Abstract:
A method for correcting reciprocity error in two port network measurements, in which an iterative algorithm is used to obtain an optimal symmetric matrix approximation to a measured 2×2 reciprocal network. The algorithm smooths measured data to obtain a low noise floor on measurements related to mode conversion in cables and connecting hardware. A geometric interpretation is used to approximate a measured data vector with the closest vector which satisfies the reciprocity constraint. An initial point in the constraint surface (reciprocity manifold) is located, and another point which is a better approximation of the measured data vector which satisfies the reciprocity constraint is generated. By repeating this process, the closest point in the manifold is quickly located. This closest point defines a reciprocal matrix which can be used to estimate the actual device under test (DUT) parameters.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention generally relates to the measurement of network parameters with a multiport network analyzer and, more particularly, to a method for correcting reciprocity error in two port network measurements. 
     2. Description of the Related Art 
     In designing cables and connecting hardware for use with high frequency signals, it is imperative to understand the electrical characteristics of the components used to link a device or cable under-test with the electronics used to transmit and receive data. All of the possible couplings between connectors and cables can be measured with multiport measurement equipment comprising a network analyzer and a switching network. The key to providing an accurate multiport measurement system is a two port calibration procedure which is used repeatedly on every combination of pairs of ports in the measurement system. 
     Errors associated with network analyzer measurements can result from either non-ideal conditions in the measurement equipment or from the effects of the mechanical fixtures, such as cables and connectors, which are used to connect a device under test (DUT) to the network analyzer or test bed. A simple way to model this error is to view the test bed and the DUT as an ideal network analyzer cascaded with an error network and followed by the DUT. Such a network representation is illustrated in FIG.  1 . The intervening network can then be used to account for the effects introduced by internal switching matrices connecting the various analyzer ports to the mechanical fixtures, as well as other sources of loss and distortion within the analyzer. The network can also be used to account for the effects of the fixtures themselves, which effects may introduce loss, crosstalk, and impedance discontinuities. Thus, the ideal network analyzer measures the cascaded network consisting of the intervening linear network followed by the DUT and obtains the measured S-parameters of the composite intervening network and DUT, here denoted as S M . (See FIG.  1 ). 
     During testing, the acquired data which is intended to characterize the DUT is oftentimes corrupted by imperfections in the measuring device. The data must therefore be processed to compensate for known errors when extracting the parameters which describe the DUT, here denoted as S A . An algorithm is used to relate the measured parameters S M  to the actual parameters S A . However, because the intervening network in the model is partially internal to the measurement equipment, S A  cannot be evaluated separately. The only way to characterize S A , therefore, is to measure various DUTs with known electrical characteristics and extract the appropriate parameters from S M . This process is further complicated when switching networks are involved because each change in the switch settings of the network analyzer results in a change in the intervening network due to differences in the signal routing. Thus, a different intervening network exists for each switch setting of the Network Analyzer, and each network must therefore be separately evaluated to compensate for measurement errors. 
     In general, the characterization of a two port reciprocal network results in a symmetric 2×2 S-parameter matrix. Measurements of this network should therefore show this symmetry but they are often corrupted by noise, resulting in asymmetrical data. This error, which is denoted as reciprocity error, affects the calculations performed on measured data which are intended to correct measurement error. Specifically, the reciprocity error directly affects the noise floor of the calculation of the scattering between modes of propagation in cables and connecting hardware. 
     It would therefore be advantageous to provide a method for correcting the reciprocity error in measured data obtained during the characterization of a 2×2 S-parameter matrix. 
     SUMMARY OF THE INVENTION 
     The above and other problems are overcome by a method for correcting reciprocity error in two port network measurements which is used to produce a matrix with an optimal symmetric approximation of a measured 2×2 reciprocal network. According to the invention, an iterative technique is used to smooth the measured data to obtain a low noise floor on measurements related to mode conversion in cables and connecting hardware. 
     The calibration procedure for network analyzer measurements relates measured data to actual system parameters. These relationships, combined with the requirement for matrix symmetry, define the explicit form of a reciprocity manifold. The closest point on the manifold surface, occurring when a normal to the surface points directly to the measured data point, is used in defining a reciprocal matrix which can be used to estimate the actual device under test (DUT) parameters. In one embodiment, the closest point to the measured data is determined utilizing a successive approximation technique which includes estimating an initial point on a reciprocity manifold, approximating the manifold as a linear surface, and moving along the surface until the normal points directly at the measured data. The data is then projected onto the reciprocity manifold to produce the next iteration in the process. 
     The method of the invention supplements and extends a conventional two port calibration algorithm to properly compensate for the reciprocity constraint of passive networks such as cables and connecting hardware. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing and other advantages and features of the invention will become more apparent from the detailed description of the preferred embodiments of the invention given below with reference to the accompanying drawings in which: 
     FIG. 1 is an illustration of a conventional error model for network analyzer measurements; 
     FIG. 2 is a graphical illustration of a reciprocity manifold; 
     FIG. 3 is a flow chart of a method for correcting reciprocity error in two port network measurements; and 
     FIG. 4 is an illustration of a 12 term error model for two port measurements. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     In making a 2 port measurement a number of calibration parameters and 4 raw measurements are needed to completely characterize a DUT. The information needed may be described as a multidimensional complex vector {right arrow over (v)}, and the actual S-parameter matrix S A  (see FIG. 1) can be determined as                S   A     =       [           s   11           s   12               s   21           s   22           ]     =     f        (     v   →     )                 (     Eq   .              1     )                                
     where s 11  is the input reflection parameter at port  1  of the network analyzer, s 12  is the reverse transmission parameter at port  2 , s 21  is the forward transmission parameter at port  2  and s 22  is the output reflection parameter at port  2 . 
     The function ƒ is defined by a method for a general intervening network, which is discussed in co-pending U.S. application Ser. No. 09/XXX,XXX, filed concurrently with this application (on XX, XX, 2000), and entitled “A Method for Calibrating Two Port High Frequency Measurements,” the entire contents of which is incorporated by reference herein. Briefly, the method for a general intervening network can be used to determine scattering parameters in the presence of an arbitrary intervening network in the forward and reverse directions. Equating s 21  and s 12  yields the scalar constraint which is used to obtain matrix symmetry 
     
