lunes, 15 de febrero de 2010

Feedback linearization of cascode amplifier configurations

What is claimed is:

1. A linearized high voltage amplifier for providing a linearly amplified high voltage output of an input signal when coupled to a first fixed potential, a second fixed potential, and a third fixed potential, with said second and third potentials having respective differences from said first potential that are relatively small and large respectively, said amplifier comprising:
a first impedance having first and second terminals and means for coupling said third potential to the first terminal of said first impedance;
a second impedance that is a fraction of said first impedance having first and second terminals and means for coupling said second potential to the first terminal of said second impedance;
means including coupling means when coupled between said first and said second potentials for generating a constant current between said coupling means, said constant current generating means including input signal circuitry to which said input signal is applied and first and second output signal circuits for providing in response to signals applied to said input signal circuitry first and second output signals having currents which sum is said constant current, said first and second output signal currents having complementary variations at said first and second output signal circuits, respectively;
a first current conducting device having an input electrode coupled to one of said first and second output signal circuits of said constant current generating means and having an output electrode coupled to the second terminal of said first impedance
means for coupling the second terminal of said second impedance to said first output signal circuit for applying said first output signal of said constant current generating means to said second impedance to produce a voltage thereacross proportional to the current of said first output signal; and
degenerative feedback means coupled to said input signal circuitry and responsive to the voltage across said second impedance for providing a signal proportional to said voltage to said input signal circuitry to thereby linearize said first and second output signals.

2. The apparatus according to claim 1 wherein said first current conducting device is a bipolar transistor having base, emitter and collector electrodes.

3. The apparatus according to claim 2 including means for coupling said second potential to the base electrode of said transistor, and wherein said means for coupling said second potential to said second impedance comprises a connection between the first terminal of said second impedance and the emitter of said bipolar transistor.

4. The apparatus according to claims 1, 2, or 3, further comprising:
a third impedance of substantially the same impedance as said first impedance having first and second terminals and means for coupling said third potential to the first terminal of said third impedance;
a second current conducting device similar to said first current conducting device having an input electrode coupled to the other of said first and second output signal circuits of said constant current generating means from that coupled to the input electrode of said first current conducting device, having an output electrode coupled to the second terminal of said third impedance, and having a control electrode; and
means for coupling said second potential to the control electrode of said second current conducting device.

5. In a high voltage amplifier of the type for providing an amplified high voltage output of an input signal when coupled to a first fixed potential, a second fixed potential, and a third fixed potential, with said second and third potentials having respective differences from said first potential that are relatively small and large respectively, said amplifier including constant current generating means coupled between said first and second potentials for providing at first and second outputs thereof first and second current signals which current sum is a constant current, said means including control input circuitry and responsive to signals applied thereto for producing complementary variations of said first and second current signals at said first and second outputs, respectively; a current conducting device having input and output electrodes; a first impedance; and means coupling said current conducting device in series with said first impedance between said third potential and said first output of said constant current generating means; a linearizing means in combination therewith comprising:
a second impedance connected between said second potential and one of said first and second outputs for producing a voltage thereacross proportional to the current through said one of said first and second outputs; and
degenerative feedback means coupled to said input signal circuitry and responsive to the voltage across said second impedance for providing a signal proportional to said voltage to said control input circuitry to thereby linearize said first and second output signals.

