# Effects of uncorrected RF performance in a vector network analyzer.

Effects of Uncorrected RF Performance in a Vector Network Analyzer

Introduction

Modern network analyzers greatly have enhanced the productivity of the microwave industry. Error-correction techniques have made it possible to use imperfect hardware and still achieve very good performance. A significant question is, "What really contributes to the accuracy and performance of a network analyzer?" Certainly, the error-correction concept of mathematically removing hardware errors has made a significant impact. New error-correction methods, like TRL, LRL, TRM and LRM, have simplified the calibration process and also provided higher accuracy. [1-4] Through close tolerance machining procedures, the quality of the calibration standards has increased the error-correction accuracy. Also, the modeling of standards has improved significantly. However, one overlooked fact is uncorrected, raw hardware performance and its effect on system accuracy.

A common misconception is that error correction does it all, and that it can calibrate out or quantify any level of uncorrected analyzer system performance. It is clear that the more stable the hardware, the better the calibration process can correct the errors. The calibration then will remain stable as a function of time and temperature, and calibrations will not need to be updated as often.

This paper focuses on the impact of uncorrected raw hardware performance on the final error-corrected performance of a network analyzer. A glossary of terms is provided in Appendix A.

The Calibration Process

There are two primary methods of error correction used in the industry today. The first method uses as many known standards as there are error terms. From the measurement of these known standards, a resultant set of simultaneous equations can be solved for the error terms. The open, short, load and through (OSLT) method is an example of this first method. It has been in use by the industry for many years. The second method takes into account redundancies that exist in many network analyzer systems that allow the system's error terms to be determined along with some of the characteristics of the calibration standards. TRL and LRM are examples of the second method. Both methods assume the error terms are systematic and stationary and that there are no nonlinearities. This means that random and nonstationary errors, such as noise, distortion, nonlinearity, frequency drift, instability, switch repeatability changes, connector repeatability changes and cable instability, are not corrected. Even he best hardware does not completely meet this stability criteria. Now one asks, "What is the effect of the hardware instability? How and to what degree do the errors degrade the final performance? Are there methods to reduce these problems and how effective are they?"

Impact of Raw Performance

on Error-Corrected Performance

Effect of Unstable Directivity,

Impedance Mismatch

and Tracking Errors

A flow graph of a one-port circuit measurement system is shown in Figure 1. These error terms correspond directly to the uncorrected, raw hardware performance. The system directivity is determined mainly by the directional coupler. The system tracking error is determined by how well the reference and test channels track each other as a function of frequency. The system port impedence match consists of the impedance match terms of all the components in the system, including the coupler, bias network, step attenuators, power splitters and switches.

The measured [[Gamma].sub.m] can be calculated in terms of the actual test device [[Gamma].sub.a] and the error terms from the flow graph shown in Figure 1. [Mathematical Expression Omitted] (1)

From Equation 1 and from the use of calibration procedure, the calculated error terms ([D.sub.c] [T.sub.c] and [M.sub.c]) can be determined. Equation 1 then can be solved for the calculated return loss [[Gamma].sub.ac] of the test device, [Mathematical Expression Omitted] (2)

Substituting for [[Gamma].sub.m] from Equation 1 and deleting second order terms yields, [Mathemathical Expression Omitted] (3)

After error correction, residual error terms can now be defined, [delta] = D-[D.sub.c]/1 + [T.sub.c] = residual system directivity (4) [tau] = T-[T.sub.c]/1 + [T.sub.c] = residual system tracking (5) [mu] = M-[M.sub.c] = residual system port impedance match (6)

It was anticipated that the calculated error terms ([D.sub.c,] [T.sub.c] and [M.sub.c]) in a perfect system are equal to the actual error terms (D, T and M) and will cancel. But, they are not equal due to an imperfect system and imperfect calibration standards. The resultant flow graph for the residual error terms ([delta], [tau] and [mu]) is shown in Figure 2. The calculated [[Gamma].sub.ac] is, [[Gamma].sub.ac [congruent with] [delta] + (1 + [tau])[[Gamma].sub.a] + [mu][[Gamma].sub.a][.sup.2] (7)

Typical values after error correction for the residual directivity [delta] are in the 35 to 60 dB range. Typical values for the residual port impedance match [mu] are from 30 to 60 dB return loss. The tracking term (1 + [tau]) normally varies from 0 to [+ or -]0.1 dB. The wide range of error-corrected performance is determined by the connector size, frequency, calibration method and the quality of the calibration standards.

