Method for calibrating two port high frequency measurements

[0002] 1. Field of the Invention

[0003] The present invention generally relates to the measurement of network parameters with a multiport network analyzer and, more particularly, to a method for calibrating the analyzer for two port high frequency measurements.

[0004] 2. Description of the Related Art

[0005] In designing cables and connecting hardware for use with high frequency signals it is imperative to understand the electrical characteristics of the components used to link a device or cable under-test with the electronics used to transmit and receive data. All of the possible couplings between connectors and cables can be measured with multiport measurement equipment comprising a network analyzer and a switching network. The key to providing an accurate multiport measurement system is a two port calibration procedure which is used repeatedly on every combination of pairs of ports in the measurement system.

[0006] Errors associated with network analyzer measurements can result from either non-ideal conditions in the measurement equipment or from the effects of the mechanical fixtures, such as cables and connectors, which are used to connect a device under test (DUT) to the network analyzer or test bed. A simple way to model this error is to view the test bed and the DUT as an ideal network analyzer cascaded with an error network and followed by the DUT. Such a network representation is illustrated in FIG. 1. The intervening network can then be used to account for the effects introduced by internal switching matrices connecting the various analyzer ports to the mechanical fixtures, as well as other sources of loss and distortion within the analyzer. The network can also be used to account for the effects of the fixtures themselves, which effects may introduce loss, crosstalk, and impedance discontinuities. Thus, the ideal network analyzer measures the cascaded network consisting of the intervening linear network followed by the DUT and obtains the measured S-parameters of the composite intervening network and DUT, here denoted as S_M. (See FIG. 1).

[0007] During testing, the acquired data which is intended to characterize the DUT is oftentimes corrupted by imperfections in the measuring device. The data must therefore be processed to compensate for known errors when extracting the parameters which describe the DUT, here denoted as S_A. An algorithm is used to relate the measured parameters S_Mto the actual parameters S_A. However, because the intervening network in the model is partially internal to the measurement equipment, S_Acannot be evaluated separately. The only way to characterize S_A, therefore, is to measure various DUTs with known electrical characteristics and extract the appropriate parameters from S_M. This process is further complicated when switching networks are involved because each change in the switch settings of the network analyzer results in a change in the intervening network due to differences in the signal routing. Thus, a different intervening network exists for each switch setting of the Network Analyzer, and each network must therefore be separately evaluated to compensate for measurement errors.

[0008] Existing calibration procedures involve very specific error models which do not account for the electrical properties of the jumpers used when the two measurement ports of the network analyzer are bridged as part of the calibration process. Although acceptable for low frequencies measurements, at high frequencies where the electrical length of the fixtures may easily approach a quarter wavelength, this failure introduces substantial errors which will not be corrected by the remaining calibration.

[0009] It would therefore be advantageous to provide a method for compensating for the inaccuracies introduced during the performance of two port calibration procedures which is used repeatedly on every combination of pairs of ports in a measurement system.

[0010] The above and other problems are overcome by a method for calibrating two port high frequency measurements which takes into account the electrical properties of jumpers which are used when the measurement ports of a network analyzer are bridged as part of the calibration process.

[0011] According to the invention, a method is used to remove systematic errors from two port network analyzer measurements. The method allows an intervening network to be completely arbitrary, and computes and factors the electrical properties of the jumpers which are used to bridge the analyzer ports of the network analyzer into a calibration process. This results in more accurate high frequency data than is possible with traditional calibration methods. This level of precision is essential when significant signal processing is applied to the measured data via software residing on a PC.

[0012] The calibration procedure consists of several stages and requires various calibration standards, typically an open circuit termination, a short circuit termination, and a reference load which is usually 50Ω or 75Ω, for example. The calibration standards are first connected to port one and are individually measured. This process is repeated with a port two of the network analyzer. Next, a jumper (for later use in bridging ports one and two of the network analyzer) is connected to port one. At the far end of the jumper, the three reference calibration standards are sequentially connected and the combination of the jumper and a termination is measured. The raw data obtained thus far is then processed to extract the two port electrical parameters of the jumper. The measured jumper is then bridged between ports one and two of the network analyzer and a series of measurements are made. Using the known electrical characteristics of the jumper, the intervening network is completely characterized and the measured S_Mand actual parameters S_Aof any DUT are related.

