Transcript Slide 1
MODELING OF RF DEVICES AND CIRCUITS
Elissaveta GADJEVA Technical University of Sofia
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
2
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 study the stability factor and gain characteristics.
The two-port S RF to the input language of the frequency range and to -parameters can be described according 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.
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1. Modeling of RF circuits
1.2. RF circuit stability investigation using PSpice simulation
A two-port is stable if the stability factor K (Rollet's stability condition): > 1
K
1
S
11 2 2
S
12
S
22 2 2
S
21 1
S
11
S
22
S
12
S
21 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) 4
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 unstable two-port ( K < 1): (MSG), defined for a potentially MAG
S
21
S
12 .
K
K
2 1 MSG The gain MSG/MAG is defined in the form: MSG/MAG MSG MAG for for
K K
1 1
S
21
S
12 5
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) 6
1. Modeling of RF circuits
1.2. RF circuit stability investigation using PSpice simulation
The gain parameter Maximum Unilateral Gain (or Mason's gain) is defined in the form: (MUG) MUG 1 2 .
K
.
S
21 /
S
21 /
S
12
S
12 Re( 1 2
S
21 /
S
12 ) A two-port is unconditionally stable if the 1 stability
S
22 1
S
* 11
S
11 2
S
21
S
12 1 7
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
8
1. Modeling of RF circuits
1.2. RF circuit stability investigation using PSpice simulation
The frequency dependence of the stability coefficient : 1 9
1. Modeling of RF circuits
1.2. 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 10
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
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1. Modeling of RF circuits
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.
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1. Modeling of RF circuits
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.
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1. Modeling of RF circuits
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
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1. Modeling of RF circuits
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 efficiency without performing a lot of additional optimizing procedures. In this paper maximal amplifier procedures for automated sizing of the class E power amplifier circuit elements such as PSpice. are presented using the possibilities of the general-purpose circuit analysis program The integration of the design and analysis stages allows reuse of the design procedure as well as fast power amplifier characteristics assessment.
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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.
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A. Automated design of class E power amplifier with small DC-feed inductance
The susceptance of the shunt capacitor C 1 is B=ωC 1 and X represents the mistuning reactance. In the classic RF C-based class E power amplifier (L 1 ∞) the design procedure consists of evaluating the three key circuit parameters: optimal load resistance
R
0 .
5768
V
2
dc
/
P out
shunt susceptance
B
0 .
1836 /
R
excessive reactance
X
1 .
152
R
V dc – supply voltage; P out – output power. The approach assumes work with a preliminary chosen value for the dc feed inductance L 1 with reactance X dc =ωL 1 . The dc resistance that circuit presents to the supply source is
R dc
V
2
dc
/
P out
17
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 recalculatedand the case normalized for finite values of dc-feed inductance.
the circuit Based parameters on the 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
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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:
n
R L
/
R X c
R L
/
n
1
X l
R n
1 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.
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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}
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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
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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}.
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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 2 Vdc+ 4 Vdc)]+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 –3 H 4 -1.5722715
10 -2 H 3 +2.78349
10 -2 H 2 +0.15350566H +0.8216771
B(H) = 3.8368045
10 -2 H 4 +0.1836775H
3 +7.567165
10 -3 H 2 - 0.994968H+0.801806.
For high supply voltage Vdc the design parameters A and B are defined by corresponding graphical dependencies on Vdc.
