Abstract:
In one embodiment, a method of calibrating a multi-port vector network analyzer (VNA) includes (i) performing two-port calibrations on pairs of ports to determine forward and reverse systematic error terms associated with each pair of ports, wherein the pairs of ports are selected such that each port&#39;s systematic error terms (directivity, source match, reflection tracking, and load match) are determined, (ii) generating a switch error correction matrix using data from the two-port calibrations, and (iii) performing unknown thru calibration for at least one pair of ports that was not utilized in step (i), wherein the unknown thru calibration comprises applying the switch error correction matrix to measurement data and determining transmission tracking error terms using the corrected measurement data.

Description:
TECHNICAL FIELD 
   The present application is generally related to calibration of a vector network analyzer (VNA). 
   BACKGROUND 
     FIG. 1  depicts VNA  100  according to a conventional design. As shown in  FIG. 1 , VNA  100  comprises switch  101  to switch between VNA ports A through D to establish a path to a single reference receiver  102 . When the RF signal path switch  101  changes position, the termination of the test port also changes. The change in termination causes the source match term to be different from the load match term. The difference is referred to as the “switch error.” The standard twelve term VNA error model derives the load match term from the through connection. Additionally, N port VNA calibration methods require a minimum of N−1 paths to be characterized. For a four-port VNA, three through paths are characterized through calibration procedures. However, most N-port devices use connector combinations that do not allow the use of a flush through connection required in typical short, open, load, through (SOLT) calibration method, Accordingly, adapter removal calibration methods are usually employed. This requires another extra calibration step. 
   The “unknown thru” calibration is ideally suitable to calibrate VNAs to test devices with non mate-able connector combinations. Like the TRL family of calibrations, it is based on the eight term error model and requires two receivers for each test port to obtain the necessary data to determine the VNA&#39;s systematic error terms. Accordingly, the usual implementation of the unknown thru calibration method cannot be applied to VNA  100 . For VNA&#39;s that only possess a single reference receiver, calibration can be unduly time consuming. 
   SUMMARY 
   Some representative embodiments are directed to a method of calibration of a VNA having a minimum of one reference receiver. When the VNA possesses an even number of ports (“N” ports), two-port calibration is applied to N/2 pairs of ports using ECal modules or other suitable configurable standards. When the VNA possesses an odd number of ports, the number of port pairs is [(N+1)/2]. The selection of the pair of ports occurs such that the directivity, source match, reflection tracking, and load match of each port, plus forward transmission tracking and reverse transmission tracking of each port pair are determined. After application of the two port calibrations, the unknown thru calibration method is applied to a minimum of (N/2−1) pairs of ports for even number of ports and a minimum of [(N−1)/2−1] for odd number of ports. For the unknown thru calibration, the pairs of ports are selected such that the pairs are different from the pairs used during the two port calibration. From the previously calculated load match terms, switch error terms can be calculated and applied to the measurement data generated by the unknown thru calibration. The transmission tracking error terms associated with the unknown through path(s) are then determined. The remaining (N(N−1)/2) transmission tracking error terms are calculated. Calibration of the multi-port VNA in this manner may occur in substantially less time than known calibration methods. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  depicts a conventional single reference receiver VNA. 
       FIG. 2  depicts a one-port VNA error model. 
       FIG. 3  depicts the “forward” half of the twelve term error model. 
       FIG. 4  depicts a signal flow graph associated with the eight term error model. 
       FIG. 5  depicts the matrix representation of the eight term error model. 
       FIG. 6  depicts the load match and transmission tracking model. 
       FIG. 7  depicts a flowchart for calibration of a multi-port VNA according to one representative embodiment. 
       FIG. 8  depicts a set-up for calibration of a VNA according to one representative embodiment. 
       FIG. 9  depicts a flowchart for calibration procedures using the set-up shown in  FIG. 8  according to one representative embodiment. 
       FIG. 10  depicts a VNA according to one representative embodiment. 
   

