ON THE LINEARIZATION OF THE STATIC CHARACTERISTIC OF A PROFILOMETER

L. Ya. Zinner, A. A. Kozlov

(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)

For monitoring the surface quality of commutators of electrical machines, profilometers are finding ever wider application; these instruments use the principle of frequency or amplitude modulation of a high-frequency generator signal as a function of the distance between the sensor and the commutator surface. Capacitive or inductive sensors are used as the measuring elements in these devices [1, 2]. A common disadvantage of such instruments is the nonlinearity of their static characteristic, due both to the nonlinearity of the sensors’ conversion characteristic and to the manner in which they are connected into the measuring circuit. If, in the static mode of measurement, this nonlinearity is easy to take into account, then when investigating a commutator in the rotation mode a measurement error arises, the magnitude of which is determined by the type of nonlinearity and by the dynamic range of measurement. In order to reduce this error, the most linear portion of the instrument’s static characteristic is usually selected, thereby narrowing the measurement interval. In this case the dynamic range of measurement may turn out to be smaller than the prescribed one, and some authors recommend using a set of sensors having different lengths of the portion of the conversion characteristic over which the sensor has constant sensitivity [3]. Since a number of contradictory requirements are imposed on the sensor, including provision of sufficient dynamic resolving power and sensitivity, a sensor satisfying the first requirement may have, over the measurement interval, significant nonlinearity, which leads to an inadmissible measurement error. Let us find an analytical expression for this error.

Let the sensor conversion characteristic have the form (Fig. 1, a):

\[ U = f(d) \]

\(U\) — signal at the sensor output;
\(d\) — distance between the sensor and the commutator surface.

Let us denote the step between two adjacent commutator bars for a small gap by \(h_1\), and for a large gap by \(h_2\) (Fig. 1, b).

Suppose that \(h_1 = h_2 = h\). When the gap changes by an amount \(h\) on the linear portion of the characteristic, the signal at the instrument output changes by the amount:

\[ \Delta U = K \cdot h, \]

where \(K\) is the slope of the linear portion of the instrument’s static characteristic.

If the function \(U=f(d)\) has a derivative in the specified interval \([0,d_{\max}]\) (which is almost always satisfied in practice), then Lagrange’s formula is valid:

\[ \frac{f(d_0+h)-f(d_0)}{h}=f'(\xi) \]

\[ d_0<\xi<d_0+h \]

Fig. 1. a — curve \(U=f(d)\); b — profile of a collector section

It may also be written as:

\[ \frac{f(d_0+h)-f(d_0)}{K}=h-\Delta h, \]

where \(\Delta h\) is the error in measuring \(h\) due to nonlinearity. Hence

\[ \Delta h=h\left[1-\frac{f'(\xi)}{K}\right]. \]

In this formula the quantity \(\xi\) is unknown. Let us write: \(\xi=d_0+\Theta h\), where \(0<\Theta<1\). If we take \(\Theta=\frac{1}{2}\), then \(\xi=d_0+\frac{h}{2}\), and the error in determining \(\Delta h\) will in the worst case be \(0.1\%\). Consequently, one may write:

\[ \Delta h=h\cdot\left[1-\frac{f'\left(d_0+\frac{h}{2}\right)}{K}\right]. \]

For constant \(K\), the error \(\Delta h\) is proportional to the measured difference \(h\) and decreases as \(f'\left(d_0+\frac{h}{2}\right)\to K\). For example, for the function \(U=\ln(d+1)\) at the point \(d_0\), where the nonlinearity is \(5\%\), the error in measuring a difference of 10 microns reaches \(20\%\). As a result, there arises a need to develop devices that make it possible to correct the nonlinearity of the static characteristic of a profilometer, with the aim of expanding the dynamic measurement range while maintaining the same resolving power. This need is especially acute in the development of profilometers for monitoring the surface of commutators of micromachines, in which the lamella width reaches small values.

Localization of the radiation field of sensors in a narrow inspection zone, accomplished by design measures, makes it possible to ensure high dynamic resolving power, but, as a rule, leads to the transformation characteristic becoming substantially nonlinear.

