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L, Pi or T: choosing a matching network and what it costs in bandwidth

An L-network's Q is whatever the transformation makes it. You do not get a say. Pi and T networks add a third component, and that third component is what buys you the choice.

The L-network has no free parameter

Two components, two things to satisfy (get the real part right, cancel the imaginary part), so the values are determined. Take 5 ohms to 50 ohms at 100 MHz:

Both have a Q of 3.00, and that is not a coincidence. For an L-network the loaded Q comes straight out of the impedance ratio:

Q = sqrt(Rhigh / Rlow - 1)

Ten to one here, so the square root of nine, so 3. Change the ratio and the Q changes with it. Keep the ratio and there is nothing you can do about the Q, because there is no third value to move.

Adding a third component

The trick in both Pi and T is a virtual resistance: an intermediate impedance that exists nowhere in the circuit, which you transform to and then from, using two L-networks back to back.

• A T transforms up to a virtual resistance higher than both ends: Rv = min(R1, R2) x (1 + Q^2)
• A Pi transforms down to one lower than both ends: Rv = max(R1, R2) / (1 + Q^2)

Pick Q, and the virtual resistance follows, and the components follow from that. Since you chose the Q, you got the bandwidth you asked for. That is the entire point of the third component.

What it costs

Ask QuickSmith for a target Q in the Auto-match dialog and it offers Pi and T networks alongside the L. The same 5 to 50 ohm transformation:

Bandwidth is roughly inversely proportional to Q, as you would expect. Note which direction the third component takes you: a Pi or a T only ever narrows things. You cannot ask for a Q below what the L-network already achieves, because there is no virtual resistance that far in. QuickSmith says so rather than offering you something that does not exist.

So why would you ever want narrower?

Three reasons, and none of them is the match.

Selectivity. If the matching network is also your filter, a higher Q means more attenuation of the neighbours. Transmitter output networks are often Pi networks chosen for exactly this: match, and knock down the second harmonic on the way.

Component values you can actually buy. A wide transformation ratio through a single L-network can land you on a 2 pF capacitor where stray capacitance is 1 pF, or a 3 nH inductor where the lead is 3 nH. Pushing the Q up moves the values somewhere sane. Look at the Q of 12 row: a 34 pF and a 95 nH are ordinary parts.

Splitting an awkward ratio. Two moderate transformations back to back are often kinder than one violent one, particularly if you have to build it on a board where parasitics matter.

Pi or T?

They are equivalent in what they achieve, so choose on practicalities.

• A T has series elements at both ends, so it is easy to get DC through, or to block DC with a series capacitor if that is what you want.
• A Pi has shunt elements at both ends, which is convenient when you have parasitic capacitance to swallow at each end anyway, such as a transistor's output capacitance. You absorb it into the network instead of fighting it.
• Look at the component values. In the example above, the Pi at Q of 8 wants a 746 pF capacitor, and the T wants 50 pF. At 100 MHz the T is the more comfortable build.

Rules of thumb

• L-network Q is sqrt(ratio - 1) and you cannot change it.
• Pi and T can only make Q higher, never lower.
• Higher Q means narrower bandwidth, more selectivity, and more loss for the same component Q. That last one is worth its own article.
• If you need a lower Q than the L-network gives, you need a multi-section network: two L-networks in cascade going through a virtual resistance between the two ends rather than outside them. QuickSmith will not design that for you automatically, but the ladder has twelve slots and nothing stops you doing it by hand.