Description:
The present invention relates to feedback linearization of cascode amplifiers.
A cascode amplifier is formed by common-emitter transistor amplifier followed in direct coupled cascade by a common-base transistor amplifier. W. M. Austin, in U.S. Pat. No. 3,449,104, issued Mar. 3, 1970 and entitled "Video Output Stage Employing Stacked High Voltage and Low Voltage Transistors," teaches the use of a relatively wide-bandwidth, low-voltage transistor in the common-emitter amplifier and a relatively narrow-bandwidth, high-voltage transistor in the common-base amplifier to secure an amplifier for providing an amplifier with both the relatively wide bandwidth and high-voltage capabilities. In his later U.S. Pat. No. 3,541,234, issued Nov. 17, 1970, and entitled "Video Circuits Employing Cascoded Combinations of Field Effect Transistors with High Voltage, Low Bandwidth Transistors," he extends his teaching to replace the common-emitter bipolar transistor amplifier with a common-source field effect transistor amplifier. Differential amplifier connections of cascode amplifiers are also known in which the common-emitter (or common-source) amplifier transistors are interconnected at their emitter (or source) electrodes. Refer, for example, to U.S. Pat. No. 3,482,177 (Sylvan), issued Dec. 2, 1969, and entitled "Transistor Differential Operational Amplifier;" U.S. Pat. No. 3,541,465 (Nagata et al.), issued Nov. 17, 1970, and entitled "Transistor Differential Amplifier Circuit;" or U.S. Pat. No. 4,121,169 (Iwamatsu), issued Oct. 17, 1978, and entitled "Amplifier Device." Such arrangements may be modified to apply input drive signal in unbalanced form to the base of only one of the "common-emitter-amplifier" transistors, the base of the other transistor being at signal ground to operate that transistor as a common-base (rather than common-emitter) amplifier transistor; and the term "cascode" as used in this disclosure is extended to include cascode of common-base amplifiers.
A differential amplifier connection of cascode amplifiers so modified was used by the present applicants for generating the deflection voltages for an electrostatically deflected cathode ray tube (CRT). The several hundred volt peak-to-peak deflection voltage requirement was met by using balanced drive from the collector circuits of the high-voltage common-base-amplifier transistors. The problem of linearizing feedback for such an amplifier had to be considered. Prior-art cascode connections used emitter degeneration resistance with the common-emitter transistor to afford current feedback, or used voltage feedback from the collector of the common-base amplifier transistor back to the base of the common-emitter transistor. The use of the former form of feedback is undesirable in differential amplifier configurations where input signal drive is of the unbalanced form described in the previous paragraph, since it causes the cascode amplifiers to have different levels of input signal coupled to them. The use of the latter form of feedback is undesirable in that it results in unequal output impedances at the common-base collector outputs and it also exposes low-voltage circuits to the hazard of application of destructive or damaging high voltage under unusual circumstances--e.g., during initial application of operating voltages or during an arc-over in the CRT.
The present invention is embodied in a cascode amplifier where the voltage to be used for linearization of a low-voltage amplifier transistor (as well as preceding amplifier circuitry, if desired) is taken across a resistor connecting its collector to the emitter of a high-voltage transistor in a following common-base-amplifier, which amplifier is inherently a linear amplifier insofar as current is concerned and will provide a linear output voltage to a fixed-resistance collector load.
In accordance with one embodiment of the present invention a linearized high voltage amplifier is disclosed which provides a linearly amplified high voltage output of an input signal. The amplifier is coupled to first, second and third potentials, wherein the difference between the first and second potentials is small and the difference between the first and third potentials is relatively larger. The apparatus includes a first impedance and the means for coupling the third potential to a first terminal of that impedance. A second impedance, smaller than the first, and means for coupling the second potential to a first terminal of the second impedance, are also provided. Means including coupling means when coupled between the first and second potentials for generating a constant current between the coupling means includes input signal circuitry to which the input signal is applied and first and second output signal circuits which provide, in response to the signals applied to the input signal circuitry, first and second output signals having currents which sum is the constant current. The currents of the signals at the first and second output circuits have complementary variations. The amplifier further includes a current conducting device having an input terminal coupled to one of the output signal circuits of the constant current generating means and having an output terminal coupled to the second terminal of the first impedance. Means for coupling the second terminal of the second impedance are provided for applying the first constant current generating means output signal to the second impedance to generate a voltage across it which is proportional to the current of the output signal. Finally, degenerative feedback means coupled to the input signal circuitry and responsive to the voltage across the second impedance provide a signal proportional to that voltage to thereby linearize the first and second output signals.
In The Drawing:
The sole FIGURE is a circuit diagram of a differential cascode amplifier providing linearizing feedback according to the present invention.
Referring to the FIGURE, the basic circuit comprises an operational amplifier 11 which drives an emitter-coupled differential amplifier 49, including transistors 12 and 13 and resistor 32, and a pair of grounded-base, high-voltage transistors 14 and 15, in cascode combination with the differential amplifier 49, which act as level shifters. A signal applied to input terminal 16 is reproduced at high amplification at output terminal 18 and the inverse of the signal at output terminal 18 appears at output terminal 20. Negative feedback responsive to the voltage developed across a resistor 31 in one cascode leg is applied via resistor 25 and variable resistor 48 to the summing point 51 input of operational amplifier 11, in accordance with the invention, assuring stable, linear gain for the frequency range of interest.
Input terminal 16 is connected via input resistor 23 to the inverting (-) input of amplifier 11. Operational amplifier 11 is suitably a type CA741 operational amplifier sold by RCA Solid State Division, Somerville, N.J., Having a non-inverting (+) input connected via resistor 24 to ground and an inverting (-) input connected to input resistor 23, feedback resistor 25 and bandwidth shaping capacitors 26 and 27. Amplifier 11 connects to a positive operating potential (e.g., +15 V) at terminal 36 and to a negative operating potential (e.g., -15 V) at terminal 47. Amplifier 11 produces an output voltage proportional to the voltage applied through the input resistors 24 and 23 connected to its non-inverting (+) and inverting (-) inputs, respectively. Because the non-inverting (+) input of amplifier 11 is tied through resistor 24 to ground, the output of amplifier 11 will respond in polarity opposite to the signal applied at the inverting (-) input.
The signal at the output of amplifier 11 is applied via resistor 28 to the base electrode of transistor 12, part of differential amplifier 49. Resistor 28 serves two functions: as a series resistor in the base circuit it acts as a parasitic suppressor, and in combination with capacitor 29 and resistor 30 it provides a break network to shape the frequency versus gain response of the overall circuit. Capacitor 29 and resistor 30 provide, at very high frequencies, a low-impedance path to ground which prevents possible high-frequency parasitic oscillators from being impressed on the signal driving the base of transistor 12. Capacitor 27, connected between the output and the inverting (-) input terminals of amplifier 11, is used to shape the high-frequency response of amplifier 11.
Emitter-coupled differential amplifier 49 comprises NPN transistors 12 and 13 and resistor 32. Transistors 12 and 13 are both suitably type 2N2219A sold by Motorola Semiconductor Products, Inc., Phoenix, Arizona. The interconnected emitters of transistors 12 and 13 connect to one end of resistor 32; the other end of resistor 32 connects via terminal 41 to a negative operating potential (e.g., -15 V). A fixed voltage maintained across resistor 32 ensures that a constant current flows through it. Hence, using Kirchhoff's Law, it can be seen that the sum of the currents through transistors 12 and 13 will be constant, thus characterizing differential amplifier as a constant current generator. A signal applied to the base of transistor 12 which results in current variations at the collector of the transistor must, by the symmetry of differential amplifier 49, cause variations in the current at the collector of transistor 13 which are complementary to those at the collector of transistor 12.
A voltage divider comprising resistors 33 and 34 connected in series between ground and the negative operating potential at terminal 41 provides a d-c voltage at the base of transistor 13. Capacitor 35, connected between the junction of resistors 33 and 34 and ground, bypasses any signal components appearing at the base of transistor 13 to signal ground. Resistor 36, between the base of transistor 13 and the junction of resistors 33 and 34, acts as a parasitic suppressor to prevent possible high frequency parasitic oscillations at that electrode.
The series combination of resistor 25 and variable resistor 48 provides negative feedback for the cascode of operational amplifier 11 and differential amplifier 49. The gain of the amplifier combination is determined by the setting of variable resistor 48. Because no substantial current flows into the inverting (-) input of amplifier 11, the current through input resistor 23 is substantially equal to the current through the feedback resistors 25 and 48. Therefore, the gain of the amplifier combination comprising operational amplifier 11 and differential amplifier 49 is equal to the sum of the resistances of resistor 25 and variable resistor 48 divided by the resistance of input resistor 23. Capacitor 26, connected between the inverting input of amplifier 11 and the output of the differential amplifier 49 at the collector of transistor 13, provides bandwidth shaping for the amplifier combination which includes operational amplifier 11 and differential amplifier 49.
HIgh-voltage NPN transistors 14 and 15 are connected in cascode combination with transistors 12 and 13, respectively, of differential amplifier 49. For the example, transistors 14 and 15 are each of the type 2N3439 sold by Semiconductor Technology, Inc., of Stuart, Florida, which have a rated breakdown voltage of 450 volts between collector and emitter. Where still larger output voltages require a larger d-c operating potential, transistors with a higher rated breakdown voltage may be selected.
Resistors 39 and 40 connect the collectors of transistors 14 and 15, respectively, to a terminal 22 to which a high-voltage (e.g., 300 V) operating potential is applied. One output of the amplifier of the instant embodiment, output B, is measured at the output terminal 18, with respect to the ground potential at terminal 19, and is one of the differential outputs of the overall amplifier. Output A which is the other differential output measured at output terminal 20 is connected to the collector of transistor 15 and is referenced to the ground potential at output terminal 21. Balanced output voltages developed across collector resistors 39 and 40 appear between terminals 19 and 18 and between terminals 21 and 20, respectively.
The base electrodes of level shifter transistors 14 and 15 receive a d-c bias voltage via parasitic supression resistors 37 and 38, respectively, from an adjustable potential divider comprising resistors 42 and 44 and potentiometer 43, in series between ground and terminal 45, to which a positive operating potential (e.g., 15 V) is applied. Capacitor 50 by-passes the tap connection of potentiometer 43 to ground for any signal components appearing at the base electrodes of transistors 14 and 15. In the example, the d-c bias voltage applied to the bases of transistors 14 and 15 is nominally +5 volts and can be used as a CRT deflection centering control.
Resistor 31 connecting the collector of transistor 13 and the emitter of transistor 15 is used to develop the feedback voltage which linearizes the cascade connection of amplifiers 11 and 49 and sets the quiescent current level through the branch of the output circuit comprising transistors 13 and 15. Resistor 31 is small in relation to collector resistor 40 and the voltage drop developed across it is proportionally smaller than the drop across resistor 40, by Ohms's Law. The voltage drop across resistor 31 is the difference between the voltage at the emitter of transistor 15 which, in the common-base configuration, is a fixed d-c bias, and the signal at the collector of transistor 13.
The degenerative feedback signal developed across resistor 25 and variable resistor 48 in response to the signal developed across resistor 31 is combined with the signal applied to input terminal 16 (via resistor 23) so as to generate a signal at the output of operational amplifier 11 which will maintain transistor 12 in its linear range. This ensures linear output signals from differential amplifier 49, resulting in linear amplification of the signal at input terminal 16 at outputs A and B.
The direct-coupled voltage feedback from the output of transistor 13 to the inverting (-) input of amplifier 11 maintains that input at the same potential (virtual ground) as applied to its non-inverting (+) input via resistor 24. So with zero voltage at input terminal 16, there is essentially no current flow through resistor 23 or into the input of operational amplifier 11, so the current through the feedback path, comprising resistor 25 and variable resistor 48, must be zero, to satisfy the requirement that the net current flow at the summing point 51 be zero. If no current flows through the feedback path, the voltage at the collector of transistor 13 must be zero volts to satisfy Ohm's Law.
Under this zero-voltage input condition, the voltage drop across resistor 31 is the voltage at the base electrode of transistor 15 less the base-emitter voltage drop across the transistor, the voltage drop across resistor 38 being negligibly small. The current flowing in the branch comprising transistors 13 and 15 is determined (in accordance with Ohm's Law) by dividing the drop across resistor 31 by its resistance, and the voltage at output terminal 20, output A, with respect to the reference ground at output terminal 21, may be determined (again in accordance with Ohm's Law) by computing the voltage drop across collector resistor 40 and subtracting that voltage drop from the voltage applied at high voltage terminal 22.
The current flowing in the branch comprising transistors 12 and 14 is the "tall" current demanded at the interconnected emitters of the transistors in differential amplifier 49 less the current in the branch comprising transistors 13 and 15, from Kirchoff's Law considerations. This "tail" current is the current flowing through resistor 32 and can be determined according to Ohm's Law by dividing the voltage drop across resistor 32 by its resistance. Voltage dividing resistors 33 and 34 provide a volage at the base of transistor 13 with reference to the voltage at the terminal 41. That base voltage, less the base-to-emitter voltage drop across transistor 13, is essentially the voltage which appears across resistor 32, the voltage drop across the parasitic suppression resistor 36 being negligible.
Knowing the current in the branch comprising transistors 12 and 14, one can calculate by Ohm's Law the potential drop across collector resistor 39. The voltage applied to high voltage terminal 22 less the voltage drop across resistor 39 yields the voltage at output terminal 18, output B, with respect to the reference ground at output terminal 19.
For positive voltages applied at input terminal 16, a current flows through input resistor 23. Because substantially no current flows into the operational amplifier 11, and because the voltage at summing point 51 is at virtual ground, the same current flowing through resistor 23 must flow through feedback resistors 25 and 48; hence the voltage at the collector of transistor 13 will drop to some negative value in response to positive voltage applied at input terminal 16. The voltage on the collector of transistor 13 may be quantitatively expressed as: V=-V in (R f /R in ),
where
V in is the input voltage,
R in is the input resistor 23, and
R f is the feedback resistance, which is the sum of resistor 25 and variable resistor 48.
As the voltage at the collector of transistor 13 drops from zero volts, under the zero input voltage condition, to a negative voltage, in response to a positive input voltage, the voltage across resistor 31 increases because the voltage at the emitter of transistor 15 is fixed. The results in increased current flow through the branch including transistors 13 and 15, resulting in a greater voltage drop across resistor 40 and hence a lower voltage appearing at output terminal 20 with respect to the ground potential at output terminal 21.
The increased current flow in the branch including transistors 13 and 15, in response to a positive voltage applied at input terminal 16, causes a decreased current flow in the branch comprising transistors 12 and 14, because of the constant-current requirement of differential amplifier 49. The decreased current flow results in a decreased voltage drop across collector resistor 39 and hence a higher voltage appearing at output terminal 18 with respect to the ground potential at output terminal 19.
A circuit as described in connection with the FIGURE was constructed and tested with components having the following values:
 ______________________________________ Resistor 23 100 K. ohms, Resistors 24 and 25 38.3 K. ohms, Resistors 28, 33, and 34 1000 ohms, Resistors 30, 37, and 38 47 ohms, Resistor 31 1330 ohms, Resistor 32 and 42 750 ohms, Resistor 36 100 ohms, Resistors 39 and 40 34 K. ohms 4 watts, Resistor 44 316 ohms, Variable resistor 48 0 to 50 K. ohms, Potentiometer 43 200 ohms, Capacitor 26 10 picofarads, Capacitor 27 47 picofarads, Capacitor 29 0.01 microfarads, Capacitors 35 and 50 1.0 microfarad. ______________________________________
Deriving the feedback signal from the low-voltage (emitter) side of level shifter transistor 15 provides the advantage of feedback stability. The voltage at the emitter of transistor 15 depends only on the d-c bias voltage applied at the base electrode. In this embodiment the common-base voltage is derived from a +15 volt supply; normally, a well-regulated power supply is used in this application. The voltage at the collector of transistor 15 depends on the high-voltage operating potential which rarely is a well-regulated voltage. Hence, greater stability is achieved in the circuit as a result of deriving the feedback signal from the more well-regulated voltage supply.
While the present invention has been described in the context of an amplifier arrangement with a pair of differentially connected cascodes to deliver balanced output signals, single-ended arrangements are also possible. Output A between terminals 21 and 20 can be dispensed with, eliminating collector resistor 40 and replacing high voltage transistor 15 with a low-voltage type with its collector returned to the +15 V supply, or dispensing with both collector resistor 40 and transistor 15 and returning the end of resistor 31 remote from the collector of transistor 13 to a positive operating potential relatively low compared to +300 V. In the alternative, output B can be dispensed with eliminating collector resistor 39 and transistor 14 and connecting the collector of transistor 12 directly to the +15 V supply.
The foregoing description has dealt with level shifter transistors 14 and 15 as bipolar transistors. However, it should be obvious to one skilled in the art to substitute field-effect transistors (FET's) for the bipolar types, with the gate, source and drain electrodes corresponding, respectively, to the base, emitter and collector electrodes. For that reason, claims which specify transistors should be construed as applying equally to either type of semiconductor device.

