Method and Apparatus for Nonlinear Compensation

1. Field of the Invention

This invention relates generally to electronic signal conditioning; and more specifically, to a method as well as circuitry for compensating for nonlinear response characteristics of loads, in particular capacitive loads, driven by an amplifier.

2. Description of Related Art

Frequently, in the control of mechanical devices, high voltage amplifiers are used to drive piezoelectric actuators or other high-capacitance devices, which serve, in turn, to apply mechanical forces to the mechanical devices being controlled. Actuators can be formed from stacks of piezoceramic wafers which appear basically as pure capacitive loads to the high-voltage (driving) amplifier.

In certain instances the nonlinear response characteristics, exhibited to varying degrees by all ceramic composition capacitors, can be ignored, particularly in those applications where the capacitive load is being operated over a small range of voltage or where the nonlinear effects are unimportant. But in those applications where the capacitive load is being utilized for its piezoelectric properties of converting electrical energy to mechanical energy, the load must be driven over a large voltage range which exacerbates its nonlinear characteristics.

A few approaches to linearizing current through such purely capacitive loads exist. U.S. Pat. No. 5,000,415 discloses the use of charge and voltage feedback loops, positioned around the load so that the nonlinear voltage vs. displacement and the charge vs. displacement characteristics of the load tend to "cancel" each other. Also, in U.S. Pat. No. 4,628,275 a single feedback loop senses the charge from a large reference capacitor in series with the capacitive load. This last approach is not practical where the load capacitance is relatively large because the reference capacitor value becomes unreasonably large.

However, both approaches have in common the drawback that one or more control values must be fed back from the load to the controlling amplifier circuit. This can have at least two disadvantages. First, if the feedback value requires filtering to remove a switching component (such as in switching amplifier designs), signal phase shifts may be introduced. These must be compensated for by the control electronics, the net effect of which is a reduced control signal frequency bandwidth. Second, the physical design of the mechanical system which utilizes the capacitive load may not lend itself to having additional sensors and wires for the feedback loops from the load back to the controlling amplifier circuit.

Accordingly, there is a need to compensate for the nonlinear response characteristics of a capacitive load without the use of feedback loops from the load to the controlling amplifier, and without the need for relative large reference capacitors.

The present invention relates to a method and apparatus for compensating for the nonlinear current response of a capacitive load by predictably non-linearizing the drive voltage waveform at the controlling amplifier in advance of the load, effectively linearizing the current through the load. Utilizing a nonlinear capacitive reference element, which exhibits the same nonlinear characteristics as the capacitive load, in a local feedback loop around a single gain stage within the controlling amplifier circuit, the nonlinear characteristics of the capacitive load are anticipated and cancelled. The local feedback loop does not physically or electrically include the capacitive load, thereby eliminating the need for load sensors and feedback loops.

This invention is particularly useful where the capacitive load comprises a "stack" of several hundred ceramic wafers, (usually electrically connected in parallel) creating the load. The reference element could then be just a single wafer of the type used in the load, thus having only a small fraction of the load's capacitance, but at the same time, exhibiting the same nonlinear capacitance vs. voltage characteristics as the load.

In the preferred embodiment, compensation is performed by an inverting, high-gain operational amplifier in a voltage multiplier configuration. The input (summing) resistor is shunted by an input capacitor. A local feedback circuit comprising a reference capacitor shunted by a feedback resistor connects the amplifier output to the amplifier input so that the entire voltage developed across the load appears across the reference element in the local feedback circuit. The reference capacitor is matched to one of the load piezo wafers or is actually a replica of a load wafer. Values of the local feedback resistor and the combination resistor and capacitor at the amplifier input are selected for proper amplifier gain, as well as pole-zero cancellation, providing a flat frequency response.

The advantages and various features of the invention will be better understood after consideration is given to the following detailed description of the preferred embodiment together with the accompanying drawings.

FIG. 1 shows a block diagram of a controlling amplifier utilizing nonlinear compensation based on a reference capacitor C.sub.ref for driving nonlinear loads.

FIG. 2 depicts a high gain operational amplifier stage showing nonlinear compensation in accordance with the invention.

FIG. 3 describes graphically the transfer function dependence of the reference capacitor C.sub.ref shown in FIG. 2.

FIG. 4 shows a typical response characteristic of a capacitive load vs. applied voltage.

FIG. 5a depicts the voltage and current waveforms, exaggerated for clarity, without compensation for a capacitive load driven by a high-voltage amplifier.

FIG. 5b shows the modified voltage and current waveforms where compensation in accordance with this invention is applied to correct the distorted current waveform of FIG. 5a.

FIG. 6 shows a typical PMN stack representative of a capacitive load.

Referring first to FIG. 4 and to FIG. 5a there is depicted the well known nonlinear capacitance vs. voltage curve response typical for ceramic capacitors. Similar nonlinear characteristics are, in fact, exhibited by all ceramic composition capacitors, to varying degrees. As the voltage increases from 0 to value V1 the capacitance of the load will decrease nonlinearly, from C0 to C1, as shown. As will be appreciated the current through the load, Ic, also changes nonlinearly in response to the changing capacitance. Specifically, FIG. 5a depicts a sinusoidal voltage, Vc (solid line) being applied across a capacitive load. The current, Ic, for a nonlinear capacitive load leads the voltage by 90.degree. (dashed line) and is asymmetrical about the vertical axis. This nonlinear effect can be described mathematically by the following relationship.

where

Ic=Current through the nonlinear capacitor

Vc=Voltage across the capacitor

C(Vc)=Voltage--dependent capacitance value of the load.

