This notes sheet looks at electric generators and motors, general electric machine properties, motoring graphs and nominal voltages.

Generators & Motors

DC Generator

DC generator diagram, electric machines, electricity generator diagram, generator diagram
  1. A torque is applied to the coil, causing it to rotate with angular velocity \omega
  2. The right-hand rule shows that a current is induced clockwise around the coil
  3. The emf can be calculated using Faraday’s law

DC Motor

DC motor, DC motor diagram, electric motor diagram, electric machine, electric engine
  1. There is an anticlockwise current in the coil, as shown
  2. Fleming’s left hand rule shows that this induces a clockwise torque couple on the coil, due to the magnetic field
  3. The motor rotates clockwise

A split-ring commutator is required where the current and emf is applied to the coil in order to maintain continuous motion (without this, the motor would change direction every half-turn).

split-ring commutator, single winding electric motor

The torque varies hugely with the position of the coil, as the Lorentz force depends on angle (it is
maximum when perpendicular, zero when parallel). This is known as the ‘cogging effect’, and can
be minimised by adding more coils:

Cogging effect, increasing torque electric motor, double winding electric motor

Despite the disadvantages, single coils are usually still used as they are cheaper and lighter.

Electric Machines

Both generators and motors can be modelled as a Thevenised generic electric machine with
internal (armature) resistance:

Electric machines, thevenised electric machine, thevenin
  • For a generator, the induced emf goes in the same direction as terminal voltage
  • For a motor, the induced emf opposes the terminal voltage (it is modelled as a load)

We express Faraday’s Law (see notes sheet on Magnetic Fields & Electromagnetism) in terms of
the angular velocity and a machine constant:


  • E is the induced emf
  • K_e is the machine constant, in Vs/rad
  • \omega is the angular velocity, in rad/s


The power of the generator is given by the mechanical input power minus the power dissipated
in the armature resistance:

    \[P=V_aI=K_e\omega I-R_aI^2\]

  • K_e\omega I is the mechanical power, T_i\omega
  • R_a I^2 is the power lost in the armature resistance

V_aI=T_i\omega - R_aI^2 for a generator

Therefore, the mechanical torque is:

    \[T_i =K_eI\]

In a generator, the mechanical torque that is applied, T, is not the same as the mechanical torque
applied to the electrical circuit. This is because some torque – knows as shaft torque – is lost as


  • T_i is the torque applied to the circuit
  • T is the total mechanical torque input
  • T_f is the torque lost as friction


Meanwhile, the power for a motor is the terminal power minus the power lost in the armature

    \[P=K_e\omega I=V_a I-R_a I^2\]

    \[V_a I = K_e \omega I + R_a I^2\]

V_a I = T_0 \omega + R_a I^2 for a motor

The mechanical torque is:

    \[T_0=K_e I\]

In a motor, the mechanical torque output is slightly less than the electrical torque output. This is
again due to friction in the system:


  • T is the mechanical torque output
  • T_0 is the torque output from the circuit (Ti in the equations above)
  • T_f is the frictional torque

Electric Motor Graphs

Torque-Speed Graphs

The relationship between torque and speed for a motor is:


torque-speed graph for an electric motor
See derivation for this relationship

    \[V_aI=K_e\omega I+R_a I^2\]






    \[T_o(\omega-\frac{V_a}{K_e})=-\frac{R_a T_0^2}{K_e^2}\]


  • The maximum speed is the no load speed, when there is no torque output, only the frictional shaft torque is overcome.
  • Stall is when the speed is zero. There output torque here is the stall torque.

Power-Speed Graphs

power-speed graph for electric motor, electric motor power speed graph
  • At stall and no load, the power output is zero.

Efficiency-Speed Graphs

efficiency graph for electric motor, electric motor efficiency-speed graph

    \[\eta=\frac{output \quad power}{input \quad power} = \frac{T\omega}{V_a I}\]

Nominal Voltage

Electric Machines will have a specified nominal voltage. This is the recommended terminal voltage
they are designed to operate at and does not have to be stuck to.

nominal voltage electric machines, motor voltage graph, voltage-speed graph, electric motor graph
  • Operating a motor above its nominal voltage gives greater performance, but reduced lifespan
  • Operating a motor below its nominal voltage reduces performance but increases lifespan.

A motor can operate anywhere in the triangle beneath its torque-speed line. Often, you want to
limit where the motor operates. This can be done in three ways:

1. Voltage Regulation

As seen above, reducing or increasing the voltage reduces or increases the area in which the
motor can operate respectively.

2. Speed Regulation

motor speed regulation, motor speed vs torque, speed torque graph, electric motor torque graph

This is when a fixed speed is set for a certain voltage range, as torque changes. However, once
the voltage passes the upper limit, the speed follows the torque-speed relation.

Torque Regulation

torque regulation, electric motor speed-torque graph, electric motor torque

This is the opposite: a constant torque is set as speed varies over a voltage rage. Once the voltage
is out of the range, the torque will decrease as speed increases.

  • The two basic electric machines are generators and motors:
    • An emf is induced in a generator – this can be calculated using Faraday’s Law, and the direction is found with the right-hand rule
    • Electric potential is converted to mechanical energy in a motor, and Fleming’s left-hand rule shows the direction of rotation.
  • It is easiest to deal with electric machines when thevanised in series with an armature resistance and a terminal p.d.
  • Faraday’s law in terms of angular velocity and machine constant: E=K_e\omega
  • For a generator:
    • V_a I = T_i \omega - R_a I^2
    • T_i = T - T_f
  • For a motor:
    • V_a I = T_0 \omega - R_a I^2
    • T = T_0 - T_f
  • All electric machines are specified with a nominal voltage – this is the recommended terminal voltage, and is a balance between lifespan and performance.
  • The operating point of the machine does not have to be at the nominal voltage, but can be regulated through voltage, speed or torque.