Abstract:
A method for determining scattering parameters of a device under test using a real-time oscilloscope. The method includes calculating a reflection coefficient of each port of a device under test with N ports, wherein N is greater than one, based on a first voltage measured by the real-time oscilloscope when a signal is generated from a signal generator. The method also includes determining an insertion loss coefficient of each port of the device under test, including calculating the insertion loss coefficient of the port of the device under test to be measured based on a second voltage measured by the real-time oscilloscope when a signal is generated from a signal generator.

Description:
BENEFIT 
       [0001]    This application claims benefit of U.S. Provisional Application No. 61/026,434, filed Jul. 18, 2014, titled S-PARAMETER MEASUREMENTS USING REAL-TIME OSCILLOSCOPES, which is incorporated herein by reference in its entirety. 
     
    
     TECHNICAL FIELD 
       [0002]    This disclosure relates to a method for measuring complete two-port and multi-port scattering parameters (S-parameters) of a device under test (DUT) using a real-time oscilloscope, a power divider and a signal source. 
       BACKGROUND 
       [0003]    Traditionally a vector network analyzer or a time-domain reflectometer (TDR) system with a sampling oscilloscope has been required to obtain S-parameter measurements for characterizations of a device under test (DUT). If a fixture is used to measure the S-parameters, then once the S-parameters of the fixture have been measured and the S-parameters of the DUT plus the fixture have been measured, a full de-embed operation can be performed to obtain only characteristics of the DUT by taking out the errors due to the fixture. However, vector network analyzers and TDR systems are expensive. 
         [0004]    As bit rate goes higher, high speed serial data link simulation and measurements require using S-parameters of the DUT to characterize the components within the simulated link for embedding and de-embedding operations. For example, as seen in  FIG. 1 , the output impedance of the transmitter  100 , the input impedance of the receiver  102 , and the full S-parameters of the channel  104  are all needed to fully characterize and simulate the link so that accurate test and measurements of the DUT may be made. 
         [0005]    Real-time oscilloscopes are widely used to do high speed serial data link debugging, testing, and measurements. It is desired to use the real-time oscilloscopes to also measure the S-parameters of the DUT and then use the measured S-parameters for other measurements and simulations, without having to use multiple instruments. 
         [0006]    Embodiments of the disclosed technology address these and other limitations in the prior art. 
       SUMMARY 
       [0007]    Certain embodiments of the disclosed technology include a method for determining scattering parameters of a device under test using a real-time oscilloscope. The method includes determining a reflection coefficient of each port of a device under test with N ports, wherein N is an integer greater than or equal to one, including terminating the ports of the device under test not being measured with a resistor, and calculating the reflection coefficient of the port of the device under test to be measured based on a first voltage measured by the real-time oscilloscope when a signal is generated from a signal generator. The method also includes determining an insertion loss and/or crosstalk coefficient between two different ports of the device under test, including calculating the insertion loss and/or a crosstalk between the ports of the device under test to be measured based on a second voltage measured by the real-time oscilloscope when a signal is generated from a signal generator. 
         [0008]    Certain embodiments of the disclosed technology also include a system, comprising a power divider, a signal generator in communication with the power divider, the real-time oscilloscope and/or a device under test, the signal generator configured to generate a signal, and a real-time oscilloscope in communication with the signal generator, the power divider and the device under test. The real-time oscilloscope is configured to calculate a reflection coefficient of each port of a device under test with N ports, wherein N is greater than or equal to one, measured based on a first voltage measured by the real-time oscilloscope when a signal is generated from a signal generator, and calculating an insertion loss and/or a crosstalk between the ports of the device under test to be measured based on a second voltage measured by the real-time oscilloscope when a signal is generated from a signal generator. A synchronized trigger between the signal generator and the real-time oscilloscope provides an absolute time reference for return loss coefficient, insertion loss and cross talk measurements. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]      FIG. 1  illustrates a serial data link system. 
           [0010]      FIG. 2  is a block diagram of a return loss coefficient measurement system according to embodiments of the disclosed technology. 
           [0011]      FIG. 3  is a signal flow diagram of the system shown in  FIG. 2 . 
           [0012]      FIG. 4  is a block diagram of an insertion loss measurement system according to embodiments of the disclosed technology. 
           [0013]      FIG. 5  is a signal flow diagram of the system shown in  FIG. 4 . 
           [0014]      FIG. 6  is a power divider of  FIG. 2 . 
       