The sensitivity of [[Gamma].sub.ac] to the uncorrected error terms (D, T and M) is determined by taking the partial derivative of [[Gamma].sub.ac,] as defined in Equation 3. The calculated error terms ([D.sub.c,] [T.sub.c] and [M.sub.c] are stationary and don't change after error correction. The partial derivative is defined as, d[[Gamma].sub.ac] = [Mathematical Expression Omitted] (8)

Taking the partial derivative of Equation 3 and dropping second order terms yields, d[[Gamma].sub.ac][congruent with] [Mathematical Expression Omitted] (9) and using the safe assumption that [T.sub.c][congruent with] T, D[[Gamma].sub.ac] = [Mathematical Expression Omitted] (10) Equation 10 clearly shows the effect of changes in directivity, tracking and impedance mismatch on the resultant error-corrected measurement. Both the stability of the error terms (dD, dT and dM) and the absolute value of (1 + T) both contribute to the stability. Also, the stability will change as a function of the test device [[Gamma].sub.a].

Adding Insertion Loss

after the Test Port

Equation 10 shows the effect if there is insertion loss caused by adapters, cables or fixtures after the directional device. The change in the error-corrected [unkeyable][sub.ac] will be dD/(1 + T), where dD is the change in directivity and (1 + T) is the tracking term, which also describes the loss in the system. Thus, an increase in the loss will magnify the impact of coupler instability. A 3 dB cable loss will cause (1 + T = 0.5), which will double the directivity sensitivity.

This also can be illustrated by referring to Equation 1. Figure 3 is a vector diagram of Equation 1 with a perfect port impedance match, showing that changes in the system directivity are more critical if there is an insertion loss. In Figure 3b if the tracking term (1 + T) is decreased, then the sensitivity is increased of [[Gamma].sub.m] to changes in the directivity error term D.

This sensitivity increase becomes more pronounced at higher freequencies where the losses are the greatest from coaxial cables and other components. The necessity for stability of the directivity term is even more important at higher frequencies. The test port of the network analyzer needs to be as close as possible to the test device.

Raw Port Impedance Match and

Effect on Transmission Tracking

The raw port impedance mismatch causes an error in the error-corrected transmission tracking. This is true even if the port impedance match is stable. The flow graph of Figure 4 shows how this occurs.

From Figure 4, the actual [S.sub.21m] can be calculated, [S.sub.21m] = b/a = [T.sub.t]/[1-M.sub.1.M.sub.2] (11) The calculated [T.sub.tc] is solved for with Equation 11, where the calculated values for [M.sub.1] and [M.sub.2], defined as [M.sub.ac] and [M.sub.2c], are used.

[T.sub.tc = [S.sub.21m]([1-M.sub.1c.M.sub.2c]) = [T.sub.t] [1-M.sub.1c.M.sub.2c]/[1-M.sub.1.M.sub.2] (12) where the calculated terms are defined as, [M.sub.1c] = [M.sub.1] + [[mu].sub.1] [[mu].sub.1] = residual port-1 impedance match [M.sub.2c] = [M.sub.2] + [[mu].sub.2] [[mu].sub.2] = residucal port-2 impedance match

The final result after substituting Equations 13 and 14 into 12 and simplifying yields, [T.sub.tc] [is congruent with] [T.sub.t] (1 - [M.sub.1][[mu].sub.2] - [M.sub.2][[mu].sub.1])

From Equation 15, the error in the transmission tracking term is [M.sub.1][[mu].sub.2] + [M.sub.2][[mu].sub.1]. The effect of this analysis can be seen in a system where the raw port impedance match [M.sub.1] and [M.sub.2] reflection coefficient is 0.316 with 10 dB return loss and the residual error-corrected port impedance match ([[mu].sub.1] and [[mu]sub.2]) reflection coefficient is 0.01 with 40 dB return loss. The error, in linear terms, will be [+ or -]0.00632, which is significant for precision, low loss measurements. Two ways to reduce this error are to achieve a better residual port impedance match using higher quality standards and error-correction methods or to imrove the uncorrected raw performance of the test ports. If the uncorrected performance is improved to 20 dB return loss, the error will only be [+ or -]0.017 dB.

