Using LC Resonators in a Match — Suppressing Harmonics While Keeping the Match

The elements of a matching network are not limited to L and C. Use an LC resonator as a single element and it can do another job while keeping the match intact.

Two kinds of resonant element

The two used here are:

Away from resonance, both behave like an ordinary L or C. It helps to think of them as elements with a special reactance only at the resonant point.

The degree of detuning can be written with one quantity:

d = 1 − (f / f_res)²

Series tank:  X = ωL / d     (denominator → 0 at f_res → open)
Shunt trap:   B = ωC / d     (denominator → 0 at f_res → short)

For f ≪ f_res, d → 1, and the tank reverts to a plain L and the trap to a plain C. In other words, put the resonance high enough and you can use them as ordinary elements.

The sign flips

d changes sign as you cross f_res. This is what makes resonant elements interesting: the same element is inductive at some frequencies and capacitive at others.

On the chart, the point appears to move in the opposite direction on either side of resonance. When designing, if you lose track of which side you are on, the direction you change the value and the direction the point moves stop agreeing, and it gets confusing.

A practical example — suppress a harmonic, keep the match

Let us do it concretely. The 200 Ω → 50 Ω network from the L-section note was

Shunt C = 1.3783 pF, series L = 13.7832 nH

Replace that shunt C with a trap. The condition is "do not change the susceptance at 1 GHz". Put the resonance at the second harmonic, 2 GHz.

The detuning at 1 GHz is

d = 1 − (1/2)² = 0.75

To produce the same susceptance, ωC/0.75 = B, so the capacitance needed is 0.75 times the original:

C = 1.0337 pF
L = 6.1259 nH   (its partner, resonating at 2 GHz)

Side by side:

The match at 1 GHz is exactly the same. |Γ| is of order 1e−16 for both; they cannot be told apart. And yet a deep notch appears at 2 GHz.

The element count has not changed. The shunt C has simply become a shunt LC: one more part, but the topology of the network is unchanged. The matching network has taken on the function of a harmonic filter.

Notch depth is set by Q

The table just says "notch" at 2 GHz because with ideal elements it is infinitely deep. The calculation produces something like −240 dB, which is not a meaningful number.

What sets the real depth is the Q of the elements. The trap's series LC always has some resistance, so even at resonance it is never a perfect short:

Impedance at resonance ≈ the element's effective series resistance

As rules of thumb in practice:

• With an inductor Q of around 50, the notch depth is 30–40 dB
• The notch center shifts with element tolerance. The deeper the notch, the sharper it is, and the more a shift hurts
• It can also drift with temperature

"Aiming for a deep notch" and "reliably suppressing" are not the same thing. It is often safer to design it a little shallower and wider, so the spec is still met when part tolerance moves the center by a few percent.

Where series tanks are useful

Where a trap "sends a particular frequency to ground", a series tank "blocks a particular frequency":

• Isolating the bias network — make the tank open at the signal frequency and it passes DC while blocking RF. A more refined choke inductor
• Out-of-band suppression between stages — put an open in series at the frequency you do not want through
• Parasitic resonance — real inductors have a self-resonance, so they may be acting as a tank without your intending it

The last point is a warning in the other direction. A part you are using as an ideal L behaves as a tank near its self-resonant frequency. If the self-resonant frequency (SRF) in the datasheet is close to your operating frequency, you have to treat it as a resonant element.

Try it

Resonant elements make sense once you watch them while moving the resonant frequency.

Smith Match — Matching Network Designer

Series tanks and shunt traps are available as elements, each with its own resonant frequency setting. Move the resonance across f0 and you can watch the direction the point moves reverse on the chart. Switch to the sweep view to see the notch above. The resonant frequency you set is drawn on the plot as a purple dashed line, so you can read which element a notch belongs to without mixing them up.