1. Multi Port Measurements
Slides from Dave Blackham and Ken Wong
At Agilent Technologies
With some additions by Doug Rytting
Dave Blackham
& Ken Wong

2. Agenda Two Port Network Analysis Multiport Network Analysis
Multiport Network Analysis Port Match Correction
Single Reference Receiver Example
Multiport Using a 2-port VNA Example
Multiport Calibration Approach How Many Connections Are Needed
Examples

3. Network Analyzer Block Diagram
This is a generic block diagram of a 4 channel network analyzer. The source can be switched to excite port-1 or port-2 of the device under test (DUT). The switch also provides a Z0 termination for the output port in each direction. Directional couplers are used to separate the incident, reflected and transmitted waves in both the forward and reverse direction. Mixers are used to down convert the RF signals to a fixed low frequency IF. The IF is digitized by A/D converters. The LO source is tuned to the frequency of the RF + IF.
The S-parameters of the DUT can be defined as follows:
S11 = b1/a1, switch in forward direction
S21 = b2/a1, switch in forward direction
S12 = b1/a2, switch in reverse direction
S22 = b2/a2, switch in reverse direction

4. 8-Term Error Model
The 8-term model can be derived from the 16-term model. First assume that the leakage terms are all zero. Or that the two primary leakage terms can be determined in a separate calibration step. Then assume that the switch is perfect and does not change the port match of the network analyzer as it is switched from forward to reverse. This assumption is valid if there are 4 measurement channels that are all on the DUT side of the switch. Then it is possible to mathematically ratio out the switch. This mathematical approach will be discussed later.

5. 8-Term Error Model
The flow graph consists of an error adapter at the input and output of the DUT. For ratio measurements of S-parameters, the number of error terms is reduced to 7 since the error terms can be normalized.

6. 8-Term Error Model
This is one of many possible mathematical formulation for the 8-term error model. Consider the error adapter as just one adapter between the perfect measurement system and the DUT. Then model this error adapter using the cascade T-parameters. This T-parameter matrix (T) can be partitioned, as described in the 16-term method, into the four diagonal sub matrixes T1, T2, T3, and T4. The 7 error terms are now defined as DX, kDY, e00, ke33, e11, ke22, and k. Which are the tracking, directivity, and match errors.

7. 8-Term Error Model
Using this approach the measured S-parameters formulation is a ‘bilinear matrix equation.’ The equation can be ‘inverted’ to solve for the actual S-parameters. And most important the relationship can be put in linear form. Expanding this matrix equation for the two-port case yields 4 equations with 4 measured S-parameters, 4 actual S-parameters, and 7 error terms. Note that these 4 equations are linear with regards to the 7 error terms.
This approach is particularly attractive for multi-port measurement systems. The matrix formulation does not change as additional ports are added.

8. 8-Term Calibration Examples
There is a number of calibration techniques that have been developed based on the 8-term error model. Seven or more independent conditions must be measured. There must be a known impedance termination or a known transmission line. And port-1 and port-2 must be connected for one of the measurements.
The list of calibration approaches can be much longer than the ones shown above. And there continues to be new and novel ways to solve for the seven error terms and calibrate the system.
The 8-term error model approach has yielded more accurate calibration methods as well as simplified the calibration process. TRL and LRL provide the best accuracy. The other methods simplify the calibration steps compared to the older TOSL 12 term model. In one case (UXYZ above) the thru standard does not need to be known as long as it is reciprocal.

9. Measuring S-parameters Removing Port Match Changes Caused by Switch
Normally we measure S-parameters by terminating one port with a Z0 impedance and then measure the input reflection and transmission coefficients then flip the switch and make the same set of measurements in the reverse direction. However, with 4 couplers we can measure both the forward and reverse terminating impedances of the switch. This allows us to “ratio out” or remove the effects of the imperfect switch.
Note that the measurement channels are all on the DUT side of the switch. This allows the measurements of the incident and reflected signals at both outputs of the switch and the switch match errors can be calculated.
Mathematically the S-parameters of the system generate 4 equations. 2 in the forward direction and 2 in the reverse direction. These 4 equations can then be solved for the 4 measured S-parameters. This general way of measuring S-parameters does not require the DUT to be terminated in a Z0 environment.

10. Measuring S-parameters
Solving the 4 previous equations yield the above results. Note that the equations are written to allow ratio measurements by the network analyzer. Typically the network analyzer is more accurate making measurements this way. Noise and other common mode errors are reduced.
Using this method for measuring the 4 S-parameters requires 6 ratio measurements. The additional two measurements are required to remove the effects of the switch. However, these two additional measurements (G1 and G2) need only be made during the calibration step since they do not change during measurement assuming the switch is stable.