Environmental Issues

It is very important that the test sets' uncorrected performance does not change with thermal effects. This requires that the samplers, splitters and cables track each other with temperature, and that the coupler, step attenuator and bias network be very stable. Also, as mechanical stress and pressure is applied to the test port, it is important that the raw performance does not change. This means that the test port must be mechanically isolated from the critical components in the test set. Figure 5 shows the stability of an older 40 GHz test set with environmental changes when measuring a 50 [Omega] termination. The new 50 GHz test set was designated with stability as a major goal. Figure 6 shows the improved performance to the same environmental changes.

Improving Performance

of Test Set Components

After years of improvements in the performance of calibration standards and the development of more powerful calibration algorithms, the key now to improving the performance of network analyzers is to improve the quality of hardware components. The front-end test set components, such as the directional couplers, bias networks, step attenuators and switches, need to be stable and have excellent raw uncorrected performance. The frontend mixers and samplers must track each other with temperature and time so that their errors are ratioed out. The IF system must be free from frequency drift and nonlinearities so they will not degrade the tracking error term. The calibration standards must be machined to the state-of-the-art accuracy and modeled accurately. Connectors must be repeatable and rugged. The test ports must be able to withstand mechanical stress without changing electrically. Cables must be as low loss and stable as they are flexed and as the temperature changes.

Time Domain View

of Raw Performance

By using time domain, some insight into theraw uncorrected performance of the test set can be determined. By putting a short at the end of an air line and connecting the air line to the test port, thevarious reflections in the system are identified. The main hardware components can be located and their design can be improved. Figure 7 shows a schematic diagram of the test set fron-end for a network analyzer with the air line and short connected to port 1.

Figure 8 shows the time domain response of a network analyzer without error correction. All of the time domain responses prior to the large response of the short are the directivity errors. The time domain responses after the short are the port impedance match erros. All of these responses need to be small and unchanging. When error correction is applied, the errors are reduced greatly and only the residual directivity and port impedance match terms are visible, as shown in Figure 9. Hoever, if a hardware component is unstable, the time domain response is degreded, as shown in Figure 10, which shows a bump in the time domaain response where the cable connects to the switch power splitter. Stress on the test port causes this connector instability. The connector instability will cause a ripple error of [+ or -] 0.3 dB. Time domain is an excellent tool for locating instabilities and pinpointing the hardware components that need to be improved.

Conclusion

The effects of raw, uncorrected hardware performance on an error-corrected network analyzer have been described. It is essential that the hardware be of highest quality to achieve the performance demands of today's measurement standards. No longer is it possible to say that the error correction will address all of the hardware-connected inaccuracies. The true quality of the measurement rests in the combination of modern error-correction techniques, high quality calibration standards and high quality hardware with a high level of performance and stability.

References

[1] G.F. Engen and C.A. Hoer, "Thru-Reflect-Line: An Improved Technique for Calibrating the Dual 6-Port Automatic Network Analyzer," IEEE Tran. on Microwave Theory and Techniques, MTT-27-2, Dec. 1979, pp. 983-987.

[2] C.A. Hoer and G.G. Engen, "Calibrating a Dual Six-Port or Four-Port for Measuring Two-Ports with any Connectors," 1986 IEEE MTT-S International Microwave Symposium Digest, June 1986, pp. 665-668.

[3] H.J. Eul and B. Schiek, "Thru-Match-Reflect: One Result of a Rigorous Theory for De-embedding and Network Analyzer Calibration," Pro. 18th European Microwave Conf., Sept. 1988, pp. 909-914.

[4] J.T. Barr and M.J. Pervere, "A Generalized Vector Network Analyzer Calibration Technique," 34th ARFTG Digest, Ft. Lauderdale, Dec. 1989.

Doug Rytting received his BSEE degree from Utah State University and his MSEE degree from Stanford University. Currently, he is research and development section manager for microwave network analyzers at Hewlett-Packard's Network Measurement Division. In 1966, Rytting joined Hewlett-Packard, where he has worked on virtually all of its high frequency network analyzer projects culminating with the new HP 8510C.

