1 / 88

MODELING OF RF DEVICES AND CIRCUITS Elissaveta GADJEVA

MODELING OF RF DEVICES AND CIRCUITS Elissaveta GADJEVA. CONTENTS. 1. Modeling of RF circuits 2. Modeling of passive elements 3. Modeling of active elements 4. Noise modeling of RF elements 5. Parameter extraction of equivalent circuits for passive and active RF elements.

avent
Download Presentation

MODELING OF RF DEVICES AND CIRCUITS Elissaveta GADJEVA

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MODELING OF RF DEVICES AND CIRCUITS Elissaveta GADJEVA

  2. CONTENTS 1. Modeling of RF circuits 2. Modeling of passive elements 3. Modeling of active elements 4. Noise modeling of RF elements 5. Parameter extraction of equivalent circuits for passive and active RF elements

  3. 1. Modeling of RF circuits1.1. Determination of S-parameters using PSpice-like simulators The S-parameter description allows to investigate the behavior of the devices at RF frequency range and to study the stability factor and gain characteristics. The two-port S-parameters can be described according to the input language of the PSpice simulator using voltage controlled voltage sources of EFREQ type with tabularly defined parameters. The S-parameters are obtained in the form of corresponding node voltages V(S_11), ... V(S_22) of the model. The stability parameters can be automatically determined using the macrodefinitions of the Probe analyzer.

  4. 1. Modeling of RF circuits 1.2. RF circuit stability investigation using PSpice simulation • A two-port is stable if the stability factorK > 1 (Rollet's stability condition): Macrodefinitions in the Probe analyzer for the stability factor K S11m = m(S11) delta = m(S11*S22-S12*S21) S12m = m(S12) K= (1-S11m*S11m-S22m*S22m S21m = m(S21) + delta*delta)/(2*S12m*S21m) S22m = m(S22)

  5. 1. Modeling of RF circuits 1.2. RF circuit stability investigation using PSpice simulation • Another important stability characteristics, based on S-parameter description, are: • The Maximum Available Gain (MAG), defined for a stable two-port (K > 1) • The Maximum Stable Gain (MSG), defined for a potentially unstable two-port (K < 1): • The gain MSG/MAG is defined in the form:

  6. 1. Modeling of RF circuits 1.2. RF circuit stability investigation using PSpice simulation MSG/MAG = (1-ena).MAG + ena.MSG, where ena =1 for K<1 and ena =0 for K>1 Macrodefinitions in the Probe analyzer: *Maximum Available Gain (MAG) MAG=S21m*(K-sqrt(K*K-1))/S12m *Mavimum Stable Gain (MSG) MSG = S21m/S12m *Gain MSG/MAG ena = pwr((1+sgn(1-K))/2,1) MSGMAG=(1-ena)*db(mag)+ena*db(msg)

  7. 1. Modeling of RF circuits 1.2. RF circuit stability investigation using PSpice simulation • The gain parameter Maximum Unilateral Gain (MUG) (or Mason's gain) is defined in the form: • A two-port is unconditionally stable if the stability coefficient :

  8. 1. Modeling of RF circuits 1.2. RF circuit stability investigation using PSpice simulation The frequency dependencies of the stability factor K and MSG/MAG K K=1 MSG/MAG

  9. 1. Modeling of RF circuits1.2. RF circuit stability investigation using PSpice simulation The frequency dependence of the stability coefficient :

  10. 1. Modeling of RF circuits1.2. RF circuit stability investigation using PSpice simulation REFERENCES [1.1] Sze, S. M., Physics of Semiconductor Devices, 2nd Edition, John Wiley, New York 1981. [1.2] Edwards, M., J. Sinsky, A New Criterion for Linear 2-Port Stability Using a Single Geometrically Derived Parameter, IEEE Trans. Microwave Theory Tech., vol. MTT-40., No. 12, December 1992. [1.3] Zinke, O., H. Brunswig, Hochfrequenztechnik 2, 4. Auflage, Springer Verlag, Berlin, 1903. [1.4] Hristov, M., M. Gospodinova, E. Gadjeva, Stability Analysis of SiGe Heterojunction Bipolar Transistors Using PSpice, IC-SPETO’2001, Gliwice, Poland, 2001 [1.5] OrCAD PSpice A/D. Circuit Analysis Software. Reference Manual, OrCAD Inc., USA, 1998

