S parameter stands for 'Scattering' Parameter. It is a kind of power ratio between components of a signal going through various different path. Strictly, each S parameter is a ratio of wave amplitudes, and its magnitude squared is the power ratio. S parameters are the language of every VNA measurement and of most RF datasheets, so let's build them up step by step. We start with the paths a signal can take through a two port device. Then we write the definition, see how a network analyzer measures each parameter, and finish with the S parameters of a few real devices.

• Where does a signal go when it hits a two port device ?
• How are S parameters defined ?
• How is each S parameter measured ?
• What do the S parameters of real devices look like ?

Where does a signal go when it hits a two port device ?

Let's suppose we have a component labeled as 'M' as shown below. Now you input (impinge) a signal toward the device at Port 1. Where the signal would go ? Ideally there would be three possible paths this signal (energy) follow through.

i) a portion that hits the device and bounce back towards the port 1

ii) a portion that goes through the device and travel towards Port 2

iii) a portion that gets lost as unusuable energy (e.g, heat)

Now you input (impinge) a signal toward the device at Port 2. Where the signal would go ? Ideally there would be three possible paths this signal (energy) follow through.

i) a portion that hits the device and bounce back towards the port 2

ii) a portion that goes through the device and travel towards Port 1

iii) a portion that gets lost as unusuable energy (e.g, heat)

The possible paths listed above can be illustrated as shown below. The only missing part in this illustration is the portion that gets lost as unusual energy.

• a₁ and a₂, drawn in green, are the waves that travel toward the device M from Port 1 and from Port 2.
• b₁ and b₂ are the waves that travel away from the device. The brackets show that each of them contains two parts: a reflected part of the wave on its own side, and a transmitted part of the wave from the other side.
• The red arrows mark the reflection and transmission of a₁, and the blue arrows mark those of a₂.

So each outgoing wave is a sum of two contributions. For example, b₂ contains the part of a₁ that passes through M and the part of a₂ that M reflects. The S parameters are simply the four weights of these contributions. The third path, the energy lost as heat, has no wave of its own. It shows up only as a shortfall: the outgoing power is smaller than the incoming power.

• Every outgoing wave mixes reflection and transmission : b₁ contains the reflection of a₁ and the transmission of a₂.
• Loss has no wave of its own : it appears as the difference between incoming and outgoing power.

How are S parameters defined ?

The definition of S parameters is defined in mathematical form as shown below. From this equation, you would get a big picture saying "S parameter(matrix) defines the relationship between the signal impinging into each port of the device and the portion that is getting away from the device".

The equation above is written with squared magnitudes, so it describes powers. The exact definition works with the complex waves themselves:

b₁ = S₁₁ a₁ + S₁₂ a₂,    b₂ = S₂₁ a₁ + S₂₂ a₂

Here a and b are normalized so that |a|² and |b|² are the powers of the waves, and each S parameter is a complex number with a magnitude and a phase. The squared form is exact for each term alone, for example when only one port is driven. It is not exact for the sum when both ports are driven, because the two contributions add with their phases. Let's check with S₁₁ = 0.2, S₁₂ = 0.9 and a₁ = a₂ = 1, with all phases zero. The complex form gives |b₁|² = (0.2 + 0.9)² = 1.21, while the squared form gives 0.04 + 0.81 = 0.85.

This is the case where a device has only two ports (one input and one output) and the S parameter matrix is 2 x 2. Before getting into the meaning of each parameter, I want you to understand the basic notation of the paramter. S parameter has two subscript and the meaning of each subscript means as shown below.

The order "to, from" follows directly from the equation for b₂. S₂₁ multiplies a₁ and produces a part of b₂, so it carries the wave from port 1 to port 2. The same rule extends to a device with N ports, which has an N x N matrix. For example, S₃₁ of a three port device is the transmission from port 1 to port 3.

