VSWR and return loss calculator: reflection coefficient, mismatch loss and uncertainty

Convert between VSWR, return loss, S11, the reflection coefficient |Γ|, reflected power and mismatch loss; find all of them from a load impedance R + jX or from forward and reflected power; and work out the mismatch uncertainty between a source and a load whose phases are not known, the way Keysight's power-measurement note does. Every conversion is exact, the figure plots the reflection on the Γ plane beside aligned scales, and the worked examples reproduce the published ones.

Fig 1 — |Γ| = 0.310: every load with a VSWR of 1.900 lies on the dashed circle, and without an impedance its phase is not known. The scales on the right are Huber+Suhner's Figure 26 redrawn from the same conversions, aligned on return loss, with 10.16 dB marked.

Start from

Quantity you have

Voltage standing wave ratio VSWR

Reference impedance Z0 (Ω)

Reflection coefficient |Γ|: 0.3103
VSWR = (1 + |Γ|)/(1 − |Γ|): 1.900 : 1
Return loss RL = −20 log|Γ| · as S11: 10.16 dB · −10.16 dB
Mismatch loss ML = −10 log(1 − |Γ|²): 0.440 dB
Power reflected |Γ|² · delivered 1 − |Γ|²: 9.63 % · 90.37 %
Reflected voltage |Γ|, of the incident: 31.0 %
Huber+Suhner Figure 26 band: poorly matched
Resistive loads with this VSWR on Z0 = 50.0 Ω: Z0·S or Z0/S: 95.0 Ω or 26.3 Ω

How this is calculated

Standard: Keysight AN 1449-3, Fundamentals of RF and Microwave Power Measurements (Part 3), 5988-9215EN, Eqs 2-2 to 2-5, 2-10, 2-25, 3-1 to 3-3; Huber+Suhner RF Connector Guide, Table 4 (Eqs 24–32), Eq 17 and Table 3; Mini-Circuits AN-70-001

$\Gamma_\ell = \frac{Z_\ell - Z_o}{Z_\ell + Z_o}, \qquad \Gamma_\ell = \rho_\ell\, e^{\,j\Phi_\ell}$: Keysight Eq 2-5: "Transmission line theory relates the reflection coefficient, Γℓ of a load to its impedance, Zℓ", with Zo "the characteristic impedance of the system". Keysight writes the magnitude as ρℓ and the phase as Φℓ. Here Z = R + jX and Z0 is real.
$SWR = \frac{|a_\ell| + |b_\ell|}{|a_\ell| - |b_\ell|} = \frac{1 + \rho_\ell}{1 - \rho_\ell}, \qquad \Gamma = \frac{VSWR - 1}{VSWR + 1}$: Keysight Eq 2-10, the ratio of the standing wave's maximum to its minimum; Huber+Suhner Eq 30 for the inverse. Keysight: "Traditionally VSWR and PSWR referred to voltage and power standing wave ratio. Since PSWR has fallen to dis-use, VSWR is shortened to SWR."
$RL = 20\log\frac{1}{\Gamma} = 20\log\left(\frac{VSWR + 1}{VSWR - 1}\right), \qquad \Gamma = \frac{1}{\operatorname{alog}\left(\frac{RL}{20}\right)}, \qquad S_{11} = 20\log|\Gamma| = -RL$: Huber+Suhner Table 4, Eqs 29, 32 and 27 (alog is the antilogarithm, 10^x). Return loss is positive: "Ideal line ⇒ RL = ∞ [dB]", "Short and open circuit ⇒ RL = 0 [dB]", and "The return loss should be as high as possible". S11 in dB is the same number with the sign a network analyser displays.
$RL = 20\log\left|\frac{Z_2 + Z_1}{Z_2 - Z_1}\right|$: Huber+Suhner Eq 17, the return loss of a step from Z1 to Z2: Eq 2-5 with Z1 as the reference. 50 Ω to 75 Ω is 20 log 5 = 13.98 dB.
$|a_\ell|^2 = P_i, \quad |b_\ell|^2 = P_r, \quad P_d = P_i - P_r, \qquad |\Gamma| = \sqrt{\frac{P_r}{P_i}}$: Keysight Eqs 2-2 to 2-4: the waves are normalised so that their squares are the incident and reflected powers, and the load takes the difference. |Γ| from two power readings follows directly.
$ML = -10\log\left(1 - |\Gamma_\ell|^2\right)$: Keysight p.16: "The second term on the right side of Equation 2-26, –10 log (1 – |Γℓ|²), is called mismatch loss. It accounts for the power reflected from the load." Positive dB, and fixed by the one port alone.
$\frac{P_{gZo}}{P_{g\ell}} = \frac{\left|1 - \Gamma_g\Gamma_\ell\right|^2}{1 - |\Gamma_\ell|^2}$: Keysight Eq 2-25: the power a generator would deliver to a Z0 load over what it delivers to this one. The numerator is the re-reflection between the two ports, and it depends on both phases. The calculator's "power into the load against a Z0 load" is this ratio, inverted, at its two extremes.
$M_{u\,max} = 10\log\left(1 + \rho_g\rho_\ell\right)^2, \qquad M_{u\,min} = 10\log\left(1 - \rho_g\rho_\ell\right)^2, \qquad \%M_u = 100\left[\left(1 \pm \rho_g\rho_\ell\right)^2 - 1\right]$: Keysight Eqs 3-1, 3-2 and 3-3, the "mismatch loss uncertainty limits": the maximum and minimum of 10 log|1 − ΓgΓℓ|² when only the magnitudes are known. Keysight's example, a source of SWR 1.9 and an 8481A sensor of SWR 1.18, gives +0.2198 dB and −0.225 dB, printed as "+0.219, –0.225 dB".
$VSWR_{max} = S_1 S_2, \qquad VSWR_{min} = \frac{S_1}{S_2}\;(S_1 \ge S_2)$: Mini-Circuits AN-70-001: "it is generally safe to assume that two VSWRs will tend to multiply rather than add", for the worst case of two mismatches in cascade. The best case, the ratio, is derived here: two reflections ρ1 and ρ2 combine to between |ρ1 − ρ2|/(1 − ρ1ρ2) and (ρ1 + ρ2)/(1 + ρ1ρ2).

