SIMPLIFIED METHOD FOR CALCULATING THERMAL PROCESSES IN DC ELECTROMAGNET WINDINGS

V. N. Gurnitskii

(Recommended by the scientific seminar of the Electromechanical Faculty)

The thermal processes that occur in windings when an electric current flows through them have the following features.

The heating and cooling times of windings are incommensurable with the duration of the electromagnetic processes taking place in them.

The time constants of heating and cooling are not equal to each other.

The thermal parameters of a winding (specific heat capacity, heat-transfer coefficient, temperature coefficient of resistance of the winding conductor material) are functions of its temperature.

The transfer of heat into the surrounding space from the surface of the winding depends on a large number of factors: the geometry, dimensions, and construction of the winding, its temperature, cooling conditions, thermal contact with the magnetic core, and the presence of other bodies in the path of the heat flux.

The temperature inside the winding is distributed nonuniformly.

The thermal regime of the winding depends significantly on the insulating materials used in it.

This variety of thermal phenomena in the windings of electrical apparatus has compelled researchers to create various physical and engineering calculation methods, applicable only in particular cases and often containing a large number of cumbersome formulas, empirical coefficients, curves, and tables [Refs. 1–6, etc.]. Below we shall consider a simplified method for calculating thermal processes in the windings of DC electromagnets, based on the linearization of the dependences of their thermal parameters on temperature.

The equation of the heat balance of an isothermal winding, for any instant of time \(dt\), having a temperature rise above the ambient temperature \(\tau^\circ\), as is known [Ref. 7], may be represented in the form

\[ P\,dt = Bc\,d\tau^\circ + S_k k_t \tau^\circ\,dt, \tag{1} \]

where \(P\) is the active power released in the winding in the form of heat;
\(B\) is the weight of the winding;
\(c\) is the specific heat capacity of the winding;
\(S_k\) is the calculated heat-transfer surface of the winding;
\(k_t\) is the heat-transfer coefficient.

If the dynamic current curve as a function of time is known, then the thermal energy released over any small time interval \(\Delta t\) under consideration will be equal to

\[ P \cdot \Delta t=\left(i_{\mathrm{p}}+\frac{\Delta i}{2}\right)^2 r \cdot \Delta t, \tag{2} \]

where \(r\) is the active resistance of the winding circuit;
\(i_{\mathrm{p}}\) is the current value at the end of the time interval preceding the one under consideration;
\(\dfrac{\Delta i}{2}\) is the average increase of current over the interval.

The value of the active resistance as a function of temperature is found from the well-known relation

\[ r=r_0\left[1+\alpha\left(t_0^\circ+\tau_{\mathrm{p}}^\circ+\frac{\Delta \tau}{2}\right)\right], \tag{3} \]

where \(r_0\) is the resistance of the winding copper at \(0^\circ\mathrm{C}\) (the resistivity of winding copper at \(0^\circ\mathrm{C}\) is \(1.75\cdot10^{-8}\ \Omega\cdot\mathrm{m}\));
\(t_0^\circ\) is the ambient temperature;
\(\alpha\) is the temperature coefficient of resistance of the winding conductor material, equal for copper to \(0.00431\ \mathrm{deg}^{-1}\);
\(\tau_{\mathrm{p}}^\circ\) is the temperature rise of the winding above the ambient at the end of the time interval preceding the one under consideration.

