ELEN 236 DC Motors

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1 DC Motors Pictures source: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/mothow.html#c1

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Some DC Motor Terms: 1. rotor: The movable part of the DC motor 2. armature: The part of the motor through which the current is applied (in all of the

above examples, the armature is the rotor, but it doesn’t have to be). 3. coil: a loop of wire in the armature carrying current through the magnetic field 4. stator: The stationary part of the DC motor consisting of the housing and the

magnets. 5. commutator/split ring: The part of the rotating coil that connects to the current

providing source. In the simplest commutator, It is two half ring shapes, one connecting to one end of the loop, and the other connected to the other end of the loop. When it rotates, it allows the current source connection to alternate between which part of the loop it connects to so the current can switch directions. More complicated commutators in motors with multiple loops will be split into more sections.

6. brushes: The connections that sweep across different parts of the commutator completing the current path.

7. magnet: Permanent or electro-magnet creating the stationary magnetic field through with the armature rotates.

8. axle: The axle running down the centre of the armature mechanically connected to whatever the motor is supposed to be turning

9. load: The thing that you are trying to turn.

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Force on Current Carrying Wire in a Magnetic Field Magnitude of force: F=BIL

F = Force in Newtons B = Magnetic Flux Density in Wb/m2 I = Current through wire in amperes L = Length of conductor in magnetic field

Direction of force: Use right hand rule – Point index finger in direction of magnetic field (N->S)

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Point Thumb in direction of current flow Point middle finger in direction of force.

Torque Torque is a tangential force that causes rotation and is equal to the tangential force times the distance the force is a way from axis of rotation:

T=Fd

1. Torque = Force times distance

2. We know how much force is acting on the current carrying wire: BILF =

3. We can figure out the torque generated by this force: BILdT =

Where d is the distance from the axis of rotation to the current carrying wire and is equal to have the width of the part of the loop of wire perpendicular to the magnetic field

4. The Ld part of the above equation is equal to ½ the total area of the loop.

We can then rewrite torque as: 2ABIT =

5. The magnetic flux density, B is equal to the total lines of flux divided by the area AB Φ= therefore the flux is equal to the flux density times the area:

BA=Φ 6. Substituting the above equation into the torque equation, we get:

IT Φ= 21

7. For any particular motor, there can be any number of windings, and any geometry, to account for motor to motor variations, we use a constant K in the equation to account for this:

IKT Φ=

In a motor, the direction of force is constant (i.e., perpendicular to the magnetic field and the current). Torque depends on the tangential component of the torque, and since the armature is rotating, the tangential component is changing. This means the torque is changing. In a single loop armature, the torque cycles between 0 and a maximum, that is one of the reasons why motors consist of multiple loops…to keep the torque more or less constant.

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Counter emf A rotating loop of wire in a constant magnetic field will still experience changing flux through it because the area of the loop perpendicular to the wire is changing as the loop rotates. Remember that while B, the magnetic flux density (Wb/m2) remains constant the flux (Wb) changes as area changes. The changing flux induces a voltage in the wire that opposes the source voltage, this induced voltage is called the counter EMF (VCEMF).

dtdANB

dtdNVCEMF =Φ

=

VCEMF is the counter EMF. N is the number of turns in the loop B is the magnetic flux density from the magnet. Ф is the magnetic flux passing through the loop of wire A is the area of the wire through which the lines of flux pass.

The counter EMF can be modeled in the following circuit:

In this circuit, we can see the following relationship from Ohm’s Law:

A

CEMFSA R

VVI −=

The VCEMF created is limited to exactly the value needed to power the motor:

CEMFAVIP =

How is the VCEMF limited? The circuit changes the rate that the armature spins.

Motor Speed Motors are generally designed to deliver a specific operating power when run at certain speeds. Speed ratings are indicated on the motor and typically given as the no load speed (i.e., the maximum running speed).