Assumptions

A real reference impedance Z0, such as 50 Ω or 75 Ω. A lossy line has a complex characteristic impedance, and the reflection coefficient against it is not given by these formulas.
One frequency. Γ, and every quantity derived from it, belongs to the frequency the impedance or the measurement was taken at; Keysight: all three quantities "are, in general, complex numbers and change with frequency".
Power readings are taken at one point through equal couplings. Loss between that point and the load makes the load look better matched: the return loss seen through a matched loss of A dB is RL + 2A (derived).
Mismatch uncertainty assumes nothing about the phases: the limits are the extremes over every phase, not a probable value.
A passive load: R ≥ 0, |Γ| < 1. A pure reactance or an open or short (|Γ| = 1) has an infinite VSWR and is refused.

What VSWR, the reflection coefficient and return loss describe

When a line of characteristic impedance Z₀ ends in a load of a different impedance, part of the wave travelling toward the load comes back. Keysight's power-measurement note describes the load not by its impedance but by that returning wave: "To characterize a passive load, Ohm's law is replaced by" the ratio of the reflected wave to the incident one, the reflection coefficient Γ. It is a complex number, a magnitude and a phase, and it follows from the load impedance as Γ = (Z − Z₀)/(Z + Z₀), Keysight's Equation 2-5. A load equal to Z₀ gives Γ = 0 and reflects nothing; a short, an open or a pure reactance gives |Γ| = 1 and reflects everything. The same note gives the reason the RF world prefers it: "As frequencies exceed 300 MHz, the concept of impedance loses usefulness and is replaced by the concept of reflection coefficient." An impedance measured down a line changes with every millimetre; the reflection coefficient keeps its magnitude and only turns in phase.

