For a quarter wave ideal transmission line of characteristic impedance of 50 Ω and load impedance of 100 Ω, the input impedance of line will be :
25 Ω
The quarter-wave transformer relation. Starting from the general lossless line equation
\(Z_{in}=Z_0\dfrac{Z_L+jZ_0\tan\beta l}{Z_0+jZ_L\tan\beta l}\)
and putting \(l=\lambda/4\), so that \(\beta l=\pi/2\) and \(\tan\beta l\to\infty\), the terms without the tangent become negligible and
\(Z_{in}=\dfrac{Z_0^{2}}{Z_L}\)
Step — substitute.
\(Z_{in}=\dfrac{50^{2}}{100}=\dfrac{2500}{100}=25\ \Omega\)
which is option 2.
The property that makes this memorable. A quarter-wave line is an impedance inverter: it reflects the load about Z0 on a logarithmic scale. Here the load is twice the characteristic impedance and the input comes out half of it. Rearranging,
\(Z_0=\sqrt{Z_{in}Z_L}\)
so Z0 is the geometric mean of the two ends — check: \(\sqrt{25\times100}=50\) ✓.
The two extreme cases worth carrying.
| Load | Zin of a λ/4 line |
|---|---|
| Short circuit, ZL = 0 | Open circuit, ∞ |
| Open circuit, ZL = ∞ | Short circuit, 0 |
| Matched, ZL = Z0 | Z0 — unchanged |
That first row is the basis of the quarter-wave stub, used as a mechanical support that is electrically invisible, and of the λ/4 choke in coaxial connectors.
Why it matters in practice. Read backwards, the geometric-mean relation is a matching recipe: to feed a 100 Ω antenna from a 50 Ω cable, insert a quarter-wave section of
\(Z_0=\sqrt{50\times100}=70.7\ \Omega\)
which is exactly why 75 Ω cable is stocked alongside 50 Ω. The one limitation is that the length is a quarter wavelength at a single frequency, so the match is inherently narrowband; multi-section transformers are used to widen it.
Where the distractors come from. 100 Ω is the load itself, which would be the answer for a half-wave line; 50 Ω would require the load to equal Z0; 175 Ω corresponds to no standard relation.
Hence, the input impedance is 25 Ω.
Similar Questions
Consider the following statements
A. Reflection coefficient is change in the magnitude of reflected wave with constant phase with respect to incident wave.
B. Transmission coefficient is change in the magnitude and phase of transmitted wave with respect to incident wave
C. On smith chart, \(\dfrac{\lambda}{2}\) distance is equivalent to 2π.
D. For distortion less line, RL = GC
E. Directivity of an antenna can be less than 1 (unity).
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