Fuente:http://www.freepatentsonline.com/4366446.html

Feedback amplifier having a voltage-controlled compensation circuit

Description

BACKGROUND OF THEINVENTION

The present invention relates generally to compensated amplifiers, and more particularly to a compensated feedback amplifier having a voltage-controlled compensation circuit.

Shunt and series feedback amplifiers are forms of amplifiers in which a resistive element is connected from output to input to establish a precise gain factor. Such feedback amplifiers are used in a wide variety of applications so as to bepervasive in the electronics industry. It is also typical to provide capacitive compensation to achieve stability and optimum transient response or frequency response. This compensation is typically implemented by placing a variable capacitor inparallel with the feedback resistor to allow optimum compensation to be adjusted. One problem associated with variable capacitors is that they typically introduce significant parasitic capacitances and inductances into the circuit to be compensated, andconsequently can be difficult to adjust for optimum transient response. Also, if the circuit involves more than one feedback capacitor, the problem is compounded. Moreover, it has been heretofore impractical to adjust feedback capacitance as a functionof temperature or voltage.

SUMMARY OF THE INVENTION

In accordance with the present invention, a compensated feedback amplifier includes a voltage-controlled compensation circuit which varies the effective feedback capacitance to provide precise compensation without the introduction of theparasitic capacitances and inductances associated with variable capacitors.

The compensation circuit includes two capacitors which are effectively connected in parallel to provide a desired nominal value of feedback capacitance from output to input of the feedback amplifier to be compensated. A first capacitor isconnected directly from output to input, which input is kept at virtual ground through operational amplifier action in the case of a shunt feedback amplifier. A second capacitor is connected from the output to a different virtual ground, but one fromwhich AC current from the capacitor is furnished in variably proportional amounts to the feedback amplifier virtual ground to complete the feedback loop. The proportionality and current steering of the current from the second capacitor is effected by amultiplier circuit, the bias of which is controlled by an adjustable DC voltage.

This compensation technique permits more than one feedback amplifier circuit or a differential amplifier circuit to be controlled with a single transient-response control voltage, which is particularly attractive for integrated circuit designswhich often incorporate more than one amplifier of similar design on a single semiconductor chip. Also, a temperature coefficient easily may be introduced in the controlling voltage to optimize amplifier transient response at various ambienttemperatures. A further benefit obtained by using a multiplier in the compensation circuit is that the multiplier is linear and provides linear control, which is not possible using other voltage-controlled capacitance methods, such as varactor networksor voltage-controlled diodes. This makes it possible to vary feedback capacitance linearly as a function of any input voltage. One example is to vary the compensation as a function of output voltage, such as z-axis drive in an oscilloscope.

It is therefore one object of the present invention to provide a novel method and apparatus for compensation of a feedback amplifier.

It is another object of the present invention to provide a voltage-controlled compensation circuit for a feedback amplifier which varies the effective feedback capacitance by routing a variable proportion of capacitor current to the amplifiersumming node.

It is a further object of the present invention to provide a compensation circuit for a feedback amplifier which permits linearly varying the effective feedback capacitance without introducing significant problem-causing parasitic capacitancesand inductances.

It is yet another object of the present invention to provide a compensation circuit capable of adjustably compensating one or more amplifiers of similar design by means of a single control voltage.

It is an additional object of the present invention to provide a compensation circuit capable of adjustably compensating differential feedback amplifiers.

Other objects, features, and advantages of the present invention will become apparent to those having ordinary skill in the art upon a reading of the following description when taken in conjunction with the accompanying drawings.

BRIEFDESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a conventional capacitive-compensated shunt feedback ampl ifier;

FIG. 2 shows a voltage-controlled compensation circuit for a shunt feedback amplifier in accordance with the present invention;

FIG. 3 shows a compensated push-pull amplifier in accordance with the present invention.

FIG. 4 shows a voltage-controlled compensation circuit for a series feedback amplifier in accordance with the present invention; and

FIG. 5 shows a shunt feedback amplifier in which compensation is a function of output voltage.

DETAILED DESCRIPTION OF THE DRAWINGS

Referring now to FIG. 1, there is shown a conventional compensated shunt feedback amplifier comprising an operational amplifier 10 with the inverting input thereof connected to an input terminal 12 and the non-inverting input thereof grounded,and the output connected to an output terminal 14. A gain-setting feedback resistor 16 is connected from the amplifier 10 output to the inverting input thereof, and a compensating capacitor 18 is connected in parallel with resistor 16. The input signalis in the form of a current input -IIN which is forced through resistor 16 to develop a voltage signal VOUT at output terminal 14. The typical method of achieving proper compensation is to apply a step function current to input terminal 12 andadjust capacitor 18 for a square corner on the leading edge of the output voltage step. With the non-inverting input connected to ground, the inverting input of amplifier 10 is held at virtual ground through operational amplifier action--that is,providing sufficient current through the feedback resistor (and capacitor under dynamic or AC conditions) to keep the inverting and non-inverting inputs balanced at the same potential.

