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

1. Modeling of RF circuits 1. 1
1. Modeling of RF circuits 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.

Macrodefinitions in the Probe analyzer for the stability factor K
1. Modeling of RF circuits RF circuit stability investigation using PSpice simulation A two-port is stable if the stability factor K > 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)

1. Modeling of RF circuits 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:

Macrodefinitions in the Probe analyzer:
1. Modeling of RF circuits 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)

1. Modeling of RF circuits 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 :

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

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

1. Modeling of RF circuits 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

1. Modeling of RF circuits
1.3. Application of Spice Simulation to Investigation of 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

1.3. Application of Spice Simulation to Investigation of 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.

1.3. Application of Spice Simulation to Investigation of 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.

1.3. Application of Spice Simulation to Investigation of 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

1.3. Application of Spice Simulation to Investigation of 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. The integration of the design and analysis stages allows reuse of the design procedure as well as fast power amplifier characteristics assessment.

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.

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

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 z2– z3 PB1=1.229–0.7171z z2– z3; PX1= z– z z3; *for 5 < z ≤ 20 R=Rdc.PR2; B=PB2/R; X=R.PX2; PR2=0.9034– z z2– z3; PB2=0.3467– z z2– z3; PX2= z– z z3; *for z > 20 R=Rdc.PR3; B=PB3/R; X=R.PX3; PR3 = ; PB3 = ; PX3 = 1.096

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.ena 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.

Computer realization of the procedure for automatic design of class E amplifier: *Input data .param Ls=1e-9 RL=50 pi= 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={ *Z *Z *Z3} + PB2={ *Z *Z E-5*Z3} + PB3=0.1999 *Polynomials R(z) .param PR={(1-ena3)*(PR1*ena1+PR2*ena2)+PR3*ena3} + PR1={ *Z *Z *Z3} + PR2={ *Z *Z E-5*Z3} PR3= PX3=1.096 *Polynomials X(z) .param PX2={ *Z *Z E-5*Z3} + PX1={ *Z *Z *Z3} + PX={(1-ena3)*(PX1*ena1+PX2*ena2)+PX3*ena3}

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) = 10–5Q5 – 10–3Q 10–2Q Q Q B(Q) = B1(Q) + B2(Q) ; B1(Q) = 10z; z = 3–9.6(Q–2.2) B2(Q) = – 10–4Q 10–2Q4 – – Q Q2 – Q

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= Rs= 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={ E-5*Q E-3*Q E-2*Q *Q *Q } .param B=pwr(10,(3-9.6*(Q-2.2))) E-4*Q E-2*Q *Q *Q2 – *Q }.

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) = 10–3H 10-2H 10-2H H B(H) = 10-2H H 10-3H H For high supply voltage Vdc the design parameters A and B are defined by corresponding graphical dependencies on Vdc.

C. Simulation results C1. 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

Comparison results for Procedure I Parameter Value in [3] Value obtained by Procedure I Rdc,  9 Xdc/Rdc 2.793 2.7925 R,  7.822 7.8225 B, S X,  5.883 5.8828 C1, pF 3.349 3.3491 Lx, nH 0.468 Cs, pF 6.33 6.3326 Lm, nH 1.44 1.4455 Cm, pF 3.67 3.6956 Pout, W 1.02 1.0094 η, % 97.4 96.9

Waveforms of the currents flowing through the switch (Isw) and through the dc-feed inductance Fig. 3 Switch voltage Fig. 1.3

C. Simulation results C2. 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

The results obtained by the second PSpice procedure are compared with the results given in [1.4]. They are presented in the table below. Parameter Q=10 H=0.001 Q=3 H=0.5 Value in [1.4] Value obtained by Procedure II A 0.933 0.9493 0.783 0.7784 B 0.356 0.4 1.034 1.0973 Lc, H 318.3u 191n n L0, H 318.3 n 318.3n 95.49n 95.493n Cs, F 10.23 n 8.831n 16.71n 15.959n C0, F 3.651n 3.532n 17.29n 17.512n R,  1 Fc, MHz 5 Vdc, V D 0.5

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. Comparison results for Procedure II Parameter Q=10 H=0.001 Value in [1.4] Value obtained by Procedure II Lc, H 663.3u u L0, H 663n n Cs, F 564p p C0, F 49.6p 49.603p

Simulation results for the output voltage Fig. 1.5 Simulation results for the drain-source voltage Fig. 1.6

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, [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, [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, [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.

