ETC AB-028

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By Jerald G. Graeme (602) 746-7412
(Originally published in EDN magazine as “Feedback Plots
Offer Insight into Operational Amplifiers” and “Bode Plots
Enhance Feedback Analysis of Operational Amplifiers” on
1/19/89 and 2/2/89, respectively.)
Feedback plots simplify the analysis of an op amp’s closedloop AC performance by showing bandwidth and stability
conditions as a function of the op amp’s gain and phase
response. These plots also provide insight into noise performance and the special feedback requirements of circuits
such as integrating converters, photodiode amplifiers, composite amplifiers and active feedback circuits.
3dB BW
|A CL |
0.1f o
10f o
For a basic voltage-gain amplifier, the location of the fp pole
determines the closed-loop bandwidth. In this case, a singlepole roll-off determines the point at which the gain magnitude goes below 3dB (equivalent to 0.707 of its low-fre-
R1 + R2
ACL ≈ 1/β = (R1 + R2)/R1
1991 Burr-Brown Corporation
A CL = e =
1 + 1/Aβ
β =
The Bode plot of Figure 1, for example, shows the interaction of the magnitude response of the open-loop gain (|A|)
and the reciprocal of the feedback factor (1/β). The fraction
of the output that feeds back to the input is β. The voltagedivider action of Figure 1’s feedback network determines
the value of β; for moderate resistance values,
β = R1/(R1 + R2). For this noninverting example, the feedback equation, ACL = A/(1 + Aβ), defines the closed-loop
voltage gain. Aβ is the loop gain, and where it is high:
Aβ represents the amplifier gain available to maintain the
ideal closed-loop response. At the point where the loop gain
no longer matches the feedback demand, the closed-loop
curve deviates from the ideal. The Bode plot graphically
defines this limit by plotting the 1/β curve with the gainmagnitude response curve of the op amp. Because the 1/β
line represents the feedback demand, closed-loop requirements will be satisfied as long as this line is below the
amplifier-gain curve. Where this condition is no longer true,
the actual response drops, following the amplifier’s openloop response downward. The rate of descent for the roll-off
is –20dB/decade (for most op amps) and is characteristic of
a single-pole response. In Figure 1, the heavier line on the
gain-magnitude plot depicts the resulting closed-loop curve.
Engineers routinely use Bode plots(1) to determine the
bandwidth and frequency stability of voltage-gain op amp
circuits. A Bode plot provides a visual representation of an
op amp’s transfer response and its potential stability. Moreover, such plots define the circuit’s pole and zero locations
at the intercepts of the response-curve extensions.
FIGURE 1. This feedback analysis provides a summary of
loop conditions in the 1/β curve and defines the
underlying poles, zeros, and phase shift.
quency level). To find this point relative to the Bode plots,
rewrite the closed-loop gain as
ACL = (1/β)/(1/Aβ + 1)
The bandwidth-defining gain error is a result of the 1/Aβ
term in the denominator. Because β is constant for the circuit
in Figure 1, the amplifier gain (A) determines the frequency
dependence of the loop gain. For a typical op amp, the gain-
Printed in U.S.A. June, 1991
In cases where the intercept is less than one decade from a
response break, the Bode approximation of the phase shift
shows a linear slope that has a maximum error of 5.7°(1). For
Figure 1, the phase-shift approximation starts at 0° one
decade before the break frequency fo. From there, it increases linearly on the log scale to 45° at the break frequency
and then to 90° one decade above it.
bandwidth product is constant after the first break frequency
occurs and A = jfc/f = j|A| where fc is the amplifier’s unitygain crossover frequency. For this common condition,
ACL = (1/β)/(1 + 1/(j|A|β)
The bandwidth is defined in terms of the absolute value
(magnitude) of ACL:
|ACL| = (1/β)/√(1 + 1/(|A|2 β2))
which, at the –3dB point, becomes
Using this approximation, you can combine the stability
criteria for loop-gain magnitude and feedback phase shift to
obtain the rate-of-closure indicator. Rather than computing
phase shifts from slopes, you can use this indicator to deal
with the slopes directly. Rate-of-closure is simply the difference in slopes of the gain-magnitude curve and the 1/β curve
when they intercept. This difference reflects the combined
phase shift around the feedback loop. For Figure 1, the rateof-closure is 20dB/decade, which corresponds to a stable
90° phase shift.
|ACL| = 0.707(1/β) = (1/β)/√2
Comparing the last two expressions, you can see that the
–3dB bandwidth occurs when |A| = 1/β. This equality is true
when the gain supply drops to the exact level of the feedback
demand. When you plot these two functions on the same
graph, they reach equality at the intersection of the two
curves. This intercept pinpoints the closed-loop pole location and defines the circuit bandwidth for the voltage-gain
In other cases, the slope of the 1/β curve is not zero, giving
a 40dB/decade rate-of-closure that indicates an oscillatory
180° of phase shift. Rate-of-closure alone is an exact stability indicator where the intercept is at least one decade away
from all other break frequencies. In still other cases, the
Bode phase approximation modifies the rate-of-closure result.
This critical intercept point also exhibits other characteristics that can help you define conditions for frequency stability. By relating the phase shift to the slopes of the gainmagnitude and 1/β curves, you can determine the loop phase
shift at this intercept. Again, the importance of the intercept
is apparent from the closed-loop-gain expression,
ACL = A/(1 + Aβ). If Aβ becomes –1, the closed-loop gain
will be infinite and will support an output signal even in the
absence of an input signal, which is a condition for oscillation. The magnitude of Aβ is unity only at the intercept point
because it is at this point that A = 1/β; a negative polarity for
Aβ only requires 180° phase shift.
