Return loss is just another way of representing Reflection Coefficient. It is just log scale of Reflection Coefficient as shown below. So for the physical meaning of Return Loss, you have to refer to Reflection Coefficient.

Return loss is the number you will read most often on a VNA screen and on the datasheet of an antenna, a filter or a connector. So it is worth being exact about it. Let's first write the formula in its complete form, and then connect return loss to VSWR and mismatch loss. The last two sections cover two traps in practice: the sign of the number, and the effect of the cable between the instrument and the load.

• How is return loss calculated from the reflection coefficient ?
• How do return loss, VSWR and mismatch loss relate ?
• Why is return loss positive when S11 in dB is negative ?
• Why does a cable in front of the load make return loss look better ?

How is return loss calculated from the reflection coefficient ?

Return loss compares two powers at one port: the power that travels toward the load, and the power that comes back from it. The reflection coefficient already describes this, but as a ratio of wave amplitudes. So the question is how to turn an amplitude ratio into a power ratio in dB, and which sign to use.

Let's build it in two steps. The reflection coefficient Γ is the ratio of the reflected voltage wave to the incident voltage wave. Power goes with the square of the amplitude, so the reflected power is |Γ|² times the incident power. Return loss is this power ratio in dB, with the sign chosen so that the result is a positive number.

RL = -10 log₁₀(P_reflected / P_incident) = -10 log₁₀|Γ|² = -20 log₁₀|Γ| dB

Compare this with the formula image in the introduction. It writes 10 Log₁₀ of |Γ|, with no minus sign. Read literally, that expression gives a negative value, and it gives only half of the dB value. For example, |Γ| = 0.1 gives -10 from the image formula, but the return loss is 20 dB. So please read the image as a short notation for the full form above. The factor is 20 because Γ is an amplitude ratio, and the minus sign makes the loss positive.

Let's take a concrete load. A 75 ohm load on a 50 ohm line has Γ = (75 - 50) / (75 + 50) = 0.2. The return loss is -20 log₁₀(0.2) = 13.98 dB, or about 14 dB. In power terms, 4 % of the incident power comes back, because 0.2² = 0.04. A 100 ohm load on the same line gives Γ = 1/3 and a return loss of 9.5 dB.

• RL = -20 log₁₀|Γ| dB : the factor 20 comes from the amplitude ratio, and the minus sign makes the value positive.
• A larger return loss means a better match : a perfect match has Γ = 0 and an infinite return loss, and a full reflection has |Γ| = 1 and 0 dB.
• A 75 ohm load on a 50 ohm line has about 14 dB return loss : 4 % of the incident power comes back to the source.

How do return loss, VSWR and mismatch loss relate ?

Datasheets describe the same mismatch in different ways. An antenna datasheet may give VSWR, a filter datasheet may give return loss, and a system engineer wants to know how much power is lost. All of these numbers come from the same |Γ|, so we can convert any one of them into the others.

We need two formulas besides the return loss formula. The first is VSWR = (1 + |Γ|) / (1 - |Γ|). The second is the mismatch loss, which is the drop in delivered power caused only by the reflection. It is ML = -10 log₁₀(1 - |Γ|²) dB. The table below lists the four quantities for some common return loss values.

Look at the last column first. Once the return loss is above 10 dB, the power lost to reflection is less than half a dB. So a return loss target of 20 dB or more is usually not about delivered power. The reason is elsewhere. A reflected wave can pull the frequency of an oscillator, change the gain and flatness of an amplifier, and add ripple to a measurement. For these effects, the size of the reflected wave matters more than the lost power.

A return loss of 10 dB, or VSWR of about 1.9, is a common limit for the usable band of an antenna. At that limit 10 % of the power reflects, and the mismatch loss is 0.46 dB. Test ports, attenuators and cables normally need much better values, because they sit inside the measurement itself. For the standing wave view of the same mismatch, see VSWR.

