Q3Computer Aided Design of Electrical Machines

(a) Cooling Time Constant

When an electrical machine that has reached a steady operating temperature is switched off (removing its internal heat generation), its temperature falls exponentially back toward the ambient temperature, governed by the same fundamental heat-balance differential equation used for heating (discussed in the corresponding companion answer), but with the heat-generation term set to zero:

Practical measurement of heating and cooling time constants: in practice, both time constants are determined experimentally rather than purely analytically, since the effective thermal capacity and cooling coefficient of a real machine (with its non-uniform mass distribution and multiple heat paths) are difficult to compute accurately from first principles alone. The standard test procedure records the machine's temperature rise (using embedded resistance-temperature detectors or thermocouples at representative points, or the winding-resistance method for average winding temperature) at successive time intervals after the machine is loaded to produce a known constant loss, plotting theta against t; the heating time constant is then read off directly as the time at which the recorded curve reaches 63.2% of its eventual steady value, or, more accurately, obtained from the slope of a semi-log plot of (theta_final-theta) against time, which should yield a straight line for a true single-time-constant exponential system, with the time constant equal to the reciprocal of the slope. An entirely analogous procedure, switching off the loss source and recording the temperature decay, is used to obtain the cooling time constant. These experimentally-determined time constants are of direct practical value in rating a machine for short-time and intermittent duty, and in setting the trip/alarm delay times of thermal-overload protection relays so that they correctly track the machine's actual thermal response rather than tripping prematurely or dangerously late.

where theta(t) is the temperature rise above ambient at time t after switch-off, theta0 is the initial temperature rise at the moment of switch-off, and tau_c is the cooling time constant. The cooling time constant is defined as the time taken for the temperature rise to fall to 1/e (approximately 36.8%) of its initial value theta0, and is given by tau_c=(thermal capacity)/(cooling coefficient) — physically, tau_c represents how quickly the machine's stored thermal energy is dissipated to the surrounding environment once internal heat generation ceases, with a machine having better ventilation/cooling exhibiting a shorter cooling time constant, and, importantly, the cooling time constant for a machine at standstill (with cooling fans stopped) is typically considerably longer than its heating time constant during normal running (with cooling fans operating), since forced ventilation substantially improves heat removal compared to natural convection alone.

(b) MMF Required for Air Gap of a Slotted Armature

The air gap of a rotating electrical machine represents the single largest contributor to the total magnetic circuit's reluctance (since air has vastly lower permeability than the iron core material), and correctly calculating the MMF required to drive the necessary flux across this air gap is therefore central to determining the machine's overall magnetizing (exciting) current requirement.

Basic (smooth-gap) MMF requirement: for an idealized smooth (unslotted) air gap of physical length lg, carrying a flux density Bg, the required MMF is simply:

Effect of slotting: in a practical rotating machine, the armature (or stator) surface facing the air gap is not smooth but contains slots housing the winding conductors, and the presence of these slot openings distorts the air-gap flux distribution — flux tends to concentrate and bulge outward (fringe) around each slot opening, effectively increasing the reluctance of the air-gap path compared to an idealized smooth gap of the same physical length, since the flux lines must travel a longer, more distorted path to cross the gap in the vicinity of each slot opening.

Carter's coefficient and effective air gap: this slotting effect is quantified using Carter's gap coefficient, Kg (also called Carter's coefficient), an empirical/analytical correction factor greater than 1, calculated from the ratio of slot opening width to air-gap length and the ratio of slot pitch to air-gap length, typically expressed via the formula:

where tau_s is the slot pitch, wo is the slot opening width, and lg is the actual physical air-gap length. The effective air-gap length, accounting for this slotting-induced reluctance increase, is then given by:

and the corrected MMF requirement for the actual slotted air gap becomes:

Meaning of effective air-gap length: the effective air-gap length (lg,eff=Kglg) represents the length of an idealized, equivalent smooth (unslotted) air gap that would present exactly the same magnetic reluctance to the flux as the actual physical slotted gap, correctly capturing the additional reluctance introduced by the slot openings without requiring detailed field-distortion analysis at every individual slot — this effective length concept allows the machine designer to continue using the simple smooth-gap MMF formula (AT=Bglg,eff/mu0) while still correctly accounting for the real, physically-slotted geometry's true magnetic behavior, and is a standard, essential correction applied whenever calculating the air-gap MMF component of a slotted rotating machine's total magnetizing characteristic — for a doubly-slotted machine (both stator and rotor surfaces slotted, as in an induction motor), separate Carter's coefficients are calculated for the stator-side and rotor-side slotting, and their combined effect (typically the product of two individual coefficients, or a suitably combined effective single coefficient) is used to determine the overall effective air-gap length for the complete magnetic circuit calculation.

Typical numeric range: for slot-opening-to-gap ratios (wo/lg) commonly encountered in practical machine design, Carter's coefficient Kg typically takes values in the range of roughly 1.1 to 1.4 for a single (one-sided) slotted surface. For example, a machine with a physical air gap of 3mm and a slot opening of 6mm has wo/lg=2, giving gamma=(2)^2/(5+2)=0.571 and Kg=tau_s/(tau_s-0.571*3); for a typical slot pitch of around 25mm this yields Kg of approximately 1.07, whereas a wider slot opening relative to the gap (say wo/lg=5, as with a larger slot opening or a smaller physical gap) pushes gamma up to 25/10=2.5 and correspondingly increases Kg further. This illustrates the general trend that a narrower slot opening (or a larger physical air gap) relative to the slot pitch reduces the slotting-induced reluctance penalty, which is one practical reason semi-closed slots (with a narrow bridge across the slot mouth) are preferred over fully open slots wherever winding insertion considerations permit, since they keep Kg closer to unity and reduce both the extra magnetizing MMF required and the associated air-gap flux pulsation (a source of additional iron loss and noise).