0.2 Chapter 3:electromechanical-energy-conversion  (Page 2/4)

–Electrical losses: ohmic losses…

–Mechanical losses: friction, windage…

Fig. 3.2(b): a simple force-producing device with a single coil forming the electric terminal, and a movable plunger serving as the mechanical terminal.

• The interaction between the electric and mechanical terminals, i.e. the electromechanical energy conversion, occurs through the medium of the magnetic stored energy.

Figure 3.2(a) Schematic magnetic-field electromechanical-energy-conversion device;

(b) simple force-producing device.

• $W_{\text{fld}}$ : the stored energy in the magnetic field

$\frac{{\text{dW}}_{\text{fld}}}{\text{dt}}=\text{ei}-f_{\text{fld}}\frac{\text{dx}}{\text{dt}}$ (3.7)

$e=\frac{\mathrm{d\lambda }}{\text{dt}}$ (3.8)

${\text{dW}}_{\text{fld}}=\text{id}\lambda -f_{\text{fld}}\text{dx}$ (3.9)

• Equation (3.9) permits us to solve for the force simply as a function of the flux and the mechanical terminal position x .
• Equations (3.7) and (3.9) form the basis for the energy method.

§3.2 Energy Balance

• Consider the electromechanical systems whose predominant energy-storage mechanism is in magnetic fields. For motor action, we can account for the energy transfer as

$\left[\begin{array}{c}\text{Energy input}\\ \text{form electric}\\ \text{source}\end{array}\right]=\left[\begin{array}{c}\text{Mechanical}\\ \text{energy}\\ \text{ouput}\end{array}\right]+\left[\begin{array}{c}\text{Increase in energy}\\ \text{stored in magnetic}\\ \text{field}\end{array}\right]+\left[\begin{array}{c}\text{Energy}\\ \text{converted}\\ \text{into heat}\end{array}\right]$ (3.10) $$

• Note the generator action.

• The ability to identify a lossless-energy-storage system is the essence of the energy method.

• This is done mathematically as part of the modeling process.
• For the lossless magnetic-energy-storage system of Fig. 3.3(a), rearranging (3.9) in form of (3.10) gives

${\text{dW}}_{\text{elec}}={\text{dW}}_{\text{mech}}+{\text{dW}}_{\text{fld}}$ (3.11)

where

${\text{dW}}_{\text{elec}}=\text{id}\lambda$ differential electric energy input

${\text{dW}}_{\text{mech}}=f_{\text{fld}}\text{dx}$ differential mechanical energy output

${\text{dW}}_{\text{fld}}$ differential change in magnetic stored energy

• Here e is the voltage induced in the electric terminals by the changing magnetic stored energy. It is through this reaction voltage that the external electric circuit supplies power to the coupling magnetic field and hence to the mechanical output terminals.

${\text{dW}}_{\text{elec}}=\text{eidt}$ (3.12)

• The basic energy-conversion process is one involving the coupling field and its action and reaction on the electric and mechanical systems.
• Combining (3.11) and (3.12) results in

${\text{dW}}_{\text{elec}}=\text{eidt}={\text{dW}}_{\text{mech}}+{\text{dW}}_{\text{fld}}$ (3.13)

§3.3 Energy in Singly-Excited Magnetic Field Systems

• We are to deal energy-conversion systems: the magnetic circuits have air gaps between the stationary and moving members in which considerable energy is stored in the magnetic field.

• This field acts as the energy-conversion medium, and its energy is the reservoir between the electric and mechanical system.

• Fig. 3.3 shows an electromagnetic relay schematically. The predominant energy storage occurs in the air gap, and the properties of the magnetic circuit are determined by the dimensions of the air gap.

Figure 3.3Schematic of an electromagnetic relay.

$\lambda =L(x)I$ (3.14)

${\text{dW}}_{\text{mech}}=f_{\text{fld}}\text{dx}$ (3.15)

${\text{dW}}_{\text{fld}}=\text{id}\lambda -f_{\text{fld}}\text{dx}$ (3.16)

• $W_{\text{fld}}$ is uniquely specified by the values of $\lambda$ and x . Therefore, $\lambda$ and x are referred to as state variables.
• Since the magnetic energy storage system is lossless, it is a conservative system. $W_{\text{fld}}$ is the same regardless of how $\lambda$ and x are brought to their final values. See Fig. 3.4 where tow separate paths are shown.