    
    
     DETAILED DESCRIPTION 
       [0015]    In the drawings, which are not necessarily to scale, like or corresponding elements of the disclosed systems and methods are denoted by the same reference numerals. 
         [0016]    The disclosed technology measures S-parameters of a DUT  206  using a real-time oscilloscope  200 , a signal generator  202 , and a power divider  204 , as shown in  FIG. 2 . A two-port DUT  206  has four S-parameters: reflection coefficients s 11   DUT , s 22   DUT  and insertion loss and/or crosstalk terms s 12   DUT , s 21   DUT . 
         [0017]    In the setup shown in  FIG. 2 , the synchronized trigger  208  between the signal generator  202  and the real-time oscilloscope  200  provides an absolute time reference for the reflection coefficients and the insertion loss and/or crosstalk coefficients. This synchronizing action establishes a zero time or zero phase reference point for the acquisition system of the oscilloscope  200 . Signal generator  202  may either output sweep sine signals or fast step signals that may cover a wide bandwidth. For example, a fast step signal can cover more than 50 GHz bandwidth. For swept sine signals, zero phase is the reference point. For fast step signals, zero time is the reference point. 
         [0018]    Power divider  204  may be any type of power divider or splitter. For example, power divider  204  may be a power divider having three 16⅔ Ohm resistors at three branches creating a three way power network, as seen in  FIG. 6 . 
         [0019]      FIG. 3  illustrates a signal flow diagram of the system shown in  FIG. 2 . 
         [0020]    The power divider  204 , as seen in the signal flow diagram of  FIG. 3 , is a three-port network and characterized by a three port S-parameters data set: 
         [0000]    
       
         
           
             
               
                 
                   s 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             s 
                             11 
                           
                         
                         
                           
                             s 
                             12 
                           
                         
                         
                           
                             s 
                             13 
                           
                         
                       
                       
                         
                           
                             s 
                             21 
                           
                         
                         
                           
                             s 
                             22 
                           
                         
                         
                           
                             s 
                             23 
                           
                         
                       
                       
                         
                           
                             s 
                             31 
                           
                         
                         
                           
                             s 
                             32 
                           
                         
                         
                           
                             s 
                             33 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0021]    The S-parameters of the power divider  204  may be measured ahead of time and saved in a memory (not shown) of the real-time oscilloscope  200 . The DUT  206  is modeled as a two port S-parameters data set s DUT . 
         [0022]    To measure the reflection coefficient s 11   DUT  of port one  210  of the DUT  206 , port two  212  of the DUT  206  is terminated with an ideal 50 Ohm resistor  214 . The signal generator  202  is modeled with its ideal voltage ν s  and has the reflection coefficient s 22   ss  at the output port. The real-time oscilloscope  200  has the reflection coefficient s 11   scope  at its input port. 
         [0023]    As the power divider is a three-port network, with port two being terminated by DUT  206 , a two-port system can be derived from port one and port three of the power divider  204  as follows: 
         [0000]    
       
         
           
             
               
                 ( 
                 2 
                 ) 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     
                       b 
                       1 
                     
                   
                 
                 
                   
                     
                       b 
                       2 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 { 
                 
                   
                     [ 
                     
                       
                         
                           
                             s 
                             11 
                           
                         
                         
                           
                             s 
                             12 
                           
                         
                       
                       
                         
                           
                             s 
                             21 
                           
                         
                         
                           
                             s 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             
                               s 
                               13 
                             
                           
                         
                         
                           
                             
                               s 
                               23 
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       
                         
                           s 
                           11 
                           DUT 
                         
                          
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 s 
                                 33 
                               
                                
                               
                                 s 
                                 11 
                                 DUT 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 s 
                                 31 
                               
                             
                             
                               
                                 s 
                                 32 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
                 } 
               
                
               
                 [ 
                 
                   
                     
                       
                         a 
                         1 
                       
                     
                   
                   
                     
                       
                         a 
                         2 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0000]    Let equation (3): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           s 
                           ~ 
                         
                         = 
                           
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     s 
                                     ~ 
                                   
                                   11 
                                 
                               
                               
                                 
                                   
                                     s 
                                     ~ 
                                   
                                   12 
                                 
                               
                             
                             
                               
                                 
                                   
                                     s 
                                     ~ 
                                   
                                   21 
                                 
                               
                               
                                 
                                   
                                     s 
                                     ~ 
                                   
                                   22 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     s 
                                     11 
                                   
                                 
                                 
                                   
                                     s 
                                     12 
                                   
                                 
                               
