DC Motors for GATE RA 2027: Principle, Back-EMF, Torque Equation & Torque-Speed Characteristics

DC motors are the classic electric actuator and a guaranteed topic in the “Actuators” line of the GATE Robotics and Automation (RA) 2027 syllabus. They convert direct-current electrical energy into rotary mechanical energy and remain the workhorse of robotics because of their simple control, high starting torque and well-behaved torque-speed characteristics. This guide explains the principle of operation, construction, back-EMF, torque equation, performance and the torque-speed characteristics of DC motors — separately excited, shunt and series — with labelled diagrams, governing equations, worked examples and the exam pitfalls to avoid.

Principle of operation

A DC motor works on the motoring action of a current-carrying conductor in a magnetic field. When a conductor carrying current I lies in a magnetic field of flux density B, it experiences a force given by the Lorentz relation F = B·I·L, whose direction is found by Fleming’s left-hand rule. In a motor, many such conductors are placed on a rotating armature inside the field of the stator; the forces on them produce a torque that spins the shaft. A commutator reverses the current in each conductor as it crosses from one pole to the other, so the torque always acts in the same direction and rotation is continuous.

Construction of a DC motor

A conventional DC motor has two main parts. The stationary stator carries the field system — either permanent magnets or field windings on salient poles — which sets up the main magnetic flux φ. The rotating armature (rotor) is a laminated iron core with slots holding the armature winding; it carries the armature current I_a. The commutator, a segmented copper ring on the shaft, together with carbon brushes, feeds current to the armature and switches its direction at the right instant. Additional features such as interpoles and compensating windings reduce sparking from armature reaction. In small robotic drives, permanent-magnet DC motors and brushless DC (BLDC) motors — where electronic commutation replaces brushes — are extremely common.

Back-EMF and the voltage equation

As the armature rotates, its conductors cut the field flux and generate an EMF that, by Lenz’s law, opposes the applied voltage. This is the back-EMF E_b:

where P is the number of poles, Z the total armature conductors, N the speed (rpm), A the number of parallel paths and φ the flux per pole. The back-EMF is proportional to flux and speed, E_b ∝ φN. The applied voltage divides between the back-EMF and the armature-resistance drop: V = E_b + I_aR_a. The back-EMF is what makes a DC motor self-regulating: if the load increases the motor slows, E_b falls, so I_a rises and more torque is produced — automatically matching the demand. At the instant of starting, N = 0 so E_b = 0 and the armature current is limited only by R_a, giving a very large inrush — the reason a starter resistance is used on large machines.

Torque equation

The electromagnetic torque developed by the armature is proportional to the product of flux and armature current:

This single relation, T ∝ φI_a, explains the whole family of DC-motor behaviour. In a shunt or separately excited motor the flux φ is held nearly constant, so torque is proportional to armature current alone, T ∝ I_a. In a series motor the field carries the armature current, so φ ∝ I_a (below saturation) and torque rises with the square of the current, T ∝ I_a² — which is why series motors give enormous starting torque and are used in traction and cranes.

Types and torque-speed characteristics

DC motors are classified by how the field winding is connected relative to the armature. The torque-speed characteristic — how shaft speed changes as the load torque changes — is the single most examined property, so learn the shapes:

• Separately excited / shunt motor: the field is fed from a constant source, so φ is constant. From N ∝ (V − I_aR_a)/φ, the speed drops only slightly as load (I_a) rises. The result is a nearly flat, gently drooping torque-speed line — a good constant-speed drive.
• Series motor: φ ∝ I_a, so speed N ∝ (V − I_a(R_a+R_se))/I_a — roughly inversely proportional to current. Speed is very high at light load and low at heavy load: a steeply falling, hyperbolic torque-speed curve. It must never be run unloaded, or it will overspeed dangerously.
• Compound motor: has both series and shunt fields. A cumulative compound motor combines a fair starting torque with a safe no-load speed — a compromise between shunt and series.

