A Novel Complete High Precision Standard Linear Element-Based Equivalent Circuit Model of CMOS Gyrator-C Active Transformer

Transcription

1 WSEAS TANSATIONS on IUITS and SYSTES A Novel omplete Hih Precision Standard Linear Element-Based Equivalent ircuit odel of OS Gyrator- Active Transformer AWID BANHUIN Department of omputer Enineerin Siam University 35 Petchakasem d., Phasi-charoen, Bankok THAILAND rawid_b@yahoo.com Abstract: - In this research, the novel equivalent circuit model of the OS yrator- active transformer has been proposed. Besides the effect of the unwanted intrinsic elements, the effect of the finite open loop bandwidth, which was nelected in previous research, is also taken into account. By this reason, it is a complete model. The proposed model is constructed based upon the simple standard linear elements and can accurately simulate the characteristics of the OS yrator- active transformer for various decades of frequency which cover the operatin rane of the on-chip monolithic transformer iven by a few GHz up to 10 GHz. The percentae of deviations between each of the parameters obtained from the equivalent circuit model and a similar one obtained from the oriinal active transformer were found to be very small, for example, the deviation between the e[z, Im[z and e[z 1 of the model and the topoloy II active transformer have been found to be lower than 0.007%, 0.703% and % respectively. The averae computational times for the simulations are sinificantly reduced by usin this model. Hence, the proposed model has been found to be a convenient tool for both manual and computer based analysis/desin of various applications involvin the OS yrator- active transformer. Furthermore, a simple procedure to minimize the effects of both major nonidealities is also discussed. Key-Wor: - On-chip monolithic transformer, OS yrator- active transformer, Equivalent circuit model, Standard linear element, Hih order element 1 Introduction The on-chip monolithic transformer has been adopted in many analo/mixed sinal applications nowadays, for example, Low Noise Amplifiers (LNA, mixers, and oscillators. These applications have been employed in many areas such as telecommunications, measurement and instrumentation. Oriinally, the passive onchip monolithic transformer was constructed based on a spiral inductor with no active elements included as proposed in many researches for example, [1-[6. However, the drawbacks of a spiral inductor such as the lare chip area and lack of tunin capability, are inherited by the cited passive on-chip transformer. Later, the Distributed Active Transformer (DAT which is constructed by usin both passive and active elements, inductive metals and transistors, respectively, was proposed in many researches for example, [7-[10. However, to the knowlede of the author, there is no application of DAT other than power combinin and impedance transformation. Finally, an active on-chip monolithic transformer with no inductive metal include, has been proposed by F. Yuan [. This active transformer, OS yrator- active transformer [, was constructed based upon the OS technoloy by usin the active couplin of the yrator- based active inductors via the transconductors. Accordin to [1, the OS yrator- active transformer has many far superior characteristics compared with the passive ones, for example, tunability of the couplin ratio, larer and tunable self and mutual inductances, hiher and tunable quality factors, and a smaller chip area. This OS yrator- active transformer has been adopted in various analo/mixed sinal applications such as quadrature oscillators [1, voltae controlled oscillators [13, current-mode phase-lock loops [14 and QPSK modulators [15 which are obviously employed in many areas in circuits and systems enineerin. Hence, it has been found to be the most interestin on-chip monolithic transformer. For convenience in the analysis and desin of any analo/mixed sinal application, a simple and precise model of any complicated element with sinificant intrinsic nonidealities such as a transconductor, an OP-AP, or on-chip monolithic transformer, is necessary. By usin the model, both E-ISSN: 4-66X 67 Issue 3, Volume, arch 01

