Nested control

Until now, we have used the following strategy for the design of negative feedback amplifiers:

1. Design the ideal gain of the amplifier using a nullor as controller

2. Relate the controller requirements to those of the amplifier

3. Select an operational amplifier that meets the controller requirements

4. Perform frequency compensation of the low-pass cut-off

5. Apply biasing

6. If negative feedback biasing is applied, perform frequency compensation of the high-pass roll-off

This is a useful strategy for amplifiers comprising only one controller.

These amplifiers only exhibit {over-all feedback}.

However, there may be situations in which it is preferable to build a controller with cascaded local feedback amplifier stages. Each of these amplifier stages has its transfer accurately fixed by means of negative feedback, which gives the controller an accurately fixed gain and pole-zero pattern. In control theory, this is often referred to as nested control loops.

PID controllers with local feedback amplifiers

We will illustrate the application and design of controllers with local feedback amplifier stages with the design of a motor driver. Fig. 441 shows the application of the motor driver. The motor driver has to drive a voice coil motor with a current that is accurately related to the output voltage of a digital-to-analog converter (DAC). One of the motor terminals is connected to ground. The following requirements apply:

Fig. 441 Half-bridge motor current driver, dirven from a voltage output DAC.

1. The motor driver should be capable of driving multiple types of motors. The dynamic response of the motor driver should not depend on the properties of the motor. The small-signal impedance of the voice-coil motors can be represented by a series \(LR\) connection. The inductance \(L_{m}\) and the resistance \(R_{m}\) of the voice coil differ for each motor.

2. Digital settings may be applied to adjust the gain and the dynamic response for each motor type.

3. A single-pole high-efficiency power voltage amplifier with a voltage gain \(A_{v}(s)=\frac{A_{0}}{1+s\tau}\) is available and should be used for driving the motor.

Over-all feedback architecture

According to the design method discussed in Chapter Design of feedback amplifier configurations, we could design a single-controller active feedback current driver as shown in Fig. 442.

Fig. 442 Conceptual design of the motor driver amplifier.

In this amplifier structure, a sense resistor with a value \(R_{s}\) has been used to convert the motor current into a (floating) voltage. A differential voltage converts this voltage into a single-ended one that can be compared with the single-ended DAC output voltage. The gain of this differential amplifier equals \(A_{d}\). A nullator sets the zero condition for this comparison, while a norator provides the dependent motor current for satisfying this condition.

At a first glance, this seems to be a perfect solution. The ideal transconductance gain of the motor driver equals \(A_{d}/R_{s}\). The bandwidth can be designed by inserting a zero into the gain of the differential amplifier. This establishes bandwidth limitation with the aid of an active phantom zero. These techniques have been discussed in sections Bandwidth limitation with phantom zeros and Active phantom zeros, respectively.

The above design approach can also be used if the single-pole high-efficiency power amplifier is applied as the final stage in the controller. Fig. 443 shows the concept in which the impedance \(R_{m}+sL_{m}\) represents the small-signal motor impedance. If the nullor were to be implemented with a single-pole operational amplifier and an active phantom zero in the differential voltage amplifier, the gain would show three poles while the servo function would be second order.

The gain of a controlled source that implements the nullor has been taken as the loop gain reference.

One extra phantom zero will be required to give the servo function an MFM characteristic.

Fig. 443 Conceptual design of the motor driver amplifier.

Nested control

The design approach sketched above cannot always be applied. This is because active phantom zero compensation and bandwidth limitation with the aid of phantom zeros are two techniques that increase the effect of non-dominant poles. Particularly if the load impedance varies over a wide range, it may become difficult to establish robust control in this way.

A more robust control can be obtained by designing a controller with an accurate and programmable dynamic performance. This can be done as follows:

1. The static error of the transfer will be zero if the controller has an integrating character at low frequencies.

2. The bandwidth of the system will not depend on the load impedance if the product of the loop gain and the dominant poles does not depend on the motor impedance.

3. The pole due to the motor inductance can be compensated for in the controller. This can be done by creating a programmable zero in the transfer of the controller.

4. The low-frequency loop gain variation due to \(R_{m}\) can be compensated for by making the controller gain programmable.

