Calculation of the Magnetic Circuit of a Transformer-Type Angle Transducer Taking into Account Fringing and Leakage Fluxes
Abstract Generated abstract
This paper develops a method for calculating the magnetic circuit of a transformer-type angle transducer operating in a low-saturation regime, where air-gap permeances dominate and steel reluctance is neglected. The approach uses field-pattern construction, following B. K. Bul’s magnetic-circuit methods, to estimate working-gap permeances while accounting for fringing and leakage fluxes, then derives expressions for magnetomotive force, flux distribution, and output emf. A numerical comparison shows that including fringing and leakage substantially changes the calculated permeances, required magnetomotive force, and output voltage. Verification on a model angle transducer used in a prototype flow-velocity sensor gave an error on the order of 10 percent.
Full Text
CALCULATION OF THE MAGNETIC CIRCUIT OF A TRANSFORMER ANGLE TRANSDUCER TAKING INTO ACCOUNT FRINGING AND LEAKAGE FLUXES
A. E. SIMAKOV, M. A. TYGYSKİN
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
For converting deflection angles in sensors of automatic-control systems, contactless transformer angle transducers are used \[1, 2, 3\]. In tracking systems such transducers are sometimes called a “tracking transformer” \[4\].
One of the stages in the calculation of such transducers is the calculation of its magnetic circuit: determination of the magnetic resistances of the steel sections, the permeances of the air gaps, and determination of the required magnetomotive force. The accuracy of calculating the static characteristic of the transducer, and its reactive forces and moments, depends on the accuracy of calculating the magnetic circuit.
A calculation taking into account the resistance of the steel can be carried out by representing the resistance of the steel in complex form \[5, 6\]. To eliminate the influence of the resistance of the steel sections of the magnetic circuit on the static characteristics, the transducers under consideration operate, as a rule, in a low-saturation mode; in this case only the air gaps offer the principal resistance to the magnetic flux.
Transformer angle transducers are used mainly as two-sided-action transducers with a symmetrical magnetic circuit. In such transducers all four pole shoes are identical \[2\]. When it is required to convert deflections of only one sign, one pole shoe may be made shorter than the other three (Fig. 1, a).
The need for a shortened pole shoe, when the ratio of the length of the normal pole to the gap is large, is due mainly to the requirement of eliminating the reactive torque in the zero position of the armature. With small pole shoes and large gaps, the difference in the lengths of the pole shoes may, in addition, lead to a large mismatch in the zero position of the armature. The influence of the length of the shortened pole shoe on the reactive torque and mismatch is considered in \[7\].
This article presents a method for calculating the magnetic circuit without taking into account the resistance of the steel, based on the works of Prof. B. K. Bul’ on the theory and calculation of magnetic circuits \[8, 9, 10\].
The transducer is considered with the excitation winding on the middle pole and with signal windings, connected in opposition, on the outer poles.
Fig. 1. Schematic diagram (a) and equivalent circuit of the magnetic circuit (b) of the transformer angle transducer without taking into account the reluctance of the steel and with the output open.
For currents, EMFs, and voltages, rms values are adopted; for induction and magnetic flux, maximum values.
Taking into account the active resistance of the wire of the excitation winding, the number of turns of this winding [10] is
\[ w_0=\frac{U_{\mathrm{in}}}{\sqrt{C_1^2+C_2^2}}, \tag{1} \]
where
$U_{\mathrm{in}}$ is the supply-source voltage,
\[ C_1=\rho l_{\mathrm{cp}} i_q,\qquad C_2=\frac{\omega \Phi_{\mathrm{p}}}{\sqrt{2}} . \tag{2} \]
Here $\rho$ is the electrical resistivity of the winding wire;
$i_q$ is the current density in the wire of the excitation coil,
$\omega$ is the frequency of the supply source.
The magnetic flux in the core of the pole of the excitation winding is
\[ \Phi_{\mathrm{p}}=B_m S, \tag{3} \]
where $S$ is its cross section,
$B_m$ is the magnetic induction in the core.
The mean length of a turn (Fig. 2a, b) is
\[ \begin{aligned} l_{\mathrm{cp}}&=2(a'+b')+\pi H,\\ a'&=a+2\Delta_1,\\ b'&=b+2\Delta_1 . \end{aligned} \tag{4} \]
To determine the magnetomotive force that provides the specified magnetic flux $\Phi_{\mathrm{p}}$, it is necessary to determine the magnetic permeances of the gaps.
These permeances can be calculated, taking fringing into account, analytically or by constructing the field pattern. Analytically, the specific fringing permeances can be determined with the aid of the experimental curves of Prof. B. K. Bul’ [9] for the corresponding ratios of the magnetic-circuit parameters [7]. It should be borne in mind, however, that direct use of the curves of B. K. Bul’, which were obtained for the case “pole—plane,” is possible without substantial error only for transducers with large radii $r_1$ (Fig. 1a), when the curvature of the armature is insignificant. For small $r_1$, the actual specific fringing permeance will be less than the specific permeance determined from the curves of B. K. Bul’.
