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- 1. Lecture 4 - EE743Electromechanical 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 SystemWE = We + WeL + WeSEnergy Energy transferred to Energy stored in thesupplied by Energy losses of the the coupling field by the electric o magnetic fieldan electric electric system. electric systemsource Basically, I2R 3
- 4. Electromechanical Energy ConversionWM = Wm + WmL + WmSEnergy Energy transferred to Energy losses of the Energy stored in thesupplied by a the coupling field from mechanical system moving member andmechanical the mechanical system compliance of thesource 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 ∂xSubstituting 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 22

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