Abstract:
Embodiments of the present invention relate to circuits to be inserted in a signal path between a signal generator and a test port of a vector network analyzer (VNA), where the circuits enable the VNA to make scattering parameter measurements for a load (RL) connected to the test port when the signal generator generates signals having frequencies that are below a low frequency limit (e.g., 2 MHz) of an actual dual directional coupler of the VNA. Embodiments of the present invention are also directed to a VNA that includes such circuits.

Description:
FIELD OF THE INVENTION  
       [0001]     Embodiments of the present invention relate to vector network analyzers (VNAs). More particularly, embodiments of the present invention relate to circuits that enable a VNA to make measurements over a large frequency range (e.g., DC to 8 GHz).  
       BACKGROUND  
       [0002]      FIG. 1  is a high level diagram of a conventional vector network analyzer (VNA)  100  which is shown as including a signal generator  102  (also known as a voltage source) having a source impedance RS, a forward coupler  104 , a reverse coupler  106 , one or more test port  108 , a receive/detector  110  and a processor/display  120 . The signal generator  102  produces signals that are transmitted to a test device, shown as a resistive load RL, which is connected to one or more test port  108  of the VNA. The resistive load RL, also known as a device under test (DUT), can be, e.g., an open, a short, or anything in-between.  
         [0003]     The forward coupler, which includes a resistive bridge having an impedance R F , provides a forward signal (also known as an incident signal) to a first input of the receiver/detector  110 . The reverse coupler  108 , which includes a resistive bridge having an impedance R R , provides a reverse signal (also known as a reflected signal) to a second input of the receiver/detector  110 . The forward signal is often referred to as signal VF hereafter. Similarly, the reverse signal is often referred to as signal VR hereafter. In the arrangement shown, the forward coupler  104  and the reverse coupler  106  together form what is known as a dual directional coupler. Preferably, the impedances of the source, the load and the couplers  104  and  106  all match (i.e., preferably R S =R L =R F =R R ). A typical value for R S , R L , R F  and R R  is 50 ohms, or 75 ohms, but other values may be used.  
         [0004]     The receiver/detector  110 , which is tuned to the frequency of the signal generator  102 , likely includes components such as a local oscillator (LO), band-pass filters (BPFs) and a synchronous detector or digital-signal processor (DSP). The LO is used to mix received RF signals down to lower intermediate frequency (IF) signals and baseband signals. The BPFs are used to filter out undesired harmonics and noise from the IF and baseband signals. The synchronous detector or DSP is used to extract magnitude and phase information from the baseband signals, which can be, e.g., about 200 KHz signals. More specifically, the synchronous detector or DSP can converts the VF and VR signals into digital signals indicative of real and imaginary components of the VF signal and real and imaginary components of the VR signal. The processor/display  120  formats the reflection and transmission data in ways that make it possible to interpret results of measurements.  
         [0005]     Conventional forward and reverse couplers  104  and  106  typically have a low frequency limit of approximately 1 or 2 MHz. This is primarily due to the balun used to create the single ended to differential signal for the resistive bridge of the coupler. More specifically, a conventional coupler typically includes a coaxial transmission line surrounded by Ferrite beads of about a quarter inch in length. A coupler with one quarter inch Ferrite bead will enable use of the coupler down to about 2 MHz. To get down to about 1 MHz, two such quarter inch Ferrite beads are needed; to get down to about 0.5 MHz, four such quarter inch ferrite beads are required; and so on. Accordingly, it can be appreciated that it would be prohibitive to produce a coupler useful down to about DC, using the above described methodology. Nevertheless, there is a desire to measure S-parameters down to DC. Accordingly, there is a need for couplers that are useful down to DC.  
       SUMMARY  
       [0006]     Embodiments of the present invention are directed to circuits to be inserted in a signal path between a signal generator and a test port of a vector network analyzer (VNA), where the circuits enable the VNA to make scattering parameter measurements for a load (RL) connected to the test port when the signal generator generates signals having frequencies that are below a low frequency limit (e.g., 2 MHz) of an actual dual directional coupler of the VNA. Embodiments of the present invention are also directed to a VNA that includes such circuits.  
         [0007]     In accordance with an embodiment of the present invention, the circuit includes an attenuator (e.g., a pi-attenuator) that is inserted in the signal path between the signal generator and the test port. The circuit also includes a monitoring circuit connected to the attenuator to monitor voltages at two nodes along the signal path that are formed by the attenuator. Additionally, the circuit include a dual directional coupler emulation circuit, made up of op-amps and resistors, connected to the monitoring circuit to emulate a dual directional coupler at frequencies that are below the low frequency limit of an actual dual directional coupler. Outputs of the dual directional coupler emulation circuit are indicative of signals that are incident on and reflected from the load (RL), and thus can be used to measure scattering parameters of the load (RL).  
         [0008]     In accordance with an embodiment of the present invention, the dual directional coupler emulation circuit includes a first circuit portion that emulates a forward coupler, as well as a second circuit portion that emulates a reverse coupler.  
         [0009]     In accordance with an embodiment of the present invention, when the circuit is inserted in a VNA that includes an actual dual directional coupler, the circuit is preferably inserted in the signal path of the VNA such that it is between the signal generator and the actual dual direction coupler of the VNA.  
         [0010]     Further and alternative embodiments, and the features, aspects, and advantages of the present invention will become more apparent from the detailed description set forth below, the drawings and the claims.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0011]      FIG. 1  is a high level block diagram of a conventional vector network analyzer (VNA) that can not make measurements below the low frequency limit of the dual directional coupler of the VNA.  
         [0012]      FIG. 2  is a high level diagram of a VNA that, in accordance with embodiments of the present invention, that can make measurements below the low frequency limit of the actual dual directional coupler of the VNA.  
         [0013]      FIG. 3  is a circuit diagram that is useful for explaining the known characteristics of a pi-attenuator.  
         [0014]      FIG. 4  is a circuit diagram that is useful for explaining how op-amps can be used to emulate characteristics of a reverse coupler and a forward coupler at relatively low frequencies (i.e., frequencies in the range of DC to a few MHz), in accordance with embodiments of the present invention.  
         [0015]      FIG. 5  is diagram that is used to illustrate the characteristics of ideal forward and reverse couplers, and to explain how embodiments of the present invention can be used to emulate such ideal couplers.  
         [0016]      FIG. 6  illustrates details of a circuit to emulate a dual directional coupler, in accordance with embodiments of the present invention.  
         [0017]      FIG. 7  illustrates how the embodiments of the present invention can be used together with an actual dual directional coupler to thereby provide a VNA capable of making measurements over a larger bandwidth than a typical VNA.  
     
