$ to the right is the continuation of the cable

Therefore the impedance looking into the left is: -

$$Z_0 = R + j\omega L + Z_0||\dfrac{1}{G + j\omega C}$$

$$= R + j\omega L + \dfrac{\frac{Z_0}{G+j\omega C}}{Z_0 + \frac{1}{G+j\omega C}}$$

$$= R + j\omega L + \dfrac{Z_0}{1 + Z_0(G+j\omega C)}$$

$$Z_0[1 + Z_0(G+j\omega C)] = [R+j\omega L][1 + Z_0(G+j\omega C)]+Z_0$$

$$Z_0 + Z_0^2(G+j\omega C) = R+j\omega L + Z_0[(R+j\omega L)(G+j\omega C)]+Z_0$$

$$Z_0^2(G+j\omega C) = R+j\omega L + Z_0[(R+j\omega L)(G+j\omega C)]$$

The important thing next is to recognize that \$(R+j\omega L)(G+j\omega C)\$ is insignificant as the "lump" approaches zero length and we are left with: -

$$Z_0^2(G+j\omega C) = R+j\omega L$$

hence $$Z_0 = \sqrt{\dfrac{R+j\omega L}{G+j\omega C}}$$

The impedance of a parallel connected resistor and capacitor is

$$Z = R\,||\frac{1}{j\omega C} = \frac{1}{\frac{1}{R}+ j\omega C} = R \,\frac{1}{1+j \omega RC} $$

This is of the form of a resistance times a dimensionless, frequency dependent quantity.

There is a particular frequency of interest,

$$\omega_0 = \frac{1}{RC}$$

which is the characteristic frequency. At this frequency, the impedance is

$$Z_0 = R \,\frac{1}{1+j \omega_0 RC} = R \,\frac{1}{1+j} = \frac{R}{\sqrt{2}}e^{-j\frac{\pi}{4}}$$