The thermal energy stored in the winding owing to its heat capacity cannot be determined exactly, since the weight and specific heat capacity of the various types of insulation are in practice difficult to take into account. However, it is clear that their product, giving the heat capacity of the insulation, is negligibly small in comparison with the heat capacity of the winding material. If the heat capacity of the insulation is neglected, it provides a certain margin with respect to heating. If one takes into account the nearly linear dependence of the specific heat capacity of the winding copper on temperature [L. 8], then the first term of equation (1), in finite differences (average value over the interval), will be approximately equal to

\[ Bc\Delta \tau \approx B_{\mathrm{m}}\Delta \tau \left[ c_{\mathrm{m}0}+\beta\left(t_0^\circ+\tau_{\mathrm{p}}^\circ+\frac{\Delta \tau^\circ}{2}\right) \right], \tag{4} \]

where \(B_{\mathrm{m}}\) is the weight of the winding copper;
\(c_{\mathrm{m}0}\) is the specific heat capacity of copper at a temperature of \(0^\circ\mathrm{C}\), equal to

\[ 3.85\cdot10^{-4}\ \frac{\mathrm{W}\cdot\mathrm{s}}{\mathrm{kg}\cdot\mathrm{deg}}; \]

\(\beta\) is the temperature coefficient of heat capacity, equal for copper [L. 8] to

\[ 1.47\cdot10^{-7}\ \frac{\mathrm{W}\cdot\mathrm{s}}{\mathrm{kg}\cdot\mathrm{deg}^2}; \]

\(\dfrac{\Delta \tau^\circ}{2}\) is the average increase of temperature rise over the interval.

The calculated heat-transfer surface of the winding is found from the formula [L. 7]

\[ S_k=S_D+\eta_{\mathrm{t}}\cdot S_d, \]

where \(S_D\) is the area of the outer surface of the winding;
\(S_d\) is the area of the inner surface of the winding;
\(\eta_{\mathrm{t}}\) is the coefficient of effectiveness of heat transfer from the inner surface of the winding, equal to [L. 4]:

\(0.9\) — for frameless windings,
\(1.7\) — for windings with a metal frame,
\(2.4\) — for windings wound directly on the core of the electromagnet.

In the course of thermal calculations of windings, the greatest difficulties are encountered in determining the coefficients of heat transfer from their surface.

However, this dependence is close to linear; then

\[ k_{\tau}=k_{\tau_0}+\gamma\left(t_1^\circ+\tau_{\mathrm{p}}^\circ+\frac{\Delta\tau^\circ}{2}\right), \tag{6} \]

where: \(k_{\tau_0}\) is the heat-transfer coefficient of the winding, depending on the size of its surface, equal to [Lit. 7], for \(1\ \text{cm}^2<S_k<100\ \text{cm}^2\),

\[ k_{\tau_0}=6\cdot10^{-7}\frac{1}{\sqrt[3]{S_k}}\, \frac{\text{W}}{\text{m}^2\cdot\text{deg}}, \tag{7} \]

for \(100\ \text{cm}^2<S_k<5000\ \text{cm}^2\)

\[ k_{\tau_0}=3\cdot10^{-7}\frac{1}{\sqrt[5]{S_k}}\, \frac{\text{W}}{\text{m}^2\cdot\text{deg}}, \tag{8} \]

\(\gamma\) is the temperature coefficient of heat transfer.

As the literature sources indicate [Lit. 6, 7, 9–11, etc.], and also certain experiments conducted by the author, the coefficient \(\gamma\) varies approximately within the limits \(\gamma=5\cdot10^{-10}\).

\[ \gamma=(4\div6)\cdot10^{-10}\, \frac{\text{W}}{\text{m}^2\cdot\text{deg}^2}. \]

If no additional information on the coefficient \(\gamma\) is available, then in calculations one should take \(\gamma=5\cdot10^{-10}\).