In amplitude modulation, various kinds of functional converters are used as correcting devices; their amplitude characteristic is selected in such a way that the overall characteristic of the instrument becomes linear. The choice of the type of such a converter and the method of connecting it into the circuit are determined by a number of requirements, which include: accuracy in reproducing the specified nonlinearity; stability of the corrector over time and with temperature; absence of frequency and phase distortions; simplicity and reliability; absence of inertia; a sufficient transmission coefficient; and others.

In instruments with frequency modulation, considerable difficulties arise in constructing corrector circuits.

The diversity of sensor characteristics does not make it possible to use known nonlinear elements directly as correctors, since the set of available nonlinear characteristics is very limited. In addition, the stability of these elements often proves insufficient for realizing a specified characteristic with a high degree of accuracy. The most universal are diode and transistor correctors. In designing such devices, the most effective method is the use of piecewise-linear approximation of the specified function, followed by reproduction of the approximating dependence in circuits using diodes or transistors, in which the latter serve as contactless switches. In this case, depending on the method of specifying the function, the approximation is calculated analytically or graphically with the prescribed accuracy.

The corrector may be connected at the output of the instrument or in the feedback circuit, which in this case becomes nonlinear. Connecting the corrector at the output of the amplifier makes it possible to use it simultaneously as a detector. A disadvantage of this method is that it is necessary to increase the gain of the circuit, since diode correctors have a low transmission coefficient. Connecting the corrector into the negative-feedback circuit is inadvisable, since at operating frequencies on the order of 1--10 MHz semiconductor elements have significant reactive conductances, which leads to phase and frequency distortions in the negative-feedback circuit. For correction purposes it is more advantageous to use an automatic gain-control circuit (AGC) with a nonlinear detector, encompassing the profilometer amplifier (Fig. 2).

Fig. 2. Diagram of a profilometer with nonlinear AGC. G -- generator; PA -- preamplifier; S -- sensor; D' -- detector; A -- amplifier

Suppose that the signal at the output of the sensor is proportional to the excitation voltage and is determined by the expression: \(U_g = U_b \cdot F(d)\).

For complete linearization it is necessary to regulate the gain coefficient of the preamplifier so that the signal at the sensor output is proportional to the gap:

\[ U_g = K \cdot d, \]

where \(K\) is the proportionality coefficient.

Whence

\[ K_{\mathrm{AGC}}=\frac{K}{f(d)}, \qquad (1) \]

where

\[ f(d)=\frac{F(d)}{d}. \]

This coefficient, in turn, depends on the control voltage \(U_{\mathrm{AGC}}\):

\[ K_{\mathrm{AGC}}=\psi(U_{\mathrm{AGC}}) \tag{2} \]

Equating (1) and (2), we obtain an implicit equation for determining the function \(U_{\mathrm{AGC}}=\varphi(d)\). This dependence is implemented by a nonlinear detector in the AGC circuit.

Since the literature contains no method for calculating AGC circuits for the purpose of linearizing the static characteristics of instruments, we shall attempt to fill this gap using a specific circuit as an example.

Fig. 3. Schematic diagram of a controlled stage with an AGC circuit.

As the controlled stage of the preamplifier, a resonant transistor amplifier is used, whose gain is varied by changing the dc operating point. The collector current \(i_k\) is used as the control parameter. The calculation of the circuit is based on representing the transistor as a four-terminal network described by a system of small-signal \(y\)-parameters. In profilometer circuits, high-frequency transistors are usually used; therefore the operating frequency turns out to be much lower than the value at which the reactive components of the admittances begin to have an effect. Consequently, the complex nature of the \(y\)-parameters may be neglected, assuming: \(y_{11}=g_{11}\); \(y_{21}=g_{21}\); \(y_{22}=g_{22}\). The conductance \(y_{12}\) is set equal to zero, assuming that the amplifier is neutralized.

