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However, in this case, the voltage corresponds to the induced emf, according to Faraday s law: e= d d =N dt dt (1616)
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Figure 168 Relationship between ux linkage, current, energy, and co-energy
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Principles of Electromechanics
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and is therefore related to the rate of change of the magnetic ux The energy stored in the magnetic eld could therefore be expressed in terms of the current by the integral Wm = ei dt = d i dt = dt i d (1617)
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It should be straightforward to recognize that this energy is equal to the area above the -i curve of Figure 168 From the same gure, it is also possible to de ne a ctitious (but sometimes useful) quantity called co-energy, equal to the area under the curve and identi ed by the symbol Wm From the gure, it is also possible to see that the co-energy can be expressed in terms of the stored energy by means of the following relationship: Wm = i Wm (1618)
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Example 161 illustrates the calculation of energy, co-energy, and induced voltage using the concepts developed in these paragraphs The calculation of the energy stored in the magnetic eld around a magnetic structure will be particularly useful later in the chapter, when the discussion turns to practical electromechanical transducers and it will be necessary to actually compute the forces generated in magnetic structures
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EXAMPLE 161 Energy and Co-Energy Calculation for an Inductor
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Problem
Compute the energy, co-energy, and incremental linear inductance for an iron core inductor with a given -i relationship Also compute the voltage across the terminals given the current through the coil
Solution
Known Quantities: -i relationship; nominal value of ; coil resistance; coil current Find: Wm ; Wm ; L ; v
Schematics, Diagrams, Circuits, and Given Data: i = + 05 2 ; 0 = 05 V s;
; i(t) = 0625 + 001 sin(400t)
Assumptions: Assume that the magnetic equation can be linearized and use the linear model in all circuit calculations Analysis:
Calculation of energy and co-energy From equation 1617, we calculate the energy as follows
Wm =
i( )d =
3 2 + 2 6
The above expression is valid in general; in our case, the inductor is operating at a nominal ux linkage 0 = 05 V-s and we can therefore evaluate the energy to be: Wm ( = 0 ) = 3 2 + 2 6 = 01458 J
=05
Part III
Electromechanics
Thus, after equation 1618, the co-energy is given by: Wm = i Wm where i = + 05 2 = 0625 A and Wm = i Wm = (0625)(05) (01458) = 01667 J 2 Calculation of incremental inductance If we know the nominal value of ux linkage (ie, the operating point), we can calculate a linear inductance L , valid around values of close to the operating point 0 : L = d di =
= 0
1 1+
= 0667 H
=05
The above expressions can be used to analyze the circuit behavior of the inductor when the ux linkage is around 05 V s, or, equivalently, when the current through the inductor is around 0625 A Circuit analysis using linearized model of inductor We can use the incremental linear inductance calculated above to compute the voltage across the inductor in the presence of a current i(t) = 0625 + 001 sin(400t) Using the basic circuit de nition of an inductor with series resistance R, the voltage across the inductor is given by: v = iR + L di = [0625 + 001 sin(400t)] 1 + 0667 4 cos(400t) dt = 0625 + 001 sin(400t) + 2668 cos(400t) = 0625 + 2668 sin(400t + 898 )
Comments: The linear approximation in this case is not a bad one: the small sinusoidal
current is oscillating around a much larger average current In this type of situation, it is reasonable to assume that the inductor behaves linearly This example explains why the linear inductor model introduced in 4 is an acceptable approximation in most circuit analysis problems
` Ampere s Law As explained in the previous section, Faraday s law is one of two fundamental laws relating electricity to magnetism The second relationship, which forms a counterpart to Faraday s law, is Amp` re s law Qualitatively, Amp` re s law states e e that the magnetic eld intensity, H, in the vicinity of a conductor is related to the current carried by the conductor; thus Amp` re s law establishes a dual relationship e with Faraday s law In the previous section, we described the magnetic eld in terms of its ux density, B, and ux To explain Amp` re s law and the behavior of magnetic e materials, we need to de ne a relationship between the magnetic eld intensity, H, and the ux density, B These quantities are related by: B = H = r 0 H Wb/m2 or T (1619)
where the parameter is a scalar constant for a particular physical medium (at least, for the applications we consider here) and is called the permeability of the medium The permeability of a material can be factored as the product of the
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