[0013] The foregoing and other advantages and features of the invention will become more apparent from the detailed description of the preferred embodiments of the invention given below, with reference to the accompanying drawings in which:

[0014] FIG. 1 is an illustration of an error model for network analyzer measurements;

[0015] FIG. 2 is a flow chart showing a method for computing the parameters of a general intervening network;

[0016] FIG. 3 illustrates a 12 term error model for two port measurements; and

[0017] FIGS. 4(a) & 4(b) are a flow chart illustrating the steps for determining S_Fand S_Rfor the 12 term error model.

[0018] FIG. 1 is an illustration of a linear intervening network (primarily switching networks and mechanical fixtures, such as cables and connectors, for example). Typically, a linear intervening network as described corrupts measured data. In such a network, relating the measured data S_Mand the actual data S_Afor a completely general intervening network is problematic. Due to internal switching, the intervening network may have completely different electrical characteristics for forward measurements (s₁₁and s₂₁) and reverse measurements (s₂₂and s₁₂). For forward measurements, when the switch is set in one direction, the intervening network is characterized by a completely general 4×4 S-parameter matrix S_Fof the form: $\begin{array}{cc}{S}_F=\left[\begin{array}{cccc}{e}_{00}& e_{01}& e_{02}& e_{03}\\ e_{10}& e_{11}& e_{12}& e_{13}\\ e_{20}& e_{21}& e_{22}& e_{23}\\ e_{30}& e_{31}& e_{32}& e_{33}\end{array}\right]& \left(\mathrm{Eq}.1\right)\end{array}$

[0019] where each e in the matrix represents the reflection value obtained when a port of the intervening network is excited by an input signal (see FIG. 3).

[0020] With reverse measurements the intervening network is characterized by a completely general S-parameter matrix S_Rof the form: $\begin{array}{cc}{S}_R=\left[\begin{array}{cccc}{p}_{00}& p_{01}& p_{02}& p_{03}\\ p_{10}& p_{11}& p_{12}& p_{13}\\ p_{20}& p_{21}& p_{22}& p_{23}\\ p_{30}& p_{31}& p_{32}& p_{33}\end{array}\right]& \left(\mathrm{Eq}.2\right)\end{array}$

[0021] where each p in the matrix represents the reflection value obtained when a port of the intervening network is excited by an input signal.

[0022] The new method for performing computations for determining the parameters of a general intervening network illustrated in the flow chart shown in FIG. 2 is as follows. First, the uncalibrated 2×2 matrix of measured data for the DUT is obtained (step 1) according to the relationship: $\begin{array}{cc}\begin{array}{c}{S}_M=\left[\begin{array}{cc}{\mathrm{sm}}_{11}& {\mathrm{sm}}_{12}\\ {\mathrm{sm}}_{21}& {\mathrm{sm}}_{22}\end{array}\right]\\ =\left[\begin{array}{cc}f& r\end{array}\right],\end{array}& \left(\mathrm{Eq}.3\right)\end{array}$

[0023] where 𝑓 and r are 2×1 vectors given by the relationship $\begin{array}{cc}f=\left[\begin{array}{c}{\mathrm{sm}}_{11}\\ {\mathrm{sm}}_{21}\end{array}\right]& \left(\mathrm{Eq}.4\right)\\ r=\left[\begin{array}{c}{\mathrm{sm}}_{12}\\ {\mathrm{sm}}_{22}\end{array}\right]& \left(\mathrm{Eq}.5\right)\end{array}$

[0024] and sm₁₁, sm₂₁, sm₁₂, and sm₂₂are the measured input reflection at port 1 of the network analyzer, the measured forward transmission at port 2, the measured reverse transmission at port 2 and the measured output reflection at port 2, respectively.