To use feedback relationships to perform circuit analysis,
you should consider the feedback network separately. This
separation parallels the nature of the op amp’s open-loop
gain, which is a characteristic of the amplifier in the absence
of the feedback network. You only need to retain the loading
effects between the amplifier and the feedback network to
determine their individual responses(2). Then, by putting the
two responses on the same plot, you can see how they will
work together.
Virtually every practical analog circuit is a minimum-phase
system. For such systems, which have only left-half-plane
poles and zeros, you can directly read the phase shift from
the gain-magnitude response(2). Although many op amps do
have a right-plane zero caused by Miller phase compensation, the effects of this zero are suppressed below the unitygain crossover. For the case of a minimum-phase system, a
pole creates a –20dB/decade response roll-off and a –90°
phase shift; a zero produces the same effects but with
opposite polarities. Additional poles and zeros simply add to
the response slope and phase shift in increments of the same
Figure 2 shows a generalized feedback condition defined by
Z1 and Z2. The equations of Figure 2a directly determine the
circuit response for high loop gain and moderate impedances. Nonetheless, the input impedance of the amplifier
alters the simplified results of these equations by shunting
the feedback network. The inclusion of this loading effect on
the feedback network completes the 1/β analysis in the circuit of Figure 2b. Here, the op amp input resistance (Ri),
differential input capacitance (Cid), and common-mode input capacitance (Cicm) all shunt impedance Z1. Except for
conditions where the feedback impedances have low values,
you need to include these amplifier characteristics in your
Relying on the feedback phase shift’s correlation with the
response slope, you can determine its value at the critical
intercept from the gain-magnitude and 1/β curves. For the
example of Figure 1, the gain-magnitude curve has a slope
of –20dB/decade and the 1/β curve has a zero slope for a net
90° feedback phase shift at the intercept. This situation
leaves a phase margin of 90° out of the 180° that would
cause oscillation. Because the intercept is well removed
from the open-loop-response break frequencies, the analysis
of this example is easier to understand. The intercept occurs
after the amplifier’s first pole develops the full 90° phase
shift, but well before the second pole has any effect.
Where there is impedance in series with the amplifier’s
noninverting input, you must add this too-along with the
shunting effect of the input’s Cicm capacitance. You can then
find the feedback factor from the divider action, ej/eo. For the
1/β curve, this result is inverted and, in the logarithmic
format of computer simulations, becomes simply
For any given feedback network, the inverting and
noninverting configurations develop signal gains that differ
in magnitude as well as in sign; nevertheless, the feedback
conditions remain the same. In both cases, your feedback
analysis is concerned with the gain-error voltage developed
between the op amp inputs. This error signal always receives
the gain of the noninverting connection, as you would see if
you performed superposition analysis. Superposition of the
signal between the amplifier inputs grounds the signal source,
producing the noninverting configuration.
eo = –
The same condition holds true for the input voltage noise of
an op amp, resulting in the noise-gain characteristic for the
1/β curve. In practice, the noise gain and the 1/β curve are
the same—until they intercept with the gain-magnitude
curve. After that, the noise gain rolls off with the amplifier
open-loop response but the 1/β curve continues on its path.
For the noninverting voltage amplifier, the noise gain and
the closed-loop gain, ACL, are the same.
ei ,
For |A| >> 1/β and Z i >> Z1 || Z 2
where Z i = Ri || Cid || Cicm
C id
1/ β =
C icm
In inverting configurations, this correspondence does not
hold true, giving rise to frequent surprises during attempts at
noise filtering. The simplest case of the inverting amplifier,
where it is common practice to bypass the feedback resistor,
serves to illustrate the inverting relationship (Figure 3).
Bypassing the feedback resistor is intended to limit noise
bandwidth, and it does indeed remove noise presented as an
input signal. However, the circuit will continue to pass
amplifier noise across the entire op amp bandwidth. Cf
shunts the signal supplied through R1 for the desired lowpass
roll-off of the op amp’s eo/ei response. To the op amp noise
voltage, en, Cf merely presents the unity feedback of a
voltage-follower. Noise gain drops to unity but continues
out to the open-loop roll-off of the op amp. This leveling off
of 1/β also shows why the op amp must be unity-gain stable,
even though the circuit gain has been rolled off well below
the amplifier response. With 1/β following the unity gain
axis, the critical intercept occurs at fc.
LOG 1/ β = VDB (o) – VDB (j)
FIGURE 2. To determine the 1/β curve for the generalized
circuit of (a), you can draw a voltage-divider
circuit that represents the feedback network and
the shunting effects of the amplifier input (b).
VDB(o) – VDB(j). By adding this curve to the plot of the
amplifier’s gain-magnitude response, you can display the
characteristics of the critical intercept for subsequent feedback interpretation.
While the continued noise gain is at a lower level, it covers
much of the amplifier bandwidth, which can result in a
dramatic increase in output noise. For example, if you’re
using the 2MHz Burr-Brown OPA111 shown and choose Cf
to obtain a 2kHz roll-off, only 0.1% of the amplifier bandwidth will be enclosed in the intended system response.
Although the logarithmic scale of the frequency axis may be
visually deceptive, the remaining 99.9% of the bandwidth is
still available to the amplifier’s voltage noise. For an initial
gain of 10, the output noise that this amplifier produces is
more than doubled by the bandwidth effect. Many activefilter configurations are subject to the same limitation.
The 1/β curve also communicates performance information
across the entire response range of the op amp. For example,
it displays loop gain, which provides an indication of gain
accuracy vs frequency and the ultimate bandwidth limit.
Furthermore, the 1/β curve demonstrates that the circuit’s
signal bandwidth can be different from its noise bandwidth.
Note that the previous feedback-network analysis returns Z1
to ground as it would in a noninverting op amp configuration, even though the op amp shown is in the inverting mode.