FIG. 2 shows a compensated shunt feedback amplifier having a voltage-controlled compensation circuit in accordance with the present invention. Here, the shunt feedback amplifier including operational amplifier 10 is shown as described above inconnection with FIG. 1; however in place of capacitor 18 is a first feedback capacitor 20 having a value CFB1 and a second capacitor 22 having a value CFB2 such that ##EQU1## the desired value for the particular amplifier design. However, theeffective capacitance is provided by controlling the amount of current from capacitor 22 to the amplifier summing node 24 as will be described. Capacitor 20 is connected from the output of amplifier 10 to the summing node 24 at the inverting inputthereof. Capacitor 22 is connected from the output of amplifier 10 to the emitters of an emitter-coupled pair of transistors 30 and 32, which together with another emitter-coupled pair of transistors 34 and 36 and diodes 38 and 40 forms a multipliercircuit of the type taught by Gilbert in U.S. Pat. No. 3,689,752. The collectors of transistors 30 and 34 are connected together to the summing node 24. The collectors of transistors 32 and 36 are connected together to ground. Connected to therespective common emitters of transistors 30-32 and 34-36 is a constant current sink 44 and 46. A constant current source 48 is connected to the summing node 24 to furnish current to the collectors of transistors 30 and 34. Current sinks 44 and 46 andcurrent source 48 are all constant current generators of equal value IE , which amplies that a standing current of IE is also pulled from ground by the collectors of transistors 32 and 36.

The bases of transistors 30 and 36 are connected together to ground. The bases of transistors 32 and 34 are connected together to a biasing network including diodes 38 and 40, a resistor 50 connected from the common anodes of diodes 38 and 40 toa suitable positive voltage source, a resistor 52, and a potentiometer 54 connected between ground and a suitable negative voltage source -V.

To understand how the DC voltage produced at the wiper arm of potentiometer 54 controls the effective feedback capacitance, consider the following condition: Suppose the wiper arm is at the top, or ground end, of the potentiometer 54. In thiscondition, transistors 32 and 34 are conducting all of the current IE in each case while transistors 30 and 36 are off. Therefore, any current produced through capacitor 22 is conducted to ground through transistor 32, and, hence, capacitor 20represents the entire feedback capacitance because capacitor 22 is effectively out of the circuit. Now consider the situation in which the wiper arm is at the bottom, or -V end of potentiometer 54. Here, transistors 30 and 36 are conducting whiletransistors 32 and 34 are off, so that any current through capacitor 22 is connected to virtual ground at summing node 24 through transistor 30. At the midpoint of potentiometer 54, then, all four transistors 30-34 conduct equally so that currentthrough capacitor 22 is split equally between transistors 30 and 32. It now becomes apparent that as potentiometer 54 is adjusted throughout its range, current through capacitor 22 is shifted between transistors 30 and 32. In conclusion, it can bediscerned that the adjustment of potentiometer 54 controls that proportion of current through capacitor 22 that reaches the summing node 24, thus controlling the effective feedback capacitance. It should be pointed out that the control provided bypotentiometer 54 in shifting current between transistors 30 and 32 (where capacitor 22 current is affected) is linear due to the linearizing function of diodes 38 and 40.

A push-pull, or differential amplifier configuration embodying the compensation circuit of the present invention is shown in FIG. 3. Where the appropriate, the reference numerals of FIG. 2 have been retained, with prime symbols added tocorresponding components on the other side of the push-pull amplifier. An input differential current signal is applied to input terminals 12--12' and an output differential voltage signal is taken across terminals 14--14'. Rather than the collectors oftransistors 32 and 36 being connected to ground as for the single-ended configuration of FIG. 2, they are connected to provide positive feedback to summing node 24' of amplifier 10'. A capacitor 22' is connected from the common emitters of transistors34 and 36 to the output of amplifier 10'. A current source 48' is provided to supply the current demanded by the collectors of transistors 32 and 36. In all other respects, amplifier 10' and feedback components 16' and 20' are substantially identicalto amplifier 10, resistor 16, and capacitor 20, respectively. At the midrange setting of the compensation circuit, the positive feedback from capacitor 22 nulls the negative feedback from capacitor 22' . Thus, the adjustment range of effective feedbackcapacitance is CFB1 -CFB2 to CFB1 CFB2. It can be seen that adjustment of potentiometer 54 adjusts the compensation and transient response equally for both sides of the push-pull amplifier, which makes it particularly attractive toimplement this amplifier configuration in integrated circuit form.

FIG. 4 shows a series feedback amplifier incorporating a voltage-controlled compensation circuit in accordance with the present invention. This configuration may be compared with the shunt feedback amplifier configuration of FIG. 2, and only thedifferences will be described. Like reference numerals are used for like components. A voltage generator 60 to represent a signal source is coupled to the non-inverting terminal of amplifier 10. A resistor 62 is connected from the inverting terminalof amplifier to ground. In operation, the inverting and non-inverting terminals of amplifier 10 attempt to remain balanced, so the input signal is developed across resistor 62 and the current therethrough flows through resistor 16 to develop the outputvoltage. Thus, the output voltage is the same polarity as the input voltage, and the amplifier gain is determined by the ratio of the resistors 16 and 62. The compensation mechanism works in substantially the same way as for FIG. 2, although it must bepointed out that the junction of resistors 16 and 62 is not a virtual ground.

FIG. 5 shows a shunt feedback amplifier again; however, in this configuration the compensation is a function of output voltage. Here, the output voltage is coupled via a resistor 70 to the bases of transistors 32 and 34 to control the balance ofcurrent through the respective emitter-coupled pairs. Transient-response current is injected via capacitor 22 into the emitters of transistors 30 and 32 as before. Diodes 38 and 40 are connected slightly differently in this configuration to permit theoutput voltage to be applied to the bases of transistors 32 and 34 as well as to provide the linearizing function. Resistor 72 is added as a bias resistor. This amplifier configuration is ideally suited for voltage level-setting applications, such as,for example, z-axis or intensity drive for cathode-ray tubes. In this type of amplifier configuration, response is significantly slowed on larger amplitude output steps. It can be discerned, then, that a peaking effect will be exhibited by delivering agreater amount of AC current from capacitor 22 to ground, improving transient response of large positive-going voltage steps. Of course, for smaller steps, more of the AC current is delivered to summing node 24 via transistor 30.

While I have shown and described the preferred embodiments of my invention, it will be apparent to those skilled in the art that many changes and modifications may be made without departing from my invention in its broader aspects.

Fuente: http://www.patentstorm.us/patents/4481480/description.html

Un amplificador es diseñado para responder a tensiones o corrientes a la entrada y para suministrar tensiones o corrientes a la salida. En un amplificador realimentado, el tipo de señal muestreada a la salida (corriente o tensión) y el tipo de señal mezclada a la entrada (tensión o corriente) dan lugar a cuatro tipos de topologías: 1 ) realimentación de tensión en serie o nudo-malla o nudo-serie, 2) realimentación de corriente en serie o malla-malla o malla-serie, 3) realimentación de corriente en paralelo o malla-nudo o malla-paralelo, y 4) realimentación de tensión en paralelo o nudo-nudo o nudo-paralelo.

La realimentación (feedback en inglés) negativa es ampliamente utilizada en el diseño de amplificadores ya que presenta múltiples e importantes beneficios. Uno de estos beneficios es la estabilización de la ganancia delamplificador frente a variaciones de los dispositivos, temperatura, variaciones de la fuente de alimentación y envejecimiento de los componentes. Otro beneficio es el de permitir al diseñador ajustar la impedancia de entrada y salida del circuito sin tener que realizar apenas modificaciones. La disminución de la distorsión y el aumento del ancho de banda hace que la realimentación negativa sea imprescindible en amplificadores de audio y etapas de potencia. S in embargo,  presenta dos inconvenientes básicos. En primer lugar, la ganancia del amplificador disminuye en la misma proporción con el aumento de los anteriores beneficios. Este  problema se resuelveincrementando el número de etapas amplificadoras para compensar esa pérdida de ganancia con el consiguiente aumento de coste. El segundo problema está asociado con la realimentación al tener tendencia a la oscilación lo que exige cuidadosos diseños de estos circuitos.

Figura 1.- Diagrama de bloques de un circuito realimentado.

Teoria basica de realimentacion

pero como Xi=Xs-Xf, A=Xo/Xi y ß=Xf/Xo, fácilmente se comprueba que

La ganancia del amplificador realimentado Af es la ganancia del amplificador básico A dividida por el factor de desensibilidad D=1+ßA. La realimentación negativa se produce cuando ßA>0, luego Af < A ya que D>1 . La realimentación positiva se produce cuando ßA<0 y da lugar a circuitos no lineales. La teoría de realimentación exige considerar una serie de suposiciones para que sean válidas las expresiones que se van a obtener seguidamente. Estas suposiciones son
• La señal de entrada se transmite a la salida a través del amplificador básico y no a través de la red de realimentación.
• La señal de realimentación se transmite de la salida a la entrada únicamente a través de la red de realimentación y no a través del amplificador básico.
• El factor ß es independiente de la resistencia de carga (RL) y de la fuentes (RS).
En las dos primeras suposiciones se aplica el criterio de unidireccionalidad: Xs→Xo a través de A, Xo→Xf a través de ß. Estas suposiciones hacen que el análisis de circuitos aplicando teoría de realimentación y sin ella difieran mínimamente. Sin embargo, la teoría de realimentación simplifica enormemente el análisis y diseño de

Las variaciones debidas al envejecimiento, temperatura, sustitución de componentes, etc. . . , hace que se produzca variaciones en el amplificador básico y, por consiguiente, al amplificador realimentado. Este efecto puede ser analizado diferenciando la ecuación

resolviendo y aplicando calculo incremental resulta

Así, por ejemplo, si D=1+ßA=100 y A sufre una variación del 10% (∆A/A=0.1 ) entonces la ganancia del amplificador realimentado sólo varía en un 0.1% (∆Af/Af=0.001 ) . Con ello, se estabiliza la ganancia del amplificador realimentado a variaciones del amplificador básico.

