How do I calculate the input impedance of a transmission line terminated in an arbitrary load?
Transmission Line Impedance Transformation
A transmission line transforms the impedance of its termination as a function of its electrical length. This impedance transformation is the basis for stub matching, quarter-wave transformers, and most microwave matching network designs. The transformation is periodic with a half-wavelength period: the impedance at any point repeats every λ/2 along the line.
| Parameter | Semi-Rigid | Conformable | Flexible |
|---|---|---|---|
| Loss (dB/m at 10 GHz) | 0.8-2.5 | 1.0-3.0 | 1.5-5.0 |
| Phase Stability | Excellent | Good | Fair |
| Bend Radius | Fixed after forming | Hand-formable | Continuous flex OK |
| Shielding (dB) | >120 | >90 | >60-90 |
| Cost (relative) | 2-5x | 1.5-3x | 1x |
Frequently Asked Questions
How do I use this for matching?
Plot the load impedance on a Smith chart. Rotate clockwise (increasing line length) until the impedance reaches the desired value. The required line length and impedance are read from the chart. Software tools (matching network synthesis) automate this process for multi-element networks.
What about multi-section transformers?
Cascading multiple quarter-wave sections of graduated impedance provides wideband matching. A two-section transformer provides 30-50% bandwidth; a three-section extends to 60-80%. The impedances are chosen using Chebyshev or maximally-flat transformer design tables.
Does the formula work for waveguide?
Yes, replacing Z0 with the waveguide impedance (typically 400-600 Ω for rectangular waveguide in the TE10 mode). The wavelength is the waveguide wavelength λg, not the free-space wavelength. All the same impedance transformation properties apply.