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The Match Component

Direct synthesis of a bandpass matching network that absorbs both terminations.

If you have used a single-frequency matching tool

Read this first if your instinct is "give me a target impedance, 50 Ω, and a part count." Match does exactly that job — it just does it over a band, and that changes what comes out in three ways that surprise people.

What Match is not. It is not an L / π / T tuner that transforms one impedance at one frequency, and it is not a Smith-chart tool that walks you round constant-R and constant-G circles element by element. Those give you the smallest network that is exact at a point and falls apart either side of it. Match gives you the widest band the load's Q allows, which is a different network with a different part count. If a single frequency is genuinely all you need, place the L and C yourself — Match is the tool for the case where a point match is not enough.

Three things that follow, and that the rest of this page assumes you have read:

• The part count is higher than a two-element L-match, and some of the parts are already on your board. The smallest network Match offers is order 2, which is four elements in every form (see Bandpass, lowpass and highpass); an L-match is two. But the ones the terminations supply are not parts you buy — a transistor's C_gs or C_ds is absorbed and becomes an element of the ladder, drawn at the end that supplies it. Read the count off the ladder, not off the order. See Absorption for why that is the whole point of the component.

• Elements to ground are normal, and they are half of every ladder. A matching ladder alternates series arms and shunt arms; the shunt arms go to ground because that is what a shunt arm is. Seeing capacitors and inductors between a node and ground is the network working, not a bug.

• Shunt DC Block does not put a capacitor in the through path. If you came here expecting the series blocking capacitor a legacy tool drops into the signal line, see Shunt DC Block — the button solves a different problem, and Match never inserts a through-path block. This is the single most-misread control in the Designer.

What Match is

Match is a two-port component that synthesises a bandpass LC matching network matching both of its ports simultaneously to the network around it. Place it, tell it what is on each side and over what band, and it produces a ladder — element by element, with values.

It is direct synthesis: closed-form, procedural, no optimiser. There is no seed, no convergence, no "run it again and see if it finds something better". The same specification always produces the same network, and when a specification has no solution at the order you asked for, you are told so with numbers rather than left watching an optimiser fail to converge.

The component stamps the ladder element by element on internal nodes — not as a lumped ABCD block — so it behaves correctly at DC and in harmonic balance, and it is identical by construction to the cell that Flatten to Cell writes.

The symbol tells you what it is

A Match puts no parameter text on the schematic — no F1, no F2, no Order. Everything about it is edited in the Match Designer, and everything about it is read there or in the Properties panel, so three numbers beside the symbol were three numbers that could not be acted on where they stood.

The glyph says the part you need at a glance. Its three stacked waves are a frequency axis, highest at the top, and a slash means that band is blocked:

It follows the design: apply a lowpass solution and the symbol on the page becomes a lowpass symbol, and switching Bands to Dual or Tri splits the wave stack into two or three smaller ones. The glyph is drawn from the component's own Form and Bands parameters, so a schematic printed for a review states the topology of every matching network on it without anyone opening a Designer.

Absorption, and why it is the whole point

Most matching networks tune out a termination's reactance: the transistor has 10 pF across its input, so you resonate it away and match what is left. That works, over a narrow band, because the resonance only cancels at one frequency.

Match absorbs each termination's reactance into the network instead. A transistor's C_gs or C_ds becomes an element of the matching filter — one of the ladder's own capacitors, in the position the filter wanted a capacitor anyway.

Two consequences worth being clear about:

• The absorbed elements are not inside the component. The external network supplies them — that is what makes the match real. The Designer always draws them immediately beside the termination that supplies them — that position, and the one-line legend under the schematic, are how you tell which elements you do not have to buy. They are drawn at full brightness like everything else: the part you are matching against is the last thing that should be hard to read.
• Q is the entire content of a termination, as far as the synthesis is concerned. At band centre ω₀ = √(ω₁ω₂) with fractional bandwidth w: a parallel R‖C has Q = ω₀RC, a series R+C has Q = 1/(ω₀RC). R re-appears later only as an impedance scale and as the Norton-transform target.

The higher-Q end drives the synthesis, because it is the binding constraint — the one Fano's bound limits. The far end falls out of the synthesis and is then reconciled against its real termination.