Fig. 2. For determining the number of turns of the excitation winding.
This article presents a method for determining the permeances of the gaps by constructing field patterns. In what follows, let us agree to denote the quantities relating to the left half of the transducer (pole of signal winding 1)—magnetic permeances, magnetic fluxes, etc.—with subscript 1, and the same quantities relating to the right half (pole of signal winding 2)—with subscript 2.
To construct the field pattern it is necessary to know the magnetic potential at which the armature is situated. In accordance with the equivalent circuit of the magnetic circuit (Fig. 1b, without taking into account the reluctance of the steel sections), the magnetic potential of the armature is
\[ F_1=\frac{G_0}{G_1+G_2+G_0}\,F_{\mathrm{k}}, \tag{5} \]
\[ G_0=\frac{1}{R_0},\qquad G_1=\frac{1}{R_1},\qquad G_2=\frac{1}{R_2}. \]
The magnetic permeances of the air gaps—the pole of the excitation winding \(G_0\) and the poles of the signal windings \(G_1\) and \(G_2\)—can here be determined approximately analytically. Without taking fringing into account, these permeances are determined only by the parameters of the main gaps. Under this condition, for \(F_1\) one may write:
\[ F_1 \approx \frac{g_{00}}{g_{01}+g_{02}+g_{00}}\,F_{\mathrm{k}}, \tag{6} \]
where \(g_{00}\), \(g_{01}\), \(g_{02}\) are the specific permeances of the main gaps, which are determined (for \(-\varepsilon \le \alpha \le \varphi\)) as:
\[ g_{00}=\frac{a}{\delta_0},\qquad g_{01}=\frac{l_0+x}{\delta_1}, \]
\[ g_{02}=\frac{l_0-x}{\delta_1}; \]
\[ a=\frac{\pi\left(r_1+\dfrac{\delta_0}{2}\right)\gamma}{180}, \]
\[ l_0=\frac{\pi\left(r_1+\dfrac{\delta_1}{2}\right)\varphi}{180}, \tag{7} \]
\[ x=\frac{\pi\left(r_1+\dfrac{\delta_1}{2}\right)\alpha}{180}. \]
Taking (7) into account, the magnitude of the magnetic potential of the armature is
\[ F_1=\frac{1}{1+\kappa l_0'}\,F_{\mathrm{k}},\qquad \kappa=2\,\frac{\delta_0}{\delta_1 a}. \tag{8} \]
The value found for the magnetic potential \(F_1\) will differ somewhat from the actual value because of the presence of fringing.
In accordance with the field patterns shown in Fig. 3a, b, c, let us determine the permeances of the working air gaps, as well as the leakage permeances. The region of armature deflection is in this case bounded by the limits: \(-\varepsilon \le \alpha \le \varphi\).
The permeance of the gap of the pole of the excitation winding is
\[ G_0=\mu_0\left[ \frac{(a-2a_1)}{\delta_0}(b-2b_1)+\varepsilon b\,g_{\mathrm{vt}}+2a\,g_{\mathrm{at}} \right], \tag{9} \]
\[ \mu_0=0.4\pi\,10^{-8}\ \frac{\mathrm{H}}{\mathrm{cm}}, \]
where \(g_{\mathrm{vt}}\) and \(g_{\mathrm{at}}\) are the specific fringing permeances from the end face of the pole and from its faces \(B\) and \(A\) (Fig. 1a). The value \(g_{\mathrm{vt}}\) is determined from the field pattern in Fig. 3a, and \(g_{\mathrm{at}}\) from the field pattern in Fig. 3b:
\[ g_{\mathrm{vt}}=\left(\frac{1}{2}+\frac{4}{4}\right), \]
\[ g_{\mathrm{at}}=\left(\frac{1}{2}+\frac{3}{4}+\frac{1}{8}\right). \tag{10} \]
The permeances of the gaps of the poles of the signal windings are
\[ \begin{aligned} G_1&=\mu_0\left\{ \frac{[(l_0-l_1-l_2)+x]}{\delta_1}\,(b-2b_2)+b\,(g_{\mathrm{tp}_1}+g_{\mathrm{tt}_1})+2(l_0+x)g_{\mathrm{tc}} \right\},\\ G_2&=\mu_0\left\{ \frac{[(l_0-l_3-l_4)-x]}{\delta_1}\,(b-2b_2)+b\,(g_{\mathrm{tp}_2}+g_{\mathrm{tt}_2})+2(l_0-x)g_{\mathrm{tc}} \right\}, \end{aligned} \tag{11} \]
where
\[ \begin{aligned} g_{\mathrm{tp}_1}\ \text{and}\ g_{\mathrm{tp}_2} &\text{ are the specific permeances of fringing from the end faces of the pole shoes,}\\ &\text{the face } N \text{ and the inner face } F \text{ of the armature;}\\ g_{\mathrm{tt}_1}\ \text{and}\ g_{\mathrm{tt}_2} &\text{ are the specific permeances of fringing from the end faces of the pole shoes}\\ &\text{and their faces } t;\\ g_{\mathrm{tc}} &\text{ is the specific permeance of fringing from the end faces of the pole shoes,}\\ &\text{as well as their faces } C \text{ and } E. \end{aligned} \]
Fig. 3. Field patterns for determining the magnetic permeances of air gaps, taking account of fringing and leakage. Parameters of the transducer: \(\gamma=18^\circ,\ \xi=75^\circ,\ r_1=3.1\ \mathrm{mm},\ e=30^\circ,\ z=7\ \mathrm{mm},\ K=2\ \mathrm{mm},\ z_{\eta}=2\ \mathrm{mm},\ e=2)\ \mathrm{mm},\ \delta_0=2\ \mathrm{mm},\ \delta_1=4\ \mathrm{mm},\ h=2)\ \mathrm{mm}\), scale 2:1.
