What is the input impedance of a terminated transmission line?
For a lossless line of characteristic impedance Z0 and electrical length βℓ terminated in ZL, the input impedance is Zin = Z0·(ZL + jZ0 tan βℓ)/(Z0 + jZL tan βℓ).[1] On the Smith chart, moving along the line rotates Γ around the chart center, one full turn per half wavelength.
What are the special cases?
- Matched (ZL = Z0): Zin = Z0 at any length.
- Half wave (ℓ = λ/2): Zin = ZL.
- Quarter wave (ℓ = λ/4): Zin = Z0²/ZL, an impedance inverter (see quarter-wave transformer).
- Short-circuited stub: Zin = jZ0 tan βℓ (inductive below λ/4).
- Open-circuited stub: Zin = −jZ0 cot βℓ (capacitive below λ/4).
How are stubs used as components?
Short stubs act as inductors or capacitors; a λ/4 shorted stub is an open circuit at its design frequency and is used as a DC feed or bias tee; a λ/4 open stub is a short at its design frequency and is used as a notch or harmonic trap. Richards’ transformation formalizes this substitution for distributed filters.[1]
How does loss change the formula?
Replace jβℓ with γℓ and tan with tanh: Zin = Z0(ZL + Z0 tanh γℓ)/(Z0 + ZL tanh γℓ). A long lossy line tends toward Z0 regardless of load, which is why cable loss masks antenna mismatch.
Frequently asked questions
Why does impedance repeat every half wavelength?
Because the round-trip phase is 360°, so the reflected wave returns in the same phase.
Is an open stub always capacitive?
Only for lengths below λ/4; between λ/4 and λ/2 it is inductive.
Does this apply to microstrip?
Yes, with the guided wavelength and ignoring dispersion and end effects.
See also
References
Online sources accessed October 21, 2026; printed works cited by edition. See all sources.
- “Microwave Engineering, 4th ed.”. D. M. Pozar, Wiley. 2012.