# Replacing Electrical System with a Gain?

I have a system containing both a mechanical and an electrical part, as shown in the block diagram below. I am interested in simplifying this block diagram by eliminating the electrical part and replacing with a suitable gain block.

The electrical system can be described by $$\V = iR + Li + αẏ\$$, where $$\α\$$ is the relationship between the magnetic field induced voltage and the velocity. • Gain $$\i\$$ = $$\1/L\$$
• Gain $$\j\$$ = $$\R \$$
• Gain $$\c\$$ = $$\α\$$ (not part of the electrical system)

I am aware of the dominant pole approximation, where poles closer to the origin will dominate the response. I plotted the poles of my whole system below and it can be seen that one of the poles can be eliminated.

This shows that the system can be reduced to a 4th order system (from a 5th order). I have performed all calculations to get the required fourth order transfer function, by eliminating the left most pole.

Since the integrator of the electrical system is linked with the derivative, this must be eliminated.

Question:

I am interested in simplifying the block diagram and not just the transfer function. What calculations must be performed to replace the whole electrical system with a gain block?

I found that simply replacing the whole electrical system with a gain of 1, retains the same exact response, but this was not done from first principles or via calculations. How can one explain the removal of the electrical system blocks with a gain intuitively?

I am looking for a general approach of how to simply block diagrams by replacing electrical constants with a gain block. Any help would be appreciated.

The electrical system can be described by $$\V=iR+Li+αẏ\$$, where α is the relationship between the magnetic field induced voltage and the velocity.
$$\dfrac{i}{ij +s}$$