A Method for Calculating and Assessing the Electrodynamic Stability of Transformers with Layer Windings at the Design Stage

I. D. Kutyavin, P. E. Azaryan

Many works have been devoted to the problem of the strength of transformer windings under the action of axial forces; their purpose has been to determine axial electromagnetic forces, the resulting mechanical stresses in conductors and insulation, and also the forces acting on the end insulation.

In these works the short-circuit current (s.c.) was regarded as constant in time, and the winding configuration as allowing displacement of turns and layers or mutual displacement of windings under the action of electromagnetic forces, which leads to a change in the forces themselves [1, 2, 3].

The electromagnetic force, varying during the short-circuit process, acts on a winding that constitutes an elastic mechanical system. To analyze the phenomena occurring in the winding, it is necessary to take into account the time-varying coefficients of elasticity of various insulating materials, the forces of inertia, elasticity, and friction [5, 6, 7, 8, 9].

Until now, almost all authors have investigated the mechanical processes of helical-coil windings of high-power transformers, where the elementary coils are separated by spacing strips which, together with the turn insulation, constitute an elastic spring (as a result of preliminary pressing) with its own natural frequency of free oscillations.

Friction forces here cannot play a significant role, and some authors [8, 9] carried out investigations in a conservative system. As for the mechanical processes in the turns or layers of layer windings, they have scarcely been considered in the specialized literature.

The purpose of this article is to investigate mechanical processes in the layers of layer windings in a dissipative system under a non-steady short-circuit condition, with the development of an engineering method for calculating and assessing the required friction force between the layers of layer windings, such that the displacement of the layers under the impact short-circuit current would not exceed the permissible deformation of the layer end insulation. To simplify the problem, the following assumptions are adopted:

1. The system performs only longitudinal oscillations. There are no distortions (bending) of axes or cross sections,

2. Between the spacers of the yoke insulation and the layers (after preliminary pressing), as well as between the turns in a layer under tight winding, there are no gaps,

1. Friction between layers is uniform along the length of the layer and proportional to the velocity of its displacement.

2. The end insulation has negligibly small mass and is regarded as an ideal spring.

3. In comparison with the electrodynamic forces, the force of gravity is insignificant.

4. The layer of turns connected to one another by springs with large stiffnesses is considered absolutely rigid.

The investigation is reduced to the known problem of forced longitudinal vibrations of an elastic rod in a dissipative system.

The equation of vibrations of the \(i\)-th layer in the case of a short circuit at the instant when the voltage passes through zero will be

\[ \ddot S_i=2l_i\dot S_i+k_i^2S_i=\frac{1}{2}H(1+2e^{-\alpha t}-4e^{-2\alpha t}\cos\omega t+\cos 2\omega t), \tag{1} \]

where

\[ H=\frac{F}{nM},\qquad l_i=\frac{\eta_i}{2M},\qquad k^2=\frac{\kappa}{M}. \tag{2} \]

For the beginning and end of the \(i\)-th layer the following initial conditions hold:

\[ t=0;\qquad S_i=0;\qquad \dot S_i=0. \tag{3} \]

Integrating equation (1), we obtain

\[ S_i=X_i+\left[\frac{H}{2k_i^2}+\frac{He^{-2\alpha t}}{\omega^2+(C_i-2\alpha)^2}\right]- \]

\[ -\frac{2He^{-\alpha t}\cos(\omega t-\alpha_i)} {\sqrt{(k_i^2-2C_i\alpha+\alpha^2-\omega^2)^2-4\omega^2(C_i-\alpha)^2}} +\frac{H\cos(2\omega t+\delta_i)} {2\sqrt{(k^2-4\omega^2)^2+16C_i^2\omega^2}}, \tag{4} \]

where

\[ \gamma_i=\operatorname{arctg}\frac{2\omega(\alpha-C_i)} {\omega^{*2}-\omega^2+(C_i-\alpha)^2}, \tag{5} \]