• S parameters are complex : the magnitude gives the amplitude ratio, and the phase gives the delay or phase shift.
• The first subscript is the output port and the second is the input port : S₂₁ is the forward transmission from port 1 to port 2.
• |S|² is a power ratio for one path at a time : with both ports driven, the waves add with their phases.

How is each S parameter measured ?

When measuring each of these S parameter, we don't put the signal to both ports at the same time. First we put a signal to one port and measure half set of S parameters and then we put a signal to the other port and measure the remaining half set of S parameters.

The equipment (e.g, Network Analyzer) measures the S11, S21 by inputting a signal from Port 1 and terminate Port 2 with a matched load. From this setting, we can get two S parameters as shown below.

• S₁₁ = b₁/a₁ is the input reflection coefficient, the V reflected at port 1 over the V coming into port 1.
• S₂₁ = b₂/a₁ is the forward transmission, the V going out of port 2 over the V coming into port 1.
• Both hold only when a₂ = 0, which means that nothing comes back into port 2.

The equipment (e.g, Network Analyzer) measures the S22, S12 by inputting a signal from Port 2 and terminate Port 1 with a matched load. From this setting, we can get two S parameters as shown below.

• S₁₂ = b₁/a₂ is the reverse transmission, and S₂₂ = b₂/a₂ is the output reflection coefficient.
• Both hold only when a₁ = 0.

Why does the condition a₂ = 0 need a matched load, and not a ground? A ground is a short circuit, and a short reflects the whole wave that reaches it, with |Γ| = 1. The wave b₂ would then return to the device as a new a₂, and the measured b₁/a₁ would include it. A load equal to the reference impedance, usually 50 ohm, absorbs b₂ completely, so a₂ is really zero.

A real VNA terminates its ports with its own 50 ohm source and receiver, and these are never perfect. So the instrument measures its own port match and directivity with known standards, such as short, open, load and thru, and it removes their effect from the result. This step is the calibration. A calibrated VNA also measures all four S parameters without reconnecting the device, because it switches the source between its two ports.

• Each measurement drives one port and terminates the other : port 1 driven gives S₁₁ and S₂₁, and port 2 driven gives S₂₂ and S₁₂.
• The unused port needs a matched load : a short or an open reflects the wave and breaks the condition a = 0.
• Calibration removes the imperfect termination : the VNA measures known standards and corrects its own errors.

What do the S parameters of real devices look like ?

The four numbers become useful when we can read a device from them. Let's look at three simple devices and see which patterns in the matrix tell us what the device does. The table uses magnitudes only, in linear form and in dB, where the dB value is 20 log₁₀|S|.

The values in the table are examples chosen to show the patterns. Let's read the rows one by one. The attenuator and the cable are passive and reciprocal, so S₂₁ = S₁₂. The amplifier is not reciprocal: it has 15 dB of gain forward, and it blocks the reverse direction by 25 dB. That reverse number is the isolation of the amplifier.

For a passive device we can also check the power balance of the first section. With port 1 driven, the power that leaves is |S₁₁|² + |S₂₁|², and the rest is lost as heat. For the cable this is 0.04 + 0.81 = 0.85, so 15 % of the incident power is dissipated. For the matched attenuator it is 0 + 0.5, so half of the power is dissipated. A lossless device would give exactly 1.

The dB values connect to the other pages of this handbook. The Return Loss is -20 log₁₀|S₁₁|, which is 14 dB for the cable. The Insertion Loss is -20 log₁₀|S₂₁|, which is 0.92 dB for the cable. S₁₁ itself is the Reflection Coefficient at port 1, and it also sets the VSWR.

• S₂₁ = S₁₂ for a reciprocal device : passive devices without magnetic materials, such as cables and attenuators, are reciprocal.
• |S₁₁|² + |S₂₁|² = 1 for a lossless device : a smaller sum means that the device dissipates power.
• Return loss and insertion loss are S₁₁ and S₂₁ in dB : both are written as positive numbers, with the sign flipped.