2 WSEAS TANSATIONS on IUITS and SYSTES manual and automated analysis/desin effort can be sinificantly reduced while the acceptable analysis/desin precision can be maintained. For example, by usin the model proposed in [16 to simulate the behavior of the diital to analo converter, the computational time can be sinificantly reduced by a factor of 874 compared to that required for the simulation of the oriinal diital to analo converter circuit where as the simulation error of below 1% can be achieved [16. This computational time reduction is obviously an important issue in the analysis/desin of analo/mixed sinal applications nowadays which the transistor level simulation of the entire system is infeasible in the computational time aspect since the nanometer OS technoloy allows the interation of a tremendous number of transistors into a sinle system on chip [17. Furthermore, the analysis and desinin of any application by usin the model can be performed in a much simpler fashion than those usin the oriinal transistor level circuit. For the on-chip monolithic active transformer of our interest, its equivalent circuit models have been proposed in [, [13 and. Obviously, the effects of both major nonidealities of the basis transconductor entitled unwanted intrinsic elements and finite open loop bandwidth must be taken into account for the model to be complete similarly to the models of the OTA-based inductors proposed in [18. However the models proposed in [ and [13 take only the effect of the unwanted intrinsic elements into account while the finite open loop bandwidth has been totally inored. So, they are incomplete. Later, an improved model has been proposed in [19. The effects of both nonidealities mentioned above have been taken into account. As such, it is a complete model. However, this model contains a super inductor which is a troublesome hih order element as discussed in [18, 19 and will be seen later. So, this model is considered to be a complicate one which seriously requires modification. Hence, the novel equivalent circuit model of the OS yrator- active transformer is proposed in this research. The effects of both nonidealities have been taken into account. So, it is also a complete model. Furthermore, the proposed model has been found to be very convenient since it has been constructed based upon the ordinary and L which can be classified as the simple standard linear elements alon with the ideal linearly controlled voltae/current sources only. There is none of any complicate hih order element included. The characteristics of the OS yrator- active transformer can be accurately simulated by the proposed model for various decades of the operatin frequency which cover the operatin rane of the on chip monolithic transformer iven by a few GHz up to 10 GHz [0-. With the acceptable accuracies, the averae computational times of the simulations usin the proposed models are sinificantly reduced compared to those for the oriinal active transformer circuits based simulations. Therefore, it has been found to be an efficient tool for both manual/automated analysis and desin of various applications involvin the OS yrator- active transformer. Furthermore, a simple procedure to minimize the effects of both major nonidealities has also been suested in this study. An Overview of OS Gyrator- Active Transformer In this section, an overview of the OS yrator- active transformer will be discussed. The yrator- active transformer can be constructed by the active couplin of the yrator- active inductor via the transconductors. They can be classified into two topoloies entitled topoloy I and topoloy II accordin to [, which can be depicted in fi.1 and fi. respectively. It should be mentioned here that mi where {i} {1, } and where {i} {1, } and {j} {1, }, denote the transconductances of the transconductors within the basis active inductors and those of the couplin transconductors, respectively. Accordin to [-15, each transconductor within the active transformer has been simply realized by a sinle OS transistor of either n or p-type. So, the resultin active transformer can be entitled OS yrator- active transformer. Both topoloies of the resultin OS yrator- active transformers have been realized as proposed in [ which can be depicted in fi. 3 where each basis transconductor has been simply realized by a sinle OS transistor. Fi. 1 Topoloy I yrator- active transformer E-ISSN: 4-66X 68 Issue 3, Volume, arch 01

3 WSEAS TANSATIONS on IUITS and SYSTES Fi. Topoloy II yrator- active transformer mismatch amon the OS transistors within the same monolithic I has been found to be very small accordin to [3, 4. Hence, f T alon with, d and s of each basic OS transistor are assumed to be identical. Note also that the sinle pole model of the frequency dependent transconductance, m (s [5 is employed in order to demonstrate the effect of f T. This sinle pole model which demonstrates the effect of the finite open loop bandwidth of the OTA and also adopted in [18 is adequate to demonstrate the effect of f T since a OS transistor is basically a transconductor like an OTA as also mentioned in [19. Hence, m (s can be defined in this research as follows m s 1+ sτ 0 m( (1 where m0 denotes the dc-transconductance of any basis OS transistor. It should be mentioned here that τ which denotes the time delay [5 can be iven as a function of f T by 1 τ ( πf T Fi. 3 ealization of OS-yrator- active transformer: Topoloy I (above and Topoloy II (below. 3 The Proposed Equivalent ircuit odel The proposed equivalent circuit model will be presented in this section. The effects of both unwanted intrinsic elements and finite open loop bandwidth have been taken into account. Since, a OS transistor is adopted as the basis transconductor, the unwanted intrinsic elements are and d. This is because s is ideally untilized as the capacitive part for the OS yrator- active inductors which serve as the primary and secondary windins of the active transformer. In this study, the finite open loop bandwidth means the finite transition frequency, f T of the OS transistor. At this point, it should be mentioned here that all transistors within the active transformer are assumed to be identical. This assumption, which has also been adopted in [19, is acceptable since the Even thouh f T,, d and s have been assumed to be identical, this assumption is not applied to m0. This is for the resultin model bein capable to reflect the tunin capability of the active transformer which is the result of the independent tunin capability of m0 of each basis OS transistor. Hence, m0 of the OS transistors within the active based primary/secondary windins are denoted by and where as those of active couplin transistors are iven by and 1 respectively. By takin both major nonidealities into account, both self and mutual impedances of the OS yrator- active transformer are not purely inductive anymore. These impedances can be commonly iven by Z ( s α s + β s + γ (3 where {i} {1, } and {j} {1, }. In other wor, the coefficients for both self impedances which are Z (s and Z (s can be denoted by α, β and γ for Z(s alon with α, β and γ for Z (s. On the other hand, the coefficients for both mutual impedances which are Z 1 (s and Z 1 (s can be denoted by α 1, β 1 and γ 1 for Z 1 (s alon with E-ISSN: 4-66X 69 Issue 3, Volume, arch 01