[0025] Next, from the matrixes S_Fand S_R, the following sub-matrices and vectors are computed (step 2): $\begin{array}{cc}{A}_{11}=\left[\begin{array}{cc}{e}_{00}& e_{03}\\ e_{30}& e_{33}\end{array}\right]A_{12}=\left[\begin{array}{cc}{e}_{01}& e_{02}\\ e_{31}& e_{32}\end{array}\right]A_{21}=\left[\begin{array}{cc}{e}_{10}& e_{13}\\ e_{20}& e_{23}\end{array}\right]A_{22}=\left[\begin{array}{cc}{e}_{11}& e_{12}\\ e_{21}& e_{22}\end{array}\right]B_{11}=\left[\begin{array}{cc}{p}_{00}& p_{03}\\ p_{30}& p_{33}\end{array}\right]B_{12}=\left[\begin{array}{cc}{p}_{01}& p_{02}\\ p_{31}& p_{32}\end{array}\right]B_{21}=\left[\begin{array}{cc}{p}_{10}& p_{13}\\ p_{20}& p_{23}\end{array}\right]B_{22}=\left[\begin{array}{cc}{p}_{11}& p_{12}\\ p_{21}& p_{22}\end{array}\right]& \left(\mathrm{Eq}.6\right)\\ {ϵ}_1=\left[\begin{array}{c}1\\ 0\end{array}\right],{ϵ}_2=\left[\begin{array}{c}0\\ 1\end{array}\right]& \left(\mathrm{Eq}.7\right)\end{array}$

[0026] where ε₁and ε₁are standard unit vectors.

[0027] Next, in step 3, the vectors F and R are computed according to the following relationship where $\begin{array}{cc}F=A_{12}^{-1}\left(f-A_{11}{ϵ}_1\right)=A_{12}^{-1}\left[\begin{array}{c}{\mathrm{sm}}_{11}-e_{00}\\ {\mathrm{sm}}_{21}-e_{30}\end{array}\right]& \left(\mathrm{Eq}.8\right)\\ R=B_{12}^{-1}\left(r-B_{11}{ϵ}_2\right)=B_{12}^{-1}\left[\begin{array}{c}{\mathrm{sm}}_{12}-p_{03}\\ {\mathrm{sm}}_{22}-p_{33}\end{array}\right].& \left(\mathrm{Eq}.9\right)\end{array}$

[0028] In step 4, the actual S-parameters for the DUT are computed according to the following relationship, where

[0029] S_A=[F R][A₂₂F+A₂₁ε₁B₂₂R+B₂₁ε₂]⁻¹  (Eq. 10)

[0030] where F and R represent a composite of processed data for forward and reverse measurements, respectively.

[0031] FIG. 3 is an illustration of a 12 term error model for two port measurements. Application of the method for the general intervening network illustrated in FIG. 2 to the network shown in FIG. 3 yields 12 unknown quantities from the intervening network and 4 uncalibrated measurements of the DUT. These 16 quantities completely define all of the information needed to compute the actual S-parameters of the DUT, and are expressed in vector form as: $\begin{array}{cc}{v}_s=\left[\begin{array}{cccccccccccccccc}{e}_{00}& e_{30}& e_{10}{e}_{01}& \frac{e_{01}}{e_{32}}& e_{11}& \frac{e_{01}{e}_{22}}{e_{32}}& p_{33}& p_{03}& p_{23}{p}_{32}& \frac{p_{32}}{p_{01}}& p_{22}& \frac{p_{11}{p}_{32}}{p_{01}}& {\mathrm{sm}}_{11}& {\mathrm{sm}}_{21}& {\mathrm{sm}}_{12}& {\mathrm{sm}}_{22}\end{array}\right].& \left(\mathrm{Eq}.11\right)\end{array}$

[0032] where ν_sis obtained by evaluating the expression in Eq. 10.