Reducion de distorsion

La realimentación negativa en amplificadores reduce las características no lineales del amplificador básico y, por consiguiente, reduce su distorsión. Como ejemplo, en la figura 4.2.a se muestra la característica de transferencia en tensión no-lineal de un amplificador que presenta dos ganancias A1 y A2. La aplicación de una realimentación negativa reduce fuertemente esa distorsión tal como se describe en la figura 4.2.b. Más aún, si se verifica ßA1 ,ßA2>>1 , entonces la ecuación 4.5 indica que la distorsión puede ser eliminada al ser independiente de la ganancia del amplificador.

Routh–Hurwitz stability criterion

The Routh–Hurwitz stability criterion is a necessary (and frequently sufficient) method to establish the stability of a single-input, single-output (SISO), linear time invariant (LTI) control system. More generally, given a polynomial, some calculations using only the coefficients of that polynomial can lead to the conclusion that it is not stable. For the discrete case, see the Jury test equivalent.
The criterion establishes a systematic way to show that the linearized equations of motion of a system have only stable solutions exp(pt), that is where all p have negative real parts. It can be performed using either polynomial divisions or determinant calculus.
The criterion is derived through the use of the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices.

Using Euclid's algorithm

The criterion is related to Routh–Hurwitz theorem. Indeed, from the statement of that theorem, we have $p-q=w(+\infty)-w(-\infty)$ where:
• p is the number of roots of the polynomial ƒ(z) located in the left half-plane;
• q is the number of roots of the polynomial ƒ(z) located in the right half-plane (let us remind ourselves that ƒ is supposed to have no roots lying on the imaginary line);
• w(x) is the number of variations of the generalized Sturm chain obtained from P0(y) and P1(y) (by successive Euclidean divisions) where f(iy) = P0(y) + iP1(y) for a real y.
By the fundamental theorem of algebra, each polynomial of degree n must have n roots in the complex plane (i.e., for an ƒ with no roots on the imaginary line, p + q = n). Thus, we have the condition that ƒ is a (Hurwitz) stable polynomial if and only if p − q = n (the proof is given below). Using the Routh–Hurwitz theorem, we can replace the condition on p and q by a condition on the generalized Sturm chain, which will give in turn a condition on the coefficients of ƒ.

Using matrices

Let f(z) be a complex polynomial. The process is as follows:
1. Compute the polynomials P0(y) and P1(y) such that f(iy) = P0(y) + iP1(y) where y is a real number.
2. Compute the Sylvester matrix associated to P0(y) and P1(y).
3. Rearrange each row in such a way that an odd row and the following one have the same number of leading zeros.
4. Compute each principal minor of that matrix.
5. If at least one of the minors is negative (or zero), then the polynomial f is not stable.
Example
• Let f(z) = az2 + bz + c (for the sake of simplicity we take real coefficients) where $c\neq 0$ (to avoid a root in zero so that we can use the Routh–Hurwitz theorem). First, we have to calculate the real polynomials P0(y) and P1(y):
f(iy) = − ay2 + iby + c = P0(y) + iP1(y) = − ay2 + c + i(by).
Next, we divide those polynomials to obtain the generalized Sturm chain:

• P0(y) = (( − a / b)y)P1(y) + c, yields P2(y) = − c,
• P1(y) = (( − b / c)y)P2(y), yields P3(y) = 0 and the Euclidean division stops.
Notice that we had to suppose b different from zero in the first division. The generalized Sturm chain is in this case (P0(y),P1(y),P2(y)) = (cay2,by, − c). Putting $y=+\infty$, the sign of cay2 is the opposite sign of a and the sign of by is the sign of b. When we put $y=-\infty$, the sign of the first element of the chain is again the opposite sign of a and the sign of by is the opposite sign of b. Finally, -c has always the opposite sign of c.
Suppose now that f is Hurwitz-stable. This means that $w(+\infty)-w(-\infty)=2$ (the degree of f). By the properties of the function w, this is the same as $w(+\infty)=2$ and $w(-\infty)=0$. Thus, a, b and c must have the same sign. We have thus found the necessary condition of stability for polynomials of degree 2.

Higher-order example

The MATLAB script can be used to determine the stability of any nth-degree characteristic equation.
A tabular method can be used to determine the stability when the roots of a higher order characteristic polynomial are difficult to obtain. For an nth-degree polynomial
• $D(s)=a_ns^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0$
the table has n + 1 rows and the following structure:
 an an − 2 an − 4 $\dots$ an − 1 an − 3 an − 5 $\dots$ b1 b2 b3 $\dots$ c1 c2 c3 $\dots$ ... ... ... $\dots$
where the elements bi and ci can be computed as follows:
• $b_i=\frac{a_{n-1}\times{a_{n-2i}}-a_n\times{a_{n-2i-1}}}{a_{n-1}}$
• $c_i=\frac{b_1\times{a_{n-2i-1}}-a_{n-1}\times{b_{i+1}}}{b_1}.$
When completed, the number of sign changes in the first column will be the number of non-negative poles.
Consider a system with a characteristic polynomial
• $D(s)=s^5+4s^4+2s^3+5s^2+3s+6.\,$
We have the following table:
 1 2 3 0 4 5 6 0 0.75 1.5 0 0 −3 6 0 3 0 6 0
In the first column, there are two sign changes (0.75 → −3, and −3 → 3), thus there are two non-negative poles and the system is unstable.

Appendix A

Suppose f is stable. Then, we must have q = 0. Since p + q = n, we find p − q = n. Suppose now that p − q = n. Since p + q = n, subtracting the two equations, we find 2q = 0, that is f is stable.
Fuente: http://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stability_criterion

Root locus

In control theory, the root locus is the locus of the roots of the characteristic equation of the closed loop transfer function as the loop gain of the feedback system is increased from zero to infinty. Traditionally the root locus is constructed with the loop gain K as the variable. However, any system parameter can be used to construct a root locus. The root locus is a useful tool for analyzing the transient response, as well as the stability of a single input single output dynamic systems. A system is stable if all of its poles are in the left-hand side of the s-plane (for continuous systems) or inside the unit circle of the z-plane (for discrete systems).

Uses:
In addition to determining the stability of the system, the root locus can be used to design for the damping ratio and natural frequency of a feedback system. Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arcs whose center points coincide with the origin. By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency a gain, K, can be calculated and implemented in the controller. More elaborate techniques of controller design using the root locus are available in most control textbooks: for instance, lag, lead, PI, PD and PID controllers can be designed approximately with this technique.
The definition of the damping ratio and natural frequency presumes that the overall feedback system is well approximated by a second order system, that is, the system has a dominant pair of poles. This often doesn't happen and so it's good practice to simulate the final design to check if the project goals are satisfied.