The only way to avoid excessive noise bandwidth is to
restrict the frequency range of the op amp. By doing so, the
control of the noise response switches from the 1/β curve to
the amplifier roll-off. Where the op amp has provision for
external phase compensation, this control is a simple matter
and permits you to remove bandwidth from signal and noise
Underlying the difference between noise and signal bandwidth is the concept of noise gain, which is the source of
some of the more common op amp application problems.
alike. However, because most op amps lack an externalphase-compensation facility, passive filtering within the
feedback loop offers a broader solution(3). Such filtering
introduces a capacitive shunt to ground following the amplifier but within the feedback loop.
intercepts. Not only does the 1/β curve intercept the gainmagnitude curve at the high-frequency extreme, but it does
so at the low-frequency end as well. Each intercept indicates
a lack of amplifier gain for support of the feedback and a
departure from the ideal response. At the high end, the 1/β
curve and noise-gain level off, leaving ACL to continue as
long as the loop gain lasts. Next, 1/β intercepts the gainmagnitude curve at fc where the noise gain rolls off.
You can also demonstrate the extended noise bandwidth of
an integrating converter using a feedback plot but, more
importantly, the curves illustrate the dynamic-range limit for
integrator-based instrumentation. In Figure 4, the integrator
1/β curve also levels off at the unity-gain line for continued
noise gain out to where the op amp rolls off. Be aware that
this action has far less noise significance for the integrator
because of its increasing gain at lower frequencies. Integrators designed for operation to 1kHz or even higher are
generally unaffected by the added noise bandwidth.
This intercept is a high-frequency 3dB point for the integrator response, which then usually rolls up rather than down.
Upward response in this region is due to signal feedthrough
caused by the feedback elements in the absence of loop
control. At the lower frequencies, the increasing gain demand encounters the DC gain limit of the op amp. This
intercept marks the second 3dB point for the integrator
response, which sets the range for accurate performance.
Both intercepts have a 20dB/decade rate-of-closure, indicating stable operation.
Nevertheless, the feedback plot for the integrator demonstrates a unique bandwidth limitation involving two critical
1/2 π R 2 Cf
|A CL |
|A CL |
FIGURE 4. Defining the dynamic range for integrating data
converters, the integrator 1/β curve displays
upper and lower intercepts with the gain-magnitude response.
FIGURE 3. Highlighting the difference between closedloop gain and noise gain, this inverting op amp
configuration demonstrates the greater bandwidth that is often available to amplifier noise.
Between the two integrator-response limits is the usable
dynamic range for dual-slope A/D and V/F converters. The
gain error limits this dynamic range; the plots are a graphic
representation of this error. The gain error is inversely
related either to a circuit’s loop gain or the difference
between the amplifier’s open-loop gain and the feedback
demand of 1/β. On the response plots, the loop gain is the
vertical distance between the two curves. For the Figure 4
integrator, this separation decreases following 1/βs encounter with the unity-gain axis. From there, the separation
finally reduces to zero at fc. The gain error then becomes the
distance between the dashed continuation of the ideal integrator response and the actual ACL response. Graphically,
this distance is the source of the large-signal limitation for
integrating converters where higher signals correspond to
the upper frequencies.
with the voltage divider formed by the feedback resistors,
you can achieve the results of the 1/β curve in Figure 5. This
curve rises at high frequencies, increasing the rate-of-closure and flagging the need for closer stability analysis. The
phase margin drops as 1/β rises and, at the limit, goes to zero
if the 1/β rise spans one decade of frequency. Generally, the
span is much smaller than that and the Bode phase approximation evaluates the actual conditions. The key to minimizing the effect on the feedback factor is the low input
capacitance that the small input FETs of the OPA128 device
provide. The net 3pF of input capacitance leaves the response undisturbed until the parallel combination of the two
resistors reaches 50kΩ.
The capacitive bypassing of R2 increases the high-frequency
feedback, which counteracts the shunting of Cia by leveling
off the 1/β curve. The selection of this capacitor is better
illustrated by Figure 6’s photodiode amplifier. You can
reduce the non-obvious bandwidth of this application to an
equation. The circuit contends with diode capacitances at the
input up to 20,000pF. As a result, the break in the 1/β curve
is generally far removed from the intercept, making the rateof-closure analysis accurate without requiring any adjustment of the phase-shift approximation.
At the other end of the converter range, lower-level signals
demand low-frequency integrator operation that encounters
a similar limitation. Below the frequency of the op amp’s
first pole, fo, the separation between the 1/β and gain-magnitude curves again drops, signaling reduced loop gain.
Moving further down in frequency, the 1/β curve finally
crosses the op amp’s DC-gain level, and the actual response
flattens again. For integrating-type converters, this action
defines a range of performance that is accurate to within 3dB
from fc down to the lower intercept. To extend the dynamic
range, you move the lower intercept downward either with
a lower integrator-time-constant or with boosted DC gain.
A higher accuracy dynamic range results from the unique
loop-gain conditions of the integrator. The loop gain is
constant for the integrator from fo to its unity-gain crossing.
The gain error in this range is constant as marked by the
uniform separation of the gain-magnitude and 1/β curves. You
can compensate for such an error by making a fixed adjustment to the feedback network, leaving gain-accuracy bounded
by the stability of the network. This limit permits you to
adjust the more restricted dynamic range to 0.01% levels.
For the OPA111 op amp and a 100kHz integrator crossover
frequency, this more precise dynamic range has a span of
C ia
C ia = C id + C icm
The previous discussion of the inverter and the integrator
considered the feedback network independent of the amplifier input shunting. Although engineers frequently use this
simplification, they often encounter unexpected results.
Because of the feedback factor, most first-time users of op
amps with large feedback-resistance values are surprised by
the response curve. Transient-response ringing or even oscillation sometimes occurs; the common cure is a capacitive
bypass of the feedback resistor. The 1/β curve can display
the problem and provide some guidance in the selection of
the bypass capacitor.