   DETAILED DESCRIPTION 
   To assist the reader&#39;s understanding of calibration of VNAs according to some representative embodiments, the following mathematical discussion is provided. In  FIG. 2 , one-port VNA error model  200  is shown. The error terms include directivity (E D ), source match (E S ) and reflection tracking (E R ). In matrix equation form, the error model is as follows: 
               [           E   D               E   S                   E   S     ⁢     E   D       -     E   R             ]     =         [         1             Γ   m1     ⁢     Γ   a1       -     Γ   a1               1             Γ   m2     ⁢     Γ   a2       -     Γ   a2               1             Γ   m3     ⁢     Γ   a3       -     Γ   a3             ]       -   1       ⁡     [           Γ   m1               Γ   m2               Γ   m3           ]               (   1   )             
 
   The twelve term error model is an extension of the one-port error model that includes four additional transmission error terms plus two cross talk terms.  FIG. 3  depicts the “forward” half of twelve term the error model  300 . The “reverse” half is the same except that S 11  is replaced by S 22  and S 21  is replaced by S 12 . The load match (E L ) and transmission tracking (E T ) terms are determined as follows: 
               E   L     =       (       S     11   ⁢   m       -     E   D       )       [         E   S     ⁡     (       S     11   ⁢   m       -     E   D       )       +     E   R       ]               (   2   )                 E   T     =         S     21   ⁢   m       ⁡     (     1   -       E   S     ⁢     E   L         )       -     E   x               (   3   )             
 
   A VNA measurement system may also be represented by the eight term error model. Signal flow graph  400  associated with the eight term error model is shown in  FIG. 4 . The matrix solution  500  is shown in  FIG. 5 . The terms “a” and “b” are used to differentiate between the applied and generated signals. The numerical subscripts refer to respective ports. The subscript “m” (a 1m , a 2m , b 2m , and b 1m ) is used to indicate the measurements made by the VNA. Matrices [A], [T], and [B] are cascade matrices where matrices [A] and [B] represent the error terms and the [T] matrix represents the S-parameters. The matrices can be represented as follows: 
                 [     T   mc     ]     =         [   A   ]     *     [   T   ]     *     [   B   ]       =       (     β   α     )     ⁢       (     1     E   RR       )     ⁡     [           Δ   A           E   DF               -     E   SF           1         ]       *     [   T   ]     *     [           Δ   B           E   SR               -     E   DR           1         ]           ⁢     
     ⁢       T   mc     =         1     S21   mc       ⁡     [           -     Δ   Sm             S11   mc               -     S22   mc           1         ]       ⁢           ⁢   where       ⁢           ⁢     
     ⁢       Δ   Sm     =         S11   mc     ⁢     S22   mc       -       S21   mc     ⁢     S12   mc           ⁢     
     ⁢       Δ   A     =         E   RF     -       E   DF     ⁢     E   SF     ⁢           ⁢     Δ   B         =       E   RR     -       E   DR     ⁢     E   SR                     (   4   )             
 
where [T mc ] is a matrix containing the switch error corrected measurement data. If each port has two receivers, four measurements can be taken in the forward and reverse direction and the switch error can be formulated as follows: 
                     [     S   m     ]     =       [             b     1   ⁢   mf         a     1   ⁢   mf                 b     1   ⁢   mr         a     2   ⁢   mr                     b     2   ⁢   mf         a     1   ⁢   mf                 b     2   ⁢   mr         a     2   ⁢   mr               ]     =         [           S     11   ⁢   m             S     12   ⁢   m                 S     21   ⁢   m             S     22   ⁢   m             ]     ⁢     
     [     S   mc     ]     =     [           S     11   ⁢   mc             S     12   ⁢   mc                 S     21   ⁢   mc             S     22   ⁢   mc             ]           ;     ⁢           [     M   sc     ]     =       [         1           a     1   ⁢   mr         a     2   ⁢   mr                     a     2   ⁢   mf         a     1   ⁢   mf             1         ]     =         [         1         L   r               L   f         1         ]     ⁢     
     [     S   m     ]     =         [     S   mc     ]     *     [     M   sc     ]     ⁢           ⁢     and   ⁢           [     S   mc     ]       =       [     S   m     ]     *       [     M   sc     ]       -   1                       (   5   )             
 
