Question

Characteristic impedance of a quarter wave transformer connected in between a load of 100 ohm and a transmission line of characteristic impedance 225 ohms is
1. 100 ohm
2. 225 ohm
3. 600 ohm
4. 150 ohm

Answer 1

Correct Answer - Option 4 : 150 ohm

Concept:

\({{Z}_{in}}={{Z}_{0}}\left( \frac{{{Z}_{L}}+j{{Z}_{0}}\tan \beta l}{{{Z}_{0}}+j{{Z}_{L}}\tan \beta l} \right)\)

\(\because \beta =\frac{2\pi }{\lambda }\), at a quarter \(\left( \frac{\lambda }{4} \right)\) distance away,

\({{Z}_{in}}={{Z}_{0}}\frac{\left( {{Z}_{L}}+j{{Z}_{0}}\tan \left( \frac{2\pi }{\lambda }.\frac{\lambda }{4} \right) \right)}{\left( {{Z}_{0}}+j{{Z}_{L}}\tan \left( \frac{2\pi }{\lambda }.\frac{\lambda }{4} \right) \right)}\)

\({{Z}_{in}}=\frac{Z_{0}^{2}}{{{Z}_{L}}}\)

Calculation:

Given, ZL = 225 Ω

Zin = 100 Ω

\({{Z}_{in}}=\frac{Z_{0}^{2}}{{{Z}_{L}}}\)

Putting on the respective values and solving for Z0, we get:

Z02 = Zin × ZL

Z02 = 100 × 225 = 22500

Z0 = 150 Ω

So the quarter-wave transformer should have a characteristic impedance of 150 Ω.