Q19Electrical Machines and Drives

Speed Expression and Speed Control of a DC Motor

For a DC motor, the back EMF developed is Eb = Kphiw (where K is a machine design constant, phi is the field flux per pole, and w is the angular speed in rad/s), and applying Kirchhoff's voltage law to the armature circuit gives Va = Eb + IaRa (where Va is applied armature voltage, Ia is armature current, and Ra is armature circuit resistance).

This is the fundamental DC motor speed equation, showing that speed depends directly on applied armature voltage Va, inversely on field flux phi, and is reduced slightly below the ideal Va/(Kphi) value by the armature-resistance voltage-drop term Ia*Ra/(Kphi) (the speed regulation, which grows with load current).

For speed control below rated (base) speed, armature voltage control is used: with the field held at its full rated value (maximum flux, hence field current is not reduced), the armature voltage Va is reduced below its rated value using a variable DC source (rheostat control historically, or modern rectifier/chopper control) - since w is directly proportional to Va (for constant flux and neglecting the small IaRa term), this smoothly reduces speed below base speed. Because flux remains at its rated (maximum) value throughout, and because Ia is limited to its rated value for safe operation, the maximum developed torque (T=KphiIa) remains constant (at its rated value) throughout this speed range - armature voltage control is therefore called constant-torque speed control, appropriate for below-base-speed operation.

For speed control above rated (base) speed, field control (field weakening) is used instead: with armature voltage held at its full rated value, the field current (and hence flux phi) is progressively reduced below its rated value using a field rheostat or field-control converter - since w is inversely proportional to phi (for constant Va), reducing flux increases speed above base speed. In this mode, since Va and Ia both remain at their rated values, the rated armature power (VaIa) remains constant, but because w increases as phi decreases, the developed torque (T=KphiIa, decreasing as phi decreases at constant Ia) falls correspondingly - field control above base speed is therefore called constant-power (or constant-horsepower) speed control.

Together, this gives a combined control strategy in which armature voltage control (constant torque, linearly rising power with speed) is used from zero up to base speed, and field control (constant power, torque falling inversely with speed) is used from base speed upward to the motor's maximum permissible speed, providing the widest possible practical speed range while always keeping the motor operating within its rated current and voltage limits - this combined scheme is the standard approach used in DC drives (including thyristor-converter-fed DC drives for rolling mills, traction, and machine tools) requiring a wide constant-power speed range beyond what pure armature voltage control alone could provide within safe rated-current limits.

The statement that DC series motors are never started at no load follows directly from the series motor's field-current-equals-armature-current relationship: at no load (very light load torque), the required armature current (and hence field current, since they are the same current in a series motor) becomes very small, and because torque in the pre-saturation (approximately linear) region is proportional to phiIa which is itself approximately proportional to Ia^2 (since flux is itself roughly proportional to current in an unsaturated series field), even a very small load torque still permits a very small Ia, but for speed, since w is proportional to (Va-IaRa)/(Kphi) and phi itself falls toward zero as Ia falls toward zero (unsaturated series field), the speed expression's denominator (Kphi) falls toward zero much faster than the numerator changes, causing the theoretical no-load speed to rise without bound (mathematically toward infinity) - in practice, this manifests as the series motor accelerating dangerously and uncontrollably (mechanically 'running away') if ever operated with little or no mechanical load connected to its shaft, since there is no natural equilibrium speed at which the (very small) developed torque can balance a (near-zero) load torque at any finite, safe speed, which is precisely why series motors (widely used in traction and crane applications for their high starting torque) must always be permanently, directly coupled to their load (through gears or a direct coupling, never through a belt or clutch that could accidentally disconnect the load) and must never be run light or unloaded.

It is useful to examine the practical implementation of armature voltage control in more detail: historically this was achieved by a variable series rheostat in the armature circuit, but this dissipates significant power as heat in the rheostat itself and gives poor efficiency at reduced speed; modern DC drives instead use a phase-controlled thyristor rectifier or a chopper (as examined elsewhere in this examination) to directly produce a variable, adjustable DC armature voltage from a fixed AC or DC source, achieving the same speed-control effect (Va proportional to firing angle or duty cycle) with far higher efficiency, since no resistive element is deliberately introduced into the main power path.

Similarly, field control in a practical drive is implemented using a much smaller-rated field-circuit converter (rectifier or chopper) than the main armature converter, since the field winding, being of much higher resistance and carrying only a small fraction of the armature current, requires comparatively little power to control - this asymmetry (a large, high-power armature converter and a small, low-power field converter) is a standard, cost-effective feature of combined armature-and-field-controlled DC drive systems used in applications such as rolling-mill main drives, where the constant-torque region below base speed handles the bulk of the mechanical loading while the constant-power field-weakened region above base speed is used mainly for lighter, higher-speed operations such as unloaded traversing or coiling.

The series-motor no-load runaway phenomenon described above is also the underlying reason that series motors used in variable-voltage or chopper-controlled DC traction and crane-hoist drives always incorporate a mechanical or electrical safeguard against complete load loss - such as a maximum-speed cutout relay, mechanical shaft coupling design that cannot accidentally disengage, or, in more modern power-electronic drives, a current or speed feedback loop specifically designed to reduce or cut off armature voltage automatically if the sensed speed rises beyond a safe threshold, precisely because the series motor's own natural speed-torque characteristic provides no inherent protection against this runaway condition, unlike a shunt or separately-excited motor whose largely load-independent field flux gives it a bounded, safe no-load speed instead.