\[ \delta_i=\operatorname{arctg}\frac{2\omega C_i}{4\omega^2-k^2}, \tag{6} \]

\[ X_i=a_i e^{C_i t}\operatorname{sh}(\omega^*t+\beta_i),\qquad \text{when } C_i<k_i, \]

with

\[ \omega^*=\sqrt{k_i^2-C_i^2}, \tag{7} \]

\[ X_i=a_i e^{-C_i t}\operatorname{sh}(\omega^*t+\beta_i),\qquad \text{when } C_i>k_i, \]

with

\[ \omega^*=\sqrt{C_i^2-k_i^2}, \tag{8} \]

\[ X_i=e^{-C_i t}(C_{i1}+C_{i2}t),\qquad \text{when } C_i=k_i, \tag{9} \]

with \(\omega^*=0\).

\(a_i,\ \beta_i,\ C_{i1}\) and \(C_{i2}\) are constants determined from the initial conditions (3).

As is known, at the beginning of the short circuit an electromagnetic force with apparent frequency \(\omega\) acts on the layer. In this case the displacement of the layer will be

\[ S=\frac{2He^{-\alpha t}\cos(\omega t+\gamma_i)} {\sqrt{(k_i^2+2\alpha C_i+\alpha^2-\omega^2)^2+4\omega^2(C_i-\alpha)^2}}. \tag{10} \]

Investigations of equation (4) in the dynamic and static regimes in a dissipative and conservative system with respect to the terms have shown,

that the displacement of the layers, expressed by equation (10), is 3–4 times greater than in the other modes, and depends strongly on the damping coefficient \(C_i\).

In the dynamic mode the displaced layer resonates weakly when

\[ C_i=\frac{\alpha\left(k_i^2+\omega^2+\alpha^2\right)}{2\left(\omega^2+\alpha^2\right)} \tag{11} \]

and resonates strongly in the static mode, when

\[ C_i=\frac{k^2+\alpha^2}{2\alpha}. \tag{12} \]

Consequently, from the interval

\[ \frac{\alpha\left(k_i^2+\omega^2+\alpha^2\right)}{2\left(\omega^2+\alpha^2\right)}<C_i<\frac{k^2+\alpha^2}{2\alpha} \tag{13} \]

one can choose such a \(C_i\) for which the amplitude at the instant \(t=0\) from (10) would not exceed the required value, that is,

\[ \frac{2H}{\sqrt{\left(k_i^2-2\alpha C_i+\alpha^2-\omega^2\right)^2+4\omega^2\left(C_i-\alpha\right)^2}}\leq \frac{[\sigma]_{\text{ins}}h}{E_{\text{ins}}}, \tag{14} \]

where \(E_{\text{ins}}\) is the modulus of elasticity of the insulation, equal for pressed electrocardboard to \(3.5\cdot10^3\ \frac{\text{kgf}}{\text{cm}^2}\) at a pressing force of \(\frac{\text{kgf}}{\text{cm}^2}\) (10); \([\sigma]_{\text{ins}}\) is the stress of the end insulation of the electrocardboard layer; it is taken as \(200\ \frac{\text{kgf}}{\text{cm}^2}\) [1]; \(h\) is the height of the end insulation.

On the other hand, the friction force is

\[ F_{\tau}=2C_iM\frac{d\delta}{dt}. \tag{15} \]

Differentiating equation (10) and substituting into (15), after some transformation we obtain

\[ F_{\tau}= \frac{ 8F\omega^2+\dfrac{4F\alpha\left(k_i^2-\omega^2\right)}{C_i} }{ 4\omega^2 n+\dfrac{n\left(k_i^2-\omega^2\right)^2}{C_i^2} }. \tag{16} \]

If the system is conservative, that is, \(C_i=0\), then \(F_{\tau}=0\). In this case the natural frequency of the system depends on the preliminary pressing of the windings; and if the system is dissipative, then with an increase in the damping coefficient \(C_i\), \(F_{\tau}\) increases, and as \(C_i\to\infty\), \(F_{\tau}\) tends to the constant value

\[ \lim_{C\to\infty}F_{\tau}=\frac{2F}{n}. \]

We take

\[ F_{\tau}=\frac{2F}{n}. \tag{17} \]

When (15) is satisfied, \(C_i\) from expression (16) assumes the finite value

\[ C_i=\frac{k^2-\omega^2}{2}. \tag{18} \]

Expression (18) is the value of the damping coefficient at which the friction force corresponds to value (17), and lies in interval (13).

