Advanced Microwave Circuits and Systems Part 2 ppt

AdvancedMicrowaveCircuitsandSystems24 S 11TH S 33TH S 33OS S 11OS S 43TH S 21TH S 21OS , S 43OS 10 20 30 40 −0.1 −0.05 0 0.05 0.1 Freq [GHz] 50 0 |S 43TH | [dB] −0.1 −0.05 0 0.05 0.1 |S 21TH | [dB] Fig. 10. Characteristics of the THRU (Fig. 9a) after performing the thru-only (S 11TH , S 33TH , S 21TH , S 43TH ) or open-short (S 11OS , S 33OS , S 21OS , S 43OS ) de-embedding (Amakawa et al., 2008). Fig. 10 shows the de-embedded characteristics of a symmetric THRU itself (Fig. 9a). The re- flection coefficients obtained by the proposed method (S 11TH and S 33TH ) stay very close to the center of the Smith chart and the transmission coefficients (S 21TH and S 43TH ) at its right end as they should. Fig. 11 shows even- and odd-mode transmission coefficients for a pair of 1 mm- long transmission lines shown in Fig. 9(b). A comparatively large difference is seen between the results from the two de-embedding methods for the even mode. One likely cause is the nonideal behavior of the SHORT (Goto et al., 2008; Ito & Masu, 2008). The odd-mode results, on the other hand, agree very well, indicating the immunity of this mode (and the differential mode) to the problem that plague the even mode (and the common mode). 6. Decomposition of a 2n-port into n 2-ports The essential used idea in the previous section was to reduce a 4-port problem to two inde- pendent 2-port problems by mode transformation. The requirement for it to work was that the 4 ×4 S matrix of the THRU dummy pattern (a pair of nonuniform TLs) have the even/odd symmetry and left/right symmetry. This development naturally leads to the idea that the same de-embedding method should be applicable to 2n-ports, where n is a positive integer, provided that the S-matrix of the THRU (n coupled nonuniform TLs) can somehow be block- diagonalized with 2 ×2 diagonal blocks (Amakawa et al., 2009). Modal analysis of multiconductor transmission lines (MTLs) have been a subject of intensive study for decades (Faria, 2004; Kogo, 1960; Paul, 2008; Williams et al., 1997). MTL equations are typically written in terms of per-unit-length equivalent-circuit parameters. Experimental characterization of MTLs, therefore, often involves extraction of those parameters from mea- sured S-matrices (Nickel et al., 2001; van der Merwe et al., 1998). We instead directly work with S-matrices. In Section 5, the transformation matrix (37) was known a priori thanks to the even/odd symmetry of the DUT. We now have to find the transformation matrices. As before, we assume throughout that the THRU is reciprocal and hence the associated S-matrix symmetric. S o2o1TH S o2o1OS S o2o1before S e2e1TH S e2e1before S e2e1OS Fig. 11. Even-mode (broken lines) and odd-mode (solid lines) transmission coefficients for a pair of transmission lines (Fig. 9b) before and after de-embedding (thru-only or open-short) (Amakawa et al., 2008). Our goal is to transform a 2n ×2n scattering matrix S into the following block-diagonal form: ˜ S  =    S m1 . . . S mn    , (62) where S mi are 2 ×2 submatrices, and the rest of the elements of ˜ S  are all 0. The port number- ing for ˜ S  is shown in Fig. 12 with primes. Note that the port numbering convention adopted in this and the next Sections is different from that adopted in earlier Sections. Once the trans- formation is performed, the DUT can be treated as if they were composed of n uncoupled 2-ports. This problem is not an ordinary matrix diagonalization problem. The form of (62) results by first transforming S into ˜ S, which has the following form: ˜ S =    . . . . . . . . . . . .    , (63) and then reordering the rows and columns of ˜ S such that S mi in (62) is built from the ith diagonal elements of the four submatrices of ˜ S (Amakawa et al., 2009). The port indices of ˜ S are shown in Fig. 12 without primes. The problem, therefore, is the transformation of S into ˜ S followed by reordering of rows and columns yielding ˜ S  . In the case of a cascadable 2n-port, it makes sense to divide the ports into two groups as shown in Fig. 12, and hence the division of S, a, and b into submatrices/subvectors: b =  b 1 b 2  =  S 11 S 12 S 21 S 22  a 1 a 2  = Sa. (64) Athru-onlyde-embeddingmethodforon-wafercharacterizationofmultiportnetworks 25 S 11TH S 33TH S 33OS S 11OS S 43TH S 21TH S 21OS , S 43OS 10 20 30 40 −0.1 −0.05 0 0.05 0.1 Freq [GHz] 50 0 |S 43TH | [dB] −0.1 −0.05 0 0.05 0.1 |S 21TH | [dB] Fig. 10. Characteristics of the THRU (Fig. 9a) after performing the thru-only (S 11TH , S 33TH , S 21TH , S 43TH ) or open-short (S 11OS , S 33OS , S 21OS , S 43OS ) de-embedding (Amakawa et al., 2008). Fig. 10 shows the de-embedded characteristics of a symmetric THRU itself (Fig. 9a). The re- flection coefficients obtained by the proposed method (S 11TH and S 33TH ) stay very close to the center of the Smith chart and the transmission coefficients (S 21TH and S 43TH ) at its right end as they should. Fig. 11 shows even- and odd-mode transmission coefficients for a pair of 1 mm- long transmission lines shown in Fig. 9(b). A comparatively large difference is seen between the results from the two de-embedding methods for the even mode. One likely cause is the nonideal behavior of the SHORT (Goto et al., 2008; Ito & Masu, 2008). The odd-mode results, on the other hand, agree very well, indicating the immunity of this mode (and the differential mode) to the problem that plague the even mode (and the common mode). 6. Decomposition of a 2n-port into n 2-ports The essential used idea in the previous section was to reduce a 4-port problem to two inde- pendent 2-port problems by mode transformation. The requirement for it to work was that the 4 ×4 S matrix of the THRU dummy pattern (a pair of nonuniform TLs) have the even/odd symmetry and left/right symmetry. This development naturally leads to the idea that the same de-embedding method should be applicable to 2n-ports, where n is a positive integer, provided that the S-matrix of the THRU (n coupled nonuniform TLs) can somehow be block- diagonalized with 2 ×2 diagonal blocks (Amakawa et al., 2009). Modal analysis of multiconductor transmission lines (MTLs) have been a subject of intensive study for decades (Faria, 2004; Kogo, 1960; Paul, 2008; Williams et al., 1997). MTL equations are typically written in terms of per-unit-length equivalent-circuit parameters. Experimental characterization of MTLs, therefore, often involves extraction of those parameters from mea- sured S-matrices (Nickel et al., 2001; van der Merwe et al., 1998). We instead directly work with S-matrices. In Section 5, the transformation matrix (37) was known a priori thanks to the even/odd symmetry of the DUT. We now have to find the transformation matrices. As before, we assume throughout that the THRU is reciprocal and hence the associated S-matrix symmetric. S o2o1TH S o2o1OS S o2o1before S e2e1TH S e2e1before S e2e1OS Fig. 11. Even-mode (broken lines) and odd-mode (solid lines) transmission coefficients for a pair of transmission lines (Fig. 9b) before and after de-embedding (thru-only or open-short) (Amakawa et al., 2008). Our goal is to transform