ELECTRICAL ENGINEERING

M. A. ROZENBLAT, O. G. KASATKIN

MAGNETIC INTEGRATION AND DIFFERENTIATION OF ELECTRICAL SIGNALS

(Presented by Academician V. A. Trapeznikov, December 24, 1963)

Recently, branched magnetic cores made of ferromagnets with a rectangular hysteresis loop have found wide application for storing analog quantities \((^{1,2})\). A typical core is shown in Fig. 1a. The right-hand rod is maintained in a state of positive saturation by the current \(i_{\text{см}}\) in the winding \(W_{\text{см}}\). The magnetic flux \(\Phi_y\)

Fig. 1

of the left-hand rod is controlled by the current \(i_y\) in the winding \(W_y\) within the limits from \(-\Phi_s\) to \(\Phi_s = B_s S\), where \(B_s\) is the saturation induction and \(S\) is the cross-sectional area of the rod.

By the current \(i^c\) in the winding \(W_{\text{сч}}\), the information recorded in the core in the form of the residual flux \(\Phi_y\) is read out. For an ideal rectangular hysteresis loop (Fig. 1b), the average value of the emf induced in the output winding \(W_{\text{в}}\) is \((^3)\):

\[ U_{\text{в}} = 2W_{\text{в}} f(\Phi_y + \Phi_s), \tag{1} \]

where \(f\) is the frequency of the readout current.

Equation (1) shows that, in order to create, on the basis of such a core, a device integrating the voltage \(u\), it is sufficient that the residual flux \(\Phi_y\) vary proportionally to the integral \(\int u\,dt\). The circuit of Fig. 1a solves this problem. For the input circuit in this circuit we have

\[ u = (R_0 + R_{\text{вх}})i + W_{\text{ос}}\frac{d\Phi_y}{dt}, \tag{2} \]

where \(R_0\) is the active resistance of the feedback winding \(W_{\text{ос}}\), \(R_{\text{вх}}\) is the input resistance of the amplifier \(y\), and \(i\) is the current at the amplifier input.

The current \(i_y\) in the winding \(W_y\) up to saturation of the rod of the core under consideration \((|\Phi_y| < \Phi_s)\) can approximately be expressed in terms of the coercive force \(H_c\) as

\[ i_y \approx \pm \frac{H_c l}{W_y}, \tag{3} \]

where \(l\) is the mean path length for the flux \(\Phi_y\).

Therefore the first term on the right-hand side of (2), which determines the error

of integration, can be expressed in the following form

\[ (R_{\mathrm{o}}+R_{\mathrm{in}})i=(R_{\mathrm{o}}+R_{\mathrm{in}})\frac{i_y}{K_I} =\pm \frac{(R_{\mathrm{o}}+R_{\mathrm{in}})H_c l}{W_y K_I} =\pm u_0, \tag{4} \]

where \(K_I\) is the current gain of amplifier \(U\).

Substituting (4) into (2) and integrating, we obtain

\[ \Phi_y=\Phi_0+\frac{1}{W_{\mathrm{oc}}}\int_0^t (u\pm u_0)\,dt. \tag{5} \]

If we set \(\Phi_0=-\Phi_s\), then from (1) and (5) we obtain

\[ v_{\mathrm{v}}=2\frac{W_{\mathrm{v}}}{W_{\mathrm{oc}}}f\int_0^t (u\pm u_0)\,dt . \tag{6} \]

Thus, even with an ideal rectangular hysteresis loop, an integration error occurs, due to the fact that changing the magnetic state of the core requires a finite value of the current \(i_y\), and consequently also of \(i\).

If we denote by \(u_m\) the maximum value of \(u\), then for the relative error we have

\[ \gamma_{(H_c)}=\frac{u_0}{u_m} =\frac{(R_{\mathrm{o}}+R_{\mathrm{in}})H_c l}{W_y K_I u_m}. \tag{7} \]

It follows from (7) that, in principle, arbitrarily small values of \(\gamma_{(H_c)}\) can be obtained by increasing \(K_I\); however, in practice, as \(K_I\) is increased, the problem arises of ensuring the stability of the circuit. Another way to reduce \(\gamma_{(H_c)}\) consists in increasing \(u_m\), but in this case the maximum integration time decreases. A reduction of \(i_y\), and consequently of \(u_0\) and \(\gamma_{(H_c)}\), can also be achieved by superposing an alternating current on \(i_y\).

The integration time for \(u=u_m\gg u_0\) is determined from (5) by substituting \(\Phi_0=-\Phi_s\) and \(\Phi_y=\Phi_s\):

\[ T=\frac{2\Phi_s W_{\mathrm{oc}}}{u_m} =\frac{2\Phi_s W_{\mathrm{oc}}\gamma_{(H_c)}}{u_0} =\frac{2W_y W_{\mathrm{oc}}\Phi_s\gamma_{(H_c)}K_I}{H_c l(R_{\mathrm{o}}+R_{\mathrm{in}})}. \tag{8} \]

Thus, as the permissible value of the relative error \(\gamma_{(H_c)}\) and \(K_I\) are increased, the integration time increases.

