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heating loss due to current flowing in the winding of the energy converter. This loss is known as the i2R loss in the resistance (R) of the winding. The field loss is the core loss due to changing magnetic field in the magnetic core. The mechanical loss is the friction and windage loss due to the motion of the moving components. All these losses 95 r 96 chapter 3 Electromechanical Energy Conversion Electrical system Electrical loss Coupling field t Field loss Mechanical system t Mechanical loss FIGURE 3.1 Electromechanical converter system. Pmech are converted to heat. The energy balance equation 3.1 can therefore be written as Electrical energy = mechanical energy + increase in stored input from source output + friction field energy + - resistance loss and windage loss core loss (3.2) Now consider a differential time interval dt during which an increment of electrical energy dWe (excluding the i 2R loss) flows to the system. During this time dt, let dWr be the energy supplied to the field (either stored or lost, or part stored and part lost) and dW m the energy converted to mechanical form (in useful form or as loss, or part useful and part as loss). In differential forms, Eq. 3.2 can be expressed as (3.3) Core losses are usually small, and if they are neglected, dWr will represent the change in the stored field energy. Similarly, if friction and windage losses can be neglected, then all of dW m will be available as useful mechanical energy output. Even if these losses cannot be neglected they can be dealt with separately, as done in other chapters of this book. The losses do not contribute to the energy conversion process. 3.2 FIELD ENERGY Consider the electromechanical system of Fig. 3.2. The movable part can be held in static equilibrium by the spring. Let us assume that the movable part is held stationary at some air gap and the current is increased from zero to a value i. Flux will be established in the magnetic system. Obviously, (3.4) and from Eqs. 3.3 and 3.4, dWe=dWr (3.5) If core loss is neglected, all the incremental electrical energy input is stored as incremental field energy. Now, dA e=-dt (3.6) Immovable part Field Energy 9 7 .,.,.,__ __ "'----11 Reference position FIGURE 3.2 Example of an electromechanical system. dw. = eidt (3.7) From Eqs. 3.5, 3.6, and 3.7, dWr =idA (3.8) The relationship between coil flux linkage A and current i for a particular air gap length is shown in Fig. 3.3. The incremental field energy dWr is shown as the crosshatched area in this figure. When the flux linkage is increased from zero to A, the energy stored in the field is (3.9) This integral represents the area between the A axis and the A-i characteris- tic, the entire area shown shaded in Fig. 3.3. Other useful expressions can also be derived for the field energy of the magnetic system. Let He= magnetic intensity in the core Hg =magnetic intensity in the air gap lc = length of the magnetic core material lg = length of the air gap FIGURE 3.3 A-i characteristic for the system in Fig. 3.2 for a particular air gap length. 98 chapter 3 Electromechanical Energy Conversion Then Also A =NcfJ =NAB where A is the cross-sectional area of the flux path B is the flux density, assumed same throughout From Eqs. 3.9, 3.10, and 3.12, Wr = I Hclc ~ Hglg NA dB For the air gap, H=B g 1'-o From Eqs. 3.13 and 3.14, Wr = I ( Hclc + !