FIGURE 5. Higher feedback resistances will react with the
op amp’s input capacitance to produce a peaking effect, which the 1/β curve anticipates.
Underlying the problem is the op amp input capacitance’s
effect on the feedback factor. By including this capacitance
The intercept with the gain-magnitude curve marks the end
of the response rise for the noise gain. This curve has a –
20dB/decade slope so, if left uncompensated, the rate-ofclosure at the intercept will be 40dB/decade. Thus, the plot
indicates two poles at that intercept frequency, fp. This intercept is the point at which there is no longer sufficient
amplifier gain for the feedback-factor demand, and it indicates response roll-off independent of the op amp function.
Any amplifier function would then roll off with a slope equal
to the rate-of-closure.
C ia
Because the rate-of-closure is 40dB/decade, you should
examine phase shift at the intercept to determine the phase
compensation necessary for stability. When the various
break frequencies are well removed from the intercept, the
rate-of-closure accurately reflects 180° of phase shift for the
uncompensated loop. To avoid oscillation and to achieve
good damping characteristics, you must reduce this phase
shift by at least 45° through roll-off of the 1/β curve.
f i ≈ 1/2π R f (C D + C ia)
C ia = C id + C icm
fp = fi fc
C f = 1/2π R f f p
BW ≈1.4f p
According to the Bode phase approximation, this phase shift
is the amount of phase introduced at a break frequency.
Choosing Cf to break with Rf at the intercept frequency, fp,
yields 45° of phase margin. Accompanying this phase condition is a 3dB peak in the signal response, which for a 2pole response pushes the –3dB bandwidth out to 1.4fp. For
the OPA111 and the feedback elements shown, the 3dB
response extends to 48kHz. (You can extend this analysis to
lower capacitance levels, and the common solution mentioned above will still suffice-even for the high-feedbackresistance case of Figure 5.)
As long as Cf breaks with Rf at the frequency of the intercept, the 1/β rise contributes no more than 45° of phase shift.
In the range where the op amp phase shift is 90°, this rise
leaves a stable 45° phase margin. Nevertheless, as the op
amp approaches its crossover frequency, fc, its contribution
to phase shift moves toward 135°. The rule of thumb for
selecting Cf remains valid, however, because any intercept
near fc must be a result of a 1/β rise of short duration. The
added phase shift of the amplifier, accompanied by a necessary decrease in feedback phase shift at the intercept, results
in a net zero effect. By simple sketching of the phase
approximations for the 1/β and gain-magnitude curves, you
can show this transition.
FIGURE 6. A photodiode amplifier’s voltage function serves
to obscure its bandwidth and stability, but you
can rely on feedback-loop conditions to define
its performance.
Unfortunately, the bandwidth for the circuit of Figure 6 is
obscured by its function. Because its function is a current-tovoltage conversion, rather than simple voltage gain, you
cannot draw the signal-gain curve on the gain-magnitude
response to estimate bandwidth. When you inspect the
circuit to find its bandwidth limitation, you’ll see that the
only inherent break frequency is that of the feedback resistance and the capacitance of the input circuit. By plotting the
1/β curve, however, you can see that the loop gain remains
to support the ideal feedback condition far beyond the fi
break frequency.
To select the compensation capacitance, it is desirable to
reduce the graphical analysis to an equation. Luckily, the
response plots provide an elegantly simple solution. Straightline extensions of the 1/β and gain-magnitude curves form a
triangle with the horizontal axis. These extensions have
equal but opposite slopes, which form an isosceles triangle.
The peak of the triangle, located over the center of its base,
lies at the average of the base end points. Mathematically,
this average point is equal to
Initially, the 1/β curve is flat at unity because of the direct
output-to-input connection of Rf. When the feedback is later
shunted by CD and Cia, 1/β rises at a 20dB/decade rate. The
transition between these regions occurs at
fi ≈ 1/2πRf(CD + Cia)
Log fp = (Log fi + Log fc)/2
where Cia = Cid + Cicm.
For very high-frequency amplifiers, like the OPA620
wideband amplifier of Figure 7, sufficient amplifier gain
exists at the resonant frequency to give the appearance of
zero gain margin. A comparison of the output signal (eo) with
that at the summing junction (ej) produces the plot’s gain
and phase responses. Following unity crossover, the gain
curve rises again above the unity axis; this rise generally
guarantees oscillation for lower gain levels. Adding to stability concerns is the phase plot, which swings wildly
through 180° during the gain peak.
C icm
C id
C icm
By adding the 1/β curve to the plot, you can see that this
curve does not intersect the gain peak but merely rides over
it. Without an intercept there is no oscillation, regardless of
the phase shift, because the loop gain is insufficient. Loopgain demand rises in synchronization with the gain peak
because the resonant circuit also alters the feedback network.
In many cases, the gain peaking results from conditions in
the amplifier output rather than from the input circuit. In
such a case, no corresponding modification of feedback
occurs, and an intercept and oscillation result. However, for
Figure 7, the gain margin remains high, as you can see by the
separation between the 1/β curve and the gain response when
the phase reaches 180°. This separation remains large
throughout the region of higher phase shift, indicating good
relative stability.
Whereas the normal op amp feedback loop involves only
one amplifier, designers often need to extend the feedback
loop to work with composite circuits that use two or more op
amps for increased gain. By adhering to conventional feedback principles, you can implement phase compensation for
the extended loop and rely on a Bode plot to provide a visual
representation of the increased gain and the opportunity for
extended bandwidth.
