EQUIVALENT THERMAL CIRCUITS OF BODIES WITH A ONE-DIMENSIONAL TEMPERATURE FIELD

D. I. Sannikov

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

The solution of various problems in calculating the heating of electrical machines, owing to the complexity of the temperature field, proves possible only when approximate methods are used. These include, for example, reducing the three-dimensional field in some part of the machine to a one-dimensional one, and the method of equivalent thermal circuits. Combining both methods makes it possible to avoid some of the shortcomings inherent in each of them separately.

Fig. 1.

Let us consider the general case of a stationary temperature field in a rod with internal heat sources, with heat transfer along the \(x\)-axis (Fig. 1). Let the cross section of the rod, \(s(x)\), the volumetric density of the heat sources \(\sigma(x)\), and the thermal conductivity \(\lambda(x)\) vary along the length of the rod and be independent of temperature. The heat flux along the rod is

\[ A=-\lambda s\frac{dt}{dx}. \tag{1} \]

On the basis of the heat balance for an element \(dx\), we have

\[ dA=\sigma s\,dx. \tag{2} \]

Eliminating \(A\) and introducing the notation:

thermal resistance per unit length

\[ r=\frac{1}{\lambda s}, \tag{3} \]

linear density of heat sources (losses)

\[ q=\sigma s, \tag{4} \]

we obtain the differential equation of heat conduction

\[ \frac{d}{dx}\left(\frac{1}{r}\frac{dt}{dx}\right)+q=0, \tag{5} \]

whose solution can be written in the form

\[ t(x)=t_q(x,q)+t_1\cdot f_1(x)+t_2\cdot f_2(x), \tag{6} \]

where

\[ f_1(x)=\frac{R(x_2)-R(x)}{R(x_2)-R(x_1)}, \tag{7} \]

\[ f_2(x)=\frac{R(x)-R(x_1)}{R(x_2)-R(x_1)}=1-f_1(x), \tag{8} \]

\[ R(x)=\int_0^x r\,dx, \tag{9} \]

\[ t_q(x,q)=-\Theta(x)+\Theta(x_1)\cdot f_1(x)+\Theta(x_2)\cdot f_2(x), \tag{10} \]

\[ \Theta(x)=\int_0^x r\left(\int_0^x q\,dx\right)dx. \tag{11} \]

Thus, the temperature is composed of three components: \(t_q(x,q)\)—depending on the losses, \(t_1\cdot f_1(x)\) and \(t_2\cdot f_2(x)\)—depending on the boundary conditions. On the basis of the general expression for the heat flux

\[ A=-\frac{1}{r}\frac{dt}{dx} =\int_0^x q\,dx+\frac{\Theta(x_1)-\Theta(x_2)}{R(x_2)-R(x_1)} +\frac{t_1-t_2}{R(x_2)-R(x_1)} \tag{12} \]

we find the fluxes emerging from the end sections,

\[ A_1=-A_{(x=x_1)}=Q_1+\frac{t_2-t_1}{R}, \tag{13} \]

\[ A_2=A_{(x=x_2)}=Q_2+\frac{t_1-t_2}{R}, \tag{14} \]

where

\[ Q_1=-\int_0^{x_1} q\,dx+\frac{\Theta(x_2)-\Theta(x_1)}{R}, \tag{15} \]

\[ Q_2=\int_0^{x_2} q\,dx-\frac{\Theta(x_2)-\Theta(x_1)}{R}. \tag{16} \]

\[ R = R(x_2) - R(x_1) \tag{17} \]

is the total thermal resistance of the rod.

\[ Q_1 + Q_2 = \int_{x_1}^{x_2} q\,dx = Q \tag{18} \]

are the total losses in the rod.

Expressions (13) and (14) correspond to the equivalent circuit (Fig. 2a), which makes it possible to determine the temperatures \(t_1\) and \(t_2\) under any boundary conditions.

Fig. 2. Equivalent circuits:
(a) \(A_1\), \(t_1\), \(Q_1\), \(R\), \(Q_2\), \(t_2\), \(A_2\);
(b) \(A_1\), \(t_1\), \(R_1\), \(Q\), \(R_2\), \(t_2\), \(A_2\);
(c) \(A_1\), \(Q_{x1}\), \(t_1\), \(R_{x1}\), \(t_x\), \(Q_x\), \(R_{x2}\), \(t_2\), \(Q_{x2}\), \(A_2\).