    
     DETAILED DESCRIPTION  
       [0018]      FIG. 2  is a high level diagram of a VNA  200  that, in accordance with embodiments of the present invention, can make measurement from DC to about 8 GHz. As shown in  FIG. 2 , embodiments of the present invention allow the use of conventional forward and reverse couplers  106  and  108  to be used for their intended frequency coverage from the low MHz (e.g., 2 MHz) to the mid GHz (e.g., 8 GHz) range. The addition of a low value (e.g., &lt;1 dB) attenuator  202  placed in the source signal path  103  before the couplers  106  and  108  allows a convenient monitoring point to derive DC to 2 MHz forward and reverse voltage parameters. More specifically, embodiments of the present invention allow S-parameters to be measured down to DC, or at least close thereto. Preferably, embodiments of the present use no switches in the source signal path  103 , which would cause additional losses as well as switching uncertainties.  
         [0019]     In accordance with an embodiment of the present invention, the attenuator  202 , which is shown as including three resistors (i.e., R 1 , R 2  and R 3 ), is a pi-attenuator, which is given that name because the configuration of resistors R 1 , R 2  and R 3  resemble the symbol pi (i.e., π). Characteristics of the attenuator  202  are described with reference to  FIG. 3 , and Equations 1-3. The insertion loss of the attenuator is preferably low, e.g., below 1 dB. However, embodiments of the present invention also encompass those situations where the attenuator has a greater insertion loss. Further, while the use of a pi-attenuator is preferred, use of other attenuators are within the scope of the present invention. For example, it is possible to string two T-attenuators together to form two shunt arms. Five resistors (instead of three) would be used to form the two T-attenuators, and the math would differ due to the five resistors. For the following discussion, it is assumed that a pi-attenuator is used.  
         [0020]     Equation 1 below is an equation for the voltage (VA) in  FIG. 3 , which is the voltage at node (N 1 ) in  FIG. 2 . In this and further equations, Vinc is the “incident” voltage generated by the voltage source  102 .  
             VA   =     Vinc   ⁢           ⁢         R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (     R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·     +   R       ⁢           ⁢     1   ·   R     ⁢           ⁢   3     )                   R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS       +               RL   ⁡     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )                         (     EQ   .           ⁢   1     )             
 
         [0021]     Equation 2 below is an equation for the voltage (VB) in  FIG. 3 , which is the voltage at node (N 2 ) in  FIG. 2 .  
             VB   =     VA   ⁢           ⁢       RL   ·     R   ⁢   3             R   ⁢   2     ·     R   ⁢   3       +     RL   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )                     (     EQ   .           ⁢   2     )             
 
