technical university of sofia modeling of rf devices and circuits elissaveta gadjeva

Technical University of Sofia

MODELING OF RF DEVICES AND CIRCUITS

Elissaveta GADJEVA

2

CONTENTS

1. Modeling of RF circuits2. Modeling of passive elements3. Modeling of active elements4. Noise modeling of RF elements5. Parameter extraction of equivalent

circuits for passive and active RF elements

3

1. Modeling of RF circuits 1.1. 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

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)

12

1

2112

2222

211

SS

SSK

21122211 SSSS

A two-port is stable if the stability factor K > 1 (Rollet's stability condition):

1. Modeling of RF circuits 1.2. RF circuit stability investigation

using PSpice simulation

Macrodefinitions in the Probe analyzer for the stability factor K

5

1 forMAG 1forMSG

MSG/MAG KK



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):

1.MAG 2

12

21 KKS

S

12

21MSGS

S

The gain MSG/MAG is defined in the form:

6



MSG/MAG = (1-ena).MAG + ena.MSG, where ena =1 for K<1 and ena =0 for K>1

*Maximum Available Gain (MAG)MAG=S21m*(K-sqrt(K*K-1))/S12m*Mavimum Stable Gain (MSG) MSG = S21m/S12m*Gain MSG/MAGena = pwr((1+sgn(1-K))/2,1)MSGMAG=(1-ena)*db(mag)+ena*db(msg)

Macrodefinitions in the Probe analyzer:

7

A two-port is unconditionally stable if the stability coefficient :

The gain parameter Maximum Unilateral Gain (MUG) (or Mason's gain) is defined in the form:

)/Re(/.

1/.

2

1MUG

12211221

21221

SSSSK

SS

1

11

12211122

211

*

SSSS

S



8

The frequency dependencies of the stability factor K and MSG/MAG

K

K=1

MSG/MAG



9

The frequency dependence of the stability coefficient :

1



10

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

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 resultsC1. Simulation Results from PSpice Procedure I C2 Simulation Results from PSpice Procedure II

1. Modeling of RF circuits1.3. Application of Spice Simulation to Investigation

of Class E Power Amplifier Characteristics

12

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

• 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

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

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.



16


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 =2f 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

outdc PVR /5768.0 2

shunt susceptance

optimal load resistance

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:

excessive reactance RX 152.1Vdc – 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

outdcdc PVR /2


RB /1836.0

18

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 ≤ 5R=Rdc.PR1; B=PB1/R; X=R.PX1; PR1=1.979–0.7783z+0.1754z2–0.01397z3PB1=1.229–0.7171z+0.1881z2–0.01672z3; PX1= -1.202+1.591z–

0.4279z2+0.03894z3; *for 5 < z ≤ 20R=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 > 20R=Rdc.PR3; B=PB3/R; X=R.PX3; PR3 = 0.6106 ; PB3 = 0.1999 ; PX3 = 1.096


19

The calculation of the output matching network of the power amplifier is set in the procedure:

RRn L / 1/ nRX Lc 1 nRX l

RL – load resistance; R – optimal load; Xc and Xl – reactances of the inductanceand 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 Ecircuit components as well as those of the output matching circuit, are defined inthe 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,whereena1 = 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

*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}

Computer realization of the procedure for automatic design of class E amplifier:


21


This procedure for automated design using PSpice is based on theapproach 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 =2f;the resonant angular frequency 0=2f0; the ratio of the resonant tooperating frequency A=f0/f; the ratio of resonant to parasitic capacitance onMOSFET transistor B=C0/Cj0; the ratio of resonant to dc-feed inductanceH=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) = 0.32928610–5Q5 – 0.22624410–3Q4 + 0.60828910–2Q3 + 0.079868Q2 + 0.513996Q -0.353653B(Q) = B1(Q) + B2(Q) ; B1(Q) = 10z; z = 3–9.6(Q–2.2) B2(Q) = –0.28224510–4Q5 + 0.201105810–2Q4 – – 0 .0552482Q3 + 0.726058Q2 – 4.546336Q +11.217525

22

A(Q) and B(Q) are defined in the PSpice model as parameters by usingPARAMETERS statement and the realization of the procedure for power amplifierdesign 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

In the case of high output Q and RF choke an equivalent linear capacitance of theMOSFET switch is defined in the form:CSequ=24VbiCj0/{12Vbi+[6Vbi(24Vbi-242Vdc+4Vdc)]+92(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 beapproximated from their graphical dependencies on the ratio of resonant todc-feed inductance H. The functions A(H) and B(H) are approximated bythe following polynomials:

A(H) = - 4.424310–3H4-1.572271510-2H3 +2.7834910-2H2 +0.15350566H

+0.8216771B(H) = 3.836804510-2H4+0.1836775H3+7.56716510-3H2- 0.994968H+0.801806.

