Q5Utilization Of Electrical Power

Tractive Effort of a Train and its Functions

Tractive effort is the net force, developed at the periphery of the driving wheels and transmitted to the rail through adhesion, that is available to propel the train. Its function is to overcome, in combination, three distinct opposing forces: the force required to accelerate the train's mass (including the effective, rotational-inertia-augmented mass of the rotating parts), the force required to overcome the component of gravity acting along the track when running on a gradient, and the force required to overcome the train's resistance to motion (rolling friction, flange friction, and air resistance).

Derivation of Tractive Effort for Acceleration

The tractive effort component required for acceleration follows directly from Newton's second law applied to the effective mass of the train, which exceeds its actual static mass because the rotating parts (wheels, axles, gear wheels, and armatures of the traction motors) must also be angularly accelerated, requiring additional torque/force beyond that needed to linearly accelerate the train's translational mass alone; this effect is captured by an empirical percentage addition to the actual dead mass, giving an effective mass Me = (W/g)(1 + %/100), where W is the dead weight of the train and the percentage addition (typically 8-15% for locomotives and 4-6% for unpowered coaching stock) accounts for the extra force needed to accelerate the rotating masses.

The gradient component of tractive effort is the component of the train's weight acting along the direction of the track when running on an incline, equal to W*sin(theta) where theta is the angle of the gradient; for the shallow gradients typical of railway practice, sin(theta) is very closely approximated by 1/G for a gradient expressed as 1 in G, giving Fg = W/G (positive when climbing, and acting as an assisting rather than opposing force when descending).

The resistance component of tractive effort is obtained by multiplying the train's weight by its specific tractive resistance (the resistance to motion per unit weight, typically expressed in kg-force per tonne, or newtons per tonne in SI units), a value that itself depends on speed (increasing with speed due to the growing contribution of air resistance) and is generally determined from empirical formulae or test data for the specific rolling stock and route conditions in question.

Summing these three components gives the total tractive effort the traction motors must develop at the wheel rim at any instant of the journey, and this combined quantity, evaluated throughout the acceleration, running, and gradient phases of a journey, directly determines the required motor torque rating, gear ratio, and (via the adhesion-limited maximum tractive effort available at the wheel-rail contact, discussed in relation to related traction calculations elsewhere in this examination) the minimum adhesive weight and number of driving axles the locomotive must possess to develop the necessary tractive effort without wheel slip.

Advantage of Series-Parallel Starting for a Pair of DC Traction Motors

In ordinary rheostatic starting of a pair of DC traction motors, both motors are connected directly in parallel across the full line voltage from the very start, and the entire voltage in excess of the motors' back-EMF (which is low or zero at standstill and low speed) must be dropped across external starting resistance to limit the starting current to a safe value; since the motors are already at full line voltage, only a single running condition (full parallel, full voltage) is available once the resistance is fully cut out, and a comparatively large amount of external resistance, and hence a comparatively large amount of wasted I^2R energy, is needed throughout the low-speed portion of acceleration to hold current within safe limits. In series-parallel starting, by contrast, the two motors are first connected in series with each other across the line voltage, so each motor initially sees only half the line voltage; since the back-EMF needed to balance a given fraction of this halved applied voitage is reached at a correspondingly lower speed, less external starting resistance is needed to limit current to the same safe value during this first (series) stage of acceleration, directly reducing the resistive energy wasted compared to starting both motors in full parallel connection from the outset. After the series stage brings the train up to an intermediate speed with all series-stage resistance cut out, the motors are switched to the parallel connection (each now receiving the full line voltage), and a further, similarly reduced amount of resistance is notched out to complete acceleration to full running speed.

The overall advantage of series-parallel starting over plain rheostatic (all-parallel) starting is therefore a substantial reduction in the total energy wasted as heat in the starting resistors during acceleration, since less resistance is needed at any given speed during the series stage than would be needed to achieve the same safe starting current with the motors already in parallel, together with the availability of two distinct, fully efficient (all resistance cut out) running speeds, one in series connection and a higher one in parallel connection, rather than only a single efficient running speed as with plain rheostatic starting of parallel-connected motors.

It is also useful to note that the tractive effort demand calculated by summing the acceleration, gradient, and resistance components as derived above represents only the instantaneous demand at a particular point in the journey; since speed, and hence resistance (which grows with speed), and gradient (which varies with track profile) both change continuously throughout a run, the complete tractive effort versus time (or versus distance) profile for a full journey is obtained by evaluating this same three-term expression repeatedly at successive points along the route, using the corresponding speed-time curve (as discussed in relation to another question in this examination) to determine the instantaneous speed, and hence resistance, applicable at each point; this complete tractive-effort profile is precisely what determines the peak and continuous power ratings required of the traction motors selected for a given locomotive and route.

This closes the requested definition of tractive effort of a train and its functions with the derivation of the expression for tractive effort developed by a train unit, together with the discussion of the advantage of series-parallel starting against ordinary rheostatic starting for a pair of DC traction motors.