Dynamical Equations of Electromechanical Systems

Dynamical Equations of Electromechanical Systems:

Dynamical Equations of Electromechanical Systems – Figure 4.19 shows an electromagnetic relay whose armature is loaded with spring K, damper B,. mass Mand a force generator F. Figure 4.20 shows the abstracted diagram of a general electromechanical system. It is easily noticed that the electromechanical device has one electrical port and one mechanics’ port (one terminal of the mechanical

Dynamical Equations of Electromechanical Systems

port being the ground) through which it is connected to the electrical source on one side and mechanical load on the other side. In general there can be more than one electrical port (multiply excited system).

Let the electromechanical device has an inductance

Dynamical Equations of Electromechanical Systems

The governing electrical equation is

Dynamical Equations of Electromechanical Systems

Mechanical power output results (i.e. electrical power is converted to mechanical form) when the current in the device flows in opposition to the speed voltage. When the current in the device is in the same direction as the speed voltage, electrical power is output, i.e., mechanical power is converted to electrical form.

The governing differential equation of the mechanical system is

Dynamical Equations of Electromechanical Systems

Now for the specific system of Fig. 4.18, when the armature is in position x, the self-inductance L is found below:

Dynamical Equations of Electromechanical Systems

Substituting in Eqs. (4.98) and (4.99), the two differential equations defining the system’s dynamic behaviour are obtained as:

Dynamical Equations of Electromechanical Systems

These are nonlinear differential equations which can be solved numerically on the digital computer. However, for small movement around the equilibrium point the following procedure can be adopted.

Let the equilibrium point be (V0, 4, xo,f0). At equilibrium the system is stationary and all derivatives are zero. Thus from Eqs (4.102) and (4.103), the following relationships between equilibrium values are obtained.

Dynamical Equations of Electromechanical Systems

Let the departure (small) from the equilibrium values be

Dynamical Equations of Electromechanical Systems

Neglecting products of small departures and small departures compared to equilibrium values, and also cancelling out equilibrium terms as per Eqs (4.104) and (4.105)

Dynamical Equations of Electromechanical Systems

Dynamical Equations of Electromechanical Systems

 

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