Impedance matching calculator: L network and pi network

What sets an L network: the resistance ratio fixes Q

A source delivers the most power into a load that is its complex conjugate. Silicon Labs' application note AN1275, Impedance Matching Network Architectures, states it as the starting point: "the maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance" (p. 6). A matching network is the lossless circuit that makes the load look like that conjugate. Built from inductors and capacitors, it can only do so at one frequency, and the note says so on the same page: "since the reactance in the circuit is frequency-dependent, the perfect impedance match between the source and the load will also occur at a particular frequency."

The smallest such network has two parts, one in series and one to ground, in the shape that gives it the name L network. AN1275 assigns each part a job: "The function of the parallel component of the L matching network is to transform a larger impedance down to smaller value equating the real part of the impedances between source and the load. Similarly, the function of the series component is to cancel out any reactive components present in the circuit by resonating with equal and opposite reactance" (p. 11). So the shunt part always sits on the side with the larger resistance: "If the load impedance is higher than the source, then the parallel component will lie on the right side of the series component". AN1275's Figure 4.1 calls the two orientations Type 1 (shunt beside the source) and Type 2 (shunt beside the load), and the calculator uses the same names.

Both parts work at one Q. §4.2 (p. 12) gives Q_S = Q_P = √(R_P/R_S − 1), where R_P is the larger resistance and R_S the smaller, and then X_S = Q·R_S and X_P = R_P/Q. Two resistances in a ratio of 2:1 give Q = 1.00; 4:1 gives 1.73; 10:1 gives 3.00. Nothing else enters, and that is the L network's limitation in AN1275's words: "the designer does not have control over the value of the Q of the network because the value of Q is a function of the value of the resistance of the series and the parallel matching components" (pp. 14–15).

Each orientation has two solutions, one of each polarity. A series inductor with a shunt capacitor is the low-pass version; a series capacitor with a shunt inductor is the high-pass one, where, in AN1275's words, "the series capacitor will block dc into the load and the inductor will act as a short at low frequencies" (p. 11). The calculator shows the one you prefer and lists every other network that matches.

Q and bandwidth against the resistance ratio: a table

The table runs §4.2 for a 50 Ω source into resistive loads from 5 Ω to 200 Ω at 2445 MHz, AN1275's frequency, with the low-pass part values and AN1275's bandwidth F/Q. Below 50 Ω the shunt capacitor sits at the source (Type 1); above it, at the load (Type 2).

Two things follow. A small ratio gives a Q well under 1, and the F/Q figure becomes larger than the frequency itself, which says only that the match is broad, not where it ends; the calculator's swept bands answer that. And a large ratio drives the Q, and with it the sensitivity to part values, up quickly: a 10:1 step costs a Q of 3 in a single L section.

Worked example: AN1275's EFR32 output into 50 Ω

AN1275's first problem statement (p. 23): "The optimum termination impedance for EFR32 Series 1 RFIC for 2.4 GHz is ZLoad_Opt = ~23 + j11.5 Ω. The center frequency of operation is 2445 MHz." The network must present that impedance to the chip, which is the same as matching to a source of its conjugate: in the note's words, "the matching network’s input impedance should be complex conjugate of the RFIC’s output impedance", so it takes Z_Source = 23 − j11.5 Ω. That is what the calculator's defaults enter. The load is the 50 Ω connector. 50 Ω is the larger resistance, so the shunt part goes at the load: a Type 2 network, which is the only kind the calculator finds.

Q        √(50/23 − 1)                     = 1.0835       AN1275: 1.08
XS       Q × 23                           = 24.92 Ω    AN1275: 24.84 Ω
XP       50 / Q                           = 46.15 Ω    AN1275: 46.29 Ω
low-pass
  C      1/(2π × 2445 MHz × 46.15)       = 1.411 pF   AN1275: 1.40 pF
  L'     24.92 / (2π × 2445 MHz)            = 1.622 nH   AN1275: 1.61 nH
  Lres   11.5 / (2π × 2445 MHz)             = 0.749 nH   AN1275: 0.74 nH
  L      L' + Lres                        = 2.371 nH   AN1275: 2.35 nH
high-pass
  L      46.15 / (2π × 2445 MHz)            = 3.004 nH   AN1275: 3.01 nH
  XC     24.92 − 11.5                     = 13.42 Ω    AN1275: 13.34 Ω
  C      1/(2π × 2445 MHz × 13.42)       = 4.851 pF   AN1275: 4.88 pF

The note works by hand with π = 3.14 and keeps two decimals at every step, after rounding Q to 1.08. Carried through exactly, the low-pass network is 2.371 nH and 1.411 pF against the note's 2.35 nH and 1.40 pF. The difference is small next to any part tolerance: the note's own values return 41.7 dB at 2445 MHz, and the high-pass pair 53.4 dB.