Speed control methods

From N ∝ (V − I_aR_a)/φ, there are three ways to control the speed of a DC motor, and each defines an operating region used in robotics:

• Armature-voltage control: vary the voltage V applied to the armature (today via a PWM chopper or controlled rectifier). Speed varies linearly with voltage from zero up to base speed at constant torque. This is the primary method in modern electronic drives.
• Field-flux control: weaken the field φ (reduce field current) to run above base speed at constant power but reduced torque. Used for the high-speed region.
• Armature-resistance control: add a series resistance to drop voltage; simple but wasteful (I²R losses) and rarely used now.

The combination of armature-voltage control below base speed (constant torque) and field weakening above it (constant power) gives the classic two-region speed envelope that servo and traction drives exploit.

Performance, losses and efficiency

The power flow in a DC motor runs from electrical input V·I to the air-gap (developed) power E_b·I_a and finally to the mechanical output at the shaft. The losses along the way are the copper losses (I_a²R_a in the armature and the field-winding loss), the iron/core losses (hysteresis and eddy currents in the rotating core, roughly proportional to flux and speed), and the mechanical losses (friction and windage). The efficiency is the ratio of shaft output power to electrical input power, η = P_out/P_in, and peaks when the variable copper loss equals the fixed (iron + friction) loss. The developed mechanical power equals the back-EMF times the armature current, P_dev = E_bI_a, and the shaft torque follows from T = P_dev/ω.

Worked examples

Key formulas

Applications

Shunt / separately excited motors, with their steady speed, drive lathes, conveyors, fans, pumps and any application needing constant speed with light load variation. Series motors, with their huge starting torque, power electric traction, cranes, hoists, and starter motors. Permanent-magnet and brushless DC motors dominate robotics: wheeled robots, joint drives, electric vehicles, drones, and precision positioning stages, where their high torque density, easy speed control and good efficiency are ideal. The linear, predictable torque-speed behaviour of the DC machine is exactly why it remains the reference actuator for control-system study.

Common mistakes to avoid

• Forgetting that at starting N = 0 so E_b = 0, giving a huge inrush current limited only by R_a.
• Writing series-motor torque as T ∝ I_a — it is T ∝ I_a² below saturation because φ ∝ I_a.
• Running a series motor at no load — it overspeeds dangerously; always couple it to a load.
• Mixing up the two speed regions: armature-voltage control = constant torque below base speed; field weakening = constant power above it.
• Confusing back-EMF with a real supply — it is an induced opposing voltage, not a source, and it vanishes at standstill.

Frequently asked questions

What is back-EMF and why does it matter?

Back-EMF is the voltage induced in the rotating armature that opposes the applied voltage (Lenz’s law), E_b = kφN. It makes the motor self-regulating: when load increases the motor slows, E_b falls, armature current rises and more torque is produced automatically. At standstill E_b = 0, which is why starting current is very high.

Why does a series DC motor have such high starting torque?

In a series motor the field carries the armature current, so flux φ ∝ I_a. Since torque T ∝ φI_a, this gives T ∝ I_a². The torque therefore rises with the square of the current, producing very large starting torque — ideal for traction, cranes and hoists.

Why must a series motor never run without load?

At no load the current is small, so the flux is small, and since speed N ∝ 1/φ the motor accelerates to a dangerously high speed that can destroy it mechanically. A series motor must always be directly coupled (gear or belt) to its load.

How is the speed of a DC motor controlled below and above base speed?

Below base speed, the armature voltage is varied (usually by a PWM chopper), giving constant-torque operation. Above base speed, the field flux is weakened, giving constant-power operation with reduced torque. Together they form the standard two-region speed envelope.

What is the difference between a brushed and brushless DC motor?

A brushed DC motor uses a mechanical commutator and carbon brushes to switch the armature current. A brushless DC (BLDC) motor replaces these with electronic commutation driven by rotor-position feedback, eliminating brush wear and sparking and giving higher efficiency and reliability — which is why BLDC motors dominate modern robotics and drones.

This guide is part of the Actuators series in the complete GATE RA 2027 Syllabus overview. Continue with Stepper Motors and Servo Motors, or review Hydraulic & Pneumatic Actuators.