The quantity \(g_{\mathrm{tc}}\) may approximately be considered constant along the beveled pole shoe [7].
In accordance with the field pattern of Fig. 3,a,
\[ \begin{aligned} g_{\mathrm{tp}_1}&=\left(\frac{2}{2}+\frac{3}{4}+\frac{1}{8}+\frac{1}{16}\right),\\ g_{\mathrm{tp}_2}&=\left(\frac{2}{2}+\frac{4}{4}+\frac{1}{16}\right),\\ g_{\mathrm{tt}_1}&=g_{\mathrm{tt}_2}=\frac{1}{2}. \end{aligned} \tag{12} \]
From the field pattern of Fig. 3, c,
\[ g_{\mathrm{tc}}=\left(\frac{1}{2}+\frac{2}{4}+\frac{1}{8}\right). \tag{13} \]
The leakage permeances to the yoke and pole pieces, respectively, are
\[ \begin{aligned} G_{\mathrm{sya}} &= \mu_0(2g_{sb}b+2g_{sa}a),\\ G_{\mathrm{sn}} &= 2\mu_0 g_{sn}b, \end{aligned} \tag{14} \]
where
\(g_{sb}\) and \(g_{sa}\) are the specific leakage permeances to the yoke, respectively from faces \(B\) and \(A\) of the pole of the excitation winding;
\(g_{sn}\) is the specific leakage permeance from face \(B\) to the pole piece.
From the field pattern of Fig. 3, a,
\[ \begin{aligned} g_{sn} &= \frac{1}{20},\\ g_{sb} &= \left(\frac{2}{5}+\frac{1}{10}\right). \end{aligned} \tag{15} \]
From the field pattern of Fig. 3, b,
\[ g_{sa}=\left(\frac{2}{5}+\frac{1}{10}\right). \tag{16} \]
The distance \(K\) between the pole of the excitation winding and the pole piece of the signal-winding pole (Fig. 1, a) was taken as 20 mm on the basis that, in this case, the leakage flux to the pole piece is sufficiently small.
Fig. 4. Dependences of the specific permeances on the distance between the pole of the excitation winding and the pole piece.
Figure 4 gives plots of the dependences of the specific permeances: of the gap of the excitation-winding pole \(g'_i\), taking into account fringing from faces \(B\); of leakage to the pole pieces \(g'_{sn}\); and of leakage from faces \(B\) to the yoke \(g'_{sya}\), on the distance \(K\). These dependences were constructed on the basis of field patterns for various \(K\). The field pattern for \(K=20\) mm is given in Fig. 3, a.
