Amplifiers – Signal feedback – Phase shift means in loop path
Reexamination Certificate
2001-01-12
2003-06-03
Pascal, Robert (Department: 2817)
Amplifiers
Signal feedback
Phase shift means in loop path
C330S109000, C330S294000
Reexamination Certificate
active
06573790
ABSTRACT:
2 FIELD OF INVENTION
The field of invention is analog electrical circuits, more particularly, this invention relates to operational amplifiers with increased gain in a wide bandwidth.
3 DESCRIPTION OF PRIOR ART
An operational amplifier, hereinafter referred to as an “opamp,” is a general building block used in numerous analog electrical circuits. The symbol used to represent a general-purpose opamp [
32
] is shown in
FIG. 1. A
differential-input single-ended-output opamp [
32
] has a non-inverting input terminal [
34
], an inverting input terminal [
36
], and an output terminal [
38
], the voltage V
out
of which is defined with respect to a ground terminal [
40
]. In the ideal case, the two input terminals [
34
] and [
36
] are high-impedance terminals, i.e., they do not conduct any current, and the output voltage V
out
is a factor of A higher than the differential input voltage V
in
, where the gain A approaches infinity. In principle, the output voltage V
out
will depend only on the voltage difference V
in
between the two input terminals [
34
] and [
36
], not on their average voltage with respect to the ground terminal [
40
], i.e., the common-mode input voltage. Hence, for an ideal infinite-gain opamp the ground terminal's [
40
] potential is somewhat arbitrary.
Opamps are generally used in feedback configurations.
FIG. 2
shows a typical feedback configuration which may be used, for example, for current-to-voltage conversion in combination with a current-mode digital-to-analog converter. The reference potential for the output voltage V
out
is obtained by connecting the non-inverting input terminal [
34
] to the ground terminal [
40
]. If the opamp's [
32
] gain A is indeed infinite and the system is stable, the inverting input terminal [
36
] will have the same potential as the ground terminal [
40
]. A linear impedance element [
42
] is connected between the inverting input terminal [
36
] and the output terminal [
38
]. The impedance element[
42
] is a one-port network, usually consisting of only resistors, capacitors, and possibly also inductors, which here is described by the Laplace-transformed impedance Z(s). Hence, the output voltage is described by the Laplace-transformed relationship
V
out
(
s
)=
Z
(
s
)·
I
in
(
s
). (1)
Physical opamps are not ideal. The gain A is not infinite, and variations in the output voltage V
out
are delayed with respect to variations in the input voltage V
in
. For small signals the opamp can be characterized by a transfer function,
V
out
(
s
)=
H
opamp
(
s
)·
V
in
(
s
). (2)
The opamp's [
32
] gain A(f) is frequency-dependent, typically with a high static value A
dc
=H
opamp
(0). When taking Equation 2 into account, Equation 1 takes the form
V
out
(
s
)
=
Z
(
s
)
·
I
in
(
s
)
-
V
in
(
s
)
=
Z
(
s
)
·
I
in
(
s
)
1
+
1
/
H
opamp
(
s
)
.
(
3
)
One of the difficulties in using non-ideal opamps is that the closed-loop system, e.g. the circuit shown in
FIG. 2
, may become unstable if the system's open-loop frequency response H
OL
(s) is not properly designed. The open-loop frequency response H
OL
(s) is the product of the opamp's frequency response H
opamp
(s) and the feedback network's frequency response &bgr;(s). The feedback network's frequency response &bgr;(s) can be evaluated as
β
(
s
)
=
-
V
in
(
s
)
V
out
(
s
)
when the opamp is removed from the circuit, provided that its input/output impedances are properly modeled. The accurate calculation of the open-loop frequency response H
OL
(s) requires some experience, but it is discussed in several textbooks and taught at most electrical engineering schools; hence the concept is well-known to those ordinarily skilled in the art. For the closed-loop circuit shown in
FIG. 2
, the feedback network's frequency response is &bgr;(s)=1, and thus, H
OL
(s)=H
opamp
(s).
3.1 Stability
Stability/instability can be determined using one of several stability criterions. Nyquist's stability criterion will be used for the following discussion. Nyquist's stability criterion states that the closed-loop system will be stable if the polar plot of H
OL
(s), s=j2&pgr;f, f∈R, does not encircle the point −1, otherwise the system will be unstable. For all real systems, H
OL
(j2&pgr;f) and H
OL
(−j2&pgr;f) are complex-conjugate values, hence it is sufficient to plot H
OL
(j2&pgr;f) for positive values of f only (and connect H
OL
(0) and lim
f→∞
H
OL
(j2&pgr;f) by a straight line).
It is unavoidable that stray capacitors will make the angle &phgr;(f) of an opamp's frequency response H
opamp
(j2&pgr;f)=A(f)·e
j·&phgr;(f)
, i.e., the phase response, uncontrollable at high frequencies, which is why it is necessary to reduce the gain A(f) to less than 1 at such high frequencies. Opamps are usually designed to have a frequency response similar to that shown in
FIGS. 3
,
4
, and
5
.
FIG. 3
shows the opamp's gain A in deci Bell (dB) versus the frequency f in Hertz on a logarithmic scale. The opamp's unit-gain frequency, which is also called the opamp's gain-bandwidth frequency f
gbw
, is an important parameter. The typical target opamp-gain characteristic is A(f)=f
gbw
/f=−20·log
10
(f/f
gbw
) dB, but the gain is generally limited at low frequencies. In other words, the opamp's frequency response has a pole at a low frequency f
pole,1
. Although the low-frequency (dominating) pole f
pole,1
is intentional, the parameters f
pole,1
and A
dc
are usually somewhat undetermined; an opamp should be designed to have a well-controlled f
gbw
, which is the most important parameter with respect to stability concerns.
FIG. 4
shows a plot of the phase response &phgr;(f) in degrees versus the frequency f. The phase margin is defined as 180°+&phgr;(f
gbw
). A general design rule is to make the phase margin at least 45°. This will generally require that f
gbw
is slightly lower that the opamp's first non-dominating non-canceled (undesired) pole/right-plane-zero. The achievable f
gbw
is dependent on the technology used and the power consumption allowed.
FIG. 5
shows a polar plot of the frequency response H
opamp
(j2&pgr;f); it is merely an alternative graphical representation of A(f) and &phgr;(f). The respective closed-loop system, i.e., the circuit shown in
FIG. 2
, is stable because the curve does not encircle the critical point −1. Clearly, a large phase margin is preferable because that will avoid close proximity of the critical point and the area enclosed by H
opamp
(j2&pgr;f), f∈R (which is the main stability concern).
3.2 Opamp Implementation
FIG. 6
shows the conceptual topology of a simple one-stage opamp (often also called an OTA). The opamp consists of only one transconductance stage [
44
] providing an output current I
out
proportional to the differential input voltage V
in
,
I
out
(
s
)=
g
m
·V
in
(
s
). (4)
Equation 4 is valid for frequencies up to a certain frequency f′ only. Hence the gain of the opamp must be less than one at frequencies higher than f′. The opamp's voltage gain A(f) is determined by the load [
46
], which is modeled as a resistor [
48
] R
load
and a capacitor [
50
] C
load
connected in parallel. The opamp's static gain is
A
dc
=g
m
·R
load
(5)
and its unity-gain frequency is
f
gbw
=
g
m
2
π
C
load
.
(
6
)
Because the transconductance g
m
cannot be made arbitrarily high (for a limited power/current consumption), this type of opamp provides only relatively little static gain when driving a resistive load [
48
]. Because the unity-gain frequency (
Esion LLC
Nguyen Khanh Van
Pascal Robert
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