The two configurations treat the chip's −j11.5 Ω differently, and the note names both methods on page 14. "Resonance—This approach resonates with any stray reactance with an equal and opposite reactance at the operating frequency": in the low-pass network an extra 0.749 nH in series cancels the source's capacitance, and the two series inductors merge into one. "Absorption—This approach absorbs any stray reactance into the impedance matching network by smartly placing each impedance matching component with the stray reactance": in the high-pass network the source's capacitance already supplies 11.5 Ω of the 24.92 Ω the series capacitor needs, so the capacitor supplies only the remaining 13.42 Ω. The calculator's closed-form solution includes both: it puts the source's series reactance into the series part, whichever sign it has.

Rounded to E24, the low-pass parts become 2.4 nH and 1.5 pF, and the match at 2445 MHz falls to 28.7 dB of return loss (VSWR 1.08). The high-pass pair rounds to 4.7 pF and 3.0 nH, 40.5 dB. Both are well inside AN1275's target on page 4: "an S11 value of –10 dB to –15 dB is recommended", because beyond that the mismatch loss barely improves.

Complex loads, and the networks that cannot exist

With reactance at either end, the comparison that decides the orientation is between one end's series resistance and the other end's parallel-equivalent resistance, the step AN1275 performs on page 31: R_P = R_S(Q² + 1) with Q = X_S/R_S. A Type 2 network exists when the source resistance is at most the load's parallel resistance; a Type 1 when the load resistance is at most the source's. When both conditions hold, up to four networks exist, and this is the only region where a network of two parts of the same kind can appear.

AN1275's Figures 4.2 and 4.3 map where that happens on the Smith chart. The same-type networks, two inductors or two capacitors, "are only valid when the load has low resistance and low conductance at the same time", and an LL pair can match only capacitive loads of that kind, a CC pair only inductive ones (p. 14). The mixed networks cover more ground: "the configuration where the two quantities are of the opposite type is valid for any kind of load". The calculator lists a same-type network when it is one of the solutions, labelled as such. If neither condition holds, no L network exists; that cannot happen with both ends having positive resistance, because the larger of the two always qualifies.

The pi network: a virtual resistance and a Q you choose

When the L network's Q gives too broad a match, AN1275 adds a third part. §5.1 builds the pi network from two L sections back to back, joined at a virtual resistance R: "Note that the virtual resistance is not a physical component present in the Pi network rather it is a virtual component only used and calculated to determine the values of the components of both individual L networks" (p. 17). Choosing Q fixes R = R_H/(Q² + 1), with R_H the higher terminating resistance; each section then transforms its own end down to R at its own Q, and "The two series components of each L network can be now combined by adding their reactance."

R must sit below both ends, which sets a floor on Q. AN1275 p. 16: "The circuit Q established while designing an L network will be the minimum circuit Q that can be chosen for a three-element network." For 50 Ω into 21 Ω that floor is 1.175; ask for Q = 1 and the calculator refuses, because the virtual resistance would have to be above 21 Ω and a shunt element can only bring a resistance down. At exactly the floor one section disappears and the pi becomes the L network.

With complex ends, each end's reactance is first converted to its parallel equivalent, and the shunt part beside it is reduced or increased by that much; AN1275 does this at the load end of its second example. Each section can independently be low-pass or high-pass, so there are four pi networks for every Q: C–L–C, L–C–L and two mixed. The calculator shows the one matching the response you choose and lists the opposite one. AN1275 also states when the pi suits better than its mirror, the T network, which this calculator does not design: "Use a T network architecture between low-value impedances (< 50 Ω)" and "Use a Pi network architecture between high-value impedances (> 50 Ω)" (p. 18).

Worked example: AN1275's chip antenna, pi network at Q = 4.89

AN1275's second problem statement (p. 28): "A chip antenna was mounted on a 4-layer PCB of 62 mil thickness and the impedance value of ~21 + j1.15 Ω was measured right at the feed point of the antenna. Design an impedance matching network to match the antenna with 50 Ω source impedance operating at 2445 MHz with 500 MHz bandwidth and passes dc current." 500 MHz at 2445 MHz asks for Q = 4.89. The L network between 50 Ω and 21 Ω has Q = 1.175, F/Q = 2.08 GHz, and the note concludes: "Thus, a two-element matching network does not satisfy the requirements of the problem statement." A low-pass pi network passes DC through its series inductor.