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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
24
C. Simulation results C1. Simulation Results from PSpice Procedure I
Comparison results for Procedure I
Parameter Rdc, Xdc/Rdc R, B, S X, C1, pF Lx, nH Cs, pF Lm, nH Cm, pF Pout, W η, % Value in [3] 9 2.793
7.822
0.04208
5.883
3.349
0.468
6.33
1.44
3.67
1.02
97.4
Value obtained by Procedure I 9 2.7925
7.8225
0.042086
5.8828
3.3491
0.46814
6.3326
1.4455
3.6956
1.0094
96.9
25
C. Simulation results C1. 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
26
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. 27 Fig. 1.4
C. Simulation results C2. 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. Parameter A B Lc, H L0, H Cs, F C0, F R, Fc, MHz Vdc, V D Q=10 H=0.001
Value in [1.4] 0.933
Value obtained by Procedure II 0.9493
0.356
318.3u
318.3 n 10.23 n 3.651n
1 5 1 0.5
0.4
318.3u
318.3n
8.831n
3.532n
1 5 1 0.5
Q=3 H=0.5
Value in [1.4] 0.783
Value obtained by Procedure II 0.7784
1.034
191n 95.49n
16.71n
17.29n
1 5 1 0.5
1.0973
190.986n
95.493n
15.959n
17.512n
1 5 1 0.5
28
C. Simulation results C2. 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. Comparison results for Procedure II Parameter Lc, H L0, H Cs, F C0, F Q=10 H=0.001
Value in [1.4] Value obtained by Procedure II 663.3u
663n 564p 49.6p
663.146u
663.146n
563.667p
49.603p
29
C. Simulation results C2. Simulation Results from PSpice Procedure II Simulation results for the output voltage Simulation results for the drain-source voltage Fig. 1.5
Fig. 1.6
30
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 House, (Artech 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.
31
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 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. accordance with the syntax of the PSpice input language. Macromodels are constructed in the form of parametrized subcircuits in Based on the possibilities of the nonlinear analysis, optimal design of the inductor can be is performed. The two-port factor are obtained in the graphical analyzer S -parameters and the Probe PSpice macros.
The model descriptions and simulation results are given.
-like circuit Q using corresponding 32
2. Modeling of passive elements
2.1. Modeling of spiral inductors
Fig. 2.1. Physical equivalent circuit of planar spiral inductor 33
2. Modeling of passive elements
2.1. Modeling of spiral inductors
Outer diameter
Dout
Inner diameter
Din
Average diameter
D avg
= 0.5 (Dout+ Din) Number of turns
n
Fill ratio (
Dout
–
Din
)/(
Dout
+
Din
) Width of spiral trace
w
Metal skin depth Metal tickness Line spacing
s
Thikness of the oxide insulator between the spiral and underpass
tM1-M2
Thikness of the oxide layer between the spiral and substrate
tox
Inductance
Ls
Metal conductivity Parameters of spiral inductors and corresponding names in the P Spice model Substrate conductance
Gsub
Substrate capacitance
Csub
Csub Length of spiral trace
l
Permitivity of the oxide Dout Din Davg n ro t w delta sp tM1M tox Ls sigma Gsub Csub L Eox 34
2. Modeling of passive elements
2.1. Modeling of spiral inductors
The circuit elements are defined by the following equations:
R s
w
.
.
.
1
l
e
t
2 o
C s
n
.
w
2 .
ox t oxM
1
M
2
C ox
1 2 .
l
.
w
.
t
ox ox C si
1 2 .
l
.
w
.
C sub R si
2
l
.
w
.
G sub
35
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 applicable for square, hexagonal and octagonal integrated spiral inductors:
L s
1
K
1 o
n
2
D avg
1
K
2 The coefficients inductors K K 1 and 1=2.34 and In accordance with the the form: K K 2 depend on the inductor layout. In the case of square 2= 2.75.
OrCAD PSpice language, the value of Ls 1 is defined in {K1*mju*(n*n*Davg)/(1+K2*ro)} 36
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 square, hexagonal, octagonal and circle integrated spiral inductors can be described by the expression: Ls 2 of
L s
2 .