When designed in this way, the controller gain \(A_{c}(s)\) becomes

\[A_{c}(s)=\frac{A}{s}\frac{R_{s}+R_{m}}{R_{s}}\left( s\frac{L_{m}}{R_{s} +R_{m}}+1\right) .\]

With this controller gain, we obtain a loop gain \(L(s)\):

The gain of a controlled source that implements the nullor has been taken as loop gain reference.

\[L(s)=\frac{A}{s}\frac{A_{0}}{1+s\tau}A_{d}.\]

If the pole of the voltage amplifier is dominant, the bandwidth \(B\) of the transfer can be obtained as

\[B=\frac{1}{2\pi}\sqrt{\frac{AA_{0}A_{d}}{\tau}}~\text{[Hz].}\]

This yields a design equation for the integrator gain. The poles of the servo function can be brought into MFM positions with the aid of a phantom zero in \(A_{d}\).

Fig. 444 Conceptual design of the motor driver amplifier.

Fig. 444 shows a more detailed concept of an inverting motor driver with nested control. The integrator gain is fixed with the aid of \(C_{i}\). The zero at \(s=-\frac{1}{R_{z}C_{i}}\) should coincide with the pole caused by the motor impedance. For this reason, \(R_{z}\) has been made adjustable. The controller gain can be adjusted with the aid of the relative potentiometer setting \(x\). A phantom zero at \(s=-\frac{1}{R_{c}C_{c}}\) brings the two poles of the servo function into MFM positions.

In terms of classical control theory, the part of the circuit that consists of the differential voltage amplifier with gain \(A_{d}\) and the nullor with \(R_{c}\), \(C_{c}\), \(C_{i}\) and \(R_{z}\) can be regarded as a PID-controller. The Proportional action has been implemented with \(R_{c}\) and \(R_{z}\), the Integration action with \(C_{i},\) and the Differentiation action with \(C_{c}\). The controller gain \(A_{c} =\frac{V_{o}}{V_{d}}\) can be obtained as

\[A_{c}=\frac{V_{o}}{V_{d}}=-A_{d}\frac{\left( 1+sC_{i}R_{z}\right) \left( 1+sC_{c}R_{c}\right) }{sC_{i}R_{c}}\frac{R_{b}+xR_{a}}{R_{b}+R_{a}} \frac{A_{0}}{1+s\tau}.\]

The gain of the amplifier can be changed by changing the value of \(R_{i}\) or of the DAC reference voltage \(V_{ref}\).

The transfer of the controller gain has been fixed accurately with the aid of local feedback amplifiers. The frequency of the zero that should cancel the pole of the load can be adjusted by making \(R_{z}\) programmable using a digitally controlled potentiometer. The controller gain can be adjusted with another digitally controlled potentiometer. Aside from the design equations that follow from the required transfer, the controller gain and the motor impedance, budgets should be defined for the influence of non-dominant poles. Non-dominant poles may be introduced by the nonzero output impedances the nonzero input capacitances of the amplifiers. The circuit should be dimensioned such that the influence of non-dominant poles is kept within acceptable limits.

Increasing bandwidth without adding dominant poles

Sometimes, it is not possible to find operational amplifiers that combine a sufficiently large gain bandwidth product with other performance aspects that are relevant for the controller. There are two approaches to solving this problem:

1. Split the over-all feedback amplifier into several cascaded over-all feedback amplifiers. This generally results in more design flexibility.

2. Use several cascaded amplifiers within the over-all feedback loop. In order to limit the number of dominant poles in this loop, apply local feedback or pole-splitting techniques in one or more of these amplifier stages.

Fig. 445 Dual stage transimpedance amplifiers:

A: A feedback transimpedance amplifier cascaded with a feedback voltage amplifier

B: An over-all feedback transimpedance amplifier with a two-stage controller.

Fig. 445 illustrates the two design approaches. If both circuits exhibit equal over-all transimpedance gain, equal second stage voltage amplifiers and equal first stage operational amplifiers, the noise performance of the circuit from Fig. 445B will be better than that of the circuit from Fig. 445A.

If, in the circuit from Fig. 445B, the bandwidth of the voltage amplifier is much larger than that of the servo function of the over-all loop, the voltage gain of this amplifier will contribute to the loop gain poles product of the over-all loop without adding dominant poles.

Interaction with other performance aspects

The application of over-all feedback with a single controller and phantom zero frequency compensation usually results in the best possible performance. However, if there are many dominant poles, this design approach may result in a too complex dynamic behavior. In those situations, nested loop control may be a useful design approach. With nested loop control, the impedances inserted into the signal path of the local feedback stages usually have a larger impact on other design aspects than the feedback impedances in the over-all design would have. Special care should be taken inserting integrators within an over-all feedback loop. Clipping of one of the other stages in the loop may result in so-called integrator windup, which may seriously degrade the overdrive recovery behavior.

Integrator windup is a term used in control theory to indicate the charge accumulated in the integrator during overdrive.