a 2n ×2n scattering matrix S into the following block-diagonal form: ˜ S  =    S m1 . . . S mn    , (62) where S mi are 2 ×2 submatrices, and the rest of the elements of ˜ S  are all 0. The port number- ing for ˜ S  is shown in Fig. 12 with primes. Note that the port numbering convention adopted in this and the next Sections is different from that adopted in earlier Sections. Once the trans- formation is performed, the DUT can be treated as if they were composed of n uncoupled 2-ports. This problem is not an ordinary matrix diagonalization problem. The form of (62) results by first transforming S into ˜ S, which has the following form: ˜ S =    . . . . . . . . . . . .    , (63) and then reordering the rows and columns of ˜ S such that S mi in (62) is built from the ith diagonal elements of the four submatrices of ˜ S (Amakawa et al., 2009). The port indices of ˜ S are shown in Fig. 12 without primes. The problem, therefore, is the transformation of S into ˜ S followed by reordering of rows and columns yielding ˜ S  . In the case of a cascadable 2n-port, it makes sense to divide the ports into two groups as shown in Fig. 12, and hence the division of S, a, and b into submatrices/subvectors: b =  b 1 b 2  =  S 11 S 12 S 21 S 22  a 1 a 2  = Sa. (64) AdvancedMicrowaveCircuitsandSystems26 1 n+1 2n-port 2 n n+2 2n 3 4 n − 1 2n − 1 n+3 S, T, S n+4 S′, S′ 1′ 2′ 3′ 4′ 5′ 6′ 7′ 8′ (2n − 3)′ (2n − 1)′ (2n − 2)′ (2n)′ ̃ ̃ a 1 b 1 a 2 b 2 Fig. 12. Port indices for a cascadable 2n-port. The ports 1 through n of S constitute one end of the bundle of n lines and the ports n + 1 through 2n the other end. This was already done in earlier Sections for 4-ports. Since our 2n-port is reciprocal by as- sumption, S is symmetric: S T = S. Then, it can be shown that the following change of bases gives the desired transformation.  a 1 a 2  =  W 1 (W T 2 ) −1  ˜ a 1 ˜ a 2  , (65)  b 1 b 2  =  (W T 1 ) −1 W 2  ˜ b 1 ˜ b 2  , (66) where the blanks represent zero submatrices. W 1 and W 2 diagonalize S −1 21 S 22 S −1 12 S 11 and S 22 S −1 12 S 11 S −1 21 , respectively, by similarity transformation: W −1 1 S −1 21 S 22 S −1 12 S 11 W 1 = Λ 1 , (67) W −1 2 S 22 S −1 12 S 11 S −1 21 W 2 = Λ 2 , (68) where Λ 1 and Λ 2 are diagonal matrices. W 1 and W 2 can be computed by eigenvalue decom- position. The derivation is similar to (Faria, 2004). ˜ S is thus given by ˜ S =  W T 1 S 11 W 1 W T 1 S 12 (W T 2 ) −1 W −1 2 S 21 W 1 W −1 2 S 22 (W T 2 ) −1  . (69) 7. Multiport de-embedding using a THRU Suppose, as before, that the device under measurement and the THRU can be represented as shown in Fig. 13. Here the DUT is MTLs. In terms of the transfer matrix T defined by  a 1 b 1  = T  b 2 a 2  =  T 11 T 12 T 21 T 22  b 2 a 2  , (70) T =  T 11 T 12 T 21 T 22  =  S −1 21 −S −1 21 S 22 S 11 S −1 21 S 12 −S 11 S −1 21 S 22  , (71) TLsL (a) (b) R L R Fig. 13. (a) Model of n coupled TLs measured by a VNA. The TLs sit between the intervening structures L and R. (b) Model of a THRU. pads4 TLs Prepare component S matrices Match? De-embed S matrix of TLs de-embed- ded 4 TLs 1 2 3 4 5 6 7 8 4 TLs Synthesize as-measured S matrices by cascading pads pads pads pads Fig. 14. Flow of validating the de-embedding method. S =  S 11 S 12 S 21 S 22  =  T 21 T −1 11 T 22 −T 21 T −1 11 T 12 T −1 11 −T −1 11 T 12  . (72) The as-measured T-matrix for Fig. 13(a) is T meas = T L T TL T R . In order to de-embed T TL from T meas , the THRU (Fig. 13(b)) is measured, and the result (T thru = T L T R ) is transformed into the block-diagonal form ˜ S  thru . Since each of the re- sultant 2 × 2 diagonal blocks of ˜ S  thru is symmetric by assumption, the method in Section 3 can be applied to determine T L and T R . Then, the characteristics of the TLs are obtained by T TL = T −1 L T meas T −1 R . Shown in Fig. 14 is the procedure that we followed to validate the thru-only de-embedding method for 2n-ports (Amakawa et al., 2009). S -parameter files of 1 mm-long 4 coupled TLs and pads were generated by using Agilent Technologies ADS. A cross section of the TLs is shown in Fig. 15. The schematic diagram representing the pads placed at each end of the bun- dle of TLs is shown in Fig. 16. Figs. 17 and 18 show the characteristics of the “as-measured” TLs and the THRU, respectively. The characteristics of the bare (un-embedded) TLs and the de-embedded results are both shown on the same Smith chart in Fig. 19, but they are indistin- guishable, thereby demonstrating the validity of the de-embedding procedure. We also applied the same de-embedding method to the TLs shown in Fig. 9, analyzed earlier by the even/odd transformation in Section 5 (Amakawa et al., 2008). The numerical values Athru-onlyde-embeddingmethodforon-wafercharacterizationofmultiportnetworks 27 1 n+1 2n-port 2 n n+2 2n 3 4 n − 1 2n − 1 n+3 S, T, S n+4 S′, S′ 1′ 2′ 3′ 4′ 5′ 6′ 7′ 8′ (2n − 3)′ (2n − 1)′ (2n − 2)′ (2n)′ ̃ ̃ a 1 b 1 a 2 b 2 Fig. 12. Port indices for a cascadable 2n-port. The ports 1 through n of S constitute one end of the bundle of n lines and the ports n + 1 through 2n the other end. This was already done in earlier Sections for 4-ports. Since our 2n-port is reciprocal by as- sumption, S is symmetric: S T = S. Then, it can be shown that the following change of bases gives the desired transformation.  a 1 a 2  =  W 1 (W T 2 ) −1  ˜ a 1 ˜ a 2  , (65)  b 1 b 2  =  (W T 1 ) −1 W 2  ˜ b 1 ˜ b 2  , (66) where the blanks represent zero submatrices. W 1 and W 2 diagonalize S −1 21 S 22 S −1 12 S 11 and S 22 S −1 12 S 11 S −1 21 , respectively, by similarity transformation: W −1 1 S −1 21 S 22 S −1 12 S 11 W 1 = Λ 1 , (67) W −1 2 S 22 S −1 12 S 11 S −1 21 W 2 = Λ 2 , (68) where Λ 1 and Λ 2 are diagonal matrices. W 1 and W 2 can be computed by eigenvalue decom- position. The derivation is similar to (Faria, 2004). ˜ S is thus given by ˜ S =  W T 1 S 11 W 1 W T 1 S 12 (W T 2 ) −1 W −1 2 S 21 W 1 W −1 2 S 22 (W T 2 ) −1  . (69) 7. Multiport de-embedding using a THRU Suppose, as before, that the device under measurement and the THRU can be represented as shown in Fig. 13. Here the DUT is MTLs. In terms of the transfer matrix T defined by  a 1 b 1  = T  b 2 a 2  =  T 11 T 12 T 21 T 22  b 2 a 2  , (70) T =  T 11 T 12 T 21 T 22  =  S −1 21 −S −1 21 S 22 S 11 S −1 21 S 12 −S 11 S −1 21 S 22  , (71) TLsL (a) (b) R L R Fig. 13. (a) Model of n coupled TLs measured by a VNA. The TLs sit between the intervening structures L and R. (b) Model of a THRU. pads4 TLs Prepare component S matrices Match? De-embed S matrix of TLs de-embed- ded 4 TLs 1 2 3 4 5 6 7 8 4 TLs Synthesize as-measured S matrices by cascading pads pads pads pads Fig. 14. Flow of validating the de-embedding method. S =  S 11 S 12 S 21 S 22  =  T 21 T −1 11 T 22 −T 21 T −1 11 T 12 T −1 11 −T −1 11 T 12  . (72) The as-measured T-matrix for Fig. 13(a) is T meas = T L T TL T R . In order to de-embed T TL from T meas , the THRU (Fig. 13(b)) is measured, and the result (T thru = T L T R ) is transformed into the block-diagonal form ˜ S  thru . Since each of the re- sultant 2 × 2 diagonal blocks of ˜ S  thru is symmetric by assumption, the method in Section 3 can be applied to determine T L and T