       
           T ( {right arrow over (v)} )=0  (Eq. 2) 
       
     
     where {right arrow over (v)} defines the vector which includes the measured data for the DUT as well as the calibration coefficients. At the moment that T({right arrow over (v)}) is equal to zero, the matrix of Eq. 1 will be symmetric. 
     The occurrence of matrix symmetry may be obtained from a geometric interpretation whereby the region where T({right arrow over (v)})=0 defines an n−1 dimensional complex surface in an n dimensional complex space. Within this constraint surface, defined herein as the reciprocity manifold, the closest point to the measured data vector defines the best symmetric matrix approximation. This point is located where the ‘normal’ to the manifold surface points directly at the measured data vector. Where {right arrow over (P)} 0  is the optimal point on the surface and {right arrow over (P)} is the measured data point, the optimal condition can be represented as: 
     
       
           {right arrow over (P)}   0   −{right arrow over (P)}=μ∇T ( {right arrow over (P)}   0 )*  (Eq. 3) 
       
     
     where μ is a scalar constant. 
     For a 12 term error model (illustrated in the network of FIG.  4 ), there are 12 calibration parameters and 4 raw data measurements, indicating that the dimension of {right arrow over (v)} is 16. In this case, 
     
       
           T ( {right arrow over (v)} )= v   5   v   10 ( v   15   −v   8 )( v   13   −v   1 )− v   4   v   11   
       
     
     
       
         ( v   16   −v   7 )( v   14   −v   2 )+( v   15   −v   8 )( v   1   v   12   +v   3   v   10   −v   
       
     
     
       
           12   v   13 )+( v   14   −v   2 )( v   6   v   16   −v   6   v   7   −v   4   v   9 ).  (Eq. 4) 
       
     
     FIG. 2 is an illustration of a reciprocity manifold, where a manifold is a surface in a vector space. The reciprocity manifold is a surface where any point within the manifold possesses the imposed reciprocity constraints. As shown in FIG. 2, the measured data point {right arrow over (P)} will generally be on the surface defined by T({right arrow over (P)})=ε, where ε generally is small and represents the difference between the measured data point and the desired point on the reciprocity manifold due to the accuracy of the new “Algorithm for a General Intervening Network” 
     FIG. 3 is a flow chart of the new method for correcting reciprocity error in two port network measurements. First, via a software program residing on a PC, data point {right arrow over (P)} is determined by moving from the measured point in the direction of the normal to the surface T({right arrow over (v)})=ε until the surface T({right arrow over (v)})=0 is reached. (Step 1). 
     The point of intersection can be defined as: 
     
       
           T ( {right arrow over (P)}+λ∇T ( {right arrow over (P)} )*)=0  (Eq. 5) 
       
     
     This expression can be evaluated via a Taylor expansion of T({right arrow over (v)}) derived from Eq. 5 which provides that 
     
       
           T ( {right arrow over (P)} )+λ|∇ T ( {right arrow over (P)} )| 2 ≈0  (Eq. 6) 
       