Order parity is not free. An end absorbing a series reactance needs a series arm there; an end absorbing a shunt reactance needs a shunt arm; and arms alternate. So one-series-one-shunt forces an even order and like-kind forces an odd one. The order picker offers only the parities that can work — {2, 4, 6} or {3, 5}, or all of 2…6 when an end is purely resistive — and changing a topology adjusts the order rather than presenting an impossible one, saying so when it does.

The Match Designer

Double-click a placed Match to open the Designer. (The properties panel for a selected Match shows a compact summary and an Open Match Designer… button; the design blob is never rendered as a text row.)

That is a freshly placed component: 50 Ω to 10 Ω over 1.8–2.2 GHz, order 4 — a real 5:1 transformation, arriving with a solution already applied so there is something to look at rather than an identity.

Four regions, left to right and top to bottom:

• Specification — the two terminations, the band, the order, the response shape and the options.
• Network — the synthesised ladder as a real circuitRF schematic, with the value grid in a scroll view underneath it. The schematic draws each element with its instance name and value, out-of-range values in red, a grounded termination at each end, and a brace under the elements each transform produced. Right-click either view for Copy, which puts the drawing on the clipboard as something a schematic page can paste and PowerPoint can paste as vector art; the grid's menu also offers Copy as CSV.
• Response — two rectangular plots, drawn by the Data Display's own plot control from an S-parameter run of the full design (ladder plus both terminations, port references R1 and R2). |S11| and |S21| on the first — S21 against the right-hand axis, because the two live decades apart and one shared scale renders the insertion loss as a flat line on the ceiling; phase and group delay on the second. Markers, their info boxes and the plot's own Copy work exactly as they do in a Data Display. The plot band defaults to the design band ±10%.
• Transforms — the linked Norton slider rack, and below everything the status strip: Q1, Q2, worst in-band return loss, insertion loss, ripple, and the achieved-versus-required Π N².

Each of the four can be put away, and the gaps between them can be dragged. The toolbar's first four buttons are one lamp per panel, in the order the window lays them out — Specification, Network, Transforms, Response. A lit button means the panel is on screen; turning one off gives its space to the panels that are left, so any one of them can have most of the window for a moment. At least one is always showing, so the last lit button is greyed rather than silently doing nothing.

Between each pair of panels is an invisible gripper: put the pointer on the gap, and the cursor changes and the boundary drags. A width you set survives hiding the panel and showing it again, and the centre column stays the one that grows when the window is resized.

Refusals appear in the status strip, with numbers. "No real root at this order", "the far end is not absorbable", "the transforms cannot reach the target" — each names the quantity that failed, and the affected termination turns red.

The specification pane

Each termination is a resistance in series or in parallel with one reactive element — C, L, or none. The little pictogram beside it shows which, at a glance: it is the library part that termination is, drawn by the same glyph a schematic would use — an R, an SRL, an SRC, a PRL or a PRC. Every numeric field is an ordinary circuitRF value-and-unit pair, so unit entry works exactly as it does elsewhere.

Probe: reading the terminations off your own circuit

Each termination carries a Probe button that looks outward from that pin into the circuit the Match is placed in, and fills in R, topology, reactance kind and value for you.

What it does, so you know what you are getting:

1. It extracts the enclosing test bench, deletes the Match instance so the probe cannot measure itself, and attaches a 50 Ω Term to the net that pin was on.
2. It keeps every DC source and bias network — the interesting case is a transistor, and a transistor's small-signal impedance is only meaningful at its operating point. If the DC solve fails, the probe refuses and reports the DC failure rather than returning an impedance computed at zero bias.
3. It runs an S-parameter sweep over the design band and converts to Z.
4. It fits all four two-element models — series R+C, series R+L, parallel R‖C, parallel R‖L — each as a linear least-squares fit in its natural domain, then scores each by mean |ΔΓ| over the band. That single bounded metric ranks all four on equal terms.

All four fits are shown with their residuals in Γ units, so you can take the second-best when you know better. The best physical fit is applied. If even the best residual is poor (mean |ΔΓ| above 0.05 by default) the result is still applied but flagged: the external network is not well described by a two-element model over this band — which is the honest answer for a network with an in-band resonance, and points you at narrowing the band.