For high supply voltage Vdc the design parameters A and B are defined bycorresponding 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 forclass E power amplifiers with small dc-feed inductance (Fig. 1.2). The input parameters are: supply voltage Vdc=3V; required output powerPout=1W; load resistance RL=50; operating frequency fc=2GHz, L1=2nH,Ls=1nH. The switch used for the procedure verification has a resistanceRON=0.1 for the ON state and ROFF=1106 for the OFF state.

Fig. 1.2

25

Comparison results for Procedure I

Parameter Value in [3]Value obtained by

Procedure I

Rdc, 9 9

Xdc/Rdc 2.793 2.7925

R, 7.822 7.8225

B, S 0.04208 0.042086

X, 5.883 5.8828

C1, pF 3.349 3.3491

Lx, nH 0.468 0.46814

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


Procedure I

26

Waveforms of the currents flowing through the switch (Isw) and through the dc-feed inductance

Fig. 3

Switch voltage

Fig. 1.3


Procedure I

27

A simulation example for the procedure II is based on the approach forinvestigation of class E amplifier with nonlinear capacitance for any output qualityfactor Q and finite dc-feed inductance. In this case the preliminary defined inputparameters are: supply voltage Vdc; loaded quality factor Q; ratio of the resonantinductance 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 angularfrequencies c and 0. Verification of the described approach is performed by using the following designspecifications: 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 isshown in Fig. 1.4.

Fig. 1.4


Procedure II

28

Parameter

Q=10 H=0.001 Q=3 H=0.5

Value in [1.4]

Value obtained by Procedure II

Value in [1.4]


A 0.933 0.9493 0.783 0.7784

B 0.356 0.4 1.034 1.0973

Lc, H 318.3u 318.3u 191n 190.986n

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 1 1 1

Fc, MHz 5 5 5 5

Vdc, V 1 1 1 1

D 0.5 0.5 0.5 0.5

The results obtained by the second PSpice procedure are compared with the results given in [1.4]. They are presented in the table below.


Procedure II

29

The element values obtained by the computer-aided design procedure arecompared 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]


Lc, H 663.3u 663.146u

L0, H 663n 663.146n

Cs, F 564p 563.667p

C0, F 49.6p 49.603p


Procedure II

30

Simulation results for the output voltage

Fig. 1.5

Fig. 1.6

Simulation results for the drain-source voltage


Procedure II

31

[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.

REFERENCES

1. Modeling of RF circuits

32

2. Modeling of passive elements2.1. 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


Fig. 2.1. Physical equivalent circuit of planar spiral inductor

34

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 betweenthe spiral and underpass tM1-M2

tM1M

Thikness of the oxide layerbetween 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


35


tsew

lR

1...

o

2

21

2 ..MoxM

oxs t

wnC

ox

oxox t

wlC

...2

1

subsi CwlC ...2

1

subsi Gwl

R..

2

The circuit elements are defined by the following equations:

36

Modelling of the inductance Ls : Wheeler formula

The simple modification of the Wheeler formula is applicable for square, hexagonal and octagonal integrated spiral inductors:


2

2

o11 1 K

DnKL

avgs

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

Using current sheet approximation [2.2,2.4], the inductance Ls2 of square, hexagonal, octagonal and circle integrated spiral inductors can bedescribed by the expression:


Modelling of the inductance Ls : Current sheet approximation

2

4321

2

2 ln2

.cc

ccDnL avg

s

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


Using the data fitted monomial expression [2.2], the inductance Ls3 is described in the form:

Modelling of the inductance Ls :Data fitted monomial expression

543213

aavgouts snDwDL

This expression is valid for square, hexagonal and octagonal integrated spiral inductors. In the case of square inductors

–0.03 ; 1.78 ;2.4

0.147,– ;1.21– ; 3-1.62x10

543

21

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

Fig. 2.2. Relative error determination of inductanceapproximations Ls1, Ls2 and Ls3


40

Modelling of the resistance Rs


Rs is presented by a voltage controlled current source of GLAPLACE type (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))))))}

Fig. 2.3. Modelling of frequency dependent resistance Rs

tsew

lR

1...

o

2

41

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)}

21

2 ..MoxM

oxs t

wnC

ox

oxox t

wlC

...2

1subsi CwlC ...

2

1

subsi Gwl

R..

