Proceedings of the 13th WSEAS International Conference on CIRCUITS
On Stability of Electronic Circuits
HASSAN FATHABADI
Electrical Engineering Department
Azad University (South Tehran Branch)
Tehran, IRAN
h4477@hotmail.com
NIKOS E. MASTORAKIS
Industrial Engineering Department
Technical University of Sofia,
Sofia 1000, BULGARIA
mastor@wseas.org
Abstract: -In this paper, a necessary and sufficient condition for robust stability of electronic circuits at high
frequency which contain differential pair (emitter-coupled pair) is proposed. In fact, when emitter-coupled pairs
are used as one input signal – one output signal, they have uncertainty in their transfer functions at high
frequency. Even as will be shown, this uncertainty can be caused instability in closed loop electronic circuits at
high frequency. The uncertainty will be modeled as multiplicative perturbation in the transfer function of the
differential pair at high frequency. Based on this uncertainty model, a necessary and sufficient condition for
robust stability of above electronic circuits at high frequency will be presented. This condition guarantees
internal stability of electronic circuits at high frequency with respect to above uncertainty.
Key- Words: - Stability, robust, uncertainty, perturbation, electronic, circuit.
1 Introduction
When Vo = VO 2 , Vi = V1 and V2 = 0 , differential
A group of electronic circuits that contain emittercoupled pairs are very important. They have many
applications in amplifiers, communication circuits
and etc [1]. The differential pair or emitter-coupled
pair is an essential building block in modern
integrated circuits (IC) amplifiers [2]. This circuit is
shown in Fig. 1.
This paper considers this group of electronic
circuits at high frequency. As we know, emittercoupled pair has two main properties:
A) It rejects most of common noises that arrive from
two bases.
B) Its transfer characteristic has linear region larger
than a stage which consists of only one BJT.
As we know, the property (A) is decreased by
increasing frequency [1]. About property (B), in fact,
for
small
difference
voltage
(
Vd
pair has one input signal( Vi ) and one output
signal ( V0 ) as shown in Fig.2. Thus, we can
consider differential pair as SISO (Single Input
Single Output) block which shown in Fig.3.
4VT ⟩ V1 − V2 = Vd in Fig.1), the differential pair
behaved as a linear amplifier [2].
differential pair which shown in Fig.1.
Consider
Corresponding author’s address: Hassan Fathabadi, No.
27-Ghazanfar Khavand Alley -30 Metry Jey BloverTehran-Iran
.P.O.Box:
13517-53593.
E-mail:
h4477@hotmail.com
ISSN: 1790-5117
Fig. 1. Emitter-coupled pair.
43
ISBN: 978-960-474-096-3
Proceedings of the 13th WSEAS International Conference on CIRCUITS
Uncertainty in systems is one of the most difficult
problems [3], [4], [5]. Almost, all systems, such as
linear, nonlinear, discrete, neural and etc have
uncertainty in their transfer function [6], [7], [8]. In
recent years, there are many researches in analysis of
circuits and robust stability [9], [10], [11]. In this
paper, we show, the block which shown in Fig. 3 has
uncertainty at high frequency. This uncertainty is in
the place of its poles. We consider only domain pole
and its uncertainty will be modeled as multiplicative
perturbation at high frequency in the transfer function
of the above block. Then base on this uncertainty
model, we will study the robust stability of the closed
loop electronic circuits which contain above block as
a section of their plant at high frequency. Finally a
necessary and sufficient condition for robust stability
and stabilization of above electronic circuits at high
frequency will be proposed.
2 Multiplicative Perturbation
Suppose that the nominal plant transfer
function is P (s ) and consider perturbed plant
transfer function of the form
~
P ( s ) = [1 + Δ ( s )W ( s )]P ( s ) .