         [0022]     Combining Equations 1 and 2 results in Equation 3 shown below.  
             VB   =     Vinc   ⁢           ⁢       R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3               R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS       +               RL   ⁡     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )                         (     EQ   .           ⁢   3     )             
 
         [0023]      FIG. 4  illustrates how a pair of op-amps A 1  and A 2  can be used to monitor the voltages VA and VB, which as mentioned above, are the voltages at nodes N 1  and N 2 . As can be seen, the non-inverting (+) input of the op-amp A 1  is connected to ground; the inverting (−) input of the op-amp A 1 , which is the summing input of the op-amp A 1 , is connected to one of the arms of the attenuator  202 ; and a feedback resistor equal to the resistor R 1  is connected between the inverting (−) input and the output of the op-amp A 1 . Similarly, the non-inverting (+) input of the op-amp A 2  is connected to ground; the inverting (−) input of the op-amp A 2 , which is the summing input of the op-amp A 2 , is connected to the other one of the arms of the attenuator  202 ; and a feedback resistor equal to the resistor R 3  is connected between the inverting (−) input and the output of the op-amp A 2 . This arrangement will result in a voltage −VA at the output of the op-amp A 1  (i.e., Node N 5 ), and a voltage −VB at the output of op-amp A 2  (i.e., Node N 6 ). In other words, in the arrangement shown, the op-amps A 1  and A 2  are use used to monitor the voltages VA and VB at nodes N 1  and N 2  of the source signal path  103 . This is only good within the bandwidth of the op-amps A 1  and A 2 , which is likely from DC to about 5 MHz. So for a limited frequency range (e.g., from DC to about 2 MHz), the outputs of the op-amps A 1  and A 2  are equal in magnitude (but opposite in sign) to the voltages at nodes N 1  and N 4 .  
         [0024]      FIG. 4  also shows how a differential amplifier A 3  can be used to convert voltages VA and VB to the form VO=Kp(VP)−Kn(VN). More specifically, Equation 4 below is an equation for VO of circuit  414 .  
             VO   =       Vp   ⁢           ⁢       R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7         ⁢     (     1   +       R   ⁢           ⁢   5       R   ⁢           ⁢   4         )       -     Vn   ⁢           ⁢       R   ⁢           ⁢   5       R   ⁢           ⁢   4                   (     EQ   .           ⁢   4     )               
         [0025]     Equation 5 below assumes R 4 =R 5 .  
             VO   =       Vp   ⁢       2   ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7           -   Vn             (     EQ   .           ⁢   5     )             
 
         [0026]     The dual directional coupler of the present invention includes a reverse coupler portion and a forward coupler portion. The reverse coupler case is first discussed, followed by a discussion of the forward coupler case.  
         [0000]     Reverse Coupler Case  
         [0027]     The reverse coupler case should have the characteristics of a reflectometer in that VO=0 when RL=Ro (i.e., the output is zero when the characteristic impedance of the coupler equals the load impedance). Equation 6 below illustrates this for the reverse coupler case. 
 
− VA ( K )−(− VB )=0   (EQ. 6) 
 
         [0028]     Equation 7 below solves for K. 
 
 K=VB/VA    (EQ. 7) 
 
         [0029]     Letting Vp=VA and Vn=VB, allows Equation 5 to be rewritten as Equation 8 below.  
             VO   =       VA   ⁢       2   ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7           -   VB             (     EQ   .           ⁢   8     )             
 
         [0030]     Now, assuming RL=Ro=50 ohms, and that VO=0 when RL=Ro, Equation 8 can be rewritten as Equation 9 below.  
             0   =       VA   ⁢       2   ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7           -   VB             (     EQ   .           ⁢   9     )             
 
         [0031]     Combining Equation 9 with Equation 2 results in Equation 10 below.  
                 2   ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7         =         RL   ·   R     ⁢           ⁢   3         R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )                   (     EQ   .           ⁢   10     )             
 
         [0032]     Solving for R 7  results in Equation 11 below.  
               R   ⁢           ⁢   7     =       R   ⁢           ⁢     3   ·   R     ⁢           ⁢     6   ·   RL           2   ⁢   R   ⁢           ⁢   2   ⁢     (       R   ⁢           ⁢   3     +   RL     )       +     R   ⁢           ⁢     3   ·   RL                   (     EQ   .           ⁢   11     )             
 
