Incomplete 2-Port Vector Network Analyzer
Calibration Methods
A. Henze 1, N. Tempone 2, G. Monasterios 3, H. Silva 4
RF Metrology Laboratory – Instituto Nacional de Tecnología Industrial (INTI) Buenos Aires, Argentina 1 ahenze@inti.gov.ar
2 ntempone@inti.gob.ar 3 guillem@inti.gov.ar 4 hsilva@inti.gov.ar
Abstract— Different types of incomplete 2-port vector
network analyzer (VNA) calibration methods are explained. All of them are particular cases of the 12term error model and a comparison, including advantages and disadvantages, between them and a Full 2-Port method, such as TOSM, is made.
Resumen— En el presente informe se explican los distintos tipos de calibración incompleta de un VNA de 2 puertos. Todos ellos son casos particulares del modelo de 12 términos de error, y se realiza una comparación, incluyendo ventajas y desventajas, entre ellos respecto a un método Full 2-Port como el método TOSM.
where forward error terms are
e00 : e11 : e10e01 : e10e32 : e30 : e22 :
Directivity (F) Port-1 Source Match (F) Reflection Tracking (F) Transmission Tracking (F) Leakage (Crosstalk)(F) Port-2 Load Match (F)
I. INTRODUCTION
When calibrating a 2-Port VNA, Full 2-Port calibration is usually employed [1]. There are different types of methods depending on the error model to be considered. The most common calibration method used for coaxial systems is TOSM (also known as SOLT) which uses the 12term error model [2]. However, this method requires an 8step procedure to get both ports calibrated.
When is not necessary to measure all four scattering parameters (i.e. S11, S12, S21 and S22) of the Device Under Test (DUT) or uncertainties are not necessary to be as small as possible, alternative calibration methods may be employed. Advantages and disadvantages must be previously considered in order to determine which one will be the best option for each particular case.
II. TOSM CALIBRATION
Before measuring any DUT S-parameters, both VNA’s ports must be first calibrated in order to calculate system errors. Most common employed method is TOSM. It consists in calculating 6 forward (F) and 6 reverse (R) error terms as shown in figures 1 and 2.
Fig. 2 Reverse 12-term error model flow chart
where reverse error terms are
e′33 : e′11 : e′23e′32 : e′23e′01 : e′03 : e′22 :
Directivity (R) Port-1 Load Match (R) Reflection Tracking (R) Transmission Tracking (R) Leakage (Crosstalk) (R) Port-2 Source Match (R)
Solving measured S-parameters from figures 1 and 2 [2]
( ) S11M =
b0 a0
=
e00
+
1−
e10e01 ⋅ e11S11 −
S11 − e22∆ S e22 S22 + e11e22∆ S
(1)
S 21M
=
b3 a0
= e30
+
e10e32 ⋅ S21
1 − e11S11 − e22 S22 + e11e22∆ S
(2)
( ) S22M
=
b3′ a3′
=
e'33
+
1
−
e'23 e'32 ⋅ S22 − e'11 S11 − e'22 S22
e'11 ∆ S + e'11 e'22
∆S
(3)
S12M
=
b0′ a3′
=
e'03
+
1
−
e'11
S11
e'23 e'01⋅S12 − e'22 S22 +
e'11
e'22
∆S
(4)
where
Fig. 1 Forward 12-term error model flow chart
SxxM: Sxx:
Measured, i.e. uncorrected, S-parameters Corrected S-parameters
∆ S = S11S22 − S12S21
(5)
A. VNA Calibration
Calibration procedure consists in measuring 7 different reference standards (2 Opens, 2 Shorts, 2 Matches and a Thru) with known reflection and/or transmission values from a TOSM calibration kit. In this paper reference standards are considered to have ideal values as follows
ΓOPEN = 1
(6)
ΓSHORT = −1
(7)
ΓMATCH = 0
(8)
0 1 STHRU =1 0
(9)
To perform a complete 2-Port calibration, an 8-step procedure must be done as follows
Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8:
Connect Open1 to Port 1 Connect Short1 to Port 1 Connect Match1 to Port 1 Connect Open2 to Port 2 Connect Short2 to Port 2 Connect Match2 to Port 2 Connect Match1 Port 1 / Match2 Port 2 Connect Thru between Port 1 and Port 2
1) Port 1 Calibration: Making steps 1 to 3, an OSM calibration [2] to Port 1 is performed and the following forward error terms are calculated from (1)
e00 = S11M (match1 )
(10)
e11
=
S11M (open1) + S11M (short1) − 2 ⋅ e00 S11M (open1) − S11M (short1)
[ ] [ ] e10e01 =
− 2⋅
S11M (open1) − e00 ⋅ S11M (short1) − e00 S11M (open1 ) − S11M (short1 )
(11) (12)
Reference
Open1 Short1 Match1
TABLE I PORT 1 CALIBRATION SUMMARY
Error to be corrected
e11 e10e01
e00
Description
Source Match (F) Reflection Tracking (F)
Directivity (F)
2) Port 2 Calibration: Making steps 4 to 6, an OSM calibration to Port 2 is performed and the following reverse error terms are calculated from (3)
e´33= S22M (match2 )
(13)
e´11=
S22M (open2 ) + S22M (short2 ) − 2 ⋅ e´33 S22M (open2 ) − S22M (short2 )
(14)
[ ] [ ] e´23 e´32=
− 2⋅
S22M (open2 ) − e´33 ⋅ S22M (short2 ) − e´33 S22M (open2 ) − S22M (short2 )
(15)
Reference
Open2 Short2 Match2
TABLE II PORT 2 CALIBRATION SUMMARY
Error to be corrected
e′11 e′23e′32
e′33
Description
Source Match (R) Reflection Tracking (R)
Directivity (R)
3) Isolation Ports Calibration: Step 7 is optionally made only when very low transmission parameters must be measured. In most cases this error term is neglected.
( ) e30 = S21M match1,2
(16)
( ) e0′3 = S12M match1,2
(17)
TABLE III ISOLATION PORTS CALIBRATION SUMMARY
Reference
Match1 Match2
Error to be corrected
e30 e’03
Description
Crosstalk (F) Crosstalk (R)
4) Calibration between Ports: When making step 8 both Load Match and Transmission Tracking error terms are calculated from (1), (2), (3) and (4) as follows
e22
=
S11M (Thru) − e00 S11M (Thru) ⋅ e11 − ∆e
(18)
[ ] e10e32 = S21M (Thru) − e30 ⋅ (1 − e11e22 )
(19)
e1′1
=
S22M (Thru) − e3′3 S22M (Thru) − ∆e′
(20)
[ ] e2′3e3′2 = S12M (Thru) − e0′3 ⋅ (1− e3′3e1′1)
(21)
where
∆e = e00.e11 − e01e10
(22)
∆e´= e´33.e´22 −e´23 e´32
(23)
TABLE IV CALIBRATION BETWEEN PORTS SUMMARY
Reference Thru
Error to be corrected
e22 e’11 e10e32 e’23e’01
Description
Load Match (F) Load Match (R) Transmission Tracking (F) Transmission Tracking (R)
Equations (10) to (21) represent the 12 error terms to be calculated.
B. DUT S-Parameters Measurement
Solving equations (1) to (4), corrected S-parameters of the DUT can be expressed as follows [1]−[4]
( ) S11 =
A11 ⋅
1+
A22
⋅ e2′2 D
− e22
⋅ A21 ⋅
A12
(24)
[ ] ( ) S21 =
A21 ⋅
1 + A22 ⋅ D
e2′2
− e22
(25)
( ) S22 =
A22 ⋅
1+
A11 ⋅ e11 D
− e1′1 ⋅ A21 ⋅ A12
(26)
[ ] ( ) S12 =
A12 ⋅
1+
A11 ⋅ D
e11
− e1′1
(27)
where
N11
=
S11M − e00 e10e01
N12
=
S12M −e0′3 e′23e0′1
N 21
=
S21M − e30 e10 e32
N 22
=
S22M − e3′3 e2′3e3′2
( ) ( ) D = 1 + A11 ⋅ e11 ⋅ 1 + A22 ⋅ e2′2 − A21 ⋅ A12 ⋅ e22 ⋅ e1′1
(28) (29) (30) (31) (32)
where Nxx are normalized S-parameters [4].
C. Full 2-Port: Advantages and Disadvantages
1) Advantages: Provides low uncertainties as all 12 error terms are calculated and all four DUT S-parameters are measured.
used [5]. All of them are partial calibrations based on TOSM method described in section II.
To simplify mathematical expressions, crosstalk error terms will be considered null valued for all cases
e30 = 0
(33)
e0′3 = 0
(34)
IV. TRANSMISSION RESPONSE
It is the simplest 2-Port calibration method and is used when only S21 (or S12) parameter is of interest. A Thru reference element is connected between ports for the calibration, so only transmission tracking error term is partially calculated. This causes the highest uncertainties in transmission S-parameter measurements.
A. VNA Calibration (between Ports)
From (2) and (33):
S 21M
=
1
−
e11S11
(Thru
e10e32 ⋅ S21 (Thru ) ) − e22S22 (Thru ) +
e11e22
∆
S
(Thru
)
(35)
Applying (9) in (30)
S 21M
=
e10 e32
⋅
1
−
1 e11e22
(36)
As neither e11 nor e22 are calculated, the correction term related to them is considered null valued.
e11e22 = 0
(37)
Replacing (37) in (36)
2) Disadvantages: Needs an 8-step procedure calibration. It is always necessary to measure all four DUT Sparameters even if only one is needed to be corrected.
III. INCOMPLETE 2-PORT VNA CALIBRATION In the past, VNAs had only a transmission/reflection (T/R) test set. This allowed only forward parameters to be measured, since Port 1 acted as a source and Port 2 as a load. Then, calibration methods used were:
• Transmission Response (TR) • 1-Port + Normalization (1-P+N) • Enhanced Response(ER) • One-Path 2-Port (1-P 2-P)
e10e32 = S21M
(38)
Similar considerations are applied for the reverse transmission tracking term
e0′1e2′3 = S12M
(39)
B. DUT S21 (or S12) Measurement
As only S21 parameter is measured and e10e32 error term is calculated, equation (25) is reduced to
S 21
=
S 21M (DUT e10 e32
)
(40)
Nowadays, most VNAs have a full S-parameter test set.
Similar considerations can be applied for S12
This allows the source to be switched to both ports, hence it
is able to measure all four S-parameters and a TOSM, i.e. complete, calibration can be done.
S12
=
S12M (DUT ) e'23 e'01
(41)
However, when it is not necessary (or convenient for
some reason) to measure all four DUT S-parameters or,
uncertainties are not necessary to be as small as possible,
above mentioned incomplete calibration methods can be
C. Transmission Response: Advantages and Disadvantages
1) Advantages: Very fast one-step calibration procedure. Only S21 needs to be measured in order to get its corrected value, so a good option for unidirectional devices.
2) Disadvantages: Only for transmission (S21 or S12) parameters. As Transmission Tracking error term is not calculated correctly, this method is not very accurate with lossy DUTs. On the other hand, it is recommended only for insertable devices as in practice this method always considers ideal Thru values as in (9).
VI. ENHANCED RESPONSE It is an improvement of the 1-P+N method for measuring Forward (or Reverse) S-parameters. It needs the same four steps as before to calibrate the VNA but, in this case, it also calculates Load Match error term. This allows Transmission Tracking error term to be correctly calculated.
A. VNA Calibration
1) Port 1 (or Port 2) Calibration: Procedure is applied in the same manner as in Section II A.1 (or section II A.2).
2) Calibration between Ports: Procedure is applied in the same manner as in Section II A.4.
V. 1-PORT + NORMALIZATION This method performs a 1-Port calibration (at Port 1 or Port 2) and, separately, a transmission response. This is usually employed when only forward parameters (S11 and S21) or reverse parameters (S22 and S12) are required.
A. VNA Calibration
1) Port 1 (or Port 2) Calibration: Procedure is applied in the same manner as in Section II A.1 (or section II A.2).
2) Calibration between Ports: Procedure is applied in the same manner as in Section IV A.
B. DUT Forward (or Reverse) Parameters Measurement As DUT reverse parameters are not measured, e22 and all
reverse error terms are not calculated, hence all correction terms related to them in (24) and (25) are considered null valued. S11 can be expressed as follows
S11=
S11M (DUT ) − e00 S11M (DUT ) ⋅ e11 − ∆e
(42)
As OSM and transmission normalization calibrations are performed separately, S21 corrected value remains the same as in section IV B.
S 21
=
S 21M (DUT e10 e32
)
(43)
B. DUT Forward (or Reverse) Parameters Measurement
Although e22 is calculated in this case, DUT reverse parameters are not measured and none of the reverse error terms is calculated. Hence, all correction terms related to them in (24) and (25) are considered null valued and S11 and S21 can be derived as follows
S11=
S11M (DUT ) − e00 S11M (DUT ) ⋅ e11 − ∆e
(46)
S 21
=
S
21M
(
DUT
)
⋅
e01e10
e10e32 e11S11M (DUT )
−
∆e
(47)
Similar considerations are applied for reverse parameters
S 22 =
S22M (DUT ) − e3′3 S22M (DUT ) ⋅ e′22 − ∆e′
(48)
S12
=
S12M (DUT e2′ 3e0′ 1
)
⋅
e2′ 2
S
e2′ 3e3′ 2 22M (DUT
)
−
∆e′
(49)
C. Enhanced Response: Advantages and Disadvantages
1) Advantages: Calculates Transmission Tracking error term correctly.
Similar considerations are applied for reverse parameters
S 22 =
S22M (DUT ) − S22M (DUT ) ⋅ e2′2
e3′3 − ∆e′
(44)
S12
=
S12M (DUT e′23e0′ 1
)
(45)
C. 1-Port + Normalization: Advantages and Disadvantages
1) Advantages: Corrects directivity, reflection tracking and source match of Port 1 (or Port 2).
2) Disadvantages: Similar as in Transmission Response method.
2) Disadvantages: As only Forward (or Reverse) Sparameters are measured, Load Match value can not be used for correcting DUT S-parameters.
VII. ONE-PATH 2-PORT
Originally named One-Path Full 2-Port, was introduced to T/R VNAs in order to measure all four S-parameters. However, DUT must be manually reversed to measure Reverse S-parameters .
At present most VNAs have this calibration option, but special care must be taken as some manufactures consider ER method as 1-P 2-P.
A. VNA Calibration
This method considers Forward and Reverse models the same as follows
e00 =e3′3 e11 = e2′2 e10e01 = e′23e3′2 e10e32 = e0′1e′23
e22 =e1′1 e30 = e0′3
(50) (51) (52)
(53) (54) (55)
Hence, it needs the same four steps as in the previous methods to calibrate the VNA and only 6 forward error terms are needed to be calculated using (10), (11), (12), (16), (18) and (19).
B. DUT Parameters Measurement
When measuring DUT device, forward parameters are measured first, and then DUT is connected backwards and reverse parameters are measured. This allows equations (24) to (27) to be used with no correction terms null valued.
C. One Path 2-Port: Advantages and Disadvantages
Fig. 3 |S11| maximal deviation for 1-Port + Normalization and Enhanced Response with respect to TOSM method.
B. S21 Deviation Results
Different maximal deviations of |S21| for TR and ER methods respect to TOSM are shown in figures 4 and 5 respectively for a DUT having a nominal attenuation value of 0 dB.
1) Advantages: All four DUT S-parameters can be measured. As Forward and Reverse error terms have same values, only a four-step procedure is needed to calibrate the VNA.
2) Disadvantages: Not recommended for VNAs using any combination of coaxial sexed port connectors due to the necessity of adapters. A series of single sweep and DUT manually change procedure must be performed in order to get all four S-parameters. In practice, uncertainties may be higher that Full 2-Port due to connector mechanical repeatability or cable flexibility.
VIII. SIMULATIONS
A series of comparisons between incomplete calibrations methods respect to TOSM were carried out. S11 and S21 measurements were simulated and maximal deviation results are shown in figures 3 to 6.
Fig. 4 |S21|dB maximal deviation for Transmission Response with respect to TOSM method when measuring a S21 value of 0 dB.
A. S11 Deviation Results
Different maximal deviations of |S11| for 1-P+N and ER methods respect to TOSM are shown in figure 3. As in both incomplete methods, S11 has the same value (see equations (42) and (46)), such deviations respect to TOSM are the same. These deviations depend on Port 2 Load Match, i.e. e22 value, and DUT´s attenuation, i.e. S21 value.
For example, if the DUT consists of a 6-dB attenuator and |e22| = 0.1, then maximal deviation respect to TOSM method will be 0.026.
If now the DUT consists of a coaxial cable with nominal 0 dB attenuation value, and |e22| remains in 0.1, then maximal deviation respect to TOSM method arises to 0.100.
Fig. 5 |S21|dB maximal deviation for Enhanced Response with respect to TOSM method when measuring a S21 value between 0 dB and 6 dB.
For example, if DUT´s parameters |S11| = |S22| = 0.1, and VNA´s error terms |e11| = |e22| = 0.1, then maximal deviation respect to TOSM method will be 0.17 dB for TR method and 0.09 dB for ER method.
According to figures 5 and 6, if now the same DUT has an attenuation value of 6 dB and all other values remain the same, then maximal deviation respect to TOSM method arises to 0.24 dB for TR method and remains in 0.09 dB for ER method.
Fig. 6 |S21|dB maximal deviation for Transmission Response with respect to TOSM method when measuring a S21 value of -6 dB.
IX. CONCLUSIONS
Different types of incomplete 2-Port VNA calibrations methods are explained in this paper. Each one of them has its own advantages and disadvantages respect to a complete 2-Port method as TOSM.
In practice, if VNA´s error terms and/or DUT´s mismatches are quite low, there will be no significant deviation between incomplete and complete calibration methods when measuring S11 or S21. On the contrary, if VNA´s source and load match error term values are considerable and also DUT is lossy, then incomplete methods are not suitable due to the significant deviations they may have.
REFERENCES
[1] AN 1287-3, “Applying error correction for VNAs”, 2nd. Ed., Santa Clara, CA: Agilent Tech., 2002.
[2] D. Rytting, “Network analyzer error models and calibration methods”, Palo Alto, CA, Hewlett Packard Inc., 1998.
[3] B. Hall, “VNA error models: Comments on EURAMET/cg-2/v.01”, ANAMET Report 051, Measurement Standards Laboratory of New Zealand Lower Hutt, New Zealand, 2010.
[4] J. Dunsmore, “Handbook of microwave component measurement”, Agilent Tech., John Wiley & Sons, Ltd., UK, 2012.
[5] M. Hiebel, “Fundamentals of vector network analysis”, Rohde & Schwarz GmbH & Co. KG, 5th Ed., 2011.
[6] G. Wübbeler, Clemens Elster, Thomas Reichel and Rolf Judaschke, “Determination of the complex residual error parameters of a calibrated one-port vector network analyzer”, IEEE Transactions, Vol. 58, No. 9, 2009.
[7] AN 1287-11, “Specifying calibration standards and kits for Agilent vector network analyzers”, Agilent Tech., 2011.
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