(1)
Here W (s ) is a fixed stable transfer function, the
weigh, and Δ (s ) is a variable stable transfer
function satisfying Δ ( s )
∞
≤ 1 further more, it
is assumed that no unstable poles of P ( s ) are
~
canceled on forming P (s ) [8]. Thus, P (s ) and
~
P ( s ) have the same unstable poles. Such a
perturbation Δ (s ) is said to be allowable. The
idea behind this uncertainty model is that
Δ ( s )W ( s ) is normalized plant perturbation
away from (1):
~
P ( s)
− 1 = Δ( s)W ( s)
P( s )
hence if Δ ( s )
∞
(2)
≤ 1 , then
~
P ( s)
− 1 ≤ W ( s) ,
P( s )
∀ω .
(3)
So W (s) provides the uncertainty profiles. The
inequality (3) describes a disk in the complex
~
P (s)
plane. In fact, at each frequency the point
P( s )
lies in the disk with center 1 and radius W ( s) .
Typically,
W (s) is a (roughly) increasing
function of ω . In other word, uncertainty
increases with increasing frequency. The main
purpose of Δ (s ) is to account for phase
uncertainty and to act as a scaling factor on the
magnitude of the perturbation. Thus, this
uncertainty model is characterized by a nominal
plant P (s ) together with a weighting function
W (s ) [12].
Fig. 2. Differential pair with one input and one output.
3 Perturbed Transfer Function of
Differential Pair
Consider differential pair which shown in
Δ
Fig.1. We define:
Vid ( s ) = V1 ( s ) − V2 ( s )
,
Δ
Vod ( s ) = VO1 ( s ) − VO 2 ( s ) ,
Fig. 3. SISO equivalence block.
ISSN: 1790-5117
44
ISBN: 978-960-474-096-3
(4)
Proceedings of the 13th WSEAS International Conference on CIRCUITS
[V1 ( s ) + V2 ( s)]
,
2
Δ [V ( s ) + V
O 2 ( s )]
Voc ( s) = O1
2
Δ
ADM ( s ) ≈
Vic ( s) =
(5)
Δ
Δ V ( s)
Vod ( s)
, ACM ( s ) = oc
Vid ( s )
Vic ( s )
(12)
As we know, domain pole p1 is computed from
following equation [1]:
and
ADM ( s ) =
− g m RC
.
s
(1 + )
p1
(6)
p1 =
where
Vid (s ) is differential-mode input
Vod (s ) is differential-mode output
Vic (s) is common-mode input
Voc (s ) is common-mode output
ACM (s ) is common-mode gain
1
R {Cπ + C μ [(1 + g m RC ) +
RC
R
] + C C cs }
R
R
(13)
where
R = ( RS + rb )
rπ .
In above Equations RS is total resistance of
differential –mode input signal source. rπ , rb ,
ADM (s) is differential-mode gain
As we saw, when Vo = VO 2 , Vi = V1 and V2 = 0 , the
Cπ , C μ and C cs are shown in Fig.4 which is
equivalence circuit of BJT.
differential pair which shown in Fig.2 is obtained.
Thus, we have
1
Vo ( s ) = − Vod ( s ) + Voc ( s )
2
so
Vo ( s ) = −
1
ADM ( s )Vid ( s ) + ACM ( s )Vic ( s )
2
(7)
1
1
ADM ( s ) + ACM ( s )]Vi ( s )
2
2
(8)
and
Vo ( s ) = [−
From (8), we can write the transfer function of the
differential pair which shown in Fig.2 as
V ( s)
A ( s) 1
~
= [1 − DM ] ACM ( s) .
P (s) = o
Vi ( s)
ACM ( s) 2
Fig. 4. Equivalence circuit of BJT.
(9)
It is reminding that, p1 also is upper (high)
half-power frequency of differential pair. Just as
we see, p1 is a multivariable function of rπ , rb ,
Also we know that [1]
RC
(10)
(1 + sRE C E )
2 RE
where C E and RE are output capacity and resistance
ACM ( s ) ≈ −
Cπ and C μ . In a BJT, rπ is input dynamic
resistance, rb is base pin resistance, Cπ is
internal capacity between internal base and
internal emitter and C μ is internal capacity
of current source, respectively. The differential-mode
gain is obtained from following equation [1]:
ADM ( s ) =
− g m RC
s
s
(1 + )(1 + )
p1
p2
between internal base and internal collector. On
the other hand, we know rπ , rb , Cπ and C μ are
(11)
related to internal properties of BJT, such as WB
where is transfer conductance of BJT . Also p1 and
( width of base ), N D (concentration of donor
p2 are poles of above transfer function. In fact,
p2 and
p2 >> p1 . Thus, we can ignore of
atoms), N A (concentration of acceptor atoms),
Dn or D p (constant distribution of minority
carriers), β 0 and etc [1], [2].