[0033] FIG. 4 is a flow chart which illustrates the steps for determinating S_Fand S_Rfor the 12 term error model. To determine the parameters which describe the intervening network, with both measurement ports terminated in the reference impedance, first measure the forward S-parameter vector 𝑓₀and determine e₀₀, and e₃₀(step 5) according to the relationship: $\begin{array}{cc}{f}_0=\left[\begin{array}{c}{e}_{00}\\ e_{30}\end{array}\right]& \left(\mathrm{Eq}.12\right)\end{array}$

[0034] Next, the reference is kept on port 2 of the network analyzer, an open termination is connected to port 1, the forward S-parameter vector 𝑓₁is measured and {tilde over (𝑓)}₁is computed (step 6) according to the relationship: $\begin{array}{cc}{\tilde{f}}_1=f_1-f_0=\left[\begin{array}{c}{f}_p\\ 0\end{array}\right]& \left(\mathrm{Eq}.13\right)\end{array}$

[0035] In step 7, the reference is kept on port 2 of the network analyzer, a short termination is connected to port 1, the forward S-parameter vector 𝑓₋₁is measured and {tilde over (𝑓)}₋₁is computed according to the relationship $\begin{array}{cc}{\tilde{f}}_{-1}=f_{-1}-f_0=\left[\begin{array}{c}{f}_n\\ 0\end{array}\right]& \left(\mathrm{Eq}.14\right)\end{array}$

[0036] where 𝑓₁represents the data which is obtained when an open termination is connected to port 1 and the reference is connected to port 2, 𝑓₀represents the data which is obtained when the a reference termination is connected to port 1 and a reference is connected to port 2, and 𝑓₋₁represents the data which is obtained when a short termination is connected to port 1 and the reference is connected to port 2.

[0037] Next the parameters e₁₁and e₁₀e₀₁are computed (step 8) according to the relationships: $\begin{array}{cc}{e}_{11}=\frac{f_p+f_n}{f_p-f_n}& \left(\mathrm{Eq}.15\right)\\ e_{10}{e}_{01}=2\frac{f_p{f}_n}{\left(f_n-f_p\right)}.& \left(\mathrm{Eq}.16\right)\end{array}$

[0038] Here, although the two port calculation is incomplete, enough information has been determined to completely correct so measurements. Denoting the raw measured s₁₁parameters as Γ_raw, the following relationship can be used to express the actual value Γ_actualas $\begin{array}{cc}{\Gamma }_{\mathrm{actual}}=\frac{{\Gamma }_{\mathrm{raw}}-e_{00}}{e_{11}{\Gamma }_{\mathrm{raw}}+e_{10}{e}_{01}-e_{00}{e}_{11}}.& \left(\mathrm{Eq}.17\right)\end{array}$

[0039] Next, with both measurement ports of the network analyzer terminated in the reference impedance, the reverse S-parameter vector r₀is measured and p₃₃and p₀₃are determined (step 9) in accordance with the relationship: $\begin{array}{cc}{r}_0=\left[\begin{array}{c}{p}_{03}\\ p_{33}\end{array}\right].& \left(\mathrm{Eq}.18\right)\end{array}$

[0040] Next, with the reference still connected to port 1 of the network analyzer, an open termination is connected to port 2, the reverse S-parameter vector r₁, is measured and {tilde over (r)}₁is computed (Step 10) according to the relationship: $\begin{array}{cc}{\tilde{r}}_1=r_1-r_0=\left[\begin{array}{c}0\\ r_p\end{array}\right].& \left(\mathrm{Eq}.19\right)\end{array}$

[0041] While leaving the reference on port 1, a short termination is connected to port 2 of the network analyzer, the reverse S-parameter vector r₋₁is measured and {tilde over (r)}₋₁is computed (step 11) according to the relationship:. $\begin{array}{cc}{\tilde{r}}_{-1}=r_{-1}-r_0=\left[\begin{array}{c}0\\ r_n\end{array}\right].& \left(\mathrm{Eq}.20\right)\end{array}$

[0042] where r₁represents the data which is obtained when an open termination is connected to port 2 and the reference is connected to port 1, r₀represents the data which is obtained when a reference termination is connected to port 2 and a reference is connected to port 1, and r₋ represents the data which is obtained when a short termination is connected to port 2 and the reference is connected to port 1.