A probed termination is a snapshot, not a live link. It records where it came from and shows a badge; editing the value by hand clears the badge to manual, and your override always wins. Changing the surrounding circuit does not silently re-synthesise the network — re-probing is always an explicit action.

The button is greyed out, with a reason, when the pin is unconnected, when its net has nothing else on it, when the schematic has unresolved errors, or when the Match is inside a cell rather than in a test bench — there is no external network to look at from inside a definition.

Shunt DC Block: why the capacitor lands in a shunt arm

A shunt inductor at a biased node is a short across the supply. If the end of your ladder is a shunt inductor and the termination behind it carries DC — a drain, a gate — the network as synthesised puts your supply on ground. The Shunt DC Block toggle on that termination's card fixes it, and fixes the thing people normally get wrong afterwards.

Click it and two things happen:

• A capacitor goes in series with that inductor. It appears in the network pane as an element of its own, named after the inductor it blocks — L1 blocks with L1blk — and it is written into the flattened cell as a real component.
• The inductor is enlarged so the branch is unchanged at band centre. The branch's reactance is now ωL′ − 1/ωC instead of ωL, so L′ = L + 1/(ω₀²C) makes it exactly what the synthesis asked for at ω₀. That is the number the ladder shows: L1 reads its compensated value, and the status line says what it was before.

The compensation is exact at band centre and second-order away from it. The branch's own series resonance sits below the band at f_s = 1/(2π√(L′C)), and between the two band edges the effective inductance runs a little either side of what the synthesis wanted. The status line states all of it:

DC block at termination 1: 1 nF in series with L1 (105.9 pH, from 99.5); branch resonates at 489.1 MHz; inductance ±1.3 % across the band. Feed the bias through L1, not through a separate choke.

Bigger is better, and the default is chosen for you. The seed puts f_s at about one tenth of band centre, which keeps the spread under a percent — and it is capped, because at a low band with a small end inductor that rule alone asks for tens of nanofarads: fine on a board, impossible on an MMIC. The cap is in Settings ▸ DC block default. Past that, the value is yours: type anything positive into L1blk and it is compensated exactly at ω₀ whatever it is. Above f_s > f₀/5 the line turns to a warning and tells you what a ten-times-larger part would buy — measured on a 20 % band, a 500 pF block costs about 3 dB of worst-case return loss where 10 nF costs none. Typing 0 removes the block.

The block goes on the first shunt inductor your bias current would reach. A series inductor in the way doesn't stop it, a real series capacitor does, and your FET's own input capacitance is not a real capacitor on the board. So a termination whose end arm is a series arm — a gate modelled as R in series with C_gs — still gets a block: the arm's capacitor is the device's own and is left out of the flattened cell, the arm's inductor passes DC, and the ladder's next shunt inductor is what would short the gate bias. The toggle names that inductor and the one the bias reaches it through, and the status line's feed rule says the same: feed the bias through L3; it reaches the termination through L4. A Norton T on the end pair, which turns the end arm into a series inductor, moves the block one product in the same way — and a Norton π of inductors, whose series product passes DC between its two shunt products, gets a block on each of them, with your one value and each inductor compensated on its own.

Where the toggle is greyed out, it says why. When Match has put a real capacitor of its own in that end's through path — a CFano or CDetune from a termination whose Q is far below what the synthesis needs — that capacitor already isolates the termination, and a block on a shunt inductor beyond it would protect nothing; feed that termination's bias on its own side of the named capacitor instead. A lowpass ladder has no shunt inductor at all — it passes DC end to end, and blocking that would need a series capacitor in the through path, which is a different network and a different compensation; Match does not offer it. Shunt inductors beyond the next real series capacitor never need one: that capacitor ends the DC path.

A block is a specification input, not something the synthesis chose. It is applied after the transforms, attached by node rather than by name, so a Norton transform that replaces L1 with a product still leaves the block on whatever shunt inductor is first on that end's DC path — and switching the end pair between π and T keeps your value and moves the block to the new host. Nothing in the solutions list, the transform ranges or the feasibility numbers changes when you set one.

Norton transforms: moving values without moving the response

This is the part of the window that makes a synthesised network buildable.

A Norton transform replaces an L-section of two like-kind elements with a π or T of three like-kind elements plus an ideal transformer of ratio N, then absorbs the transformer by scaling the rest of the network (impedances by N², so L·N² and C/N²).