consider only domain pole p1 . So we have
ISSN: 1790-5117
45
ISBN: 978-960-474-096-3
Proceedings of the 13th WSEAS International Conference on CIRCUITS
ADM ( jω ⟨⟨ ACM ( jω
From (13) and above notes, we conclude that,
domain pole p1 is changed when a BJT in emittercoupled pair is replaced by another BJT (for example,
when we repair electronic circuits which contain this
differential pair). Thus, domain pole p1
has
uncertainty.
By replacing (10) and (12) in equation (9), it is not
difficult to verify that
and from (8), it follows that
− RC
1
~
P ( s ) ≈ ACM ( s ) =
(1 + sRE C E ) ,
2
4 RE
this is the same result which has been obtained in
(17). Even, reader can note that, although at low
~
frequency, the phase of P ( jω ) is about 0° , at
high frequency, its phase is about 180 ° .This
causes to change negative feedback to positive
feedback and thus, instability in closed loop
electronic circuits at high frequency.
As we see, in (15), Δ (s ) is variable (because
2gm RE
R
−1
~
P(s) = [1+ (
)(
)](− C )(1+ sRECE )
s
4RE
(1+ ) 1+ sRECE
p1
(14)
So equation (14) is the transfer function of the
differential pair which shown in Fig.2. In (14), we
define
of uncertainty of p1 ) stable transfer function
Δ( s) =
is fixed stable transfer function. Also in (17),
P (s ) is the nominal transfer function with no
−1
,
s
(1 + )
p1
2 g m RE
.
W (s) =
1 + sRE C E
with Δ( s )
(15)
~
~
replacing (15), (16), (17) in (14), P ( s ) is become
(16)
~
as the form P ( s ) = [1 + Δ ( s )W ( s )]P ( s ) ,which
is the same equation (1) and as we saw, is
multiplicative perturbed transfer function.
Thus, the differential pair which shown in
Fig. 2 or its equivalence SISO block diagram
which shown in Fig.3 at high frequency, has
multiplicative perturbed transfer function
~
P ( s ) which has been presented in (14) with
Δ (s ) , W (s ) and P ( s ) which have been
obtained in (15), (16) and (17), respectively .
− RC
~
lim P ( jω ) = lim
(1 + jωRE C E ) .
ω →∞ 4 R
E
ω →∞
So at high frequency ( ω ⟩⟩ p1 ), we have
RC
(1 + sRE C E ) ,
4 RE
(17)
~
and P (s ) is the nominal transfer function of P ( s ) .
Consider equation (8), in fact, at low frequency
(ω ≤
1
), we have
RE C E
4 Internal Stability
Consider a basic feedback loop as shown in
Fig. 5. The signals shown in Fig.5, have the
following interpretations:
r
reference or command input
y output and measured signal
n
sensor noise
d1 , d 2
external disturbances
ADM ( jω ⟩⟩ ACM ( jω ,
in the other word
CMRR( jω ) =
ADM ( jω
⟩⟩1 ,
ACM ( jω
where CMRR ( jω )
is "Common Mode Reject
x1 , x2 , x3 , x4 state variables
P1 , P , P2 forward plants
F feedback transfer function or the transfer
1
, the
RE C E
and
CMRR ( jω ) is decreased with slope -20 dB
Dec
for ω ⟩ p1 , the CMRR ( jω ) is decreased with slope
Ratio". When ω is increased, at ω ⟩
-40 dB
Dec
function of sensor.
Note that, all of above signals and transfer
functions are in Laplace domain. For example
P1 = P1 ( s) , x1 = x1 ( s) and etc.