/g) A dB = I (He dB Ale + : 0 dB lgA) = I He dB X volume of magnetic material B2 + - 2 X volume of air gap 1'-o (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) =wrcXVc+WrgXVg (3.17) = Wrc + Wrg (3.18) where Wrc = J He dEc is the energy density in the magnetic material Wrg = B 2/2f.J-o is the energy density in the air gap Vc is the volume of the magnetic material Vg is the volume of the air gap Wrc is the energy in the magnetic material Wrg is the energy in the air gap Normally, energy stored in the air gap (Wrg) is much larger than the energy stored in the magnetic material (Wrc). In most cases Wrc can be neglected. For a linear magnetic system H =Be c 1'-c (3.19) Field Energy 99 Therefore (3.20) The field energy of the system of Fig. 3.2 can be obtained by using either of Eqs. 3.9 and 3.16. EXAMPLE 3.1 The dimensions of the actuator system of Fig. 3.2 are shown in Fig. E3.1. The magnetic core is made of cast steel whose B-H characteristic is shown in Fig. 1.7. The coil has 250 turns, and the coil resistance is 5 ohms. For a fixed air gap length g = 5 mm, a de source is connected to the coil to produce a flux density of 1.0 tesla in the air gap. (a) Find the voltage of the de source. (b) Find the stored field energy. Solution (a) From Fig. 1.7, magnetic field intensity in the core material (cast steel) for a flux density of 1.0 T is 5 em 5 em He= 670At/m Length of flux path in the core is Zc = 2(10 + 5) + 2(10 + 5) em = 60cm The magnetic intensity in the air gap is Bg 1.0 Hg = J.to = 477 1 o-7 At/m = 795.8 X 103 At/m Depth= 10 em FIGURE E3.1 100 chapter 3 Electromechanical Energy Conversion The mmf required is Ni = 670 X 0.6 + 795.8 X 103 X 2 X 5 X 10-3 At = 402 + 7958 = 8360 At . _ 8360A l- 250 = 33.44A Voltage of the de source is Vdc = 33.44 X 5 = 167.2 V (b) Energy density in the core is II.O d Wrc = 0 H B This is the energy density given by the area enclosed between the B axis and the B-H characteristic for cast steel in Fig. 1.7. This area is Wrc =~X 1 X 670 = 335 11m3 The volume of steel is Vc = 2(0.05 X 0.10 X 0.20) + 2(0.05 X 0.10 X 0.10) = 0.003 m 3 The stored energy in the core is Wrc = 335 X 0.003 J = 1.005 J The energy density in the air gap is - 1.02 J/ 3 Wfg- 2 X 41T X 10-7 m = 397.9 X 103 J/m3 The volume of the air gap is Vg = 2(0.05 X 0.10 X 0.005) m 3 = 0.05 X 10-3 m 3 The stored energy in the air gap is Wrg = 397.9 X 103 X 0.05 X 10-3 = 19.895 joules 1 um 1 B 1 B u oT E c A 1 r a A 1 I > Field Energy 101 The total field energy is Wr = 1.005 + 19.895 J = 20.9 J Note that most of the field energy is stored in the air gap. \u2022 3.2.1 ENERGY, COENERGY The A-i characteristic of an electromagnetic system (such as that shown in Fig. 3.2) depends on the air gap length and the B-H characteristics of the magnetic material. These A-i characteristics are shown in Fig. 3.4a for three values of air gap length. For larger air gap length the characteristic is essentially linear. The characteristic becomes nonlinear as the air gap length decreases. For a particular value of the air gap length, the energy stored in the field is represented by the area A between the A axis and the A-i characteristic, as shown in Fig. 3.4b. The areaB between the i axis and the A-i characteristic is known as the coenergy and is defined as w; = t A.di (3.21) This quantity has no physical significance. However, as will be seen later, it can be used to derive expressions for force (or torque) developed in an electromagnetic system. From Fig. 3.4b, Wi +Wr=Ai (3.22) Note that Wi > Wr if the A.-i characteristic is nonlinear and Wi = Wr if it is linear. (a) air gap length (b) FIGURE 3.4 (a) A.-i characteristics for different air gap lengths. (b) Graphical representation of energy and coenergy. 102 chapter 3 Electromechanical Energy Conversion 3.3 MECHANICAL FORCE IN THE ELECTROMAGNETIC SYSTEM Consider the system shown in Fig. 3.2. Let the movable part move from one position (say x = x 1) to another position (x = x 2) so that at the end of the movement the air gap decreases. The A.-i characteristics of the system for these two positions are shown in Fig. 3.5. The current (i =viR) will remain the same at both positions in the steady state. Let the operating points be a when x = x 1 and b when x = x 2 (Fig. 3.5). If the movable part has moved slowly, the current has remained essentially

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