RL = root locus
Suppose there is a plant (process) with a transfer function expression P(s), and a forward controller with both an adjustable gain K and a transfer function expression C(s). A unity feedback loop is constructed to complete this feedback system. For this system, the overall transfer function is given by
$T(s) = \frac{K C(s)P(s)}{1+K C(s)P(s)}.$
Thus the closed-loop poles (roots of the characteristic equation) of the transfer function are the solutions to the equation 1+ KC(s)P(s) = 0. The principal feature of this equation is that roots may be found wherever KCP = -1. The variability of K (that's the gain you can choose for the controller) removes amplitude from the equation, meaning the complex valued evaluation of the polynomial in s C(s)P(s) needs to have net phase of 180 deg, wherever there is a closed loop pole. We are solving a root cracking problem using angles alone! So there is no computation per-se, only geometry. The geometrical construction adds angle contributions from the vectors extending from each of the poles of KC to a prospective closed loop root (pole) and subtracts the angle contributions from similar vectors extending from the zeros, requiring the sum be 180. The vector formulation arises from the fact that each polynomial term in the factored CP,(s-a) for example, represents the vector from a which is one of the roots, to s which is the prospective closed loop pole we are seeking. Thus the entire polynomial is the product of these terms, and according to vector mathematics the angles add (or subtract, for terms in the denominator) and lengths multiply (or divide). So to test a point for inclusion on the root locus, all you do is add the angles to all the open loop poles and zeros. Indeed a form of protractor, the "spirule" was once used to draw exact root loci.
From the function T(s), we can also see that the zeros of the open loop system (CP) are also the zeros of the closed loop system. It is important to note that the root locus only gives the location of closed loop poles as the gain K is varied, given the open loop transfer function. The zeros of a system can not be moved.
Using a few basic rules, the root locus method can plot the overall shape of the path (locus) traversed by the roots as the value of K varies. The plot of the root locus then gives an idea of the stability and dynamics of this feedback system for different values of k.
The method is due to Walter R. Evans

Sketching root locus

• Mark open-loop poles and zeros
• Mark real axis portion to the left of an odd number of poles and zeros
• Find asymptotes
The asymptotes intersect the real axis at the point:
$\alpha = \frac{\sum_P - \sum_Z}{P - Z}$
where
 ∑ P
is the sum of all the locations of the poles, and
 ∑ Z
is the sum of all the locations of the explicit zeros.
and P is the number of poles and Z is the number of zeros
• Phase condition on test point to find angle of deparature
• Compute breakaway/break-in points
The breakaway points are located at the roots of the following equation:
$\frac{-G(s)H(s)}{ds} = 0\text{ or }\frac{\overline{GH}(z)}{dz} = 0$
Once you solve for z, the real roots give you the breakaway/reentry points. Complex roots correspond to a lack of breakaway/reentry.
The break-away (break-in) points are obtained by solving a polynomial equation

z-plane versus s-plane

The root locus can also be computed in the z-plane, the discrete counterpart of the s-plane. An equation (z = esT) maps continuous s-plane poles (not zeros) into the z-domain, where T is the sampling period. The stable, left half s-plane maps into the interior of the unit circle of the z-plane, with the s-plane origin equating to z = 1 (because e0 = 1). A diagonal line of constant damping in the s-plane maps around a spiral from (1,0) in the z plane as it curves in toward the origin. Note also that the Nyquist aliasing criteria is expressed graphically in the z-plane by the x-axis, where (wnT = π). The line of constant damping just described spirals in indefinitely but in sampled data systems, frequency content is aliased down to lower frequencies by integral multiples of the Nyquist frequency. That is, the sampled response appears as a lower frequency and better damped as well since the root in the z-plane maps equally well to the first loop of a different, better damped spiral curve of constant damping. Many other interesting and relevant mapping properties can be described, not least that z-plane controllers, having the property that they may be directly implemented from the z-plane transfer function (zero/pole ratio of polynomialls), can be imagined graphically on a z-plane plot of the open loop transfer function, and immediately analyzed utilizing root locus.
Since root locus is a graphical angle technique, root locus rules work the same in the z and s planes.
The idea of a root locus can be applied to many systems where a single parameter K is varied. For example, it is useful to sweep any system parameter for which the exact value is uncertain, in order to determine its behavior.

Control Clasico

Hasta bien entrado el siglo XX las únicas herramientas analíticas que poseía el especialista en control eran la utilización de ecuaciones diferenciales ordinarias junto con criterios algebraicos para determinar la posición de las raíces de la ecuación característica asociada. Aplicando el criterio de Routh y Hurwitz el ingeniero determinaba la estabilidad o no de los sistemas, pero para esto se debía obtener el modelo matemático operando mediante ecuaciones diferenciales. Esto suponía un arduo trabajo. Además ahí que destacar que el criterio de Routh y Hurwitz no ofrece información de cómo mejorar la estabilidad del sistema.
Desde el punto de vista teórico, la Ingeniería de Control se empieza a consolidar cuando se produce el traslado y aplicación de los conocimientos adquiridos en los problemas de amplificación de señales a los problemas de control industrial.
Estos estudios desembocan en la llamada Teoría Clásica de Control, en la cual se utililizaban como herramientas matemáticas los métodos de Transformación de Laplace y Fourier y la descripción externa de los sistemas.
Dos trabajos de singular importancia son los desarrollados por Minorsky y Hazen. En el trabajo de Minorsky "Directional Stability of Automatic Steered Bodies" [Thaler 74] de 1922, se reconoce la no-linealidad de los sistemas y aplica la linealización mediante el desarrollo en serie de Taylor a sistemas no-lineales correspondientes al movimiento angular de un buque. Estudia la estabilidad y los efectos de los retrasos de la información sobre las salidas de los Sistemas.

Hazen en su publicación "Theory of Servomechanism" (1934) [Thaler 74], analiza el funcionamiento de los servomecánismos utilizando en su análisis entradas típicas de escalón y rampa. Aparte de proponer un marco conceptual, Hazen utiliza herramientas matemáticas como el cálculo operacional de Heaviside. En sus trabajos estudia el diseño de servomecanismos para posicionar ejes.

