Characteristics of DC Shunt Motors - Electrical Machines - Electrical Engineering

Introduction - modes of operation

A DC machine can operate either as a generator or as a motor depending on the relative magnitudes of the induced emf and the terminal supply voltage. The induced emf in the armature is E = k φ ω, where k is a machine constant, φ is the flux per pole and ω is the angular speed. If the prime mover drives the armature so that E > Vt (terminal voltage), the machine delivers power to the external circuit and behaves as a generator. If E < Vt, current flows into the armature and the machine behaves as a motor.

Electrical relations in generator and motor modes

For the armature circuit the voltage relation differs by sign depending on the mode of operation.

• Generator mode: Vt = E - Ia Ra.
• Motor mode: Vt = E + Ia Ra.

The field current in a shunt winding is If = Vt / Rf, and the flux φ is a nonlinear function of If (magnetisation or saturation characteristic). In many analyses we assume φ is constant for small variations of If around the operating point, but the saturation curve must be kept in mind for larger changes.

Power flow and loss components

The components of power and losses differ in generator and motor modes but the basic elements are the same:

• Electrical input (motor) or electrical output (generator): Vt Il (where Il is line/load current).
• Armature copper loss: Ia² Ra.
• Field copper loss: If² Rf.
• Developed electrical power converted to mechanical power: Ed Ia (often written E Ia).
• Mechanical losses: friction, windage and iron losses (collectively shown as mechanical loss). The net mechanical power available at the shaft is the developed electrical power minus these mechanical losses.

When the machine works as a generator, the prime mover provides mechanical power T ω and electrical power is delivered to load or returned to a supply. When it works as a motor the electrical supply provides input power and the shaft delivers mechanical output T ω to the load.

Armature current and equivalent expressions

From the voltage equations, armature current can be written in terms of speed or induced emf:

• Generator mode: Ia = (E - Vt) / Ra.
• Motor mode: Ia = (Vt - E) / Ra.

Using E = k φ ω gives expressions linking speed, armature current and terminal voltage. The armature current determines electromagnetic torque since Te = k φ Ia. The sign of Te follows the chosen sign convention: torque that assists motion is taken positive (motoring), torque opposing motion is negative (generating).

Torque-speed characteristic (shunt machine)

For a shunt machine with approximately constant field flux φ, one can derive a linear torque-speed relation.

Start from the motor voltage equation and torque expression and eliminate Ia.

• From Vt = E + Ia Ra and E = k φ ω, we get Ia = (Vt - k φ ω) / Ra.
• Using Te = k φ Ia gives Te = (k φ / Ra) (Vt - k φ ω).
• Rearranging for speed: ω = Vt / (k φ) - (Ra / (k φ)²) Te.

This is the equation of a straight line in the ω-Te plane: the intercept on the speed axis (Te = 0) is the no-load speed ωnl = Vt / (k φ), and the slope is negative with magnitude Ra / (k φ)². The corresponding armature current axis is proportional to torque because Te ∝ Ia. In generator mode the same straight line applies with the appropriate sign convention for torque.

Two machines mechanically coupled - determining motoring and generating roles

Consider two separately supplied DC machines coupled on a common shaft and running at the same speed ω. Let the applied terminal voltages be Vt1 and Vt2, with field fluxes φf1 and φf2, armature resistances Ra1, Ra2, and armature currents Ia1, Ia2. The torque expressions are:

• Te1 = k φf1 Ia1, with Ia1 = (Vt1 - k φf1 ω) / Ra1.
• Te2 = k φf2 Ia2, with Ia2 = (Vt2 - k φf2 ω) / Ra2.

Since the two machines are mechanically coupled and rotate together, the algebraic sum of torques must balance the load torque and shaft inertias. For steady state with no net accelerating torque the condition is:

• Te1 + Te2 = 0 (if only internal machine torques act and shaft damping/inertia are not accelerating).

Graphically, plotting Te1(ω) and -Te2(ω) on the same axes yields the intersection point giving the operating speed and the sign of torque for each machine. If Te is positive (assisting motion) the machine is motoring; if Te is negative (opposing motion) the machine is generating. Thus, whichever machine's torque-speed line lies above the other will tend to drive the coupled system and the other will act as a generator.

Interpretation of torque sign and mechanical power

By convention used here, torque that assists motion (in the direction of rotation) is positive; torque opposing motion is negative. If a machine produces a positive developed torque its electrical input is partly converted to mechanical output and the machine behaves as a motor. If the developed torque opposes motion, mechanical power is being delivered to the machine and it acts as a generator.

Characteristic parameters and how they change

Key features of the torque-speed characteristic for a shunt machine with field flux φ fixed:

• No-load speed ωnl = Vt / (k φ) (neglecting mechanical losses).
• Rated armature current limits the maximum torque available: Te,max = k φ Ia,rated.
• Speed drop under load is proportional to Ra and torque, and inversely proportional to φ². A larger armature resistance gives a steeper slope (more speed drop for the same torque).
• Raising armature voltage Vt shifts the characteristic upward (increases no-load speed) without changing slope if Ra and φ are unchanged.
• Reducing field flux increases no-load speed and increases the magnitude of the slope; the characteristic becomes steeper and the machine can reach higher speeds for the same terminal voltage.

