Download Modeling of RF Devices and Circuits

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 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.
3
1. Modeling of RF circuits
1.2. RF circuit stability investigation using
PSpice simulation

A two-port is stable if the stability factor K > 1
(Rollet's stability condition):
2
K
2
1  S11  S22  
2 S12 S21
2
1
  S11S22  S12S21
Macrodefinitions in the Probe analyzer for the stability factor K
S11m = m(S11)
S12m = m(S12)
S21m = m(S21)
S22m = m(S22)
delta = m(S11*S22-S12*S21)
K= (1-S11m*S11m-S22m*S22m
+ delta*delta)/(2*S12m*S21m)
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 (MSG), defined for a potentially
unstable two-port (K < 1):
S 21 
MAG 
. K  K 2  1 

S12 
S 21
MSG 
S12
 The gain MSG/MAG is defined in the form:
 MSG for K  1
MSG/MAG  
 MAG for K  1
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 (MUG)
(or Mason's gain) is defined in the form:
2
S21 / S12  1
1
MUG  .
2 K . S21 / S12  Re( S21 / S12 )
 A two-port is unconditionally stable if the stability
coefficient   1 :

1  S11
2
S 22  S11*   S 21S12
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
11
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 voltagecurrent product is minimized even if the switch has a finite
switch-on time.

12
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.
13
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
14
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 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.
15
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.
16
A. Automated design of class E power amplifier
with small DC-feed inductance
The susceptance of the shunt capacitor C1 is B=ωC1 and X
represents the mistuning reactance. In the classic RF C-based class E power
amplifier (L1∞) the design procedure consists of evaluating the three key
circuit parameters:
2
R

0
.
5768
V
dc / Pout
 optimal load resistance

shunt susceptance

excessive reactance
B  0.1836 / R
X  1.152R
Vdc – supply voltage; Pout – output power.
The approach assumes work with a preliminary chosen value for the dcfeed inductance L1 with reactance Xdc=ωL1. The dc resistance that circuit
presents to the supply source is
R dc  Vdc2 / Pout
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 the case for finite dc-feed inductance. Based on the
recalculatedand normalized values of the circuit parameters interpolation
polynomials are composed, giving the explicit values for circuit parameters.
According to the parameter z value, there are groups of polynomials:
*for z ≤ 5
R=Rdc.PR1; B=PB1/R; X=R.PX1; PR1=1.979–0.7783z+0.1754z2–0.01397z3
PB1=1.229–0.7171z+0.1881z2–0.01672z3; PX1= -1.202+1.591z–
0.4279z2+0.03894z3;
*for 5 < z ≤ 20
R=Rdc.PR2; B=PB2/R; X=R.PX2; PR2=0.9034–0.04805z+
0.002812z2–5.707.10-5z3;
PB2=0.3467–0.02429z+0.001426z2–2.893.10-3z3;
PX2=0.6784+0.006641z–0.003794z2+7.587.10-5z3;
*for z > 20
R=Rdc.PR3; B=PB3/R; X=R.PX3; PR3 = 0.6106 ; PB3 = 0.1999 ; PX3 = 1.096
18
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:
Xc  RL / n  1
Xl  R n  1
n  RL / R
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.
19
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}
20
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
21
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}.
22
B. Automated design of class E power amplifier
with nonlinear capacitance
In the case of high output Q and RF choke an equivalent linear capacitance of the
MOSFET switch is defined in the form:
CSequ=24VbiCj0/{12Vbi+[6Vbi(24Vbi-242Vdc+4Vdc)]+92(2+4)VbiVdc]1/2},
where Vbi is the built-in potential, with a typical value Vbi = 0.7.
In the case of finite dc-feed inductance the coefficients A and B can be
approximated from their graphical dependencies on the ratio of resonant to
dc-feed inductance H. The functions A(H) and B(H) are approximated by
the following polynomials:
A(H) = - 4.424310–3H4-1.572271510-2H3 +2.7834910-2H2 +0.15350566H
+0.8216771
B(H) = 3.836804510-2H4+0.1836775H3+7.56716510-3H2- 0.994968H+0.801806.
For high supply voltage Vdc the design parameters A and B are defined by
corresponding graphical dependencies on Vdc.
23
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
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
25
C. Simulation results
C1. Simulation Results from PSpice Procedure I
Waveforms of the currents flowing through the switch (Isw) and through the dcfeed 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.
Fig. 1.4
27
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
Q=10
H=0.001
Q=3
H=0.5
Value in Value obtained Value in Value obtained
[1.4]
by Procedure II [1.4]
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
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
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
Q=10
H=0.001
Value in
[1.4]
Value obtained
by Procedure II
Lc, H
663.3u
663.146u
L0, H
663n
663.146n
Cs, F
564p
563.667p
C0, F
49.6p
49.603p
29
C. Simulation results
C2. Simulation Results from PSpice Procedure II
Simulation results for the output voltage
Fig. 1.5
Simulation results for the drain-source voltage
Fig. 1.6
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, E89A(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.
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 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.
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
Parameters of spiral inductors
and corresponding names
in the PSpice model
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
34
2. Modeling of passive elements
2.1. Modeling of spiral inductors
The circuit elements are defined by the following equations:
Rs 