2


42


.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

Parametrized PSpice model of spiral inductor

43

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

The dependence of the inductance Ls on trace width w with parameter the number of turns


45

The dependence of trace width w on the number of turns n for a given inductance Ls


46

2. Modeling of passive elements2.2. Modeling of planar transformers

47


48

i b

T

tt

sb

L

eW

Rb

ib

)1(

1

otovtoxt CAAC )(2

1 obboxb CAC

2

1 oovov CAC

2

1

optovcptMttpt CAWWNC )( _1 opbMbbpb CWWNC 1

ti

ti MTLT

1

bi

ti MTLT

1

0

2

21

ILOLA

2OLA

2/)()/())(4( 22 ILOLDWDWDNILOLL tttt

2/)()/())(4( 22 ILOLDWDWDNILOLL bbbb

t

tt N

DNILOLW

)1()(5,0

;


49

Parameter description

;


50

2 1

P A R A M E T E R S :Y 1 1 I H = 8 5 . 4 7 mY 1 2 I H = -7 5 . 7 4 mY 2 2 I H = 1 3 2 . 0 8 1 mF H = 3 5 G H zC b = {(Y 2 2 I H +Y 1 2 I H )/ (2 * p i* F H )}C o v = {-Y 1 2 I H / (2 * p i* F H )}C t = {(Y 1 1 I H +Y 1 2 I H )/ (2 * p i* F H )}

C 2 a{C t }

R s t a{R s t }

P A R A M E T E R S :z1 2 R L = -1 . 2 8 9 mZ 1 2 I L = 4 . 2 1

Z 2 2 R L = 2 . 2 4 8Z 2 2 I l = 7 . 7 9 3

M = {-Z 1 2 I L / (2 * p i* F L )}F L = 5 0 e 6p i = 3 . 1 4 1 5 9 6 5L s t = {Z 1 1 I L / (2 * p i* F L )}L s b = {Z 2 2 I L / (2 * p i* F L )}Z 1 1 I L = 2 . 7 9 8Z 1 1 R L = 7 . 2 6 6R s t = {Z 1 1 R L }R s b = {Z 2 2 R L }

R 2 a1 e 5 0

R s b a{R s b }

1 1

R 1 a1 e 5 0

I 1 a0 A d c{1 -p a r}

TX1 a

C O U P L I N G = 0 . 9L 1 _ V A L U E = {L s t }L 2 _ V A L U E = {L s b }

0

C 4 a{C b }

C 1 a{C o v }

I 2 a0 A d c{p a r}

0

PSpice model

;


51

[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. 155-158.[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. 1398-1400.[2.4] Rosa, E. B., “Calculation of the self-inductances of single-layer coils, Bull. Bureau Standards, vol. 2, n. 2, 1906, pp. 161-187.[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, 26-28 June 2003, Poland.

2. Modeling of passive elementsREFERENCES

52

;

3. Modeling of active elements3.1. Modeling of heterojunction bipolar

transistors

Fig. 3.1. Small-signal equivalent circuit of heterojunction bipolar transistor (HBT)

0

0

1

j

e j

53Fig. 3.2. Simplified small-signal RF NMOSFET equivalent circuit

;

3. Modeling of active elements3.2. Modeling of RF NMOSFET

)1( jgy mm

54

Fig. 3.3. Modified equivalent circuit of MOSFET (a) and PSpice model (b)

a) b)

;

3. Modeling of active elements3.2. Modeling of RF NMOSFET

55

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.

4. Noise modeling of RF elements

56


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.

CII nb 22

ba CCC j

Fig. 4.1. Correlated current noise sources

222 1 CII na

Fig. 4.2. Simplified equivalent circuitof correlated current noise sources

57

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


58

.subckt In_cor a b c d PARAMS: Ca=1, Cb=1, In1=1p, In2=1pR1 1 0 16.56kV1 1 0 DC 0R2 2 0 16.56kV2 2 0 DC 0*noise source I1GI1 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


PSpice modelThe 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

59

5. Parameter extraction of equivalent circuits for passive and active RF

elements5.1. 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.

60

The HBT small-signal S-parameters:

2

1

2221

1211

2

1 .a

a

SS

SS

b

b

mm

mm

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 elements5.1. Extraction of HBT small-signal parameter

values

61Fig. 5.1. Small-signal equivalent circuit of the HBT



values

62

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:

)22(;)21(

)12(;)11(

2221

1211

SVSSVS

SVSSVS

mm

mm



values

63

They are tabularly defined using dependent sources of EFREQ type in accordance with the input language of the PSpice simulator:

E_S11m S11 0 FREQ = {V(1,0)} mag + (1G,0.671,-63.4) (2G,0.615,-102.5)....E_S22m S22 0 FREQ = {V(1,0)} mag+ (1G,0.816,-37.36) (2G,0.6,-59.47)....V1 1 0 ac 1

Fig. 5.2. A behavioral PSpice model for description of the measured S-parameters



values

64

The parameter values of extrinsic elements are usedto deembed the two-port parameters of subcircuit Na.