                               
                                 
                                   
                                     s 
                                     21 
                                   
                                 
                                 
                                   
                                     s 
                                     22 
                                   
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       s 
                                       13 
                                     
                                   
                                 
                                 
                                   
                                     
                                       s 
                                       23 
                                     
                                   
                                 
                               
                               ] 
                             
                              
                             
                               
                                 
                                   
                                     s 
                                     11 
                                     DUT 
                                   
                                    
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         
                                           s 
                                           33 
                                         
                                          
                                         
                                           s 
                                           11 
                                           DUT 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         s 
                                         31 
                                       
                                     
                                     
                                       
                                         s 
                                         32 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    denote equation (4): 
         [0000]        {tilde over (s)}   11   DUT   =s   11   DUT [1− s   33   s   11   DUT ] −1   (4)
 
         [0000]    Then equation (3) can be re-written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               s 
                               ~ 
                             
                             11 
                           
                         
                         
                           
                             
                               s 
                               ~ 
                             
                             12 
                           
                         
                       
                       
                         
                           
                             
                               s 
                               ~ 
                             
                             21 
                           
                         
                         
                           
                             
                               s 
                               ~ 
                             
                             22 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               s 
                               11 
                             
                             + 
                             
                               
                                 s 
                                 13 
                               
                                
                               
                                 
                                   s 
                                   ~ 
                                 
                                 11 
                                 DUT 
                               
                                
                               
                                 s 
                                 31 
                               
                             
                           
                         
                         
                           
                             
                               s 
                               12 
                             
                             + 
                             
                               
                                 s 
                                 13 
                               
                                
                               
                                 
                                   s 
                                   ~ 
                                 
                                 11 
                                 DUT 
                               
                                
                               
                                 s 
                                 32 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               s 
                               21 
                             
                             + 
                             
                               
                                 s 
                                 23 
                               
                                
                               
                                 
                                   s 
                                   ~ 
                                 
                                 11 
                                 DUT 
                               
                                
                               
                                 s 
                                 31 
                               
                             
                           
                         
                         
                           
                             
                               s 
                               22 
                             
                             + 
                             
                               
                                 s 
                                 23 
                               
                                
                               
                                 
                                   s 
                                   ~ 
                                 
                                 11 
                                 DUT 
                               
                                
                               
                                 s 
                                 32 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0024]    From equation (2), the transfer function from the voltage source of the signal generator  202  to the input to the real-time oscilloscope can be derived as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       b 
                       2 
                     
                     
                       v 
                       s 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     · 
                     
                       
                         1 
                         - 
                         
                           s 
                           22 
                           ss 
                         
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   s 
                                   22 
                                   ss 
                                 
                                  
                                 
                                   
                                     s 
                                     ~ 
                                   
                                   11 
                                 
                               
                             
                             ) 
                           
                            
                           
                             
                               
                                 s 
                                 ~ 
                               
                               21 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       s 
                                       ~ 
                                     
                                     22 
                                   
                                    
                                   
                                     s 
                                     11 
                                     scope 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           
                             s 
                             22 
                             ss 
                           
                            
                           
                             
                               s 
                               ~ 
                             
                             12 
                           
                            
                           
                             s 
                             11 
                             scope 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The terms of equation (5) may be inserted into equation (6), and equation (6) can be rewritten as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       b 
                       2 
                     
                     
                       v 
                       s 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     · 
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               s 
                               22 
                               ss 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               s 
                               21 
                             
                             + 
                             
                               
                                 s 
                                 23 
                               
                                
                               
                                 
                                   s 
                                   ~ 
                                 
                                 11 
                                 DUT 
                               
                                
                               
                                 s 
                                 31 
                               
                             
                           
                           ) 
                         
                       
                       
                         
                           
                             
                               1 
                               - 
                               
                                 
                                   s 
                                   22 
                                   ss 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       s 
                                       11 
                                     
                                     + 
                                     
                                       
                                         s 
                                         12 
                                       
                                        
                                       
                                         
                                           s 
                                           ~ 
                                         
                                         11 
                                         DUT 
                                       
                                        
                                       
                                         s 
                                         31 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       s 
                                       22 
                                     
                                     + 
                                     
                                       
                                         s 
                                         23 
                                       
                                        
                                       
                                         
                                           s 
                                           ~ 
                                         
                                         11 
                                         DUT 
                                       
                                        
                                       
                                         s 
                                         32 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   s 
                                   11 
                                   scope 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 s 
                                                 11 
                                               