     
     where              λ   ≈     -                  T        (     P   →     )                ∇     T        (     P   →     )              2                 (     Eq   .              7     )                                
     {right arrow over (P)}+λΔT({right arrow over (P)})* in Eq. 5 represents a straight line from the measured data point {right arrow over (P)} in the direction of the normal on the surface T({right arrow over (P)})=ε. 
     The starting point for the iteration is the projection of {right arrow over (P)} 1  onto the reciprocity manifold, where {right arrow over (P)} 1  is given by the relationship                  P   →     1     ≡       P   →     -                      T        (     P   →     )       ′              ∇     T        (     P   →     )              2            ∇     T        (     P   →     )                     (     Eq   .              8     )                                
     This equation can be obtained by substituting Eq. 6 for λ in Eq. 5. This projection is achieved by modifying one of the components of {right arrow over (P)} 1  to force T({right arrow over (P)} 1 ) to be identically zero. 
     FIG. 4 is an illustration of a 12 term error model for two port network measurements wherein an error term is associated with each connection path in the network. Application of the method for the general intervening network to the network shown in FIG. 4 yields 12 unknown quantities from the intervening network and 4 uncalibrated measurements of the DUT. These 16 quantities completely define all of the information needed to compute the actual S-parameters of the DUT, and are expressed in vector form as:                v   s     =       [                      e   00           e   30             e   10          e   01               e   01       e   32             e   11               e   01          e   22         e   32             p   33           p   03             p   23          p   32               p   32       p   01             p   22               p   11          p   32         p   01             sm   11           sm   21           sm   12           sm   22           ]     .             (     Eq   .              9     )                                
     where each e in Eq. 9 represents an appropriate component in a S-parameter characterization of the intervening network in the forward direction, each p represents the appropriate component in the S-parameter characterization of the intervening network in the reverse direction, and sm 11 , sm 21 , sm 12 , and sm 22  are the measured input reflection at port  1  of the network analyzer, the measured forward transmission at port  2 , the measured reverse transmission at port  2  and the measured output reflection at port  2 , respectively. 
     For the 12 term error model, the projection is achieved by modifying the first component of {right arrow over (P)} 1  which is denoted as v 1 , where                v   1     =       v   13     +         v   3          v   10             v   5          v   10       -     v   12         +         v   4            v   9          (       v   14     -     v   2       )             (       v   8     -     v   15       )          (         v   5          v   10       -     v   12       )         +         (       v   16     -     v   7       )          (       v   14     -     v   2       )          (         v   4          v   11       -     v   6       )           (       v   8     -     v   15       )          (         v   5          v   10       -     v   12       )                   (     Eq   .              10     )                                
     and v 1  through v 12  represent the 12 calibration coefficients and v 13  through v 16  represent the four the raw data points sm 11 , sm 12 , sm 21 , sm 22  of Eq. 9. This projection {right arrow over (Q)} 1  of {right arrow over (P)} 1  may be represented by the functional relationship {right arrow over (Q)} 1 =proj({right arrow over (P)} 1 ). 
     Next, given the n th  iterant {right arrow over (Q)} n  (where {right arrow over (Q)} 1  is the starting point), the next iterant {right arrow over (Q)} n+1  is computed (step 2), according to the relationship 
     
       
           {right arrow over (P)}   n+1   ={right arrow over (P)} +( {right arrow over (Q)}   n   −{right arrow over (P)} )· {right arrow over (u)} ( {right arrow over (Q)}   n ) {right arrow over (u)} ( {right arrow over (Q)}   n )  (Eq. 11) 
       
     
     where                  u   →          (       Q   →     n     )       ≡         ∇       T        (       Q   →     n     )       *                ∇   T          (       Q   →     n     )              .             (     Eq   .              12     )                                
     These equations are derived by approximating T({right arrow over (v)}) as a linear surface which passes through {right arrow over (Q)} n . In Eq. 11, {right arrow over (P)} n+1  is the point on the linear surface having a normal which points directly to {right arrow over (P)} 1 . {right arrow over (P)} n+1  is subsequently projected onto the reciprocity manifold. This is accomplished by modifying the first component of {right arrow over (P)} n+1  in accordance with Eq. 9. As a result of the modification, the projected point becomes equal to {right arrow over (Q)} n+1 . 
     Subsequently, convergence is checked by examining the cosine of the angle between {right arrow over (Q)} n −{right arrow over (P)} and ∇T({right arrow over (Q)} n )*. (Step 3) This is achieved by computing the inner product of                      Q   →     n     -     P   →                  Q   →     n     -     P   →              ·       ∇       T        (       Q   →     n     )       *                ∇   T            (       P   →     n     )     *                      (     Eq   .              13     )                                
     If this quantity is sufficiently close to 1, the process is terminated. Otherwise, a return to step 2 occurs and the process is repeated. A value of 0.9999999999, for example, has been achieved in less than five iterations and provides acceptable results. However, the required closeness to 1 depends on the desired level of accuracy which is dependent to some extent on the frequency being evaluated. 
     S-parameter measurements of reciprocal networks using a network analyzer result in symmetric matrices. Errors associated with the network analyzer measurements can result from either non-ideal conditions in the measurement equipment, resulting in a non-symmetrical matrix. Using the method according to the invention, a best symmetric approximation is achieved via processing on a PC. 
     Although the invention has been described and illustrated in detail, it is to be clearly understood that the same is by way of illustration and example, and is not to be taken by way of limitation. The spirit and scope of the present invention are to be limited only by the terms of the appended claims.