VSWR is the older way of stating the same magnitude. The incident and reflected waves interfere along the line, and in Keysight's words "The ratio of the maximum to the minimum is called the standing-wave ratio (SWR, sometimes referred to as voltage-standing-wave-ratio, VSWR)." The maximum is where the two waves add, |a| + |b|, and the minimum where they subtract, so SWR = (1 + |Γ|)/(1 − |Γ|), Equation 2-10: 1 for a perfect match and without limit for total reflection. Return loss puts the reflection on a decibel scale, RL = 20 log(1/|Γ|), Huber+Suhner's Equation 29. Huber+Suhner call it "a logarithmic measure of the reflection coefficient": an ideal line has an infinite return loss, a short or open 0 dB, and "The return loss should be as high as possible".

Mismatch loss is what the reflection costs the load. The power that comes back is |Γ|² of the incident power, from Keysight's Equations 2-2 and 2-3, so the load gets 1 − |Γ|² of it. In decibels, and in Keysight's words: "The second term on the right side of Equation 2-26, –10 log (1 – |Γℓ|²), is called mismatch loss. It accounts for the power reflected from the load." Unlike return loss, which grows as the match improves, mismatch loss shrinks toward zero.

The sign convention on this page. Return loss and mismatch loss are positive numbers of decibels, as Huber+Suhner and Keysight write them: a return loss of 20 dB means the reflection is 20 dB below the incident wave. A network analyser plots the same measurement as S11 in dB, 20 log|Γ|, which is negative: −20 dB. The calculator takes either, and refuses a negative return loss rather than guessing which was meant.

VSWR to return loss conversion table

Every column is the same reflection in a different unit, computed from the VSWR with Huber+Suhner's Equations 30 and 32 and Keysight's mismatch loss. The last column is the band Huber+Suhner's Figure 26 puts each return loss in: "not matched" below 10 dB, "poorly matched" to 14 dB, "matched" to 20 dB, "well matched" to 30 dB and "very well matched" beyond.

VSWR	Return loss (dB)	\|Γ\|	Reflected power	Mismatch loss (dB)	Figure 26 band
1.05	32.26	0.024	0.06 %	0.003	very well matched
1.10	26.44	0.048	0.23 %	0.010	well matched
1.15	23.13	0.070	0.49 %	0.021	well matched
1.20	20.83	0.091	0.83 %	0.036	well matched
1.25	19.08	0.111	1.23 %	0.054	matched
1.30	17.69	0.130	1.70 %	0.075	matched
1.40	15.56	0.167	2.78 %	0.122	matched
1.50	13.98	0.200	4.00 %	0.177	poorly matched
1.60	12.74	0.231	5.33 %	0.238	poorly matched
1.70	11.73	0.259	6.72 %	0.302	poorly matched
1.80	10.88	0.286	8.16 %	0.370	poorly matched
1.90	10.16	0.310	9.63 %	0.440	poorly matched
2.00	9.54	0.333	11.11 %	0.512	not matched
2.50	7.36	0.429	18.37 %	0.881	not matched
3.00	6.02	0.500	25.00 %	1.249	not matched
5.00	3.52	0.667	44.44 %	2.553	not matched
10.00	1.74	0.818	66.94 %	4.807	not matched

Three things are worth reading off it. First, mismatch loss is small until the match is poor: a VSWR of 2 reflects 11.1 % of the power and costs 0.51 dB, and a VSWR of 1.5 costs only 0.18 dB. What a mismatch costs in practice is rarely the power it reflects; it is the uncertainty it adds to every measurement, the standing wave on the cable, and the power a transmitter or amplifier has to absorb. Second, the familiar pairing of a VSWR of 2 with 10 dB of return loss is close but not exact: VSWR 2 is 9.54 dB, and 10 dB is a VSWR of 1.925. Third, the scale is steep near a match: going from VSWR 1.2 to 1.05 buys 11.4 dB of return loss, which is why return loss, not VSWR, is the figure specified for well-matched parts.

Huber+Suhner's Table 3 gives the same relation as reflected power and reflected voltage. The power column is |Γ|² and agrees with the calculator to the table's two figures: 50.1 % at 3 dB, printed 50 %. The voltage column is |Γ|, and three of its entries are printed a little low: 70 % at 3 dB, where 20 log gives 70.8 %; 31.5 % at 10 dB for 31.6 %; and 3.1 % at 30 dB for 3.16 %. The differences are small, but they are why the calculator is tested against the table's power column only.

