TRANSFORMATIONS OF THERMAL EQUIVALENT CIRCUITS OF ELECTRIC MACHINES

D. I. SANNIKOV

(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)

In analyzing thermal equivalent circuits of electric machines in the course of developing methods for thermal calculation, various transformations of these circuits prove useful.

The present work considers three types of transformations of steady-state heating circuits and gives examples of their use.

A thermal circuit is constructed on the basis of the heat balance of the individual elements of an electric machine, or on the basis of the solution of differential equations of heat conduction [2], and consists of nodes with heat sources \(P\), connected by thermal conductances \(G\). It is assumed that the nodes representing elements of the cooling flow have a fixed temperature and that the circuit is linear.

The heat balance for the \(i\)-th node of the circuit is

\[ P_i = \sum_{\substack{k=a \\ k \ne i}}^{n} (\theta_i - \theta_k)G_{ik}, \tag{1} \]

where

\[ \begin{aligned} \theta &\text{— temperatures of the nodes,}\\ k &\text{— ordinal numbers of nodes, varying from } a \text{ to } n,\\ G_{ik} &\text{— mutual conductances,} \end{aligned} \]

leads to the equation

\[ -\theta_i \sum_{\substack{k=a \\ k \ne i}}^{n} G_{ik} + \sum_{\substack{k=a \\ k \ne i}}^{n} \theta_k G_{ik} + P_o = 0. \tag{2} \]

The circuit as a whole is described by a system of analogous equations [4]

\[ \|G_{ik}\| \times \theta_i + P_i = 0. \tag{3} \]

Each diagonal term of the conductance matrix, according to (2), is equal to

\[ G_{ii} = -\sum_{\substack{k=a \\ k \ne i}}^{n} G_{ik} \tag{4} \]

and may be called the self-conductance of the node, taken with a minus sign.

Elimination of a Node from the Circuit

Let us single out from the circuit a node with index \(O\) (Fig. 1, a). Eliminating the equation of this node ...

Fig. 1. Elimination of a node from a thermal circuit

\[ \theta_0 \sum_{i=1}^{n} G_{0i} = \sum_{i=1}^{n} \theta_i G_{0i} + P_0 \tag{5} \]

from system (3), by the method of substitution, we obtain a system that corresponds to a new circuit with the number of nodes reduced by one (Fig. 1, b). Thus, the star is transformed into an equivalent polygon with the following parameters:

a) the increment of the sources of the remaining nodes

\[ \Delta P_i = P_0 k_{0i}, \tag{6} \]

where

\[ k_{0i} = \frac{G_{0i}}{(-G_{00})}, \tag{7} \]

is the coefficient for reducing the source of the eliminated node to the remaining node,

\[ (-G_{00}) = \sum_{k=1}^{n} G_{0i} \tag{8} \]

is the self-conductance of the eliminated node;

b) the increment of conductances between the remaining nodes

\[ \Delta G_{ik} = \frac{G_{0i}G_{0k}}{(-G_{00})}; \tag{9} \]

c) the increment of the self-conductances

\[ \Delta G_{ii} = -\,\frac{G_{0i}^{2}}{(-G_{00})}. \tag{10} \]

It should be noted that the total power of the heat sources of the circuit does not change, since

\[ \sum_{i=1}^{n} \Delta P_i = P_0 \frac{\sum_{k=1}^{n} G_{0i}}{(-G_{00})} = P_0, \tag{11} \]

that is,

\[ \sum_{i=1}^{n} K_{oi}=1. \tag{12} \]

This transformation is used in simplifying thermal circuits; however, up to now it has not been clearly formulated in the literature. Moreover, on its basis other types of transformation can be developed, such as, for example, the inclusion of a new node in the circuit and the merging of two nodes of the circuit.

Unlike the method of [1], the proposed transformation methods are fundamentally exact, but are applicable only to a stationary process.

Inclusion of a New Node in the Circuit

If, to an existing circuit with \(n\) nodes \((1 \leq i \leq n)\), it is necessary to add one more node, for which the power of the source \(P_o\) and the dependence of the temperature on the temperatures of the other nodes are known,

\[ \theta_o=\sum_{i=1}^{n} k_{oi}\theta_i+C_o, \tag{13} \]

(where the condition \(\sum_{i=1}^{n} k_{oi}=1\) must be satisfied, which can always be achieved by using a node with zero temperature), then it is first necessary to reduce this dependence to the form (5) by multiplying (13) by the scale factor

\[ k_m=\frac{P_o}{C_o}. \tag{14} \]

As a result, the equation of the node will be obtained, that is, a star of conductances

\[ G_{oi}=k_{oi}\cdot k_m=k_{oi}\frac{P_o}{C_o}. \tag{15} \]

with a source at the vertex equal to \(P_o\) and with self-conductance

\[ (-G_{oo})=k_m. \tag{16} \]

Direct connection of this star to the nodes of the circuit leads to a violation of the existing connections, which can be compensated by the simultaneous connection of a second star with the sign of the parameters changed, that is, with conductances \((-G_{oi})\) and source \((-P_o)\). The correctness of such a transformation is proved by the fact that elimination of both connected nodes by the method considered earlier leads to the original circuit.