The rack has one row per applied transform: a π/T selector, a numeric box, a slider, and a lock. Which two elements a transform acts on is read off the schematic above it, where a brace spans exactly the elements that transform produced. + add lists the pairs currently available by element name; − remove removes the last.

• The slider ends where the maths does. Past a positivity threshold the transform produces negative elements, so the range is bounded at N = 1 + Z₁/Z₂ when stepping up and N = Z₂/(Z₁+Z₂) when stepping down. Turning on Allow negative components widens it; the values then go negative, which is occasionally what a designer wants and usually not.
• Which pairs are transformable is structural: two like-kind elements at most three positions apart, of opposite orientation, with matching intervening elements — and never an end element carrying an absorbed termination reactance, because moving it would break the absorption.
• Link (on by default) keeps the product on target. The transforms must satisfy Π N² = R_far,target / R_{far,synthesised}. Drag one slider and the unlocked others re-solve so the product stays on target, each clamped to its own range. Locked rows are never touched. With a single transform and link on, N is fully determined and its slider is disabled. When the product cannot reach the target within the ranges, the offending termination goes red and the design is flagged as matched at one end only.

The solutions list

Solutions ▸ slides out a docked list: every valid transform set for this specification, so you can click through candidates and watch the ladder and the response change live. Each row shows a badge (current / previously applied / never applied), the transform count, the element pairs each transform acts on, the Q-adjust value when non-zero, and the response type.

Ordering is by transform count, then by position, then by Q-adjust — the simplest realizable solution first. Within one form, order and family the response is the same for every row; what differs is the element values you would have to build.

Bandpass, lowpass and highpass

The list covers three network forms, and the filter's first group turns each on and off. A bandpass network is two-element arms resonant at band centre. A lowpass network is series inductors in the through path and shunt capacitors to ground; a highpass network is the other way round. All three are matched between f1 and f2, and all three use 2n elements at order n (2n + 1 when both terminations are the same topology) — the lowpass form is not the cheaper one.

What the lowpass and highpass forms buy is tame element values at wide bandwidth — there are no resonators, so the L's and C's stay within a factor of a few of each other instead of spreading over decades — plus a DC path (lowpass) or a DC block (highpass), which is what a bias network wants. What they cost is return loss that depends on the impedance ratio: the ladder is transparent at DC, so |Γ(0)| = (r−1)/(r+1) is fixed by the two resistances and comes out of the same budget as the in-band match. At a 2:1 band, four elements, that is −22.2 dB into a 2:1 ratio and −10.5 dB into a 10:1 one, against a bandpass order-2 network's −16.4 dB at any ratio. Which is better depends on your numbers, so the panel lists all three and you read the return loss off the cards.

Two consequences worth knowing before you go looking for them:

• A lowpass or highpass network has no Norton transforms. Every like-kind element in it sits in the same orientation, so there is no pair to transform; the ratio is already on target and the transform rack says so instead of showing an empty list.
• Each form absorbs half of the termination kinds, and the impedance ratio decides which end takes which. A lowpass ladder puts its series inductor against the lower-impedance port and its shunt capacitor against the higher-impedance one. So a shunt capacitance on the high side of a step-down absorbs; the same capacitance on the low side of a step-up does not, and the refusal says so and points you at bandpass form.
• Two terminations of the same topology are fine here too, at one element more. The ladder then has an odd element count, both of its ends face the same way, and both are absorbed. Its two ends flip together, so a parallel pair has to be analysed from the higher resistance and a series pair from the lower; if you have it the other way round the refusal says which end to analyse from. Odd counts are Chebyshev only.

The math behind the three forms

All three forms come out of one two-parameter prototype family, evaluated in the squared frequency variable u = Ω² with the band mapped onto [-1, 1]. With a = F1/F2:

x(u) = (2u − 1 − a²) / (1 − a²)             the band [a², 1] mapped onto [−1, 1]

Φ(u) = T_n(x(u))²        Chebyshev          T_n = the nth Chebyshev polynomial
Φ(u) = x(u)^(2n)         Butterworth

|Γ(u)|² = (K + ε²·Φ(u)) / (1 + ε²·Φ(u))

Φ has maximum 1 in band, so the worst in-band reflection is |Γ|²_worst = (K + ε²)/(1 + ε²) — which is why the two free parameters are exactly K and ε², and why the panel can quote a return loss before it has drawn an element.

