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## 16 April 2020

### Six Ways Not to be Confused by S-parameters (Part II)

In Part I, we discussed three causes of confusion when working with S-parameters and what you can do to avoid them.  In this post, we’ll discuss three more ways not to be confused by S-parameters.

### 4. Know the difference between return loss and reflection coefficient Figure 1. A 2-port device has two distinctly interesting S-parameters, S11 and S21.  S11 is the reflection coefficient of Port 1 (also called return loss), and S21 is the transmission coefficient of Port 2 (also called insertion loss).
Let’s start by looking at a 2-port device illustrated in Figure 1.

There are two S-parameters of interest in a 2-port device.  The first is the reflection coefficient, S11, that measures the ratio of the reflection from Port 1 to the drive signal at that port.  The second is the transmission coefficient, S21, that is the ratio of the output of Port 2 to the drive signal into Port 1.  Confusion arises because historical measurements of return loss and insertion loss are often used interchangeably with reflection coefficient and transmission coefficient, respectively.

The source of this confusion is the method used to make this measurement before the availability of vector network analyzers (VNAs).  Return loss was measured by driving a signal source into an open circuit and measuring the reflected signal.  Of course, since 100% of the signal was reflected back to the source by the open, this was used as a reference power level.  Next, the device under test was inserted and the test repeated (Figure 2).  Return loss was calculated using the ratio of power reflected from the open (the reference level) compared to the power reflected from the input port with the DUT connected:

Return Loss = 10 * Log ((P reference)/(P reflected)) Figure 2. Return loss is measured by comparing power reflected from the DUT with power reflected from an open circuit. Return loss will always be a positive, scalar value.
Since the reference power level will always be greater or equal to the reflected power, the return loss will always be a positive, scalar value.  S11, on the other hand, is a vector value with a magnitude and a phase. The magnitude of S11 will always be a negative number in dB.

The return loss in dB is -1 times the magnitude of the reflection coefficient S11.

The smaller the S11 magnitude, the greater the return loss and vice versa. So, the better the impedance match at Port 1, the greater the return loss and the smaller S11, the reflection coefficient.

You will hear S11 referred to as the “return loss”, but when you hear S11, you should always think “reflection coefficient” to avoid confusion.

### 5. Know the difference between insertion loss and transmission coefficient Figure 3. Insertion loss measures the power at Port 2 without the DUT and then with the DUT inserted.
There is a similar confusion when discussing insertion loss and transmission coefficient due to the pre-VNA measurement process.  Insertion loss measurement looks at the power at Port 2 (load) without the DUT and then the power with the DUT inserted (Figure 3).

Insertion loss is then the loss in the signal due to insertion of the DUT.

Since the power passed to load without the DUT is greater or equal to the power with the DUT inserted, the insertion loss will always be positive in dB.  A larger insertion loss means that less power is delivered with the DUT installed. S21 will always be a negative dB value and the insertion loss will be -1 times the transmission coefficient magnitude.

Insertion Loss = 10 * Log ((P witout DUT )/(P with DUT))

When you hear “insertion loss”, think “S21 transmission coefficient is -1* insertion loss in dB”.

### 6. Know S-parameters are ratios of voltages described as ratios of power (dB)

When we look at S-parameters, we calculate them as the ratios of voltages. If they are expressed in dB, we have to be a little careful because dB is basically a ratio of measured powers.We have to look at the underlying voltage associated with that power. Power is related to the square of the voltage resulting in the following:

A_dB = 10 * Log P_Out/P_In = 10 * Log (V_Out)^2/(V_In)^2 = 20 * Log  V_Out/V_In

Note that when we compute dB with voltages, there is a factor of 2 that accounts for the squaring of the voltage measurements to convert them to power.  In converting from dB to voltage ratios, the factor of 20 must be used:

V_Out/V_In = 10^(A_dB/20)

Using these equations, we can compute some significant voltage dB ratios:

Ratio of Voltage
Amplitudes (%)     Value in dB
100                               0
90                              -1
80                              -2
70                              -3
50                              -6
30                            -10
10                            -20
5                            -26
3                            -30
1                            -40 Figure 4. Comparing the S11 and S21 for a high bandwidth K connector and a lower bandwidth SMA connector. Vertical scale is negative dB matching the S-parameters. Return loss and insertion loss will be positive in dB.

Let’s take a look at a simple example of S-parameter measurements comparing two different connectors (Figure 4). The K connector has a rated bandwidth of 40 GHz, while the SMA connector’s bandwidth is 18 GHz.  The transmission coefficient of the K connector is near 0 dB, while that of the SMA connector drops below 0 dB, to about -3 dB, between 20 and 32 GHz.  Remember that the insertion loss is -1 times the transmission coefficient, so that the maximum insertion loss of the SMA connector is about 3 dB.  At that frequency, only about 70% of the applied voltage is reaching the output port.

Likewise, the reflection coefficient of the SMA at around 25 GHz is -4 dB.  This means that the return loss is 4 dB.

Future posts in this series will look at how to understand commonly seen patterns that occur in plots of S-parameters. In the meantime, watch the on-demand webinar Reading S-Parameters Like a Book by Dr. Eric Bogatin, Signal Integrity Evangelist, for a more in-depth treatment of this topic.