MEASUREMENT OF THE CURRENT RATIO IN SENSORS OF A PERMALLOY MAGNETOMETER

B. N. Kalinin

To measure the distribution of instantaneous values of alternating magnetic fields in accelerator electromagnets, a permalloy magnetometer is used [1, 2]. The total measurement error should not exceed 0.01–0.1%. A significant error in measuring the field with a permalloy magnetometer may be introduced by the error in determining the ratio of direct currents in the magnetizing windings of the measuring sensor and the zero-field-level sensor. The requirements for simultaneous instability of the currents in the magnetizing windings of the sensors can be reduced by a factor of 10–100, since a joint change in the currents will lead only to a change in the field level. The character of the field distribution in accelerator electromagnets depends only weakly on the level of the measured field. A relative change in the field level by \(\delta H\) leads to an error in determining the field distortions

\[ \Delta H = H_i \delta H, \tag{1} \]

where \(H_i\) is the field distortion due to eddy currents, the residual field, and the finite value of the magnetic permeability of the steel.

Fig. 1. Schematic diagram for measuring the current ratio with a voltage stabilizer

In [2] a circuit is described for measuring the ratios of currents in the magnetizing windings of the sensors of a permalloy magnetometer, using one source of stabilized voltage (Fig. 1).

As analysis shows, this circuit has the essential disadvantage that temperature changes in the resistance of the sensor windings lead to a considerable error in measuring the current ratio:

\[ \delta\left(\frac{I_1}{I_2}\right) = \gamma l \frac{R_k}{U_0}\Delta T, \tag{2} \]

where \(\gamma\) is the temperature coefficient of resistance of the sensor winding,
\(I\) is the current in the sensor winding,
\(R_k\) is the resistance of the sensor winding,
\(U_0\) is the stabilized voltage of the power supply,
\(\Delta T\) is the temperature difference between the windings of the measuring sensor and the field-level sensor.

For example, at \(I = 0.1\ \text{A}\), \(R_k = 100\ \Omega\), \(U_0 = 400\ \text{V}\), \(\gamma = 0.004\ \text{deg}^{-1}\text{C}\), and \(\Delta T = 30^\circ\text{C}\), the measurement error is \(0.3\%\).

To measure the ratio of the currents in the sensors of a permalloy magnetometer, a circuit may be used in which each sensor is supplied from a separate adjustable DC stabilizer. In this case, two current stabilizers with instability up to \(0.005\%\) are required, which complicates the circuit as a whole.

Fig. 2. Schematic circuit for measuring the current ratio with a current stabilizer

We have proposed a circuit (Fig. 2) that makes it possible to measure the ratio of the currents in the sensors of a permalloy magnetometer with an error of up to \(0.01\%\). In this circuit, one adjustable current stabilizer with an instability of \(0.5\text{–}1.0\%\) is used. The magnetizing windings of both sensors \(K_{\mathrm{u}}\) and \(K_{\mathrm{y}}\) are connected in series. The current \(I_{\mathrm{u}}\) in the winding of the measuring sensor is regulated by the shunting decade box \(R_6\). By means of the resistance decade box \(R_5\), equilibrium is established in the bridge formed by the resistances \(R_1\), \(R_2\), \(R_3\), and \(R_5\). The ratio of the currents is then proportional to the value of the resistance set on the decade box \(R_5\):

\[ \frac{I_{\mathrm{u}}}{I_{\mathrm{y}}} = R_5 \frac{R_1}{R_2 R_3}. \tag{3} \]

The measurement error, equal to the difference between the current ratio obtained from formula (3) and the true current ratio, can be divided into the circuit error caused by the fact that the current ratio is not exactly proportional to \(R_5\) because \(R_1\) is shunted by the resistance \(R_2\), the temperature error—caused by the temperature change in the resistance of the winding of the measuring sensor—and the error of the resistance decade box \(R_5\).

Solving the system of loop currents (Fig. 2) under the condition of bridge equilibrium, we obtain the ratio of currents in the sensor windings:

\[ \frac{I_i}{I_y} = \frac{R_1}{R_3}\cdot \frac{R_5^2 a + R_5 b + c}{R_5 d + f}, \tag{4} \]

where

\[ a=(R_3+R_4)(R_2+R_2+R_1); \]

\[ b=(R_3+R_4)\bigl[(R_1+R_2)(R_3+R_{\Gamma})+R_3R_{\Gamma}\bigr] +R_3R_4(R_1+R_2+R_{\Gamma}); \]

\[ c=R_3R_4\bigl[(R_1+R_2)(R_3+R_{\Gamma})+R_3\cdot R_{\Gamma}\bigr]; \]

\[ d=(R_1+R_2)(R_3+R_4)(R_1+R_2+R_{\Gamma}); \]

\[ f=R_2R_3R_{\Gamma}(R_1+R_2) +R_4\bigl[(R_1^2+R_2^2)(R_3+R_{\Gamma})+2R_1R_2R_3+ \]

\[ \qquad\qquad\qquad\qquad +R_{\Gamma}(R_1R_2+R_1R_3+R_2R_3)\bigr]. \]