Introduction

Modern network analyzers greatly have enhanced the productivity of the microwave industry. Error-correction techniques have made it possible to use imperfect hardware and still achieve very good performance. A significant question is, "What really contributes to the accuracy and performance of a network analyzer?" Certainly, the error-correction concept of mathematically removing hardware errors has made a significant impact. New error-correction methods, like TRL, LRL, TRM and LRM, have simplified the calibration process and also provided higher accuracy. [1-4] Through close tolerance machining procedures, the quality of the calibration standards has increased the error-correction accuracy. Also, the modeling of standards has improved significantly. However, one overlooked fact is uncorrected, raw hardware performance and its effect on system accuracy.

A common misconception is that error correction does it all, and that it can calibrate out or quantify any level of uncorrected analyzer system performance. It is clear that the more stable the hardware, the better the calibration process can correct the errors. The calibration then will remain stable as a function of time and temperature, and calibrations will not need to be updated as often.

This paper focuses on the impact of uncorrected raw hardware performance on the final error-corrected performance of a network analyzer. A glossary of terms is provided in Appendix A.

The Calibration Process

There are two primary methods of error correction used in the industry today. The first method uses as many known standards as there are error terms. From the measurement of these known standards, a resultant set of simultaneous equations can be solved for the error terms. The open, short, load and through (OSLT) method is an example of this first method. It has been in use by the industry for many years. The second method takes into account redundancies that exist in many network analyzer systems that allow the system's error terms to be determined along with some of the characteristics of the calibration standards. TRL and LRM are examples of the second method. Both methods assume the error terms are systematic and stationary and that there are no nonlinearities. This means that random and nonstationary errors, such as noise, distortion, nonlinearity, frequency drift, instability, switch repeatability changes, connector repeatability changes and cable instability, are not corrected. Even he best hardware does not completely meet this stability criteria. Now one asks, "What is the effect of the hardware instability? How and to what degree do the errors degrade the final performance? Are there methods to reduce these problems and how effective are they?"

Impact of Raw Performance

on Error-Corrected Performance

Effect of Unstable Directivity,

Impedance Mismatch

and Tracking Errors

A flow graph of a one-port circuit measurement system is shown in Figure 1. These error terms correspond directly to the uncorrected, raw hardware performance. The system directivity is determined mainly by the directional coupler. The system tracking error is determined by how well the reference and test channels track each other as a function of frequency. The system port impedence match consists of the impedance match terms of all the components in the system, including the coupler, bias network, step attenuators, power splitters and switches.

The measured [[Gamma].sub.m] can be calculated in terms of the actual test device [[Gamma].sub.a] and the error terms from the flow graph shown in Figure 1. [Mathematical Expression Omitted] (1)

From Equation 1 and from the use of calibration procedure, the calculated error terms ([D.sub.c] [T.sub.c] and [M.sub.c]) can be determined. Equation 1 then can be solved for the calculated return loss [[Gamma].sub.ac] of the test device, [Mathematical Expression Omitted] (2)

Substituting for [[Gamma].sub.m] from Equation 1 and deleting second order terms yields, [Mathemathical Expression Omitted] (3)

After error correction, residual error terms can now be defined, [delta] = D-[D.sub.c]/1 + [T.sub.c] = residual system directivity (4) [tau] = T-[T.sub.c]/1 + [T.sub.c] = residual system tracking (5) [mu] = M-[M.sub.c] = residual system port impedance match (6)

It was anticipated that the calculated error terms ([D.sub.c,] [T.sub.c] and [M.sub.c]) in a perfect system are equal to the actual error terms (D, T and M) and will cancel. But, they are not equal due to an imperfect system and imperfect calibration standards. The resultant flow graph for the residual error terms ([delta], [tau] and [mu]) is shown in Figure 2. The calculated [[Gamma].sub.ac] is, [[Gamma].sub.ac [congruent with] [delta] + (1 + [tau])[[Gamma].sub.a] + [mu][[Gamma].sub.a][.sup.2] (7)

Typical values after error correction for the residual directivity [delta] are in the 35 to 60 dB range. Typical values for the residual port impedance match [mu] are from 30 to 60 dB return loss. The tracking term (1 + [tau]) normally varies from 0 to [+ or -]0.1 dB. The wide range of error-corrected performance is determined by the connector size, frequency, calibration method and the quality of the calibration standards.