Worked example: Keysight's signal generator and power sensor

Keysight's note works a measurement at 2.4 GHz, "the RF frequency for Bluetooth™ and IEEE 802.11b wireless LAN radio systems": a signal generator driving an 8481A power sensor. "The output SWR of the E4433B at 2.4 GHz is 1.9, with an electronically switched attenuator, or 1.35, with a mechanically switched attenuator." And: "The specified SWR of an 8481A sensor at 2.4 GHz is 1.18, with a reflection coefficient of 0.0826. The 1.9 SWR of the signal generator is equivalent to ρg = 0.310, so the mismatch uncertainty is +0.219, –0.225 dB." The left column is the calculator's arithmetic; the right is what Keysight prints. The calculator's defaults are this example.

sensor     ρℓ = (1.18 − 1)/(1.18 + 1)               = 0.0826           Keysight: 0.0826
source     ρg = (1.9 − 1)/(1.9 + 1)                 = 0.3103           Keysight: 0.310
product    ρg·ρℓ                                    = 0.02562         
upper      10 log(1 + ρg·ρℓ)²                       = +0.2198 dB       Keysight: +0.219 dB
lower      10 log(1 − ρg·ρℓ)²                       = −0.2255 dB       Keysight: –0.225 dB
per cent   100[(1 ± ρg·ρℓ)² − 1]                    = +5.19 / −5.06 % 

source     ρg = (1.35 − 1)/(1.35 + 1)               = 0.1489           Keysight: 0.149
upper      10 log(1 + ρg·ρℓ)²                       = +0.1062 dB       Keysight: +0.106 dB
lower      10 log(1 − ρg·ρℓ)²                       = −0.1075 dB       Keysight: –0.107 dB

ML         −10 log(1 − ρℓ²), the sensor             = 0.0297 dB

Every figure agrees. The upper limit with the SWR 1.9 source is +0.2198 dB, which rounds to 0.220; Keysight prints +0.219, the same number cut at the third decimal rather than rounded. Its own rounded reflection coefficients, 0.310 and 0.0826, give the same result. With the mechanical attenuator the limits fall to +0.106 dB and −0.107 dB, 48 % of the first pair, which is Keysight's "reduced by half".

The two numbers are not a tolerance to be split evenly. With only the two SWRs known, the reading can sit anywhere between the limits, and which end depends on the phase of the re-reflection between generator and sensor, which moves with frequency and cable length. The sensor's own mismatch loss, 0.030 dB, is a separate and fixed quantity; in Keysight's words it "is usually taken into account when correcting for the calibration factor of the sensor". Put together through Keysight's Equation 2-25, the power the sensor absorbs lies between +0.196 dB and −0.249 dB of what the generator would put into a perfect Z₀ load. Keysight draws the practical lesson: "Note that the manufacturer's accuracy specification cannot include mismatch uncertainty because the load SWR is unknown and variable."

Worked example: a 75 Ω cable in a 50 Ω system

Keysight's first example is a mistake that is easy to make: "You pull out an unmarked cable with BNC connectors from a drawer, or borrow one from a colleague. Unknowingly, you connect a 75-Ω cable into your 50-Ω test-system." Its simulation of a 75 Ω line with a 1 ns delay between a 50 Ω generator and a 50 Ω sensor shows the power in the load rising and falling with frequency: "The peak-to-peak variation is about 0.7 dB, and can be calculated from the mismatch uncertainty limits." Seen from inside the cable, each 50 Ω end is a mismatch against 75 Ω, and the two ends play the parts of source and load:

each end   ρ = (75 − 50)/(75 + 50)                  = 0.2000          
VSWR       (1 + ρ)/(1 − ρ)                          = 1.500           
step RL    20 log|(75 + 50)/(75 − 50)|              = 13.98 dB        
upper      10 log(1 + ρ·ρ)²                         = +0.341 dB       
lower      10 log(1 − ρ·ρ)²                         = −0.355 dB       
spread     upper − lower                            = 0.70 dB          Keysight: about 0.7 dB

The spread between the limits is 0.70 dB, Keysight's "about 0.7 dB". The return loss of the step itself is Huber+Suhner's Equation 17, 20 log|(Z₂ + Z₁)/(Z₂ − Z₁)|, which gives 13.98 dB, the same as a 75 Ω resistor measured in a 50 Ω system. At low frequencies the cable is too short to matter; Keysight notes that below about 10 MHz "the system behaves as if the source and load were connected directly together." The loss appears as the cable becomes a sizeable fraction of a wavelength, and repeats "When the two-way transit time of the cable is equal to one cycle of the generator frequency", every 500 MHz for 1 ns of delay. To reproduce it, choose the mismatch uncertainty mode, specify the ports as |Γ|, and enter 0.2 for both.