• Return loss, |Γ|, VSWR and mismatch loss carry the same information : each one is a function of |Γ| alone.
• Mismatch loss becomes small quickly : 10 dB return loss costs 0.46 dB, and 20 dB return loss costs only 0.04 dB.
• A high return loss target protects the circuit, not the power budget : oscillator pulling, gain ripple and measurement error depend on the reflected wave itself.

Why is return loss positive when S11 in dB is negative ?

On a VNA you normally display S11 in dB, and the trace sits below 0 dB. A datasheet, however, gives return loss as a positive number. Both describe the same port, so the difference is only a sign convention. Let's make the rule explicit, because mixing the two is a very common mistake in reports.

S11 is the reflection coefficient at port 1 when port 2 is matched, so |S11| = |Γ| for a one port load. The VNA plots 20 log₁₀|S11|. For a passive load |S11| is at most 1, so this value is 0 dB or lower. Return loss is the same number with the sign flipped: RL = -20 log₁₀|S11|. So an S11 trace at -15 dB means a return loss of 15 dB.

The word "loss" already says that a positive value means attenuation. So a phrase like "return loss of -15 dB" is, strictly, a claim that the reflected wave is 15 dB stronger than the incident wave. A passive load cannot do that. In practice the writer usually means an S11 of -15 dB, but it is better to write either "S11 = -15 dB" or "RL = 15 dB". Some instruments label the S11 trace as return loss and still show negative values, so check the axis before copying a number into a report.

There is one case where the reflected wave can be larger than the incident wave. The input of an active device, such as a negative resistance oscillator or an unstable amplifier, can show |S11| greater than 1. Then S11 in dB is positive and the return loss is negative. A negative return loss is therefore a warning of possible instability, not a typing error.

• RL in dB is S11 in dB with the sign flipped : an S11 of -15 dB is a return loss of 15 dB.
• A passive load always has a return loss of 0 dB or more : |S11| cannot exceed 1 without a power source inside the load.
• A negative return loss on an active port points to instability : the port returns more power than it receives.

Why does a cable in front of the load make return loss look better ?

You rarely measure a load directly at the port of the instrument. There is usually a cable, a connector or an attenuator in between. So the question is what that extra loss does to the return loss you read, and whether the number still describes the load.

The key point is that the reflected wave crosses the lossy part twice. The incident wave loses the cable loss on the way to the load. The load reflects a part of what arrives. Then the reflected wave loses the cable loss again on the way back. So the measured return loss is the return loss of the load plus two times the one way loss of the cable. Figure 1 shows this with an antenna of 10 dB return loss behind a cable of 3 dB loss.

Figure 1. Return loss measured through a lossy cable. The VNA reads 16 dB, but the antenna itself has only 10 dB, because the reflected wave loses 3 dB in each direction.

• The blue arrow is the incident wave. It leaves the VNA at 0 dBm and reaches the antenna at -3 dBm.
• The red arrow is the reflected wave. The antenna reflects 10 dB below its input, so -13 dBm, and the wave reaches the VNA at -16 dBm.
• The line at the bottom gives the reading at the VNA: 16 dB, which is 10 dB plus two times 3 dB.

This effect works in both directions, and it is easy to misuse. A long lossy cable makes even a bad antenna look well matched. In the limit, an open cable end with |Γ| = 1 behind 10 dB of cable still shows 20 dB return loss at the instrument. So a return loss measured through a cable describes the load only after we remove the cable. The usual fix is to calibrate the VNA at the end of the cable, so that the reference plane sits at the load. Engineers also use the same effect on purpose. They place an attenuator in front of a poorly matched device, and a 3 dB attenuator adds 6 dB to the return loss that the source sees.

• Measured RL = load RL + 2 x one way loss : the reflected wave crosses the lossy part in both directions.
• Cable loss can hide a bad match : an open circuit behind 10 dB of cable still reads 20 dB return loss.
• Calibrate at the end of the cable : then the reference plane sits at the load, and the reading describes the load alone.