If we take \(R_4=R_{\Gamma}=0\), then from (4) we obtain

\[ \frac{I_i}{I_y} = \frac{R_1}{R_3}\cdot \frac{R_5+R_3}{R_1+R_2}. \tag{5} \]

The reduced circuit error in measuring the current ratio, defined as the ratio of the absolute measurement error \(\Delta I_i\) to the current \(I_y\) in the winding of the field-level sensor, can be written in the form

\[ \frac{\Delta I_i}{I_y} = \frac{R_5}{R_{5\max}} - \frac{I_i}{I_y}, \tag{6} \]

where \(R_{5\max}\) is the maximum value of \(R_5\). Substituting (5) into (6) and taking into account the equality \(R_1R_{5\max}=R_2R_3\), we obtain

\[ \frac{\Delta I_i}{I_y} = \frac{R_1}{R_1+R_2} \left( \frac{R_5}{R_{5\max}}-1 \right) \approx \frac{R_1}{R_2} \left( \frac{R_5}{R_{5\max}}-1 \right). \tag{7} \]

As follows from (7), the reduced circuit error increases linearly from 0 (at \(R_5=R_{5\max}\)) to \(R_1/R_2\) (at \(R_5=0\)). To reduce the circuit error it is necessary to decrease the ratio \(R_1/R_2\).

It should be noted that in this circuit the current in the winding of the measuring sensor is always less than the current in the winding of the field-level sensor. Therefore, in order to measure magnetic fields whose magnitude is greater than the field level, it is necessary that the bias constant of the measuring sensor winding be greater than the constant of the field-level sensor. The ratio of the magnetic fields at the locations of the measuring sensor and the field-level sensor is equal to

\[ \frac{H_i}{H_y} = \frac{I_i q_i}{I_y q_y}, \tag{8} \]

where \(q_i\) and \(q_y\) are the bias constants of the measuring sensor and the field-level sensor, respectively. The reduced error in measuring the magnetic-field ratio, taking (6) and (8) into account, can be written in the form

\[ \frac{\Delta H_i}{H_y} = \frac{q_i}{q_y} \left( \frac{R_5}{R_{5\max}}-1 \right). \tag{9} \]

The current constant of the galvanometer must be no less than

\[ \Delta I_{\Gamma} = I_{y\min} \frac{R_1R_{5\max}^{2}\delta R_{5\min}} {(R_{5\max}+R_{\Gamma})^{2}R_2} \approx H_{y\min} \frac{R_1}{q_yR_2}\delta R_{5\min}, \tag{10} \]

where \(H_{y\min}\) is the minimum value of the level of the measured field that can be measured with the given circuit with an error not exceeding \(R_1/R_2\); \(\delta R_{5\min}=\Delta R_5/R_{5\max}\) is the permissible error in reading the current ratio, equal to the minimum value of the step of \(R_5\), referred to the maximum value \(R_5=R_{5\max}\).

The temperature error of measurement can be determined from expression (4) as the difference of the current ratios at the temperature of the measuring-sensor winding \(T_0\) and \(T_0+\Delta T\).

As an example, a calculation is given of the circuit and temperature errors in measuring the current ratio for \(R_1=R_3=10\ \Omega\), \(R_2=100\ \text{k}\Omega\), \(R_4=100\ \Omega\), \(R_{5\max}=100\ \text{k}\Omega\), \(R_r=10\ \text{k}\Omega\), \(\Delta I_r=1.8\cdot 10^{-9}\ \text{A/division}\), \(I_{y\min}=0.2\ \text{A}\), \(\delta R_{5\min}=1\cdot 10^{-4}\), and \(\Delta T=30^\circ\text{C}\). The calculation results are given in Table 1.

Table 1

Column 4 (Table 1) gives the results of calculating the reduced circuit error by the exact formulas (4) and (6), and column 5 by the approximate formula (7). As is seen from comparing columns 4 and 5, the error in determining the circuit error by the approximate formulas is no more than \(7.7\cdot 10^{-3}\%\). Column 6 gives the reduced temperature error, calculated by formula (4), for a change in the temperature of the measuring-sensor winding by \(30^\circ\text{C}\). Temperature changes in the resistance of the field-level sensor winding do not lead to error, since the circuit is powered from a current stabilizer. As is seen from Table 1, the reduced error in measuring the current ratio is no more than \(0.015\%\). By choosing the ratio \(R_1/R_2\), the reduced circuit error can be reduced to \(0.01\%\).

Conclusions

The proposed circuit is rather simple and makes it possible to measure the ratio of currents in the sensors of a permalloy magnetometer with a reduced error down to \(0.01\%\). The sensors are powered from a single current stabilizer of modest stability. For approximate calculations of the circuit elements one may use formulas (3), (7). With an appropriate choice of the circuit elements, the measurement error is determined only by the error of the resistance box \(R_5\).

References

1. K. N. Shorin, Yu. N. Metelnikov, M. V. Eremin, PTE, No. 4, 1958.
2. G. M. Tumaikin, V. N. Eponeshnikov, P. P. Krasnososenskikh. Proceedings of the IV Interuniversity Conference on Electron Accelerators. Izd. “Vysshaya shkola,” Moscow, 1964.