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.

2.1. Modeling of spiral inductors Fig Physical equivalent circuit of planar spiral inductor

2.1. Modeling of spiral inductors Outer diameter Dout Dout Inner diameter Din Din Average diameter Davg = 0.5 (Dout+ Din) Davg Number of turns n n Fill ratio (Dout– Din)/(Dout+ Din) ro Width of spiral trace w w Metal skin depth delta Metal tickness t Line spacing s sp Thikness of the oxide insulator between the spiral and underpass tM1-M2 tM1M Thikness of the oxide layer between the spiral and substrate tox tox Inductance Ls Ls Metal conductivity sigma Substrate conductance Gsub Gsub Substrate capacitance CsubCsub Csub Length of spiral trace l L Permitivity of the oxide Eox Parameters of spiral inductors and corresponding names in the PSpice model

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

2.1. Modeling of spiral inductors Modelling of the inductance Ls : Wheeler formula The simple modification of the Wheeler formula is applicable for 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)}

2.1. Modeling of spiral inductors Modelling of the inductance Ls : Current sheet approximation Using current sheet approximation [2.2,2.4], the inductance Ls2 of square, hexagonal, octagonal and circle integrated spiral inductors can be described by the expression: In the case of square inductors c1=1.27, c2= 2.07, c3= 0.18 and c4= [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)}

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}

2.1. Modeling of spiral inductors Fig Relative error determination of inductance approximations Ls1, Ls2 and Ls3

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

2.1. Modeling of spiral inductors Modelling of the elements Cs, Cox, Csi and Rsi The values of the elements Cs ,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)}

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 ind 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} Cox {0.5*L*w*Eox/tox} Cox {0.5*L*w*Eox/tox} Rsi {2/(L*w*Gsub)} Csi {0.5*L*w*Csub} Csi {0.5*L*w*Csub} Rsi {2/(L*w*Gsub)} .ends

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.

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

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

2.2. Modeling of planar transformers

2.1. Modeling of planar transformers

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

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

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

REFERENCES [2.1] Yue, C. P., C. Ryu, J. Lau, T. H. Lee and S. S. Wong, “A Physical model for planar spiral iductors on silicon”, Proc. IEEE Int. Electron Devices Meeting Tech. Dig. San Francisco, CA, Dec. 1996, pp [2.2] Mohan, S. S., M. M. Hershenson, S. P. Boyd and T. H. Lee, “Simple Accurate Expressions for Planar Spiral Inductances”, IEEE Journal of Solid-State Circuits, October 1999. [2.3] Wheeler, K.A., “Simple Inductance formulas for radio coils”, Proc. IRE, vol. 16, no. 10, Oct. 1928, pp [2.4] Rosa, E. B., “Calculation of the self-inductances of single-layer coils, Bull. Bureau Standards, vol. 2, n. 2, 1906, pp [2.5] OrCAD PSpice and Basics. Circuit Analysis Software. OrCAD Inc., USA, 1998 [2.6] M. Hristov, E. Gadjeva, D. pukneva, Computer Modelling and Geometry Optimization of Spiral Inductors for RF Applications using Spice,10-th International Conference “Mixed Design of Integrated Circuits and Systems” - MIXDES’2003, Lodz, June 2003, Poland.

3. Modeling of active elements
; 3. Modeling of active elements 3.1. Modeling of heterojunction bipolar transistors Fig Small-signal equivalent circuit of heterojunction bipolar transistor (HBT)

; 3. Modeling of active elements 3.2. Modeling of RF NMOSFET Fig Simplified small-signal RF NMOSFET equivalent circuit

; 3. Modeling of active elements 3.2. Modeling of RF NMOSFET a) b) Fig Modified equivalent circuit of MOSFET (a) and PSpice model (b)

4. Noise modeling of RF elements
The computer-aided noise modeling and simulation of electronic circuits at RF is based on adequate noise models of electronic components [3.4,3.5]. Parametrized macromodels for the noise analysis of RF electronic circuits are used, which enhance the possibilities for noise analysis using general-purpose circuit analysis programs. The parametrized macromodels are included in the model and symbol libraries of the OrCAD PSpice simulator. They allow to construct user-defined noise models at RF, which are not implemented in the standard PSpice simulator and give the opportunity for noise characteristic investigation in the design process.

