Thermal network of an induction motor

= −

εσ . (4.26)

4.3 Thermal network of an induction motor

Because there is a full analogy between electric and thermal circuits, the heat transfer can be solved by using similar circuits, where thermal resistance corresponds to electric resistance, heat flux corresponds to current, and temperature difference corresponds to voltage. Also, thermal capacitances are required in the model, as the main focus is on the dynamic modelling. Different thermal network models for induction machines in the literature have been introduced for example by Kylander (1995) and Mellor (1991). Thermal network is constructed by first dividing the motor into separate geometrical sections, which are connected to neighbouring sections through thermal resistances. As the main interest of the study is in the transient heat transfer, thermal capacitances are added to all the sections comprised from the solid material. Internal heat generation is added for all those components, in which the losses are generated. Individual sections that comprise the model are

1. Frame (Cth) 6. End winding (Cth, Ploss) 2. Stator yoke (Cth, Ploss) 7. End cap air

3. Stator teeth (Cth, Ploss) 8. Rotor winding (Cth, Ploss) 4. Stator winding (in slots) (Cth, Ploss) 9. Rotor iron (Cth)

5. Air gap 10. Shaft (Cth)

Cth after the component denotes the thermal capacitance and Ploss the internal heat generation due to the losses. The modelling is carried out by presenting these 10 components as cylindrical components, symmetrical to the shaft and the radial plane of the machine. Cylindrical components are modelled with the so-called two-node configuration, where the two nodes represent the radial and axial heat flow. In such a way, the independent modelling of radial and axial heat flows is most straightforward. The simplified thermal model presented at the end of this chapter is constructed by using a common single-node configuration. Four assumptions are required to justify the modelling of individual machine sections as cylindrical components (Mellor 1991):

1. Heat flows in radial and axial directions are independent.

2. A mean temperature Tave defines the heat flow in both axial and radial directions.

3. There is no circumferential heat flow (except from stator teeth to stator iron).

4. Thermal capacitance and heat generation are uniformly distributed.

Figure 4.4 a) depicts the simplified cylindrical component, its dimensions and temperatures, and Fig. b) the equivalent thermal circuit.

Figure 4.4. General cylindrical component and its equivalent thermal circuit.

In Fig. 4.4, the temperatures in the radial directions Tr,out and Tr,in are the temperatures on the outer and inner surface of the cylinder, respectively, and the temperatures in the axial directions Tax,1 and Tax,2 are the temperatures at both ends. Rth,rad denotes the thermal resistance for the radial heat flow, and accordingly Rth,ax for the axial heat flow. The model therefore takes in the account the different heat flow characteristics in radial and axial directions, for example, in the iron due to laminations. Thermal resistance in radial direction can be expressed as

λL

and the thermal resistance in axial direction

( )

The upper and lower part in Fig. 4.4 b) represent the one-dimensional heat flow in the axial and radial directions, respectively, and as these two one-dimensional models are connected from the node where the heat is generated, a two-dimensional model for the cylinder results. If there is internal heat generation in the element, a negative-value resistance must be added between the

point of the heat generation and the central node (black spots in Fig 4.4 b)) of each one-dimensional model. This is due to fact that without this negative interconnecting resistance, superposition of internal heat generation would result in average temperature of the element Tave

that is lower than the value given by the central node (Mellor 1991). Values for the thermal capacitances can be calculated

m c

C_th = _th , (4.29)

where cth is the specific heat capacitance and m the mass. Two-dimensional thermal networks for each ten elements are built, and these separate networks are then connected through thermal resistances of each boundary section. The thermal network model for the whole motor is shown in Fig. 4.5, and the explanations of different thermal resistances are listed in Table 4.1.

R_th1

Figure 4.5. Thermal network of an induction motor based on a cylindrical component representation

Table 4.1 Thermal resistances and their explanations of the network Comp. Explanation

Rth1 Measured resistance from frame to ambient

Rth2 Measured radial resistance from frame to stator yoke Rth3 Axial resistance from stator yoke to end cap air Rth4 Radial interconnecting resistance of the stator yoke Rth5 Radial resistance from the stator yoke to frame Rth6 Radial resistance from the stator yoke to stator teeth Rth7 Axial resistance from stator teeth to end cap air

Rth8 Radial/circumferential resistance from stator teeth to stator winding Rth9 Radial interconnecting resistance of the stator teeth

Rth10 Radial resistance to from the stator teeth to stator yoke Rth11 Radial resistance to from the stator teeth to air gap

Rth12 Radial/circumferential resistance from the stator coils to stator teeth Rth13 Axial resistance to from the stator coils to end-winding

Rth14 Radial resistance from the stator coils to stator yoke Rth15 Radial resistance from the stator coils to air gap Rth16 Radial resistance from the air gap to stator teeth Rth17 Radial resistance from the air gap to stator coils Rth18 Radial resistance from the air gap to rotor bars Rth19 Axial resistance from the end-winding to stator coils Rth20 Resistance from the end-winding to end cap air (“legs”) Rth21 Resistance from the end-winding to end cap air (“toroid”) Rth22 Axial resistance from the end cap air to frame

Rth23 Axial resistance from the end cap air to stator yoke Rth24 Resistance from the end cap air to stator teeth Rth25 Resistance from the end cap air to end-winding Rth26 Resistance from the end cap air to rotor end-rings Rth27 Resistance from the end cap air to rotor iron

Rth28 Resistance from the end cap air to cooling holes (if present) Rth29 Axial resistance from the rotor bars to end cap air

Rth30 Radial interconnecting resistance of the rotor bars Rth31 Radial resistance from the rotor bars to air gap Rth32 Radial resistance from the rotor bars to rotor iron Rth33 Axial resistance from the rotor iron to end cap air Rth34 Radial interconnecting resistance of the rotor iron Rth35 Radial resistance from the rotor iron to rotor bars Rth36 Radial resistance from the rotor iron to shaft Rth37 Radial resistance from the shaft to rotor iron

Rth38 Axial resistance from the shaft to frame through bearings Rrad Radiation resistance from the frame to the ambient