Loop gain

The product AVOLβ which occurs in the above equations, is called loop gain, a well-known term in feedback theory. The improvement in closed-loop performance due to negative feedback is, in nearly every case, proportional to loop gain.
The term "loop gain" comes from the method of measurement. This is done by breaking the closed feedback loop at the op amp output, and measuring the total gain around the loop. In Fig. 1-6 for example, this could be done between the amplifier output and the feedback path (see arrows). Approximately, closed-loop output impedance, linearity, and gain instability errors reduce by the factor AVOLβ , with the use of negative feedback.
Another useful approximation is developed as follows. A rearrangement of Eq. 1-9 is:
image
So, for high values of AVOLβ,
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Consequently, in a given feedback circuit the loop gain, AVOLβ, is approximately the numeric ratio (or difference, in dB) of the amplifier open-loop gain to the circuit closed-loop gain.
This loop gain discussion emphasizes that indeed, loop gain is a very significant factor in predicting the performance of closed-loop operational amplifier circuits. The open-loop gain required to obtain an adequate amount of loop gain will, of course, depend on the desired closed-loop gain.
For example, using Equation 1-14, an amplifier with AVOL = 20,000 will have an AVOLβ ≈ 2000 for a closed-loop gain of 10, but the loop gain will be only 20 for a closed-loop gain of 1000. The first situation implies an amplifier-related gain error the order of ≈0.05%, while the second would result in about 5% error. Obviously, the higher the required gain, the greater will be the required open-loop gain to support an AVOLβ for a given accuracy.
Frequency Dependence of Loop Gain
Thus far, it has been assumed that amplifier open-loop gain is independent of frequency. Unfortunately, this isn't the case at all. Leaving the discussion of the effect of open-loop response on bandwidth and dynamic errors until later, let us now investigate the general effect of frequency response on loop gain and static errors.
Figure 1-7: Op amp closed-loop gain and loop gain interactions with typical open-loop responses
Figure 1-7: Op amp closed-loop gain and loop gain interactions with typical open-loop responses
The open-loop frequency response for a typical operational amplifier with superimposed closed-loop amplifier response for a gain of 100 (40dB), illustrates graphically these results, in Figure 1-7. In these Bode plots, subtraction on a logarithmic scale is equivalent to normal division of numeric data (The log-log displays of amplifier gain (and phase) versus frequency are called Bode plots. This graphic technique for display of feedback amplifier characteristics, plus definitions for feedback amplifier stability were pioneered by Hendrick W. Bode of Bell Labs (Reference 6: Hendrick W. Bode, "Relations Between Attenuation and Phase In Feedback Amplifier Design," Bell System Technical Journal, Vol. 19, No. 3, July, 1940. See also: "Amplifier," US Patent 2,173,178, filed June 22, 1937, issued July 12, 1938)). Today, op amp open-loop gain and loop gain parameters are typically given in dB terms, thus this display method is convenient.
A few key points evolve from this graphic figure, which is a simulation involving two hypothetical op amps, both with a DC/low frequency gain of 100dB (100kV/V). The first has a gain-bandwidth of 1MHz, while the gain-bandwidth of the second is 10MHz.
  • The open-loop gain AVOL for the two op amps is noted by the two curves marked 1 and 10MHz, respectively. Note that each has a –3dB corner frequency associated with it, above which the open-loop gain falls at 6dB/octave. These corner frequencies are marked at 10 and 100Hz, respectively, for the two op amps.
  • At any frequency on the open-loop gain curve, the numeric product of gain AVOL and frequency, f, is a constant (10,000V/V at 100Hz equates to 1MHz). This, by definition, is characteristic of a constant gain-bandwidth product amplifier. All voltage feedback op amps behave in this manner.
  • AVOLβ in dB is the difference between open-loop gain and closed-loop gain, as plotted on log-log scales. At the lower frequency point marked, AVOLβ is thus 60dB.
  • AVOLβ decreases with increasing frequency, due to the decrease of AVOL above the open-loop corner frequency. At 100Hz for example, the 1MHz gain-bandwidth amplifier shows an AVOLβ of only 80–40 = 40dB.
  • AVOLβ also decreases for higher values of closed-loop gain. Other, higher closed-loop gain examples (not shown) would decrease AVOLβ to less than 60dB at low frequencies.
  • GCL depends primarily on the ratio of the feedback components, ZF and ZG, and is relatively independent of AVOL (apart from the errors discussed above, which are inversely proportional to AVOLβ). In this example 1/β is 100, or 40dB, and is so marked at 10Hz. Note that GCL is flat with increasing frequency, up until that frequency where GCL intersects the open-loop gain curve, and AVOLβ drops to zero.
  • At this point where the closed-loop and open-loop curves intersect, the loop gain is by definition zero, which implies that beyond this point there is no negative feedback. Consequently, closed-loop gain is equal to open-loop gain for further increases in frequency.
  • Note that the 10MHz gain-bandwidth op amp allows a 10× increase in closed-loop bandwidth, as can be noted from the –3dB frequencies; that is 100kHz versus 10kHz for the 10MHz versus the 1MHz gain-bandwidth op amp.
Fig. 1-7 illustrates that the high open-loop gain figures typically quoted for op amps can be somewhat misleading. As noted, beyond a few Hz, the open-loop gain falls at 6dB/octave. Consequently, closed-loop gain stability, output impedance, linearity and other parameters dependent upon loop gain are degraded at higher frequencies. One of the reasons for having DC gain as high as 100dB and bandwidth as wide as several MHz, is to obtain adequate loop gain at frequencies even as low as 100Hz.
A direct approach to improving loop gain at high frequencies other than by increasing open-loop gain is to increase the amplifier open-loop bandwidth. Figure 1-7 shows this in terms of two simple examples. It should be borne in mind however that op amp gain-bandwidths available today extend to the hundreds of MHz, allowing video and high-speed communications circuits to fully exploit the virtues of feedback.
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