         [0033]     In other words, by selecting a value for R 7  that satisfies Equation 11, then the circuit  414 , shown in  FIG. 4 , will emulate a reverse coupler. Since this is the reverse coupler case, the term R 7   R  will be used, with the R subscript signifying the reverse coupler case.  
         [0034]     The pi attenuator  202  should be designed to have low insertion loss, so that minimum power is wasted. This is especially important at microwave frequencies where power is at a premium. Accordingly, it is preferred that the pi attenuator have an attenuation of less than 1 dB. For the following discussion, a pi attenuator with a value of 0.869 dB is selected, e.g., which results when R 1  and R 3  are 1 kilo-ohms. This is a good value for the wide range of op-amps used in developing −VA and −VB. Also assume that R 2  is approximately 5 ohms. More specifically, letting R 1 =R 3 =1K, and R 2 =5.11, the actual insertion loss =0.885 dB, and Zo=50.088 ohms. For an ideal insertion loss of 0.869 and Z0=50, R 1  and R 3  would be 1K and R 2  would be 5.012531. For the following examples, it is assumed that R 1 =R 2 =1K and R 2 =5.11 and RL=50. It is also assumed that R 6 =511 ohms.  
                     R   ⁢           ⁢     7   R       =       ⁢       R   ⁢           ⁢     3   ·   R     ⁢           ⁢     6   ·   RL             2   ·   R     ⁢           ⁢   2   ⁢     (       R   ⁢           ⁢   3     +   RL     )       +     R   ⁢           ⁢     3   ·   RL                       =       ⁢       1   ⁢     K   ⁡     (   511   )       ⁢     (   50   )           2   ⁢     (   5.11   )     ⁢     (       1   ⁢   K     +   50     )       +     1   ⁢     K   ⁡     (   5   )                         =       ⁢   420.7077111                               
 
         [0035]     VO is now determined for RL other than 50 ohms.  
         [0036]     Equation 9 can be rewritten as Equation 12 shown below.  
             VB   =     VA   ⁢       2   ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7                   (     EQ   .           ⁢   12     )             
 
         [0037]     Combining Equations 8 and 12 leads to Equation 13 shown below.  
             VO   =     VA   ⁡     (         2   ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7         -         RL   ·   R     ⁢           ⁢   3         R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )             )               (     EQ   .           ⁢   13     )             
 
         [0038]     Combining Equations 1 and 13 leads to Equation 14 (EQ. 14) shown below.  
               VO   R     =     Vinc   (         R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2     +     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3       )                   R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS       +               RL   ⁡     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )               )                 (         2   ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7         -         RL   ·   R     ⁢           ⁢   3         R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )             )             
 
         [0039]     Assuming R 1 =R 3 =1K, R 2 =5.11, RS=50, R 6 =511 and R 7 =420.7077111, then Equation 14 becomes:  
         VO   R     =       Vinc   ⁡     (       50   -   RL       49.85704395   +   RL       )       ⁢     (   0.10202053   )           
 
         [0040]     Table 1 below illustrates VO for various values for RL, assuming Vinc=2V.  
                                                         TABLE 1                                   RL   VO(mV)   dBV   Phase(deg)                                         0   +167.475   −15.521   0            0.5   +164.154   −15.645   0            5.0   +136.989   −17.266   0           25   +55.772   −25.072   0           45   +8.803   −41.108   0           50   0   −180.637   0             55.5555   −8.801   −41.109   −180           100    −55.718   −25.080   −180           500    −136.669   −17.287   −180           5000    −163.694   −15.719   −180           Infinity   −166.996   −15.546   −180                      
 
         [0041]     In Table 1, the phase of VO is 180 degrees out of phase with the standard definition of rho, where rho=(Z−Zo)/(Z+Zo), for rho&lt;Zo, phase=180; for rho&gt;Zo, phase=0. Table 1 is used to illustrate that circuit  414 , with appropriate values for the resistors R 4 , R 5 , R 6  and R 7 , as defined above in Equation 11, will indeed act as a reverse coupler, where RL is a SHORT (i.e., RL=0), an OPEN (i.e., RL=infinity), or any value there-between.  
         [0042]     The reverse coupler case expressed in Equation 14 above can be rewritten as Equation 15 (EQ. 15) shown below.  
               VO   R     =       ⁢     -     Vinc   (         R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2     +     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3       )                   R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS       +               RL   ⁡     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )               )                       ⁢     (           RL   ·   R     ⁢           ⁢   3         R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )           -       2   ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7           )               
 