4 WSEAS TANSATIONS on IUITS and SYSTES α 1, β 1 and γ 1 for Z 1 (s. For the topoloy I active transformer, these coefficients can be iven by ( + s d α (4 ( 1 τ ( s + d + 4τ β (5 ( 1 [ + ( + τ s d α (19 ( 1 [ s + ( + d + τ β (0 ( 1 ( 1 γ (1 γ (6 ( 1 ( + s d α (7 ( 1 τ ( s + d + 4τ β (8 ( 1 [ + ( + τ s 1 d α 1 ( ( 1 [ s + ( + 1 d + τ β 1 (3 ( 1 ( 1 γ 1 (4 γ (9 ( 1 ( + τ s d α 1 (10 ( 1 ( s + d + 4τ β 1 ( ( 1 and [ + ( + τ 1 s d α 1 (5 ( 1 1[ s + ( + d + τ β 1 (6 ( 1 1 ( 1 γ 1 (7 and γ 1 (1 ( 1 ( + 1 s d α 1 (13 ( 1 τ 1( s + d + 4τ β 1 (14 ( 1 1 γ 1 (15 ( 1 On the other hand, for the active transformer of topoloy II, these coefficients can be iven by [( + τ + s d 1 d α (16 ( 1 [( s + d + τ + 1 d β (17 ( 1 ( 1 γ (18 Based on the concept of usin the ideal controlled sources to realize the mutual couplin of the transformer which is adopted in [1 and [6, the prototype model for both topoloies of the active transformer can be depicted as in fi.4. It can be seen from fi.4 that each Z (s can be realized as a series combination of the super inductor (L s of the size α, the ordinary L of the size β and the ordinary of the size γ. For both topoloies, the analytical expressions of the size of these elements for any Z (s can be found from (4- (7. This L s based equivalent circuit model has been proposed in [19. However, as proposed in [18, 19, the constitutive relation of L s ( f L S ( i, φ can be iven by 1 L ( i, φ : i φ ( τ dτ L (8 f S where i and ø denote the current and manetic flux respectively. They are both physically meaninful variables. Since this constitutive relation is not a strictly linear alebraic function of i and ø because it contains the interation, L s cannot be S t 0 E-ISSN: 4-66X 70 Issue 3, Volume, arch 01

5 WSEAS TANSATIONS on IUITS and SYSTES classified as a standard linear element. This is because the constitutive relations of any standard linear element must be a strictly linear alebraic function of their correspondin physically meaninful variables only [18, 19. Hence, as also proposed in [18, there can be only three elements which can be classified as standard linear ones. These elements are the ordinary, L and respectively since their constitutive relations have been found to be strictly linear alebraic function of their correspondin physically meaninful variables as mentioned in [18. Furthermore, the impedance function of L s is iven by [18, 19 L S Z ( s s (9 From this impedance function, L s can be classified as a hih order element since the order of s is larer than 1 [18, 19. As a hih order element, L s is far complicate to, L and. It has been found to be an inferior buildin block for the modelin purposes compared to those standard linear elements as proposed in [18, 19. Hence, this prototype L s based model requires much improvement in order to avoid the troublesome L s while maintainin the accuracy as stated in [19. By the similar mathematical methodoloy to that used in [18, each Z (s can be similarly approximated by Z ( s L s + (30 f L ( i, φ : ø Li (34 Obviously, these constitutive relations are strictly linear functions of voltae (v, i and ø which are physically meaninful. Based upon this realization of Z (s by the standard linear elements which are ordinary and L in this srudy, the proposed standard linear element based equivalent circuit model for both topoloies of OS-yrator- active transformer can be constructed as shown in fi.5. By usin (4-(15 alon with (31 and (3, the analytical expressions of the elements of the model for topoloy I active transformer can be iven by L L ( [ ( [ ( d [ [ ( d + ( 1 s 4τ 1 d + ( s 4τ ( + d + ( s 1 s + 4τ d + ( s + 4 τ ( + ( 1 + d s (35 (36 (37 (38 d s 1 (39 ( 1 where [(4 s + (4 d L β 4α γ ( τ ( 1 ( d + s (40 and (3 Hence, each Z (s can be now realized as a series combination of the ordinary L of the size L and the ordinary of the size. As mentioned earlier, both ordinary and L are the standard linear element since their constitutive relations ( f ( i, v and f L ( i, φ can be iven respectively by and 1 (α {4α β β [ β β 4α γ } f ( i, v : v i (33 and ( [ ( 1 [ d + ( s 4τ 1 1 d + ( s 4τ ( + d s (41 (4 On the other hand these expressions can be alternatively iven for topoloy II by usin (16- (7, (31 and (3 as follows: E-ISSN: 4-66X 71 Issue 3, Volume, arch 01