For this purpose the measured S-parameters are converted into Y-parameters using the following expressions:



values

65

A

SSSSY mmmm

m21121122

11

11

A

SY m

m12

12

2

A

SY m

m21

21

2

A

SSSSY mmmm

m21122211

22

11

211222110 11 SSSSRA



values

66

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)/AFig. 5.3. Macrodefinitions for two-port parameter

conversion in the Probe analyzer



values

67

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)

qpbma CCjYY 1111 qpcma CCjYY 2222

qma CjYY 1212 qma CjYY 2121



values

68

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. 5.4. Macros for deembedding of Ya two-port parameters of subcircuit Na



values

69

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:

y

aa D

YZ 22

11 y

aa D

YZ 12

12

y

aa D

YZ 21

21

y

aa D

YZ 11

22

aaaay YYYYD 21122211



values

70

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:

exzba ,ZZZ

exz ,Z

ebab ZZZZ 1111 eab ZZZ 1212

eab ZZZ 2121 ecab ZZZZ 2222



values

71

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. 5.5. Macros for deembedding of Ya two-port parameters of subcircuit Na



values

72

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:

z

b

D

ZY 22

11 z

b

D

ZY 12

12

z

b

D

ZY 21

21 z

b

D

ZY 11

22

bbbbz ZZZZD 21122211



values

73

The transistor parameters Yij, i,j = 1,2, are expressed

by the model parameters using the relationships:

B

YYYY bebc

ex

)1(11

B

YYYY bcbe

ex

21

BRYYYY bbebcex )1( 222

])1[(1 2 bcbeb YYRB exex CjY bcbcbc CjGY

bebebe CjGY bcbc RG 1 bebe RG 1

where



values

B

YYY bc

ex 12

74

The extraction procedure consists of the following steps:

Step 1. Determination of the current gain

The parameter is the current gain at low frequencies:

j

e

YY

YY j

1

0

2211

1221

0

)( min0 f



values

75

The cutoff frequency and the base transit time are obtained from the magnitude response and phase response at high frequencies:

f

farg

1/

22

max0

max

f

f

max

maxmax

2

arg2

f

ffarctg



values

76

The corresponding macros for determination of parameters ,

and , defined in Step 1 of the extraction algorithm, are

presented in Fig. 5.6.

0

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



values

77

The modeled frequency characteristic of the current

gain is presented in Fig. 5.7.

Fig. 5.7. Frequency dependence of the modeled current gain



values

78

Step 2. Determination of Cex, Rbe,

Cbe and Rb2

The product is obtained in the

form:

Yex can be determined approximatelyat higher frequencies:

2bbcRY

2111

12112 YY

YYRY bbc

12, YY aex at fmax



values

79

As a result the parameters , Ybe , Rb2 and Ybc are approximately determined:

112,

22,2

YY

YYRY

aex

aexabbe

222121211, 1 YYRYYYY abbeabe

abeabebeab YRYR ,,2 / abbbcabc RRYY ,22,

abcabeab

abeabcex YYR

YYYjY

,,,2

,,11 11

1Im



values

80

Finally, the parameters Cex, , Ybe ,Gbe, Cbe and Rb2 are obtained more precisely:

max)Im( ffYC exex 112

222

YY

YYRY

ex

exbbe

222121211 1 YYRYYYY bbebe

bebe YG Re /Im bebe YC bebebeb YRYR /2



values

81

Step 3. Determination of parameters Rbc and Cbc

2

2

b

bbcbc R

RYY bcbc YG Re

bcbc GR /1 /Im bcbc YC



values

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.

82

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%.

Cpb, fF Cpc, fF Cq. fF Rb. RъRеLb, pH Lc, pH Lc, pH

Rb2Cbe, pFRbe,Cbc, fFRbc, Cex, fFps

f0 GHz

Table 5.1



values

83The simulated phasor S11



values

84

EXTRACTION OF HBT SMALL-SIGNAL PARAMETER VALUES

The simulated phasor S12

85The simulated phasor S21



values

86

The simulated phasor S22



values

87

REFERENCES[5.1] Rudolph, M., R. Doener, P. Heymann, “Direct extraction

of HBT Equivalent-Circuit Elements”, IEEE Trans.Microwave Theory Tech., vol. 47, pp. 82-84, Jan. 1999.

[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. 2035-2039, Sept. 1995.

[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. 149-153, Jan. 1997.



values

88

[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.



values

technical university of sofia modeling of rf devices and circuits elissaveta gadjeva

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