If it is more convenient to assign the losses to a single node, the circuit is transformed to the form shown in Fig. 2b. Here

\[ R_1 = R\frac{Q_2}{Q}, \tag{19} \]

\[ R_2 = R\frac{Q_1}{Q}. \tag{20} \]

The temperature in any section \(x\) can be found either by means of expression (6), or directly from the circuit by dividing the rod into two parts by the section \(x\) (Fig. 2c). The parameters of the left-hand part of the circuit are determined from expressions (15–17) by replacing \(x_2\) with \(x\); for the right-hand part, by replacing \(x_1\) with \(x\). The mean heat source is

\[ Q_x = Q - Q_{x1} - Q_{x2}. \tag{21} \]

The mean temperature of the body can also be determined by means of equivalent circuits (Fig. 3).

\[ \bar t = \frac{\displaystyle \int_{x_1}^{x_2} t \cdot s \cdot dx}{V} = \bar t_q + t_1 \cdot \bar f_1 + t_2 \cdot \bar f_2, \tag{22} \]

where

\[ V = \int_{x_1}^{x_2} s \cdot dx. \tag{23} \]

Introducing the notation

\[ \overline{\Theta}=\frac{1}{V}\int_{x_1}^{x_2} s\cdot \Theta(x)\cdot dx, \tag{24} \]

\[ \overline{R}=\frac{1}{V}\int_{x_1}^{x_2} s\cdot R(x)\cdot dx, \tag{25} \]

one may write

\[ \overline{t}_q=-\overline{\Theta}+\Theta(x_1)\cdot \overline{f}_1+\Theta(x_2)\cdot \overline{f}_2, \tag{26} \]

\[ \overline{f}_1=\frac{R(x_2)-\overline{R}}{R}, \tag{27} \]

\[ \overline{f}_2=\frac{-R(x_1)+\overline{R}}{R}=1-\overline{f}_1. \tag{28} \]

Having, according to the circuit (Fig. 3a), formed expressions for \(A_1\), \(A_2\), and \(\overline{t}\), and equating the coefficients in these expressions and in (13), (14), and (22), we determine the parameters of the circuit:

\[ R'_1=-R(x_1)+\overline{R}, \tag{29} \]

\[ R'_2=R(x_2)-\overline{R}=R-R'_1, \tag{30} \]

\[ Q'_1=-\int_{0}^{x_1} q\,dx+\left[\overline{\Theta}-\Theta(x_1)\right]\frac{1}{R'_1}, \tag{31} \]

\[ Q'_2=\int_{0}^{x_2} q\,dx+\left[\overline{\Theta}-\Theta(x_2)\right]\frac{1}{R'_2}, \tag{32} \]

\[ Q'=Q-Q'_1-Q'_2. \tag{33} \]

The second variant of the circuit for determining the mean temperature of the rod is obtained by assigning the total losses to one node (Fig. 3b). Its parameters are determined in an analogous way:

\[ R_0=\frac{\overline{t}_q}{Q}-\frac{R'_1\cdot R'_2}{R}, \tag{34} \]

\[ Q''=Q'_1-Q\cdot \overline{f}_1=-Q'_2+Q\cdot \overline{f}_2. \tag{35} \]

The third variant of the circuit is obtained by a star-to-delta transformation (Fig. 3c):

\[ \overline{R}_1=\frac{\overline{t}_q}{Q\cdot f_1}, \tag{36} \]

\[ \overline{R}_2=\frac{\overline{t}_q}{Q\cdot f_2}, \tag{37} \]

\[ R''=\frac{1}{\dfrac{1}{R}-\dfrac{1}{\overline{R}_1+\overline{R}_2}}. \tag{38} \]

Let us consider the application of this method to specific bodies.

1. Solid cylinder (per unit length)

\[ s=2\pi x,\qquad \sigma=\text{const},\qquad \lambda=\text{const}. \]

\[ r=\frac{1}{2\pi\lambda x}, \tag{39} \]

\[ q=2\pi\sigma x. \tag{40} \]

[Diagram with equivalent thermal circuits labeled \(a\), \(б\), and \(в\), containing elements \(A_1\), \(A_2\), \(t_1\), \(t_2\), \(\bar t\), \(Q_1'\), \(Q'\), \(Q_2'\), \(Q''\), \(-Q''\), \(Q\), \(R_1'\), \(R_2'\), \(R_0\), \(R''\), \(\bar R_1\), and \(\bar R_2\).]