6 WSEAS TANSATIONS on IUITS and SYSTES L [( [( m 01 m 0 + ( + ( [( ( m 01 + m 01 s m 01 1 m 0 + d 1 d + τ m 01 + ( m 01 d s 1 s τ (43 (44 4 The odel Verification The accuracy verification of the proposed model will be performed in this section. In order to do so, the comparisons between the frequency responses of e[z (ω, ω -1 Im[Z (ω, e[z 1 (ω, ω - 1 Im[Z 1 (ω and voltae transfer ratio (n, obtained from the model and the oriinal active transformer have been performed for both topoloies. athematically, n can be defined as follows L [( + ( d + ( s 1 τ (45 V n V 1 ( and 1 [( + (( + 3 d + s τ (46 4 ( [( 1 1 [( ( [( + [( [( ( + ( + ( [( ( s 1 + ( ( d 1 + d d d + d τ + d + d s ( 1 s ( s 1 s 1 ( τ s d τ (47 τ 1 d s + s (48 (49 τ (50 So, it can be seen from (35-(50 that the effect of the finite open loop bandwidth is captured by the proposed model due to the inclusion of τ which is the function of f T. It can also be seen from these equations that the effect of unwanted intrinsic elements is also captured due to the inclusion of and d. Hence, the proposed standard linear element based equivalent circuit model has been found to be complete since the effects of both major nonidealities of the basis transconductor have been included. where V 1 and V denote the voltaes at the primary and secondary terminal respectively. The chosen frequency rane is iven by 0.1 khz up to 10 GHz which cover the operatin rane of the on chip monolithic transformer accordin to [0-. The realization of the oriinal active transformers has been performed based upon the 90 nm OS technoloy due to the rise of the nanometer OS technoloy. For the model parameterizations, the values of, d, s and f T at 90 nm level have been used where as those electronically tunable dc transconductances can be iven for both topoloies in Table 1 In order to verify the accuracy of the proposed model quantitatively, the percentae of deviation in the parameter of interest has been used. Let the parameter of interested be x which can be either e[z (ω, ω -1 Im[Z (ω, e[z 1 (ω, ω - 1 Im[Z 1 (ω or n, this percentae of deviation can be defined as follows: x δ x model x x oriinal oriinal 100% (5 where x model denotes any parameter of interest obtained from the model and x oriinal denotes the similar parameter of the oriinal active transformer respectively. Hence, there are five deviations of interested for each topoloy which are δ e[z(ω, δ ω- 1Im[Z(ω, δ e[z1(ω, δ ω-1im[z1(ω and δ n. At this point, the comparative frequency responses of e[z (ω, ω -1 Im[Z (ω, e[z 1 (ω, ω -1 Im[Z 1 (ω and n obtained from the oriinal active transformer of topoloy I (Fi. and the proposed model can be respectively plotted in Fi.(6-(10. On the other hand, the similar comparative frequency responses between the oriinal active transformer of topoloy II (Fi. 3 and the model can be plotted in Fi.(-(15. For both topoloies, the frequency responses of e[z (ω, ω -1 Im[Z (ω, e[z 1 (ω, ω - 1 Im[Z 1 (ω and n obtained from the proposed E-ISSN: 4-66X 7 Issue 3, Volume, arch 01