K is not free: it is pinned at DC, and that is the whole cost of the lowpass and highpass forms. A ladder of single elements is transparent at u = 0 — a lowpass ladder's inductors are shorts and its capacitors are opens — so the network's reflection there is whatever the two resistances make it:

Γ(0) = (r − 1)/(r + 1)        r = R_far / R_analysis        K = Γ(0)²

That number comes out of the same budget as the in-band match, which is the whole reason a large impedance ratio costs return loss in these forms and costs nothing in bandpass form — the figures quoted above are this expression. (K = 0 exactly is a trap rather than the ideal case — the numerator's roots then sit in double pairs on the jω axis and there is no well-defined spectral factor — so circuitRF floors it at 1e-12, a −120 dB ceiling that is past anything a matching network means. The equal- resistance case r = 1 drops the pin entirely and makes ε the free parameter instead.)

With K and ε² chosen, the reflection's numerator and denominator are both of the form c + ε²Φ, so their roots are written down rather than searched for — one arccosine (Chebyshev) or one root of unity (Butterworth), mapped x → u → s — and the ladder falls out of a continued-fraction (Cauer) expansion. Order n gives 2n elements — T_n(x) is degree n in x and x is degree 1 in u, so Φ is degree 2n in u — or 2n + 1 when both terminations are the same topology, where the family gains a third parameter: an extra pole at u = −u_R, outside the band.

Denormalising is where lowpass and highpass part company, and it is the only place they differ. The prototype g-values become elements at a single reference frequency — the top of the band for a lowpass network, the bottom for a highpass one, because that is the edge each form is matched up to:

lowpass   ω_ref = 2π·F2      shunt C:  g = ω_ref·R·C        series L:  g = ω_ref·L / R
highpass  ω_ref = 2π·F1      shunt L:  g = R / (ω_ref·L)    series C:  g = 1 / (ω_ref·R·C)

Read the highpass row as the lowpass row with ω → −1/ω, which is the classical lowpass-to-highpass transformation: every series inductor becomes a series capacitor and every shunt capacitor a shunt inductor. Both ends normalise at the analysis end's resistance, not each at its own — in this prototype the terminating resistance is a ratio and element values do not rescale at the far port.

Two consequences fall straight out of those four expressions:

• Which reactance each end can absorb is fixed by the form and by the ratio. A lowpass ladder holds a series L or a shunt C and nothing else; a highpass ladder a series C or a shunt L. The low-impedance port takes the series arm and the high-impedance port the shunt arm, so a shunt capacitance absorbs on the high side of a step-down and is refused on the low side of a step-up.
• The values stay tame. There are no resonators, so nothing is set by a fractional bandwidth: at a wide band a bandpass arm's two elements are pushed apart by roughly 1/w while these stay within a factor of a few of each other.

A highpass design additionally requires F1 > 0 — it is matched between F1 and F2 and pinned at infinity, so a zero lower edge is the lowpass form's degenerate case and is refused by name.

Multiband: two or three bands at once, and the gaps left alone

Set Bands to Dual and a second pair of edges appears. The network is then matched over f1–f2 and f3–f4 together, and the region between them is deliberately not matched. Tri adds a third pair, f5–f6, and two such gaps.

That is the whole idea rather than a compromise. The Fano bound is a fixed budget spread over all frequency; a single wide match from f1 to f4 spends it across the whole span, gap included, and everything spent between f2 and f3 is wasted if your application does not use those frequencies. A dual-band network spends the budget in the two bands and leaves the gap reflecting. For 20 Ω ‖ 2.5 pF into 50 Ω over 2.4 GHz and 5 GHz Wi-Fi, eight elements reach −31.8 dB in both bands, against −18.8 dB for a single-band eight-element network covering the whole span.

The gap mismatch is the design working, so the status strip states it. Beside the worst in-band return loss you get a line like gap 2.5–5.15 GHz: max |S11| 0.445 (−7.0 dB), and that number rises with order — a higher-order network is bigger in the gap, and that is exactly where the extra in-band return loss comes from.