Fig. 3.

For the circuits (Figs. 2б, 3б)

\[ Q=\pi\sigma\left(x_2^2-x_1^2\right). \tag{41} \]

\[ Q''=0, \tag{42} \]

\[ R_1=R_1'=\frac{1}{2\pi\lambda} \left( \frac{x_2^2}{x_2^2-x_1^2}\ln\frac{x_2}{x_1}-\frac{1}{2} \right), \tag{43} \]

\[ R_2=R_2'=\frac{1}{2\pi\lambda} \left( -\frac{x_1^2}{x_2^2-x_1^2}\ln\frac{x_2}{x_1}+\frac{1}{2} \right), \tag{44} \]

\[ R_0=-\frac{1}{2\pi\lambda\left(x_2^2-x_1^2\right)} \left[ \frac{1}{4}\left(x_2^2+x_1^2\right) -\frac{x_1^2x_2^2}{x_2^2+x_1^2}\ln\frac{x_2}{x_1} \right]. \tag{45} \]

2. Rod of constant cross section or a plane wall with a uniform distribution of losses

\[ s=\mathrm{const},\quad \sigma=\mathrm{const},\quad \lambda=\mathrm{const}. \]

\[ x_1=0,\quad x_2=l. \]

For Figs. 2, a and 2, b we obtain the usual expressions:

\[ R=rl, \tag{46} \]

\[ Q=ql, \tag{47} \]

\[ Q_1=Q_2=\frac{Q}{2}, \tag{48} \]

\[ R_1=R_2=\frac{R}{2}. \tag{49} \]

For Fig. 3, a

\[ R'_1=R'_2=\frac{R}{2}, \tag{50} \]

\[ Q'_1=Q'_2=Q'=\frac{Q}{3}. \tag{51} \]

For Fig. 3, b

\[ R_0=-\frac{R}{6}, \tag{52} \]

\[ Q''=0. \tag{53} \]

For Fig. 3, c

\[ \bar R_1=\bar R_2=\frac{R}{6}, \tag{54} \]

\[ R''=-\frac{R}{2}. \tag{55} \]

With the simplest boundary conditions, on the basis of these circuits one obtains the well-known expressions for the internal resistance of a body with a parabolic temperature distribution:

for symmetric cooling (Fig. 3, c)

\[ A_1=A_2,\quad t_1=t_2, \]

\[ \bar t-t_2=Q\frac{\bar R_1\cdot \bar R_2}{\bar R_1+\bar R_2}=Q\frac{R}{12}; \]

for one-sided cooling

\[ A_1=0, \]

\[ t-t_2=Q\frac{\bar R_2(\bar R_1+R'')}{\bar R_1+\bar R_2+R''}=Q\frac{R}{3}; \]

in the absence of internal heat sources

\[ Q=0,\quad -A_1=A_2, \]

\[ \bar t-t_2=A_2\frac{R''\cdot \bar R_2}{\bar R_1+\bar R_2+R''}=A_2\frac{R}{2}. \]

The circuits obtained make it possible to calculate the mean temperature and the temperature at the end points of the body in the most general case—when the ...

under symmetric cooling and boundary conditions of any kind—is simpler in comparison with existing methods.

Analogous circuits can be constructed for a rod cooled along its length, taking into account the dependence of losses on temperature. Let, for the rod (Fig. 4),

\[ r=\mathrm{Const}, \]

\[ q=q_0(1+\alpha_0 t), \tag{56} \]

where \(q_0\) are the losses at the temperature \(\Theta\), taken as the reference level, and \(\alpha_0\) is the temperature coefficient of losses.

Fig. 4.

