Basic Design Relations in a Shielded Electromagnetic Vibrator
Abstract Generated abstract
This paper develops basic design relations for a shielded electromagnetic vibrator intended for sealed chemical apparatuses where mechanical motion must be transmitted through a protective wall under vacuum, high temperature, aggressive media, or explosion hazard. The authors formulate coupled mechanical and electrical equations that account for eddy currents in metallic shielding elements, magnetic resistance of the air gap, leakage fluxes, and practical resonance operation with sinusoidal coil current. Under simplifying assumptions appropriate to small armature displacements, they derive expressions for the exciting force, dynamic magnetic resistance, and armature displacement amplitude. The resulting formulas are reported to agree satisfactorily with experiments and are proposed for calculating both shielded and unshielded electromagnetic vibrators, including effects of magnetic nonlinearity and eddy currents.
Full Text
BASIC CALCULATION RELATIONSHIPS IN A SHIELDED ELECTROMAGNETIC VIBRATOR
V. Ya. UFIMTSEV, N. S. SIUNOV, B. V. ZHDANOV
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
In the modern chemical industry, technological processes often require equipment to operate under conditions of high vacuum, elevated temperatures, chemically aggressive media, explosion hazard, etc. A weak point of many apparatuses operating under these conditions is the sealing of moving parts, by means of which mechanical energy is introduced from outside into the process zone. Experience in operating chemical apparatuses shows that, at high speeds of moving parts, it is practically impossible to achieve complete hermetic sealing with a gland seal \[1, 2, 3\].
The search for a way out of the situation that had arisen led to the creation of a whole series of devices employing the introduction of mechanical motion into an apparatus by transmitting the energy of an electromagnetic field through a solid wall separating the apparatus from the external environment. Examples of such mechanisms, directly coupled with the working machine, are shielded asynchronous motors, magnetic couplings, reducers, and other mechanisms in which the magnetic flux is closed through the separating partition, serving as a conductor of energy into the technological apparatus from outside. In all the cases mentioned, the apparatus is distinguished by reliability, absolute hermeticity, and high efficiency.
Promising for many chemical technologies (and analogous ones) is the creation of a shielded electromagnetic vibrator (SEV), which imposes no restrictions on the designer in terms of power, pressure, temperature, or medium. An uncomplicated and simple part is introduced into the apparatus—the vibrator armature—which is very easy to protect from the medium of the apparatus.
As an example, Fig. 1 shows a diagram of an SEV. The vibrator stator 1, placed outside the chemical apparatus 2, acts through the separating partition 3 on the armature 4, which is located in the protective casing 5. Such a design of an SEV built into a chemical apparatus is typical, despite the infinite variety of kinematic schemes of vibration-transmission mechanisms. It is obvious that the separating partition, or simply the partition, and the protective casing may be made of any material, either metallic or nonmetallic, capable of performing protective functions in the given medium. In the event that the named structural elements are made of metal, then, when the magnetic flux \(\Phi\) is closed through the magnetic circuit of the vibrator (stator and armature steel), they become shielding (SS), i.e.
in the protective shell of the armature and in the sealing partition, eddy currents are induced, creating a magnetic flux that weakens the main flux \(\Phi\). It is in this sense that the term “shielded vibrator” is used.
Fig. 1. Diagram of the SEV
The system of differential equations of the SEV in general form may be represented as follows:
\[ f(t)=m\frac{d^{2}x}{dt^{2}}+\gamma\frac{dx}{dt}+kx, \tag{1} \]
\[ U_{1}(t)=i_{1}r_{1}+\frac{d}{dt}(L_{1}i_{1})+\frac{d}{dt}(M_{12}i_{2})+\frac{d}{dt}(M_{13}i_{3})+\ldots+\frac{d}{dt}(M_{1n}i_{n}) \tag{2} \]
\[ 0=i_{2}r_{2}+\frac{d}{dt}(L_{2}i_{2})+\frac{d}{dt}(M_{21}i_{1})+\frac{d}{dt}(M_{23}i_{3})+\ldots+\frac{d}{dt}(M_{2n}i_{n}), \tag{3} \]
\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]
\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]
\[ 0=i_{n}r_{n}+\frac{d}{dt}(L_{n}i_{n})+\frac{d}{dt}(M_{n1}i_{1})+ \]
\[ +\frac{d}{dt}(M_{n2}i_{2})+\ldots+\frac{d}{dt}(M_{n(n-1)}i_{n-1}), \tag{4} \]
where
\(f(t)\)—the force applied to the armature;
\(m\)—the mass of the armature and of all moving parts of the vibrator, reduced to the armature;
\(x\)—the linear displacement of the armature, measured from the position of static equilibrium;
\(\gamma\)—the coefficient of velocity (linear) friction, i.e., friction proportional to the first power of velocity (the damping coefficient);
\(k\)—the stiffness coefficient of the armature suspension;
$u_1(t)$—the voltage applied to the terminals of the vibrator coil;
$t$—time;
$i_1, i_2 \ldots i_n$—currents in the circuits;
$r_1, r_2 \ldots r_n; L_1 \ldots L_n$—active resistances and inductances in the circuits;
$M_{ks}$—coefficients of mutual induction between the $k$-th and $S$-th circuits.
The system of equations (1—4) describes all electromechanical processes in the SEV, taking into account eddy currents in the SE core of the vibrator, hysteresis, and the nonlinearity of the magnetization curve [4].
