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

Question

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?

Answer 1

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.

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