In real integrating devices there are also the following sources of error: 1) variation of the frequency of the readout current, which, according to (6), causes a proportional change in \(U_{\mathrm{v}}\); 2) variation of the residual flux \(\Phi_y\) after the absolute value of \(i_y\) becomes less than \(H_c l/W_y\), because of the nonideal rectangularity of the hysteresis loop; 3) variation of \(\Phi_y\) with temperature (at \(u=0\)), due to the dependence of the residual magnetization of ferromagnets on temperature; 4) variation of the amplitude of the readout current, which, because of the nonideal rectangularity of the hysteresis loop, causes a change in the mean value \(\overline{U}_{\mathrm{v}}\). To eliminate this source of error one can use output-voltage gating [3], which consists in performing rectification by means of a controlled switch that is closed when \(i_{\mathrm{sc}}>0\) and open when \(i_{\mathrm{sc}}<0\). In this case the mean value of the output voltage is determined by formula (1), independently of the amplitude and shape of the readout-current curve, provided \(i_{\mathrm{sc}}>i_{\mathrm{sc\,min}}\), where \(i_{\mathrm{sc\,min}}\) is the minimum value of the readout current required for complete remagnetization of the magnetic core around the small aperture of the core.

A feature of the described circuit is the absence of feedback between the output and the input of the device. Switching off the readout current does not disturb the integration process. Therefore, the output quantity can be read out either continuously or at separate intervals of time. Moreover, if the zero drift of amplifier \(У\) does not exceed

\[ |i_{у0}| < \frac{H_c l}{W_y}, \]

then there is no drift with time of the output voltage of the integrating device.

The dynamic characteristics of the circuit are determined by the frequency of the readout current, and also by the bandwidth of the output smoothing filter and of amplifier \(У\).

Fig. 2

Experimental investigations were carried out on a core made of magnetic material 50 NP with a lamination thickness of 0.05 mm. Circuit data: \(B_s = 1.45\) T, \(H_c = 0.2\) A/cm, \(S = 0.29\ \text{cm}^2\), \(l = 11\) cm, \(W_y = 500\), \(W_{\mathrm{oc}} = 2500\), \(R_o = 100\ \Omega\), \(K_I = 1.6 \cdot 10^5\), \(R_{\text{in}} = 1000\ \Omega\), \(f = 400\) Hz. The investigations showed that, at \(\gamma_{(H_c)} = 1\%\) and \(u = u_m\), \(T = 70\) s; the change in the output voltage caused by decreasing \(i_y\) from \(\dfrac{H_c l}{W_y}\) to zero does not exceed 1%; as the temperature increases, the output voltage decreases by \(0.08\%/^\circ\text{C}\), but by applying temperature compensation this error can be reduced to \(0.01\%/^\circ\text{C}\).

A change in the amplitude of the readout current by \(\pm 20\%\) does not cause a change in the mean value of the sampled \(U_{\text{out}}\). The frequency characteristic of the circuit, with smoothing by an \(RC\) circuit with \(T = 3 \cdot 10^{-3}\) s, is shown in Fig. 2.

If, during the integration process, the flux \(\Phi_y\) reaches the value \(+\Phi_s\) or \(-\Phi_s\) and the input signal corresponds to a change in \(\Phi_y\) in the same direction in which the core is already saturated, then the current \(i_y\) at the amplifier output rises sharply, assuming the value

\[ i_y = K_I \frac{u}{R_o + R_{\text{in}}}\operatorname{sign}\Phi_y . \]

Fig. 3

This current pulse \(i_y\) can be used to feed a signal indicating that the integrator is full. If this same current pulse \(i_y\) is applied to the input of a reversible pulse counter and, at the same time, the magnetization of the left-hand rod is quickly changed to the opposite direction, then an analog-digital integrating device is obtained for any integration time.

The described multi-aperture core can serve as the basis of a differentiating device (Fig. 3). In winding \(W_d\), placed on the left-hand rod of the cores, a voltage is induced

\[ u_d = W_d \frac{d\Phi_y}{dt}. \tag{9} \]

From equation (9) it is clear that, to construct a differentiating device on the basis of such a core, it is sufficient that \(\Phi_y\) vary proportionally to \(u(t)\). This problem is solved by the circuit of Fig. 3. The difference of voltages \(u(t) - U_{\text{out}}\) is applied to the input of amplifier \(У\). The output of the amplifier is connected

to the winding \(W_y\), and the magnetomotive force \(i_y W_y\) changes \(\Phi_y\) in such a way that the difference \(u(t) - U_{\mathrm{в}}\) decreases. If the voltage \(u(t)\) varies sufficiently slowly, so that the lag in the readout circuit may be neglected, and the amplifier \(У\) has a sufficiently high gain, then the circuit maintains

\[ U_{\mathrm{в}} = u(t). \tag{10} \]

Substituting (1) into (9) and taking (10) into account, we obtain:

\[ u_{\mathrm{д}} = \frac{W_{\mathrm{д}}}{2 W_{\mathrm{в}} f}\,\frac{du(t)}{dt}. \]

Institute
of Automation and Telemechanics

Received
23 XII 1963

REFERENCES

¹ C. L. Bovajian, Proc. Special Techn. Conf. on Nonlinear Magnetics and Magnetic Amplifiers, September, 1959. ² G. F. Haass, Nachrichtentechn. Zs., H. 8 (1961). ³ О. Г. Касаткин, М. А. Розенблат, Automation and Telemechanics, No. 4 (1964).