The thermal conductance between the rod and the surrounding medium per unit length is \(\Lambda=\mathrm{const}\); the temperature of the surrounding medium varies linearly along the length, for example as a result of heating of the cooling flow,

\[ t_0=t_{01}\left(1-\frac{x}{l}\right)+t_{02}\frac{x}{l}. \tag{57} \]

On the basis of the heat balance for the element \(dx\), we obtain the heat-conduction equation

\[ \frac{1}{r}\frac{d^2 t}{dx^2}-\Lambda(t-t_0)+q=0. \tag{58} \]

Substituting expressions (56) and (57), we reduce the equation to the form

\[ \frac{d^2 t}{dx^2} = r\left[ (\Lambda-\alpha_0 q_0)t -\Lambda\cdot t_{01}\frac{l-x}{l} -\Lambda\cdot t_{02}\frac{x}{l} -q_0 \right], \tag{59} \]

whose solution is the expression

\[ t=(t_1-t_q-kt_{01})\frac{\operatorname{sh}\beta(l-x)}{\operatorname{sh}\beta l} +(t_2-t_q-kt_{02})\frac{\operatorname{sh}\beta x}{\operatorname{sh}\beta l} + \]

\[ +\,kt_{01}\frac{l-x}{l}+kt_{02}\frac{x}{l}+t_q, \tag{60} \]

where

\[ \beta=\sqrt{\Lambda' r};\qquad \Lambda'=\Lambda-\alpha_0 q_0; \tag{61, 62} \]

\[ t_q=\frac{q_0}{\Lambda'};\qquad k=\frac{\Lambda}{\Lambda'}. \tag{63, 64} \]

The mean temperature of the rod is

\[ \bar t=(t_1+t_2)\zeta+\left[(t_{01}+t_{02})\frac{k}{2}+t_q\right](1-2\zeta), \tag{65} \]

where

\[ \zeta=\frac{\operatorname{ch}\beta l-1}{\beta l\,\operatorname{sh}\beta l}. \tag{66} \]

A possible variant of the circuit corresponding to this case is shown in Fig. 5. The conductances of the circuit are determined by equating the coefficients in the expressions for \(\dot Q\), \(A_1\), and \(A_2\), as formed from the circuit and obtained on the basis of expressions (60) and (65):

Fig. 5.

\[ G_0=\Lambda l, \tag{67} \]

\[ G_q=-\alpha_0 q_0 l, \tag{68} \]

\[ G_r=\Lambda' l r, \tag{69} \]

\[ G_l=\frac{\beta}{r}\,\frac{1}{\operatorname{sh}\beta l}-G_r\zeta, \tag{70} \]

\[ G_0'=\frac{k}{2rl\gamma}, \tag{71} \]

\[ Q=q_0l. \tag{72} \]

Here

\[ \gamma=\frac{\zeta}{1-2\zeta}. \tag{73} \]

For the purpose of simplifying expressions (69–71), they may be represented in the form of series. Taking the first two terms of the series, we obtain:

\[ G_r=\frac{6}{rl}+\Lambda'\frac{l}{10} \tag{74} \]

\[ G_l=-\frac{2}{rl}+\Lambda'\frac{l}{30}, \tag{75} \]

\[ G'_o=\frac{\Lambda l}{12}\left(1-\frac{\Lambda' r l^2}{60}\right) \tag{76} \]

When calculating the temperature distribution along the length of the windings of electrical machines, in the case \(l<60\ \text{cm}\) the error of the simplified formulas usually does not exceed 5%; for a length \(l<25\ \text{cm}\) one may confine oneself to the first term. In this case the form and parameters of the circuit correspond to Fig. 3, c for \(q=\text{const}\), \(r=\text{const}\), with the addition of conductances \(G_o\) and \(G_q\), assigned to the middle node and accounting for heat removal along the length of the rod—

Fig. 6.

—and the increase in losses with rising temperature, and the conductances \(G'_o\) and \((-G'_o)\), assigned to the end nodes and accounting for the effect of heating of the cooling flow along the length of the rod.

As an example, let us consider the use of the equivalent circuits obtained for calculating the temperature distribution along the length of the armature winding of a direct-current machine with an axial ventilation system and with no ventilation ducts in the armature. The thermal connections between the individual bodies entering into the active part of the armature can be represented in the form of a circuit (Fig. 6, a) using the circuits (Fig. 3, b) and formulas (43—47, 50, 52, 53).

\(Q_{\mathrm{mo}}\) are the losses in the slot part of the winding at ambient temperature \(\Theta\), taken as the reference base for the temperature rise \(t\),

\(Q_z\) and \(Q_a\) are the losses in the steel of the teeth and of the yoke.