n
2
D avg c
1 2 ln
c
2
c
3
c
4 2 In the case of square inductors [2.2]. In accordance with the in the form: c 1=1.27, c 2= 2.07, c 3= 0.18 and c 4= 0.18 OrCAD PSpice language, the Ls 2 value is defined {0.5*mju*n*n*davg*c1*(log(c2/ro)+c3* ro+ c4*ro*ro)} 37
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 in the form:
L s
3
D
1
out w
2
D
3
avg n
4
s a
5 Ls3 is described This expression is valid for square, hexagonal and octagonal integrated spiral inductors. In the case of square inductors 3 1.62x10
2.4
; 4 3 ; 1 1.78
– ; 5 1.21
; 2 –0.03
– 0.147, The description in accordance with the has the form: OrCAD PSpice language of the Ls 3 value {beta*pwr(Dout*1e6,al1)*pwr(w*1e6,al2)* pwr (Davg*1e6,al3)*pwr(n,al4)*pwr(sp*1e6,al5)*1e-9} 38
2. Modeling of passive elements
2.1. Modeling of spiral inductors
Fig. 2.2. Relative error determination of inductance approximations Ls1, Ls2 and Ls3 39
2. Modeling of passive elements
2.1. Modeling of spiral inductors
Modelling of the resistance Rs
R s
w
.
.
.
1
l
e
t
2 o Fig. 2.3. Modelling of frequency dependent 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))))))} 40
2. Modeling of passive elements
2.1. Modeling of spiral inductors
Modelling of the elements Cs, Cox, Csi and Rsi
The values of the elements are defined in the form: Cs , Cox , Csi and Rsi of the equivalent circuit
C s
n
.
w
2 .
ox t oxM
1
M
2
C ox
1 2 .
l
.
w
.
t
ox ox C si
1 2 .
l
.
w
.
C sub R si
2
l
.
w
.
G sub
Capacitance Cs: {n*pwr(w,2)*Eox/toxM1M2} Capacitance Capacitance Resistance Cox Csi Rsi : {0.5*L*w*Eox/tox} : {0.5*L*w*Csub} : {2/ (L*w*Gsub)} 41
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
42
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. 43
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 44
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 45
2. Modeling of passive elements
2.2. Modeling of planar transformers
46
2. Modeling of passive elements
2.1. Modeling of planar transformers
47
R sb
1
i W t
t
( 1
L b e
T
b i b
) ;
2. Modeling of passive elements
2.2. Modeling of planar transformers
C oxt
1 2 (
A t
A ov
)
C ot C oxb
1 2
A b C ob C pt
(
N t W t W M
1
A cpt
_
ov
)
C opt C ov
1 2
A ov C o C pb
N b W b W M
1
C opb A
1
OL
IL
2
A
OL
2
T ti
1
MTL ti T ti
1
MTL bi
2 0
L t
(
OL
2
IL
2 4
DN t
(
W t
D
)) /(
W t
D
) (
OL
IL
) / 2
L b
(
OL
2
IL
2 4
DN b
(
W b
D
)) /(
W b
D
) (
OL
IL
) / 2
W t
0 , 5 (
OL
IL
) (
N t N t
1 )
D
48
;
2. Modeling of passive elements
2.2. Modeling of planar transformers
Parameter description 49
;
2. Modeling of passive elements
2.2. Modeling of planar transformers
PSpice model C1a {Cov } I1a 0Adc {1-par} 11 R2a 1e50 PARAMETERS: z12RL = -1.289m
Z12IL = 4.21
Z22Il = 7.793
Z22RL = 2.248
M = {-Z12IL/(2*pi*FL)} FL = 50e6 pi = 3.1415965
Lst = {Z11IL/(2*pi*FL)} Lsb = {Z22IL/(2*pi*FL)} Z11IL = 2.798
Z11RL = 7.266
Rst = {Z11RL} Rsb = {Z22RL} Rsta {Rst} Rsba {Rsb} TX1a C2a {Ct} COUPLING = 0.9
L1_VALUE = {Lst} L2_VALUE = {Lsb} 0 PARAMETERS: Y 11IH = 85.47m
Y 12IH = -75.74m
Y 22IH = 132.081m
FH = 35GHz Cb = {(Y 22IH+Y 12IH)/(2*pi*FH)} Cov = {-Y 12IH/(2*pi*FH)} Ct = {(Y 11IH+Y 12IH)/(2*pi*FH)} C4a {Cb} 0 R1a 1e50 I2a 0Adc {par} 21 50
2. Modeling of passive elements
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. 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.