It is expedient to transform the second connected star into a polygon in order to eliminate the unnecessary node. Thus, inclusion of a new node in the circuit is carried out by connecting a star with parameters \(G_{oi}\) and \(P_o\) and simultaneously adding compensating increments to the sources of the circuit

\[ \Delta P_i=-\frac{P_oG_{oi}}{(-G_{oo})}=-P_ok_{oi}, \tag{17} \]

and to the conductances—compensating increments

\[ \Delta G_{ik}=-\frac{G_{oi}G_{ok}}{(-G_{oo})}=-k_{oi}\cdot k_{ok}\cdot k_m. \tag{18} \]

The use of this transformation can be illustrated by the following example. Suppose it is necessary to take into account the heat removal from the end surface of the core of an armature or stator that has no radial ducts (Fig. 2, a). Owing to the low thermal conductivity of the stack in the direction across the sheet \((\lambda'_{\mathrm{cm}})\), the axial heat flow in the core has an appreciable magnitude only in the immediate vicinity of the end face, creating here a significant temperature drop of the core \(\theta_c\) along the armature axis (Fig. 2, b). In comparison with the variation of \(\theta_c\), in this zone one may assume

\[ \theta_m = \mathrm{const}, \qquad \theta_f = \mathrm{const}. \]

Fig. 2. Inclusion in the thermal circuit of an armature of a node accounting for cooling of the core end faces.

and, considering the core as a heat-conducting rod oriented along the \(x\)-axis, connected with the winding and the air by conductances \(\Lambda_{\mathrm{cm}}\) and \(\Lambda_{\mathrm{cf}}\) per unit length and having losses distributed along its length with density \(\rho_c\), obtain an exponential character of the distribution of \(\theta_c\) along the armature.

Since the propagation constant of the exponent is

\[ L = \frac{1}{\sqrt{(\Lambda_{\mathrm{cm}} + \Lambda_{\mathrm{cf}}) r_c}} \ll l_a, \]

where \(r_c\) is the axial thermal resistance per unit length of the stack, the influence of the opposite end face may be neglected.

The expression for the heat flow passing from the end face of the stack into the air,

\[ Q_{co} = \frac{1}{r_c}\,\frac{d\theta_c}{dx\,(x=0)} = (\theta_{co} - \theta_f) G_{ct}, \]

leads to an equation of the form (5) for the end-face temperature

\[ \theta_{co}(\Lambda_{\mathrm{cm}} + \Lambda_{\mathrm{cf}})L = \theta_m \Lambda_{\mathrm{cm}} L + \theta_f \Lambda_{\mathrm{cf}} L + \rho_c L - Q_{co}. \]

Thus, on the basis of the expression obtained and of the previously developed method for transforming circuits, it is necessary, in order to take account of the thermal

of the flow \(Q_{\mathrm{co}}\), include in the initial thermal circuit of the armature a node with temperature \(\theta_{\mathrm{co}}\) (Fig. 2, b), a source

\[ P'_c = p_c L \]

and conductances

\[ G_{\mathrm{cm}} = \Lambda_{\mathrm{cm}} L;\qquad G_{\mathrm{cf}} = \Lambda_{\mathrm{cf}} L, \]

then attach the compensating sources and conductances

\[ \Delta P_{\mathrm{cm}} = - P'_c \frac{\Lambda_{\mathrm{cm}}}{\Lambda_{\mathrm{cm}}+\Lambda_{\mathrm{cf}}}; \qquad \Delta P_{\mathrm{cf}} = - P'_c \frac{\Lambda_{\mathrm{cf}}}{\Lambda_{\mathrm{cm}}+\Lambda_{\mathrm{cf}}}, \]

\[ \Delta G_{\mathrm{mf}} = - \frac{G_{\mathrm{cm}}G_{\mathrm{cf}}}{G_{\mathrm{cm}}+G_{\mathrm{cf}}}, \]

after which the conductance \(G_{\mathrm{ct}}\) should be introduced, taking into account the conditions of heat transfer from the end face of the core.

