# How can a quarter-wave transmission line transformer be implemented with lumped elements?

Sometimes designs call for quarter-wave sections of transmission lines. There are some situations when using actual transmission lines would not be practical:

• The necessary characteristic impedance is something not easily obtained
• A quarter wavelength of transmission line would be too physically large

Can I accomplish the same thing with lumped elements like inductors and capacitors? If so, how?

It can be done! Remember that a transmission line consists of some self-inductance per unit length, and some capacitance per unit length, and the ratio of these determines the line's characteristic impedance:

$$Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}}$$

Intuitively then, a lumped element implementation might look like some series inductance and some parallel capacitance. We need something that's symmetrical, so a pi network1 like this should do it: simulate this circuit – Schematic created using CircuitLab

All we have to do is determine the appropriate values. Since transmission lines work in both directions this arrangement is going to be symmetrical, so C1 = C2. And as it turns out, the values are very elegant: the reactance of the components is equal to the characteristic impedance. Put mathematically:

$$X_{L1} = Z_0 \\ X_{C1} = Z_0$$

And while we're at it, let's review the formulae for reactance:

$$X_L = 2 \pi f L \\ X_C = 1 / (2 \pi f C)$$

Putting those together, with a bit of algebra, you get:

$$L = {Z_0 \over 2 \pi f}$$

$$C = {1 \over 2 \pi f Z_0}$$

Let's say we want a characteristic impedance of 86.6 ohms so we can realize a 3-way Wilkinson power splitter. We'll use the frequency 435 MHz for the middle of the 70cm band.

$$L = { 86.6\:\Omega \over 2\pi \cdot 435\:\mathrm{MHz} } = 31.5\:\mathrm{nH}$$

$$C = {1 \over 2\pi \cdot 435\:\mathrm{MHz} \cdot 86.6\:\Omega } = 4.22 \:\mathrm{pF}$$

So now the circuit with real values, terminated with an 86 ohm load: simulate this circuit

If this is indeed a quarter-wave transmission line, we should see R1 90 degrees out of phase with the input. Let's run a time-domain simulation:  Bingo! The impedance transforming properties of a quarter wave transmission line are also preserved: if the output is open, the source will see a short, and so on. So this circuit can be used in place anywhere that calls for a quarter-wave transmission line.

1 This pi network makes a low pass filter, which can be a nice side-effect since it reduces harmonic distortion.

• As a bonus, this circuit works as a Low Pass filter with a -3 dB cut off at about 620 MHz. -10 dB at 880 MHz. -20 dB at 1.3 GHz. May 18, 2016 at 20:56
• That looks suspiciously like the symbol Pi. :) May 19, 2016 at 17:10
• How do we know that this provides Zo ohms impedance? Aug 1, 2017 at 14:40
• @NourhanElsayed Not sure what you mean. It doesn't provide any impedance: it provides an impedance transformation equivalent to a quarter-wave section of some transmission line of some impedance. If you are wondering how we know what "some impedance" is, the math is in the answer. Aug 1, 2017 at 16:04