Time Constant of Reactors as Functions of Volume

The time constant of an inductive circuit is defined as the ratio L/R where L is the inductance and R the resistance of the circuit and both L and R are assumed to be constant. If a reactor has an air gap in its core and the reluctance of the iron, hysteresis, and eddy currents are neglected, then the inductance is constant; and if the winding is at a constant temperature, its resistance is constant also. Consider, on that basis, an iron-core reactor with an air gap of length g as shown in Fig. 5-16. Let the effective cross-sectional area of the air gap be A_g and the net area occupied by the conductors in the winding be k_wA_w, where k_w is a space factor and A_w is the area of the window in the core, i.e., the area of the space between the center leg and one outer leg of the core in Fig. 5-16. The space factor k_w is less than unity because not all of the window area A_w is occupied by conductor material of the winding since some space is occupied by electrical insulation and voids between turns, winding, and core.

Figure 5-16. Three-legged reactor with air gap. (a) Top view; (b) side view

When the reluctance of the iron and magnetic leakage are neglected, the self-inductance is, according to Eq. 4-11

where N is the number of turns in the reactor winding.

The resistance of the winding, which is a conductor of uniform cross section, is given by

[5-79]

Where p is the resistivity of the conductor material, usually copper, l the length of the winding, and A the cross-sectional area of the conductor material. The length is

[5-80]

Where l_t is the mean length of the turns and the area

[5-81]

From Eqs. 5-79, 5-80, and 5-81 it follows that the resistance of the winding is

[5-82]

so that the time constant

[5-83]

If all linear dimensions of the reactor core and air gap are increased by a factor k, the areas A_w and A_g are each increased by k², whereas the lengths l_t and g are increased directly as k, so that the time constant L/R for a given configuration of core and winding increases as k² or as the 2/3 power of the volume, i.e.

[5-84]

It should be noted that the time constant is independent of the number of turns because the inductance and resistance of the winding both vary as the square of the turns for a given space factor k_w.