For instance, with two op amps in the same loop as in Figure
8, you can achieve increased gain without incurring any
added offset and noise error. The input-error effects of the
second amplifier are divided by the open-loop gain of the
first amplifier. The net open-loop gain of this composite
circuit becomes the product of the individual op amp gains
and greatly reduces the overall gain error and nonlinearity.
FIGURE 7. Although amplifier gain and phase plots suggest instability, the 1/β curve shows stable
conditions for a circuit with input-lead inductance.
Given the expressed logarithmic nature of the frequency
axis, you can reduce this relationship to the simple geometric mean of the two characteristic frequencies:
In Figure 8, the two op amps are those of the dual OPA2111,
which imposes only a modest cost increase over a single
device. You could, of course, select individual op amps to
provide specific performance characteristics. In the latter
case, you might select the input amplifier for good DC and
noise performance and the output amplifier for its loaddriving and slewing performance. For example, the output
amplifier could handle the load current and the resulting
power dissipation, thus producing no thermal feedback to
the input of the composite circuit. Moreover, it could also
fulfill the high-slew-rate demands of the application. The
input amplifier in this case would only swing through small
fp = √fi fc
where fi = 1/2πRf(CD + Cid + Cicm), and fc equals the unitygain bandwidth of the op amp.
For a third input-circuit effect, the 1/β curve demonstrates
stable conditions where typical gain and phase plots would
point to oscillation. In addition to input capacitance, op
amps have input inductance; this combination produces a
high-frequency resonance. The inductance is small but inescapable, being associated with internal input wires and being
compounded by external wiring.
3,000,000:1. Nevertheless, this restraint is a 30:1 improvement over a single op amp and moves the focus of measurement accuracy to other factors.
In the op amp test-loop application(4), the addition of the
second amplifier removes signal swing from the output of
the tested device. The extra gain transfers that voltage swing
to an isolated output and removes any gain error from the
signal detected at the tested amplifier’s inputs. This gainerror removal permits the discernment of other input-error
signals for the measurement of parameters such as powersupply and common-mode rejection. If these parameters
were to approach the level of the amplifier’s open-loop gain,
gain-error signals at the input would cloud the effects of the
measured parameter.
BW = f p =
With a composite op amp structure, you must include the
roll-off characteristics of both amplifiers in your AC analysis and have some means of providing phase compensation
for the loop. Two op amps in a common loop invite oscillation; the individual amplifier poles combine for a composite
2-pole roll-off. As shown in Figure 8, the logarithmic scale
makes the initial composite-response curve the linear sum of
the two individual responses. The upper, dashed response
curve, which has a –40dB/decade slope, shows this result.
Two methods are available for compensating the composite
loop. One modifies the gain-magnitude response and the
other alters the 1/β curve. The more usual of the two approaches is to reduce the slope of the gain-magnitude curve
in the vicinity of the intercept, as Figure 8 does. After
forcing the compensated response to roll off earlier, the
gain-magnitude curve returns with a more gentle slope to the
boundary of the uncompensated response. This action serves
the general-purpose requirements of voltage-gain applications and produces a stable range that you can place almost
anywhere in the total composite-gain range.
1/2 π R 4 C1
|A CL |
FIGURE 8. To utilize the boosted gain of the composite
amplifier, traditional phase-compensation techniques tailor the gain-magnitude slope to obtain
a stable region with a 1/β intercept.
Figure 8 achieves this compensation by creating a modified
integrator response for IC1B. Because this integrator is an
inverting circuit, the inputs of IC1A are reversed to retain
only one phase inversion in the loop. Capacitor C1 blocks the
local DC feedback, and the overall gain is still the product
of the two open-loop gains. The integrator response that R3
and C1 established for IC1B rolls off this composite gain.
Next, the first open-loop pole of IC1A returns the compensated response slope to –40dB/decade. At a higher frequency, a response zero provides the region of reduced slope
thanks to the inclusion of R4. Above the break frequency of
R4 and C1, R4 transforms the response of IC1B from an integrator to an inverting amplifier with a gain of –R4/R3.
An integrator and a common op amp test loop can demonstrate the benefits of using a composite amplifier. By extending the composite open-loop gain to higher levels, you can
expand the dynamic range for integrating analog functions.
The low-frequency intercept moves back by a factor equal to
the added amplifier gain. This change is so extreme that
other error effects will surface well before the gain error
from the intercepts. For two op amps in the same loop each
having 100dB open-loop gain, the composite gain is 200dB.
At this gain level, an input error of 1nV will develop the full
10V output swing. Long before the circuit reaches that limit,
noise becomes a prime AC constraint and typically restrains
lower-level accuracy over a dynamic range of about
Where this gain is unity, the compensated response drops to
and follows the open-loop response of IC1A as shown. For
gain levels other than unity, you have different options,
which you can explore by using other response plots and
defining the particular stable conditions you have in mind.
Having control of this gain becomes particularly useful as
the 1/β intercept approaches the uncompensated unity-gain
crossover point. In this region, the second poles of the two
op amps increase the phase shift. In such cases, you have to
make the magnitude of the internal R4/R3 gain less than unity
to force the compensated response to cross over earlier.
Generally, when you have two op amps of the same type,
making R4 = R3/3 will yield a unity-gain stable composite
The net phase correction that you can achieve with this
technique depends on the frequency-response range for
which you maintain the –20dB/decade slope. This span
begins with the R4C1 break frequency and ends with the
intercept of the composite open-loop response. After this
intercept, the lack of open-loop gain returns the response to
that of the uncompensated composite amplifier. To ensure a
phase margin of 45° or more, you can use the guidance that
the Bode phase approximation provides; the plot shows that
this reduced slope region must last for three decades of
frequency and must intercept the 1/β curve after running for
at least a decade.