To ensure the stability of the windings in the same regime at the value \(C_i\), it is necessary and sufficient that the displacement of the layers in the unsteady short-circuit regime satisfy inequality (14).

Substituting value (18) into (14) and solving the inequality with respect to \(k^2\), we obtain

\[ k^2 \geq \omega^2 + \frac{E_{\mathrm{iz}} a F_{\mathrm{t}}}{M |\sigma|_{\mathrm{iz}} h \sqrt{\omega^2+a^2}} . \tag{19} \]

Expression (19) shows that the greater the friction forces between the layers in the dynamic regime, the greater the stiffness of the winding.

Consequently, in a dissipative system the stiffness of the winding depends not only on the preliminary pressing, but also on the friction force between the layers.

In the particular case where \(\omega = 0\) and the supporting surface

\[ Q=\frac{F}{n[v]_{\mathrm{iz}}}, \qquad \text{then} \qquad K=\frac{E_{\mathrm{iz}}Q}{h}. \]

We have obtained the well-known formula of mechanics for determining the stiffness of a spring in the static regime.

From expression (19), the supporting surface \(Q'\) in the dynamic regime will be

\[ Q' \geq \frac{4\omega^2 h M}{E_{\mathrm{iz}}}+\frac{4aF_{\mathrm{t}}}{[v]_{\mathrm{iz}}\omega}. \tag{20} \]

For transformers of sizes I and II, with \(M\) in \(\mathrm{kgf}\cdot \mathrm{s}^2/\mathrm{m}\), and \(h\) in meters, expression (20) takes the following form:

\[ Q' \geq \left(1.6M+64aF_{\mathrm{t}}10^{-6}\right)\ \mathrm{cm}^2 . \tag{21} \]

In this case the cyclic frequency of the free oscillations of a layer in the dissipative system will be

\[ \omega=\sqrt{C_i^2-h^2}, \tag{22} \]

where \(C_i\) and \(k^2\) are from (18) and (19).

Expression (22) shows that the motion of the layer in this case does not have an oscillatory character. In this case the free oscillations of the system turn out to be completely eliminated, and the displacement of the layers in the unsteady short-circuit regime does not exceed the permissible deformation of the end insulation.

The essence of the methodology for calculating the electrodynamic stability of transformers with layer windings is as follows. For the specified transformer parameters, [1] is calculated:

\[ F=7.65a_0D_{\mathrm{avg}}\left(\frac{YW_{\mathrm{sc}}}{h}\right)^2 k^2 k_{\mathrm{ud}}^2\ \mathrm{kgf}. \]

After this, \(F_{\mathrm{t}}\) is determined by formula (17); \(\sigma_{\mathrm{pr}}\), from curve (10); and the required supporting surface \(Q'\), by formula (20).

If

\[ \sigma_{\mathrm{pr}} \leq [v]_{\mathrm{pr}} Q' \leq [Q], \]

then the transformer is electrodynamically stable. Here \([Q]\) is the possible supporting surface of the windings obtained during design.

transformers; \([\gamma]_{\mathrm{pr}} = 3.5 \dfrac{\mathrm{kgf}}{\mathrm{mm}^2}\) for APBBO-grade wire, and for APB-grade wire \([\sigma]_{\mathrm{pr}} = 4.2 \dfrac{\mathrm{kgf}}{\mathrm{mm}^2}\).