Thermal resistances:

\(R_a\) — from the armature surface to the cooling air,

\(R_z\) — over the tooth height,

\(R_{\mathrm{i}}\) — through the thickness of the winding insulation,

\(R_{a1}, R_{a2}\) and \(R_{a0}\) — over the yoke height,

\(R_\Delta\) — in the gap between the core and the shaft,

\(R_{\mathrm{v}}\) — shaft resistance over the active length of the armature \(l\),

\(R_{\mathrm{vv}}\) and \(R_{\mathrm{bb}}\) — resistances on the path of heat removal from the shaft through the surfaces of the fan and of the balancing ring,

\(t_{\mathrm{ok}}\) and \(t_{\mathrm{ov}}\) — temperature rise of the internal air at the surface of the commutator and at the fan.

The dependence of the winding losses on temperature is taken into account by the conductance

\[ G_q=-\frac{Q_{\mathrm{mo}}}{235+\Theta}. \tag{77} \]

The circuit is reduced to the simplest form (Fig. 6, b) with one heat source assigned to the winding,

\[ Q=Q_{\mathrm{mo}}+ \left[ Q_z\left(\frac{R_{\mathrm{i}}}{R_{zm}}-\frac{R_z}{6}G_z\right) + Q_a\left(1-\frac{0.5R_z+R_{az}}{R_{zv}}\right) \right]\frac{1}{R_{zm}G_z} \tag{78} \]

and with parameters

\[ G_0=\frac{1}{(R_a+0.5R_z)R_{zm}G_z}; \tag{79} \]

\[ G_{\mathrm{v}}=\frac{1}{R_{zm}R_{zv}G_z}\cdot \frac{R_{\mathrm{v}}''}{R_{\mathrm{v}}'+R_{\mathrm{v}}''}; \tag{80} \]

\[ G_{\mathrm{v}}''=\frac{1}{R_{zm}R_{zv}G_z}\cdot \frac{R_{\mathrm{v}}'}{R_{\mathrm{v}}'+R_{\mathrm{v}}''}, \tag{81} \]

where

\[ R_{zm}=R_{\mathrm{i}}-\frac{R_z}{6}, \tag{82} \]

\[ R_{zv}=0.5R_z+R_{a2}+R_{a1}+R_\Delta-\frac{R_{\mathrm{v}}}{6} +\frac{R_{\mathrm{v}}'\cdot R_{\mathrm{v}}''}{R_{\mathrm{v}}'+R_{\mathrm{v}}''}, \tag{83} \]

\[ R_{\mathrm{v}}'=0.5R_{\mathrm{v}}+R_{\mathrm{bb}}, \tag{84} \]

\[ R_{\mathrm{v}}''=0.5R_{\mathrm{v}}+R_{\mathrm{vv}}, \tag{85} \]

\[ G_z=\frac{1}{R_a+0.5R_z}+\frac{1}{R_{zm}}+\frac{1}{R_{zv}}. \tag{86} \]

Further, the armature may be represented in the form of a rod with uniformly distributed losses \(q=Q/l\), cooled along its length, having a resistance per unit length \(r\), equal to the axial re-

to the resistance of the winding, and connected in series with two rods representing the end parts. On the basis of the circuits (Fig. 5) and (Fig. 6, b), the complete equivalent circuit of the armature is constructed (Fig. 7). The conductances \(G'_o\) of the circuit (Fig. 5) in this case may be disregarded.

Fig. 7.

Parameters of the end parts:

\(Q_l\) — losses in one end part at temperature \(\Theta\);

\(G_{rl}\), \(G_{ll}\), \(G_{fl}\) are calculated analogously to the corresponding parameters of the active part through \(Q_l\), \(l_l/2\), and \(c\);

\(G'_{ol}\) and \(G''_{ol}\) — conductances between the end parts and the air—are calculated taking into account the temperature drop in the turn insulation over \(1/6\) of the winding thickness and in the insulation of the bands; in addition, \(G'_{ol}\) accounts for heat removal from half of the surface formed by the section leads, while the conductance \(G''_{ol}\) accounts for heat removal from the inner surface of the winding holder;

\(G_l\) — conductance from the end face of the end parts to the air.

Since the circuit takes into account the thermal connection between the winding and the commutator through the section leads—\(G_{lk}\), it includes such parameters as \(Q_k\) — commutator losses, and \(G_{ko}\) — conductance from the commutator surface and half of the surface of the connecting leads to the air.