It is known that the voltage gain of a stage with a single tuned circuit is

\[ K=\frac{m_1m_2}{g_k}\cdot\frac{1}{1+\dfrac{g_{\mathrm{in}}}{g_k}}, \tag{3} \]

where

\[ g_k=\frac{1}{R_k} \]

is the intrinsic resonant conductance of the circuit;

\[ g_{\mathrm{in}}=m_1^2 g_n + m_2^2 g_{22} \]

is the introduced conductance;

\(m_1\) and \(m_2\) are coupling coefficients determined by the coupling elements and by the circuit for connecting the transistor into the loop [4].

\(g_{\mathrm{n}}\) is the load conductance.

The dependence of the conductance \(g_{21}\) on the collector current, with accuracy sufficient for engineering calculation, can be represented by the expression [5]:

\[ g_{21}=\frac{a i_{\mathrm{k}}}{1+a b i_{\mathrm{k}}}, \tag{4} \]

\[ a=\frac{q}{kT}, \]

where \(q\) is the electron charge;

\(k\) is Boltzmann’s constant;

\(T\) is the temperature in Kelvin;

The coefficient \(b\) is defined as the ratio of the distributed base resistance \(r_{\mathrm{b}}\) to the mean value of the static current gain \(h_{21\mathrm{e}}\).

The dependence \(g_{22}(i_{\mathrm{k}})\) is expressed by the relation [6]

\[ g_{22}=C\cdot i_{\mathrm{k}}, \tag{5} \]

\(C\) is a coefficient that depends on the limiting frequency and has the dimension \(\mu\text{mho}/\text{mA}\).

\[ C \approx \frac{h_{22\mathrm{b}}}{(1-h_{21\mathrm{b}})\cdot i_{\mathrm{en}}}, \quad [7] \]

\(h_{21\mathrm{b}}\) is the current gain in the common-base circuit.

For effective regulation of the gain of the stage, the current \(i_{\mathrm{k}}\) must be varied over wide limits. However, its maximum and minimum values are constrained by a number of conditions. Thus, \(i_{\mathrm{kmax}}\) is limited by the power dissipated at the transistor collector. Even slight heating of the transistor due to the action of \(i_{\mathrm{k}}\) can cause instability of the APC. At large \(i_{\mathrm{k}}\), the power consumed by the stage from the nonlinear detector increases, which requires additional amplification in the APC circuit. At large collector currents, those transistors that have a small value of \(b\) are regulated better.

The value \(i_{\mathrm{kmin}}\) is selected taking into account the requirement of minimal nonlinear distortions, which are determined mainly by the nonlinearity of the input characteristics of the transistor and by the amplitude of the high-frequency voltage \(U_{\mathrm{c}}\). As \(i_{\mathrm{k}}\) decreases, the influence of the uncontrolled collector current \(I'_{\mathrm{ko}}\) on the operation of the circuit increases. In this case the temperature stability of the stage deteriorates significantly.

The range of variation of the current \(i_{\mathrm{k}}\), selected with these conditions in mind, is realized by changing the voltage supplied from the nonlinear detector to the input of the transistor. The position of the operating point in the initial mode, and its stabilization when the temperature changes, when the transistor is replaced, etc., is accomplished by the resistances \(R_1, R_2, R_3\), which are calculated by known formulas. In this case, the instability of the operating-point position can be characterized by the quantity [8].

\[ \frac{\Delta i_{\mathrm{k}}}{i_{\mathrm{k}}} = \frac{R_1+R_{\mathrm{e}}+\dfrac{R_1R_{\mathrm{e}}}{R_2}} {h_{21\mathrm{b}}\cdot E_{\mathrm{k}}} \cdot I_{\mathrm{ko}}(0)\cdot e^{\gamma t}; \tag{6} \]

\(t\) is the temperature,

\(I_{\mathrm{ko}}(0)\) is the reverse current of the collector junction at \(t=0^\circ\text{C}\),

\(\gamma\) is the temperature coefficient.

