# DC electric motors with independent excitation

As already indicated, Figure 8 illustrates the electromechanical outline of a DC motor with independent excitation. Figure 9 represents the same model in a Bond-Graph using the Sources of Effort "SE", with ratios "U_{e}" for the stator and "U_{r}" for the rotor. The electric

current voltage "U_{e}", is used to overcome the ohmic resistances "R_{e}" in the stator circuit and to generate a magnetic field "®" in the winding. The ohmic resistances are represented by the Resistance port "R" with parameter "R_{e}". The behaviour of the winding is frequently represented by an Inertial port. However, in this case, in order to be able to consider the magnetic losses produced in the air-gap and in the gap between the stator and the rotor, the electrical energy reaching the winding is firstly converted into magnetic energy. The equations governing the transformation of electrical energy into magnetic energy in the stator winding are:

=^{U}* d°)

M =N_{b}/_{e} (11)

Where "®" is the magnetic flux generated in the stator winding, "U_{b}" is the voltage to which it is subjected and "N_{b}" is its number of turns. "M" is the induced magnetomotive force and "I_{e}" is the strength of the electric current in the winding. This transformation of electrical into magnetic energy is represented in the Bond Graph by the "GY" port with a "1/N_{b}" ratio. The magnetic field generated by the stator winding is represented in the Bond Graph by a Compliance port "C", in ratio to the reluctance of the magnetic field "R", in such a way that the relationship between the flux of the magnetic field and the magnetomotive force "M" is given by the expression:

jC ^{dt}=^{M} (^{12})

The resistance port, R, with ratio "P" represents the losses of the magnetic field produced in the air-gap of the stator and in the gap between the stator and rotor.

We will now model the electrical circuit of the rotor. In this case, the electrical energy is used to overcome the ohmic resistances represented by the resistance port "R : R_{r}", in establishing the magnetic field of the winding represented by the Inertial port "I" with an inductance parameter of "L_{r}", and in overcoming the counter-electromotive force "E" induced by the stator's magnetic field and which causes the rotor to rotate. All these elements described are subjected to the same voltage as the rotor circuit "I_{r}", for which reason they are connected to a "1" Junction. Due to the movement of the current "I_{r}" in the rotor at the core of the magnetic field generated by the stator "®", mechanical forces appear that cause the rotor to rotate. The equations governing the transformation of electrical into mechanical energy at the core of the rotor (Karnopp, 2005), are:

t = K l_{e} l_{r} (13)

E = K l_{e} w (14)

Where "K" is a constant, "t" is the motor torque generated in the rotor and "co" is its angular velocity. These equations for the transformation of electrical into mechanical energy are represented in the Bond Graph by the "MGY" port with variable ratio "K Ie". The connection between the Bond Graph of the stator and the rotor is produced through the current intensity "I_{e}". This connection is represented by a conventional arrow with a broken line.

Fig. 9. Bond-Graph model of a DC motor with independent excitation.

In the mechanical field, part of the energy is inverted to overcome the rotational inertia of the rotor, represented by the Inertial port "I", the ratio of which is the moment of inertia of the rotor "J"; it is also inverted to overcome the friction losses in the rotor shaft supports by means of the Resistance port "R", whose ratio is the viscous or Coulomb coefficient "i". In this case, these losses will be cancelled out as they are already taken into account in equation (2). As a result, what is left is the useful energy that will be inverted to power the train through the drive wheels. The useful rotational energy generated by the motor is associated with the bond marked with the letter "A" in Figure 9, which fulfils the flow "o", the flow of the adjoining junction "1", and the effort is the useful torque "t_{u}". The motor is connected to the locomotive's drive wheels which convert the rotational mechanical energy into linear energy, in accordance with the following expressions:

Where "V" is the longitudinal speed of the train and "F_{t}" is the total tractive effort supplied by the wheels in contact with the rails. This effort is the same as that already mentioned in section 2.1 and which is shown in Figure 2. The energy transformation represented by equations (15), (16) and (17), is modelled in the Bond Graph by the Transformer port "TF:1/r", which takes the useful torque "t_{u}" from the electric motor output of the bond marked with an "A" in Figure 9, and converts it into the tractive effort "F_{t}". In this case the energy conversion is produced without losses, since the mechanical losses produced are taken into account in the Bond-Graph in Figure 2, without doing anything other than eliminate the Source of Effort "SE : F_{t}" which originally supplied the tractive effort "F_{t}", and then connect the bond marked with a "B" in Figure 9 to junction "1" of the Bond-Graph in Figure 2.

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