Combination of Two Nodes of the Circuit

Let the circuit contain two nodes \(a\) and \(b\) (Fig. 3, a), giving the average temperatures \(\theta_a\) and \(\theta_b\) of two elements of the machine, having volumes, respectively, \(V_a\) and \(V_b\), where

\[ a = \frac{V_a}{V_a+V_b};\qquad b = \frac{V_b}{V_a+V_b}, \tag{19} \]

\[ (a+b=1). \]

Fig. 3. Combination of two nodes of the thermal circuit

The distribution of heat sources between the nodes in the general case is not proportional to volume,

\[ P_a = P_o(a+\Delta);\qquad P_b = P_o(b-\Delta) \tag{20} \]

\[ P_o = P_a + P_b \tag{21} \]

It is required to replace both nodes by one node having the mean temperature

\[ \theta_o = a\theta_a + b\theta_b \tag{22} \]

and the source \(P_o\) (Fig. 3, b).

The equations for nodes \(a\) and \(b\) can be represented as the system

\[ \begin{cases} \theta_a(G_a+G_{ab})-\theta_bG_{ab}=\displaystyle\sum_{i=1}^{n}\theta_iG_{ai}+P_a,\\[6pt] -\theta_aG_{ab}+\theta_b(G_b+G_{ab})=\displaystyle\sum_{i=1}^{n}\theta_iG_{bi}+P_b, \end{cases} \tag{23} \]

where

\[ G_a=\sum_{i=1}^{n}G_{ai}; \qquad G_b=\sum_{i=1}^{n}G_{bi}. \tag{24} \]

Solving (23) with respect to \(\theta_a\) and \(\theta_b\) and substituting the expressions obtained into (22), we have

\[ \theta_o=\sum_{i=1}^{n}\theta_i\frac{A_i}{D}+P_o\frac{C}{D}+P_o\frac{\Delta(aG_b-bG_a)}{D}. \tag{25} \]

Here

\[ D=(G_a+G_{ab})(G_b+G_{ab})-G_{ab}^{2} \tag{26} \]

is the determinant of system (23),

\[ A_i=G_{ai}(aG_b+G_{ab})+G_{bi}(bG_a+G_{ab}) \tag{27} \]

\[ C=a^{2}G_b+G_{ab}+b^{2}G_a. \tag{28} \]

Multiplying (25) by \(D/C\), we obtain the equation of the combined node “\(O\)”

\[ \frac{D}{C}\theta_o=\sum_{i=1}^{n}\theta_i\frac{A_i}{C}+P_o+P_\delta. \tag{29} \]

In view of the presence of the additional heat source

\[ P_\delta=P_o\frac{\Delta}{C}(aG_b-bG_a), \tag{30} \]

the initially formulated transformation condition is not fully satisfied. The reason for this is the nonproportional distribution of sources, characterized by the coefficient \(\Delta\). Any attempt to eliminate \(P_\delta\) by replacing the multiplier \(D/C\) with the multiplier

\[ \frac{D}{C+\Delta(aG_b-bG_a)} \]

leads to a dependence of the circuit conductances on the heat sources and excessively complicates the calculation. Therefore expressions (29, 30) should be adopted finally.

In accordance with the foregoing, the transformation of the circuit consists of the following three operations:

1) elimination of nodes “\(a\)” and “\(b\),” which leads to a change in the sources and mutual conductances of the remaining nodes by \(\Delta P_i'\) and \(\Delta G_{ik}'\);

2) connection of node “\(O\)” (29) with source \((P_o+P_\delta)\) and conductances

\[ G_{oi}=\frac{A_i}{C}, \tag{31} \]

whose self-conductance is

\[ G_o=\frac{D}{C}; \tag{32} \]

3) connection of a compensating polygon which, in accordance with the rules for including a new node in the circuit, has the parameters \(\Delta P_i''\) and \(\Delta G_{ik}''\) (17, 18).

The total change in the sources and mutual conductances of the circuit nodes as a result of the first and third operations is

\[ \Delta P_i=\Delta P_i' + \Delta P_i'' = P_o\frac{\Delta}{C}\left(bG_{ai}-aG_{bi}\right), \tag{33} \]

\[ \Delta G_{ik}=\Delta G_{ik}' + \Delta G_{ik}'' = \frac{\left(bG_{ai}-aG_{bi}\right)\left(bG_{bk}-aG_{ak}\right)}{C}. \tag{34} \]

In this case the balance of the heat sources is satisfied:

\[ \sum_{i=1}^{n}\Delta P_i=P_o. \]

This type of transformation, despite a certain complexity of the formulas, proves effective in those cases when the parameters of the nodes being excluded are mutually dependent.