Parametrized macromodels of correlated noise sources In the process of development of heterojunction bipolar transistor macromodel, correlated noise sources have to be created (Fig. 4.1). The correlated noise source I2 is divided into two parts – independent part I2a and dependent part I2b. A significant feature of the model is that the correlation coefficient C is a complex number. Fig Correlated current noise sources Fig Simplified equivalent circuit of correlated current noise sources

A standard noisy resistor Rref is used for generating of reference noise current In ref =1pA (Fig. 4.3). Fig.4.3. Equivalent circuit of current noise source In=Inref=1pA

PSpice model The currents I2a and I2b are defined by PSpice sources of GLAPLACE type. Subcircuit description of the parametrized correlated current noise sources according to the input language of the PSpice simulator .subckt In_cor a b c d PARAMS: Ca=1, Cb=1, In1=1p, In2=1p R k V DC 0 R k V DC 0 *noise source I1 GI1 a b VALUE={I(V1)*1e12*In1} *noise source I2 : independent part GI2a c d LAPLACE {I(V2)} = {1e12*sqrt + (1-Ca*Ca +Cb*Cb+ (-2*Ca*Cb)*s/sqrt(-s*s))*In2} *noise source I2 : dependent part GI2b c d LAPLACE {I(V1)}={1e12*(Ca+Cb*s/sqrt(-s*s))*In2} .ends In_cor

5.1. Extraction of HBT small-signal parameter values
5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values Computer-aided extraction algorithm of parameter values of small-signal HBT equivalent circuit can be developed using standard circuit simulator OrCAD PSpice. A good agreement between the measured and modeled values of S-parameters is achieved. The calculated maximal relative error is 4%. The algorithm is realized using the rich possibilities for postprocessing and definition of macros in the Probe analyzer. The proposed approach is characterized by flexibility and gives the opportunity for modification, extension and improvement of extraction procedure.

The HBT small-signal S-parameters:
5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The HBT small-signal S-parameters: S11m – input reflection coefficient; S21m – forward transmission coefficient; S12m – reverse transmission coefficient: S22m – output reflection coefficient.

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values Fig Small-signal equivalent circuit of the HBT

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values A behavioral PSpice model is constructed to introduce the measured S-parameters. The phasors Sijm , i,j=1,2 are represented in the form of corresponding node voltages of the model:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values They are tabularly defined using dependent sources of EFREQ type in accordance with the input language of the PSpice simulator: E_S11m S FREQ = {V(1,0)} mag + (1G,0.671,-63.4) (2G,0.615,-102.5) .... E_S22m S FREQ = {V(1,0)} mag + (1G,0.816,-37.36) (2G,0.6,-59.47) V ac 1 Fig A behavioral PSpice model for description of the measured S-parameters

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The parameter values of extrinsic elements are used to deembed the two-port parameters of subcircuit Na. For this purpose the measured S-parameters are converted into Y-parameters using the following expressions:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values A=R0*((1+S11m)*(1+S22m)-S12m*S21m) Y11m=(((1+S22m)*(1-S11m)+S12m*S21m)/A Y12m=((S12m*(-2))/A Y21m=((S21m*(-2))/A Y22m=((1+S11m)*(1-S22m)+S12m*S21m)/A Fig Macrodefinitions for two-port parameter conversion in the Probe analyzer

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values Using the relationship Ym = Ya + Ycex (5.2) where Ym is the Y-matrix of external elements Cq, Cpb and Cpc, the parameters Yija of the subcircuit Na are obtained in the form: (5.3)

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values j =V(j1) Y11a = Y11m-j*(Cpb+Cq)*2*pi*frequency Y12a = Y12m+j*Cq*2*pi*frequency Y21a = Y21m+j*Cq*2*pi*frequency Y22a = Y22m-j*(Cpc+Cq)*2*pi*frequency Fig Macros for deembedding of Ya two-port parameters of subcircuit Na