51
;
3. Modeling of active elements
3.1. Modeling of heterojunction bipolar transistors
1 0
e
j
j
0 Fig. 3.1. Small-signal equivalent circuit of heterojunction bipolar transistor (HBT) 52
;
3. Modeling of active elements
3.2. Modeling of RF NMOSFET
y m
g m
( 1
j
) Fig. 3.2. Simplified small-signal RF NMOSFET equivalent circuit 53
;
3. Modeling of active elements
3.2. Modeling of RF NMOSFET
a) b) Fig. 3.3. Modified equivalent circuit of MOSFET (a) and PSpice model (b) 54
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 allow to construct user-defined noise models at RF, which are not implemented in the standard design process.
PSpice simulator. They simulator and give the opportunity for noise characteristic investigation in the 55
4. Noise modeling of RF elements
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 I 2 is divided into two parts – independent part I 2 a and dependent part I 2 b . A significant feature of the model is that the correlation coefficient C is a complex number.
C
C a
j
C b
Fig. 4.1. Correlated current noise sources
I
2
a
I n
2 1
C
2 Fig. 4.2. Simplified equivalent circuit of correlated current noise sources
I
2
b
I n
2
C
56
4. Noise modeling of RF elements
A standard noisy resistor current I n ref = Rref 1pA (Fig. 4.3).
is used for generating of reference noise Fig.4.3. Equivalent circuit of current noise source I n = I nref =1pA 57
4. Noise modeling of RF elements
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 R1 1 0 16.56k
V1 1 0 DC 0 R2 2 0 16.56k
V2 2 0 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 58
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 analyzer.
Probe The proposed approach is characterized by flexibility and gives the opportunity for modification, extension and improvement of extraction procedure.
59
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:
b b
2 1
S S
11
m
21
m S
12
m S
22
m
.
a a
2 1
S
11
m
–
S
21
m
input reflection coefficient; – forward transmission coefficient;
S
12
m
– reverse transmission coefficient:
S
22
m
– output reflection coefficient. 60
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values Fig. 5.1. Small-signal equivalent circuit of the HBT 61
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
S ijm
,
i,j
=1,2 are represented in the form of corresponding node voltages of the model:
S
11
m
S
21
m V
V
(
S
11 ) (
S
21 ) ; ;
S
12
m S
22
m
V
(
S
12 )
V
(
S
22 ) 62
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 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 63
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
N
a
.
For this purpose the
measured S-parameters are converted into Y-parameters
using the following expressions: 64
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values
Y
11
m
1
S
22
m
1
S
11
m
S
12
m S
21
m A Y
12
m
2
S
12
m A Y
21
m
2
S
21
m A Y
22
m
1
S
11
m
1
S
22
m
S
12
m S
21
m A A
R
0 1
S
11 1
S
22
S
12
S
21 65
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. 5.3. Macrodefinitions for two-port parameter conversion in the Probe analyzer 66
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values Using the relationship
Y
m
=
Y
a
+
Y
cex where
Y
m (5.2) is the Y-matrix of external elements
C q , C pb
subcircuit and
N a C pc ,
the parameters are obtained in the form:
Y ija
of the
Y
11
a
Y
11
m
j
C pb
C q
Y
22
a
Y
22
m
j
C pc
C q
Y
12
a
Y
12
m
j
C q Y
21
a
Y
21
m
j
C q
(5.3) 67
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. 5.4. Macros for deembedding of
Y
a two-port parameters of subcircuit N a 68
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
N b
) are deembedded from
Y
a and parameters values of external elements
, i=b,e,c
. For this purpose the
Y
a -parameters are converted into
Z
a -parameters using the following expressions:
Z
11
a
Y
22
a D y Z
21
a
Y
21
a D y Z
12