In Fig.5, we have
[1]. So for ω ⟩⟩ p1 , we have
ISSN: 1790-5117
≤ 1 and in (16), W (s ) (the weigh)
canceled unstable pole in forming P ( s ) . By
As we see in (14),
P( s) = −
∞
46
ISBN: 978-960-474-096-3
Proceedings of the 13th WSEAS International Conference on CIRCUITS
⎧ x1 = r − Fx4
⎪x = d + P x
⎪ 2
1
1 1
.
⎨
x
d
Px
=
+
3
2
2
⎪
⎪⎩ x4 = n + P2 x3
( P1 PP2 F in Fig. 5). Then, at least one internal
signal of this electronic circuit is unbounded.
This means that, at least one signal is clipped.
Consequently, this electronic circuit can not
operate as linear circuit or amplifier (about the
operating point).
(18)
5 Robust Stability
Suppose that, P (s ) in Fig. 5 is replaced by
~
P ( s ) which has uncertainty and belongs to a set
such as M . This is shown in Fig. 6.
Definition 2: The system shown in Fig. 6 has
robust stability if it provides internal stability for
~
every P ( s ) ∈ M [12].
~
Assume that, in Fig. 6 P ( s ) has uncertainty
as multiplicative perturbation form. In other
~
word, P ( s ) is described with equation (1).
Theorem 1: The closed loop system which
shown in Fig. 6 has robust stability iff
Fig. 5. Basic feedback loop.
In matrix form these are
⎡ 1
⎢− P
⎢ 1
⎢ 0
⎢
⎣ 0
0
1
0
0
1
−P
0 P2
F ⎤⎛ x1 ⎞
⎛r⎞
⎜ ⎟
⎜ ⎟
⎥
0 ⎥ ⎜ x2 ⎟
⎜d ⎟
= ⎜ 1 ⎟.
⎟
⎜
0 ⎥ x3
d
⎜ 2⎟
⎥⎜⎜ ⎟⎟
⎟
⎜
1 ⎦ ⎝ x4 ⎠
⎝n⎠
P1 PP2 ( s)
⟨ 1.
1 + P1 PP2 F ( s) ∞
Proof:
Assume
(⇐)
W ( s) F ( s)
(19)
W ( s) F ( s)
⎡ 1
⎢ P
⎢ 1
⎢ P1 P
⎢
⎣ P1 PP2
− PP2 F
1
P
PP2
− P2 F
− P1 P2 F
1
P2
− F ⎤⎛ r ⎞
⎜ ⎟
− P1 F ⎥⎥⎜ d1 ⎟
− P1 PF ⎥⎜ d 2 ⎟
⎥⎜ ⎟
1 ⎦⎜⎝ n ⎟⎠
P1 PP2 ( s)
⟨ 1 . Construct the
1 + P1 PP2 F ( s) ∞
P2 ( s) in Re( s ) ≥ 0 . Fix an
allowable Δ (s ) . Construct the Nyquist plot of
~
P1 P P2 F ( s ) = [1 + Δ ( s )W ( s )]L( s ) .
P1 ( s) and
(20)
Definition 1: If the sixteen transfer functions in
(20) are stable, then the feedback system which
shown in Fig. 5 is said to be internally stable [12].
As a consequence, if the exogenous inputs are
bounded in magnitude, so too are x1 , x2 , and x3 , and
hence u , y and v . So, for an electronic circuit,
internal stability is very important, because it
guarantees bounded internal signal for all points of
electronic circuit. Suppose that, a electronic circuit
has BIBO stability but don’t have internal stability. In
other word, there is at least one pole–zero
cancellation in Re( s ) ≥ 0 when the closed loop
transfer function of the electronic circuit is formed
ISSN: 1790-5117
that
Nyquist plot of L( s) = P1 PP2 F ( s) , indenting
D to the left around poles on the imaginary axis.