Three things to know before you type four frequencies:

• Type the edges in any order. f1…f6 are one ordered list of passband boundaries, so if you put a number in the wrong field the Designer sorts them back into increasing order and renumbers the fields in front of you. What it cannot fix is two edges that are equal — a zero-width band — and that stays a refusal.
• The bands must be geometric mirrors of each other — f1·f4 = f2·f3, which is the same as equal ratio bandwidths f2/f1 = f4/f3. Yours will not be. circuitRF keeps the wider band exactly, widens the narrower one away from the gap until the ratios match, and says so in one line under the fields: "Band 1 widened to 2.201–2.5 GHz to mirror band 2 about 3.588 GHz." The design is always to the effective bands, never silently to the ones you typed. Where the two ratio bandwidths differ a lot the stretch is severe and this is the wrong tool — a 0.9–1.0 GHz band paired with 3–6 GHz would be designed as 0.5–1.0 GHz, which is not what you asked for.
• Order is match points per band, so orders 1, 2 and 3 give 4, 8 and 12 elements for a mixed termination pair — the same twelve a single-band order-6 network gives, which is the fair comparison. Two bands stop there. Three bands go on to orders 4, 5 and 6 — 16, 20 and 24 elements — for the reason the Feasibility section below explains: a tri-band spec with a narrow middle band is not three bands at all until order 4. Higher is not automatically better; the solutions list spans every order and ranks by return loss, so pick by the number on the card rather than by the order.
• Two terminations of the same topology get one arm more, not a refusal. A shunt-C-to-shunt-C interstage needs a ladder whose two ends face the same way, which is an odd number of arms, so those orders give 6, 10 and 14 elements instead. You do not choose this — your terminations do, and the order picker's tooltip says which count you are getting. The extra arm buys the absorption, not return loss: at the same order the odd ladder measures within 0.01 dB of the even one.

Everything else is unchanged. The terminations are read at ω₀ — which is now the gap centre, where every arm of the ladder is transparent — the ladder is an ordinary bandpass ladder with twice as many arms, so Norton transforms, absorption, the excess-element rule, the solutions list and Flatten all work exactly as they do for a single band.

Three bands

Tri works the same way with one difference in the rule above: the middle band is the one that is kept, because a three-band response has to be symmetric about its own centre and only the middle band can straddle it. Bands 1 and 3 are widened onto each other's mirror image about ω₀ = √(f3·f4), and the note names every band that moved. Switching from Dual to Tri therefore moves your existing second band out to f5–f6 and seeds a new middle band between them, rather than hanging a third band off the end where it would immediately be mirrored on top of the second.

The status strip shows two gap lines, one per gap. On a spec that already mirrors they come out equal; they separate as soon as the middle band sits off centre.

Three bands are Chebyshev only. Butterworth means maximally flat at the middle of one passband, and three bands do not have a single middle to be flat at — the equal-ripple answer is the only one there is, so that is what is offered. For 50 Ω ‖ 4 pF over 0.5–0.6, 0.9–1.1 and 1.65–1.98 GHz, eight elements reach −12.0 dB in all three bands, twelve reach −14.5 dB and sixteen reach −18.9 dB.

Multiband is bandpass only, and the solutions filter says so in place of its form group: the lowpass and highpass forms of the previous section have no multiband version yet. Nor do asymmetric bands — bands whose ratio bandwidths are genuinely different, matched as requested rather than widened. Both are recorded as future work.

The math behind two and three bands

There is no separate multiband synthesis. A multiband network is the same single-element prototype family the previous section describes, pushed through the same bandpass transformation an ordinary Match uses — so what comes out is an ordinary alternating bandpass ladder of 2n two-element arms, and Norton transforms, absorption, the excess-element rule, Flatten and the stamp all handle it with no multiband case anywhere.

Step 1 — the bands are made mirror images. A real network's |Γ(jΩ)|² is an even function of Ω, so the passbands a resonated ladder produces are mirror images about ω₀ in log frequency. That is one equation, and it is what your four (or six) numbers have to satisfy:

two bands    f1·f4 = f2·f3                    ⇔  f2/f1 = f4/f3   (equal ratio bandwidths)
three bands  f1·f6 = f2·f5 = f3·f4            the MIDDLE band is the one kept

Yours will not satisfy it. circuitRF keeps the wider band exactly, widens the narrower one away from the gap until the ratios match, and states the result in one line under the fields. Everything from here uses the effective bands.