         [0043]     The inversion can be addressed in post processing.  
         [0000]     Forward Coupler Case  
         [0044]     For a forward coupler case, the circuit  414  should have the characteristics VO=(Vinc)K for all values of RL. An equation is now determined that produces a constant output VO for all values of RL from 0 to infinity.  
         [0045]     Equation 16 below illustrates the forward coupler case, with the subscript “o” OPEN, and the subscript “s” indicating a SHORT.  
               -       (     KVBo   -   VAo     )     OPEN       =     -       (     KVBs   -   VAs     )     SHORT               (     EQ   .           ⁢   16     )             
 
         [0046]     Equation 16 can be rewritten as Equation 17 below.  
             K   =       VAo   -   VAs       VBo   -   VBs               (     EQ   .           ⁢   17     )             
 
         [0047]     Equations 18 and 19 shown below are for VA and VB for OPEN.  
             VAo   =     Vinc   ⁢         R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2     +     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3           R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )                     (     EQ   .           ⁢   18     )               VBo   =     Vinc   ⁢       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3         R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )                     (     EQ   .           ⁢   19     )             
 
         [0048]     Equations 20 and 21 shown below are for VA and VB for SHORT.  
             VAs   =     Vinc   ⁢       R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3         R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS                     (     EQ   .           ⁢   20     )               VBs   =   0           (     EQ   .           ⁢   21     )             
 
         [0049]     Equations 17 and 21 can be combined for SHORT, as shown in Equation 22 below.  
             K   =       VAo   -   VAs     VBo             (     EQ   .           ⁢   22     )             
 
         [0050]     Combining equations 18, 19 and 22 leads to Equation 23, shown below.  
             K   =       ⁢         Vinc   ⁢         R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2     +     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3           R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )             -     
     ⁢     Vinc   ⁢       R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3         R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS                 Vinc   ⁢       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3         R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )                       (     EQ   .           ⁢   23     )             
 
         [0051]     Equation 23 can be simplified to Equation 24 shown below.  
             K   =           R   ⁢           ⁢   2     +     R   ⁢           ⁢   3         R   ⁢           ⁢   3       -       R   ⁢           ⁢   2   ⁢     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )                 R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS                     (     EQ   .           ⁢   24     )             
 
         [0052]     Letting Vp=VB and Vn=VA, allows Equation 5 to be rewritten as Equation 25 below.  
             VO   =       VB   ⁢           ⁢       2   ⁢           ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7           -   VA             (     EQ   .           ⁢   25     )             
 
         [0053]     From  FIG. 4 , Equation 5 can be rewritten as Equation 26 below. 
 
 VO =( VB ) K−VA    (EQ. 26) 
 
         [0054]     Combining equations 25 and 26 leads to Equation 27 below.  
                 VB   ⁢           ⁢       2   ⁢           ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7           -     V   ⁢           ⁢   A       =         (   VB   )     ⁢   K     -   VA             (     EQ   .           ⁢   27     )             
 
         [0055]     Solving for K leads to Equation 28 below.  
             K   =       2   ⁢           ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7                 (     EQ   .           ⁢   28     )             
 
         [0056]     Combining Equations 24 and 28 (i.e., Let K=K) leads to Equation 29 below.  
                 2   ⁢           ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7         =           R   ⁢           ⁢   2     +     R   ⁢           ⁢   3         R   ⁢           ⁢   3       -       R   ⁢           ⁢   2   ⁢     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )                 R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS                     (     EQ   .           ⁢   29     )             
 
         [0057]     Solving for R7F results in Equation 30 below.  
               R   ⁢           ⁢     7   F       =     R   ⁢           ⁢   6   ⁢     (                 (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )     ⁢     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   S       )       -               (     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3   ⁢     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +     R   ⁢           ⁢   S       )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )                           (       R   ⁢           ⁢   3     -     R   ⁢           ⁢   2       )     ⁢     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   S       )       +               (     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3   ⁢     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +     R   ⁢           ⁢   S       )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )                 )               (     EQ   .           ⁢   30     )             
 
         [0058]     In other words, by selecting a value for R 7  that satisfies Equation 30, then the circuit  414 , shown in  FIG. 4 , will emulate a forward coupler. Since this is the forward coupler case, the term R 7   F  will be used, with the F subscript signifying the reverse coupler case.  
         [0059]     Still assuming that R 1 =R 2 =1K, R 2 =5.11, R 6 =511 and RS=50, then R 7   F =420.3065369.  
         [0060]     Combining Equations 26 and 28 with Equations 1 and 3 leads to Equation 31 below.  
                     VO   F     =         Vinc   ⁡     (       2   ⁢           ⁢   R   ⁢           ⁢   7         R   ⁢           ⁢   6     +     R   ⁢           ⁢   7         )       ⁢     (         RL   ·   R     ⁢           ⁢     1   ·   R     ⁢           ⁢   3               R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS       +               RL   (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +                   RS   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )       )             )       -                   -   Vinc     ⁢           ⁢         R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2     +     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3       )                   R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS       +               RL   ⁡     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )                             (     EQ   .           ⁢   31     )             
 