The value \(R_\mathrm{e}\) must be chosen somewhat smaller than in ordinary stages, since at a large value of it the effectiveness of the AGC circuit decreases owing to the action of the current negative feedback. In stages with AGC, in [9] it is recommended to choose

\[ \frac{\Delta i_k}{i_k} \approx \frac{\Delta i_\mathrm{e}}{i_\mathrm{e}} = 0.2 \div 0.4. \]

The influence of the control signal on the control parameter is estimated by the control coefficient: \(N=\dfrac{\Delta i_k}{\Delta U_y}\). To determine this quantity, the stage is represented as a set of four-terminal networks. Thus, the parameters of the four-terminal network equivalent to the transistor and the resistance \(R_1\) are characterized by the matrix:

\[ [y]= \begin{bmatrix} y_{11}+\dfrac{1}{R_1} & y_{12}-\dfrac{1}{R_1}\\[6pt] y_{21}-\dfrac{1}{R_1} & y_{22}+\dfrac{1}{R_1} \end{bmatrix}. \]

Next, passing to \(z\)-parameters, we determine the matrix of the four-terminal network obtained from the preceding one with the addition of \(R_\mathrm{e}\).

\[ [z]= \begin{bmatrix} \dfrac{y_{22}+\dfrac{1}{R_1}}{\Delta y}+R_\mathrm{e} & \dfrac{-y_{12}+\dfrac{1}{R_1}}{\Delta y}+R_\mathrm{e}\\[10pt] \dfrac{-y_{21}+\dfrac{1}{R_1}}{\Delta y}+R_\mathrm{e} & \dfrac{y_{11}+\dfrac{1}{R_1}}{\Delta y}+R_\mathrm{e} \end{bmatrix} \]

where \(\Delta y\) is the determinant of the matrix \([y]\);
\(\Delta z\) is the determinant of the matrix \([z]\).

Omitting the transformations involving the intermediate matrix \([a]\), we obtain:

\[ N=\frac{\Delta i_k}{\Delta U_y} \approx \frac{ y_{21}-\dfrac{1}{R_1}-R_\mathrm{e}\cdot \Delta y }{ \left(1+\dfrac{R_\mathrm{f}}{R_2}\right)\cdot \Delta z\cdot \Delta y + R_\mathrm{f}\left(y_{11}+\dfrac{1}{R_1}+R_\mathrm{e}\cdot \Delta y\right) }. \]

\(R_\mathrm{f}\) is the resistance of the nonlinear detector filter. Substituting the determinants and assuming that \(y_{21}\gg y_{11}\); \(y_{21}\gg y_{22}\); \(\dfrac{1}{R_1}\ll \dfrac{R_\mathrm{e}\cdot y_{21}}{R_1}\), and also neglecting the complexity of the \(y\)-parameters, after transformations we obtain:

\[ N \approx \frac{ g_{21}\left(1-\dfrac{R_\mathrm{e}}{R_1}\right) }{ 1+R_\mathrm{e}g_{21} + R_\mathrm{f}\left[ g_{11}+\dfrac{1}{R_2} + \dfrac{R_\mathrm{e}(R_1+R_2)}{R_1R_2}\cdot g_{21} \right] }. \tag{7} \]

In the process of regulation, the collector current changes according to the law

\[ i_k=i_{k\min}+N\cdot K_\mathrm{p}\cdot U_2, \tag{8} \]

where \(K_\mathrm{p}\) is the transfer coefficient of the nonlinear detector,

\[ K_\mathrm{p}=\frac{U_y}{U_2}\geq 0. \]

The voltage \(U_y = K_{\mathrm{d}} \cdot U_2 - U_{\mathrm{CM}1}\), then

\[ K_{\mathrm{P}} = K_{\mathrm{d}} - \frac{U_{\mathrm{CM}1}}{U_2}. \tag{9} \]

\(K_{\mathrm{d}}\) is the detection coefficient of diode \(D_1\).