The sensitivity of [[Gamma].sub.ac] to the uncorrected error terms (D, T and M) is determined by taking the partial derivative of [[Gamma].sub.ac,] as defined in Equation 3. The calculated error terms ([D.sub.c,] [T.sub.c] and [M.sub.c] are stationary and don't change after error correction. The partial derivative is defined as, d[[Gamma].sub.ac] = [Mathematical Expression Omitted] (8)

Taking the partial derivative of Equation 3 and dropping second order terms yields, d[[Gamma].sub.ac][congruent with] [Mathematical Expression Omitted] (9) and using the safe assumption that [T.sub.c][congruent with] T, D[[Gamma].sub.ac] = [Mathematical Expression Omitted] (10) Equation 10 clearly shows the effect of changes in directivity, tracking and impedance mismatch on the resultant error-corrected measurement. Both the stability of the error terms (dD, dT and dM) and the absolute value of (1 + T) both contribute to the stability. Also, the stability will change as a function of the test device [[Gamma].sub.a].

Adding Insertion Loss

after the Test Port

Equation 10 shows the effect if there is insertion loss caused by adapters, cables or fixtures after the directional device. The change in the error-corrected [unkeyable][sub.ac] will be dD/(1 + T), where dD is the change in directivity and (1 + T) is the tracking term, which also describes the loss in the system. Thus, an increase in the loss will magnify the impact of coupler instability. A 3 dB cable loss will cause (1 + T = 0.5), which will double the directivity sensitivity.

This also can be illustrated by referring to Equation 1. Figure 3 is a vector diagram of Equation 1 with a perfect port impedance match, showing that changes in the system directivity are more critical if there is an insertion loss. In Figure 3b if the tracking term (1 + T) is decreased, then the sensitivity is increased of [[Gamma].sub.m] to changes in the directivity error term D.

This sensitivity increase becomes more pronounced at higher freequencies where the losses are the greatest from coaxial cables and other components. The necessity for stability of the directivity term is even more important at higher frequencies. The test port of the network analyzer needs to be as close as possible to the test device.

Raw Port Impedance Match and

Effect on Transmission Tracking

The raw port impedance mismatch causes an error in the error-corrected transmission tracking. This is true even if the port impedance match is stable. The flow graph of Figure 4 shows how this occurs.

From Figure 4, the actual [S.sub.21m] can be calculated, [S.sub.21m] = b/a = [T.sub.t]/[1-M.sub.1.M.sub.2] (11) The calculated [T.sub.tc] is solved for with Equation 11, where the calculated values for [M.sub.1] and [M.sub.2], defined as [M.sub.ac] and [M.sub.2c], are used.

[T.sub.tc = [S.sub.21m]([1-M.sub.1c.M.sub.2c]) = [T.sub.t] [1-M.sub.1c.M.sub.2c]/[1-M.sub.1.M.sub.2] (12) where the calculated terms are defined as, [M.sub.1c] = [M.sub.1] + [[mu].sub.1] [[mu].sub.1] = residual port-1 impedance match [M.sub.2c] = [M.sub.2] + [[mu].sub.2] [[mu].sub.2] = residucal port-2 impedance match

The final result after substituting Equations 13 and 14 into 12 and simplifying yields, [T.sub.tc] [is congruent with] [T.sub.t] (1 - [M.sub.1][[mu].sub.2] - [M.sub.2][[mu].sub.1])

From Equation 15, the error in the transmission tracking term is [M.sub.1][[mu].sub.2] + [M.sub.2][[mu].sub.1]. The effect of this analysis can be seen in a system where the raw port impedance match [M.sub.1] and [M.sub.2] reflection coefficient is 0.316 with 10 dB return loss and the residual error-corrected port impedance match ([[mu].sub.1] and [[mu]sub.2]) reflection coefficient is 0.01 with 40 dB return loss. The error, in linear terms, will be [+ or -]0.00632, which is significant for precision, low loss measurements. Two ways to reduce this error are to achieve a better residual port impedance match using higher quality standards and error-correction methods or to imrove the uncorrected raw performance of the test ports. If the uncorrected performance is improved to 20 dB return loss, the error will only be [+ or -]0.017 dB.