   By applying a one port calibration method to port 1 and a one port calibration method to port 2, six of the error terms (E DF , E RF , E SF , E SR , E RR , and E DR ) can be determined. The α and β terms remain to be determined. 
   By connecting a thru between port-1 and port-2 and measuring the S-parameters, four more equations are defined. After correcting the measured data for switch errors, equation (4) is defined. If S 21 , and S 12  of the thru are equal, the determinant of [T] is equal to one and the determinants of equation (4) can be simplified as follows: 
                      T   mc          =            A        *        T        *        B          =       k   2     ⁢          A   ′          *          B   ′                ⁢     
     ⁢                 k   =       (     β   α     )     ⁢     (     1     E   RR       )         ;     ⁢           [     A   ′     ]     =     [           Δ   A           E   DF               -     E   SF           1         ]       ;     ⁢           [     B   ′     ]     =     [           Δ   B           E   SR               -     E   DR           1         ]               (   6   )                          T   mc          =       S     12   ⁢   mc         S     21   ⁢   mc           ;           ⁢            A   ′          =     E   RF       ;           ⁢            B   ′          =     E   RR         ⁢     
     ⁢         k   2     =              T   mc                   A   ′          *          B   ′              =       S     12   ⁢   mc           S     21   ⁢   mc       ⁢     E   RF     ⁢     E   RR             ,           ⁢     k   =     ±         S     12   ⁢   mc           S     21   ⁢   mc       ⁢     E   RF     ⁢     E   RR                         (   7   )             
 
   In order to determined the correct root of K, some knowledge of the phase or electrical length of the thru is typically employed. After calculating the correct value of K, the value α/β can be determined as is known in the art. Additionally, load match and transmission tracking model  600  is shown in  FIG. 6 . The load match and transmission tracking error terms can then be calculated: 
                 E   LF     =       E   SR     +         E   RR     ⁢     Γ   f         1   -       E   DR     ⁢     Γ   f               ,       E   LR     =       E   SF     +         E   RF     ⁢     Γ   f         1   -       E   DF     ⁢     Γ   r                       (   8   )                   E   TF     =       (     α   β     )     ⁢     (       E   RR       1   -       E   DR     ⁢     Γ   f           )         ,       E   TR     =       (     β   α     )     ⁢     (       E   RF       1   -       E   DF     ⁢     Γ   r           )                 (   9   )             
 
   As previously discussed, the switch error correction terms can be readily calculated when the VNA includes two receivers per port. However, using conventional techniques, the single reference receiver design imposes a significant complication to calibration of a VNA using the 8-term error model. Some representative embodiments overcome this difficulty by the realization that the preceding mathematical derivation can be employed in “reverse” order. Specifically, if the load match term (E L ) has been determined and the other 1-port error terms are known (E D , E S  and E R ) are known, the switch error terms (Γ f  and Γ r ) can be calculated. The switch error terms are unique to each port (provided that only one switch is involved) as given by: 
                 Γ   f     =       (       E   LF     -     E   SR       )         E   RR     +       E   DR     ⁡     (       E   LF     -     E   SR       )             ,       Γ   r     =       (       E   LR     -     E   SF       )         E   RF     +       E   DF     ⁡     (       E   LR     -     E   SF       )                     (   10   )             
 
   The switch error terms can be used to correct the switch error in the measurement data as follows: 
                     [     S   mc     ]     =         [     S   m     ]     ⁡     [     M   SC     ]         -   1         ;     ⁢           [     M   SC     ]     =     [         1         Γ   r               Γ   f         1         ]             (   11   )             
 