With complete linearization, the voltage \(U_2\) is proportional to the gap \(d\), \(U_2 = Md\), where \(M\) is the proportionality coefficient, and expression (8) is written as

\[ i_{\mathrm{k}} = i_{\mathrm{k}min} + N K_{\mathrm{P}} M d. \tag{10} \]

Linearization of the static characteristic is carried out by an appropriate choice of the dependence of the resistance of the nonlinear-detector filter \(R_{\mathrm{f}}\) as a function of the gap, \(R_{\mathrm{f}} = f(d)\). To find this dependence, we equate expressions (1), (3) and obtain an equation in implicit form

\[ F(i_{\mathrm{k}}, d) = 0. \tag{11} \]

It is now necessary to represent expression (11) in explicit form:

\[ i_{\mathrm{K}} = \varphi(d). \tag{12} \]

If dependence (12) cannot be expressed through elementary functions, the transformation is carried out graphically. Substituting equation (12) into (7) and (10), we obtain:

\[ R_{\mathrm{f}} = \psi(d). \tag{13} \]

This function is formed by selecting the resistances \(R'_1,\ R'_2,\ldots,\ R'_n,\ R''_1,\ R''_2,\ldots,\ R''_n\) and the bias voltages \(U_{\mathrm{CM}1},\ldots,\ U_{\mathrm{CM}2}\). For example,

\[ R_{\mathrm{f}1} = R'_1 + R''_1;\quad U_{\mathrm{CM}1} < U_2 < U_{\mathrm{CM}2}; \]

\[ R_{\mathrm{f}2} = R''_1 \parallel (R'_2 + R''_2) + R'_1;\quad U_{\mathrm{CM}2} < U_2 < U_{\mathrm{CM}3}. \]

In the calculation it is necessary to take into account that the filter \(R_{\mathrm{f}max} C_{\mathrm{f}}\) will not reduce the dynamic accuracy of the instrument if its time constant

\[ \tau_{\mathrm{f}} \ll t_{min}; \tag{14} \]

\(t_{min}\) is the duration of the signal corresponding to the interlamellar section.

In the calculation it was assumed that the constant voltage at the collector \(E_{\mathrm{k}}\) remains unchanged. However, in a real circuit it changes due to ohmic resistances in the transistor supply circuit, reducing the effectiveness of the AGC. Therefore it is necessary to choose \(E_{\mathrm{k}}\) of sufficiently large magnitude and to reduce the resistances in the transistor supply circuit to a minimum.

All of the foregoing makes it possible to propose the following method for calculating the controlled stage for the purpose of linearizing the static characteristic of the profilometer:

1. Find the control law \(K_{\mathrm{PU}}\) according to expression (1).

2. Choose \(i_{\mathrm{k}min}\) and \(i_{\mathrm{k}max}\), and determine

\[ \sigma = \frac{K_{\mathrm{PU}max}}{K_{\mathrm{PU}min}}, \]

where \(K_{\mathrm{PU}}\) is determined by formula (3). If the value found is

\[ \sigma > \frac{K}{f'(d_{max})}, \]

then the circuit will ensure linearization of the characteristic. If this quantity is smaller, then it is necessary to reduce \(d_{max}\) and perform linearization over a smaller section.

1. Calculate the temperature stabilization of the operating point of the stage, proceeding from the specified temperature difference. The magnitude of the instability—

of the collector current is determined by formula (6) for the nominal value $i_k$.

1. The parameters obtained as a result of the calculation are substituted into expression (13). We approximate the resulting dependence, based on the specified accuracy, and carry out the calculation of the nonlinear-detector filter.

Fig. 4. Static characteristic of the profilometer: 1 — before linearization, 2 — after linearization

The developed method for calculating the controlled stage with an AGC circuit was tested experimentally on a profilometer mock-up. A resonant stage on a P416B transistor was used as the controlled stage. The experiment showed that an AGC circuit with a nonlinear detector can be successfully used if the nonlinearity of the sensor reaches 25–30%. The calculation error is about 20% and is readily eliminated in the process of adjusting the profilometer. The test results for the mock-up are presented in Fig. 4.

REFERENCES

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