S-parameter measurements using real-time oscilloscopes

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.

Claims

1. A method for determining scattering parameters of a device under test using a real-time oscilloscope, comprising: determining a reflection coefficient for a first port of a device under test including: terminating a second port of the device under test with a resistor, sending, via a signal generator, a first signal to a first port of a power divider, measuring, based on a synchronized trigger that provides an absolute time reference between the signal generator and the real-time oscilloscope, a voltage level of the first signal by the real-time oscilloscope at a second port of the power divider while a device under test is connected, via the first port of the device under test, to a third port of the power divider, and calculating the reflection coefficient of the first port of the device under test utilizing the measured voltage level of the first signal; and determining a reflection coefficient for the second port of the device under test including: terminating the first port of the device under test with a resistor, sending, via the signal generator, a second signal to the first port of the power divider, measuring, based on the synchronized trigger, a voltage level of the second signal by the real-time oscilloscope at a second port of the power divider while a device under test is connected, via the second port of the device under test, to the third port of the power divider, and calculating the reflection coefficient of the second port of the device under test utilizing the measured voltage level of the second signal; and determining an insertion loss coefficient of the first port of the device under test, including: sending, via the signal generator, a third signal to the second port of the device under test and to the real-time oscilloscope; measuring, based on the synchronized trigger, a voltage level of the third signal by the real-time oscilloscope at the first port of the device under test; calculating, by the real-time oscilloscope, the insertion loss coefficient of the first port utilizing the measured voltage level of the second signal and the determined reflection coefficients of the first port and the second port of the device under test; and utilizing, by the real-time oscilloscope, the reflection coefficient and the insertion loss coefficient of the first port of the device under test to perform a de-embed operation.

2. The method of claim 1, wherein the resistor is a 50 Ohm resistor.

3. The method of claim 1, wherein the reflection coefficient of the first port of the device under test is calculated using the equations: $\frac{b_2}{v_{s1}}=\frac{1}{2}.Math.\frac{\left(1-s_{22}^{\mathrm{ss}}\right)\left(s_{21}+s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}\right)}{1-s_{22}^{\mathrm{ss}}{s}_{12}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}-s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{32}{s}_{11}^{\mathrm{scope}}}$ $\mathrm{and}$ ${\tilde{s}}_{11}^{\mathrm{DUT}}={s_{11}^{\mathrm{DUT}}\left[1-s_{33}{s}_{11}^{\mathrm{DUT}}\right]}^{-1}$ where b.sub.2 is the measured voltage level of the first signal, .sub.s1 is the first signal, s.sub.22.sup.ss is an impedance of the signal generator, s.sub.21, s.sub.12, s.sub.31, s.sub.23, s.sub.32, s.sub.33 are the scatter parameter terms of the power divider, s.sub.11.sup.scope is an input impedance of the real-time oscilloscope, and s.sub.11.sup.DUT is the reflection coefficient.

4. The method of claim 1, wherein the reflection coefficients are calculated for the second port of the device under test using the equations: $\frac{b_2}{v_{s2}}=\frac{1}{2}.Math.\frac{\left(1-s_{22}^{\mathrm{ss}}\right)\left(s_{21}+s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}\right)}{1-s_{22}^{\mathrm{ss}}{s}_{12}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}-s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{32}{s}_{11}^{\mathrm{scope}}}$ $\mathrm{and}$ ${\tilde{s}}_{11}^{\mathrm{DUT}}={s_{11}^{\mathrm{DUT}}\left[1-s_{33}{s}_{11}^{\mathrm{DUT}}\right]}^{-1}$ where b.sub.2 is the measured voltage level of the second signal, .sub.s2 is the second signal, s.sub.22.sup.ss is an impedance of the signal generator, s.sub.21, s.sub.12, s.sub.31, s.sub.23, s.sub.32, s.sub.33 are the scatter parameter terms of the power divider, s.sub.11.sup.scope is an input impedance of the real-time oscilloscope, and s.sub.22.sup.DUT is the reflection coefficient.

5. The method of claim 4, wherein the insertion loss coefficient is calculated for each of the first port and second port of the device under test using the equation: $\frac{b_2}{v_s}=\frac{1}{2}.Math.\frac{\left(1-s_{22}^{\mathrm{ss}}\right)s_{21}^{\mathrm{DUT}}}{1-s_{22}^{\mathrm{ss}}{s}_{11}^{\mathrm{DUT}}-s_{22}^{\mathrm{DUT}}{s}_{11}^{\mathrm{scope}}}$ where s.sub.21.sup.DUT is the insertion loss coefficient.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 illustrates a serial data link system.

(2) FIG. 2 is a block diagram of a return loss coefficient measurement system according to embodiments of the disclosed technology.

