# ELECTRICAL AND DYNAMIC EQUATIONS

### IDIOMA (LANGUAGE) : **ESPAÑOL**

The most representative electric motor model to obtain the aforementioned equation is shown below, it should be taken into account that everything mentioned in this topic basic concepts. If you use the formulas, you must take into consideration that the speed-torque radio of a constant field excitation DC motor will have a variation in its characteristic curve depending on the type of field connection.

## ELECTRICAL EQUATIONS FOR DC MOTOR

The impedance of the DC motor armature can be indicated as a resistor *R* in series with the parallel combination of an inductance *L* and a second resistor *RL*. As the conductive armature begins to rotate in the magnetic field produced by the stator, a voltage called contra-fem V*fem* is induced in the armature windings as opposed to the applied voltage. The contra-fem is proportional to the motor speed ω in rad/s:

*V*

_{fem}= k_{e}ωWhere the constant of proportionality *ke* is called the electric constant of the motor. The resistive loss in the magnetic circuit (*RL*) is usually in order of magnitude greater than *R*, the resistance of the windings, and can be neglected. If the voltage applied to the armature is *Vent*, the current through the armature is lent, and if it is assumed that *RL*=0, the electric equation for the motor is:

*V*= L

_{ent}*I*d

_{ent}*t*

*RI*+

_{ent}*k*

_{e}ω**DYNAMIC EQUATIONS OF A DC MOTOR**

For this case a permanent magnet DC motor will be analyzed since it is easier to understand and analyze, its governing equations can be observed in greater detail. Due to the interaction between the stator field and the armature current, the torque generated by a DC motor PM is directly proportional to the armature current:

*T = k*

_{t}I_{ent}Where *ki* is defined as the motor torque constant.

The electric constant *ke* and the torque constant *kt* of a permanent magnet motor are very important parameters and are often reported in the manufacture’s specifications. When considering the dynamics of the motor and its charge, the torque generated by the motor T is given by:

*T*= (

*J*)

_{a}+ J_{L}*ω*d

*t*

*T*+

_{f}*T*

_{L}Where *Ja* and *JL* are the moments of polar inertia of the armature and the attached charge, Tf is the fractional torque that opposes the rotation of the armature and TI is the torque dissipated by the charge.

By applying some voltage to a permanent magnet DC motor, the rotor is accelerated until a steady operating condition is achieved. The equation in steady state would be:

*V*=

_{ent}*RI*+

_{ent}*k*

_{e}ωThe power delivered by the motor at different speeds can be expressed as:

*P(ω)*=

*Tω*=

*ωT*(1-

_{s}*ω*

*ω*

_{máx}The maximum output power occurs when we derive and equalize our equation to zero:

*P*d

*ω*

*T*(1-

_{s}*ω*

*ω*

_{max}The speed to achieve maximum output power of a permanent magnet motor is half the speed of no charge.

*ω*=

*ω*

_{máx}The critical current *Is*, in terms of the armature resistance and voltage supply is:

*I*=

_{s}*V*R

_{ent}This equation for current is valid only when the motor rotor does not rotate; otherwise, the rotor current is affected by the counter-fem induced in the motor windings. The critical current is the maximum current trough the motor for a given voltage source.