R . Then, the characteristics of the TLs are obtained by T TL = T −1 L T meas T −1 R . Shown in Fig. 14 is the procedure that we followed to validate the thru-only de-embedding method for 2n-ports (Amakawa et al., 2009). S -parameter files of 1 mm-long 4 coupled TLs and pads were generated by using Agilent Technologies ADS. A cross section of the TLs is shown in Fig. 15. The schematic diagram representing the pads placed at each end of the bun- dle of TLs is shown in Fig. 16. Figs. 17 and 18 show the characteristics of the “as-measured” TLs and the THRU, respectively. The characteristics of the bare (un-embedded) TLs and the de-embedded results are both shown on the same Smith chart in Fig. 19, but they are indistin- guishable, thereby demonstrating the validity of the de-embedding procedure. We also applied the same de-embedding method to the TLs shown in Fig. 9, analyzed earlier by the even/odd transformation in Section 5 (Amakawa et al., 2008). The numerical values AdvancedMicrowaveCircuitsandSystems28 1, 5 2, 6 3, 7 4, 8 30 200 210 30 25 25 310 340 560 15 30 30 30 Fig. 15. Schematic cross section of the 1 mm-long 4 coupled TLs (not to scale), labeled with port numbers. Dimensions are in µm. Relative dielectric permittivity is 4. Metal conductivity is 5.9 ×10 7 (Ω ·m) −1 . tan δ = 0.04 (Amakawa et al., 2009). 1Ω 0.5Ω 0.1nF 70fF k = 0.15 k = 0.15 k = 0.15 0.15nH Fig. 16. Model of the left half of the THRU including pads (Amakawa et al., 2009). of even/odd transformed and fully block-diagonalized S-matrix of the THRU (Fig. 9(a)) at 10 GHz are, respectively, S  e/o =    0.050 − 0.064j 0.828 − 0.369j 0.001 − 0.000j 0.001 −0.000j 0.828 −0.369j 0.051 − 0.064j −0.000 + 0.002j 0.001 + 0.000j 0.001 −0.000j −0.000 + 0.002j −0.030 −0.124j 0.904 −0.322j 0.001 −0.000j 0.001 + 0.000j 0.904 − 0.322j −0.030 − 0.123j    , ˜ S  =    0.051 − 0.064j 0.828 −0.369j 0.000 + 0.000j 0.000 + 0.000j 0.828 −0.369j 0.050 −0.064j 0.000 + 0.000j 0.000 + 0.000j 0.000 + 0.000j 0.000 + 0.000j −0.030 −0.124j 0.903 −0.324j 0.000 + 0.000j 0.000 + 0.000j 0.903 − 0.324j −0.032 −0.123j    . The reference impedance matrices are Z  0e/o = ˜ Z  0 = 50 · 1 4 . The upper diagonal block in S  e/o is the even-mode S matrix and the lower diagonal block is the odd-mode S matrix. The residual nonzero off-diagonal blocks in S  e/o , representing the crosstalk between the even and odd modes, were ignored in Section 5 (Amakawa et al., 2008). The transformation (69) can better block-diagonalize the S-matrix. 8. Conclusions We have reviewed the simple thru-only de-embedding method for characterizing multiport networks at GHz frequencies. It is based on decomposition of a 2n-port into n uncoupled S 11 S 51 S 33 S 73 S 33 S 11 S 51 S 73 Fig. 17. Characteristics of the 4 coupled TLs from 100 MHz to 40 GHz before de-embedding (Amakawa et al., 2009). S 11 S 51 S 33 S 73 S 11 S 33 S 73 S 51 Fig. 18. Characteristics of the THRU from 100 MHz to 40 GHz (Amakawa et al., 2009). 2-ports. After the decomposition, the 2-port thru-only de-embedding method is applied. If the DUT is a 4-port and the THRU pattern has the even/odd symmetry, the transformation matrix is simple and known a priori (Amakawa et al., 2008). If not, the S-parameter-based decomposition proposed in (Amakawa et al., 2009) can be used. While the experimental results reported so far are encouraging, the validity and applicabil- ity of the de-embedding method should be assessed carefully. It is extremely important to ascertain the validity of the 2-port de-embedding method because the validity of the multi- port method