Worked example: a reactive load

Keysight's examples are resistive. For a load with reactance, the impedance mode computes Γ as a complex number. Take 50 + j50 Ω on a 50 Ω line, an illustrative load rather than a published one: Γ = j50/(100 + j50) = 0.200 + j0.400, a magnitude of 0.447 at 63.4°. That is a VSWR of 2.618 and a return loss of 6.99 dB, the same reflection as a resistor of 130.9 Ω or 19.1 Ω. The capacitive load 50 − j50 Ω has the same magnitude, 0.447, at −63.4°: VSWR and return loss cannot tell the two apart, and neither can a power meter. Only the phase, which the figure plots above the real axis for an inductive load and below it for a capacitive one, says which way a matching network has to go.

Mismatch uncertainty, and why it is a bound

When a source with reflection Γ_g drives a load with reflection Γ_ℓ, the wave the load reflects is reflected again by the source and comes back to the load, and again, without end. Keysight's Equation 2-25 sums that series: the power delivered depends on |1 − Γ_gΓ_ℓ|², and that depends on the phases of both reflections. But "Γℓ and Γg are seldom completely known for both magnitude and phase. Only the magnitudes ρℓ and ρg are usually measured or specified." With only the magnitudes, the term can only be bounded: it is largest when the re-reflection adds in phase, 10 log(1 + ρ_gρ_ℓ)², and smallest when it subtracts, 10 log(1 − ρ_gρ_ℓ)². Those are Keysight's Equations 3-1 and 3-2, and "The minimum limit will always be a negative number. It is also true that the magnitude of the minimum limit will be greater than the magnitude of the maximum limit, but usually by a very small amount."

The limits in decibels, upper and lower, for a range of source and load VSWRs:

Source VSWR	Load 1.05	Load 1.10	Load 1.20	Load 1.50
1.20	+0.019 / −0.019	+0.038 / −0.038	+0.071 / −0.072	+0.157 / −0.159
1.50	+0.042 / −0.042	+0.082 / −0.083	+0.157 / −0.159	+0.341 / −0.355
2.00	+0.070 / −0.071	+0.137 / −0.139	+0.259 / −0.267	+0.561 / −0.599

The product ρ_gρ_ℓ is what matters, so improving either port helps equally, and improving the better one is often cheaper. Keysight gives two rules of thumb from its chart of the product: "if the source and load both had a ρ of 0.1, the approximate mismatch uncertainty would be approximately 0.09 dB", and the formulas give +0.086 and −0.087 dB; and with one port at ρ = 0.05, "even if the ρ2 reflection coefficient goes up to 0.5 (SWR = 3.0), the mismatch uncertainty only increases to about 0.2 dB", where the formulas give +0.214 and −0.220 dB. That is the argument for a well-matched power sensor, and for the pad Keysight suggests when neither port can be improved: "the return loss of the attenuator is better than the original source or load."

Mini-Circuits' application note AN-70-001 tabulates the same limits against pairs of VSWRs, alongside the worst-case VSWR of the pair: "it is generally safe to assume that two VSWRs will tend to multiply rather than add. For example, when connecting a component with a VSWR of 3.0:1 to a second component with a VSWR of 1.50:1, the resulting maximum VSWR will be 4.50:1, with a corresponding return loss of 3.93 dB." The calculator gives 4.50 and 3.93 dB. Every row of the table but two reproduces to the digits printed. Those two have a mislabelled second VSWR. The row printed as 2, 1.5, 3.5 has a maximum VSWR, return loss and uncertainty (3.5, 5.11 dB, +0.756/−0.828 dB) that belong to 2 and 1.75, not to 2 and 1.5, which give 3.0 and +0.561/−0.599 dB and appear correctly in the row above it. The row printed as 2, 1.75, 4 belongs to 2 and 2 (4.0, 4.44 dB, +0.915/−1.023 dB). The calculator's tests hold the other rows as printed and these two with the corrected labels.