As an example, one may consider the thermal circuit for substitution of the end part of the armature winding of a direct-current machine. If the winding is made of flexible sections, has no grating in the end parts, and the winding supports are not cooled by air (Fig. 4, a), then practically all the heat released in the lower layer of the end parts is transferred to the upper layer, spreading along the conductors. It can be shown that heat transfer through the interlayer insulation is negligibly small in this case.

Fig. 4. Combination of the upper and lower layers of the end parts in transforming the thermal circuit of an armature

The thermal circuit, compiled on the basis of [2], is shown in Fig. 4, b. The lengths of the section portions located in the upper and lower layers are considered identical; therefore the heat sources \(p_l\) for both layers and the axial thermal resistances \(R\) are equal.

\(G_{lo}\) is the conductance from the upper layer to the air.

The axial conductances are expressed with sufficient accuracy by the formulas:

for the upper layer

\[ g'_r=\frac{6}{R}\left(1+\frac{m}{60}\right);\qquad g'_1=-\frac{2}{R}\left(1-\frac{m}{60}\right), \tag{35} \]

where

\[ m=RG_{\text{lo}} \tag{36} \]

for the lower layer

\[ g''_r=\frac{6}{R};\qquad g''_1=-\frac{2}{R}, \tag{37} \]

\(G'\) is the conductance that takes into account the connection between the layers of the winding in the active part of the armature.

In the thermal calculation of the armature it is inadvisable to divide the winding into two layers, but at the same time it is necessary to take into account the effect of the poorly cooled lower layer of the end-winding parts on the average temperature of the winding. Pairwise combining nodes 1 and 2, representing the average temperature of the layers, and nodes 3 and 4, which correspond to the boundary sections between the end-winding and slot parts, makes it possible to solve this problem quite simply.

When replacing nodes 1 and 2 by a single node “L”,

\[ a=b=0.5;\qquad \Delta=0;\qquad G_{ab}=G_{12}=0, \tag{38} \]

\[ G_a=G_1=G_{\text{lo}}+2g'_r=\frac{12}{R}\left(1+\frac{m}{10}\right), \]

\[ G_b=G_2=2g''_r=\frac{12}{R}, \tag{39} \]

\[ C=0.25(G_1+G_2)=\frac{6}{R}\left(1+\frac{m}{20}\right), \tag{40} \]

\[ P_L=2p_L;\qquad P_d=0;\qquad \Delta P_i=0. \tag{41} \]

Using, in the calculation by (31), the formulas of simplified multiplication and division, because of the smallness of the second terms in expressions (35–40), we obtain

\[ G'_{\text{lf}}=G_{\text{lo}}\left(1-\frac{m}{20}\right), \tag{42} \]

\[ G_{\text{l3}}=\frac{6}{R}\left(1-\frac{m}{30}\right);\qquad G_{\text{l4}}=\frac{6}{R}\left(1+\frac{m}{20}\right);\qquad G_{\text{lo}}=\frac{12}{R}\left(1+\frac{m}{20}\right). \tag{43} \]

The expressions for the mutual conductances of the remaining nodes acquire a similar form.

In an analogous manner, nodes 3 and 4 are combined into node “C,” and the circuit acquires its final form (Fig. 4, c); moreover, to simplify the circuit, the conductances \(G_{of}\) and \(G_{cf}\), which have a negligible value, are added to the conductance \(G_{rf}\).

The parameters of the circuit are:

\[ G_{\text{lf}}=G_{\text{lo}}\left[1-\frac{m}{24}\left(1+\frac{2.85}{1+2G'R}\right)\right], \tag{44} \]

\[ G_r=\frac{12}{R};\qquad G_1=-\frac{4}{R}. \tag{45} \]

The conductance \(G_{\text{lf}}\) takes into account, with sufficient accuracy, the influence of the different cooling conditions of the upper and lower layers of the winding on the average temperature of the end-winding parts and can be recommended for use in various methods of thermal calculation of the armature.

Conclusions

1. In constructing and analyzing thermal equivalent circuits and developing methods for their calculation, such types of transformations as eliminating a node from the circuit, introducing a new node into the circuit, and combining two nodes of the circuit can be used effectively.

2. These transformations are, in principle, exact; however, the use of various methods for simplifying the formulas substantially increases their effectiveness, that is, ultimately, it helps improve the accuracy of thermal calculations without significantly complicating them.

References

1. I. P. Voldev. Increasing the accuracy of the thermal calculation of an electrical machine by the method of equivalent thermal circuits with a small number of elements. Elektromekhanika, No. 1, 1965.

2. D. I. Sannikov. Equivalent thermal circuits of bodies with a one-dimensional temperature field. Izvestiya TPI, vol. 138, 1965.