With the resonance tuning of vibration machines that is practically used, the current of the coil of an electromagnetic vibrator without constant bias magnetization is practically sinusoidal [6]; consequently, the equilibrium equations of the e.m.f. (2—4) may be represented in symbolic form as follows:
\[ \dot U_1=(r_1+jx_1)\dot I_1+jx_\mu\dot I_\mu , \tag{5} \]
\[ 0=r_2'\dot I_2' + jx_\mu\dot I_\mu , \tag{6} \]
\[ 0=r_n'\dot I_n' + jx_\mu\dot I_\mu , \tag{7} \]
where
$\dot U_1, \dot I_1, \dot I_2', \ldots \dot I_n'$—the voltage applied to the vibrator coil and the currents in the corresponding circuits;
$x_1$—the inductive reactance due to leakage fluxes that close around the coil and do not link the other circuits;
$x_\mu$—the inductive reactance corresponding to the main magnetic flux $\Phi_\delta$, closing through the air gap of the SEV.
In the system of equations (5—7) it is taken into account that the leakage inductive reactances of the SE are small in comparison with their active resistances, as is the case in low-power electrical machines [7], and, in addition, that the inductive reactances caused by mutual leakage inductance between the circuits are small.
In the system of equations (5—7), the magnetizing current of the SEV is equal to
\[ \dot i_\mu=\dot i_1+\dot i_2' + \ldots + \dot i_n' . \]
Here and below, symbols with primes denote quantities reduced to the number of turns of the SEV coil.
Since the SE are penetrated only by the main magnetic flux and, to an insignificant degree, by leakage fluxes, the coupling coefficient between the circuits is equal to
\[ k_{rs}=1. \tag{9} \]
The complex magnetic resistance (CMR) to the main magnetic flux in the SEV may be represented in the form
\[ \dot z_\mu=\sum_{k=2}^{n}\dot z_{\mu k}, \tag{10} \]
where
$k=1$ corresponds to the circuit of the SEV coil;
$z_{\mu k}$—the CMR of the $k$-th circuit.
For electromagnets satisfying equalities (8—10), the general expression for the disturbing force is valid [4]:
\[ f(t)=\frac{(iw)^2}{2}\cdot \frac{R_{\mu\delta}^{2}}{z_\mu^{2}}\cdot \frac{d}{dx}\cdot \frac{1}{R_{\mu\delta}}, \tag{11} \]
where
\((Iw)\) — the ampere-turns of the EEV coil required to produce the main magnetic flux;
\(R_\delta\) — the active magnetic resistance, AMR, of the air gap of the EEV to the main magnetic flux;
\(z_\mu\) — the CMR to the main magnetic flux.
It can be shown [8] that the leakage magnetic fluxes of the coil do not affect the electrodynamic processes in the electromagnet; however, their magnitude is
\[ \Phi_s = \frac{(Iw)}{R_{\mu s} + j x_{\mu s}}, \tag{12} \]
where
\(R_{\mu s}\) — the AMR to the leakage magnetic flux,
\(x_{\mu s}\) — the reactive magnetic resistance (RMR) to the leakage magnetic fluxes, caused by the presence of eddy currents in the core of the magnetic circuit and in the ferromagnetic EEs.
Since, for electromagnetic vibrators, the following equality is valid [9],
\[ \frac{x_m}{\Delta} < 1, \tag{13} \]
then for the EEV it may be assumed with great accuracy that
\[ \left( \frac{x_m}{\Delta} \right)^2 = 0, \tag{14} \]
where
\(x_m\) — the peak value of the armature vibration displacement;
\(\Delta\) — the mean air gap, taking into account the thickness of the nonferromagnetic EEs.
Taking the expression for the vibrator inductance, due to the time-varying air gap, equal to (10),
\[ L_\delta = - \frac{L_\Delta}{1 - \beta \frac{x}{\Delta}}, \tag{15} \]
where the modulation coefficient \(\beta\) is equal to
\[ \beta = \frac{L_\Delta}{2} \cdot \left( \frac{1}{L_{\min}} - \frac{1}{L_{\max}} \right), \tag{16} \]
and taking into account equality (14), and also assuming
\[ \beta \frac{R_{\mu \Delta}^{2}}{R_{\mu \Delta}^{2} + X_{\mu}^{2}} = 0, \tag{17} \]
we obtain the expression for the exciting force of the EEV in the form
\[ f(t) = \frac{(Iw)^2}{2\Delta} \cdot \frac{R_{\mu \Delta}}{R_{\mu \Delta}^{2} + X_{\mu}^{2}}, \tag{18} \]
where
\(R_{\mu \Delta}\) — the AMR of the mean air gap of the EEV,
\(X_\mu\) — the RMR to the main magnetic flux in the EEV.
Taking, according to (6), the current in the EEV coil as equal to
\[ i_1 = I_m \sin(\omega_c t + z), \tag{19} \]
obtain the amplitude of displacement of the armature of the electromagnetic vibrator
\[ x_m=\frac{I_1^2 w_1^2 \lambda}{2\Delta R_{\mu f}\cdot m\omega_0^2}, \tag{20} \]
where
\(I_1\) — rms value of the current in the coil of the electromagnetic vibrator;
\(\lambda\) — dynamic coefficient of the vibration mechanism according to [6];
\(R_{\mu f}\) — dynamic active resistance of the electromagnetic vibrator, equal to
\[ R_{\mu f}=\frac{R_{\mu \Delta}^{2}+X_{\mu}^{2}}{R_{\mu \Delta}}. \tag{21} \]
Expression (20) may be used for vibrators with nonmetallic protective elements, as well as for electromagnetic vibrators with large air gaps. The results of calculation by (20) show satisfactory agreement with experimental results.
Conclusions
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Basic calculation relationships have been obtained for a shielded electromagnetic vibrator.
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The relationships obtained show satisfactory agreement with experimental results.
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The results obtained in this work may be used in calculating unshielded vibrators, taking into account the nonlinearities of the magnetization curve of the magnetic core and eddy currents.
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