Operating zones: constant-torque and constant-power

Because armature current is limited by rating, the machine cannot be operated beyond the vertical limits corresponding to Ia,max. Two important operating regions are:

• Constant-torque zone: when armature voltage is reduced below rated value, the torque limit is set by the maximum armature current and remains approximately constant with speed. The machine behaves as a constant torque device up to the current limit.
• Constant-power zone: when field flux is reduced (field weakening) but armature current remains limited, the product Te,max ω (mechanical power) remains about constant. Therefore, reducing the field flux increases speed but reduces torque such that maximum developed mechanical power remains essentially constant - this defines the constant-power operating zone (rectangular hyperbola in Te-ω plane).

The practical operating point of a shunt machine must lie within the current limits so that armature heating is controlled and equipment ratings are not exceeded.

Effect of armature reaction

Armature reaction is the distortion and partial demagnetisation of the main field produced by the armature mmf when armature current flows. The net effect is that the effective field flux per pole φeq is reduced as armature current increases. A simple representation is:

• φeq = φf - φar, where φar is the flux reduction due to armature reaction and is approximately proportional to Ia for moderate currents.

Consequences of armature reaction:

• For a given terminal voltage and higher armature current the effective flux falls, so the induced emf E = k φeq ω reduces and more current flows, partially compensating torque production.
• The current-speed and torque-speed curves including armature reaction lie above the curves that neglect armature reaction for the same load, because the fall of flux tends to reduce the drop in speed for a given current.
• Armature reaction is more significant at high load currents and must be accounted for in accurate performance prediction; compensating windings or interpoles are used in larger machines to reduce its effect.

Example - estimation of flux reduction due to armature reaction

Problem statement (preserved facts): A DC machine rated 8 kW, 230 V, 1200 RPM has armature resistance Ra = 0.7 Ω. The no-load terminal voltage set is 250 V. The motor runs at no-load speed Nnl = 1250 rpm and draws armature current Ia,nl = 1.6 A. Under a particular load the armature current becomes Ia,load = 40 A and the speed falls to Nload = 1150 rpm. Find the percentage reduction in flux per pole Δφf / φf (due to armature reaction).

Solution:

Use the relation between induced emf, terminal voltage and armature drop:

E = Vt - Ia Ra (motor equation).

Also E = k φ N if N is speed in rpm and k includes the conversion factor between ω and N; here k is the same constant in both cases.

Compute flux at no load (φ_nl):

V_t = 250 V, Ia,nl = 1.6 A, Ra = 0.7 Ω, Nnl = 1250 rpm.
E_nl = V_t - Ia,nl Ra
E_nl = 250 - (1.6 × 0.7)
E_nl = 250 - 1.12 = 248.88 V
Therefore φ_nl = E_nl / (k Nnl) = 248.88 / (k × 1250) = 0.199104 / k ≈ 0.1991 / k
Compute flux under load (φ_load):
Ia,load = 40 A, Nload = 1150 rpm.
E_load = V_t - Ia,load Ra
E_load = 250 - (40 × 0.7)
E_load = 250 - 28 = 222 V
Therefore φ_load = E_load / (k Nload) = 222 / (k × 1150) = 0.1930435 / k ≈ 0.1930 / k
Change in flux and percentage reduction:
Δφ = φ_nl - φ_load
Δφ = (0.199104 / k) - (0.193044 / k) = 0.00606 / k
Percentage reduction = (Δφ / φ_nl) × 100%
Percentage reduction = (0.00606 / 0.199104) × 100% ≈ 3.0% (rounded)

Note: Using the rounded values reported in the source calculation gives approximately 3.5%. Small differences arise from rounding of the intermediate numerical values or interpretation of the given no-load set voltage; the important point is that the flux reduction due to armature reaction is small (a few percent) for this loading but not negligible for accurate performance prediction.

Practical implications and applications

Key practical points for DC shunt motors:

• Shunt motors give a fairly steep torque-speed characteristic and are suitable where approximately constant speed is required under varying load (e.g., lathes, fans, machine tools).
• Speed can be controlled by varying armature voltage (V_t) or by weakening the field (reducing If) - the latter gives higher speed but reduced torque (field weakening / constant-power operation).
• Armature current limits and heating determine safe operating regions (constant-torque zone bounded by Ia,max and constant-power zone under field weakening).
• Armature reaction and saturation limit how much the flux can be increased; in practice fields are designed to operate near rated flux and field weakening is used carefully for speed control.
• When two machines are connected on the same shaft with separate supplies, either machine may operate as motor or generator depending on relative voltage, flux and speed characteristics; torque equilibrium determines the steady operating point.

Summary

For a DC shunt machine with essentially constant field flux, the torque-speed characteristic is approximately linear with slope determined by armature resistance and field flux. The no-load speed is proportional to armature voltage and inversely proportional to flux. Armature reaction reduces the effective flux at higher armature currents, slightly modifying the ideal linear characteristics. Limits on armature current create distinct operating zones (constant-torque and constant-power) that determine safe and practical machine operation. Careful selection of armature voltage, field current and added resistances allows control of speed and torque to meet application requirements.