Transmission Lines & Impedance

At low frequencies (DC, Audio), a wire is just a connector. At RF, a wire is a Transmission Line—a complex distributed network of inductance and capacitance. The voltage is not the same everywhere on the wire at the same time.

1. Distributed Element Theory (Telegrapher's Equations)

If you zoom in on an infinitesimal slice of a coaxial cable, it's not a perfect conductor. It looks like a circuit:

• L' (Inductance/m): Series inductance due to magnetic flux around the center conductor.
• C' (Capacitance/m): Shunt capacitance between center conductor and shield.
• R' (Resistance/m): Series resistance (copper loss + skin effect).
• G' (Conductance/m): Shunt leakage through the dielectric.

Characteristic Impedance ($Z_0$)

If you send a pulse down an infinite transmission line, it will charge the C's and flow through the L's. The ratio of the Voltage wave to the Current wave is constant.

$Z_0 = \sqrt{\frac{R' + j\omega L'}{G' + j\omega C'}}$

For a Lossless Line (R=0, G=0), this simplifies to:

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

• Key Insight: $Z_0$ depends only on geometry (dimensions) and material constants. A 50$\Omega$ cable measures 50$\Omega$ regardless of whether it's 1 meter long or 1 km long. You cannot measure $Z_0$ with a DC multimeter (it will read Open).

2. Reflections & The Reflection Coefficient ($\Gamma$)

When a traveling wave hits a Load ($Z_L$) that doesn't match the line's impedance ($Z_0$), not all energy can be absorbed. To conserve energy, some must reflect.

$\Gamma = \frac{V_{reflected}}{V_{incident}} = \frac{Z_L - Z_0}{Z_L + Z_0}$

The Three Extremes

1. Matched Load ($Z_L = Z_0$):
   • $\Gamma = 0$. No reflection. All power absorbed.
2. Short Circuit ($Z_L = 0$):
   • $\Gamma = -1$. Total reflection, $180^\circ$ phase flip.
3. Open Circuit ($Z_L = \infty$):
   • $\Gamma = +1$. Total reflection, $0^\circ$ phase shift.

3. Measurable Metrics: VSWR & Return Loss

In the real world, we rarely measure $\Gamma$ directly. We use Vector Network Analyzers (VNAs) to measure Return Loss or VSWR.

Return Loss (RL)

The ratio of Reflected Power to Incident Power, in dB.

• $RL_{dB} = -20 \log_{10} |\Gamma|$
• Sign Convention: Usually positive number (e.g., "20dB Return Loss" means the reflection is 20dB down from the signal).
• Good Value: >15dB is usually acceptable. >20dB is good.

Voltage Standing Wave Ratio (VSWR)

The interaction of the Forward wave and Reflected wave creates a "Standing Wave" pattern of voltage peaks and nulls along the line.

• $VSWR = \frac{V_{max}}{V_{min}} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$
• 1:1: Perfect Match.
• $\infty$:1: Total Reflection (Open/Short).
• Typical Spec: Antennas are often specified as "VSWR < 2:1".

4. Impedance Transformation

This is the most non-intuitive part of RF: The impedance of a load looks different depending on how far away you are standing.

The Input Impedance ($Z_{in}$) looking into a lossless line of length $l$:

$Z_{in}(l) = Z_0 \frac{Z_L + jZ_0 \tan(\beta l)}{Z_0 + jZ_L \tan(\beta l)}$

The Quarter-Wave Transformer

If length $l = \lambda/4$, then $\tan(\beta l) \to \infty$. The equation simplifies beautifully:

$Z_{in} = \frac{Z_0^2}{Z_L}$

• Magic: You can turn a Short Circuit ($Z_L=0$) into an Open Circuit ($Z_{in}=\infty$) just by adding a $\lambda/4$ transmission line!
• Application: This is how we match disparate impedances (e.g., matching a 10$\Omega$ transistor PA to a 50$\Omega$ system).