(3) FIG. 3 is a signal flow diagram of the system shown in FIG. 2.

(4) FIG. 4 is a block diagram of an insertion loss measurement system according to embodiments of the disclosed technology.

(5) FIG. 5 is a signal flow diagram of the system shown in FIG. 4.

(6) FIG. 6 is a power divider of FIG. 2.

DETAILED DESCRIPTION

(7) 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.

(8) 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.sub.11.sup.DUT, s.sub.22.sup.DUT and insertion loss and/or crosstalk terms s.sub.12.sup.DUT, s.sub.21.sup.DUT.

(9) 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.

(10) Power divider 204 may be any type of power divider or splitter. For example, power divider 204 may be a power divider having three 16 Ohm resistors at three branches creating a three way power network, as seen in FIG. 6.

(11) FIG. 3 illustrates a signal flow diagram of the system shown in FIG. 2.

(12) 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:

(13) $\begin{array}{cc}s=\left[\begin{array}{ccc}{s}_{11}& s_{12}& s_{13}\\ s_{21}& s_{22}& s_{23}\\ s_{31}& s_{32}& s_{33}\end{array}\right]& \left(1\right)\end{array}$

(14) 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.sup.DUT.

(15) To measure the reflection coefficient s.sub.11.sup.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 .sub.s and has the reflection coefficient s.sub.22.sup.ss at the output port. The real-time oscilloscope 200 has the reflection coefficient s.sub.11.sup.scope at its input port.

(16) 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:

(17) $\left(2\right)\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]=\left\{\left[\begin{array}{cc}{s}_{11}& s_{12}\\ s_{21}& s_{22}\end{array}\right]+\left[\begin{array}{c}{s}_{13}\\ s_{23}\end{array}\right]s_{11}^{\mathrm{DUT}}\left[1-s_{33}{s}_{11}^{\mathrm{DUT}}\right]\left[\begin{array}{cc}{s}_{31}& s_{32}\end{array}\right]\right\}\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]$
Let equation (3):

(18) $\begin{array}{cc}\begin{array}{c}\tilde{s}=\left[\begin{array}{cc}{\tilde{s}}_{11}& {\tilde{s}}_{12}\\ {\tilde{s}}_{21}& {\tilde{s}}_{22}\end{array}\right]\\ =\left[\begin{array}{cc}{s}_{11}& s_{12}\\ s_{21}& s_{22}\end{array}\right]+\left[\begin{array}{c}{s}_{13}\\ s_{23}\end{array}\right]{s_{11}^{\mathrm{DUT}}\left[1-s_{33}{s}_{11}^{\mathrm{DUT}}\right]}^{-1}\left[\begin{array}{cc}{s}_{31}& s_{32}\end{array}\right]\end{array}& \left(3\right)\end{array}$
denote equation (4):
{tilde over (s)}.sub.11.sup.DUT=s.sub.11.sup.DUT[1s.sub.33s.sub.11.sup.DUT].sup.1(4)
Then equation (3) can be re-written as:

(19) $\begin{array}{cc}\left[\begin{array}{cc}{\tilde{s}}_{11}& {\tilde{s}}_{12}\\ {\tilde{s}}_{21}& {\tilde{s}}_{22}\end{array}\right]=\left[\begin{array}{cc}{s}_{11}+s_{13}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}& s_{12}+s_{13}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{32}\\ s_{21}+s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}& s_{22}+s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{32}\end{array}\right]& \left(5\right)\end{array}$

(20) 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:

(21) $\begin{array}{cc}\frac{b_2}{v_s}=\frac{1}{2}.Math.\frac{1-s_{22}^{\mathrm{ss}}}{\left(1-s_{22}^{\mathrm{ss}}{\tilde{s}}_{11}\right){\tilde{s}}_{21}^{-1}\left(1-{\tilde{s}}_{22}{s}_{11}^{\mathrm{scope}}\right)-s_{22}^{\mathrm{ss}}{\tilde{s}}_{12}{s}_{11}^{\mathrm{scope}}}& \left(6\right)\end{array}$
The terms of equation (5) may be inserted into equation (6), and equation (6) can be rewritten as:

(22) $\begin{array}{cc}\frac{b_2}{v_s}=\frac{1}{2}.Math.\frac{\left(1-s_{22}^{\mathrm{ss}}\right)\left(s_{21}+s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}\right)}{\begin{array}{c}1-s_{22}^{\mathrm{ss}}\left(s_{11}+s_{12}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}\right)-\left(s_{22}+s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{32}\right)s_{11}^{\mathrm{scope}}+\\ \left[\begin{array}{c}\left(s_{11}{s}_{22}-s_{21}{s}_{12}\right)+\\ \left(s_{13}{s}_{31}{s}_{22}+s_{23}{s}_{32}{s}_{11}-s_{23}{s}_{31}{s}_{12}+s_{13}{s}_{32}{s}_{21}\right){\tilde{s}}_{11}^{\mathrm{DUT}}\end{array}\right]s_{22}^{\mathrm{ss}}{s}_{11}^{\mathrm{scope}}\end{array}}& \left(7\right)\end{array}$

