1. Lecture 4 - EE743
Electromechanical Energy
Conversion
Professor: Ali Keyhani

2. Electromechanical Energy Conversion
The electromechanical energy conversion
theory allows the representation of the
electromagnetic force or torque in terms of
device variables, such as the currents and the
displacement of the mechanical systems.
An electromechanical system consists of an
electric system, a mechanical system, and a
means whereby the electric and mechanical
systems can interact.
2

3. Electromechanical Energy Conversion
Consider the block diagram depicted below.
Coupling
Electric Field Mechanic
System System
WE = We + WeL + WeS
Energy Energy transferred to Energy stored in the
supplied by Energy losses of the
the coupling field by the electric o magnetic field
an electric electric system.
electric system
source Basically, I2R
3

4. Electromechanical Energy Conversion
WM = Wm + WmL + WmS
Energy Energy transferred to
Energy losses of the Energy stored in the
supplied by a the coupling field from
mechanical system moving member and
mechanical the mechanical system compliance of the
source
mechanical system
The energy transferred to the coupling
field can be represented by
WF = We + Wm
Total energy Energy transferred to Energy transferred to the
transferred to the coupling field by coupling field from the
the coupling field the electric system mechanical system
WF = Wf + WfL
Energy stored in the Energy dissipated as heat
electric system (I2R)
4

5. Electromechanical Energy Conversion
The electromechanical systems obey the
law of conservation of energy.
WF = Wf + WfL = We + Wm
Energy Balance in an Electromechanical
System
WeL WfL WmL
WE
∑ ∑ ∑ WM
WeS Wf WmS
5

6. Electromechanical Energy Conversion
If the losses are neglected, we will obtain
the following formula,
WF = We + Wm
Energy transferred to Energy transferred to
the coupling field by the coupling field from
the electric system the mechanical system
6

7. Electromechanical Energy Conversion
Consider the electromechanical system
given below, φ
k
r L
i f
+ + m
ef N fe
V
- - D
x
x0
7

8. Electromechanical Energy Conversion
The equation for the electric system is-
di
V = ri + L + e f
dt
The equation for the mechanical system is-
2
dx dx
f = m 2 + D + K ( x − x0 ) − fe
dt dt
8

9. Electromechanical Energy Conversion
The total energy supplied by the electric
source is -
 di 
WE = ∫ V i dt = ∫  ri + L + e f  i dt
 dt 
The equation for the mechanical system is-
dx
WM = ∫ f dx = ∫ f dt
dt
9

10. Electromechanical Energy Conversion
Substituting f from the equation of motion-
 
 
 
 
 dx 2 dx 
WE = ∫ f dx = ∫  m 2 + D + K ( x − x0 ) − fe
 dx
 dt dt  
  
  Potential Energy Total energy
 Kinetic energy Heat loss
due the friction
stored in the spring transferred to the 
 stored in the mass (Wall) coupling field
from the 
 mechanical 
 system 
10

11. Electromechanical Energy Conversion
WM = − ∫ f e dx
* Recall
W f = We + WM
W f = ∫ e f idt − ∫ f e dx
dW f = e f idt − f e dx
11

12. Electromechanical Energy Conversion
If dx=0 is assumed, then
dλ
W f = WE = ∫ e f idt = ∫ i dt
dt
W f = ∫ idλ
dx =0
12

13. Electromechanical Energy Conversion
Recalling the normalized magnetization
curve, W = idλ
λ f ∫
λ = λ (i, x)
dλ
Wc = ∫ λ di
i
13

14. Electromechanical Energy Conversion
λ = λ (i, x)
∂λ (i, x) ∂λ (i, x)
dλ = di + dx
∂i ∂x
∂λ (i, x)
Wf = ∫ i di
∂i dx = 0
14

15. Electromechanical Energy Conversion
i = i (λ , x )
∂i (λ , x) ∂i (λ , x)
di = dλ + dx
∂λ ∂x
 ∂i (λ , x) 
Wc = ∫ λ di = ∫ λ  dλ 
 ∂λ 
  dx =0
15

16. Electromechanical Energy Conversion
From the previous relationship, it can be
shown that for* one coil,
i
Wf = ∫i dλ λ = L( x) i
0
i* 
 
W f = ∫ i ( L( x)di )
0
For a general case,
W f = ∫ ∑ i j dλ j
j =1 dx =0
16

17. Electromechanical Energy Conversion
For two coupled coils,
1 1
W f = L11i 1 + L12i1i2 + L22i 2 2
2
2 2
For the general case with n-coupled coils,
1 n n
W f = ∑ ∑ L pq i p iq
2 p =1q =1
17

18. Electromagnetic Force
Recalling,
φ
k
r L
i f
+ + m
ef N fe
V
- - D
x
x0
18

19. Electromagnetic Force
W f = We + WM
W f = ∫ e f idt − ∫ f e dx
f e dx = dWe − dW f
dλ
ef =
dt
19

20. Electromagnetic Force
dλ
dWe = e f idt = i dt = i dλ
dt
f e dx = i dλ − dW f
∂λ (i, x) ∂λ (i, x)
dλ = di + dx
∂i ∂x
∂W f (i, x) ∂W f (i, x)
dW f = di + dx
∂i ∂x
Substituting for dλ and dWf in fedx=id λ dWf, it can be shown
∂λ
f e ( i, x ) = i − dW f
∂x
20

21. Electromagnetic Force
W f = ∫ idλ
Recall, λ
dλ
Wc = iλ − W f Wc = ∫ λ di
∂Wc ∂λ ∂W f
=i − i
∂x ∂x ∂x
∂λ ∂W f
f e (i, x) = i −
∂x ∂x
21

22. Electromagnetic Force
λ i = W f + Wc W f = λ i − Wc
∂λ ∂W f
f e (i, x) = i −
∂x ∂x
∂λ ∂λ  ∂Wc 
f e (i, x) = i −i −− 
∂x ∂x  ∂x 
∂Wc
f e (i, x) = +
∂x
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