  11. 1. Modeling of RF circuits 1.3. Application of Spice Simulation to Investigationof Class E Power Amplifier Characteristics A. Automated design of class E power amplifier with small DC-feed inductance B. Automated design of class E power amplifier with nonlinear capacitance C. Simulation results C1. Simulation Results from PSpice Procedure I C2 Simulation Results from PSpice Procedure II

  12. 1. Modeling of RF circuits 1.3. Application of Spice Simulation to Investigationof Class E Power Amplifier Characteristics • The increasing application of the wireless communications requires design and optimization of power amplifiers, which are the most power consuming part in the transceivers. The class E power amplifier is widely used as it provides a large value of the output power with high efficiency, working in switch mode. • A power amplifier could be defined as class E if a few criteria are fulfilled. • First of them is that the voltage across the switch remains low when the switch turns off. • When the switch turns on, the voltage across the switch should be zero. • Finally, the first derivative of the drain voltage with respect to time is zero, when the switch turns on • The first two conditions suggest that the power consumption by the switch is zero. The last condition ensures that the voltage-current product is minimized even if the switch has a finite switch-on time.

  13. 1. Modeling of RF circuits 1.3. Application of Spice Simulation to Investigationof Class E Power Amplifier Characteristics • Procedures for fast and accurate sizing of class E power amplifier circuit elements are developed using the PSpice circuit simulator. • Verification of the obtained results is performed. • The implementation of the design procedure in the simulation model gives the possibility for modification, comparison of variants and performance optimization.

  14. 1. Modeling of RF circuits 1.3. Application of Spice Simulation to Investigationof Class E Power Amplifier Characteristics A typical configuration of a class E power amplifier is shown in Fig. 1.1. Class E power amplifiers achieve 100% efficiency theoretically in the expense of poor linearity performance. Fig. 1.1

  15. 1. Modeling of RF circuits 1.3. Application of Spice Simulation to Investigationof Class E Power Amplifier Characteristics • The rapid development of the wireless communications requires minimizing the design process for all the blocks building the communication systems. • Basic task in the class E power amplifier design consists in sizing the circuit elements to achieve the maximal amplifier efficiency without performing a lot of additional optimizing procedures. • In this paper procedures for automated sizing of the class E power amplifier circuit elements are presented using the possibilities of the general-purpose circuit analysis program such as PSpice. • Theintegration of the design and analysis stages allows reuse of the design procedure as well as fast power amplifier characteristics assessment.

  16. A. Automated design of class E power amplifier with small DC-feed inductance • The procedure for automated design of class E power amplifier using the analysis program PSpice is based on the approach giving explicit design equations for class E power amplifiers with small dc-feed inductance (procedure I). • The operation is analyzed in two discrete states: OFF state (0<ωt<π) – the switch is open, and ON state (π<ωt<2π) – the switch is closed, where =2f is the operation frequency of the circuit. • The assumption is made that the loaded Q-factor of the series resonator LsCs is very high so only sinusoidal current at the carrier frequency is allowed to flow through the load resistance R.

  17. A. Automated design of class E power amplifier with small DC-feed inductance The susceptance of the shunt capacitor C1 is B=ωC1 and X represents the mistuning reactance. In the classic RF C-based class E power amplifier (L1∞) the design procedure consists of evaluating the three key circuit parameters: • optimal load resistance • shunt susceptance • excessive reactance Vdc – supply voltage; Pout – output power. The approach assumes work with a preliminary chosen value for the dc-feed inductance L1 with reactance Xdc=ωL1. The dc resistance that circuit presents to the supply source is