         [0061]     Equation 31 can be simplified to Equation 32, shown below.  
               VO   F     =     Vinc   ⁡     (         RL   ·     (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3   ⁢     (         R   ⁢           ⁢   7     -     R   ⁢           ⁢   6           R   ⁢           ⁢   6     +     R   ⁢           ⁢   7         )       -     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2       )       -     R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3                 R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS       +               RL   ⁡     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )               )               (     EQ   .           ⁢   32     )             
 
         [0062]     Assuming R 1 =R 2 =1K, R 2 =5.11, RS=50, R 6 =511 and R 7   F =420.3065369, then Equation 32 becomes:  
               VO   F     =     -     Vinc   ⁡     (       5.11   +     RL   ⁡     (   0.10249304   )           55.1102555   +     RL   ⁡     (   1.1053655   )           )                     =       -     Vinc   ⁡     (       49.85704395   +   RL       49.85704395   +   RL       )         ⁢     (   0.09272321   )                   =     -     Vinc   ⁡     (   0.09272321   )                   
 
         [0063]     Table 2 below illustrates the VO for values for RL from 0 to infinity, assuming Vinc=2 V.  
                                                         TABLE 2                                   RL   VO(mV)   dBV   Phase(deg)                                        0 to inf   −185.446   −14.636   −180                      
 
         [0064]     In Table 2, the phase of VO is 180 degrees out of phase with the standard definition of Vinc. Table 2 is used to illustrate that circuit  414 , with appropriate values for the resistors R 4 , R 5 , R 6  and R 7 , as defined above in Equation 30, will indeed act as a forward coupler, when RL is a SHORT (i.e., RL=0), when RL is an OPEN (i.e., RL=infinity), or any value there-between.  
         [0065]     Since the phase of VOF is 180 degrees from the source Vinc, the forward coupler case expressed in Equation 32 above can be rewritten as Equation 33 shown below.  
               VO   F     =     -     Vinc   ⁡     (         R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3     -     RL   ⁡     (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3   ⁢     (         R   ⁢           ⁢   7     -     R   ⁢           ⁢   6           R   ⁢           ⁢   6     +     R   ⁢           ⁢   7         )       -     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2       )                   R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢     3   ·   RS       +     R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3       )       +     R   ⁢           ⁢     2   ·   RS       +               RL   ⁡     (       R   ⁢           ⁢   1   ⁢     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3     +   RS     )       +     RS   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )         )               )                 (     EQ   .           ⁢   33     )             
 
         [0066]     The inversion can be addressed in post processing.  
         [0000]     Scattering Parameters  
         [0067]     A reflectometer can be described by scattering parameters, where scattering parameter S 11  represents the value rho. More specifically, S 11  (which is referred to as the “forward reflection” coefficient) is equal to the signal leaving port  1  relative to the signal being injected into port  1 , or simply S 11 =Reflected Voltage/Incident Voltage. Accordingly, the use of Reverse and Forward Couplers can be combined to produce S 11  by dividing VO R  by VO F , as shown in Equation 34 below.  
               S   ⁢           ⁢   11     =       VO   R       VO   F               (     EQ   .           ⁢   34     )             
 
         [0068]     Combining Equation 34 with Equations 15 and 33, and letting D=R 1 (R 3 ·RS+R 2 ·R 3 )+R 2 ·RS+RL(R 1 (R 2 +R 3 +RS)+(RS(R 2 +R 3 )), leads to Equation 35 below.  
               S   ⁢           ⁢   11     =           -   Vinc     D     ⁢     (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2     +     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3       )         )     ⁢     (           RL   ·   R     ⁢           ⁢   3         R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )           -       2   ⁢   R   ⁢           ⁢     7   R           R   ⁢           ⁢   6     +     R   ⁢           ⁢     7   R             )             -   Vinc     D     ⁢           ⁢     (       R   ⁢           ⁢     1   ·           ⁢   R     ⁢           ⁢     2   ·           ⁢   R     ⁢           ⁢   3     -           ⁢     RL   (           ⁢       R   ⁢           ⁢     1   ·           ⁢   R     ⁢           ⁢   3   ⁢           ⁢     (         R   ⁢           ⁢     7   F       -     R   ⁢           ⁢   6           R   ⁢           ⁢   6     +     R   ⁢           ⁢     7   F           )       -           ⁢     R   ⁢           ⁢     1   ·           ⁢   R     ⁢           ⁢   2       ⁢           )       ⁢           )                 (     EQ   .           ⁢   35     )             
 