How are speed and load related? Intuitively we can make the guess that as the load increases, the speed decreases, but let’s follow the reasoning through logically.

1. As the load increases, the current demand increases, and current into the load is

determined by A

CEMFSA R

VVI −=

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2. The terms VS and RA are constants in the above equation, therefore, the only way for IA to increase is for VCEMF to decrease.

3. dtdANB

dtdNVCEMF =Φ

= therefore VCEMF will decrease if the dA/dt decreases which can

occur if the armature slows down.

From the above derivation, we can also determine an equation for speed of rotation:

1. CEMFVSpeed ∝

2. Φ∝Speed

3. Putting the above to proportions together: Φ

=n

CEMF

KVSpeed

Where Kn is the speed constant and varies from motor to motor.

4. We also know VCEMF in terms of the source voltage, armature current and armature resistance: aaSCEMF RIVV −=

5. Substituting the VCEMF into the speed equation, we get:

Φ−

=n

aaS

KRIVSpeed

Speed Regulation Speed regulation is the ability of a motor to maintain its speed when a load is applied. It is a comparison of no-load speed to full-load speed:

100__

____Re% ×−

=SpeedLoadFull

SpeedLoadFullSpeedLoadNogulationSpeed

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Characteristics of a motor

1. Nominal voltage: The recommended voltage across the motor terminals, and the voltage for which the speed-torque curve (below) is plotted. The motor can be powered with less or more voltage, but higher voltage should be used with care, to prevent the motor coils from overheating due to ohmic (resistive) heating. 2. No load speed,: The speed of the motor powered by the nominal voltage when the motor provides zero torque. 3. No load current: The current required to spin the motor at the no load condition (i.e., the current needed to provide the torque necessary to overcome friction). 4. Nominal speed: The speed of the motor at the maximum continuous torque. 5. Nominal torque (max continuous torque): The maximum torque the motor can provide continuously. 6. Nominal current (max continuous current): The current that yields the maximum continuous torque. This maximum is determined by thermal characteristics of the motor. The power dissipated by the motor as heat is i2R. Larger currents are acceptable intermittently, but large continuous currents may cause the motor to overheat. 7. Stall torque,: The maximum torque achievable by the motor at the nominal voltage. This torque is achieved at zero velocity (stall). 8. Starting current,: The current through the motor at zero velocity, equal to the nominal voltage divided by the terminal resistance. Also called the stall current. 9. Max efficiency: The maximum efficiency of the motor in converting electrical power to mechanical power. This maximum efficiency typically occurs at high speed and low torque; the efficiency is zero at zero speed and zero torque, since the mechanical power is τω. 10. Terminal resistance : The resistance of the motor windings. 11. Terminal inductance : The inductance of the motor windings.

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12. Torque constant, : The constant of proportionality relating current to torque. In SI units (Newton-meters per amp), the torque constant is equivalent to the inverse of the speed constant in SI units (radians per volt-second). 13. Speed constant, : The constant of proportionality relating speed to voltage. Equivalent to the inverse of the torque constant when expressed in SI units. 14. Speed/torque gradient: A representation of the slope of the speed-torque curve (see graph below), approximately equal to the no load speed divided by the stall torque. 15. Mechanical time constant: The time it takes the unloaded motor to reach 63% of its no load speed under a constant voltage, starting from rest. Proportional to the inertia of the rotor and inversely proportional to the square of the the torque constant. 16. Rotor inertia: The inertia of the rotating element (the rotor) about the axis of rotation. Torque Vs. Speed Plot

One commonly used method of displaying motor characteristics graphically is the use of torque-speed curves as shown above. These curves show how the torque changes as a function of speed. One other characteristic that can be obtained straight from the graph is the power over the range of torques. Power can be calculated by the following equation:

STP ••= 1047.0 P is the power in Watts T is the torque in Nm S is the speed of rotation in RPMs 0.1047 is a conversion factor when using Nm for torque and RPMs for speed

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