                                                
                                               
                                                 s 
                                                 22 
                                               
                                             
                                             - 
                                             
                                               
                                                 s 
                                                 21 
                                               
                                                
                                               
                                                 s 
                                                 12 
                                               
                                             
                                           
                                           ) 
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 s 
                                                 13 
                                               
                                                
                                               
                                                 s 
                                                 31 
                                               
                                                
                                               
                                                 s 
                                                 22 
                                               
                                             
                                             + 
                                             
                                               
                                                 s 
                                                 23 
                                               
                                                
                                               
                                                 s 
                                                 32 
                                               
                                                
                                               
                                                 s 
                                                 11 
                                               
                                             
                                             - 
                                             
                                               
                                                 s 
                                                 23 
                                               
                                                
                                               
                                                 s 
                                                 31 
                                               
                                                
                                               
                                                 s 
                                                 12 
                                               
                                             
                                             + 
                                             
                                               
                                                 s 
                                                 13 
                                               
                                                
                                               
                                                 s 
                                                 32 
                                               
                                                
                                               
                                                 s 
                                                 21 
                                               
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           
                                             s 
                                             ~ 
                                           
                                           11 
                                           DUT 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                                
                               
                                 s 
                                 22 
                                 ss 
                               
                                
                               
                                 s 
                                 11 
                                 scope 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0025]    In equation (7), ν s  is the voltage source value of the signal generator  202  and may be obtained during a calibration process and saved in the memory of the real-time oscilloscope  200 . b 2  is the voltage acquired by the oscilloscope  200  when ν s  is generated, so it is known. As discussed above, all the S-parameter terms for the power divider  204  may be measured ahead of time, and stored in the memory of the oscilloscope  200 . s 22   ss , corresponding to the signal source output impedance, may also be measured ahead of time and stored in memory. s 11   scope , corresponding to the real-time oscilloscope  200  input impedance, may be measured ahead of time and stored in the memory as well. The only unknown, therefore, is {tilde over (s)} 11   DUT  which may be computed from equation (7). Once {tilde over (s)} 11   DUT  is obtained from equation (7), s 11   DUT , the reflection coefficient of the DUT  206  may be calculated using equation (4) above. 
         [0026]    Equation (7) may be simplified with some assumptions. First, the S-parameters of an ideal power divider is: 
         [0000]    
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               s 
                               11 
                             
                           
                           
                             
                               s 
                               12 
                             
                           
                           
                             
                               s 
                               13 
                             
                           
                         
                         
                           
                             
                               s 
                               21 
                             
                           
                           
                             
                               s 
                               22 
                             
                           
                           
                             
                               s 
                               23 
                             
                           
                         
                         
                           
                             
                               s 
                               31 
                             
                           
                           
                             
                               s 
                               32 
                             
                           
                           
                             
                               s 
                               33 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             0 
                           
                           
                             
                               1 
                               2 
                             
                           
                           
                             
                               1 
                               2 
                             
                           
                         
                         
                           
                             
                               1 
                               2 
                             
                           
                           
                             0 
                           
                           
                             
                               1 
                               2 
                             
                           
                         
                         
                           
                             
                               1 
                               2 
                             
                           
                           
                             
                               1 
                               2 
                             
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    For an ideal signal generator and an ideal real-time oscilloscope, the signal source impedance and oscilloscope input impedance are zero: 
         [0000]      s 22   ss =0, s 11   scope =0.  (9)
 
         [0027]    Assuming the S-parameters of the power divider  204 , signal generator  202 , and real-time oscilloscope  200  are their ideal models, then equation (7) can be simplified as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       b 
                       2 
                     
                     
                       v 
                       s 
                     
                   
                   = 
                   
                     
                       1 
                       4 
                     
                      
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             1 
                             2 
                           
                            
                           
                             s 
                             11 
                             DUT 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0028]    However, in the real world, the S-parameters of the power divider  204 , signal generator  202  and real-time oscilloscope  200  will not be ideal, but the following may be assumed to be true: 
         [0000]      i s 22   ss   s   11   scope &lt;&lt;1,  s   22   ss   s   11 &lt;&lt;1,  s   22   s   11   scope &lt;&lt;1  (11)
 
         [0000]    Then equation (7) can be approximated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       b 
                       2 
                     
                     
                       v 
                       s 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     · 
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               s 
                               22 
                               ss 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               s 
                               21 
                             
                             + 
                             
                               
                                 s 
                                 23 
                               
                                
                               