depends entirely on it. One Fig. 1(b) In particular, hardly any justification has been given for the validity of the Π-equivalent-based bisecting of the THRU (Ito & Masu, 2008; Laney, 2003; Nan et al., 2007; Song et al., 2001; Tretiakov et al., 2004a). There are other possible ways of bisecting the THRU. T-equivalent-based bisection is one example (Kobrinsky et al., 2005). Once the foundations of the 2-port method are more firmly established, the multiport method can be used with greater confidence. Athru-onlyde-embeddingmethodforon-wafercharacterizationofmultiportnetworks 29 1, 5 2, 6 3, 7 4, 8 30 200 210 30 25 25 310 340 560 15 30 30 30 Fig. 15. Schematic cross section of the 1 mm-long 4 coupled TLs (not to scale), labeled with port numbers. Dimensions are in µm. Relative dielectric permittivity is 4. Metal conductivity is 5.9 ×10 7 (Ω ·m) −1 . tan δ = 0.04 (Amakawa et al., 2009). 1Ω 0.5Ω 0.1nF 70fF k = 0.15 k = 0.15 k = 0.15 0.15nH Fig. 16. Model of the left half of the THRU including pads (Amakawa et al., 2009). of even/odd transformed and fully block-diagonalized S-matrix of the THRU (Fig. 9(a)) at 10 GHz are, respectively, S  e/o =    0.050 − 0.064j 0.828 − 0.369j 0.001 − 0.000j 0.001 −0.000j 0.828 −0.369j 0.051 − 0.064j −0.000 + 0.002j 0.001 + 0.000j 0.001 −0.000j −0.000 + 0.002j −0.030 −0.124j 0.904 −0.322j 0.001 −0.000j 0.001 + 0.000j 0.904 − 0.322j −0.030 − 0.123j    , ˜ S  =    0.051 − 0.064j 0.828 −0.369j 0.000 + 0.000j 0.000 + 0.000j 0.828 −0.369j 0.050 −0.064j 0.000 + 0.000j 0.000 + 0.000j 0.000 + 0.000j 0.000 + 0.000j −0.030 −0.124j 0.903 −0.324j 0.000 + 0.000j 0.000 + 0.000j 0.903 − 0.324j −0.032 −0.123j    . The reference impedance matrices are Z  0e/o = ˜ Z  0 = 50 · 1 4 . The upper diagonal block in S  e/o is the even-mode S matrix and the lower diagonal block is the odd-mode S matrix. The residual nonzero off-diagonal blocks in S  e/o , representing the crosstalk between the even and odd modes, were ignored in Section 5 (Amakawa et al., 2008). The transformation (69) can better block-diagonalize the S-matrix. 8. Conclusions We have reviewed the simple thru-only de-embedding method for characterizing multiport networks at GHz frequencies. It is based on decomposition of a 2n-port into n uncoupled S 11 S 51 S 33 S 73 S 33 S 11 S 51 S 73 Fig. 17. Characteristics of the 4 coupled TLs from 100 MHz to 40 GHz before de-embedding (Amakawa et al., 2009). S 11 S 51 S 33 S 73 S 11 S 33 S 73 S 51 Fig. 18. Characteristics of the THRU from 100 MHz to 40 GHz (Amakawa et al., 2009). 2-ports. After the decomposition, the 2-port thru-only de-embedding method is applied. If the DUT is a 4-port and the THRU pattern has the even/odd symmetry, the transformation matrix is simple and known a priori (Amakawa et al., 2008). If not, the S-parameter-based decomposition proposed in (Amakawa et al., 2009) can be used. While the experimental results reported so far are encouraging, the validity and applicabil- ity of the de-embedding method should be assessed carefully. It is extremely important to ascertain the validity of the 2-port de-embedding method because the validity of the multi- port method depends entirely on it. One Fig. 1(b) In particular, hardly any justification has been given for the validity of the Π-equivalent-based bisecting of the THRU (Ito & Masu, 2008; Laney, 2003; Nan et al., 2007; Song et al., 2001; Tretiakov et al., 2004a). There are other possible ways of bisecting the THRU. T-equivalent-based bisection is one example (Kobrinsky et al., 2005). Once the foundations of the 2-port method are more firmly established, the multiport method can be used with greater confidence. AdvancedMicrowaveCircuitsandSystems30 S 11 S 33 S 51 S 73 S 11d S 51d S 33d S 73d S 11r S 51r S 33r S 73r Fig. 19. Reference characteristics of the 4 coupled TLs (with a subscript r) and the de- embedded results. (with a subscript d). Actually, those two are indistinguishable on the Smith chart (Amakawa et al., 2009). 