         [0069]     Equation 35 can be simplified to Equation 36 shown below.  
               S   ⁢           ⁢   11     =         (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢     1   ·   R     ⁢           ⁢   2     +     R   ⁢           ⁢     1   ·   R     ⁢           ⁢   3       )         )     ⁢     (           RL   ·   R     ⁢           ⁢   3         R   ⁢           ⁢     2   ·   R     ⁢           ⁢   3     +     RL   ⁡     (       R   ⁢           ⁢   2     +     R   ⁢           ⁢   3       )           -       2   ⁢   R   ⁢           ⁢     7   R           R   ⁢           ⁢   6     +     R   ⁢           ⁢     7   R             )         (       R   ⁢           ⁢     1   ·           ⁢   R     ⁢           ⁢     2   ·           ⁢   R     ⁢           ⁢   3     -           ⁢     RL   (           ⁢       R   ⁢           ⁢     1   ·           ⁢   R     ⁢           ⁢   3   ⁢           ⁢     (         R   ⁢           ⁢     7   F       -     R   ⁢           ⁢   6           R   ⁢           ⁢   6     +     R   ⁢           ⁢     7   F           )       -           ⁢     R   ⁢           ⁢     1   ·           ⁢   R     ⁢           ⁢   2       ⁢           )       ⁢           )               (     EQ   .           ⁢   36     )             
 
         [0070]     It can be appreciated from Equation 36 that S 11  has no dependency on the Source Voltage (Vinc) or the Source Impedance (RS). Referring to  FIG. 4 , since the pi attenuator (made up of resistors R 1 , R 2  and R 3 ) and the amplifiers (A 1  and A 2 ) are used for both forward and reverse couplers, only one RF element is needed, which reduces the Through Pass Loss for the high frequency mode, e.g., greater than 2 MHz.  
         [0071]     Assuming R 1 =R 3 =1K, R 2 =5.11, RS=50, R 6 =511, R 7 R=420.7077111 and R 7   F =420.3065369, then Equation 36 becomes:  
         S   ⁢           ⁢   11     =       (       RL   -   50       RL   +   49.85704395       )     ⁢     (   0.90050742   )           
 
         [0072]     Table 3 below illustrates S 11  for values for RL from 0 to infinity.  
                                                         TABLE 3                                   RL   S11   20 Log[S11]   Phase(deg)                                        0   −0.90770   0.84111   −180           0.5   −0.88930   −1.01904   −180           5.0   −0.74060   −2.60830   −180           25   −0.30100   −10.4286   −180           45   −0.04749   −26.4677   −180           50   0   −inf   −180           55.5555   +0.04748   −26.4689   0           100   +0.30053   −10.4423   0           500   +0.73701   −2.65055   0           5000   +0.88271   −1.08369   0           Inf   +0.90051   −0.91025   0                      
 
         [0073]     The results for the example show a constant multiplier 0.90050742 and a slight difference from 50 Ohms in the denominator. The constant multiplier is due to the attenuator through Insertion Loss, and will always represent that value. The denominator error is due to the approximate values used for R 1 , R 2  and R 3  for the 50 Ohm attenuator values. These approximations also affect the ideal Insertion Loss value. Both errors are removed when a standard Open/Short/Load (OSL) calibration is made. Table 4 below illustrates that results that would be expected for an ideal case, where for an ideal reflectometer  
         S   ⁢           ⁢   11     =         RL   -   50       RL   +   50       .         
 
                                                         TABLE 4                                   RL   S11   20 Log[S11]   Phase(deg)                                        0   −1.00000   0   −180           0.5   −0.98020   −0.1737   −180           5.0   −0.81818   −1.7430   −180           25   −0.33333   −9.5424   −180           45   −0.05263   −25.5751   −180           50   0   −inf   −180           55.5555   +0.05263   −25.5751   0           100   +0.33333   −9.5424   0           500   +0.81818   −1.7430   0           5000   +0.98020   −0.1737   0           Inf   +1.00000   0   0                      
 
         [0074]     A comparison of Tables 3 and 4 shows that S 11  parameters measured using embodiments of the present invention are close to what is ideal, but with some errors. But, as mentioned above, such errors can be removed using a standard Open/Short/Load (OSL) calibration.  
         [0075]      FIG. 5  illustrates a practical reflectometer using forward and reverse couplers  104  and  106  with insertion loss and coupling factor values IL and CPL in dB. From  FIG. 5  it can be seen that PF and PR can be represented, respectively, by Equations 37 and 38 below.  
             PF   =     PINC   -     CPL   F               (     EQ   .           ⁢   37     )               PR   =     PINC   -     IL   F     -     IL   R     -     CPL   R     +     20   ⁢           ⁢   Log   ⁢            RL   -   50       RL   +   50                        (     EQ   .           ⁢   38     )               
         [0076]     Equation 39, below, shows scattering parameter S 11  in terms of PF and PR. 
 