11. Agenda Two Port Network Analysis Multiport Network Analysis
Multiport Network Analysis Port Match Correction
Single Reference Receiver Example
Multiport Using a 2-port VNA Example
Multiport Calibration Approach How Many Connections Are Needed
Examples

12. Multiport error correction
Is multiport error correction hard?
Dave Blackham
& Ken Wong

13. Multiport error correction
Is multiport error correction hard?
No, multiport error correction with constant match is as easy as single port error correction.
Dave Blackham
& Ken Wong

14. Multiport error box diagram
Dave Blackham
& Ken Wong

15. Multiport error box diagram
Dave Blackham
& Ken Wong

16. Multiport error box diagram
Dave Blackham
& Ken Wong

17. Multiport error box diagram
Dave Blackham
& Ken Wong

18. Multiport error box diagram
Dave Blackham
& Ken Wong

19. Multiport error box diagram
Dave Blackham
& Ken Wong

20. Multiport error box diagram
Dave Blackham
& Ken Wong

21. Multiport error box diagram with “12 term” crosstalk
Dave Blackham
& Ken Wong

22. Multiport error box diagram with full leaky model
Dave Blackham
& Ken Wong

23. Agenda Two Port Network Analysis Multiport Network Analysis
Multiport Network Analysis Port Match Correction
Single Reference Receiver Example
Multiport Using a 2-port VNA Example
Multiport Calibration Approach How Many Connections Are Needed
Examples

24. Multiport error correction
Models presented thus far assume a constant port match
similar to 8 term two-port model for non-leaky case
similar to 16 term two-port model for leaky case
Due to switching, port match is not constant
similar to 12 term two-port model
Dave Blackham
& Ken Wong

25. What Is Switch Correction?
TRL and unknown thru algorithms belong to a class that assumes a constant match at each test port.
In reality, the match at each test port will vary as the source is switched from port to port.
Switch correction is the process of characterizing the match difference then factoring it out of the calibration process
Generalized s-parameters factor out match differences during raw measurements for receivers that have dual couplers at each port (reference receiver at each port).
Two-tier calibration approaches characterize match differences with a first tier calibration using SOLT. This allows the use of generalized s-parameters approach for systems that have a single reference receiver.
Dave Blackham
& Ken Wong

26. Ideal S-Parameters Ideal s-parameters
Non-source ports terminated in perfect match—incident signal only from source port
Dave Blackham
& Ken Wong

27. Use Generalized S-Parameters
Ideal s-parameters
Non-source ports terminated in perfect match—incident signal only from source port
Generalized s-parameters
Uses incident signals from all ports & removes port match error
Dave Blackham
& Ken Wong

28. Agenda Two Port Network Analysis Multiport Network Analysis
Multiport Network Analysis Port Match Correction
Single Reference Receiver Example
Multiport Using a 2-port VNA Example
Multiport Calibration Approach How Many Connections Are Needed
Examples

29. Single Reference Receiver
Dave Blackham
& Ken Wong

30. S-parameter measurement (two-port, ideal)
Forward s-parameters Source at port 1
Reverse s-parameters
Source at port 2
Dave Blackham
& Ken Wong

31. S-parameter measurement (two-port, ideal)
Dave Blackham
& Ken Wong

32. S-parameter measurement (two-port, non-ideal)
Generalized s-parameters
Dual reflectometers at each testport allow measurement of all signals required to determine s-parameters.
Using this method will correct for the changing port match caused by the switch.
Dave Blackham
& Ken Wong

33. S-parameter measurement (two-port, non-ideal)
Generalized s-parameters
Dual reflectometers at each testport allow measurement of all signals required to determine s-parameters.
Benefit allows constant match to be assumed for error correction (eight term model)
Match variations tracked by incident wave measurements
Dave Blackham
& Ken Wong

34. S-parameter measurement (two-port, non-ideal)
Non dual reflectometer analyzers can’t measure signals reflected from switch in off position.
Requires mathematical equivalent computed from difference between source and load match at each port (delta match)
Generalized S-parameter in ratio form:
Dave Blackham
& Ken Wong

35. S-parameter measurement (two-port, non-ideal)
Dave Blackham
& Ken Wong

36. Calculate GF and GR For Single Reference Receiver
Error terms were measured during the first tier calibration using SOLT.
With GF and GR determined the generalized s-parameters can be used to remove the port match variations.
Also TRL or unknown thru, etc. can be used in a second tier calibration.
Dave Blackham
& Ken Wong

37. Agenda Two Port Network Analysis Multiport Network Analysis
Multiport Network Analysis Port Match Correction
Single Reference Receiver Example
Multiport Using a 2-port VNA Example
Multiport Calibration Approach How Many Connections Are Needed
Examples

38. Multiport Using a 2-port VNA Example
Switches Terminated
in off state
Dave Blackham
& Ken Wong

39. Multiport Using a 2-port VNA
Let:
Smi:j = measured S-parameters between ports i and j.
Rmi:j = Port impedance normalized Scattering Matrix
Gi:j = Diagonal matrix of reflection coefficient of imperfect port terminations at ports i and j. [Gi..N values must not change when signal paths are changed.]
Dave Blackham
& Ken Wong

40. Multiport Using a 2-port VNA
Let:
Rn = Composite port impedance normalized N-port Scattering Matrix
[Rn] matrix
Fill Rn matrix with calculated Rm sub-matrices
i=1, j=2
Dave Blackham
& Ken Wong