[0043] Next, the parameters p₂₂and p₂₃p₃₂are computed (Step 12) according to the following relationships: $\begin{array}{cc}{p}_{22}=\frac{r_p+r_n}{r_p-r_n}& \left(\mathrm{Eq}.21\right)\\ p_{23}{p}_{32}=2\frac{r_p{r}_n}{\left(r_n-r_p\right)}& \left(\mathrm{Eq}.22\right)\end{array}$

[0044] Next, a semi-rigid coaxial cable is connected to port one of the network analyzer; port 2 is terminated in a reference. One port measurements, s₁₁are made (step 13) on the coax to determine its 2 port S-parameter matrix having the form: $\begin{array}{cc}{s}_{\mathrm{coax}}=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right].& \left(\mathrm{Eq}.23\right)\end{array}$

[0045] A reference is then connected to the far end of the coax and the s₁₁parameter is measured. With the procedure described in step 4, the raw data is then corrected in order to determine the actual s₁₁parameter of the coax when terminated in the reference Γ_r. Step 13 is then repeated to obtain the corrected s, parameter of the coax terminated in an open circuit Γ₀and the corrected s₁₁parameter of the coax terminated in a short Γ_s(step 14). The coax parameters a, b, c are determined (step 15) according to the relationships: $\begin{array}{cc}a={\Gamma }_r& \left(\mathrm{Eq}.24\right)\\ b^2=2\frac{\left({\Gamma }_r-{\Gamma }_s\right)\left({\Gamma }_o-{\Gamma }_r\right)}{\left({\Gamma }_o-{\Gamma }_s\right)}& \left(\mathrm{Eq}.25\right)\\ c=\frac{{\Gamma }_o+{\Gamma }_s-2{\Gamma }_r}{{\Gamma }_o-{\Gamma }_s}& \left(\mathrm{Eq}.26\right)\end{array}$

[0046] where a is the amount of signal which is reflected from the coax cable, b is the amount of signal which is passed through the cable, and c is amount of signal which is reflected when the measurement process is reversed.

[0047] Next, the coax is connected between ports 1 and 2 of the network analyzer and a full two port measurement is performed, which yields a 2×2 matrix of S_rawdata defined as STOW, and the following matrix is computed (step 16) according to the relationship: $\begin{array}{cc}\left[\begin{array}{cc}{f}_a& r_b\\ f_b& r_a\end{array}\right]=S_{\mathrm{raw}}-\left[\begin{array}{cc}{e}_{00}& p_{03}\\ e_{30}& p_{33}\end{array}\right]& \left(\mathrm{Eq}.27\right)\end{array}$

[0048] Finally, the remaining parameters necessary to complete the two port calibration are computed (step 17) according to the relationships: $\begin{array}{cc}\frac{e_{01}{e}_{22}}{e_{32}}=\frac{\left(1-{\mathrm{ae}}_{11}\right)f_a-{\mathrm{ae}}_{10}{e}_{01}}{{\mathrm{bf}}_b}& \left(\mathrm{Eq}.28\right)\\ \frac{e_{01}}{e_{32}}=\frac{\left(b^2-ac\right)\left(e_{11}{f}_a+e_{10}{e}_{01}\right)+{\mathrm{cf}}_a}{{\mathrm{bf}}_b}& \left(\mathrm{Eq}.29\right)\\ \frac{p_{11}{p}_{32}}{p_{01}}=\frac{\left(1-{\mathrm{cp}}_{22}\right)r_a-{\mathrm{cp}}_{23}{p}_{32}}{{\mathrm{br}}_b}& \\ \frac{p_{32}}{p_{01}}=\frac{\left(b^2-ac\right)\left(r_a{p}_{22}+p_{23}{p}_{32}\right)+{\mathrm{ar}}_a}{{\mathrm{br}}_b}.& \left(\mathrm{Eq}.30\right)\end{array}$

[0049] Once all of the electrical characteristics are determined using the method according to the invention, the intervening network can be completely characterized. Using the method, the electrical properties of jumpers which are used to bridge the analyzer ports of the network analyzer are computed and factored into the calibration process. In this manner, systematic errors from two port network analyzer measurements can be removed via processing on a PC.

[0050] Although the invention has been described and illustrated in detail, it is to be clearly understood that the same is by way of illustration and example, and is not to be taken by way of limitation. The spirit and scope of the present invention are to be limited only by the terms of the appended claims.