7 WSEAS TANSATIONS on IUITS and SYSTES model keep accurately trackin their counterparts obtained from the oriinal active transformer alon the chosen frequency rane. This can be quantitatively stated that δ e[z(ω, δ ω-1im[z(ω, δ e[z1(ω, δ ω-1im[z1(ω and δ n which iven in Table, have been found to be very small. So, the proposed model has been verified in both qualitative and quantitative aspects to be hihly accurate and applicable for the entire typical operatin rane of the on-chip transformer accordin to the chosen frequency rane. Furthermore, the averae computational times obtained from 100 simulations for each topoloy have also been recorded as shown in Table 3. It can be seen that sinificant reductions between the averae computational times obtained from the model based simulations and those from the oriinal active transformers based simulations can be observed. In the quantitative aspect, the factor of computational time reduction can be mathematically iven by t F (53 oriinal t model where t model denotes the averae computational time obtained from the model based simulations and t oriinal denotes obtained from the simulations of the oriinal active transformer respectively. Accordin to the satisfied verification results the proposed model has been found to be a convenient tool for the analysis and desin of various applications involvin the OS yrator- active transformer. The model is applicable in various fiel of analo/mixed sinal circuits and system enineerin. Table 1. D transconductances of the basis OS transistor Topoloy I Topoloy II ms ms ms ms ms ms ms ms Table. δ e[z(ω, δ ω-1im[z(ω, δ e[z1(ω, δ ω- 1Im[Z1(ω and δ n Topoloy I Topoloy II δ e[z(ω 1.7% 0.007%, δ ω-im[z(ω 1.59% 0.703% δ e[z1(ω.04% % δ ω-im[z1(ω 1.69% 0.34% δ n 4% 3.09% Table 3. Averae computational times and the reductions for both topoloies t model (ms Topoloy I t oriinal (ms F t model (ms Topoloy II t oriinal (ms Discussion The procedure to minimize the effects of both major nonidealities will be discussed in this section. In the ideal situation, the effects of these major nonidealities are not taken into account. So, both self and mutual impedances of the OS yrator- active transformer are purely inductive. This means that for both topoloies. Furthermore, the manitudes of deviation from the desired values of both self and mutual inductances are not existed for both topoloies. eardless to the topoloy, these manitudes of deviation can be defined for self and mutual inductances respectively as follows Ld L F Δ L (54 Δ L (55 Ld L Δ (56 1 1d 1 Δ (57 1 1d 1 E-ISSN: 4-66X 73 Issue 3, Volume, arch 01

8 WSEAS TANSATIONS on IUITS and SYSTES Fi. 4. The prototype equivalent circuit model of OS-yrator- active transformer. Fi. 5. The proposed standard linear element based equivalent circuit model of OS-yrator- active transformer. E-ISSN: 4-66X 74 Issue 3, Volume, arch 01

9 WSEAS TANSATIONS on IUITS and SYSTES Fi. 6. The comparative frequency responses of e[z for topoloy 1: The proposed model ( and the oriinal active transformer (. Fi. 7. The comparative frequency responses of ω -1 Im[Z for topoloy 1: The proposed model ( and the oriinal active transformer (. Fi.8. The comparative frequency responses of e[z 1 for topoloy 1: The proposed model ( and the oriinal active transformer (. E-ISSN: 4-66X 75 Issue 3, Volume, arch 01

10 WSEAS TANSATIONS on IUITS and SYSTES Fi. 9. The comparative frequency responses of ω -1 Im[Z 1 for topoloy 1: The proposed model ( and the oriinal active transformer (. Fi. 10. The comparative frequency responses of n for topoloy 1: The proposed model ( and the oriinal active transformer (. Fi.. The comparative frequency responses of e[z for topoloy : The proposed model ( and the oriinal active transformer(. E-ISSN: 4-66X 76 Issue 3, Volume, arch 01

11 WSEAS TANSATIONS on IUITS and SYSTES. Fi. 1. The comparative frequency responses of ω -1 Im[Z for topoloy : The proposed model ( and the oriinal active transformer(. Fi.13. The comparative frequency responses of e[z 1 for topoloy : The proposed model ( and the oriinal active transformer(. Fi. 14. The comparative frequency responses of ω -1 Im[Z 1 for topoloy : The proposed model ( and the oriinal active transformer(. E-ISSN: 4-66X 77 Issue 3, Volume, arch 01