Where the model stops being valid

One frequency at a time. A reflection coefficient belongs to a frequency. Keysight says of Γ and the waves that define it that "All three quantities are, in general, complex numbers and change with frequency." A single VSWR on a datasheet is usually a maximum over a band; an antenna's impedance is valid at the frequency it was measured at.

A real reference impedance. The formulas take Z₀ as a resistance, 50 Ω or 75 Ω. A lossy line has a complex characteristic impedance, and at low frequencies a coax or a PCB trace drifts away from its nominal value; the reflection against it is then not what these formulas give. Keysight notes the distinction the other way round too: in a mixed system "where 75 Ω transmission lines are used in systems with a 50 Ω reference impedance, another symbol, such as Zr, should be used for reference impedance."

A Z₀ match is not a conjugate match. A VSWR of 1 means the load equals Z₀. It does not mean the load takes the most power a particular source can deliver, which needs the load to be the complex conjugate of the source impedance. Keysight: "The use of the single word "match" should be dropped in favor of "Zo match" to describe a load of zero reflection coefficient, and in favor of "conjugate match" to describe the load that provides maximum power transfer." Mismatch loss on this page is against a Z₀ source.

The reading is where the meter is. Forward and reflected power describe the reflection at the coupler, not at the load. A matched cable or pad between them attenuates the incident wave on the way out and the reflection on the way back, so the return loss seen at the meter is the load's plus twice the loss, a derivation rather than a quoted figure. An antenna with a VSWR of 3 (6.02 dB) fed through 3 dB of cable reads 12.02 dB at the transmitter, a VSWR of 1.67. The cable has not improved the antenna; it has hidden it. A coupler's finite directivity also leaks some forward power into the reflected reading, which sets a floor under the smallest reflection a meter can resolve.

Mismatch loss is not all the loss. Mismatch loss counts only the power sent back. Keysight's net power into the load "includes not only power converted to heat, but also power radiated to space and power that leaks through accessory cables to other pieces of equipment." A dummy load and an antenna of the same VSWR have the same mismatch loss, and very different uses for the rest.

The worst case of two VSWRs assumes the worst phase.Mini-Circuits' product S₁·S₂ is the combination when the two reflections add; the best case, derived here, is their ratio. Across a band the phase between two discontinuities turns with frequency, and a measurement will find both.

Common VSWR and return loss mistakes

Writing return loss as a negative number. Return loss is positive, and larger is better; S11 in dB is its negative. A "return loss of −20 dB" is an S11 of −20 dB, a return loss of 20 dB. The calculator's S11 input takes the analyser's sign.
Reading reflected power as reflected voltage. At 10 dB of return loss 10 % of the power comes back but the reflected voltage is 31.6 %of the incident, the two columns of Huber+Suhner's Table 3. At 20 dB it is 1 % of the power and 10 % of the voltage.
Confusing mismatch loss with mismatch uncertainty. Mismatch loss belongs to one port and is fixed: 0.030 dB for Keysight's 8481A at SWR 1.18. Mismatch uncertainty belongs to a pair of ports and is a range, +0.220 to −0.225 dB in the same example, several times larger.
Adding VSWRs. Two mismatches in cascade can combine to the product of their VSWRs, not the sum: 3 and 1.5 give up to 4.5, as Mini-Circuits notes.
Taking VSWR 2 as 10 dB. It is 9.54 dB; 10 dB is VSWR 1.92. At a specification limit the difference decides pass or fail.
Trusting an SWR reading taken through a long, lossy cable as the antenna's own. Measure at the antenna, or add twice the cable loss back to the return loss before converting.
Quoting a mismatch uncertainty as ± one number. The lower limit is always the larger in magnitude, as Keysight points out; at small products the difference is negligible, at large ones it is not: VSWRs of 3 and 3 give +1.938 and −2.499 dB.