9. Acknowledgments The authors thank H. Ito, T. Sato, T. Sekiguchi, and K. Yamanaga for useful discussions. This work was partially supported by KAKENHI, MIC.SCOPE, STARC, Special Coordina- tion Funds for Promoting Science and Technology, and VDEC in collaboration with Agilent Technologies Japan, Ltd., Cadence Design Systems, Inc., and Mentor Graphics, Inc. 10. References Amakawa, S., Ito, H., Ishihara, N., and Masu, K. (2008). A simple de-embedding method for characterization of on-chip four-port networks, Advanced Metallization Conference, pp. 105–106; Proceedings of Advanced Metallization Conference 2008 (AMC 2008), pp. 99– 103, 2009, Materials Research Society. Amakawa, S., Yamanaga, K., Ito, H., Sato, T., Ishihara, N., and Masu, K. (2009). S-parameter- based modal decomposition of multiconductor transmission lines and its application to de-embedding, International Conference on Microelectronic Test Structures, pp. 177– 180. Bakoglu, H. B., Circuits, Interconnections, and Packaging for VLSI, Addison Wesley, 1990. Bauer, R. F. and Penfield, Jr, P. (1974). De-embedding and unterminating, IEEE Transactions on Microwave Theory and Techniques, vol. 22, no. 3, pp. 282–288. Bockelman, D. E. and Eisenstadt, W. R. (1995). Combined differential and common-mode scat- tering parameters: theory and simulation, IEEE Transactions on Microwave Theory and Techniques, vol. 43, no. 7, pp. 1530–1539. Daniel, E. S., Harff, N. E., Sokolov, V., Schreiber, S. M., and Gilbert, B. K. (2004). Network analyzer measurement de-embedding utilizing a distributed transmission matrix bi- section of a single THRU structure, 63rd ARFTG Conference, pp. 61–68. Faria, J. A. B. (2004). A new generalized modal analysis theory for nonuniform multiconductor transmission lines, IEEE Transactions on Power Systems, vol. 19, no. 2, pp. 926–933. Goto, Y., Natsukari, Y. , and Fujishima, M. (2008). New on-chip de-embedding for accurate evaluation of symmetric devices, Japanese Journal of Applied Physics, vol. 47, no. 4, pp. 2812–2816. P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, 5th edition, Wiley, 2009. Han, D H., Ruttan, T. G., and Polka, L. A. (2003), Differential de-embedding methodology for on-board CPU socket measurements, 61st ARFTG Conference, pp.37–43. Issakov, V., Wojnowski, M., Thiede, A., and Maurer, L. (2009). Extension of thru de-embedding technique for asymmetrical and differential devices, IET Circuits, Devices & Systems, vol. 3, no. 2, pp. 91–98. Ito, H. and Masu, K. (2008). A simple through-only de-embedding method for on-wafer S- parameter measurements up to 110 GHz, IEEE MTT-S International Microwave Sym- posium, pp. 383–386. Kobrinsky, M. J., Chakravarty, S., Jiao, D., Harmes, M. C., List, S., and Mazumder, M. (2005). Experimental validation of crosstalk simulations for on-chip interconnects using S- parameters, IEEE Transactions on Advanced Packaging, vol. 28, no. 1, pp. 57–62. Kogo, H. (1960). A study of multielement transmission lines, IRE Transactions on Microwave Theory and Techniques, vol. 8, no. 2, pp. 136–142. Kolding, T. E. (1999). On-wafer calibration techniques for giga-Hertz CMOS measurements, International Conference on Microelectronic Test