  18. A. Automated design of class E power amplifier with small DC-feed inductance Using the ratio z=Xdc/Rdc the values of the circuit parameters R, B and X are recalculated in the case for finite dc-feed inductance. Based on the recalculatedand normalized values of the circuit parameters interpolation polynomials are composed, giving the explicit values for circuit parameters. According to the parameter z value, there are groups of polynomials: *for z ≤ 5 R=Rdc.PR1; B=PB1/R; X=R.PX1; PR1=1.979–0.7783z+0.1754z2–0.01397z3 PB1=1.229–0.7171z+0.1881z2–0.01672z3; PX1= -1.202+1.591z–0.4279z2+0.03894z3; *for 5 < z ≤ 20 R=Rdc.PR2; B=PB2/R; X=R.PX2; PR2=0.9034–0.04805z+0.002812z2–5.707.10-5z3; PB2=0.3467–0.02429z+0.001426z2–2.893.10-3z3; PX2=0.6784+0.006641z–0.003794z2+7.587.10-5z3; *for z > 20 R=Rdc.PR3; B=PB3/R; X=R.PX3; PR3 = 0.6106 ; PB3 = 0.1999 ; PX3 = 1.096

  19. A. Automated design of class E power amplifier with small DC-feed inductance The calculation of the output matching network of the power amplifier is set in the procedure: RL – load resistance; R – optimal load; Xc and Xl – reactances of the inductance and capacitor of the matching network. The following parameters of the procedure are defined as input data by the designer: the supply voltage, the desired output power, the operating frequency and the load resistance. The polynomials and the equations used for the calculation of the basic class E circuit components as well as those of the output matching circuit, are defined in the PSpice model as parameters with the statement PARAMETERS. The following expressions are used in order to evaluate R, B and X for a given value of z: R=Rdc.(PR1.ena1+PR2.ena2+PR3.ena3);X=R.(PX1.ena1+PX2.ena2 + + PX3.ena3); B=(PB1.ena1+PB2.ena2+PB3.ena3)/R, where ena1 = 1 if z ≤ 5, otherwise ena1 = 0; ena2 = 1 if 5 < z ≤ 20, otherwise ena2 = 0; ena3 = 1 if z > 20, otherwise ena3 = 0.

  20. A. Automated design of class E power amplifier with small DC-feed inductance Computer realization of the procedure for automatic design of class E amplifier: *Input data .param Ls=1e-9 RL=50 pi=3.141592654 Pout=1 Vdc=3 Fc=2e9 L1=2e-9 *Design equations .param Cs={1/(Wc*Wc*Ls)} B={PB/R} C1={B/Wc} Lx={X/Wc} R={PR*Rdc} Rdc={Vdc*Vdc/Pout} + n={RL/R} Wc={2*pi*Fc} X={PX*R} Xdc={Wc*L1} Z={Xdc/Rdc} Cm={(sqrt(n-1))/(Wc*RL)} + Lm={(R*sqrt(n-1))/Wc} *Polynomial description .param ena1={if(z>5,0,1)} ena3={if(z>20,1,0)} ena2={if(z<5,0,if(z<=20,1,0))} Z2={Z*Z} Z3={Z2*Z} *Polynomials B(z) .param PB={(1-ena3)*(PB1*ena1+PB2*ena2) + PB3*ena3} + PB1={1.229-0.7171*Z+0.1881*Z2-0.01672*Z3} + PB2={0.3467-0.02429*Z+ 0.001426*Z2-2.893E-5*Z3} + PB3=0.1999 *Polynomials R(z) .param PR={(1-ena3)*(PR1*ena1+PR2*ena2)+PR3*ena3} + PR1={1.979-0.7783*Z+0.1754*Z2-0.01397*Z3} + PR2={0.9034-0.04805*Z+0.002812*Z2-5.707E-5*Z3} PR3=0.6106 PX3=1.096 *Polynomials X(z) .param PX2={0.6784+0.006641*Z-0.003794*Z2 +7.587E-5*Z3} + PX1={-1.202+1.591*Z-0.4279*Z2+0.03894*Z3} + PX={(1-ena3)*(PX1*ena1+PX2*ena2)+PX3*ena3}