     FIG. 7  depicts a flowchart for calibration of a multi-port VNA by calculating switch error correction terms using the preceding mathematical analysis of VNA error models. In step  701 , respective two-port calibrations are performed for a subset of the ports. The two-port calibrations determine the load match terms and the other one port error terms. The pairs of ports for the two-port calibrations are selected such that all the systematic errors of each port are determined. In step  702 , the switch error terms are calculated and one or several switch error correction matrices are generated depending upon the number of ports of the VNA. 
   In step  703 , one or several unknown thru calibrations are applied to pairs of ports of the VNA depending upon the number of ports of the VNA. Each pair of ports selected for the unknown thru calibration(s) is different than the pairs of ports selected for the two-port calibrations. Also, the total number of pairs of ports including both two-port and unknown thru calibrations should equal N−1, where N represents the number of ports of the VNA. In step  704 , the switch error correction matrix/matrices are applied to the measurement data generated by the unknown thru calibration(s). In step  705 , the forward and reverse transmission tracking errors for the unknown through path(s) are determined using the corrected measurement data. After step  705 , the transmission tracking error terms have been determined for N−1 ports. In step  706 , the remaining [N(N−1)/2] transmission tracking error terms are determined from the calculated transmission tracking error using known methods. 
   Reference is now made to system  800  of  FIG. 8  to illustrate application of the calibration process to four port VNA  801  according to one representative embodiment. System  800  includes ECal module  801  that is an module that electronically steps through a plurality of impedance states (such as short, open, load, and thru) to facilitate a suitable calibration method. Such modules are commercially available from Agilent Technologies, Inc. ECal module  801  is adapted to couple to the connector types associated with ports A and B. ECal module  802  is similar to ECal module  801  except ECal module  802  is adapted to couple to the connector types associated with ports C and D. Adapter BC  803  is essentially a line between connectors adapted to couple to ports B and C. In many cases, a mechanical calibration kit can be substituted for the ECAL modules. 
   A flowchart for calibrating VNA  701  is shown in  FIG. 9 . In step  901 , a two-port calibration is performed on ports A and B using ECal module  801 . The following error terms are calculated using the two-port calibration: E DA , E SA , E RA , E DB , E SB , E RB , E LA , E LB , E TAB , E TBA , Γ AB , Γ BA . In step  902 , a two-port calibration is performed on ports C and D using ECal module  802 . The following error terms are calculated using the two-port calibration: E DC , E SC , E RC , E DD , E SD , E RD , E LC , E LD , E TCD , E TDC , Γ CD , Γ DC.    
   In step  904 , error terms E TAC  and E TCA  are determined from error terms E TAB , E TBA , E TBC , and E TCB . In step  905 , error terms E TBD  and E TDB  are determined from error terms E TBC , E TCB , E TCD , and E TDC . In step  906 , error terms E TAD  and E TDA  are determined from error terms E TAB , E TBA , E TBD , and E TDB . All transmission path errors of VNA  701  are calculated upon completion of step  905 . 
     FIG. 10  depicts VNA  1000  according to one representative embodiment. In one embodiment, VNA  1000  implements the operations shown in  FIG. 7 . The operations can be implemented using any suitable logic such code or software instructions and a suitable processor. Alternatively, the operations can be implemented using integrated circuitry. VNA  1000  comprises 2-port calibration logic  1001  that determines each port&#39;s systematic errors. VNA  1000  further comprises switch error correction calculation logic  1002  that generates switch error correction matrices from the systematic error terms. Unknown error correction logic  1003  calculates transmission tracking error correction terms for a subset of ports. 
   By appropriately selecting ports and by utilizing the load match terms to calculate a switch correction matrix, some representative embodiments enable the calibration of a VNA to occur more efficiently. A fewer number of ECal modules are used than would otherwise be required by known methodologies. Additionally, for VNAs with two reference receivers for each test port, (provided that the reference receivers measure the a 3   f  and a 0   r  terms correctly) the unknown thru calibration method can be used to perform 2-port calibrations on all the port pairs. For non mate-able ports, this method eliminates the need to perform adapter removal calibration. The omission of adapter removal calibration saves a substantial amount of time and reduces calibration error.