Cf =
10A CL
2 π R2 f c
BW = f´p =
10A CL
Although most engineers are familiar with this type of phase
compensation, it is too restrictive of bandwidth at higher
gains. For applications requiring higher gains, you can
greatly extend the bandwidth and reduce the settling time by
40:1 by using a different phase-compensation technique.
The general-purpose R4 = R3 case of Figure 8 sets a constant
closed-loop gain-bandwidth product. Looking at the curves,
you can see that the closed-loop bandwidth is the same as
that for IC1A itself when BW = fp = fc/ACL. Even so, the large
separation between the compensated and uncompensated
responses shows a significant sacrifice in bandwidth—expressly for the accommodation of phase compensation.
Uncompensated, the gain-magnitude response has a gainbandwidth product that increases with closed-loop gain and
that provides a potential bandwidth of fp = fc/√ACL. Comparing the last two expressions shows that the potential for
bandwidth improvement equals √ACL, which is significant at
higher gains.
f´p =
π R 2 Cf
|A CL |
FIGURE 9. For greater bandwidth in high-gain circuits, you
can provide phase compensation for the 1/β
response to retain a smooth open-loop response
for fast settling.
An integrator configuration, on the other hand, with its
special characteristics, inherently produces the –20dB/decade slope for 1/β and achieves optimum bandwidth and
dynamic range.
You can take advantage of quite a bit of this bandwidthimprovement opportunity by compensating the 1/β curve
instead of the gain-magnitude response curve. By referring
back to the rate-of-closure stability criteria discussed previously, you would see that both curves contribute to the rateof-closure parameter even though the gain-magnitude curve
is generally the focus of phase-compensation efforts. To
satisfy the rate-of-closure criteria, all that is necessary is to
control the difference between the slopes, regardless of the
slopes of the individual curves. So, instead of reducing the
gain-magnitude slope, increase the 1/β slope (Figure 9). A
simple capacitive bypass of feedback resistor R2 accomplishes
this slope increase for a final 20dB/decade rate-of-closure.
Two factors distinguish this feedback-factor compensation
technique for higher gains. Greater bandwidth is open for
reclaiming, and the associated 1/β curves are well above the
unity-gain axis. From higher levels, the 1/β roll-off is developed well before its intercept with the gain-magnitude
curve. Starting this roll-off a decade ahead of the final
intercept produces a 45° phase adjustment for a like amount
of phase margin. The slopes of the two curves show that, in
order to accomplish this phase adjustment, Cf must break with
R2 one-half decade below the initial intercept frequency, fp.
Then, the 2:1 difference in slopes will place the new intercept one-half decade above fp for the required full decade of
the 1/β roll-off.
Again, the design equations for the required value of Cf and
the resulting bandwidth are obvious from the logarithmic
nature of the frequency axis. Setting fp’ at one-half decade
below fp implies that
Log fp’ = (Log fp + Log fp/10)/2
for which fp’ = fp/√10. From before, you’ll remember that fp
= fc/√ACL describes the uncompensated curve’s bandwidth.
The compensated bandwidth is
BW = fp’ = fc/√(10ACL)
Here, fc is the unity-gain crossover frequency of the composite gain-magnitude response. As becomes obvious when
you examine this expression, the improved bandwidth falls
short of the total potential by √10. However, it is better than
the Figure 8 result by √(ACL/10), or a factor of 10, for a gain
of 1000. Setting Cf to break with R2 at fp’ defines the value
of this capacitor as
Cf = √(10ACL)/2πR2fc
For the op amps of the dual OPA2111 shown, the gain-of1000 bandwidth becomes 20kHz as compared with the 2kHz
you’d realize if you used just one of the op amps.
Settling time also improves when you choose the composite
amplifier’s 1/β curve for phase compensation. The improvement is a result of both the increased bandwidth and the
retained constant gain-magnitude slope. For a single amplifier of the OPA2111 type, for a gain of 1000, the settling
time would be 700µs to 0.01%. Because the Figure 9
amplifier has 10 times the bandwidth of a single amplifier,
the settling time drops by the same factor to 70µs. This
improvement would not be possible without the smooth and
continuous slope of the compensated-amplifier response. A
response having an intermediate pole and zero, such as
Figure 8 does, has low-frequency response terms that are
slow to settle following a transient. Known as an integrating
frequency doublet, this pole/zero combination is notorious
for its poor settling time(5). By providing phase compensation for the 1/β curve, you ensure that the smooth gainmagnitude curve is left undisturbed, therefore achieving the
optimum settling time.
making Cf too large will cause instability. As Cf increases,
the resulting intercept moves toward fc and encounters the
added phase shift of the secondary-amplifier poles. Even
greater values of Cf will drop the 1/β curve to its limit at the
unity-gain axis. From there, it proceeds along the axis to the
magnitude-curve intercept that guarantees oscillation. Only
a range of compensation-capacitor values provides stability
with this second approach; the 1/β curves display this range
for sensitivity-analysis purposes. Because of the capacitor’s
window of stable values, a random selection of Cf followed
by a stability test is likely to miss the bandwidth opportunity
of this technique.
Another concept fundamental to op amp feedback in composite-amplifier circuits becomes apparent when you examine phase shift and stability. Composite amplifiers such as
the one in Figure 10 produce a –40dB/decade slope over
wide ranges both before and after the 1/β intercept. Because
this slope corresponds to a 180° phase shift, frequent concern over stability conditions arises at points other than that
of the critical intercept. Beyond the 1/β intercept, the loop
gain is less than 1 and therefore it is easy to see that the
circuit cannot sustain oscillation. Yet, prior to the intercept,
the gain of the feedback loop is very high and would seem
capable of causing the circuit to oscillate.