Q        2445 MHz / 500 MHz                 = 4.89
R        50 / (4.89² + 1)                  = 2.0071 Ω   AN1275: 2.007 Ω
source   XP1 = 50 / 4.89                   = 10.225 Ω   AN1275: 10.22 Ω
         XS1 = 4.89 × 2.0071             = 9.815 Ω    AN1275: 9.78 Ω
load     RP = 21 × (0.0548² + 1)            = 21.063 Ω   AN1275: 21.06 Ω
         XP = (21² + 1.15²)/1.15            = 384.6 Ω   AN1275: 421.05 Ω
         Q2 = √(21.063/2.0071 − 1)        = 3.0813      AN1275: 3.087
         XP2 = 21.063 / 3.0813         = 6.836 Ω    AN1275: 6.822 Ω
         XS2 = 3.0813 × 2.0071         = 6.184 Ω    AN1275: 6.174 Ω
parts    CP1  1/(2π F × 10.225)            = 6.366 pF   AN1275: 6.37 pF
         L    (9.815 + 6.184)/(2π F)       = 1.041 nH   AN1275: 1.04 nH
         C'P2 1/(2π F × 6.836)             = 9.523 pF   AN1275: 9.54 pF
         CP2  C'P2 − 0.169 pF              = 9.353 pF   AN1275: 9.38 pF

Two of the printed figures do not follow from the note's own inputs. The load's parallel reactance is printed as 421.05 Ω, but the line above it divides 21.06 by 0.054, which is 390.0 Ω, and exactly it is (21² + 1.15²)/1.15 = 384.6 Ω. The load's stray capacitance is therefore 0.169 pF, not the printed 0.154 pF (which is what 421.05 Ω gives at 2445 MHz, 0.155 pF), and the load-side capacitor comes out at 9.35 pF rather than 9.38 pF. The note's own simulation schematic, Figure 7.12 on page 35, uses 9.35 pF.

The second is a sign. The antenna measures 21 + j1.15 Ω, a slightly inductive load. The note says "the output impedance towards the load side for designing the matching network is 21 – j1.15 Ω", which is right for the network's output, and then treats that conjugate as the load itself: "As the reactance has a negative sign, it denotes that the load has a capacitive reactance" (p. 31). Figure 7.6 labels the load "ZLoad = 21 − j1.15 Ω" and the simulation terminates in 21 − 1.15j Ω, so the printed design is an exact match to 21 − j1.15 Ω. For the antenna as measured, the load's parallel susceptance is inductive and the shunt capacitor grows instead: 9.69 pF, with 6.37 pF and 1.04 nH unchanged. The practical effect is small: the note's values into the measured 21 + j1.15 Ω still return 25.4 dB at 2445 MHz. For the L network the same sign turns the note's 1.68 nH series inductor into 1.53 nH (1.68 nH against 21 − j1.15 Ω). The calculator takes a load as its own impedance: enter 1.15 for the measured antenna, −1.15 to reproduce the printed design.

The note's simulation (Figure 7.13, p. 36) is reproducible from its schematic values. It reports that "the S11 value of the L network is below –12 dB for the entire 2 to 3 GHz frequency range"; with 1.53 pF and 1.68 nH into 21 − j1.15 Ω the worst point in that range is −12.9 dB. For the pi network, "the signals below 2.35 GHz and signals above 2.52 GHz have S11 values above –12 dB": with 6.37 pF, 1.04 nH and 9.35 pF the −12 dB crossings are 2.357 GHz and 2.528 GHz, and at 2 GHz S11 is −2.9 dB, where the plotted curve sits near −3 dB.

Bandwidth: what F/Q does and does not tell you

AN1275 gives one bandwidth formula, BW = F/Q, and defines it as "BW = –3 dB Bandwidth" (p. 6). It is an estimate from the network's Q, and the calculator shows it, but it also sweeps the network and reports two bands from the response itself: where the return loss stays at 10 dB or better, and where at least half the source's available power arrives, |Γ|² ≤ 0.5, the −3 dB point of mismatch loss. AN1275's Table 2.1 (p. 4) gives the second its meaning: "if the S11 value is –3 dB, only 50% of the power is delivered to the load".

• The pi network above, Q = 4.89, F/Q = 500 MHz: return loss ≥ 10 dB from 2.33 GHz to 2.55 GHz, 220 MHz wide; half power from 2.01 GHz to 2.72 GHz, 714 MHz wide.
• The L network between the same ends, Q = 1.175: return loss ≥ 10 dB from 1.26 GHz to 3.21 GHz.
• AN1275's first example, Q = 1.083, F/Q = 2.26 GHz: return loss ≥ 10 dB from 1.34 GHz to 3.15 GHz, and the half-power band reaches up to 4.19 GHz and does not end on the low side: at DC the low-pass network connects 23 Ω to 50 Ω directly, which already delivers more than half the power.

