Review of SPARQ Series Network Analyzers Part 4: About S Parameters (Part 2)

Not all S parameter files can be used directly in simulation software. One difference between SPARQ and VNA is that the S parameters measured by SPARQ can be used directly in simulation software. We know that S parameters that can be used directly in simulation software need to have the following characteristics: 1. Follow the three major S parameter characteristic principles: passivity, reciprocity, and causality. Since the S parameters generated by VNA do not follow the principles of these three characteristics, other software is needed to check and verify these three principles before they can be used for simulation. 2. There is a DC point. The S parameters generated by VNA do not have a DC point, and other methods are needed to measure the S parameter value at DC. 3. For differential signal systems, mixed-mode S parameters are required. VNA cannot directly generate mixed-mode S parameters. 4. S parameters are saved in touchstone file format. As the world's first signal integrity network analyzer, the S parameters generated by SPARQ have the above characteristics and can be used directly for simulation.

Passivity

For a lossless network, the S matrix is ​​an identity matrix, so the following relationship exists for a two-port network:

 
 

Since there is no loss, the total amount of all scattering should be 100%. When S21 (S11) is large, S11 (S21) will be smaller, which can be seen from the previous S parameter curve.
For a passive two-port network , therefore, the S parameter of a passive device will not be greater than 1 (0dB). If the S parameter results measured by VNA are not verified by software for passivity, the S parameter value will appear to be greater than 0dB and cannot be directly used in simulation software.
  It is expressed as the power scattering ratio. The smaller this value is, the greater the loss.

Passivity

If a device is used in an interchangeable direction rather than a single-phase device such as an isolator or circulator, the S matrix is ​​symmetrical, so Sij=Sji.

Causality

The so-called causality means that there is output only after the stimulus. For the passive system S parameters, since the transmission of the signal will definitely produce a certain delay, the S parameters of the passive system should comply with the principle of causality, but the actual measured S parameters often produce a certain degree of non-causality due to various reasons. Many signal integrity simulation software require S parameters that meet the causal characteristics, otherwise divergence may occur during simulation, resulting in incorrect simulation results.
 
4. Mixed mode S parameters

Differential transmission systems have long been the mainstream of high-speed signal system transmission. If the distance between differential transmission lines is very close, the differential lines can be well coupled, and the differential signal is completely symmetrical, any introduced noise will have the same effect on the two differential transmission lines. Then, at the receiving end of the chip, due to the subtraction operation, the introduced common-mode noise is eliminated. However, the actual differential system is not perfect. The imbalance of the two single-ended signals that constitute the differential signal, the unequal lengths of the two channels, and the loose coupling will all cause energy to be converted from differential mode to common mode. Since the actual differential signal is always composed of differential mode signals and common mode signals ( ), the single-ended four-port S parameter matrix cannot provide insightful information about differential mode and common mode matching and transmission. Therefore, the mixed-mode S parameters proposed in 1995 have become an important tool for evaluating differential transmission systems.
I often say that various serial data standards describe the story of "two wires". If it is not used to transmit differential signals, these "two wires" form a single-ended four-port network. The single-ended four-port S parameter matrix describes what kind of response each port has when stimulated. If it is used to transmit differential signals, this single-ended four-port network can be understood as a differential two-port network, as shown in Figure 10. The mixed-mode S parameters, from a physical sense, describe the responses of the two paired wires to the sum of the two signals (common mode) and the difference between the two signals (differential mode).

Figure 10 Mixed-mode S-parameter measurement

The single-ended four-port S parameters and the mixed-mode S parameters can be converted to each other, as shown in Figure 11. Therefore, the mixed-mode S parameters can be derived by measuring the single-ended four-port S parameters.

The relationship between the output and input of a two-port differential system expressed using mixed-mode S parameters is as follows: bd1 represents the differential output of port 1, and ad1 represents the differential input of port 1.

5. S parameter measurement method

There are two methods for measuring S parameters, one is based on the principle of swept frequency measurement (VNA), and the other is based on the principle of fast-edge step response (TDR, SPARQ).

Figure 12 is a block diagram of the VNA, which mainly includes the following parts: (1) Excitation signal source: provides an incident signal within the frequency range of interest; (2) Signal separation device: contains a power divider and a directional coupler to separate the incident, reflected and transmitted signals; (3) Receiver: tests the incident, reflected and transmitted signals of the device under test; (4) Processing and display unit: processes and displays the test results.

Figure 12: Principle block diagram of VNA

 There are six major system errors in the VNA measurement process: (1) Directional error related to signal leakage; (2) Crosstalk error related to signal leakage; (3) Source mismatch related to reflection; (4) Load impedance mismatch related to reflection; (5) Frequency response error caused by reflection inside the test receiver; (6) Frequency response error caused by transmission tracking inside the test receiver. Therefore, strict calibration is required before use. Correct calibration is a difficulty in using VNA. Whether the S parameters measured by VNA are wrong cannot be directly checked by VNA. Only when the simulation results are imported into the simulation software and found to have problems, it may be suspected that there is a problem with the S parameter measurement, and then return to check the VNA calibration and whether there are any errors in the VNA measurement operation. However, due to its time domain analysis capability, SPARQ can immediately check whether the time domain response of the currently measured S parameters is reasonable.

Theoretically, any signal has a one-to-one correspondence in the time domain and frequency domain, and can be converted to each other. This makes it possible to measure S parameters using the time domain TDR/TDT method based on step response. Figure 13 shows the principle of measuring S21 and S12 using the TDR/TDT method. ST-20 is a TDR module on the LeCroy sampling oscilloscope device WE100H, which can generate ps-level fast edges and can be used as a sampling head with a bandwidth of 20GHz. Assume that Channel 2 is port 1 and Channel 3 is port 2. Channel 1 generates a fast edge signal as the incident wave and Channel 3 receives the signal after passing through the PCB trace. The incident fast edge signal and the sampled signal can be decomposed into signals in a certain frequency range through FFT transformation, and the S parameters in the frequency domain are obtained through calculation.

Figure 13 S-parameter measurement based on TDR/TDT method

In fact, when talking about the difference between VNA and TDR in measuring S parameters, we will naturally associate it with the measurement method of the front-end frequency response curve of the oscilloscope. We can measure the frequency response curve by the traditional frequency sweeping and plotting method (adjusting the frequency of the sine wave signal source, and then measuring the peak-to-peak value measured by the oscilloscope at different frequencies), but we can also quickly and easily measure the frequency response curve by inputting the fast edge signal into the oscilloscope and performing FFT on the sampled fast edge signal. The difference in principle between these two methods of measuring the frequency response curve of the oscilloscope is the same as the difference between the two methods of measuring S parameters.
 
In recent years, the practice of three instrument manufacturers measuring S parameters based on the TDR principle has proved that the two measurement methods have a very high degree of compliance. As shown in Figure 14, the results of the S parameters measured by the two methods are compared. However, the method based on TDR has the disadvantage of not having a high dynamic range. SPARQ is based on the principle of TDR for measuring S parameters, but has achieved breakthroughs in improving the dynamic range through patented algorithms. It also has innovations in achieving automated calibration with one-click operation. It has time domain analysis capabilities and S parameter files can be directly called by SI simulation software, making SPARQ the preferred instrument for signal integrity engineers to measure S parameters.

Figure 14: The S parameters measured by VNA and TDR methods are very consistent