RF Electronic Devices and Systems

# Quarter-wave impedance transformer The quarter-wave impedance transformer is a device that matches the transmission line and the impedance and is shown in Figure 1. Let’s consider the random transmission line with the characteristic impedance Z0, and the load with resistance R.

The characteristic impedance of the quarter-wave transformer is Z1, the length is $\frac{\lambda }{4}$. The matching transmission lines are assumed to be lossless. The input impedance here is

${Z}_{i}n={Z}_{1}\frac{R+j{Z}_{1}\mathrm{tan}\beta l}{{Z}_{1}+jR\mathrm{tan}\beta l}$. For the  . And the characteristic impedance of the arbitrary transmission line is , for the lossless transmission line.

The real quarter-wave impedance transformer experiences a variety of reflected and propagated waves. Let’s consider the case of multiple reflection of the waves. Take a look at Figure 2. There is two transmission lines made with blue dashed circles  There is a wave that falls from the side of the Z0.

When it reaches the quarter-wave transformer, part of the wave reflects with the reflection coefficient ${\Gamma }_{1}$another part of the wave propagate with the coefficient T1. The propagated wave passes the length of the quarter-wave transformer reflecting on the load and passing the load partially.

The reflected part of the wave has the reflection coefficient ${\Gamma }_{2}$ , the propagated wave has the coefficient T2. The reflected part of the wave goes the length of the quarter-wave transformer $\frac{\lambda }{4}$ again and partially reflects on the Z0 with reflection coefficient ${\Gamma }_{3}$, partially propagates Z0 with the coefficient T3, and so on.  Figure 2 depicts the reflected and propagated waves.