Environmental Issues

It is very important that the test sets' uncorrected performance does not change with thermal effects. This requires that the samplers, splitters and cables track each other with temperature, and that the coupler, step attenuator and bias network be very stable. Also, as mechanical stress and pressure is applied to the test port, it is important that the raw performance does not change. This means that the test port must be mechanically isolated from the critical components in the test set. Figure 5 shows the stability of an older 40 GHz test set with environmental changes when measuring a 50 [Omega] termination. The new 50 GHz test set was designated with stability as a major goal. Figure 6 shows the improved performance to the same environmental changes.

Improving Performance

of Test Set Components

After years of improvements in the performance of calibration standards and the development of more powerful calibration algorithms, the key now to improving the performance of network analyzers is to improve the quality of hardware components. The front-end test set components, such as the directional couplers, bias networks, step attenuators and switches, need to be stable and have excellent raw uncorrected performance. The frontend mixers and samplers must track each other with temperature and time so that their errors are ratioed out. The IF system must be free from frequency drift and nonlinearities so they will not degrade the tracking error term. The calibration standards must be machined to the state-of-the-art accuracy and modeled accurately. Connectors must be repeatable and rugged. The test ports must be able to withstand mechanical stress without changing electrically. Cables must be as low loss and stable as they are flexed and as the temperature changes.

Time Domain View

of Raw Performance

By using time domain, some insight into theraw uncorrected performance of the test set can be determined. By putting a short at the end of an air line and connecting the air line to the test port, thevarious reflections in the system are identified. The main hardware components can be located and their design can be improved. Figure 7 shows a schematic diagram of the test set fron-end for a network analyzer with the air line and short connected to port 1.

Figure 8 shows the time domain response of a network analyzer without error correction. All of the time domain responses prior to the large response of the short are the directivity errors. The time domain responses after the short are the port impedance match erros. All of these responses need to be small and unchanging. When error correction is applied, the errors are reduced greatly and only the residual directivity and port impedance match terms are visible, as shown in Figure 9. Hoever, if a hardware component is unstable, the time domain response is degreded, as shown in Figure 10, which shows a bump in the time domaain response where the cable connects to the switch power splitter. Stress on the test port causes this connector instability. The connector instability will cause a ripple error of [+ or -] 0.3 dB. Time domain is an excellent tool for locating instabilities and pinpointing the hardware components that need to be improved.

Conclusion

The effects of raw, uncorrected hardware performance on an error-corrected network analyzer have been described. It is essential that the hardware be of highest quality to achieve the performance demands of today's measurement standards. No longer is it possible to say that the error correction will address all of the hardware-connected inaccuracies. The true quality of the measurement rests in the combination of modern error-correction techniques, high quality calibration standards and high quality hardware with a high level of performance and stability.

References

[1] G.F. Engen and C.A. Hoer, "Thru-Reflect-Line: An Improved Technique for Calibrating the Dual 6-Port Automatic Network Analyzer," IEEE Tran. on Microwave Theory and Techniques, MTT-27-2, Dec. 1979, pp. 983-987.

[2] C.A. Hoer and G.G. Engen, "Calibrating a Dual Six-Port or Four-Port for Measuring Two-Ports with any Connectors," 1986 IEEE MTT-S International Microwave Symposium Digest, June 1986, pp. 665-668.

[3] H.J. Eul and B. Schiek, "Thru-Match-Reflect: One Result of a Rigorous Theory for De-embedding and Network Analyzer Calibration," Pro. 18th European Microwave Conf., Sept. 1988, pp. 909-914.

[4] J.T. Barr and M.J. Pervere, "A Generalized Vector Network Analyzer Calibration Technique," 34th ARFTG Digest, Ft. Lauderdale, Dec. 1989.

Doug Rytting received his BSEE degree from Utah State University and his MSEE degree from Stanford University. Currently, he is research and development section manager for microwave network analyzers at Hewlett-Packard's Network Measurement Division. In 1966, Rytting joined Hewlett-Packard, where he has worked on virtually all of its high frequency network analyzer projects culminating with the new HP 8510C.

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Title Annotation: | radio frequency |
---|---|

Author: | Rytting, Doug |

Publication: | Microwave Journal |

Date: | Apr 1, 1991 |

Words: | 2372 |

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