The conductances removing heat to the cooling air flow are connected in the circuit to points with fixed temperature potential, which is calculated under the assumption of linear variation of the flow temperature as a result of its heating while passing through the machine.

It is convenient to calculate the circuit by the nodal-potential method. After determining the mean temperature rises \(\bar t\), \(\bar t'_l\), and \(\bar t''_l\), and the temperature rises at the ends of the sections, the curve of temperature-rise distribution can be constructed by means of formula (60) or the formula of the equivalent parabola

\[ t = t_1 \frac{l - x}{l} + t_2 \frac{x}{l} + 3(2\bar t - t_1 - t_2)\frac{x}{l}\cdot\frac{l - x}{l}. \tag{87} \]

Fig. 8 gives the calculated curve of the distribution of temperature rise \(t\) along the length of the armature winding of a direct-current motor, \(4.5\) kW, \(1500\) rpm. The heat-transfer and thermal-conductivity coefficients are taken from [1]; the losses and the temperature rise of the internal air were determined by calculation. The curve obtained is in satisfactory agreement with the readings of thermocouples (points in Fig. 8) placed at various locations in the cross-section of the slot and end portions of the winding during thermal tests of the motor. The scatter of the thermocouple readings is explained by the nonuniformity of the temperature distribution over the cross-section, and also among individual slots.

Fig. 8.

Calculations carried out for various modifications of the circuit show the following:

1) taking into account the conductances \(G'_o\) (Fig. 5) gives a reduction of the temperature rise on the commutator side by no more than \(0.5^\circ\) and an increase on the opposite side by no more than \(0.7^\circ\); consequently, omitting \(G'_o\) is justified;

2) neglecting the thermal coupling between the winding and the commutator leads to a considerable increase in the calculated temperature rise of the winding (on the average by \(30^\circ\)) and to its equalization;

3) calculation under the assumption of a constant air temperature rise along the length gives a decrease in the difference between the temperature rises of the end portions by \(4^\circ\); thus, the nonuniformity of the temperature-rise distribution along the length of the winding is explained primarily by the influence of the commutator, while the influence of air heating is secondary;

4) neglecting heat transfer through the shaft (the conductances \(G'_a\) and \(G''_a\)) leads to an increase in the calculated temperature rise by an average of \(13^\circ\); consequently, it is necessary to take into account heat removal through the shaft, the fan, and the balancing ring;

5) omission of the conductances \(G_l\) and \(G_{la}\) leads to a circuit that is used by some authors to account for the axial resistance of the winding; however, in the present case the maximum calculated temperature rise decreases by \(8^\circ\);

6) if, in addition, the resistances \((-R_z/6)\) and \((-R_v/6)\) in Fig. 6 are discarded and \(G_r = 2/rt\) is adopted, this will be equivalent to the frequently used assumption that the losses in each body are concentrated at its center; this leads to an increase in the calculated overheating for this machine by \(4^\circ\);

7) with an increase in the power, dimensions, and rotational speed of machines, the distortion of the calculation results due to the above-considered simplifications of the circuit increases substantially.

Conclusion

1. A body with internal heat sources and a one-dimensional temperature field is represented by an equivalent thermal circuit that is independent of the boundary conditions.

2. The form and parameters of the thermal circuits are determined on the basis of the solution of the differential equations of heat conduction. The expressions for the parameters in the cases of greatest practical importance have a sufficiently simple form.

3. The construction of replacement thermal circuits for electrical machines according to this principle ensures the accuracy and completeness of the calculation results characteristic of a method based on the solution of the differential equations of heat conduction, while retaining the simplicity and clarity inherent in the method of thermal circuits.

References

1. I. M. Postnikov. Design of Electrical Machines. Gostekhizdat of the Ukrainian SSR, 1960.

2. J. Hak. Method of equivalent thermal circuits. Part 1, Technical report of the Moravian Experimental Plant, translation of the Central Bureau of Technical Information of the Research Institute of the Electrical Industry, P-4937, 1959.

3. J. Hak. Heat transfer in electrical machines. Parts 1–3, Technical report of the Moravian Experimental Plant, translation of the Central Bureau of Technical Information of the Research Institute of the Electrical Industry, P-4929–4931, 1959.