DC Motor – Fundamentals

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DC Motor.

The DC motor is not so used nowadays as it was in the past. For most application, it has been replaced by the solid-state rectifiers. Figure 1 shows an elementary machine equipped with a field winding wound on the stator poles, a rotor coil, and a commutator:

Figure 1. Elementary two-pole DC Machine [1]

The commutator is made up of two semi-circular copper segments mounted on the shaft at the end of the rotor.These segments are insulated from one another as well as from the iron of the rotor. Each terminal of the rotor coil is connected to a copper segment. Stationary carbon brushes ride upon the copper segments whereby the rotor coil is connected to a stationary circuit.

The voltage equations for the field winding and rotor coil are:

Equations 1

The flux linkage is expressed as:

Equations 2

Rf and Ra are the resistances of the field winding and the armature coil. The armature is the term used to refer to the rotor, so both mean the same. The mutual inductance between the field winding and the armature coil is expressed in term of a sinusoidal function of θr as::

Equations 3

where L is a constant. As the rotor revolves, the function of the commutator is to switch the stationary terminals from one terminal of the rotor coil t the other. This commutation occurs at θr=0, Π, 2Π. At the instant of the switch, the brushes are in contact with both copper segments, so the rotor coil is short-circuited.

The way form of the voltage induced in the open-circuit armature coil during constant-speed operation with a constant field winding current may be determined by setting Ia-a´=0 and If=constant. Using the expression from equations 1, 2 and 3, we obtain:

Equations 4

Note that Va-a´=0 at θr=0, Π, 2Π because at this stage is happening the commutation. The next Figure illustrates the commutation:

Figure 2. Commutation of the Elementary DC Machine [1]

Note now that the form of Va makes this configuration an impracticable machine. It could not work effectively as a motor supplied from a voltage source due to the short-circuiting of the armature coil at each commutation.

A more useful machine with 4 pairs of parallel windings is shown in Figure 3, where the rotor is equipped with four a windings and with four A windings, yielding rectified coil voltages.

Figure 3. A DC Machine with parallel windings [1]

Now we have that the form of Va looks like this:

Figure 4. Rectified voltage for a DC Machine with parallel windings [1]

Usually, the number of the rotor coils is more than four reducing by this way the harmonic content of the open-circuit armature voltage Va. In this case, the rotor coil may be approximated as a uniformly distributed winding. So, the rotor winding is considered as current sheets that are fixed in space due to the action of the commutator and which establish a magnetic axis positioned orthogonal to the magnetic axis of the field winding. This configuration looks as follow:

Figure 5. Idealized DC Machine with uniformly distributed rotor winding [1]

Another look for the DC Machine is presented in Figure 6:

Figure 6. Basic parts of the DC Machine [2]

We can now approximate the equivalent circuit for the idealized DC Machine as:

Figure 7. Equivalent circuit for an Idealized DC Machine [1]

From here, we can derive the field and armature voltages which in matrix form look like this:

Equations 5

LFF and LAA self-inductances of the field and armature windings respectively; p is a notation for d/dt; Wr is the rotor speed and LAF the mutual inductance between the field and the armature. The product Wr.LAF.If is called back emf (electromotriz force) Vem. In this last equation IAF.If is frequently substituted by a constant called Kv:

Equations 6

This substitution is far convenient since even in the case of a permanent-magnet dc machine which has not field circuit, the constant field flux produced by the permanent magnet is analogous to a dc machine with a constant Kv.

We obtain through this important expression for describing the dynamic of DC Motor:

Vem=Kv.Wr

Equations 7

The above equation dictates that the voltage across the idealized power transducer is proportional to the angular velocity.

For a DC Machine with a field winding, the electromagnetic torque can be expressed as:

Equations 8

Here again we can substitute LAF.If by the constant Kv. So,

Te=Kv.Ia

Equations 9

The electromagnetic torque Te and the rotor speed are related by:

Equations 10

J is the moment of inertia of the rotor and TL the load torque, positive for the shaft of the rotor. Te acts to turn the rotor in the direction of increasing θr. The constant Bm is a damping coefficient associated with the mechanical rotational system of the machine.

DC Motors in Control System

The variables and parameters that matter in most of the control system designs are resumed in the following table:

Figure 8. Variables and Parameters for a DC Machine [3]

The mode commonly used to represent dc motors in control system literature is as follow:

Figure 9. Model for a DC Machine [3]

A variant is presented in Figure 10:

Figure 10. Model for a DC Machine [4]

With Figure 9 as a reference, the cause and effect equations for the DC Motor are:

Equations 11

According to Equations 11, a Block Diagram for a DC motor should be like this:

Figure 11. Model for a DC Machine [3]

Basic Types of DC Machines.

1. Separate Winding Excitation (Figure 7)
2. Shunt-Connected dc Machine

Figure 12. Shunt-Connected DC Machine [1]

1. Series-Connected dc Machine

Figure 13. Series-Connected DC Machine [1]

1. Compound-Connected dc Machine

Figure 14. Compound-Connected DC Machine [1]

Figure 15. Other notations for DC Machine Types [2]

Transfer Function of a DC Motor.

Consider the model presented in Figure 10:

                                          Figure 10. Model for a DC Machine [4]

Let’s determine the Transfer Function of the DC Motor from Figure 10. Since the current-carrying armature is rotating in a magnetic field, its voltage is proportional to its speed. That is the back electromotive force as it was established in equation 7:

Equations 12

Taking the Laplace Transform we get:

Equations 13

The torque developed by the motor is proportional to the armature current, as it was said in Equations 9:

Equations 14

Transforming every impedances of Figure 10 into their Laplace Transform equivalent , we find the voltage equation for the loop around the armature circuit:

Equations 15

Now, we substitute Equations 13 y 14 en 15:

Equations 16

We need Tm in terms of in order to find . That can be get using the equivalent model for mechanical loading on a motor as shown in Figure 11:

                        Figure 11. Typical equivalent mechanical loading for a DC Machine [4]

Where Jm and Dm are mechanical constant which can be derived from a typical configuration such as:

                        Figure 12. A DC Motor driving a rotational mechanical load [4]

Considering Figure 12, Jm and Dm are:

Equations 17

Now, from Figure 11 we can find the relationship between Tm and :

Equations 18

Substituting Equations 18 in 16 we get:

Equations 19

In the most cases La is too small compared with Ra, so Equations 19 can be simplified and rearrange as:

Equations 20

Now from Equations 20 we obtain the Transfer Function for a DC Motor as follow:

Equations 21

The electrical constants of the motor Kt y Kb can be found with the following relations:

Equations 22

Where Tstall, Ea y Wno-load, use to be derive from a Graphic Speed Vs Torque such as:

        Figure 13. Torque-speed curves with an armature voltage Ea as a parameter [4]

As an example, consider the case of Figure 14:

                   Figure 14. Torque-speed curves and system example [4]

Hence:

And using the gear ratio N1/N2=1/10:

Bibliography

[1] Analysis of Electric Machinery and Drive Systems

[2] Dynamic simulation of Electric Machinery using MATLAB

[3] Sistemas de Control Automatico, Benjamin Kuo

[4] Control Systems Engineering, Norman Nise