l
w... 1  e t 
Cs  n.w .
2

 ox
t oxM 1 M 2
1
C si  .l.w.C sub
2

Cox
2
 o 
 ox
1
 .l.w.
2
tox
2
Rsi 
l.w.Gsub
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:
Ls1  K1 o
n 2 Davg
1  K 2
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)}
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 Ls2 of
square, hexagonal, octagonal and circle integrated spiral inductors can be
described by the expression:
Ls 2
.n 2 Davg c1  c2
2
 ln  c3  c4 

2
 

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)}
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 Ls3 is described
in the form:
1  2 3  4 a5
Ls3  Dout
w Davg n s

This expression is valid for square, hexagonal
and octagonal integrated spiral
inductors. In the case of square inductors
  1.62x10 - 3 ; 1  – 1.21; 2  – 0.147,
3  2.4 ; 4  1.78 ; 5  –0.03
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}
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
Rs 

l
w... 1  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 Cs ,Cox , Csi and Rsi of the equivalent circuit
are defined in the form:
Cs  n.w .
2
 ox
t oxM 1 M 2
Cox
Capacitance Cs:
Capacitance Cox:
Capacitance Csi:
Resistance Rsi:
 ox
1
 .l.w.
2
tox
1
C si  .l.w.C sub
2
2
Rsi 
l.w.Gsub
{n*pwr(w,2)*Eox/toxM1M2}
{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
b
;
2. Modeling of passive elements
2.2. Modeling of planar transformers
Coxt
1
1
1
 ( At  Aov )Cot Coxb  Ab Cob Cov  AovCo
2
2
2
C pt  ( N tWtWM 1  Acpt _ ov )Copt
1
Tti 
MTLti
OL  IL
A1 
2
C pb  N bWbWM 1Copb
1
Tti 
MTLbi

A  OL2
2
 0
Lt  (OL2  IL2  4DNt (Wt  D)) /(Wt  D)  (OL  IL) / 2
Lb  (OL2  IL2  4DNb (Wb  D)) /(Wb  D)  (OL  IL) / 2
0,5(OL  IL)  ( N t  1) D
Wt 
Nt
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 }
Rsta
{Rst}
11
I1a
0Adc
{1-par}
Rsba
{Rsb}
21
TX1a
R2a
1e50
C2a
{Ct}
C4a
{Cb}
R1a
1e50
I2a
0Adc
{par}
COUPLING = 0.9
L1_VALUE = {Lst}
L2_VALUE = {Lsb}
0
PARAMETERS:
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}
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)}
0
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
0e  j
 

1 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
ym  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 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.
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 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.
C  Ca  jCb
I 2a  I n 2 1  C 2
I 2b  I n 2C
Fig. 4.1. Correlated
current noise sources
Fig. 4.2. Simplified equivalent circuit
of correlated current noise sources
56
4. Noise modeling of RF elements
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
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 Probe
analyzer.
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:
 b1   S11m
b   S
 2   21m
S12m   a1 
. 