 S 11 dB   =PR−PF    (EQ.39) 
 
         [0077]     Combining Equation 39 with Equations 37 and 38 leads to Equation 40 below.  
               S   ⁢           ⁢     11   dB       =     (     PINC   -     IL   F     -     IL   R     -     CPL   R     +     20   ⁢           ⁢   Log   ⁢            RL   -   50       RL   +   50              -     (     PINC   -     CPL   F       )                 (     EQ   .           ⁢   40     )             
 
         [0078]     Letting CPL F =CPL R  and IL F =IL R , leads to Equation 41 below.  
               S   ⁢           ⁢     11   dB       =       20   ⁢           ⁢   Log   ⁢            RL   -   50       RL   +   50              -     IL   R     -     IL   F               (     EQ   .           ⁢   41     )             
 
         [0079]     Letting CPL F =CPL R =14.436 dB and IL F =IL R =0.910254/2=0.455127 dB, then Equation 41 becomes:  
         S   ⁢           ⁢     11   dB       =       20   ⁢           ⁢   log   ⁢            RL   -   50       RL   +   50              -     0.90050742   ⁢           ⁢     dB   .             
 
         [0080]     Converting this to linear leads to:  
         S   ⁢           ⁢   11     =       (       RL   -   50       RL   +   50       )     ⁢       (   0.90050742   )     .           
 
         [0081]     This ideal example shows close agreement with measurements that can be obtained using embodiments of the present invention.  
         [0082]      FIG. 6  illustrates details of the circuit  205  to emulate a dual directional coupler, in accordance with embodiments of the present invention. Resistors R 4   F -R 7   F  on the left emulate a forward coupler, while resistor R 4   R -R 7   R  on the right emulate a reverse coupler.  
         [0083]      FIG. 7  illustrates how the embodiments of the present invention can be used together with a conventional dual directional coupler to thereby provide a VNA  200  capable of making measurements from the DC to multi-GHz range. In this arrangement, the circuit  205  to emulate a dual directional coupler is used when the frequency of the incident signal Vinc produced by the signal source  102  is within a first range (e.g., DC-2 MHz); and the conventional dual directional coupler made up of actual forward and reverse couplers  104  and  106  are used when the frequency of the incident signal Vinc produced by the signal source  102  is within a higher second range (e.g., 2 MHz-8 GHz).  
         [0084]     Referring back to  FIG. 2 , a detector/receiver  210  that accepts the VF and VR signals from the circuit  205 , as well as from couplers  104  and  106 , preferably includes a baseband converter for each pair of signals. For example, a baseband converter  212  may use both upconversion and downconversion to provide baseband signals (e.g., of about 200 KHz) to a synchronous detector or DSP  216 . A further baseband converter  214  will likely only used downconversion to provide baseband signals of the same frequency (e.g., of about 200 KHz) to the synchronous detector or DSP  216 . In accordance with an embodiment, through the use of switches S 1  and S 2 , only one of the baseband converters  212  and  214  provides the baseband forward and reverse signals to the synchronous detector or DSP  216 . More specifically, when the frequency of the incident signal Vinc produced by the signal source  102  is within the first range (e.g., DC-2 MHz), the switches S 1  and S 2  provide the output of the baseband converter  212  to the synchronous detector or DSP  216 . When the frequency of the incident signal Vinc produced by the signal source  102  is within the second higher range (e.g., 2 MHz to 8 GHz), the switches S 1  and S 2  provide the output of the baseband converter  214  to the synchronous detector or DSP  216 . In this manner, the VNA can make measurements over a larger bandwidth (e.g., DC to 8 GHz).  
         [0085]     While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example, and not limitation. It will be apparent to persons skilled in the relevant art that various changes in form and detail can be made therein without departing from the spirit and scope of the invention.  
         [0086]     The present invention has been described above with the aid of functional building blocks illustrating the performance of specified functions and relationships thereof. The boundaries of these functional building blocks have often been arbitrarily defined herein for the convenience of the description. Alternate boundaries can be defined so long as the specified functions and relationships thereof are appropriately performed. Any such alternate boundaries are thus within the scope and spirit of the invention.  
         [0087]     The breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments.