S-Parameter Measurements

Transcription

1 Progress In Electromagnetics Research, Vol 144, , 014 Influence of SOLT Calibration Standards on Multiport VNA S-Parameter Measurements Wei Zhao *, Jiankang Xiao, and Hongbo Qin Abstract For the GSOLT calibration algorithm of n-port vector network analyzers (VNA), the sensitivity coefficients for the S-parameters of the n-port device under test (DUT) are developed as functions of the S-parameter deviations of SOLT standards By introducing the generalized flow graph of the 3n-term error model, analytic formulas for the S-parameter deviations of the n-port DUT with respect to the error terms have been deduced In addition, expressions for the deviations of the error terms in regard to the nonideal calibration elements are given by a series of matrix operations Finally, the analytic expressions of the sensitivity coefficients are concluded, which can be used for establishing the type-b uncertainty budget for S-parameter measurements 1 INTRODUCTION With the development of microwave technology, multiport devices, whose S-parameters are typically measured by multiport vector network analyzers (VNA), are becoming more widespread in RF systems [1] To achieve high precision, the calibration procedure should be implemented before the measurement [, 3] For the n-port VNA with n1 measurement channels, the general short-openload-thru (GSOLT) procedure is now widely used [4 6] In practical application, the S-parameters associated with the calibration standards short (S), open (O), load (L) and thru (T) are not ideal as expected Through the GSOLT calibration based on such non-ideal standards, the error terms and, consequently, the S-parameters of the DUT will deviate from their true values, so it is necessary to investigate the impact that the S-parameter deviations of SOLT standards have on the uncertainty of the S Although some techniques have been developed for the estimation of uncertainties in twoport VNA measurements, they are inapplicable to the multiport VNA due to the lack of use of matrix formalisms [7 9] To solve this, the sensitivity coefficients for the S-parameters of n-port DUT have been deduced in matrix form by using the concept of general node equation for the GSOLT calibration [10] However, it is still unclear how the nonideal SOLT standards affect the calibrated S In this paper, the generalized flow graph of the 3n-term error model, where nodes and branch gains are expressed by column vectors and square matrixes respectively, is proposed for the GSOLT calibration of the n-port VNA Based on this flow graph, the dependence of the S-parameter deviations on the error term deviations is solved in the error correction procedure Then the error term deviations associated with the non-ideal standards are calculated in the process of error calibration Finally, the analytic expressions for the sensitivity coefficients, which use the S-parameters of SOLT standards as input quantities and the S-parameters of the n-port DUT as output quantities, can be further concluded THEORY The 3n-term error model of the n-port VNA with n1 measurement channels is shown in Fig 1, where the number 1 represents the source port [6] In Fig 1, S mj1 (j = 1 n) are defined as the raw Received 3 December 013, Accepted 16 January 014, Scheduled 30 January 014 * Corresponding author: Wei Zhao (zhaowei @163com) The authors are with the School of Electronical & Mechanical Engineering, Xidian University, Xi an , China

2 304 Zhao, Xiao, and Qin Figure 1 Error model of the n-port VNA with the resource connected to port 1 Figure Transformations of the flow graph of the 3n-term error model Use of the node splitting rule Use of the GSM concept scattering parameters, meanwhile, the power waves at port j of the DUT are defined as a j1 and b j1 As compared with the model given in [4], this model also includes n-1 leakage errors between the excited port and the unexcited ports Before the error correction procedure, all the error coefficients E D, E S, E R, E X, E L and E T (i, j = 1 n, i j) are solved by the GSOLT procedure [4] Then when the S-parameter measurements of an n-port DUT are performed, the error correction will be applied to compute the actual S-parameters [6] To further simplify the error correction algorithm, the following transformations are made for the flow graph of the 3n-term error model in Fig 1 The flow graph (Fig ) is constructed from Fig 1 by using the node splitting rule and then further converted into the generalized flow graph (Fig ) by the concept of the generalized scattering matrix (GSM) [11 13] In Fig, the generalized nodes and branch gains are defined as 1 [E] = 1 1, [S m1 ] = S m11 S mj1 S mn1, [a 1 ] = a 11 a j1 a n1, [b 1 ] = b 11 b j1 b n1, (1)

3 Progress In Electromagnetics Research, Vol 144, E D11 1 [E DX1 ] = E Xi1, [I 1 ] = 0, 0 E Xn1 [E SL1 ] = E S11 E Li1, [E RT 1 ] = E R11 E T i1 E Ln1 Finally, the generalized flow graph of the 3n-term error model for port i (i = 1 in Fig 1) excitation can be summarized below: Table 1 Relationship between error terms and error matrixes E T n1 Error Matrix Error Term Source [E DXi ] E D Directivity E X Leakage [E SLi ] E S Source match E L Load match [E RT i ] E R Reflection tracking Transmission tracking E T The relationship between error terms and error matrixes is described in Table 1 For the error matrix [E UV i ], the subscript UVi means that the (i, i)th element is E U and the (j, i)th element is E V (i j) The generalized branch gain [I i ] has only one nonzero element 1 at the (i, i)th position According to the generalized 3-term error model, we can obtain the matrix equation as { } { } { } [Smi ] [EDXi ] [E = RT i ] [E] () [a i ] [I i ] [E SLi ] [b i ] 1 where the generalized power waves [E] = 1 1 [b i ] = b 1i b b ni, [S mi ] = S m1i S m S mni on the VNA side and [a i ] = on the DUT side From Eq (1), the reflected wave vector [b i ] and the incident wave vector [a i ] can be deduced as follows: a 1i a a ni [b i ] = [E RT i ] 1 {[S mi ] [E DXi ][E]} (3) [a i ] = [I i ][E] [E SLi ][b i ] (4) Once the reflected wave vectors [b i ] (i = 1 n) and the incident wave vectors [a i ] (i = 1 n) are calculated by Eq () and Eq (3), the S-matrix of the n-port DUT can be concluded as, [S] = [B][A] 1 (5)

4 306 Zhao, Xiao, and Qin where [A] = {[a 1 ], [a i ], [a n ]} and [B] = {[b 1 ], [b i ], [b n ]} However, nonideal calibration elements will cause the deviation of [S] from its true value In this paper, the effect of nonideal calibration elements (S, O, L and T) on the S-parameter measurements is investigated by the following steps: 1 Deviations of S-parameters In the first step, the dependence of [δs] of a DUT on the deviations of the error matrixes [E DXi ], [E RT i ] and [E SLi ] is derived According to the self loop rule [1], the generalized flow graph in Fig 3 can be simplified into that in Fig 4 Then, the series rule and the parallel rule are successively used to obtain the ratio between [E] and [S mi ] in Fig 4 [1] Figure 3 Generalized flow graph of the 3n-term error model Figure 4 Simplification of the generalized flow graph Removal of self-loop Combination of paths [I] is the identity matrix and the branch gain in Fig 4 can be also expressed as From Fig 4, we can get the following equation {[I] [S][E SLi ]} 1 [S] = [S]{[I] [E SLi ][S]} 1 (6) [E DXi ][E] [E RT i ]{[I] [S][E SLi ]} 1 [S][I i ][E] = [S mi ] (7) Assumed that the deviation [δs] is not affected by the variations of the raw values [S mi ], the incorrectly defined error terms will lead to the incorrect [S] [δs] [E DXi δe DXi ][E] [E RT i δe RT i ]{[I] [S δs][e SLi δe SLi ]} 1 [S δs][i i ][E] = [S mi ] (8) To solve the deviation matrix [δs], we need find the inverse of matrix in Eq (7) Let [D δd] 1 = [D] 1 [δx] where the deviation [δx] is underdetermined, when the deviation [δd] of a random nonsingular matrix [D] is very close to zero By using the definition of the inverse matrix [D δd]{[d] 1 [δx]} = [I], and omitting the product of small terms [δd][δx], we can get [δx] = [D] 1 [δd][d] 1 and the equation given below [14, 15]: {[D] [δd]} 1 = [D] 1 [D] 1 [δd][d] 1 (9)

5 Progress In Electromagnetics Research, Vol 144, Using Eq (8) and omitting the smaller term [δs][δe SLi ] can obtain {[I] [S δs][e SLi δe SLi ]} 1 = {[I] [S][E SLi ]} 1 {[I] [S][E SLi ]} 1 {[δs][e SLi ] [S][δE SLi ]}{[I] [S][E SLi ]} 1 (10) Substituting Eq (9) into Eq (7) and subtracting Eq (6) form the result, we can deduce the following equation by omitting the products of deviations {[I] [S][E SLi ]}[E RT i ] 1 [δe DXi ][E] {[I] [S][E SLi ]}[E RT i ] 1 [δe RT i ][S]{[I] [E SLi ][S]} 1 [I i ][E] [S][δE SLi ][S]{[I] [E SLi ][S]} 1 [I i ][E] = [δs]{[i] [E SLi ][S]} 1 [I i ][E] (11) Because of the dependence among error terms, raw measured S-parameters and actual S-parameters, we can rewrite vectors [b i ] and [a i ] by substituting Eq (5) and Eq (6) into Eq () and Eq (3) [b i ] = [S]{[I] [E SLi ][S]} 1 [I i ][E] (1) [a i ] = {[I] [E SLi ][S]} 1 [I i ][E] (13) Substituting Eq (11) and Eq (1) into Eq (10) can obtain the equation given below: [δs][a i ] = [C i ][δe DXi ][E] [S][δE SLi ][b i ] [C i ][δe RT i ][b i ] (14) where the matrix [C i ] = {[I] [S][E SLi ]}[E RT i ] 1 With the subscript i in Eq (13) varied from 1 to n, we can establish an equation for the deviation [δs] and consequently derive [δs] = {[C 1 ][δe DX1 ][E], [C i ][δe DXi ][E], [C n ][δe DXn ][E]}[A] 1 {[S][δE SL1 ][b 1 ], [S][δE SLi ][b i ], [S][δE SLn ][b n ]}[A] 1 {[C 1 ][δe RT 1 ][b 1 ], [C i ][δe RT i ][b i ], [C n ][δe RT n ][b n ]}[A] 1 (15) It is observed that the deviation matrix [δs] is developed as a function of [δe DXi ], [δe SLi ] and [δe RT i ] Finally, the scattering parameter deviation δs kl can be concluded as δs kl = d il [c T ik ][δe DXi] d il [S k1, S kn ][diag(b i )][δe SLi ] d il [c T ik ][diag(b i)][δe RT i ](16) where [c T ik ] is the kth row of [C i] and d il is the (i, l)th element of [A] 1 For the diagonal matrix [diag(b i )] the diagonal elements are composed of [b i ] Meanwhile, the deviations [δe DXi ], [δe SLi ] and [δe RT i ] are defined as n 1 column vectors: [δe DXi ] = δe X1i δe D δe Xni, [δe SLi ] = δe L1i δe S δe Lni, [δe RT i ] = δe T 1i δe R δe T ni To explore how the deviations of errors [δe DXi ], [δe SLi ] and [δe RT i ] affect the S-parameter deviation, we need further deduce the analytical expressions for [c T ik ], d il and [diag(b i )] Because the error terms are E D 0, E S 0, E X 0, E L 0, E R 1 and E T 1 in the high-performance VNA, E D, E S, E X and E L can be regarded as small terms and their products will be neglected in the following calculation As mentioned before, the matrix [C i ] = {[I] [S][E SLi ]}[E RT i ] 1 and so we can get 1 E L1i S 11 E E T 1i S S 1i E E R Lni S 1n E T ni [C i ] = E L1i S i1 1 E E T 1i S S E E R Lni S in E T ni (17) E L1i S n1 E T 1i E S S ni E R 1 E Lni S nn E T ni

6 308 Zhao, Xiao, and Qin As the kth row of [C i ], the vector [c T ik ] is given below: [c T ik ] = [ ε1 k E L1i S k1 E T 1i, ε i k E S S ki, ε n k E Lni S kn E R E T ni ] (18) where ε i k = 1 for i = k and ε i k = 0 for i k To deduce the variable d il defined as the (i, l)th element of [A] 1, we first represent [a i ] with {[I] [E SLi ][S]}[I i ][E] by using Eq (8) and Eq (1), and thus obtain an approximate expression for [A] E S11 S 11 E L1i S 1i E L1n S 1n [A] = [I] E Li1 S i1 E S S E Lin S in (19) E Ln1 S n1 E Lni S ni E Snn S nn Then inverting the matrix [A] with Eq (8), we can deduce E S11 S 11 E L1i S 1i E L1n S 1n [A] 1 = [I] E Li1 S i1 E S S E Lin S in E Ln1 S n1 E Lni S ni E Snn S nn From Eq (19), the variable d il is finally described as { 1 ESll S d il = ll i = l E Lil S il i l (0) (1) Based on the expression for [a i ] in Eq (18) and the S-matrix definition [b i ] = [S][a i ], we can have j i S 1i E S S 1i S E L S 1j S j=1n j i [b i ] = S E S S S E L S S () j=1n j i S ni E S S ni S E L S nj S j=1n Based on Eq (1), the diagonal matrix [diag(b i )] can be easily obtained Form the above analysis, the coefficients [c T ik ], d il and [diag(b i )] are solved, and then substituting them into Eq (15) can derive the analytical expression for δs kl in the following two cases: 1) When k = l, the deviation δs kl is expressed as δs ll = E SllS ll 1 E Rll δe Dll S ll (S ll E Lil S li E T il δe Xil E Lil S li S il ) δe Sll E Lil S il δe Xli E T li E Lil S li S S il δe S

7 Progress In Electromagnetics Research, Vol 144, S li S il S ll E Sll S ll j l j=1n E Ljl S S jl δe Lil E Lil S li S il E Rll δe Rll ) When k l, the deviation δs kl is expressed as j i j=1n E Lil S li S il E T il δe T il E Lil S lj S S il δe L E Lil S li S il E T li δe T li (3) δs kl = E SllS kl δe Dll E LklS kl δe Dkk E SllS ll E Lkl S kk 1 δe Xkl E Rll E Rkk E T kl i k,l E Lil S il δe Xki S kl E T ki S ki j i j=1n S kl i k,l S il j l j=1n ( S ll E Ljl S S jl δe Lil E Lil S kj S S il δe L E SllS kl S ll E Rll E Lil S ki S il E T kl δe T kl i k,l E Lil S li S il ) δe Sll i k,l δe Rll E LklS kk S kl δe Rkk E Rkk E Lil S ki S il E T il δe T il i k,l E Lil S ki δe Xil E T il E Lil S ki S S il δe S E Lil S ki S il E T ki δe T ki (4) From Eq () and Eq (3), we can calculate the sensitivity coefficients with respect to the error terms as input quantities and the S-parameters of the DUT as output quantities Deviations of Error Terms In the second step, the deviations of error terms associated with SOLT calibration standards are solved By measuring standards SOL at port i, we can use the 3-term error model in Fig 5 to deduce the system errors E D, E S and E R by the linear equations below: 1 S m (S)Γ Si Γ Si E D 1 S m (O)Γ Oi Γ Oi E S 1 S m (L)Γ Li Γ Li E R E D E S = S m (S) S m (O) S m (L) (5) Figure 5 Error model of SOL calibration

8 310 Zhao, Xiao, and Qin where S m (X) is the raw reflection coefficient of the standard X at port i The S-parameter deviations of standards SOL (δγ Si, δγ Oi and δγ Li ) will cause the error deviations (δe D, δe S and δe R ) 1 S m (S)Γ Si Γ Si 0 S m (S)δΓ Si δγ Si 1 S m (O)Γ Oi Γ Oi 0 S m (O)δΓ Oi δγ Oi 1 S m (L)Γ Li Γ Li 0 S m (L)δΓ Li δγ Li (6) E D δe D S m (S) E S δe S = S m (O) E R E D E S E S E D 1 δe R S m (L) Subtracting (5) from (6) and substituting S m (X) = E D E R Γ Xi /(1 E S Γ Xi ) into the result, we can get the expression for the deviation vector [δe D, δe S, δe R ] T via a series of matrix operations [10] Finally, we can deduce [ ] Γ Oi Γ Li δe D = E R (Γ Oi Γ Si )(Γ Si Γ Li ) δγ Γ Li Γ Si Si (Γ Li Γ Oi )(Γ Oi Γ Si ) δγ Γ Si Γ Oi Oi (Γ Si Γ Li )(Γ Li Γ Oi ) δγ Li δe S = (1 E SΓ Oi )(1 E S Γ Li ) (Γ Oi Γ Si )(Γ Si Γ Li ) δγ Si (1 E SΓ Li )(1 E S Γ Si ) (Γ Li Γ Oi )(Γ Oi Γ Si ) (7) δγ Oi (1 E SΓ Si )(1 E S Γ Oi ) δγ Li (Γ Si Γ Li )(Γ Li Γ Oi ) (8) δe R = E R [ ES Γ Oi Γ Li Γ Oi Γ Li (Γ Oi Γ Si )(Γ Si Γ Li ) δγ Si E SΓ Li Γ Si Γ Li Γ Si (Γ Li Γ Oi )(Γ Oi Γ Si ) δγ Oi E SΓ Si Γ Oi Γ Si Γ Oi (Γ Si Γ Li )(Γ Li Γ Oi ) δγ Li It is worth noting that δe X is independent on the S-parameter deviations of SOLT standards, that is to say δe X = 0 Measuring standard T between port i and port j, we can use the flow graph combined with the transmission matrix (T -matrix) to represent the measurements S m (T ) and S m (T ) in Fig 6 The T -matrix of the error box at port i and that of the -port network of standard T, both selected by dash in Fig 6, are defined as [T i ] and [T ] respectively Cascading the T -matrixes [T i ] and [T ] can deduce the equation as follows: S m (T ) S m (T ) E X 1 = [T i][t t ] S m (T ) E X E L E T 1 E T ] (9) (30) Figure 6 Error model of T calibration

9 Progress In Electromagnetics Research, Vol 144, If the left hand of (30) is invariable, the deviations [δt i ], [δt t ], δe L and δe T occur S m (T ) E L S m (T ) E X δ E L 1 = {[T i] [δt i ]}{[T t ] [δt t ]} E T 1 E T δ 1 S m (T ) E X E T E T Subtracting (30) from (31) and omitting the product of small terms, we can deduce [ ] [ ] [ ] [ ] [ ] δel 1 EL = [T 0 E t ] 1 [T i ] 1 EL 1 EL [δt i ][T t ] [T T 1 0 E t ] 1 EL [δt t ] (3) T 1 δe T The right hand of (3) is composed of two terms The first term contains the deviation [δt i ], which associates with the deviations δγ Si, δγ Oi and δγ Li of standards SOL The second term contains the deviation [δt t ], which associates with the deviations δs T, δs T, δs T and δs T of the -port network of T connection From Fig 6, we can get the expressions for [T i ] and [T ] given below: [ ] ER E [T i ] = D E S E D (33) E S 1 [ ] [T ] = 1 S T S T S T S T S T S T S T (34) 1 By taking the total differential of Eq (3) and Eq (33), we can get [ ] ES δe [δt i ] = D E D δe S δe R δe D δe S δe R [δt ] = 1 (S T ) ST S T δst S T S T S T δst δst ( S T S T ) δs T δs T S T S T δs T S T δs T S T δs T δs T Substituting Eqs (3) to (35) into Eq (31) can determine the deviations δe L and δe T as follows: (31) (35) (36) δe L = M N Γ Oi Γ Li MN(Γ Oi Γ Li ) S T S T δγ Si M N Γ Li Γ Si MN(Γ Li Γ Si ) (Γ Si Γ Oi )(Γ Si Γ Li ) S T S T (Γ Oi Γ Li )(Γ Oi Γ Si ) δγ Oi M N Γ Si Γ Oi MN(Γ Si Γ Oi ) S T S T δγ Li (Γ Li Γ Si )(Γ Li Γ Oi ) N S T S T δs T NE L S T δs T NE L S T δs T ELδS T (37) δe T = ME T S T (Γ Oi Γ Li ) P E T Γ Oi Γ Li MQE T S T S T δγ Si (Γ Si Γ Oi )(Γ Si Γ Li ) ME T S T (Γ Li Γ Si ) P E T Γ Li Γ Si MQE T S T S T δγ Oi (Γ Oi Γ Li )(Γ Oi Γ Si ) ME T S T (Γ Si Γ Oi ) P E T Γ Si Γ Oi MQE T S T S T δγ Li (Γ Li Γ Si )(Γ Li Γ Oi ) NE T S T S T S T δs T NE T S T δs T E T E L S T S T δs T E T E L δs T (38)

10 31 Zhao, Xiao, and Qin where the variables M = S T Q = S T S T S T S T E L Q, N = 1 E L S T, P = ST E S S T S T E L S T ST and From the above analysis in part B, the deviations of error terms are expressed by the S-parameter deviations of SOLT standards, which implies that the sensitivity coefficients with respect to the S-parameters of SOLT standards as input quantities and the error terms as output quantities can be determined 3 Combination of Part A and Part B In the last step, combining the results of part A and B can obtain the sensitivity coefficients of S- parameters for arbitrary n-port DUT with respect to the nonideal calibration standards In general, the system errors are solved in the ideal condition (Γ Si = 1, Γ Oi = 1, Γ Li = 0, S T = S T = 0, S T = S T = 1), therefore the results in Part B can be simplified as δe D = E R δγ Li (39) δe S = E S 1 δγ Si 1 E S δγ Oi δγ Li (40) δe R = E R δγ Si E R δγ Oi E R E S δγ Li (41) δe L = E L δγ Si E L δγ Oi δγ Li δs T E L δs T E L δs T (4) δe T = E T E L δγ Si E T E L δγ Oi E T (E S E L )δγ Li E T δs T E T E L δs T (43) Substituting Eqs (38) (4) into Eq () and Eq (3), we can finally conclude the sensitivity coefficients for the S-parameters as follows: 1) When k = l, the deviation δs kl is expressed as ) S ll (1 S ll E Lil S li S il δs ll = S ll (1 S ll δγ Ll S li S li S il S il j l j=1n E Lil S li S il ) j l j=1n E Ljl S S jl δγ Sl δγ Ol E Ljl S S jl δγ Ll δs T il ll ) When k l, the deviation δs kl is expressed as E Lil S li S S il δγ Si E Lil S li S S il δγ Oi j=1n j i j=1n E Lil S lj S S il δγ Li E Lil S lj S S il δs T (44) δs kl = S kl (S ll E Lkl E Lil S li S il ) E Lil S ki S S il δγ Si S kl (S ll E Lkl E Lil S ki S il δγ Sl E LklS kk S kl δγ Sk ) E Lil S li S il E Lil S ki S il δγ Ol

11 Progress In Electromagnetics Research, Vol 144, E LklS kk S kl δγ Ok S il ( S kl S ki S ki S il j l j=1n j l j=1n E Lil S ki S il ) E Lil S ki S S il δγ Oi E Lkl S kl δγ Ll E Ljl S S jl δγ Ll E Lkl S kl δγ Lk E Ljl S S jl δs T kl kl δs T il ll j=1n E Lkl S kl δskk T kl E Lil S ki S il δs T il li j i j=1n i k,l E Lil S kj S S il δγ Li E Lil S kj S S il δs T E Lil S ki S il δs T ik ki (45) From Eq (43) and Eq (44), we can find out how the S-parameter deviations of SOLT standards affect the uncertainty of S 3 MEASUREMENT RESULTS To verify the proposed method, three steps are needed: first solving the system errors by GSOLT calibration, then determining the corrected S-parameters of the multiport DUT, and finally calculating the sensitivity coefficients of the S-parameters During the whole analysis process, because the test port connectors are of the same sex, the reflection coefficients satisfy the relationship of Γ Si = Γ S, Γ Oi = Γ O and Γ Li = Γ L Meanwhile, it is established that S T = S11 T, ST = S1 T, ST = S1 T and ST = S T with the port number j larger than the port number i In these cases, there are totally 7 uncertainty sources associated with the nonideal SOLT standards: Γ S, Γ O, Γ L, S11 T, ST 1, ST 1 and ST Based on the above and by applying the GSOLT calibration (Fig 5 and Fig 6), all the errors in Table 1 can be deduced from Eq (4) and Eq (9) After that, measurements are performed on a four-port dual directional coupler using the commercial four-port VNA (Agilent E5071B) Once the system errors and the raw measured data are obtained in the ideal condition (Γ S = 1, Γ O = 1, Γ L = 0, S11 T = ST = 0, S1 T = ST 1 = 1), the S-parameters of the four-port dual directional coupler are calculated by Eq (4) and compared with the Agilent E5071B results in Figs 7 10 Experiment results in Figs 7 10 can attest the good performance of the calibration method based on the generalized flow graph of the 3n-term error model However, because the results by this method Figure 7 Comparison of S 11 Magnitude Phase

12 314 Zhao, Xiao, and Qin Figure 8 Comparison of S 1 Magnitude Phase Figure 9 Comparison of S 31 Magnitude Phase Figure 10 Comparison of S 41 Magnitude Phase

13 Progress In Electromagnetics Research, Vol 144, Figure 11 Comparison of δs 11 (δγ S ) Magnitude Phase Figure 1 Comparison of δs 11 (δγ l ) Magnitude Phase is based on the assumption that Γ S = 1, Γ O = 1, Γ L = 0, S11 T = ST = 0 and ST 1 = ST 1 = 1, there will be the S-parameter deviations of the DUT caused by the nonideal standards, such as the phase deviation of corrected S 11 in Fig 7 To explore how the SOLT standards affect the S-parameter deviations of the DUT, the sensitivity coefficients are calculated by Eq (43) and Eq (44) The following results (Fig 11 Fig 18) show that the sensitivity coefficients calculated by this paper are in very good agreement with those given in [10], demonstrating the correctness of proposed method The sensitivity coefficients δs ll (δs1 T ) and δs ll(s1 T ), which are ignored in Eq (43) due to the approximation, are assigned a value of 10e-6 (eg, δs 1l (δs1 T ) in Fig 14) In fact, the modulus of the actual sensitivity coefficients δs ll (δs1 T ) and δs ll(s1 T ) are mostly less than 10e-5 for this four-port dual directional coupler In addition, because the sensitivity coefficient δs ll (δγ l ) contains a constant term 1 in Eq (43), the modulus of δs ll (δγ l ) generally has a large value, which can explain the phase deviation of S 11 in Fig 7 The great deviation of S 1 in Fig 8 is also caused by the nonideal L standard, whose impact can be estimated by the expression δs kl (δγ L ) = S ki S il in this experiment In conclusion, these results demonstrated that the analytical expressions for the sensitivity coefficients are correct and useful in the multiport DUT measurement

14 316 Zhao, Xiao, and Qin Figure 13 Comparison of δs 11 (δs11 T ) Magnitude Phase Figure 14 Comparison of δs 11 (δs1 T ) Magnitude Phase Figure 15 Comparison of δs 11 (δγ S ) Magnitude Phase

15 Progress In Electromagnetics Research, Vol 144, Figure 16 Comparison of δs 31 (δγ l ) Magnitude Phase Figure 17 Comparison of δs 31 (δs11 T ) Magnitude Phase Figure 18 Comparison of δs 31 (δs1 T ) Magnitude Phase

16 318 Zhao, Xiao, and Qin 4 CONCLUSION In this paper, the analytical expressions for the sensitivity coefficients of the S-parameters are developed for the GSOLT calibration of the n-port VNA Using the generalized flow graph of the 3n-term error model can conveniently deduce the S-parameters of the n-port DUT To investigate the influence of SOLT standards on the S, the following steps are involved: Firstly, the S-parameter deviations of the n-port DUT with respect to the error terms are solved during the error correction procedure Then, expressions representing the deviations of the error terms in regard to the nonideal SOLT calibration standards are determined in the process of error calibration Finally, the sensitivity coefficients, with respect to the S-parameters of SOLT standards as input quantities and the S-parameters of the n-port DUT as output quantities, are derived and can be used for establishing the type-b uncertainty budget for S-parameter measurements ACKNOWLEDGMENT This work was supported by the Fundamental Research Funds for the Central Universities REFERENCES 1 Ferrero, A, V Teppati, E Fledell, B Grossman, and T Ruttan, Microwave multiport measurements for the digital world, IEEE Microwave Magazine, Vol 1, No 1, 61 73, Feb 011 Rumiantsev, A and N Ridler, VNA calibration, IEEE Microwave Magazine, Vol 9, No 3, 86 99, Jun Rytting, D, Network analyzer error models and calibration methods, RF & Microwave Measurement for Wireless Application (ARFTG/NIST Short Course Notes), Heuermann, H, GSOLT: The calibration procedure for all multiport vector network analyzers, MTT-S International Microwave Symposium, , Zhao, W, Y J Zhao, and H B Qin, Calibration of the three-port VNA using the general 6-term error model, Journal of Electromagnetic Wave and Application, Vol 4, Nos 3, , Zhao, W, Y J Zhao, H B Qin, et al, Calibration algorithm of the n-port vector network analyzer using the general node equation, IET Science, Measurement & Technology, Vol 4, No 6, 98 30, Stumper, U and T Schrader, Influence of different configurations of nonideal calibration standards on vector network analyzer performance, IEEE Transactions on Instrumentation and Measurement, Vol 61, No 7, , Jul 01 8 Stumper, U, Influence of nonideal calibration items on S-parameter uncertainties applying the SOLR calibration method, IEEE Transactions on Instrumentation and Measurement, Vol 58, No 4, , Apr Stumper, U, Influence of TMSO calibration standards uncertainties on VNA S-parameter measurements, IEEE Transactions on Instrumentation and Measurement, Vol 5, No, , Apr Zhao, W, X D Yang, J K Xiao, et al, Uncertainties of multiport VNA S-parameter measurements applying the GSOLT calibration method, IEEE Transaction on Instrument and Measurement, Vol 61, No 1, , Dec Mason, S J, Feedback theory: Further properties of signal flow graphs, Proceedings of The Institute of Radio Engineers, Vol 44, No 7, 90 96, Da Silva, E, High Frequency and Microwave Engineering, Newnes, Kurokawa, K, Power waves and the scattering matrix, IEEE Transactions on Microwave Theory and Techniques, Vol 13, No, 194 0, Mar Henderson, H V and S R Searle, On deriving the inverse of a sum of matrices, Siam Review, Vol 3, No 1, 53 60, Petersen, K B and M S Pedersen, The Matrix Cookbook, Technical University of Denmark, 006