12 WSEAS TANSATIONS on IUITS and SYSTES Fi. 15 The comparative frequency responses of n for topoloy : The proposed model ( and the oriinal active transformer(. where L d, L d, 1d and 1d denote the desired values of self and mutual inductances, on the other hand, L, L, 1 and 1 denote the practically deviated ones. So, it can be mathematically stated that Δ L Δ L Δ 1 Δ 1 0 in the ideal situation. However, this is not the case in practice since the effects of both major nonidealities are taken into account. Unlike the ideal situation,,, 1 and 1 are more than zero for both topoloies with their analytical expressions as iven earlier. Furthermore, the deviations from the desired values in both self and mutual inductances for both topoloies are also more than zero. This can be stated mathematically for both topoloies that Δ L > 0, Δ L > 0, Δ 1 > 0 and Δ 1 > 0. In order to evaluate these manitudes of deviation L d, L d, 1d and 1d must be firstly determined. For topoloy 1, they are iven in this study accordin to those in [ as follows s L d (58 D s L d (59 D s 1 d (60 D 1 s 1 d (61 D On the other hand, they can be iven in this study for topoloy accordin to those in [ as follows s L d (6 D s L d (63 D s 1 d (64 D 1 s 1 d (65 D It should be mentioned here that, D can be similarly defined for both topoloies accordin to [ as follow D 1 1 (66 At this point, Δ L, Δ L, Δ 1 and Δ 1 for topoloy 1, can be derived by usin (35-(4, (54-(57 and (58-(61 as follows ΔL ΔL Δ 1 ( ( ( ( ( ( d 4 τ 1 d + 4 τ 1 1 d + 4 τ (67 (68 (69 E-ISSN: 4-66X 78 Issue 3, Volume, arch 01

13 WSEAS TANSATIONS on IUITS and SYSTES Δ 1 1 ( ( d 4 τ 1 (70 where as those for topoloy can be found by usin (43-(50, (54-(57 and (6-(65 as follows ΔL ΔL Δ Δ 1 1 [( ( [( ( 1 [( ( [( ( + 1 d + τ d 1 1 τ d τ 1 + d τ (71 (7 (73 (74 From these manitudes of deviation, it can be concluded for both topoloies that 1 Δ L ( Δ L ( Δ 1 ( Δ 1 (78 1 Furthermore, it can also be concluded for both topoloies from the observation of (36, (38, (40, (4, (44, (46, (48 and (50 that both major nonidealities can be vastly reduced if >> (83 1 Hence, it has been found that the procedure to minimize the effects of both major nonidealities is to let and which are the dc transconductances of the transistors within the active based primary/secondary windins, be much larer than and 1 which are the dc transconductances the active couplin transistors. This can be easily accomplished since these dc transconductances are electronically controllable. 6 onclusion In this research, the novel standard linear element based equivalent circuit model of the OS yrator- active transformer has been proposed. The proposed model has been found to be complete since the effects of both unwanted intrinsic elements and finite open loop bandwidth are included. The proposed model has been found to be very simple since it is composed only of the standard linear elements. It can simulate the characteristics of the taret active transformer with hih accuracy for various decades of frequency from 0.1 khz up to 10 GHz which cover the typical operatin rane of the on chip monolithic transformer. Furthermore, the averae computational times of the simulations can be sinificantly reduced by usin the proposed models. Finally, the procedure to minimize the effects of both major nonidealities has been discussed. Hence, the proposed model has been found to be a convenient tool for the analysis and desin of various applications involvin the OS yrator- active transformer as mentioned earlier. 1 ( (80 1 Acknowledements The author would like to acknowlede ahidol University, Thailand, for the online database service ( (8 1 Obviously, these unwanted manitudes of deviation and resistances which are the effects eferences: [1 J.. Lon, onolithic Transformers for Silicon F I Desin, IEEE Journal of Solid-State ircuits, Vol. 35, No. 9, 000, pp [ J. J. Zhou and D. J. Allstot, onolithic Transformers and Their Application in A differential OS F Low-Noise Amplifier, E-ISSN: 4-66X 79 Issue 3, Volume, arch 01

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15 WSEAS TANSATIONS on IUITS and SYSTES [6 J. Zhen, Y.-. Hahm, V. K. Tripathi, and A. Weisshaar, AD-Oriented Equivalent ircuit odelin ofon-hip Interconnects on Lossy Silicon Substrate, IEEE Transaction on icrowave Theory and Techniques, Vol. 48, No. 9, 000, pp E-ISSN: 4-66X 81 Issue 3, Volume, arch 01