In reality, the high loop gain is a protection against, rather
than a promoter of, oscillation. Sustained oscillation depends on the op amp’s gain-error signal. In Figure 10, the
gain error, eo/A, appears between the op amp inputs and
receives amplification from the closed-loop gain, ACL. Here,
ACL is that of the noninverting configuration, the noise gain
that reacts with any input-referred error signal. To sustain
oscillation, the amplified error signal must independently
deliver the output signal. This action requires that (–eo/A)ACL
= eo. Note that eo appears on both sides of this equation; it
should therefore be obvious that any solution must conform
to very specific constraints. This equation expresses both
polarity and magnitude constraints; the composite amplifier’s
180° phase shift satisfies the sign change.
At lower gains, the benefit of the 1/β compensation technique diminishes as does its control of phase. Because lower
gains have 1/β curves closer to the unity-gain axis, they have
less room for 1/β roll-off. To produce an intercept with the
gain-magnitude curve after a decade of 1/β roll-off requires
a minimum closed-loop gain of 10. Op amp phase shifts
impose further limits by growing from 90° to 135° as they
approach the unity-gain crossover frequency. In the practical
case, this phase-compensation method needs gains of 30 or
more for good stability.
For the magnitude constraint, two possible solutions exist.
The first is eo = 0, which is the stable state for the composite
amplifier in the questioned region. There, the loop gain
makes the signal eo/A too small to independently support an
output signal. In the plots of Figure 10, eo/A starts at a very
low level due to the high loop gain at low frequencies. As
you move up in frequency, the gain-error signal rises while
the amplifier-response slope signals its polarity inversion
through the 180° phase shift. This inversion increases the
output signal but cannot sustain it until the gain-error signal
reaches a sufficient level. This critical level is a prerequisite
for oscillation.
This type of phase compensation does have an unusual
aspect: Too great a compensating capacitance will have a
surprising effect. Whereas increasing such capacitance normally yields more damping and a more stable response,
This level applies to the second solution for the magnitude
constraint. At this level, A/ACL has unity magnitude and
maintains the balance for the previous feedback equation’s
magnitude requirement. Unity loop gain occurs at the 1/β
eo = ACL(–eo /A)
eo = –10
R2 e1
R1 e2
|A CL |
FIGURE 11. For this common analog divider, a variable
feedback magnitude allows a range of conditions that define bandwidth and stability.
e o /A
frequency characteristics of 1/β become variables. Fortunately, the gain- and feedback-response curves offer a means
of quickly evaluating the range of conditions resulting from
the changing feedback.
FIGURE 10. A phase shift of 180° causes oscillation only
where the gain-error, eo/A, is capable of independently supporting the output signal.
The most common way to provide magnitude variation in
the feedback factor is to use a low-cost analog divider
realization. Placing a multiplier in the feedback loop of an
op amp (Figure 11) makes feedback a function of a second
signal and therefore produces divider operation. With signal
dependent feedback, the bandwidth and stability conditions
also become variables.
intercept where the open-loop and noise-gain curves meet.
Without phase-compensation intervention, this intercept satisfies both the phase and magnitude requirements for oscillation. Beyond this point, eo and A fall off together, leaving
the eo/A signal constant and unable to support oscillation
with the reduced gain. At the point where the magnitude of
the gain error and the feedback phase shift must both reach
specific levels to support oscillation, the intercept becomes
critical. Before or after the intercept, the loop phase shift can
be at any level and the gain-error magnitude will not be
sufficient to cause instability.
Figure 11 shows the divider connection(6) and demonstrates
the effect of voltage-controlled feedback on 1/β. The
amplifier’s feedback inverts the function of the multiplier by
placing the feedback signal under the control of the e2 signal.
Then, the multiplier’s transfer function of XY/10 delivers
eo(e2/10) to R2. This action scales the feedback signal by
comparing e2 to 10V reference level to obtain
Unfortunately, despite the composite amplifier’s very specific requirements for oscillation, the greatly varied applications of op amps make this critical condition all to easy to
encounter. To contend with this problem, you can rely on the
1/β curve to present a visual prediction of the problem and
provide insight into a solution.
β = (e2/10) R1/(R1 + R2)
Some applications demand that you include a second active
element in the feedback loop to produce a varying feedback
factor. In these applications, both the magnitude and the
With the feedback factor under control of this signal, the
1/β curve moves across the full range of the gain-magnitude
response. As e2 nears zero, the 1/β curve approaches infinity,
leaving the op amp essentially in an open-loop configuration. At the other extreme, a full-scale 10V value for e2
delivers a feedback signal to R2 that equals eo almost as if the
multiplier were not present. Then, the net response is that of
a simple inverting amplifier with a feedback factor of
R1/(R1 + R2) and an inverting gain of –R2/R1.
Between the extremes, the variation of e2 moves the 1/β
curve from as low as the unity-gain axis to above the upper
reaches of the amplifier’s gain-magnitude curve. This variation moves the critical intercept and requires attention to the
rate-of-closure over the entire span of the gain-magnitude
response. If no significant multiplier phase shift exists, the
feedback will always resemble that of an inverting amplifier
for a zero 1/β slope, and you can ensure stability by just
using a unity-gain-stable op amp. You can then read the
range of bandwidth for the divider operation directly from
the moving 1/β intercept. For a given e2 range, the intercept
moves linearly with the signal, defining the corresponding
bandwidth range.