41. Multiport Using a 2-port VNA
Let:
Rn = Composite port impedance normalized N-port Scattering Matrix
[Rn] matrix
Fill Rn matrix with calculated Rm sub-matrices
i=2, j=3
Dave Blackham
& Ken Wong

42. Multiport Using a 2-port VNA
Let:
Rn = Composite port impedance normalized N-port Scattering Matrix
[Rn] matrix
Fill Rn matrix with calculated Rm sub-matrices
i=1, j=3
Dave Blackham
& Ken Wong

43. Multiport Using a 2-port VNA
Let:
Rn = Composite port impedance normalized N-port Scattering Matrix
[Rn] matrix
Fill Rn matrix with calculated Rm sub-matrices
Do N(N-1)/2 2-port measurements to fill
Dave Blackham
& Ken Wong

44. Multiport Using a 2-port VNA
Let:
Rn = Composite port impedance normalized N-port Scattering Matrix
Gn = Diagonal matrix of reflection coefficient of imperfect port terminations at ports 1 to N.
Sn = S-parameters of corrected N-port
Normalize Result back to System Impedance
Dave Blackham
& Ken Wong

45. Agenda Two Port Network Analysis Multiport Network Analysis
Multiport Network Analysis Port Match Correction
Single Reference Receiver Example
Multiport Using a 2-port VNA Example
Multiport Calibration Approach How Many Connections Are Needed
Examples

46. Multiport calibration Approach
Use all of the same calibration standards used by two port calibrations.
Brute force method: calibrate all possible two-port pairs
This will get tedious very quickly as the number of ports increases n 2 3 4 6 8 12
Cn:2 1 15 28 66
Dave Blackham
& Ken Wong

47. Multiport calibration error terms
Port terms
Path terms
Dave Blackham
& Ken Wong

48. Minimizing Connections During Multiport calibration
Characterize each set of port terms once (n).
Characterize (n-1) thru standards to characterize (n) load match terms and 2x(n-1) sets of transmission tracking terms. Compute the other (n-1)x(n-2) transmission tracking terms.
If desired, connect loads to each port then characterize n x (n-1) sets of crosstalk terms.
Full leaky model would connect multiple permutations of one port reflection standards to the ports and measure n x (n-1) paths for each permutation.
Dave Blackham
& Ken Wong

49. Required Number of Thrus
Connect (n-1) thru connections and characterize 2x(n-1) transmission tracking terms.
The other (n-1)x(n-2) terms can be calculated.
Port 1
Port 2
Port 3
Port N
Required Thrus
Dave Blackham
& Ken Wong

50. Compute Transmission Tracking
Characterize transmission tracking between ports i and j
Characterize transmission tracking between ports i and k
Compute transmission tracking between ports j and k
Accuracy of computed transmission tracking terms less than characterized transmission tracking terms.
Actual equation includes compensation for varying port match (source match not equal to load match at port i).
Dave Blackham
& Ken Wong

51. Agenda Two Port Network Analysis Multiport Network Analysis
Multiport Network Analysis Port Match Correction
Single Reference Receiver Example
Multiport Using a 2-port VNA Example
Multiport Calibration Approach How Many Connections Are Needed
Examples

52. Multiport Mechanical Cal
Mechanical Cal Method
Precision Mechanical 2-port Cal (SOLT or TRL)
Port 3
Port 4
Port 1
AND
Port 2
Finish 4-port Cal
Using Unknown Thru.
Only Transmission tracking
needs to be determined.
Port 3
Port 4
Unknown Thrus (adapters)
Dave Blackham
& Ken Wong

53. Only Transmission tracking
Multiport ECal Cal
Port 1
ECal 1
Port 2
ECal Method
Port 3
ECal 2
Port 4
Port 1
AND
Port 2
Finish 4-port Cal
Using Unknown Thru.
Only Transmission tracking
needs to be determined.
Port 3
Port 4
Unknown Thrus (adapters)
Dave Blackham
& Ken Wong

54. Multiport Unknown Thru Cal
Can have different
connector on
Each Port
1-Port Calibrations, ECal or Mech
Port 3
Port N
Port 1
Port 2
Finish 4-port Cal
Using Unknown Thru.
Only Transmission tracking
needs to be determined.
Port 3
Port N
Unknown Thrus (Adapters)
Dave Blackham
& Ken Wong

55. Multiport On-Wafer Cals
Straight Thrus
TRL on Wafer Cal
Port 3
Port 4
Port 1
AND
Port 2
Finish 4-port Cal
Using Unknown Thru.
Only Transmission tracking
needs to be determined.
Port 3
Port 4
Imperfect Unknown Thrus
Dave Blackham
& Ken Wong

56. Advantages of Unknown Thru Calibration in Multiport Systems
Unknown Thru is very convenient for right-angle or not-in-line thru calibrations.
S-parameters of the thru standard need not to be characterized.
Eliminates the need to move test ports and cables or probes.
Passive DUTs may be used as the unknown thru.
Noninsertable cal (mix connectors, transitions, F-F or M-M combinations) is just as easy as an insertable cal
Dave Blackham
& Ken Wong