  21. B. Automated design of class E power amplifier with nonlinear capacitance This procedure for automated design using PSpice is based on the approach for investigation of class E amplifier with nonlinear capacitance for any output quality factor Q and finite dc-feed inductance (procedure II). The basic input parameters are: the operating angular frequency =2f; the resonant angular frequency 0=2f0; the ratio of the resonant to operating frequency A=f0/f; the ratio of resonant to parasitic capacitance on MOSFET transistor B=C0/Cj0; the ratio of resonant to dc-feed inductance H=L0/Lc; the loaded quality factor Q=L0/R; the switch-on duty ratio of the switch. The values of A and B are defined using the graphical dependencies of these coefficients on the quality factor Q for H=0.001 and supply voltage 1V. The functions A(Q) and B(Q) can be approximated by the following polynomials: A(Q) = 0.32928610–5Q5 – 0.22624410–3Q4 + 0.60828910–2Q3 + 0.079868Q2 + 0.513996Q -0.353653 B(Q) = B1(Q) + B2(Q) ; B1(Q) = 10z; z = 3–9.6(Q–2.2) B2(Q) = –0.28224510–4Q5 + 0.201105810–2Q4 – – 0 .0552482Q3 + 0.726058Q2 – 4.546336Q +11.217525

  22. B. Automated design of class E power amplifier with nonlinear capacitance A(Q) and B(Q) are defined in the PSpice model as parameters by using PARAMETERS statement and the realization of the procedure for power amplifier design is as follows: *Input data .param Vdc=1 D=0.5 Q=10 R=1 Vbi=0.7 pi=3.141592654 Rs=0.01 + H=0.001 Fc=5Meg *Design equations .param Wc={2*pi*Fc} m=0.5 Lo={Q*R/Wc} Lc={Lo/H} Q2={Q*Q} + Q3={pwr(Q,3)} Q4={pwr(Q,4)} Cs ={Cjo} Q5={pwr(Q,5)} + Cjo={Co/B} Fo={A*Fc} Wo={2*pi*Fo} Co={1/{Wo*Wo*Lo}} *Polynomial description .param A={0.329286E-5*Q5-0.226244E-3*Q4 + 0.6082893E-2*Q3 + + 0.079868*Q2+ 0.513996*Q-0.353653} .param B=pwr(10,(3-9.6*(Q-2.2)))- 0.282245E-4*Q5+ + 0.20110578E-2*Q4-0.0552482*Q3+0.726058*Q2 – + 4.546336*Q+11.217525}.

  23. B. Automated design of class E power amplifier with nonlinear capacitance In the case of high output Q and RF choke an equivalent linear capacitance of the MOSFET switch is defined in the form: CSequ=24VbiCj0/{12Vbi+[6Vbi(24Vbi-242Vdc+4Vdc)]+92(2+4)VbiVdc]1/2}, where Vbi is the built-in potential, with a typical value Vbi = 0.7. In the case of finite dc-feed inductance the coefficients A and B can be approximated from their graphical dependencies on the ratio of resonant to dc-feed inductance H. The functions A(H) and B(H) are approximated by the following polynomials: A(H) = - 4.424310–3H4-1.572271510-2H3 +2.7834910-2H2 +0.15350566H +0.8216771 B(H) = 3.836804510-2H4+0.1836775H3+7.56716510-3H2- 0.994968H+0.801806. For high supply voltage Vdc the design parameters A and B are defined by corresponding graphical dependencies on Vdc.

  24. C. Simulation resultsC1. Simulation Results from PSpice Procedure I A simulation example for the first procedure, using explicit design equations for class E power amplifiers with small dc-feed inductance (Fig. 1.2). The input parameters are: supply voltage Vdc=3V; required output power Pout=1W; load resistance RL=50; operating frequency fc=2GHz, L1=2nH, Ls=1nH. The switch used for the procedure verification has a resistance RON=0.1 for the ON state and ROFF=1106 for the OFF state. Fig. 1.2

  25. C. Simulation resultsC1. Simulation Results from PSpice Procedure I Comparison results for Procedure I

  26. C. Simulation resultsC1. Simulation Results from PSpice Procedure I Waveforms of the currents flowing through the switch (Isw) and through the dc-feed inductance Fig. 3 Switch voltage Fig. 1.3

  27. C. Simulation resultsC2. Simulation Results from PSpice Procedure II A simulation example for the procedure II is based on the approach for investigation of class E amplifier with nonlinear capacitance for any output quality factor Q and finite dc-feed inductance. In this case the preliminary defined input parameters are: supply voltage Vdc; loaded quality factor Q; ratio of the resonant inductance to the dc-feed inductance H; resistive load R; switch-on resistance Rs; grading coefficient of the diode junction m; switch-on duty ratio D of the switch; operating and resonant frequencies fc and f0; operating and resonant angular frequencies c and 0. Verification of the described approach is performed by using the following design specifications: Vdc=40V, Q=10, H=0.001, R=12.5, Rs=0.4, m=0.5, D=0.5, fc=30MHz and MOSFET model parameters given in [4]. The examination circuit is shown in Fig. 1.4. Fig. 1.4

  28. C. Simulation resultsC2. Simulation Results from PSpice Procedure II The results obtained by the second PSpice procedure are compared with the results given in [1.4]. They are presented in the table below.