Since the nominal feedback system is internally
stable, we know this form the Nyquist criterion:
The Nyquist plot of L does not pass through -1
and
its
number
of
counterclockwise
encirclements equals the number of poles of
P (s ) in Re( s ) ≥ 0 plus the number of poles of
Thus, the system is well-posed if and only if the
above 4× 4 matrix is nonsingular, that is, the
determinant 1+ P1 PP2 F is not identically equal to
zero [12]. From (19), it follows that
⎛ x1 ⎞
⎜ ⎟
1
⎜ x2 ⎟
⎜ x ⎟ = 1 + P PP F
1
2
⎜ 3⎟
⎜x ⎟
⎝ 4⎠
(21)
No additional indentations are required since
Δ ( s )W ( s ) introduces no additional imaginary
axis poles. We have to show that the Nyquist plot
of [1 + Δ ( s )W ( s )]L ( s ) does not pass through -1
and
its
number
of
counterclockwise
encirclements equals the number poles of
[1 + Δ ( s )W ( s )]P ( s ) in Re( s ) ≥ 0 plus the
number of poles of P1 P2 F ( s) in Re( s ) ≥ 0 ;
equivalently,
the
Nyquist
plot
of
[1 + Δ ( s )W ( s )]L( s ) does not pass through -1
and encircles it exactly as many times as does the
47
ISBN: 978-960-474-096-3
Proceedings of the 13th WSEAS International Conference on CIRCUITS
Nyquist plot of L (s ) . We must show, in other words,
that the perturbation does not change the number of
encirclements. The key equation is
6 Robust Stability of the Circuit
Consider an electronic circuit that its block
diagram at high frequency shown in Fig. 7. As
we see, this circuit contains a differential pair,
which has one input signal and one output signal.
As we obtained in (14), the transfer function of
this differential pair at high frequency is
~
P ( s ) which has uncertainty as multiplicative
perturbation with Δ (s ) and W ( s ) that were
derived in (15) and (16). Also the nominal
~
transfer function of P ( s ) , is P ( s ) which was
obtained in (17).
1+ [1+ Δ(s)W (s)]L(s) = [1+ L(s)][1+ Δ(s)W (s)
P PP (s)
]
F(s) 1 2
1+ P1 PP2 F(s)
(22)
Since
Δ(s)W(s)F(s)
P1PP2 (s)
PPP(s)
≤ W(s)F(s) 1 2
⟨1
1+ P1PP2F(s) ∞
1+ P1PP2F(s) ∞
(23)
P1 PP2 ( s )
always
The point 1 + Δ ( s )W ( s ) F ( s )
1 + P1 PP2 F ( s )
lies in some closed disk with center 1, radius ⟨ 1 , for
all points s on D . Thus from (22), as s goes once
around D , the net change in the angle of
1 + [1 + Δ ( s )W ( s )]L( s ) equals the net change in the
angle of 1 + L ( s ) . This gives the desired result.
P1 PP2 ( s)
≥ 1.
1 + P1 PP2 F ( s) ∞
We will construct an allowable Δ (s ) that destabilizes
P1 PP2 ( s )
is
the feedback system. Since F ( s )
1 + P1 PP2 F ( s )
(⇒ ) Suppose, W ( s ) F ( s)
Fig. 6. Basic feedback loop with uncertainty.
In Fig. 7, P1 ( s) and P2 ( s) are the total
transfer functions of all stages which are located
before and after differential pair, respectively.
Also F (s ) is the transfer function of negative
feedback in circuit.
Theorem 2: The electronic circuit which
shown in Fig. 7, has robust stability with respect
~
to uncertainty of P ( s ) ( the transfer function of
this differential pair) at high frequency, iff
strictly proper, at some frequency ω , we have
W ( jω ) F ( jω )
P1 PP2 ( jω )
= 1.
1 + P1 PP2 F ( jω )
Suppose that ω = 0 , then
W ( 0) F ( 0 )
(24)
P1 PP2 (0)
is a real number, either +1
1 + P1 PP2 F (0)
or -1.
If Δ (0) = −W (0) F (0)
P1 PP2 (0)
, then Δ (0)
1 + P1 PP2 F (0)
P1 ( s ) P2 ( s ) F ( s)
1
g m RC
⟨1
R
2
1 − C (1 + sRE C E ) P1 ( s) P2 ( s ) F ( s)
4 RE
∞
is allowable and
1 + Δ (0)W (0) F (0)
P1 PP2 (0)
=0.
