Return Loss, VSWR, |S11|, and Mismatch Loss

A $10\,\mathrm{dB}$ return-loss specification is not a $10\,\mathrm{dB}$ power loss. It is $|\Gamma| = 0.316$, a VSWR of $1.92$, and $0.46\,\mathrm{dB}$ of mismatch loss. One tenth of the incident power comes back. The load still receives $90\%$ of the available power, which is $0.46\,\mathrm{dB}$ down, not $10$.

Return loss, VSWR, and $|S_{11}|$ are the same reflection coefficient in three units. Mismatch loss is a different question: how much available power failed to arrive. The table below is computed from those definitions. The two worked rows are a $25 + j50\,\Omega$ load and the synthetic $2.45\,\mathrm{GHz}$ chip antenna, both into a real $50\,\Omega$ source.

What does each name measure?

For a real reference impedance $Z_0$, the reflection coefficient is

$$\Gamma = \frac{Z - Z_0}{Z + Z_0}$$

$|S_{11}|$ at that same $Z_0$ is $|\Gamma|$. The chart, the VNA marker, and the calculator agree on this when the source is real and equal to the reference. A complex source uses the power-wave form $(Z_{\mathrm{in}} - Z_s^*)/(Z_{\mathrm{in}} + Z_s)$ instead. Do not mix the two in one sentence.

Return loss, in decibels, is the ratio of incident power to reflected power:

$$\mathrm{RL} = -20\log_{10}|\Gamma|$$

A perfect match has infinite return loss. A short or an open has $0\,\mathrm{dB}$: all of the incident power comes back. The sign convention in RF is already the positive number. If a tool reports $S_{11}$ in decibels, that trace is negative, and return loss is its negation. $|S_{11}| = -10\,\mathrm{dB}$ and “$10\,\mathrm{dB}$ return loss” are the same marker.

VSWR is the standing-wave ratio on a lossless line of impedance $Z_0$:

$$\mathrm{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$

It runs from $1$, a perfect match, upward without a ceiling. VSWR $3$ is $|\Gamma| = 0.5$.

Mismatch loss is the fraction of available power that is not delivered when the source is a real $Z_0$ and there is no other network:

$$\mathrm{ML} = -10\log_{10}\left(1 - |\Gamma|^2\right)$$

That expression is the transducer gain of a direct connection, with the sign flipped. It is small until $|\Gamma|$ is large, because the power that fails to arrive is $|\Gamma|^2$, and a moderate reflection squared is still modest.

What are the numbers?

Read the $10\,\mathrm{dB}$ row as a spec you have probably written. $|\Gamma| = 0.316$ means $10\%$ of the incident power is reflected ($|\Gamma|^2 = 0.10$). Delivered power is $1.00 - 0.10 = 0.90$, and $-10\log_{10}(0.90) = 0.46\,\mathrm{dB}$. A radio that misses a $10\,\mathrm{dB}$ return-loss mask by being at $9\,\mathrm{dB}$ has not lost an extra decibel of link budget. It has lost a few hundredths of a decibel more than the $0.46\,\mathrm{dB}$ it was already giving up.

The $6\,\mathrm{dB}$ row is the other familiar mask. Return loss $6.02\,\mathrm{dB}$ is $|\Gamma| = 0.501$ and VSWR $3.01$. A quarter of the incident power comes back. Mismatch loss is $1.26\,\mathrm{dB}$. People call this “a VSWR of $3$” and treat it as a cliff. It is $1.26\,\mathrm{dB}$.

The $20\,\mathrm{dB}$ row is why chasing return loss past a point stops paying. $|\Gamma| = 0.100$, VSWR $1.22$, mismatch loss $0.044\,\mathrm{dB}$. The next $10\,\mathrm{dB}$ of return loss, from $10$ to $20$, buys $0.41\,\mathrm{dB}$ of power. Worth it when the spec says $20$. Not worth a coil change when the spec says $10$ and you are at $14$.

