平成18年度電気関係学会四国支部連合大会

312

A Spice-Oriented Designing Algorithm ofBipolar Transistor Amplifiers

17 - 16

Kenji Matsumoto1
Masayoshi Oda1
Yoshihiro Yamagami1
Yoshifumi Nishio1
Junji Kawata2
Akio Ushida2(Tokushima University1 Tokushima Bunri University2 )1. Introduction
RF power amplifiers are sometimes composed of bipolar transistor
circuits [1]. For attaining the maximum gain, we need to set thesuitable bias voltages, and choose appropriate load impedances
which can be practiced by setting the resistance values. Usually,
the optimization can be done by trial and error process using
Spice simulator. It is really time-consuming when the design parameters are so many. In this paper, we propose a Spice-oriented
optimization algorithm based on the steepest descent algorithm[2-3] whose circuit is realized by ABMs (analog behavior Models)of Spice, and the equilibrium point corresponds to the optimum
solution. Firstly, we need to define the objective function as follow:
Φ(x, p), x ∈ Rn , p ∈ Rk(1)wherex: circuit variables such as voltages and currents.
p: optimization parameters such as bias voltages, resistor’s values
and so on. The gradient direction is decided by the solutions ofsensitivity circuits [3], and the steepest descent algorithm can berealized by the equivalent RC circuits combining with nonlinear
controlled current sources.
2. Transistor’s sensitivity modules
2.1 Sensitivity analysis: The steepest descent method is themost basic optimization approach, where the gradient direction isdecided by the solutions of sensitivity circuit. Using the Tableau
approach, we have the following Tableau equation to calculate thesensitivities:"Ki0AKv
10"−1ES R2 = −(3)R1 + R2(R1 + R2 )2These results are equal to the direct calculation results of the sensitivity circuits.
2.2 Sensitivity modules of bipolar transistor: For simplicity, we consider NPN transistor modeled by Ebers-Moll model asshown in Fig. 3(a), where
Si =0−AT0#"

Si,pi
Sv,pi
Svn ,pi¯¯∂g(v,i)Si,pi∂i
v0 ,i0#"−#¯¯∂g(v,i)Sv,pi∂v
v0 ,i0¯¯"∂g(v,i)δ(i)∂pv0 ,i0=δ(i)Aδ(i)
where δ(i) means a delta function satisfying
δ(i) =1[0,20,...
...,00i100...
...,

##(2)

k0]T

id = Is (exp(vd /vT ) − 1),R¯

1∂id ¯=¯re∂vd vd =vd00Voltage-controlled resistor
G(v0 )e(i 0) = dr(i)dp i=iCurrent-controlled resistor^
v=r(i)

0R(i0 )

Fig. 1 Sensitivity elements for resistors.
Example : Now, consider a simple example for deriving the sensitivity circuit shown in Fig. 2(a). Then, the sensitivity circuitsfor E and resistor R2 are given by Figs. 2(b) and (c), respectively.
R1EI

R1

R2 V

1Si

R1Si

R2 Sv

R2 S
R2I(a)(b)

RL

R1CCic

Q

Sid

B

B

idid

B
ib

C

id

re
Sid

E

E

E

(a)

(b)

(c)

R2e(t)

(5)

R1
VDD

vout

Sid

1[V]
Q

R2RE

(a)

Fig. 3 (a) NPN Transistor,(b) Ebers-Moll Model,(c) The Sensitivity module.

RLSr e vout
Sid
RE(b)

Fig. 4 (a) Common-emitter amp.,(b) The sensitivity circuit
for calculating the gain.

3. Steepest descent method attaining maximum gain
Now, we consider Spice-oriented optimization technique for attaining the maximum gain of Fig. 4, where resistors R1 , R2 , REshould be optimized for setting the suitable bias voltage. We needto introduce the numerical differentiation technique as shown inFig. 5 for the optimization of the gain, because the gain is obtained by the sensitivity circuit shown in Fig. 4(b). The output
sensitivities Ss are obtained by Fig. 4(b), and the output pi controls the resistive value using ABM. The simulation results areshown at our presentation of the conference.
+S (p+ p )-S (p)-ipCi

i=1,2, ... , k

R0 pi-4. Conclusions and remarks
In this paper, we proposed a Spice-oriented designing algorithmof bipolar amplifiers, the optimization technique is based on thesteepest descent method, whose gradient is calculated by the sensitivity circuits. The equivalent steepest descent circuit consistsof ABMs of Spice, and the optimum point can be found by thesolution of the transient analysis.

i0j(v 0) = dg(v )
dpv=v^i=g(v)(4)

Fig. 5 Spice-oriented optimizing method.

Sensitivity elements

Linear resistor
v=Riα = 0.99

The sensitivity module is shown by Fig. 3(c), where the incremental resistor is given byThus, the sensitivity circuit configuration is equal to the original one except for the nonlinear elements being replaced by thelinear incremental resistors at the operating points V0 , I0 . The
elementary wise transformations are shown in Fig. 1.
Resistive elementsfor vT = 0.026,(c)Fig. 2 (a) A resistive circuit, (b) Sensitivity circuit for E and (c)resistor R2 .

References[1] D. Leenaerts, J. van der Tang and C. Vaucher, Circuit
Design for RF Transivers, Kluwer Academic Pub., 2001.
[2] J.A. Snyman, Practical Mathematical Optimization: An
Introduction to Basic Optimization Theory and Classicaland New Gradient-Based Algorithm, Springer, 2005.
[3] L.O. Chua and P.-M. Lin, Computer-Aided Analysis ofElectronic Circuits: Algorithms and Computational Techniques, Prentice-Hall, 1975.