Step 2 — the bandpass transformation. With ω₀ the geometric centre and w the outer fractional bandwidth:

two bands    ω₀ = 2π·√(f1·f4)      three bands  ω₀ = 2π·√(f3·f4)   (the middle band's centre)
w = (f_high − f_low) / √(f_low·f_high)          the outer pair, gap included

Ω(f) = ( f/f0 − f0/f ) / w                      the standard bandpass map

Each single prototype element becomes a two-element resonant arm at ω₀, exactly as it does for one band. The gap is where Ω is small: the mapping folds the region between the passbands onto the middle of the prototype axis, so a prototype that has a stopband there is a network that has an unmatched gap there. That is the whole trick.

Step 3 — where the passbands land in the prototype variable. In u = Ω², and with the mirror relations above collapsing every square root into a ratio of frequency differences over the outer span:

one band     u ∈ [0, 1]

two bands    u ∈ [a², 1]                    a = (f3 − f2)/(f4 − f1)
             Φ(u) = T_n(x(u))²              x(u) = (2u − 1 − a²)/(1 − a²)

three bands  u ∈ [0, a²] ∪ [b², 1]          a = (f4 − f3)/(f6 − f1)
             Φ(u) = p(u)²                   b = (f5 − f2)/(f6 − f1)

Two bands is therefore literally the lowpass-form family of the previous section, on the interval [a², 1] instead of [0, 1] — written down by arccosine, no root-finding. Three bands is the only place a new object appears: the passband is a union of two intervals, and p is the equal-ripple polynomial on that union, produced by a Remez exchange rather than by a formula. max Φ = 1 in band either way, so the worst in-band reflection is still (K + ε²)/(1 + ε²) and the parameter search does not know which case it is looking at.

This is why Butterworth exists for two bands and does not exist for three. The Butterworth member is x(u)^(2n) — maximally flat at the centre of one interval — and a union of intervals has no single centre to be flat at. The equal-ripple answer is the only member of the family that lives on a union, so tri-band is Chebyshev only and the others are refused by name rather than silently substituted.

The two free parameters are chosen exactly as §Feasibility describes. The near end's absorbed element is pinned by the termination — g₁ = Q_analysis · w, with Q read at ω₀, which for a multiband spec is the gap centre where every arm is transparent — and the remaining freedom minimises the worst in-band |Γ|². K is scanned rather than solved, because the near element is not monotone in it (it rises and then falls); ε² is then bracketed and bisected at each K, where it is monotone. The optimum is flat — the worst return loss moves by at most 0.1 dB across a whole decade of K — so a 64-point log scan plus a bounded refinement is ample and there is no root-finding subtlety to tune.

Element counts, and where the +2 comes from. 2n arms of two elements is 4n, and one arm more — 4n + 2 — when both terminations are the same topology. Orders count match points per band, which is why dual-band stops at 3: order 3 is twelve elements, the same twelve a single-band order-6 network gives, and that is the fair comparison. Tri-band goes on to 6 because a narrow middle band is not three bands at all until order 4 — before that the middle passband has no ripple of its own to speak of. A design may then carry two more elements than the arithmetic above: an excess element where the synthesised end value exceeds what the termination supplies, and one extra arm per Norton transform that was applied.

Feasibility: what is possible before you synthesise anything

A lossless network cannot match a reactive termination arbitrarily well over an arbitrary bandwidth. There is a hard ceiling, it depends only on the terminations and the bands, and the Designer shows it beside every result and before any of them — the status strip's line reads

Fano ceiling 6.4 dB (termination 2, over the bands)

quoted positive like the return loss above it, and naming which of the two ends is the one setting it. Hover it for both ends, the ceiling over the bands as you typed them, the ceiling over the whole span from your lowest edge to your highest, and — for a multiband spec — how much of the ceiling the mirror widening cost. The line is there even when the synthesis refuses, which is when it usually matters most: a refusal and a ceiling of −3 dB are the same fact, and only one of them tells you what to change.

When the line ends "— at the ceiling", the design is within a dB of what physics allows and no amount of searching will improve it. Anything else is headroom.

