Optimum Feedback Amplifier Design For Control Systems

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There is an optimum error amplifier transfer function for any given loop crossover frequency and phase margin.
The optimum design is shown to be the one with the highest gain below
crossover and the lowest gain above crossover. There is a unique pole-zero
placement to achieve optimum performance. This pole-zero placement and
resultant circuit component values can be determined without trail-and-error.

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Abstract:
Many designers think of control system feedback loop compensation only in
terms of stable or unstable. In reality, there is an optimum error amplifier
transfer function for any given loop crossover frequency and phase margin.
The optimum design is shown to be the one with the highest gain below
crossover and the lowest gain above crossover. There is a unique pole-zero
placement to achieve optimum performance. This pole-zero placement and
resultant circuit component values can be determined without trail-and-error.
In addition, there is an optimum open loop crossover frequency (loop
bandwidth) or any given phase margin.

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Optimizing the Performance
Optimizing the performance of a feedback control loop can be done with techniques that are
simple, effective, and easy to apply. These techniques can be used on single or multiple loop
systems of any type, and will yield optimum results without the usual trial-and-error process.
Three new concepts are introduced: the concept of phase boost as the single variable of
importance in loop stability, K-Factor as a convenient method of defining both phase boost and
the shape of the Bode gain curve, and the concept of a Figure of Merit for evaluating the relative
performance of various loop compensations.

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Stability Criteria
A feedback loop will oscillate when there is a frequency at which the loop gain is unity and total
phase lag equals 360°. Stability is usually measured by two factors: Phase margin, which is the
difference between actual phase lag and 360° when the loop gain is unity (usually expressed in
degrees); and gain margin, which is the amount the gain has fallen below unity when the total
phase lag is 360° (usually expressed in dB).

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How to Stabilize a Loop
Almost any loop can be divided into two major portions. One is the power processing portion,
which takes a control signal as an input and outputs the variable to be controlled, which may be
voltage, current, torque, speed, position, temperature, etc. This portion is typically called the
modulator or plant. The other portion is the error amplifier, which compares a sample of the
controlled output to a reference, amplifies the difference, and outputs a control signal to the
modulator or plant. This portion is usually called the amplifier. Figure 3 shows the definition of
these two blocks in a typical switch-mode power supply.

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Coincident Zero-Pole Pair Placement
For Type 2 amplifiers, there is little question about placement of the zero-pole pair. Required
phase boost sets the spread, and the zero and pole are symmetrically placed on either side of
desired unity gain frequency of the overall loop. For a Type 3 amplifier, optimum zero-pole
placement is not so obvious. To understand why the two zeroes need to be at the same frequency
and the two poles need to be at the same frequency, consider the Bode gain plot of Figure 10 and
the optimum design consideration that the low frequency gain should be as high as possible. If the
two zeroes or the two poles are spread apart in frequency, the phase boost curve in Figure 10
becomes wider and the amount of phase boost is reduced. To obtain the same amount of phase
boost (required to maintain phase margin), the two –1 gain slope regions must be spread farther
apart, reducing the low frequency gain of the amplifier and increasing the high frequency gain.
Since the overall loop gain is the product of the amplifier gain and plant or modulator gain, the
overall loop gain will be affected in the same way.

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Conclusion
A simple, straightforward method exists for achieving optimum control system performance
without trial-and-error. A new concept, the K-Factor, can be used to conveniently describe the
shape of the transfer function of any error amplifier. K-Factor can also be used to calculate actual
component values to use for system frequency response compensation. And finally, K-Factor can
be used in a Figure of Merit calculation to determine the optimum bandwidth for any control
system.

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lhlbluesky

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lhlbluesky

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