In practice it has been found that the distortion characteristics of the mechanical displacement of the ceramic loads follow closely the distortion characteristics of the load current. In other words, if the current through the ceramic load is distorted, the mechanical displacement of the capacitve ceramic load will also be distorted.

FIG. 1 illustrates the invention working with a power gain amplifier, 10, shown in block diagram form. Amplifier 10 is connected to the output of preamplifier, 11, where nonlinear compensation, in accordance with this invention, is added. Preamplifier, 11 develops the entire range of voltage across a single gain stage, 12. Preamplifier 11 delivers a predictably shaped waveform to "power gain" stage, 10 which, in turn, drives a capacitive load, C.sub.load identified by numeral 14. The compensation network pre-conditions the signal delivered to power gain amplifier 10 so that the non-linear impacts of the load are cancelled, leaving current and voltage waveforms as depicted in FIG. 5b.

A capacitive reference element, C.sub.ref 15 with nonlinear capacitance vs. voltage characteristics identical to that exhibited by the load, is shown connected across the single gain stage 12 in preamplifier 11. In this configuration reference element 15 has significantly lower capacitance than the capacitive value of load 14, allowing the use of readily-available circuit components. For example, capacitive load 14 could consist of a "stack" of several hundred ceramic wafers (see FIG.6), electrically connected in parallel, creating a large capacitive load. Reference element 15 could then consist of just one wafer of the type used in the reference load.

For the nonlinear characteristics of the load to be properly canceled, reference element 15 must have the same voltage across it as does load 14. To achieve this, reference element 15 is placed "across" the single gain stage, 12, of the preamplifier 11 thus the entire voltage range, developed by the single gain stage 12 within the preamplifier 11, and eventually developed across the load, is also developed across reference element 15.

Power gain stage 10 does not create any voltage gain or loss within the frequency bandwidth of interest.

FIGS. 2 and 3 provide a more detailed illustration of the nonlinear compensation circuit in accordance with the present invention. In FIG. 2 capacitor 20 is the reference element, C.sub.ref. The combination of resistor 21(R2) and capacitor 20(C2) creates a frequency pole (fp) at

Where R2=value of resistor 21

C2=value of capacitor 20

This pole, by itself, would result in a frequency response corresponding to the downward-sloping dashed line in FIG. 3. The effect of this pole can be canceled with resistor 22(R1) and capacitor 23(C1) shown at the input of amplifier 24. This network (R1 & C1) creates a frequency zero (fz) at

Where R1=value of resistor 22

C1=value of capacitor 23

This zero, by itself, would result in a frequency response corresponding to the upward-sloping dashed line in FIG. 3. When the values of networks R1/C1 and R2/C2 are chosen such that fp=fz, the frequency response of the pole and zero acting together cancel or compensate for each other. The solid horizontal line in FIG. 3 corresponds to this overall frequency response.

Some restrictions on the selection of R1, R2, C1 and C2 exist. The transfer function of the circuit (in the Laplace domain) in FIG. 2 is

Since the DC gain of the stage is determined by choosing (non-reactive elements) resistors 21(R2) and 22(R1) these values are usually determined by load driving considerations. The value of the reference element capacitor 20(C2), is restricted because its key contribution to this arrangement is derived from its nonlinear characteristics, which must be as close as practical to the nonlinear characteristics of the load. These characteristics, along with its capacitance are, in large part, determined by the physical dimensions of the reference element. This leaves the choice of capacitor 23(C1), which is the circuit element value needed to make fp=fz. Consider now a negative capacitance change in capacitor 20(C2), .DELTA.C2, caused by the development of a high voltage from the input to the output of amplifier 24 (which appears almost entirely across reference capacitor 20). Decreasing C2 by .DELTA.C2 causes the pole, fp, to move higher in frequency to fp':

The corresponding frequency response is shown in FIG. 3 as the dotted line ("-change in C2"). Note that the frequency response still compensates, but at a higher frequency f2, and at a higher magnitude, H2, than previously. Practically, the pole and zero are made to cancel at a very low frequency so as to be out of the operating bandwidth of the driving amplifier. In summary, the gain of this nonlinear compensation stage is dynamic and dependent upon the input voltage to the circuit.

The Table below provides component values for a working arrangement driving a ceramic capacitive load.

FIGS. 5a and 5b are illustrative of the voltage Vc across and current Ic through a 60 uF ceramic capacitive load, before and after the preferred embodiment of nonlinear compensation is applied. In the working arrangement, a 22 Hz sinusoidal waveform was applied to the input of the amplifier such that the output voltage across the load was 900 V. In FIG. 5a the amplifier output voltage (with no nonlinear compensation) is sinusoidal, as is the voltage across the load. The current through the load, however, is asymmetrical about the vertical axis, indicating severe even-harmonic distortion, resulting from the nonlinear capacitance vs. voltage characteristic of the ceramic capacitive load. In FIG. 5b, with nonlinear compensation applied, the gain of the amplifier changes with the sinusoidal input signal, distorting the output voltage waveform, but linearizing the current waveform, significantly reducing the even-harmonic distortion. These are identified in FIG. 5b as voltage Vc and current Ic.

As was discussed above, the harmonic distortion characteristics of the mechanical displacement of the ceramic capacitive load are largely determined by the harmonic distortion content of the current waveform through the same. In FIG. 5a, with no nonlinear compensation applied, the mechanical displacement of the load would exhibit severe even-harmonic distortion, similar to the current waveform shown. With nonlinear compensation applied, the even-harmonic distortion component of the mechanical displacement of the load would be significantly reduced, as it is in the current waveform in FIG. 5b.