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The two-port parameters of intrinsic transistor (subcircuit Nb) are deembedded from Ya and parameters values of external elements , i=b,e,c. For this purpose the Ya-parameters are converted into Za-parameters using the following expressions:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values Using the relationship , (5.4) where is the Z-matrix of external elements Zb, Ze and Zc, the parameters Zijb of subcircuit Nb are obtained in the form:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The corresponding macros in Probe are shown in Fig. 5.5. Z11b = Z11a-Rb+Re+j*2*pi*(Lb+Le)*frequency) Z12b = Z12a-(Re+j*2*pi*Le*frequency) Z21b = Z21a-(Re+j*2*pi*Le*frequency) Z22b = Z22a-(Re+Rc+j*2*pi*(Le+Lc)*frequency) Fig Macros for deembedding of Ya two-port parameters of subcircuit Na

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The parameter extraction procedure of intrinsic transistor (subcircuit Nb) is based on two-port Y-parameter representation. For this purpose the Y Z parameter conversion is performed using the following expressions:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The transistor parameters Yij, i,j = 1,2, are expressed by the model parameters using the relationships: where

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The extraction procedure consists of the following steps: Step 1. Determination of the current gain The parameter is the current gain at low frequencies:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The cutoff frequency and the base transit time are obtained from the magnitude response and phase response at high frequencies:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The corresponding macros for determination of parameters , and , defined in Step 1 of the extraction algorithm, are presented in Fig. 5.6. ALPHA = (Y21-Y12)/(Y11+Y21) ALPHA0 = max(m(ALPHA)) Wal=2*pi*Fmax/sqrt((ALPHA0*ALPHA0)/(min(ALPHAm)*min(ALPHAm))-1) TAU=(-atan(2*pi*Fmax/Wal)-min(p(alpha))*pi/180)/(2*pi*Fmax) Fig. 5.6

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The modeled frequency characteristic of the current gain is presented in Fig. 5.7. Fig Frequency dependence of the modeled current gain

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values Step 2. Determination of Cex, Rbe, Cbe and Rb2 The product is obtained in the form: Yex can be determined approximately at higher frequencies: at fmax

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values As a result the parameters , Ybe , Rb2 and Ybc are approximately determined:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values Finally, the parameters Cex, , Ybe ,Gbe, Cbe and Rb2 are obtained more precisely:

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values Step 3. Determination of parameters Rbc and Cbc Example The approach is illustrated by extraction of parameter values of HBT small-signal equivalent circuit. The measured S-parameters [5.1] are used. The results are automatically obtained using OrCAD PSpice and Probe analyzer.

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The extracted values are presented in Table 5.1. A good agreement between the measured and modeled values of S-parameters is achieved. The calculated maximal relative error is 4%. Table 5.1 Cpb, fF Cpc, fF Cq. fF Rb. W Rъ, W Rе, W Lb, pH Lc, pH Lc, pH Rb2, W Cbe, pF Rbe, W Cbc, fF Rbc,kW Cex, fF , ps ,GHz

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values The simulated phasor S11

EXTRACTION OF HBT SMALL-SIGNAL PARAMETER VALUES
The simulated phasor S12

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values REFERENCES [5.1] Rudolph, M., R. Doener, P. Heymann, “Direct extraction of HBT Equivalent-Circuit Elements”, IEEE Trans. Microwave Theory Tech., vol. 47, pp , Jan [5.2] Wei, C.-J., and J. C. M. Hwang, “Direct extraction of equivalent circuit parameters for heterojunction bipolar transistors”, IEEE Trans.Microwave Theory Tech., vol. 43, pp , Sept [5.3] Y. Gobert, P. J. Tasker, and K. H. Bachem, “A physical, yet simple, small-signal equivalent circuit for the heterojunction bipolar transistor”, IEEE Trans. Microwave Theory Tech., vol. 45, pp , Jan

5. Parameter extraction of equivalent circuits for passive and active RF elements 5.1. Extraction of HBT small-signal parameter values [5.4] Farchy, S., S. Papasov, Theoretical Electrical Engineering, Tehnika, Sofia, 1992. [5.5] Gadjeva, E., T. Kouyoumdjiev, S. Farchy, “Computer Modelling and Simulation of electronic and electrical circuits by OrCAD PSpice”, Sofia, 2001 [5.6] OrCAD PSpice Application Notes, OrCAD Inc., USA, 1999.

MODELING OF RF DEVICES AND CIRCUITS Elissaveta GADJEVA

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