Structures, pp.105–110. Kolding, T. E. (2000a). Impact of test-fixture forward coupling on on-wafer silicon device mea- surements, IEEE Microwave Guided Wave Letters, vol. 10, no. 2, pp. 73–74. Kolding, T. E. (2000b). A four-step method for de-embedding gigahertz on-wafer CMOS mea- surements, IEEE Transactions on Electron Devices, vol. 47, no. 4, pp. 734–740. Koolen, M. C. A. M., Geelen, J. A. M., and Versleijen, M. P. J. G. (1991). An improved de- embedding technique for on-wafer high-frequency characterization, Bipolar Circuits and Technology Meeting, pp.188–191. Kurokawa, K. (1965). Power waves and the scattering matrix, IEEE Transactions on Microwave Theory and Techniques, vol. 13, no. 2, pp. 194–202. Laney, D. C. (2003). Modulation, Coding and RF Components for Ultra-Wideband Impulse Radio, PhD thesis, University of California, San Diego, San Diego, California. Magnusson, P. C., Alexander, G. C., Tripathi, V. K., and Weisshaar, A., Transmission Lines and Wave Propagation, 4th edition, CRC Press, 2001. Mangan, A. M., Voinigescu, S. P., Yang, M T., and Tazlauanu, M. (2006). De-embedding trans- mission line measurements for accurate modeling of IC designs, IEEE Transactions on Electron Devices, vol. 53, no. 2, pp. 235–241. Mavaddat, R. (1996). Network Scattering Parameters, World Scientific. Nan, L., Mouthaan, K., Xiong, Y Z., Shi, J., Rustagi, S. C., and Ooi, B L. (2007) Experimental characterization of the effect of metal dummy fills on spiral inductors, Radio Frequency Integrated Circuits Symposium, pp. 307–310. Nickel, J. G., Trainor, D., and Schutt-Ainé, J. E. (2001). Frequency-domain-coupled microstrip- line normal-mode parameter extraction from S-parameters, IEEE Transactions on Elec- tromagnetic Compatibility, vol. 43, no. 4, pp. 495–503. Paul, C. R. (2008). Analysis of Multiconductor Transmission Lines, 2nd edition, Wiley- Interscience. Pozar, D. M., Microwave Engineering, 3rd edition, Wiley, 2005. Athru-onlyde-embeddingmethodforon-wafercharacterizationofmultiportnetworks 31 S 11 S 33 S 51 S 73 S 11d S 51d S 33d S 73d S 11r S 51r S 33r S 73r Fig. 19. Reference characteristics of the 4 coupled TLs (with a subscript r) and the de- embedded results. (with a subscript d). Actually, those two are indistinguishable on the Smith chart (Amakawa et al., 2009). 9. Acknowledgments The authors thank H. Ito, T. Sato, T. Sekiguchi, and K. Yamanaga for useful discussions. This work was partially supported by KAKENHI, MIC.SCOPE, STARC, Special Coordina- tion Funds for Promoting Science and Technology, and VDEC in collaboration with Agilent Technologies Japan, Ltd., Cadence Design Systems, Inc., and Mentor Graphics, Inc. 10. References Amakawa, S., Ito, H., Ishihara, N., and Masu, K. (2008). 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A measurement of bal- anced transmission line using S-parameters, IEEE Instrumentation and Measurement Technology Conference, pp.866–869. [...]... . defined by  a 1 b 1  = T  b 2 a 2  =  T 11 T 12 T 21 T 22  b 2 a 2  , (70) T =  T 11 T 12 T 21 T 22  =  S −1 21 −S −1 21 S 22 S 11 S −1 21 S 12 −S 11 S −1 21 S 22  , (71) TLsL (a) (b) R. defined by  a 1 b 1  = T  b 2 a 2  =  T 11 T 12 T 21 T 22  b 2 a 2  , (70) T =  T 11 T 12 T 21 T 22  =  S −1 21 −S −1 21 S 22 S 11 S −1 21 S 12 −S 11 S −1 21 S 22  , (71) TLsL (a) (b) R. submatrices/subvectors: b =  b 1 b 2  =  S 11 S 12 S 21 S 22  a 1 a 2  = Sa. (64) Advanced Microwave Circuits and Systems2 6 1 n+1 2n-port 2 n n +2 2n 3 4 n − 1 2n − 1 n+3 S, T, S n+4 S′, S′ 1′ 2 3′ 4′ 5′ 6′ 7′ 8′ (2n − 3)′ (2n − 1)′ (2n − 2) ′ (2n)′ ̃ ̃ a 1 b 1 a 2 b 2 Fig.