El desarrollo de las técnicas frecuenciales:
El estudio de los servomecanismos y los reguladores en el dominio frecuencial se realiza al obtenerse resultados sobre el diseño de amplificadores de señal realimentados. Destacan los trabajos de Nyquist (1932), Black (1934) y Bode (1940).
El suceso que realmente marca época en el desarrollo de los métodos de respuesta en frecuencia es la aparición de trabajo clásico de Nyquist sobre la estabilidad de amplificadores realimentados. Nyquist presenta en este trabajo "Regeneration Theory" [Thaler 74], su celebre criterio de estabilidad. Su investigación surge de los problemas que presentaba la atenuación y distorsión de la señal en la telefonía a grandes distancias.
En 1915 la Bell System había finalizado un enlace telefónico experimental entre New York y San Francisco. Este enlace utilizó una línea aérea de cobre que pesaba 500 Kg/milla y fue cargado inductivamente para tener una frecuencia de corte de 1000 Hz. La atenuación de la señal a lo largo de las 3000 millas era de 60 dB, se redujo a 18dB utilizando seis amplificadores con una ganancia total de 42 dB.
Sin embargo el cambio a operaciones mediante cable, planteó serios problemas técnicos. Debido a la escasa sección de los cables la atenuación era grande y se requerían muchos amplificadores repetidores. Esto suponía que la señal al pasar por múltiples etapas amplificadoras, cada una con sus no-linealidades, se iba distorsionando. Para mantener la inteligibilidad de la señal de audio transmitida en distancias intercontinentales se requería una linealidad efectiva del amplificador muy lejos de la que la tecnología era capaz de dar ( una distorsión del orden del 0.005%).
Esta dificultad sólo se pudo vencer con el magnífico invento desarrollado por H. Black de los laboratorios Bell quien propuso la idea de un amplificador realimentado, en su trabajo "Stabilized Feedback Amplifiers" [Thaler 74] en 1934. El descubrimiento importante de Black fue que la elevada ganancia en un dispositivo amplificador no lineal y cuyos parámetros eran variables con el tiempo se podía negociar para conseguir una reducción en la distorsión no lineal de manera que el sistema se comportase como una ganancia lineal, estable y precisa. Black utiliza el criterio de Nyquist y llega a interpretar una serie de fenómenos que se producen en los sistemas realimentados.
El mecanismo era simplemente utilizar componentes pasivos lineales apropiados de gran precisión en el lazo de realimentación de un amplificador no lineal de elevada ganancia. Hacia 1932 Black y su equipo podían construir amplificadores que funcionaban razonablemente bien. Sin embargo presentaban una tendencia a inestabilizarse. Algunos lo hacían cuando aumentaba la ganancia del lazo del amplificador realimentado, lo cual se podía esperar, pero otros manifestaban estas características cuando la ganancia se disminuía y esto si que era completamente inesperado.
La situación era muy parecida a la asociada con los reguladores de velocidad del siglo XIX, que presentaban oscilaciones en la velocidad y cuya conducta no se podía explicar con las herramientas de análisis disponibles.
Los amplificadores realimentados de la época podían contener del orden de 50 elementos independientes almacenadores de energía (tales como condensadores, autoinducciones, etc.). Su descripción en términos de un conjunto de ecuaciones diferenciales, como en el análisis clásico de los sistemas de control automático de origen mecánico era casi una tarea imposible a la vista de las rudimentarias facilidades disponibles en esos años para la solución por computador de tales ecuaciones.
El famoso trabajo de Nyquist resolvió este misterio, abrió totalmente nuevas perspectivas en la teoría de los mecanismos realimentados y por lo tanto comenzó una nueva era en el Control Automático. Antes de 1932 el enfoque basado en las ecuaciones diferenciales había sido la gran herramienta del ingeniero del control; en la década que siguió a la contribución de Nyquist estas técnicas fueron casi completamente reemplazadas por métodos basados en la teoría de variable compleja los cuales fueron la consecuencia natural y directa de su nuevo planteamiento.
La solución del problema de la estabilidad de un sistema realimentado propuesta por Nyquist se basaba en la forma de la respuesta en frecuencia de la ganancia en lazo abierto y esto era de un valor práctico inmenso ya que se formulaba en términos de una cantidad (la ganancia) que era directamente medible. Este enlace directo con medidas experimentales era un desarrollo completamente nuevo en trabajos dinámicos de tipo aplicado.
La aplicación del criterio de estabilidad de Nyquist no dependía de la disponibilidad de un modelo del sistema en la forma de una ecuación diferencial. Más aún, el contorno del lugar de Nyquist daba una indicación inmediata de cómo se podía mejorar la conducta de un sistema realimentado que estaba muy poco amortiguado o que incluso era inestable simplemente modificando de una manera apropiada su característica de ganancia en lazo abierto en función de la frecuencia.
Con la perspectiva de hoy día puede resultarnos demasiado fácil subestimar la magnitud de la invención de Black y el logro teórico de Nyquist, sin embargo las cosas parecían muy diferentes en su tiempo. La concesión de una patente a Black por su amplificador tardó más de 9 años. La oficina de patentes de EEUU citaba trabajos técnicos que decían que la salida de un amplificador no se podía conectar a la entrada y permanecer estable a menos que la ganancia del lazo fuese menor que uno. La oficina de patentes británica, en palabras de Black, trató la aplicación "como si se tratase de una máquina de movimiento continuo".
El trabajo de Nyquist dejaba sin resolver como estaban relacionadas la amplitud y la fase en función de la frecuencia de la función de transferencia de la ganancia en lazo abierto. En otro de los trabajos clásicos que están en los fundamentos de la Teoría del Control, H. W. Bode realizó este análisis, extendiendo resultados previos de Lee y Wiener.
En el trabajo de Bode "Relations Between Attenuation and phase in Feedback Amplifier Design" [Thaler 74] de 1940, se presenta la definición de margen de fase y margen de ganancia y la definición de los diagramas logarítmicos de Bode.
Bode demostró que dada cualquier función de respuesta en frecuencia A (w) siendo A la amplitud de la ganancia en lazo abierto se le puede asociar una función F (w) siendo la fase mínima de dicha función de respuesta en frecuencia. De esta forma fue capaz de dar reglas para obtener la forma óptima de la ganancia del lazo en función de la frecuencia para un amplificador realimentado.
En la industria de los procesos químicos la introducción del control por realimentación tendió en un principio a desarrollarse de forma aislada de los desarrollos mecánicos y eléctricos. En estos procesos la evolución de la variable controlada era tan lenta ( y lo sigue siendo) que el control se hacia mediante realimentación manual. Los primeros pasos que se dan para controlar estos procesos son la incorporación de instrumentos para supervisar la operación y registradores de plumilla. El desarrollo natural fue utilizar el movimiento de la plumilla del registrador para efectuar una acción de realimentación sobre las válvulas de control en la planta utilizando líneas de transmisión, amplificadores y transductores neumáticos.
Los primeros controladores de temperatura, ofrecían una acción de control de tipo on-off por medio de un simple mecanismo conmutador o relé que pronto se reveló insuficiente para las exigencias planteadas en los procesos industriales, como por ejemplo en la industria láctea, el proceso de pasteurización de la leche. El siguiente desarrollo fueron los primeros reguladores con acción proporcional. En estos reguladores se manifestaba claramente el dilema de la automática: precisión frente estabilidad, si se desea un error estacionario pequeño, se debía aumentar la ganancia del regulador, o lo que es lo mismo disminuir la banda proporcional. Pero esto conllevaba que el proceso era sometido a fuertes oscilaciones en el transitorio. Y si se aumentaba la banda proporcional, disminuían las oscilaciones pero en caso de cambios en la carga aparecía un error estacionario apreciable. El máximo valor recomendado entonces para la banda proporcional era del cinco por ciento.

Durante los años 30 se desarrollaron completamente estos reguladores neumáticos y se transfirió a este campo del control la idea de utilizar el término de acción integral que se venía empleando desde tiempo en los sistemas mecánicos. El primer regulador de temperatura con acción proporcional integral fue el Foxboro Stabilog patentado por Mason en 1931. En este regulador neumático, se incorporaba amplificación lineal, basada en el principio de la realimentación negativa (al igual que Black en los amplificadores de señal realimentados) y acción integral (reset). Hay que hacer constar que en un principio el Stabilog no tuvo mucho éxito comercial, debido entre otras cosas a su precio y a que no era comprendido su funcionamiento.
A finales de los años 30 se introdujo la acción derivativa en estos controladores neumáticos dando lugar así al regulador PID de 3 términos (Proporcional, Integral y Derivativo).
En 1942 Ziegler y Nichols, ingenieros de Taylor Instruments hicieron un estudio importante que condujo a fórmulas empíricas para sintonizar el regulador PID al proceso. Este estudio " Optimum Settings for Automatic Controllers" [Thaler 74] fue presentado en el "ASME Winter Anual Meeting". Los coeficientes de las distintas acciones proporcional, integral y derivada, se podían determinar de valores medidos experimentalmente del proceso que se deseaba controlar. La importancia de estas reglas de ajuste óptimo de controladores es enorme, siguen siendo vigentes y profusamente usadas en el ámbito del control de procesos industriales.
El trabajo de Ziegler y Nichols es pionero en el desarrollo de la idea de control óptimo, aunque su criterio de optimización, que consiste en minimizar la superficie de error absoluto, no se puede tratar analíticamente.
Un paso crucial en la transferencia de las técnicas utilizadas en el análisis de los amplificadores realimentados de los sistemas de telefonía a otras clases de sistemas fue realizada por H. Harris del MIT en su trabajo "The analisys and design of servomechanics" [Harris 42], en el cual introduce el uso de funciones de transferencia en el análisis de un sistema realimentado general. Esto permitió que un servomecanismo mecánico o un sistema de control de un proceso químico se representasen mediante diagramas de bloques y utilizasen las técnicas del dominio frecuencial.

Avances durante la Segunda Guerra Mundial:
Un gran estímulo para el desarrollo de la técnica lo constituyen las guerras. La Segunda Guerra Mundial supuso un gran impulso al desarrollo teórico y mucho más al desarrollo práctico, dada la fuerte necesidad de sistemas de control que funcionarán como los servos de los radares y el posicionamiento de cañones.
La Segunda Guerra Mundial creó una necesidad urgente para diseñar servomecanismos de altas prestaciones y condujo a grandes avances en la forma de construir sistemas de control realimentados. Las exigencias de la guerra enfocaron la atención sobre un problema importante: el llamado problema de control de tiro, proporcionando una cadena automática de órdenes entre la detección del blanco, el apuntamiento del arma y el disparo. Este problema tiene tres etapas:
1. Detección y seguimiento del blanco.
2. Predicción.
3. Colocación del cañón en posición de disparo.

En el comienzo de la guerra, aunque cada etapa requería algunos operadores, cada uno efectuando operaciones de seguimiento manual, había una considerable controversia en cuanto al valor operacional del control automático. Esto no es sorprendente ya que los predictores que estaban en uso tenían un error medio de 2-3 grados que eran del mismo orden que el error medio de un operador de batería bien entrenado que efectuase un seguimiento manual. Cuando la guerra progresó, aumentó la velocidad de los blancos, el personal entrenado comenzó a escasear y la aparición de los radares de seguimiento mejoró notablemente la capacidad de predicción: era pues el momento para que el control automático se hiciese notar.
Con el objetivo fundamental de investigar y avanzar en los problemas de control del radar y de control de tiro, en marzo de 1942, de una manera informal se constituyó un grupo que posteriormente sería denominado el "Servo-Panel". Su principal función consistió en organizar encuentros, proporcionar información y servir de nexo de comunicación entre diferentes grupos de investigación.
El gobierno americano al intentar desarrollar los sistemas de control automático de tiro se enfrentó con el problema de que aunque había una considerable experiencia en temas de control, ésta se encontraba dispersa entre muchas ramas de la ingeniería y faltaba el atributo unificador de una terminología en común. La reacción no se hizo esperar, con la formación en 1940 bajo la dirección del Dr. Vannevar Bush del Comité de Investigación de Defensa Nacional (NDRC).
Entre los muchos comités del NDRC estaba el de Contro1 de Tiro que bajo el liderazgo de Warren Weaver coordinó el trabajo de los servicios, laboratorios industriales y universidades. El comité era responsable de la dirección de la investigación y de la circulación de informes reservados a los grupos apropiados.
Los informes de Brown, Harris, Hall, Wiener, Phillips y Weis entre otros, fueron emitidos bajo los auspicios del NDRC y su contenido no fue conocido hasta finales de los años 40. Los siguientes trabajos han sido recogidos en la colección [Thaler 74]:
• Brown, G.S., and A.C. Hall: Dynamic Behavior and Design of Servomechanism. [Brown 46].
• Harris, H. JR: The frecuency Response of Automatic Control. [Harris 46].
• Hall, A.C: Aplication of Circuit Theory to the Design of Servomechanisms. [Hall 46].
• Weiss, H.K.: Constant Speed Control Theory. [Weiss 39].

