Practical use of a quarter wave transformer

[FrankJ777]

Hi guys. I have some questions about the practical use of a quarter wave transformer to match loads. For my question please refer to the figure below...
According to my reading the input inpedance fo a quarter wave transformer is:
Z_in = Z₁²/Z_L
and to properly match the line with characteristic impedance Z₀ to the load Z_L, Z₁ must be chosen so that:
Z₁ = √Z₀Z_L

So it seems that for the quarter wave transformer to be useful I wave to be able to choose a transmission line of the appropriate impedance Z₁. When would that be of practical use? If i have only coax of 50Ω or75Ω etc., it seems I would have a difficult time finding an instance where I could use a quarter wave transformer. Obviously I'm missing something.

 

[skeptic2]

First, coax comes in many different impedances so you needn't stick to 50 or 75 ohms.

Second, quarter wave transformers are usually used with microstrip at UHF and microwave frequencies where it is easy to design a length of transmission line at nearly any impedance.

Third, you don't say if you are familiar with the Smith Chart but if you are, you can use it to match not just resistive values but also complex impedances with a just transmission line.

Fourth, if you can't find a coax of the appropriate impedance and if microstrip isn't appropriate, you can create a network that works the same as a quarter wave transformer with discrete components. You would use either a PI or T network made of two capacitors and one inductor or vice versa in which each component has a reactance equal to the value of the transmission line you would need.

Example: If you are trying to match 50 ohms to 100 ohms and you need a 71 ohm transmission line, you could use anyone of 2 PI or 2 T networks of L-C-L or C-L-C where each L and each C has a reactance of 71 ohms.

 

[Enthalpy]

If your lines are symmetric it's easy to create yours with the needed impedance, using rods. Commonly done for antennas.

 

[vk6kro]

Quarter wave lines are also useful for driving multiple antennas.

Suppose you have two antennas which have 50 ohm input impedance and you want to drive both of them in phase with a 50 ohm feedline from a transmitter.

One way of doing this is to get a quarter wavelength of 75 ohm coax attached to each antenna and then drive the two ends in parallel.

This works by stepping up the impedance of each antenna to 103 ohms according to the formula:
Z_o² = Z₁ * Z₂

so, 75² = 50 * Z₂
Z₂ = 75² / 50 = 103 ohms

Placing these in parallel gives a close match for a 50 ohm feedline.

 

[yungman]

Like Skeptic 2 said, using microstrip or stripline to realize the λ/4 line of Z1 is the easiest as you can design the impedance to Z1 easily as long as it is within reason. Or used discrete components in Pi network will work. If you are familiar with matching network design using Smith Chart, it would be very easy to find the value of the components.