  29. C. Simulation resultsC2. Simulation Results from PSpice Procedure II The element values obtained by the computer-aided design procedure are compared with the values published in [1.4]. They are shown in table below.

  30. C. Simulation resultsC2. Simulation Results from PSpice Procedure II Simulation results for the output voltage Fig. 1.5 Simulation results for the drain-source voltage Fig. 1.6

  31. 1. Modeling of RF circuits REFERENCES [1.6] N. O. Sokal and A.D. Sokal, Class E – A new class of high efficiency tuned single-ended switching power power amplifiers, IEEE Journal of Solid State Circuits,10(6), June 1975, 168-176. [1.7] H. Krauss, Solid State Radio Engineering(John Wiley & Sons, 2000). [1.8] D. Milosevic, J. Tang, A. Roermund, Explicit design equations for class-E power amplifiers with small DC-feed inductance, Conference on Circuit Theory and Design, Ireland, 2005,vol.III, 101-105. [1.9] H. Sekiya, at al, Investigation of class E amplifier with nonlinear capacitance for any output Q and finite DC-feed inductance, IEICE Trans. Fundamentals, E89-A(4), 2006, 873-881. [1.10] Cripps, S., RF Power Amplifiers for Wireless Communications (Artech House, 1999). [1.11] E. Gadjeva, M. Hristov, O. Antonova, Application of Spice Simulation to Investigation of Class E Power Amplifier Characteristics, International Scientific Conference Computer Science’2006, Istanbul, 2006.

  32. 2. Modeling of passive elements 2.1. Modeling of spiral inductors • Computer macromodels of planar spiral inductors for RF applications are developed in accordance with the input language of the PSpice-like circuit simulators. • Approximate expressions for the inductance value are used in the macromodels based on the monomial expression, modified Wheeler formula, as well as current sheet approximation. Two-port inductor computer model is constructed taking into account the parasitic effects. The elements of the equivalent circuit are defined by geometry dependent parameters. • Macromodels are constructed in the form of parametrized subcircuits in accordance with the syntax of the PSpice input language. • Based on the possibilities of the nonlinear analysis, optimal design of the inductor can be is performed. The two-port S-parameters and the Q factor are obtained in the graphical analyzer Probe using corresponding macros. • The model descriptions and simulation results are given.

  33. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Fig. 2.1. Physical equivalent circuit of planar spiralinductor

  34. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Parameters of spiral inductors andcorresponding names in the PSpice model

  35. 2. Modeling of passive elements 2.1. Modeling of spiral inductors The circuit elements are defined by the following equations:

  36. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Modelling of the inductance Ls: Wheeler formula The simple modification of the Wheeler formula is applicablefor square, hexagonal and octagonal integrated spiral inductors: The coefficients K1 and K2 depend on the inductor layout.In the case of square inductors K1=2.34 and K2= 2.75. In accordance with the OrCAD PSpice language, the value of Ls1 is defined in the form: {K1*mju*(n*n*Davg)/(1+K2*ro)}

  37. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Modelling of the inductance Ls: Current sheet approximation Using current sheet approximation [2.2,2.4], the inductance Ls2of square, hexagonal, octagonal and circle integrated spiral inductors can bedescribed by the expression: In the case of square inductors c1=1.27, c2= 2.07, c3= 0.18 and c4= 0.18 [2.2]. In accordance with the OrCAD PSpice language, the Ls2 value is defined in the form: {0.5*mju*n*n*davg*c1*(log(c2/ro)+c3* ro+ c4*ro*ro)}

