148 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1951 



If r' is thus determined, and K is known from measurements of similar 

 coils, V can be determined by means of equation (4) from the values of T\ 

 or Ti given by equations (8). If desired, the mean temperature T can then 

 be determined by equation (6). 



Conclusions 



In most relay and switch magnet coils, of the type used in telephone 

 apparatus, the heat flow under steady state conditions can be considered as 

 wholly radial, and the temperature distribution conforms approximately to 

 equation (4) above. In this expression, T' is the maximum temperature, and 

 r' the radius at which this occurs, so that heat generated inside the surface 

 of radius r' passes to the core, and that generated outside this radius passes 

 to the cover. The rate of heat removal per unit area at either of these surfaces 

 is found experimentally to be approximately proportional to the difference 

 between the surface and ambient temperatures (for the temperature range 

 of normal operation). The proportionality constant is the heat removal 

 coefficient (^i or ^2 of equations (8)). Under conditions typical of telephone 

 apparatus, this coefficient is of the order of 0.002 watts per cm^ per °C for 

 a cover surface, and 0.005 watts per cm^ per °C for an inside surface in close 

 proximity to the metal core. The heat removal coefficient for an inner 

 surface is much more variable than that for a cover surface. 



The coil temperature distribution [equation (4)] depends upon the rate of 

 heat supply per unit volume Q, and upon the effective heat conductivity 

 K. Q may be taken as equal to the total steady state power input divided 

 by the coil volume. Correction might be made for the radial variation in Q 

 resulting from the variation in copper resistivity with temperature. The 

 relatively small temperature range observed in practice, as illustrated by 

 the results of Fig. 2, makes this an unnecessary refinement. The heat con- 

 ductivity constant K is an effective average value, applying to the coil as 

 though it were a homogeneous structure. It is conveniently evaluated by 

 measurements of actual coils, and is approximately a constant for a given 

 wire size and type of insulation. 



By measurement of the resistance changes in tapped coils, the internal 

 temperature distribution can be determined. From this, values of the 

 effective heat conductivity K and of the heat removal coefficients k\ and 

 ki can be determined by the use of equation (4), as described above. Con- 

 versely, if K and the heat removal coefficients are known, equation (4) may 

 be used to estimate the internal temperature distribution, and thus the 

 mean and maximum coil temperatures. 