1 + P1 PP2 F (0)
(25)
From (22) with respect to (25), we conclude that, the
Nyquist plot of [1 + Δ ( s )W ( s )]L ( s ) passes through
the critical point, so the perturbed feedback system is
not internally stable.
ISSN: 1790-5117
(26)
Proof: (⇔ ) As we see, the block diagram of
electronic circuit which shown in Fig. 7 and the
block diagram of the closed loop system which
shown in Fig. 6, both are the same.
48
ISBN: 978-960-474-096-3
Proceedings of the 13th WSEAS International Conference on CIRCUITS
g m = 0.01 Ω −1 , rπ 2 = rπ 3 =
R = 3.76
β
gm
= 10 K and
10 K = 2.73 K .
By replacing above values in (13), we obtain
that, p1 ≈ 6.06 M rad
sec
.
For BJT ( T1 ), we have
I C = 1 mA , g mT =
Fig. 7. Block diagram of electronic circuit at high
frequency.
rπ =
β
g mT
1
Ω −1 ,
25
= 2 .5 K ,
R LT = 4.7 K 22 K 128 K = 3.76 K ,
In fact, the block with multiplicative perturbed
~
transfer function P ( s ) in Fig. 6 has been replaced by
differential pair which has multiplicative perturbed
~
transfer function P ( s ) , as was obtained in (14) with
Δ (s ) , W (s ) and P (s ) that were derived in (15) , (16)
and (17). So, from theorem 1, by replacing (16) and
(17) in (21), we have
Rπ 0 = [(0.05K 68K 68 K ) + rb ] rπ = 0.31K
and
AV =
− g mT .RLT
= −1.27 .
1 + g mT (6.8 K 5.2 K )
By replacing above values in following equation
[1]:
− RC
(1+ sRECE )P1 (s)P2 (s)F (s)
2gm RE
4RE
⟨1 .
(1+ sRECE ) 1− RC (1+ sR C )P (s)P (s)F (s)
E E 1
2
4RE
∞
1
p ≈
Rπ 0 {Cπ + Cμ [(1 + g mT RLT ) +
(27)
the domain pole of
p ≈ 73 M rad
It is not difficult to verify (26). This completes the
proof.
sec
RLT RLT
]+
Ccs }
Rπ 0 Rπ 0
(28)
T1 , is computed as
.
Comparing simulated circuit in Fig. 8 and the
block diagram in Fig. 7, it follows that
Example 1: Consider the simulated circuit which
shown in Fig. 8 with following parameters.
C μ = Cπ = C cs = 0.25 pF ,
For
BJT
( T1 ):
β = 100 , rb ≈ 0 Ω and VT ≈ 25 mV .
For BJTs ( T2 and
T3 ): Cμ = Cπ = Ccs = 1 pF ,
β = 100 , rb ≈ 0 Ω and VT ≈ 25 mV .
We have (
is the symbol of the parallel
resistances)
R S = 4 .7 K
RC = 10 K
22 K
128 K = 3.76 K ,
12 K = 5.45 K .
For biasing of the BJTs ( T2 and T3 ), we have
I CT = I CT = 0.25 mA .
2
3
So we have
ISSN: 1790-5117
Fig.8. Simulated electronic circuit.
49
ISBN: 978-960-474-096-3
Proceedings of the 13th WSEAS International Conference on CIRCUITS
− 1.27
6 .8 K
, F ( s) =
≈ 0.57
s
6 .8 K + 5 .2 K
1+
73 × 10 6
and P2 ( s ) = 1 .
P1 ( s ) =
1
P1 ( s ) P2 ( s ) F ( s )
⟨1
g m RC
RC
2
1−
(1 + sRE C E ) P1 ( s ) P2 ( s ) F ( s )
4 RE
∞
and thus, simulated circuit has robust stability
and
also
it
is
unstable
for
ω < 488.82M rad sec .
The input of the simulated circuit is small
signal sinusoidal form. As we see, the results of
simulation which shown in Fig. 9, validate the
presented proposal.
