Characteristic Impedance Calculation

The characteristic impedance ($Z_0$) of a transmission line is a crucial parameter that relates the voltage and current waves propagating along it. For a lossless transmission line, it is defined as the ratio of the square root of inductance per unit length ($L$) to capacitance per unit length ($C$):

$$Z_0 = \sqrt{\frac{L}{C}}$$

Parallel Plate Line Parameters

For a parallel plate transmission line constructed with strips of width $w$ and separated by a distance $d$, the inductance per unit length ($L$) and capacitance per unit length ($C$) can be approximated. These approximations are particularly relevant when the width $w$ is significantly larger than the separation $d$ ($w \gg d$):

• Capacitance per unit length ($C$): $$C \approx \frac{\epsilon w}{d}$$
• Inductance per unit length ($L$): $$L \approx \frac{\mu d}{w}$$

In these formulas, $\epsilon$ represents the permittivity and $\mu$ represents the permeability of the dielectric material situated between the conducting strips.

Calculating $Z_0$ for Parallel Plates

By substituting these approximate expressions for $L$ and $C$ into the formula for characteristic impedance, we obtain:

$$Z_0 \approx \sqrt{\frac{\mu d/w}{\epsilon w/d}}$$

Simplifying this equation gives:

$$Z_0 \approx \sqrt{\frac{\mu}{\epsilon} \frac{d^2}{w^2}} = \sqrt{\frac{\mu}{\epsilon}} \frac{d}{w}$$

The term $\sqrt{\frac{\mu}{\epsilon}}$ denotes the intrinsic impedance of the dielectric medium filling the space between the plates.

Effect of Doubling Line Dimensions

Let's now analyze the effect of doubling both the width ($w$) and the separation ($d$) of the parallel strips. The new dimensions will be $w' = 2w$ and $d' = 2d$. The new characteristic impedance, denoted as $Z_0'$, can be calculated using these scaled dimensions:

$$Z_0' \approx \sqrt{\frac{\mu}{\epsilon}} \frac{d'}{w'} = \sqrt{\frac{\mu}{\epsilon}} \frac{2d}{2w}$$

After simplification, the expression for $Z_0'$ becomes:

$$Z_0' \approx \sqrt{\frac{\mu}{\epsilon}} \frac{d}{w}$$

$Z_0$ Result After Dimension Change

Comparing the new characteristic impedance ($Z_0'$) with the original characteristic impedance ($Z_0$), it is evident that:

$$Z_0' = Z_0$$

This indicates that doubling both the width ($w$) and the separation ($d$) of the parallel plate transmission line does not alter its characteristic impedance.