Other ways of providing variable feedback are also available. For example, you can have the signal control the
frequency-rather than the magnitude-characteristics of the
feedback. The result is a variable slope at the intercept, as is
the case with the voltage-controlled lowpass filter in Figure
12. The basic elements of the lowpass filter are the op amp,
the resistors, and the capacitor. If you replace the multiplier
with a short circuit, these elements form a fixed-frequency
roll-off. Essentially, this shorted condition is established
when e2 = 10V and when the gain through the multiplier is
unity. Capacitor C1 then breaks with R2 to define the filter
roll-off just as if the resistor and capacitor were directly in
For levels of e2 below full scale, the multiplier serves as a
voltage-controlled attenuator to effectively alter the filter
time constant. Attenuating the feedback voltage to R2 lowers
the signal current to the summing node, which has the same
effect as increasing the resistor’s value. Increased effective
resistance corresponds to a decrease in the resistor’s break
frequency with C1. This break defines the variable filter rolloff when
fp = e2/20πR2C1
The maneuvering of the 1/β curve through this operation
deserves closer inspection. The circuit exhibits a signaldependent transition between the two different loops, which
alternately control the feedback. At low frequencies, C1 is
effectively an open circuit, and the controlling feedback path
The multiplier also introduces phase shift that alters the net
phase shift around the feedback loop. Poles in the response
of the multiplier circuit are zeros in the inverse 1/β function,
causing the curve to rise at high frequencies. This rise moves
toward the critical intercept when the multiplier control
voltage, e2, increases. This rise has an impact on the rate-ofclosure, and the op amp must introduce a dominant pole to
maintain stability. For the components shown, the OPA111
dominates the circuit roll-off because of its 2MHz unity-gain
crossover frequency. This frequency is well below the 10MHz
bandwidth of the MPY634 multiplier, placing the op amp in
control. Other options that use a separate feedback path to
restrict the op amp bandwidth are also available(3).
fp =
R1 + R2
20π R2 C
FIGURE 12. In this filter circuit, variations in the frequency characteristic of 1/β make possible a
changing 1/β slope at the intercept.
is through the op amp and the multiplier. This composite
structure has resistive feedback that defines a signal gain of
–R2/R1 and a noise gain of (R1 + R2)/R1. The latter relationship equals 1/β at low frequencies and the curve of
interest starts at this level with a zero slope. At the highfrequency end, the composite structure is overridden when
C1 acts as a short circuit, which results in a unity feedback
factor around the op amp. This short circuit absorbs all
feedback current from R2 without any corresponding change
in the amplifier output voltage. The feedback loop of the
composite structure is then disabled, switching feedback
control to just the op amp. With C1 then providing a unity
feedback factor to the op amp, the 1/β curve follows the unitygain axis at high frequencies.
Once the 1/β levels are fixed at the extremes, the multiplier
determines the nature of the transition between the two. In
the transition region, feedback currents from R2 and C1
compete for control of the summing node of the op amp
input. The contest for dominance is analogous to the frequency-dependent control of impedance with a parallel RC
circuit. In both cases, the 3dB point, where each element
carries the same magnitude of current, defines the transition
of control. The Figure 12 filter achieves equal element
currents when the impedance of C1 and the effective impedance of R2 are equal. This equality defines the voltage12
controlled roll-off frequency of the filter as previously
expressed. At this frequency, 1/β also rolls off and drops at
–20dB/decade to the high-frequency limit of the unity-gain
operating range. In addition, however, you may sometimes
require a multiplier having a bandwidth much greater than
that of the op amp, as the two previous examples demonstrate. Without a wide-bandwidth multiplier, 1/β would
begin to rise near the higher-frequency intercepts and increase the rate-of-closure. The OPA111 avoids this complication when using the MPY634 multiplier by maintaining a
dominant op amp pole.
The stability conditions of the Figure 12 circuit depend on
the particular feedback loop or the combination of elements
that are in control at the intercept point. For the lowerfrequency filter cutoff frequencies illustrated, the op amp’s
bypass capacitor takes control before the intercept and
defines the relevant feedback conditions. Because the 1/β
curve follows the unity axis at the upper end, you can
guarantee stability by ensuring that the op amp be unity-gain
stable. For higher-frequency cutoff frequencies, the 1/β
transition moves toward the gain-magnitude curve of the op
amp. Circuit response cannot move beyond this limit, so the
op amp roll-off becomes the upper boundary of filter operation.
Other applications may involve feedback peaking and op
amps that are not unity-gain stable-log amps and active
filters, for example. For these and other variations requiring
feedback analysis, the test remains the same. Look for the
critical condition where the rate-of-closure is 40dB/decade.
Where conditions approach this level, conduct further analysis and compare phase-compensation alternatives for optimization.
1. Tobey, G. E., Graeme, J. G., and Huelsman, L. P.,
Operational Amplifiers: Design and Applications, McGrawHill, 1971.
When the cutoff frequency approaches this boundary, the
intercept rate-of-closure varies, prompting stability analysis.
First, the zero of the 1/β curve approaches the intercept,
where it increases the slope of the curve. Because this action
reduces the rate-of-closure, stability is improved and a more
detailed analysis is unnecessary. A continued increase in the
cutoff frequency moves the 1/β curve further to the right
where its pole interacts at the intercept. This break frequency
returns the rate-of-closure to 20dB/decade, thus retaining
stability. Beyond this point, the intercept occurs at the flat
lower end of the 1/β curve, and no further change in the rateof-closure takes place.
2. Bower, J. L., and Schultheis, P. M., Introduction to the
Design of Servomechanisms, Wiley, 1961.
3. Stitt, R. M., and Burt, R. E., “Moglichkeiten zur
Rauschunter-druckung,” Elektronik, December 1987, pg 112.
4. Lewis, E. D., “Compensation of Linear IC Test Loops,”
Electronics Test, May, 1979, pg 83.
5. Dostal, E. J., Operational Amplifiers, Elsevier Scientific
Publishing, Amsterdam, Holland, 1981.
6. Wong, Y. J., and Ott, W. E., Function Circuits: Design
and Applications, McGraw-Hill, New York, NY, 1976.
Utilizing these various feedback conditions and a unitygain-stable op amp, you can design a composite circuit that
fulfills its primary stability requirement over the entire
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any BURR-BROWN product for use in life support devices and/or systems.