S22m  a2 
S11m – input reflection coefficient;
S21m – forward transmission coefficient;
S12m – reverse transmission coefficient:
S22m – 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 Sijm , i,j=1,2 are represented in the
form of corresponding node voltages of the
model:
S11m  V ( S11) ; S12m  V ( S12)
S 21m  V ( S 21) ; S 22m  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 Na.
 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
Y11m

1  S 22m 1  S11m   S12m S 21m

A
Y12m
Y22m
 2S12m

A
Y21m
 2 S 21m

A

1  S11m 1  S 22m   S12m S 21m

A
A  R0 1  S11 1  S22   S12 S21 
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
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:
Y11a  Y11m  j C pb  Cq 
Y12a  Y12m  j Cq
Y22a  Y22m  j C pc  Cq 
Y21a  Y21m  j Cq
(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 Ya
two-port parameters of subcircuit Na
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 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:
Z11a
Y22a

Dy
Z 21a
Y21a

Dy
Z12a
Y12a

Dy
Z 22a
Y11a

Dy
Dy  Y11aY22a  Y12aY21a
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
Zb, Ze and Zc, the parameters Zijb of subcircuit
Nb are obtained in the form:
Z11b  Z11a  Zb  Ze 
Z12b  Z12a  Z e
Z 21b  Z 21a  Z e
Z 22b  Z 22a  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 Ya
two-port parameters of subcircuit Na
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 Nb) is based on two-port Y-parameter
representation. For this purpose the Y  Z parameter
conversion is performed using the following expressions:
Y11 
Z 22b
Dz
Z 21b
Y21  
Dz
Y12  
Y22 
Z12b
Dz
Z11b
Dz
Dz  Z11b Z 22b  Z12b Z 21b
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 Yij, i,j = 1,2, are expressed
by the model parameters using the relationships:
Ybc  (1  ) Ybe
Ybc
Y11  Yex 
Y12  Yex 
B
B
 Ybe  Ybc
Y21  Yex 
B
Y22  Yex  Ybc (1  Ybe Rb 2 ) B
where
Ybc  Gbc  jCbc
Yex  jCex
B  1  Rb 2 [(1  )Ybe  Ybc ]
Ybe  Gbe  jCbe
Gbc  1 Rbc
Gbe  1 Rbe
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
Y21  Y12  0 e  j


Y11  Y22 1  j 

The parameter  0 is the current gain  at low
frequencies:
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 
are obtained from the magnitude response  f 
and phase response arg f  at high frequencies:
 
2f max
 0 /  f max   1
2
 arctg 2f max    arg  f max 

2f max
75
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  0 ,
 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
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 Cex, Rbe,
Cbe and Rb2
The product Ybc Rb 2  is obtained in the
form:
Y11  Y12
Ybc Rb 2  
Y11  Y21
Yex can be determined approximately
at higher frequencies:
Yex ,a  Y12
at fmax
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 , Ybe , Rb2 and Ybc are
approximately determined:
Yex ,a  Y22
Ybe Rb 2 a 
1
Yex ,a  Y12
Ybe,a  Y11  Y12 1  Ybe Rb 2 a   Y21  Y22
Rb 2,a  Ybe Rbe a / Ybe,a
Ybc,a  Ybc Rb 2  Rb 2,a


Ybc,a  1   Ybe,a

Yex  j Im Y11 





1

R
1


Y

Y
b 2,a
be, a
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 Cex, , Ybe ,Gbe, Cbe and Rb2 are
obtained more precisely:
Cex  Im( Yex ) f f max
Ybe Rb 2 
Yex  Y22

1
Yex  Y12
Ybe  Y11  Y12 1  Ybe Rb 2   Y21  Y22
Gbe  ReYbe 
Cbe  ImYbe  / 
Rb 2  Ybe Rbe  / Ybe
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 Rbc and Cbc
Ybc 
Ybc Rb 2 
Gbc  Re Ybc 
Rb 2
Rbc  1 / Gbc
Cbc  ImYbc  / 
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.
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. 
R ъ, 
Rе, 
Lb, pH
Lc, pH
Table 5.1
Lc, pH
35.6
70.4
10.2
1.5
5.0
4.3
47.5
52.1
2.1
Rb2, 
Cbe, pF
Rbe, 
Cbc, fF
Rbc,k
Cex, fF
0
f ,GHz
16.983
0.388
1.8003
33.633
53.748
90.767
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
S11
83
EXTRACTION OF HBT SMALL-SIGNAL PARAMETER
VALUES
The simulated phasor
S12
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
S21
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
S22
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