a
Y
12
a D y Z
22
a
Y
11
a D y
Y
11
a Y
22
a
Y
12
a Y
21
a D y
69
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values Using the relationship
Z
a
Z
b
,
Z
z
,
ex
(5.4) where
Z
z
,
ex
is the
Z
-matrix of external elements
Z b , Z e
and
Z c ,
the parameters
Z ijb
of subcircuit
N b
are obtained in the form:
Z
11
b
Z
11
a
Z b
Z e
Z
12
b
Z
12
a
Z e Z
21
b
Z
21
a
Z e Z
22
b
Z
22
a
Z c
Z e
70
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. 5.5. Macros for deembedding of Y a two-port parameters of subcircuit N a 71
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
N b
) is based on two-port
Y
-parameter representation. For this purpose the parameter conversion is performed using the following expressions:
Y
11
Z
22
b D z Y
21
Z
21
b D z Y
12
Z
12
b D z Y
22
Z
11
b D z D z
Z
11
b Z
22
b
Z
12
b Z
21
b
72
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values The transistor parameters
Y ij
,
i,j
= 1,2, are expressed by the model parameters using the relationships:
Y
11
Y ex
Y bc
( 1
B
)
Y be Y
12
Y ex
Y bc B Y
21
Y ex
Y be B
Y bc Y
22
Y ex
Y bc
( 1
Y be R b
2 )
B
where
Y bc
G bc
j
C bc Y ex
j
C ex B
1
R b
2 [( 1 )
Y be
Y bc
]
Y be
G be
j
C be G bc
1
R bc G be
1
R be
73
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
Y
21
Y
11
Y
12
Y
22 1 0
e
j
j
frequencies: 0 0 (
f
min ) 74
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 0 / 2
f
max
f
max 2 1
arctg
2
f
max 2
f
max arg
f
max 75
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values presented in Fig. 5.6.
Step 1
of the extraction algorithm, are 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 76
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. 5.7. Frequency dependence of the modeled current gain 77
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 C ex
,
C be
and
R b
2 The product
Y bc R b
2
R be,
is obtained in the form:
Y bc R b
2
Y
11
Y
11
Y
12
Y
21
Y
ex
can be determined approximately at higher frequencies:
Y ex
,
a
Y
12 at
f max
78
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 ,
Y be
approximately determined:
Y be R b
2
a
Y ex
,
a Y ex
,
a
Y
22
Y
12
, R b
2 and
Y bc
1
Y be
,
a
Y
11
Y
12
Y be R b
2
a
Y
21
Y
22 are
R b
2 ,
a
Y be R be
a
/
Y be
,
a Y bc
,
a
Y bc R b
2
R b
2 ,
a Y ex
j
Im
Y
11 1
Y bc
,
R b
2 ,
a
a
1 1
Y be
,
Y be
,
a
a Y bc
,
a
79
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values Finally, the parameters
C ex
, ,
Y be
obtained more precisely:
,G be, C be
and
R b
2 are
C ex
Im(
Y ex
)
f f
max
Y be
Y
11
Y
12
Y be R b
2
Y ex Y ex
Y
22
Y
12 1 1
Y be R b
2
Y
21
Y
22
G be
Re
be C be
Im
be
/
R b
2
Y be R be
/
Y be
80
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 R bc and C bc Y bc
Y bc R b
2
R b
2
G bc
Re
bc R bc
1 /
G bc C bc
Im
bc
/
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. 81
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.
C pb
, fF 35.6
A good agreement between the measured and modeled values of
S
-parameters is achieved. The calculated maximal relative error is 4%.
Table 5.1
C pc
, fF 70.4
C q
. fF 10.2
R b
. 1.5
R ъ
, 5.0
R е
, 4.3
L b
, pH 47.5
L c
, pH 52.1
L c
, pH 2.1
R b2
, 16.983
C be ,
pF 0.388
R be ,
1.8003
C bc , fF 33.633
R bc
, k 53.748
C ex
, fF 90.767
0
f
, GHz 0.979 135 , ps 3.5
82
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values The simulated phasor
S
11 83
EXTRACTION OF HBT SMALL-SIGNAL PARAMETER VALUES
The simulated phasor
S
12 84
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values The simulated phasor
S
21 85
5. Parameter extraction of equivalent circuits for passive and active RF elements
5.1. Extraction of HBT small-signal parameter values The simulated phasor
S
22 86
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. 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. 87
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.
88