Which end limits you, and what makes it worse

Two of the four terminations are limited by total bandwidth, and two by your lowest band edge:

That difference matters. With a series capacitance, moving your lowest band edge up by a few hundred megahertz can be worth more than everything else put together; with a shunt capacitance it is the total width that counts and where the bands sit hardly matters at all.

The hints

When the ceiling is genuinely what is stopping you — the search came back empty, or what it found is already against the wall — a line appears under the solutions list saying so and offering up to four one-variable changes that would reach −15 dB:

The best any lossless network can do here is -6.4 dB, set by termination 2 (1.25 Ω + 5 pF series) over 2.25–3 / 4.5–5 / 7.5–10 GHz. To reach -15 dB: termination 2's capacitance at or above 11.7 pF; or band 1 starting at 2.86 GHz instead of 2.25; or without band 1 the ceiling over bands 2 and 3 is -32.1 dB; or band 1 as 2.25–2.5 GHz mirrors band 3 without widening (ceiling -13.8 dB).

Each clause holds everything else fixed and solves for the one thing it names, so they are alternatives rather than a recipe. They are ceilings, not designs: reaching one still needs an order and a family that fit, and the ladder that gets there may be a longer one than you wanted. It is a hint and never a refusal — solutions that exist are still listed beside it.

Does your order actually use the gaps?

A multiband network buys its in-band return loss by leaving the gaps alone. At low order and with a narrow middle band, the prototype may not exclude them at all — the equal-ripple polynomial on your bands turns out to be the same polynomial as a single wide match over the whole span, and the result is one broad mediocre match instead of two or three good ones. This is not a fault in the synthesis; at that order no polynomial does better.

The Frequency Band card says so when it happens:

At order 2 the tri-band prototype does not exclude the gaps — this is a single-band match over 2.25–10 GHz (ceiling -3.1 dB). The gaps open at order 4 (rise ×2.9).

and each gap line in the status strip carries the same measurement as a prototype rise factor: ×1 means the gap is not being excluded, and the larger it grows the more of the budget is being reclaimed from the gap and spent in your bands. If the note says no offered order opens them, the band geometry itself is the problem — widen the middle band, or move the outer bands closer together.

Worked example: a two-stage FET interstage match

The interstage problem between two amplifier stages: stage 1's output, 200 Ω ‖ 0.125 pF, into stage 2's input, 1.25 Ω + 10 pF, over 3.3–5.0 GHz. A 160:1 impedance transformation with a reactance at both ends — exactly the case where tuning-out gets you a few percent of bandwidth and absorption gets you forty.

Reading the figure, in the order you would work:

1. The two terminations, as specified. Parallel 200 Ω ‖ 125 fF on the left; series 1.25 Ω + 10 pF on the right. Band 3.3–5.0 GHz, order 4, Chebyshev/Fano.
2. The Q values the status strip reports: Q1 = 0.638, Q2 = 3.134. The series end is the higher-Q end, so it is the analysis end and the synthesis prescribes its absorbed element there.
3. The ladder, from the analysis end. C4 = 10 pF is the load's own 10 pF, exactly — that is the absorption, visible as an element you did not have to add. Then L4 ≈ 154 pH, C3 ≈ 82.9 pF, L3 ≈ 18.5 pH and onward.
4. The far end does not land on 200 Ω by itself. Synthesised bare, it comes out at 1.68 Ω, so the required Π N² is 119.03 — and the status strip says so. That is what the transforms are for.
5. Two Norton transforms applied — the first-ranked solution from the list — bring the product to 119.027 / 119.027 and the strip reads ✔ matched.
6. The achieved response: worst in-band return loss 16.66 dB, insertion loss 0.095 dB, ripple 0.036 dB, over a 42% fractional bandwidth into a 160:1 transformation with a capacitive load at both ends. That is the number to compare against whatever you were doing before.

The element values shown are what the synthesis computes, and they are the same values the component stamps and the same values Flatten to Cell writes out.

Worked example: the same stages, matched over two bands

The same two stages as the previous section: 200 Ω ‖ 0.125 pF into 1.25 Ω + 10 pF. This time the radio only ever operates in two narrow bands with 200 MHz of nothing between them, so there is no reason to spend the Fano budget on the gap. Everything below is reproducible — place a Match, open the Designer, and type these numbers.

What to type