                                 
                                   s 
                                   ~ 
                                 
                                 11 
                                 DUT 
                               
                                
                               
                                 s 
                                 31 
                               
                             
                           
                           ) 
                         
                       
                       
                         1 
                         - 
                         
                           
                             s 
                             22 
                             ss 
                           
                            
                           
                             s 
                             12 
                           
                            
                           
                             
                               s 
                               ~ 
                             
                             11 
                             DUT 
                           
                            
                           
                             s 
                             31 
                           
                         
                         - 
                         
                           
                             s 
                             23 
                           
                            
                           
                             
                               s 
                               ~ 
                             
                             11 
                             DUT 
                           
                            
                           
                             s 
                             32 
                           
                            
                           
                             s 
                             11 
                             scope 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equation (12) provides a working practical approximation of the reflection coefficient of the DUT  206 . 
         [0029]    To measure s 22   DUT , the port two reflection coefficient of the DUT  206 , port two  212  of DUT  206  is connected to port one of the power divider  202  and port one  210  of the DUT  206  is terminated with an ideal 50 Ohm resistor  214 . Then, the same procedure used to measure the port one  210  reflection coefficient s 11   DUT  discussed above may be repeated to calculate the port two  212  reflection coefficient s 22   DUT . 
         [0030]    The insertion loss and/or crosstalk terms s 12   DUT , s 21   DUT  may be measured using the system shown in  FIG. 4 . This system includes the signal generator  202 , the DUT  206 , the real-time oscilloscope  200  and the synchronized trigger  208 . The synchronized trigger  208  provides the time reference so the phases of the measured S-parameters are correct. The signal flow diagram of the system in  FIG. 4  can be seen in  FIG. 5 . 
         [0031]    The transfer function from the signal generator&#39;s  202  voltage source to the input to the real-time oscilloscope can be derived in the same way as equation (6): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       b 
                       2 
                     
                     
                       v 
                       s 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     · 
                     
                       
                         1 
                         - 
                         
                           s 
                           22 
                           ss 
                         
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   s 
                                   22 
                                   ss 
                                 
                                  
                                 
                                   s 
                                   11 
                                   DUT 
                                 
                               
                             
                             ) 
                           
                            
                           
                             
                               s 
                               11 
                               
                                 DUT 
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     s 
                                     22 
                                     DUT 
                                   
                                    
                                   
                                     s 
                                     11 
                                     scope 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           
                             s 
                             22 
                             ss 
                           
                            
                           
                             s 
                             12 
                             DUT 
                           
                            
                           
                             s 
                             11 
                             scope 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using the assumptions of equation (11), equation (13) can be approximated as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       b 
                       2 
                     
                     
                       v 
                       s 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     · 
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               s 
                               22 
                               ss 
                             
                           
                           ) 
                         
                          
                         
                           s 
                           21 
                           DUT 
                         
                       
                       
                         1 
                         - 
                         
                           
                             s 
                             22 
                             ss 
                           
                            
                           
                             s 
                             11 
                             DUT 
                           
                         
                         - 
                         
                           
                             s 
                             22 
                             DUT 
                           
                            
                           
                             s 
                             11 
                             scope 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since s 11   DUT  and s 22   DUT  have been calculated as discussed above, there is only one unknown variable, s 21   DUT , in equation (14). Therefore, equation (14) can be solved for s 21   DUT . b 2  in equation (14) is the voltage acquired by the oscilloscope  200  when ν s  is generated when the devices are configured as in  FIG. 4 . 
         [0032]    To measure s 12   DUT , port two of DUT  206  is connected to the signal generator  202 , port one of DUT  206  is connected to the real-time oscilloscope  200 , and the same procedure that measures s 21   DUT  may be repeated to measure s 12   DUT . 
         [0033]    The method described above may be expanded to measure the reflection coefficients and insertion loss and crosstalk terms for a multi-port DUT. To measure an N-port network, the reflection coefficients can be measured and calculated by terminating all the other ports of the DUT, except the port connected to the power divider, with an ideal 50 Ohms, using the method described above. Then, the insertion loss and crosstalk terms may be measured by terminating all other ports except two ports connected to the signal generator and to the real-time oscilloscope, using the above-discussed insertion loss and crosstalk measurement method. 
         [0034]    Having described and illustrated the principles of the disclosed technology in a preferred embodiment thereof, it should be apparent that the disclosed technology can be modified in arrangement and detail without departing from such principles. We claim all modifications and variations coming within the spirit and scope of the following claims.