Two loads, unmatching and then matched

$25 + j50\,\Omega$ on a $50\,\Omega$ chart is the reading example in how to read a Smith chart. $|\Gamma| = 0.620$, return loss $4.15\,\mathrm{dB}$, VSWR $4.27$, mismatch loss $2.11\,\mathrm{dB}$. More than a third of the available power is reflected ($0.620^2 = 0.38$). An ideal L-section brings the transducer gain back to $0\,\mathrm{dB}$ at the design frequency. The return loss goes to a numerical infinity. The $2.11\,\mathrm{dB}$ comes back, at that frequency, and nowhere else for free. Which of the four sections to build is which L-network to build.

The synthetic chip antenna at $2.45\,\mathrm{GHz}$ is $10.64 + j5.56\,\Omega$. Return loss $3.70\,\mathrm{dB}$ looks “only a few dB” if you are used to reading return loss as the loss. The mismatch loss is $2.41\,\mathrm{dB}$. That is the number that belongs in a link budget for the bare antenna. After the ideal low-pass section, transducer gain at $2.45\,\mathrm{GHz}$ is $0\,\mathrm{dB}$ again, and the $2.41\,\mathrm{dB}$ is recovered at that frequency only. The $\ge 10\,\mathrm{dB}$ width of that recovery is $273\,\mathrm{MHz}$ on the antenna model. Details and the reference-plane warning are in matching the chip antenna.

Finite component Q spends some of the recovery on heat. Transducer gain then sits a few tenths of a decibel below zero even at the perfect frequency. That is insertion loss, not residual mismatch, and it is tabulated separately.

Which number belongs in the spec?

Use return loss or VSWR for the match mask. They are sensitive near the center of the chart, which is what you want when you are deciding whether a network landed. Use mismatch loss, or transducer gain if there is a network, for the power. A cascade budget that subtracts “$10\,\mathrm{dB}$ return loss” as if it were $10\,\mathrm{dB}$ of attenuation is wrong by $9.5\,\mathrm{dB}$.

On a VNA, read the marker you intended. A return-loss marker and an $S_{11}$ magnitude marker differ by a sign. A Smith marker and a $|\Gamma|$ marker differ by a conversion. VSWR blows up as $|\Gamma|$ approaches $1$, so it is a poor way to compare two bad matches and a fine way to compare two good ones.

These identities assume the reference you named. $|\Gamma|$ against $50\,\Omega$ is not $|\Gamma|$ against $75\,\Omega$. And they assume a real source equal to that reference when they are used as mismatch loss. A complex source is the conjugate-match case, and the power ratio is the transducer gain the solver reports, not the $50\,\Omega$ formula with the chart-center $\Gamma$ plugged in.

References

• D. M. Pozar, Microwave Engineering, the definitions of reflection coefficient, return loss, and VSWR in the transmission-line chapters.
• K. Kurokawa, “Power Waves and the Scattering Matrix,” IEEE Transactions on Microwave Theory and Techniques, 1965. Mismatch loss above is the real-$Z_0$ case. The power-wave form is the one to use when $Z_s$ is complex.

Related

• How to read a Smith chart
• How to design an L-network
• Smith chart calculator

FAQ

What is the difference between return loss and mismatch loss?

Return loss is −20 log10|Γ|, the ratio of incident power to reflected power. Mismatch loss is −10 log10(1 − |Γ|²), the fraction of available power that never reaches the load when the source is a real Z0. A 10 dB return loss is |Γ| = 0.316, VSWR 1.92, and only 0.46 dB of mismatch loss.

How do VSWR and |S11| relate?

For a real reference impedance, |S11| is |Γ|. VSWR is (1 + |Γ|) / (1 − |Γ|). VSWR 3 is |Γ| = 0.5 and return loss 6.02 dB. The numbers in the table are computed from those definitions.

Is 10 dB return loss a 10 dB power loss?

No. Ten decibels of return loss means one tenth of the incident power is reflected. The power delivered is 1 − |Γ|², which is 0.46 dB below the available power. The unmatched synthetic chip antenna, at 3.70 dB return loss, gives up 2.41 dB.

Which number should a matching spec use?

Use return loss or VSWR for the match specification, and mismatch loss or transducer gain for the power you actually lost. A lossless L-network that is conjugate-matched has essentially 0 dB transducer gain at the design frequency even though the bare load had several decibels of mismatch loss.