(23) In equation (7), .sub.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.sub.2 is the voltage acquired by the oscilloscope 200 when .sub.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.sub.22.sup.ss, corresponding to the signal source output impedance, may also be measured ahead of time and stored in memory. s.sub.11.sup.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)}.sub.11.sup.DUT which may be computed from equation (7). Once {tilde over (s)}.sub.11.sup.DUT is obtained from equation (7), s.sub.11.sup.DUT, the reflection coefficient of the DUT 206 may be calculated using equation (4) above.

(24) Equation (7) may be simplified with some assumptions. First, the S-parameters of an ideal power divider is:

(25) $\begin{array}{cc}s=\left[\begin{array}{ccc}{s}_{11}& s_{12}& s_{13}\\ s_{21}& s_{22}& s_{23}\\ s_{31}& s_{32}& s_{33}\end{array}\right]=\left[\begin{array}{ccc}0& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& 0& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& 0\end{array}\right]& \left(8\right)\end{array}$
For an ideal signal generator and an ideal real-time oscilloscope, the signal source impedance and oscilloscope input impedance are zero:
s.sub.22.sup.ss=0, s.sub.11.sup.scope=0.(9)

(26) 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:

(27) $\begin{array}{cc}\frac{b_2}{v_s}=\frac{1}{4}\left(1+\frac{1}{2}{s}_{11}^{\mathrm{DUT}}\right)& \left(10\right)\end{array}$

(28) 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:
s.sub.22.sup.sss.sub.11.sup.scope<<1, s.sub.22.sup.sss.sub.11<<1, s.sub.22s.sub.11.sup.scope<<1(11)
Then equation (7) can be approximated as:

(29) $\begin{array}{cc}\frac{b_2}{v_s}=\frac{1}{2}.Math.\frac{\left(1-s_{22}^{\mathrm{ss}}\right)\left(s_{21}+s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}\right)}{1-s_{22}^{\mathrm{ss}}{s}_{12}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{31}-s_{23}{\tilde{s}}_{11}^{\mathrm{DUT}}{s}_{32}{s}_{11}^{\mathrm{scope}}}& \left(12\right)\end{array}$
Equation (12) provides a working practical approximation of the reflection coefficient of the DUT 206.

(30) To measure s.sub.22.sup.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.sub.11.sup.DUT discussed above may be repeated to calculate the port two 212 reflection coefficient s.sub.22.sup.DUT.

(31) The insertion loss and/or crosstalk terms s.sub.12.sup.DUT, s.sub.21.sup.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.

(32) The transfer function from the signal generator's 202 voltage source to the input to the real-time oscilloscope can be derived in the same way as equation (6):

(33) 0$\begin{array}{cc}\frac{b_2}{v_s}=\frac{1}{2}.Math.\frac{1-s_{22}^{\mathrm{ss}}}{\left(1-s_{22}^{\mathrm{ss}}{s}_{11}^{\mathrm{DUT}}\right)s_{11}^{\mathrm{DUT}-1}\left(1-s_{22}^{\mathrm{DUT}}{s}_{11}^{\mathrm{scope}}\right)-s_{22}^{\mathrm{ss}}{s}_{12}^{\mathrm{DUT}}{s}_{11}^{\mathrm{scope}}}& \left(13\right)\end{array}$
Using the assumptions of equation (11), equation (13) can be approximated as

(34) $\begin{array}{cc}\frac{b_2}{v_s}=\frac{1}{2}.Math.\frac{\left(1-s_{22}^{\mathrm{ss}}\right)s_{21}^{\mathrm{DUT}}}{1-s_{22}^{\mathrm{ss}}{s}_{11}^{\mathrm{DUT}}-s_{22}^{\mathrm{DUT}}{s}_{11}^{\mathrm{scope}}}& \left(14\right)\end{array}$
Since s.sub.11.sup.DUT and s.sub.22.sup.DUT have been calculated as discussed above, there is only one unknown variable, s.sub.21.sup.DUT, in equation (14). Therefore, equation (14) can be solved for s.sub.21.sup.DUT. b.sub.2 in equation (14) is the voltage acquired by the oscilloscope 200 when .sub.s is generated when the devices are configured as in FIG. 4.

(35) To measure s.sub.12.sup.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.sub.21.sup.DUT may be repeated to measure s.sub.12.sup.DUT.

(36) 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.

(37) 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.