Substituting the values of its terms into equation (1), we then obtain

\[ \left(i_{\mathrm{p}}+\frac{\Delta i}{2}\right)^2 r_0 \left[ 1+4.31\cdot10^{-3} \left(t_0^\circ+\tau_{\mathrm{p}}^\circ+\frac{\Delta t^\circ}{2}\right) \right]\Delta t = \]

\[ = B_{\mathrm{m}}\Delta\tau^\circ \left[ 3.85\cdot10^{-4} +1.47\cdot10^{-7} \left(t_0^\circ+\tau_{\mathrm{p}}^\circ+\frac{\Delta\tau^\circ}{2}\right) \right]+ \]

\[ +S_k\left[ k_{\tau_0}+5\cdot10^{-10} \left(t_0^\circ+\tau_{\mathrm{p}}^\circ+\frac{\Delta\tau^\circ}{2}\right) \right]\cdot \left(\tau_{\mathrm{p}}^\circ+\frac{\Delta\tau^\circ}{2}\right)\cdot\Delta t . \tag{9} \]

Solving the last equation with respect to the increment in the excess temperature of the winding,

\[ \Delta\tau^\circ=\frac{\sqrt{Y^2-4XZ}-Y}{X}, \tag{10} \]

where the following are denoted:

\[ X=2\cdot1.47\cdot10^{-7}B_{\mathrm{m}}+5\cdot10^{-10}S_k\cdot\Delta t, \tag{11} \]

\[ Y=2B_{\mathrm{m}}\left[3.85\cdot10^{-4} +1.47\cdot10^{-7}\left(t_0^\circ+\tau_{\mathrm{p}}^\circ\right)\right]+ \]

\[ +S_k\left[k_{\tau_0}+5\cdot10^{-10} \left(t_0^\circ+2\tau_{\mathrm{p}}^\circ\right)\right]\cdot\Delta t- \]

\[ -4.31\cdot10^{-3} \left(i_{\mathrm{p}}+\frac{\Delta i}{2}\right)^2 r_0\cdot\Delta t, \tag{12} \]

\[ Z=S_k\tau_{\mathrm{p}}^\circ \left[k_{\tau_0}+5\cdot10^{-10} \left(t_0^\circ+\tau_{\mathrm{p}}^\circ\right)\right]\cdot\Delta t- \]

\[ -\left(i_{\mathrm{p}}+\frac{\Delta i}{2}\right)^2 r_0 \left[1+4.31\cdot10^{-3} \left(t_0^\circ+\tau_{\mathrm{p}}^\circ\right)\right]\cdot\Delta t . \tag{13} \]

Thus, having the dynamic current curve, one can, using equation (10), construct by intervals the curve of the excess temperature of the winding.

Equation (1) assumes the winding to be isothermal; therefore all the calculation formulas are valid for the temperature of the outer surface of the actual winding. And since the temperature difference between the hottest layer of the winding and its outer surface does not exceed \(15^\circ\text{C}\) [12], it may be considered that the permissible heating temperature of the insulation of the winding, determined by its class and the required service life, is

\[ \tau_{\text{dop}}^\circ \geq \tau^\circ + 15^\circ\text{C}. \tag{14} \]

When constructing cooling curves, in the formulas one must set

\[ \left(i_n+\frac{\Delta i}{2}\right)=0. \]

The method set forth was checked in calculating thermal processes in the windings of Ш-shaped electromagnets with a transversely moving armature, which are elements of a linear electromagnetic motor [13]. The winding temperature was measured by the resistance method. Comparison of the calculated and experimental heating and cooling characteristics of the windings of three electromagnet specimens, operating in long-term and intermittent-duty modes at different \(PV\%\), indicates the possibility of using the method in thermal calculations.

Conclusions

1. The proposed simplified method for calculating thermal processes in the windings of DC electromagnets makes it possible, in a simple way from the dynamic characteristic or current diagram, knowing only the dimensions and having almost no information about the winding design, to construct the characteristics of its heating and cooling.

2. The method gives satisfactory results in calculating both intermittent and long-term current-carrying modes of electromagnet windings.

3. The coefficient \(\beta\), at small temperature rises, has only a slight effect on the thermal characteristics of the winding and may be taken equal to zero.

4. By correcting the values of the coefficients \(\gamma\) for specific winding designs, the accuracy of the calculations can be increased.

5. In the case of an aluminum winding, the coefficients \(\alpha\) and \(\beta\) must be recalculated for winding aluminum.

References

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