The values of the specific permeances in the plots of Fig. 4 were determined as:
\[ \begin{aligned} g'_i &= (a-2a_1)+2g_{\mathrm{v}i},\\ g'_{sn} &= 2g_{sn},\\ g'_{sya} &= 2g_{sb}. \end{aligned} \tag{17} \]
To provide the specified induction in the core of the excitation winding \(B_m\), the required value of the magnetomotive force is
\[ F_k=\frac{\Phi_n}{\sqrt{2}\,G_{\mathrm{ekv}}}, \tag{18} \]
where \(G_{\mathrm{eq}}\) is the equivalent permeance of the magnetic circuit, which is determined by
\[ G_{\mathrm{eq}}=\frac{G_0(G_1+G_2)}{G_0+G_1+G_2}. \tag{19} \]
The magnetic flux in the gap of the pole of the excitation winding (in accordance with the equivalent circuit, Fig. 1, b):
\[ \Phi_0=\Phi_{\mathrm{p}}-(\Phi_{\mathrm{sp}}+\Phi_{\mathrm{sy}}), \tag{20} \]
where \(\Phi_{\mathrm{sp}}\) and \(\Phi_{\mathrm{sy}}\) are the leakage fluxes at the pole shoes and the yoke. These fluxes are determined by
\[ \Phi_{\mathrm{sp}}=\sqrt{2}\,F_{\mathrm{k}}G_{\mathrm{sp}}, \]
\[ \Phi_{\mathrm{sy}}=\sqrt{2}\,F_{\mathrm{k}}F_{\mathrm{sy}}. \tag{21} \]
The emfs induced in the signal windings are
\[ E_1=\frac{\omega w_{\mathrm{c}}}{\sqrt{2}}\,\Phi'_1, \]
\[ E_2=\frac{\omega w_{\mathrm{c}}}{\sqrt{2}}\,\Phi'_2, \tag{22} \]
where \(\Phi'_1\) and \(\Phi'_2\) are the magnetic fluxes in the cores of signal windings 1 and 2.
The magnitude of these fluxes is determined by
\[ \Phi'_1=\frac{G_1}{G_1+G_2}\,\Phi_0+\Phi_{\mathrm{sp}_1}, \]
\[ \Phi'_2=-\frac{G_2}{G_1+G_2}\,\Phi_0+\Phi_{\mathrm{s}}, \tag{23} \]
\[ \Phi_{\mathrm{sp}_1}=\Phi_{\mathrm{sp}_2}=-\frac{\Phi_{\mathrm{sp}}}{2}. \]
The emf at the converter output is
\[ E_{\mathrm{out}}=E_1-E_2=\frac{\omega w_{\mathrm{c}}}{\sqrt{2}}\Phi_0\,\frac{G_1-G_2}{G_1+G_2}. \tag{24} \]
Table 1 gives the results of calculating the magnetic circuit when fringing and leakage are not taken into account. For comparison, the same table gives the results of the calculation with fringing and leakage taken into account.
Table 1
| Calculation method | \(G_0 \times 10^{-8}\) | \(G_1 \times 10^{-8}\) | \(G_2 \times 10^{-8}\) | \(G_{\mathrm{eq}} \times 10^{-8}\) | \(F_{\mathrm{k}}\) | \(E_{\mathrm{out}}\) |
|---|---|---|---|---|---|---|
| henry | henry | henry | henry | |||
| Without allowing for fringing and leakage | 12.56 | 36.8 | 16.6 | 10.2 | 560 | 13.4 |
| With fringing and leakage taken into account | 28.8 | 53 | 28 | 17.0 | 330 | 8.3 |
The induction \(B_m\) in the core of the pole of the excitation winding was taken equal to \(0.4\cdot 10^{-4}\ \mathrm{Wb/cm^2}\); the number of turns of the signal winding —
2000 turns; the frequency of the power supply is 50 Hz; the armature position is \(\alpha = 28^\circ\). The parameters of the transducer are given in the caption to Fig. 3.
The calculation method presented was verified on a model of an angle transducer used in a prototype flow-velocity sensor with a sensing element—deflecting vane [3, 7], manufactured at the Bakhrushev Mine in Kuzbass. The calculation error in this case was on the order of 10%.
REFERENCES
-
G. N. Ageikin, ed. Sensors of Automatic Monitoring and Control Systems. Mashgiz, 1959.
-
N. N. Chistyakov. Electrical Aviation Instruments. Oborongiz, 1950.
-
M. A. Tyryshkin. Flow-velocity sensors for ventilation streams with a sensing element—a deflecting vane—and a transformer angle transducer. Izvestiya vysshikh uchebnykh zavedenii, Gornyi zhurnal, No. 12, 1963.
-
V. A. Besekerskii et al. Design of Tracking Systems. Sudpromgiz, 1958.
-
N. N. Shumilovskii. A complex method for calculating magnetic circuits. Avtomatika i telemekhanika, No. 4, 1940.
-
B. K. Bul’. Calculation of magnetic circuits with a concentrated magnetizing force. Avtomatika i telemekhanika, No. 6, 1952.
-
M. A. Tyryshkin. Velocity Sensors for Mine Ventilation Flows. Candidate dissertation, 1962.
-
B. K. Bul’. A graphoanalytic method for calculating the magnetic permeances of electrical apparatus. Elektrichestvo, No. 3, 1961.
-
B. K. Bul’. Calculation of the magnetic permeances of air gaps for circular and rectangular poles. Design manual. Correspondence Power Engineering Institute, 1961.
-
B. K. Bul’. A method for calculating a coil with steel under alternating current. Elektrichestvo, No. 8, 1954.