As an illustration, let us give an example. A \(630/6 = 0.4\) transformer, with the following parameters: \(F = 6000\ \mathrm{kgf}\), number of HV layers \(n = 6\), layer mass \(M = 0.62 \dfrac{\mathrm{kgf}\,\mathrm{s}^2}{\mathrm{m}}\),

\[ a = 25\ \frac{1}{\mathrm{s}} . \]

To ensure the electrodynamic stability of transformer windings, it is necessary to create a friction force between the layers:

\[ F_\tau = \frac{2 \cdot 6000}{6} = 2000\ \mathrm{kgf}. \]

In this case the required tension for APBBO winding wire will be \(\sigma_{\mathrm{pr}} = 3.0 \dfrac{\mathrm{kgf}}{\mathrm{mm}^2}\). The bearing surface is

\[ Q' \geq 1.6 \cdot 0.62 + 64 \cdot 25 \cdot 2000 \cdot 10^{-6} = 4.2\ \mathrm{cm}^2 . \]

In this case the preliminary pressing force must be \(50 \dfrac{\mathrm{kgf}}{\mathrm{cm}^2}\).

Conclusions

1. Formulas (19) and (20) are generalized and theoretically substantiated, which makes it possible to determine the stiffness and the necessary bearing surface for ensuring the stability of transformers with layered windings under the static and dynamic regime of a short circuit.

2. The proposed procedure, as a method for designing electrodynamically stable transformers, is generalized and theoretically substantiated. It is not opposed to existing methods, but supplements them.

Mathematical notation

\(S_i\) — displacement of the \(i\)-th layer;
\(C_i\) — damping coefficient;
\(k\) — frequency of free oscillation in a conservative system;
\(H\) — acceleration during the motion of a layer, \(a = \dfrac{R}{L}\);
\(R\) — winding resistance;
\(L\) — transformer inductance;
\(\omega\) — frequency of the disturbing force;
\(n\) — number of layers in the winding;
\(M\) — layer mass;
\(\omega^*\) — frequency of free oscillation of a layer in a dissipative system;
\(h\) — height of the end insulation;
\([\sigma]_{\mathrm{pr}}\) — permissible tensile stress of conductor materials;

\([\sigma]_{\mathrm{ins}}\) — permissible crushing stress of insulating materials;
\(E_{\mathrm{ins}}\) — modulus of elasticity of insulating materials;
\(Q, Q'\) — supporting surfaces of the windings.

References

1. S. I. Lur’e, E. I. Levitskaya. Electrodynamic strength of transformers of frame sizes I–II with layer windings. Elektrotekhnika, No. 4, 1967.

2. P. M. Tikhomirov. Calculation of Transformers. Energiya, Moscow, 1968.

3. N. I. Bulgakov. Calculation of Transformers. Gosenergoizdat, Moscow, 1950.

4. P. E. Azaryan, Yu. A. Oganesyan. Magnetosymmetric circuits of layer windings of transformers. Promyshlennost’ Armenii, No. 4, 1967.

5. M. S. Libkind. Mechanical forces in transformer windings. Elektrichestvo, Nos. 9, 12, 1945.

6. I. S. Nayashkov, S. I. Lur’e. Experimental studies of the mechanical strength of transformers under short-circuit conditions. Vestnik elektropromyshlennosti, No. 2, 1962.

7. G. N. Petrov, I. S. Nayashkov. Electrodynamic forces in transformers. Elektrichestvo, No. 9, 1955.

8. T. Tournier, Ct. Ebersahl, A. Ciniero, S. Vyakov, A. D. Madin, T. D. Whittaker. A study of the dynamic behavior of transformer windings under short-circuit conditions. CIGRE, 1962, Nos. 143, 143a.

9. E. Pirkl, G. M. Spallanzani. Recent advancements in the field of short-circuit forces in transformer coils. CIGRE, 1962, No. 122.

10. S. I. Lur’e, L. I. Milman. Mechanical characteristics of insulating materials of transformer windings. Elektrotekhnika, No. 4, 1964.