F/Q orders networks correctly, and the higher Q gives the narrower match, but the band a given return loss holds over is a property of the network and its ends, which is why the calculator computes it rather than quoting F/Q alone. The sweep holds the source and load at their entered impedances, as AN1275's simulation does; a real antenna's impedance moves with frequency, often faster than the network's.

Going the other way, toward a wider match, AN1275 cascades two L sections through a virtual resistance between the two ends; its broadest case is "when the value of the virtual resistor is the mean of the two impedances being matched" (p. 21), the geometric mean √(R_Source × R_Load). For the first example that is 33.9 Ω, and each section's Q falls to 0.689 from the single L network's 1.083. The calculator does not design that four-part network; the Q is shown for resistive ends.

Where the lumped model stops being valid

Ideal parts. AN1275 p. 27: "the component values calculated above are for the ideal lossless components". A chip inductor has resistance and self-capacitance, a capacitor has series inductance, and at 2.4 GHz a few tenths of a nanohenry or picofarad of pad and trace are a sizeable fraction of the values in the examples above. The reactance calculator shows a real part's impedance with its parasitics; the note points to Silicon Labs' device matching guides for the board-level corrections.

A known load. The match is to the impedance entered, at the frequency entered. For antennas, AN1275 p. 38 warns that "it is hard to determine the exact load impedance without practically measuring it on the custom PCB", and recommends a footprint for all three parts even where two will do: "it is always a good idea to add a three-element Pi shaped antenna matching network placeholder so that the designer can later tune the antenna by adding an appropriate matching component."

High Q is fragile. AN1275 p. 34: "Designing a High-Q (narrowband) network can lead to some problems such as difficulty in tuning the network as the network can become extremely sensitive to component value variations." Rounding the Q = 4.89 pi network to E24 leaves 12.6 dB of return loss at 2445 MHz; to E96, 35.1 dB. The L network of the first example rounded to E24 kept 28.7 dB. On page 6 the note gives the same reason to keep Q down: a high-Q network "can lead to performance issues that are caused by the technical spreading of the matching components".

Harmonics are not guaranteed. A narrow match does not by itself filter. AN1275 p. 34: "designing a high-Q matching network does not always guarantee that lower harmonics will be obtained at the output", because the source impedance at a harmonic differs from the one at the fundamental.

Lumped, not distributed. Each part is treated as a point, and the connections between them as having no length. Where the traces or cable are a meaningful fraction of a wavelength, their own impedance transforms the load before the network sees it; thecoax impedance calculator gives a line's characteristic impedance.

Common impedance matching mistakes

• Entering the chip's optimum load as the source impedance. A datasheet that gives the load a transmitter wants to see, 23 + j11.5 Ω for AN1275's EFR32, means a source of 23 − j11.5 Ω. Entered with the wrong sign, the calculator designs for the mirror image and the chip sees the wrong reactance.
• Entering the conjugate of a load instead of the load. The network presents the conjugate; the load is what was measured. AN1275's own second example does this, at a cost of a few tenths of a picofarad.
• Rounding Q before computing the parts. AN1275's 1.08 for 1.0835 and its truncated intermediate steps move the first example's inductor from 2.37 nH to 2.35 nH.
• Forgetting the stray reactance at either end. A source capacitance left out of the sum is a series reactance error of the same size; the resonance and absorption methods both start by counting it.
• Asking a pi network for a Q below the L network's. The virtual resistance would have to be higher than one of the ends, and no shunt part can do that.
• Reading F/Q as the matched band. It is an estimate; the band where the return loss is good enough comes from the response.
• Buying the nearest standard values without checking. At low Q the match barely notices; at Q = 4.89 the E24 values leave 12.6 dB of return loss. The calculator recomputes the match with the rounded parts so the cost is visible.

Further reading

• Silicon Labs AN1275, Impedance Matching Network Architectures — the L network's design formulas (§4.2, p. 12), the pi and T networks (§5), wideband networks (§6), and the two worked problems with their simulation (§7, pp. 23–38).
• VSWR and return loss calculator — turning the reflection coefficient this page reports into VSWR, return loss and mismatch loss, and back.
• Reactance calculator — X_L and X_C at any frequency, and the parasitics that move a real part away from its ideal reactance.
• LC resonance calculator — the resonance that the resonance method of cancelling stray reactance relies on.
• Coax impedance calculator — the characteristic impedance of the line that usually sits on the 50 Ω side of the network.
• E-series calculator — the preferred values the parts are rounded to.