Taking these and other values in (26), we have
− 1.27
× 0.57
s
)
6
1
73 ×10
× 0.01× 5.45 × 103
5.45
− 1.27
2
[1 −
(1 + 33 × 27 × 10−9 s)
× 0.57]
s
132
(1 +
)
73 × 106
(1 +
sup
19.7263
1.0299 + (0.0403 × 10 −6 ω ) 2
2
So, for ω > 488.82 M rad
sec
=
∞
.
>> p1
7 Conclusion
In this paper, a group of electronic circuits
which contain an emitter-coupled pair with one
input and one output signal, has been considered.
We showed that, the transfer function of this
differential pair at high frequency has
uncertainty. This uncertainty has been modeled
as multiplicative perturbation. Base on this
multiplicative uncertainty model, the necessary
and sufficient condition for robust stability at
high frequency has been presented in (26). This
condition guarantees internal stability of
electronic circuit with respect to the uncertainty
of the transfer function of differential pair at high
frequency. A direct benefit of the result in this
paper is a condition which must be satisfied,
when we design integrated circuits at high
frequency.
It is an interesting research topic to obtain
similar conditions for other differential pairs
which contain MOSFET and JFET.
Acknowledgment: Implementation of the
project has been done in Azad University (South
Tehran Branch).
References:
[1] P. E. Gray, P. J. Hurst, S. H. Lewis and R. G.
Meyer, Analysis and Design of Analog
Integrated Circuits, New York: Wiley, 2001.
[2] J. Millman and A. Grabel, Microelectronics,
2nd ed. New York: Mc Graw-Hill, 1988.
[3] C. Yuhua, M. J. Deen and C. Chih-Hung,
MOSFET modeling for RF IC design, IEEE
Trans. Electron Devices, vol. 52, 2005, pp. 1286
- 1303.
[4] M.N.A. Parlakci, Robust stability of
uncertain time-varying state-delayed systems,
Fig.9. The results of simulated circuit.
a)
ω = 666.7 M rad sec
b)
ω = 490.0 M rad sec
c)
ω = 266.7 M rad sec .
ISSN: 1790-5117
50
ISBN: 978-960-474-096-3
Proceedings of the 13th WSEAS International Conference on CIRCUITS
IEE Proc. Control Theory and Applications, vol. 153,
2006, pp. 469-477.
[5] Sanqing Hu and Jun Wang, Global robust stability
of a class of discrete-time interval neural networks,
IEEE Trans. Circuit and Systems I, vol. 53, 2006, pp.
129-138.
[6] A. Schmid and Y. Leblebici, Robust circuit and
system design methodologies for nanometer-scale
devices and single-electron transistors, IEEE Trans.
Very Large Scale Integration (VLSI) Systems, vol. 12,
2004, pp. 1156 – 1166.
[7] N. Ozcan and S. Arik, Global robust stability
analysis of neural networks with multiple time delays,
IEEE Trans. Circuit and Systems I, vol. 53, 2006, pp.
166 - 176.
[8] L. Liu and J. Huang, Adaptive robust stabilization
of output feedback systems with application to Chua's
circuit, IEEE Trans. Circuit and Systems II, vol. 53,
2006, pp. 926-930.
ISSN: 1790-5117
[9] A. Buonomo and A. Lo Schiavo, Analysis of
emitter (source)-coupled multivibrators, IEEE
Trans. Circuit and Systems I, vol. 53, 2006, pp.
1193-1202.
[10] A. Ghulchak and A. Rantzer, Robust control
under parametric uncertainty via primal-dual
convex analysis, IEEE Trans. Automatic Cont.,
vol. 47, 2002, pp. 632-636.
[11] S. Dasgupta, P. J. Parker, B. D. O.
Anderson, F. J. Kraus and M. Mansour,
Frequency domain conditions for the robust
stability of linear and nonlinear dynamical
systems, IEEE Trans. Circuit and Systems, vol.
38, 1991, pp. 389 - 397.
[12] J. C. Doyle, B. A. Francis and A. R.
Tannenbaum, Feedback Control Theory, New
York:Macmillan,1992.
51
ISBN: 978-960-474-096-3