2006 IEEE Workshop on Nonlinear Circuit Networks (NCN'06)
Tokushima University, Tokushima, Japan, Dec. 9, 2006

Designing Bipolar Transistor Amplifiers Using SpiceKenji 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 impedancesthat 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.

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.

Kv
1
00
−AT0#"¯¯"∂g(v,i)Si,pi∂i
v0 ,i0

−

#

Si,pi
Sv,pi
Svn ,pi

"

−#00

¯¯"

[V]

R1
10.0

0.0

1
[0,20,...
...,i
1...
...,

for vT = 0.026,0.18#ic

idibQSidBB

Bid

reSid

id

EEE

(a)

(b)

(c)

R2e(t)

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

(a)

vout

10

R 1 = 20[kΩ]
R 2 = 50[kΩ]

0.00150.00100.0005010

1102103104

100.145106107810 10[Hz]90.0

10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9[Hz]

RL
Sid

R2

R 1 = 43.12[kΩ]R 2 = 50[kΩ]

0.10
0.08
0.06
0.04
0.02
0.0

10 010 110 210 310 410 510 610 710 8 10 9[Hz](c)Fig. 7 Simulation result for V in=0.001[V] (a) optimal resistors(R1=43.12[ｋΩ], R2=50[ｋΩ]), (b) non-optimal resistors (R1=20 [ｋΩ], R2=50[ｋΩ]), (c) non-optimal resistors (R1=43.12[ｋΩ], R2=20[ｋΩ]).
We optimized R1 and R2 in the circuit of Fig. 4(b). And, theoptimal solutions are found as Fig. 6. Time for optimization was7.72[s]. When the value of R1 and R2 was substituted and the ACanalysis was done, the gain became 177. The gain will decreaseif either value of R1 or R2 is changed as Fig. 7(b)(c).

1[V]
Q(b)

[V]

0.12

(4)

S voutreRE[V]

(a)

k0]T

R1

2.00.0020

0.04

0.0

α = 0.99[sec]

0.0025

0.06

(2)

(5)

VDD

0.00300.02

RLC

0.00350.08¯C

1.50.0040

R 1 = 43.12[kΩ]R 2 = 50[kΩ]

0.10∂id ¯1=¯re∂vd vd =vd0C

1.0

[V]

0.16

The sensitivity module is shown by Fig. 3(c), where the incremental resistor is given by

R1

1.91

0.5

Fig. 6 Simulation result for optimization.

#Thus, the sensitivity circuit configuration is equal to the original
one except for the nonlinear elements being replaced by the linear
incremental resistors at the operating points V0 , I0 .
2.2 Sensitivity modules of bipolar transistor: For simplicity, we consider NPN transistor modeled by Ebers-Moll model asshown in Fig. 3(a), whereid = Is (exp(vd /vT ) − 1),

43.12[kΩ]GAIN0.0

where δ(i) means a delta function satisfying
δ(i) =50[kΩ]
R20.1200i-15.00.14∂g(v,i)δ(i)∂pv0 ,i0=

δ(i)
Aδ(i)

¯
¯R0 p

4. Simulation result

5.0∂g(v,i)Sv,pi∂v
v0 ,i0CiFig. 5 Spice-oriented optimizing method.

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:
Ki0Aipi=1,2, ... , k

2. Transistor’s sensitivity modules"+S (p+ p )-S (p)-SidRE(b)Fig. 4 (a) Common-emitter amp.,(b) The sensitivity circuit
for calculating the gain.

5. Conclusion and remarks
In this paper, we obtained the optimal result. Thus, we wereable to optimize an amplifier by using steepest descent methodon Spice. As a problem in the future, we will try to optimize themultistage amplifier on Spice as the next stage.

- 23 -