  38. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Modelling of the inductance Ls:Data fitted monomial expression Using the data fitted monomial expression [2.2], the inductance Ls3 is described in the form: This expression is valid for square, hexagonal and octagonal integrated spiral inductors. In the case of square inductors The description in accordance with the OrCAD PSpice language of the Ls3 value has the form: {beta*pwr(Dout*1e6,al1)*pwr(w*1e6,al2)*pwr (Davg*1e6,al3)*pwr(n,al4)*pwr(sp*1e6,al5)*1e-9}

  39. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Fig. 2.2. Relative error determination of inductanceapproximations Ls1, Ls2 and Ls3

  40. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Modelling of the resistance Rs Fig. 2.3. Modelling of frequency dependent resistance Rs Rs is presented by a voltage controlled current sourceof GLAPLACEtype (Fig.2.3): G_Rs 1 2 LAPLACE {V(1,2)}={l/(sigma*w* sqrt(2/(sqrt(-s*s)*mju*sigma))*(1-exp(-t/(sqrt(2/(sqrt(-s*s)* mju*sigma))))))}

  41. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Modelling of the elements Cs, Cox, Csi and Rsi The values of the elementsCs ,Cox , Csi and Rsi of the equivalent circuit are defined in the form: Capacitance Cs: {n*pwr(w,2)*Eox/toxM1M2} Capacitance Cox: {0.5*L*w*Eox/tox} Capacitance Csi: {0.5*L*w*Csub} Resistance Rsi: {2/ (L*w*Gsub)}

  42. 2. Modeling of passive elements 2.1. Modeling of spiral inductors Parametrized PSpice model of spiral inductor .PARAM Dout={Din+2*(n*(sp+w)-sp)}Davg={Dout-n*(sp+w)+sp} subckt ind3 1 2 6 params: beta={beta} al1={al1} al2={al2} al3={al3} + al4={al4} al5={al5} L={L}Dout={Dout} mju={mju} sigma={sigma} + w={w} Eox=3.45e-11 toxM1M2={toxM1M2} tox={tox} sp={sp} n={n} + Gsub={Gsub} Csub={Csub} t={t} Ls 1 3 {beta*pwr(Dout*1e6,al1)*pwr(w*1e6,al2)* pwr(Davg*1e6,al3)*pwr(n,al4)*pwr(sp*1e6,al5)*1e-9} G_Rs 3 2 LAPLACE {V(3,2)}={l/(sigma*w*sqrt(2/(sqrt(-s*s)*mju*sigma))*(1-exp(-t/(sqrt(2/(sqrt(-s*s)* mju*sigma))))))} Cs 1 2 {n*pwr(w,2)*Eox/toxM1M2} Cox1 1 4 {0.5*L*w*Eox/tox} Cox2 2 5 {0.5*L*w*Eox/tox} Rsi1 4 6 {2/(L*w*Gsub)} Csi1 4 6 {0.5*L*w*Csub} Csi2 5 6 {0.5*L*w*Csub} Rsi2 5 6 {2/(L*w*Gsub)} .ends

  43. 2. Modeling of passive elements 2.1. Modeling of spiral inductors • Application of parametric analysis to geometry design and optimization • The possibilities of the PSpice-like simulator to define one or more independent variables as simulation parameters can be effectively applied to geometry design of planar spiral inductors. • Using the ABM blocks from the analog behavioral modeling library, the geometry and electrilal inductor parameters (Din, Dout, w, n, Ls, etc.) can be defined, changed and investigated using behavioral computer model of the spiral inductor.

  44. 2. Modeling of passive elements 2.1. Modeling of spiral inductors The dependence of the inductance Ls on trace width w with parameter the number of turns

  45. 2. Modeling of passive elements 2.1. Modeling of spiral inductors The dependence of trace width w on the number of turns n for a given inductance Ls

  46. 2. Modeling of passive elements 2.2. Modeling of planar transformers

  47. 2. Modeling of passive elements 2.1. Modeling of planar transformers

  48. Параметрите на модела се изчисляват по формулите [4][8]: ; ; ; ; ; 2. Modeling of passive elements 2.2. Modeling of planar transformers

  49. ; 2. Modeling of passive elements 2.2. Modeling of planar transformers Parameter description

  50. ; 2. Modeling of passive elements 2.2. Modeling of planar transformers PSpice model

More Related