Albores de la era espacial:
Desde siempre los procesos más complejos comandados por computador han sido las aplicaciones de control de vuelo aerospaciales. Sin disponer de las tecnologías del control automático y los computadores, hubiera sido imposible que el hombre hubiera viajado al espacio. Los pioneros en esta aplicación fueron, además de otros, el ruso Constantin E. Tsiolkovsky (1857-1935), y el alemán Hermann Ganswindt (1856-1934) que criticaron a los astrónomos y matemáticos de la época que aseguraban que nunca jamás el ser humano poseería los medios para conseguir el control, la precisión y la velocidad necesaria para los vuelos en el espacio.
Uno de los primeros trabajos en este campo se debe al alemán Hermann Orberth, que en su articulo "Die Rakete zu den Planetenräumen" (Cohetes en el espacio interplanetario) publicado en 1923, afirma que para poder efectuar vuelos en el espacio, el hombre debe acceder a técnicas de control automático mucho más sofisticadas que las disponibles entonces. En su trabajo de 1929 "Wege zur Raumschiffahrt" (Métodos para volar en el espacio) predice que el desarrollo de cohetes que dispongan de la suficiente fuerza propulsiva llevará largo tiempo y lo mismo sucederá con la necesaria tecnología de control automático. Asimismo un elemento fundamental en la navegación espacial será la precisión a la hora de maniobrar dado que las velocidades y las distancias implicadas son enormes (evidentemente astronómicas). Para colocar un satélite orbitando sobre la tierra es necesaria una velocidad mínima de 7,904 kilómetros por segundo. Para poder escapar de la tierra y navegar por el espacio interestelar es necesaria una velocidad mínima de 11,178 Km./seg. conocida como la velocidad de escape. Estas velocidades resultaban difíciles de imaginar para la época cuando un coche que circulaba a 100 Km./hora necesitaba cinco minutos para recorrer la distancia de ocho kilómetros. En otras palabras el cohete debía ir a una velocidad trescientas veces superior a la del coche.
La falta de oxigeno en el espacio exterior conllevaba la imposibilidad de realizar la combustión en las turbinas de los cohetes. Robert H. Goddard publica en 1919 el primer trabajo "A Method of Reaching Extreme Altitudes" donde se describen cohetes cuya combustión se basaba en oxígeno líquido.
Las mayores contribuciones al campo de la navegación espacial que posibilitaron que el hombre llegara a la luna en 1969 se realizaron en la base alemana de Peenemünde situada en la isla de Usedom del mar Báltico. La base fue construida entre 1936 y 1940. Las investigaciones y desarrollos realizados ahí constituyen uno de los capítulos más excitantes de la historia de la ciencia y la técnica.
Las primeras unidades desarrolladas para el ejercito alemán, las denominadas A1 y A2, fueron destinadas principalmente al ensayo de sistemas de propulsión y control de cohetes. Una vez se dispuso de unidades en funcionamiento, enseguida se observo que el principal problema a solucionar era mantener el sistema estable. Según palabras de Willy Ley [Willy 44] los conocimientos que se poseían entonces sobre la estabilidad de los cohetes "se podían escribir en una postal, dejando alguna parte en blanco".
Para el desarrollo del tercer ingenio, la A3, la marina Alemana envió a un reconocido especialista en el problema de estabilización y alineamiento de las torretas de tiro, el clásico problema del control de la segunda guerra mundial. Sin embargo este ingenio no se terminó de construir dado que el mecanismo de control se revelo inadecuado. Después de lo cual se desarrolla un nuevo sistema de control bastante avanzado para la época. Este sistema utilizaba giróscopos y acelerómetros como elementos sensores y disponía de servomotores eléctricos que podían efectuar pequeños y precisos movimientos, construidos en molibdeno, un material resistente a altas temperaturas, y encargados de controlar el suministro de gas a las turbinas del cohete.
Para estudiar la dinámica del sistema se construyo un simulador mecánico, cuyo diseño se basó en los registros obtenidos de los vuelos de los primeros ensayos mediante radiotelemetría (otro desarrollo pionero). En este momento, Willy Ley hubiera necesitado al menos doce docenas de postales.
Durante las últimas fases de la segunda guerra mundial en la base de Peenemünde se llegaron a realizar proyectos sobre misiles transatlánticos (la unidad A6)…Incluso la Gestapo llego a arrestar a Wernher Von Braun por haber hablado abiertamente de la posibilidad de enviar objetos al espacio. Fue liberado gracias a la mediación del director de la base de Peenemünde, que explicó a altos oficiales de la Gestapo que las ideas de von Braun contribuían a la creación de nuevos y más potentes misiles de justo castigo. Cuando Alemania esta ya prácticamente derrotada, en mayo de 1945, la base de Peenemünde junto con todo su arsenal de cohetes cayó en manos de los aliados, y en Julio de ese mismo año trescientos vagones de tren cargados de cohetes A4 llegaron a una base de Nuevo Méjico. También se traslado allí todo el equipo científico alemán que continuó con su labor de investigación.
El resto de la historia es de sobra conocida por todos nosotros. ¿Llegará alguno de nuestros hijos al Planeta Rojo?

Los años clásicos: 1945-1955:
Desde el punto de vista del desarrollo de las técnicas de diseño de control automático, el principal resultado de este gran esfuerzo y experiencia fue extender rápidamente la utilización de las ideas de respuesta en frecuencia a todos los campos y producir así una teoría unificada y coherente para los sistemas realimentados con un único lazo.
Coincidiendo con la segunda guerra mundial, el matemático Wiener desarrolla la teoría estocástica clásica, la cual tuvo su inicio en el estudio del problema de automatización de un cañón aéreo. En este trabajo se da un enfoque radicalmente distinto del estudio del problema del control, y supone el inicio de la conocida como teoría estocástica clásica. Las aportaciones de Wiener consisten en considerar la presencia de ruidos en las señales, e introduce también el concepto de control óptimo, cuyo objetivo consiste en minimizar un determinado criterio que define la calidad del control, en este caso minimiza la superficie de error cuadrático [Wiener 49].
Wiener también establece la relación entre estos ingenios autogobernados y determinados procesos que suceden en los seres vivos. Todo ello, conduce a la formulación de lo que se denominaría cibernética en su trabajo "Cybernetics" de 1948 publicado por el MIT press [Wiener 48].
A finales de la década de los cuarenta, surgen otras dos vías de desarrollo de la teoría de control: el Método del modelo de Truxal [Truxal 54] y el método del lugar de las Raíces, de Evans. Se presentan también aportaciones como la extensión de los métodos frecuenciales a sistemas no-lineales y a sistemas estocásticos.
El método del modelo es una adaptación del método de Guillemin desarrollado inicialmente para el diseño de redes pasivas. Partiendo de las especificaciones deseadas se obtiene la función de transferencia que debe seguir el sistema de control. El cálculo de la función de transferencia del regulador se realiza fácilmente por medio de operaciones álgebraicas. Este método resultaba atractivo dado que no utiliza la técnica de prueba y error. Pero se manifestaban en él algunas dificultades prácticas como podían ser la complejidad de los correctores que se obtienen, que dejaban de tener la estructura clásica PID.
Los trabajos de Evans:
"Graphical Analysis of Control Systems" [Evans 48].
"Control System Synthesis by Root Locus Method" [Evans 50].
ambos recogidos en [Thaler 74], constituyen la última gran contribución a la teoría clásica de control. En palabras del propio autor "el lugar de las raíces determina todas las raíces de la ecuación diferencial de un sistema de control por medio de una representación gráfica, la cual permite una síntesis rápida de la respuesta transitoria o frecuencial deseada".
El método de Evans cuenta con el handicap de no poder abordar el análisis de sistemas con retraso puro y la difícil estimación de la respuesta temporal de sistemas con distribuciones dispersas de polos y ceros. A su favor, aporta un método gráfico de estimar la influencia de variaciones en los parámetros del sistema o del regulador sobre la estabilidad y el comportamiento dinámico de los sistemas.