Inductor equation

The Inductor Charging Circuit with R = 0, and the "Inductor Equation ... 

Note When working with the inductor equation, for simplicity, we usually plug in only the magnitudes of all the quantities involved (though we do mentally keep track of what is really happening — i.e. current rising or falling). 

To analyze what happens in Figure 1-6 we must first learn the capacitor equation — analogous to the inductor equation derived previously. If the duality principle is correct,... 

There is another way to describe a steady state, by bringing in the inductor equation V = LAI/At. 

Note The very use of the inductor equation V = Ldl/dt actually implies we are ignoring its parasitic resistance, DCR. The inductor equation is an idealization, applying only to a perfect inductor. That is why we had to put R = 0 when we derived it previously. 

When the switch turns ON, the current ramps up in the inductor according to the inductor equation Von = L x AIon/Ion- The current increment during the on-time is AIon =... 

Von x 1on)/L. When the switch turns OFF, the inductor equation Voff = L x AIon/Ioff leads to a current decrement AIqff = (Voff x toFF)/L. 

We have seen that the ac component (Iac = AI/2) is derivable from the voltseconds law. From the basic inductor equation V = Ldl/dt, we get... 

Note that the ratio r is defined for CCM (continuous conduction mode) operation only. Its valid range is from 0 to 2. When r is 0, AI must be 0, and the inductor equation then implies a very large (infinite) inductance. Clearly, r = 0 is not a practical value If r equals 2, the converter is operating at the boundary of continuous and discontinuous conduction modes (boundary conduction mode or BCM ). See Figure 2-5. In this so-called boundary (or critical ) conduction mode, Iac = Idc by definition. Note that readers can refer back to Chapter 1, in which CCM, DCM, and BCM were all initially introduced and explained. 

As mentioned previously, from the inductor equation V = Ldl/dt, we can derive another useful relationship that we are calling the L x I equation... 

To determine the corresponding inductance we use the definition r = AI/Il, along with the inductor equation, to get... 

The electric analogy is one of the most extensively used methods for flow and species transport modeling in channel-based microfluidic systems. A microfluidic network is equivalent to an electric circuit, of which each component can be individually described by resistors, ccmductors, and inductors. Equations 1 and 2 show the RLC circuit models in electric and fluidic domain ... 

This equation is useful for determining how close to saturation an inductor or transformer is operating, which could avoid a catastrophe. 

January 2001 I used the virtual bench design tool to verily my comp selection for 12V to 23V at 500mA step-up converter using the 2577. The virtual bench tool came back with a much smaller inductor (68uH) than the charts and equations in the datasheet seem to indicate. What s the deal ... 

An alternative method for the epoxidation of enones was developed by Jackson and coworkers in 1997 , who utilized metal peroxides that are modified by chiral ligands such as diethyl tartrate (DET), (5,5)-diphenylethanediol, (—)-ephedrine, ( )-N-methylephedrine and various simple chiral alcohols. The best results were achieved with DET as chiral inductor in toluene. In the stoichiometric version, DET and lithium tert-butyl peroxide, which was generated in situ from TBHP and n-butyllithium, were used as catalyst for the epoxidation of enones. Use of 1.1 equivalent of (-l-)-DET in toluene as solvent afforded (2/f,35 )-chalcone epoxide in 71-75% yield and 62% ee. In the substo-ichiometric method n-butyllithium was replaced by dibutylmagnesium. With this system (10 mol% Bu2Mg and 11 mol% DET), a variety of chalcone-type enones could be oxidized in moderate to good yields (36-61%) and high asymmetric induction (81-94%), giving exactly the other enantiomeric epoxide than obtained with the stoichiometric system (equation 37). 

Very recently, Lattanzi and coworkers reported on the use of enantiomericaUy pure camphor derived hydroperoxide 61 for the Ti(OPr-/)4 catalyzed chemoselective asymmetric oxidation of aryl methyl sulfides (equation 59) . The corresponding sulfoxides could be obtained in moderate yields (39-68%) and ee values up to 51%. The sulfoxidation to the sulfoxides is accompanied by further oxidation of the sulfone (kinetic resolution, yields of sulfone up to 9%). This process is stereodivergent with respect to the sulfoxidation step, which was found for the first time. Although the obtained enantioselectivities for the sulfoxides were only moderate, they proved to be among the best reported at that time with the use of enantiopure hydroperoxides as the only asymmetric inductor. The... 

The enantioselective oxidation of prochiral sulfides with DMD has been achieved by using bovine serum albumin (BSA) as the chiral inductor Moderate to good enan-tioselectivities have been reported in the presence of this protein, for which a typical example is shown in equation 22 . As yet, however, no enantioselective oxidation of a prochiral sulfide has been documented by employing an optically active dioxirane. We have tried the enantioselective oxidation of methyl phenyl sulfide with the dioxirane generated from the ketone 7 (Shi s ketone), but an ee value of only ca 5% was obtained. One major hurdle that needs to be overcome with such enantioselective dioxirane oxidations is the suppression of the background oxidation of the sulfide substrate by Caroate, an unavoidable feature of the in-situ mode. 

Dayhoff [50] suggested that one might measure a rest mass of photon by designing a low-frequency oscillator from an inductor-capacitor (LC) network. The expected frequency can be calculated from Maxwell s equations, and this may be used to give an effective wavelength for photons of that frequency. He claimed that one would have a measure of the dispersion relationship at low frequencies. Williams [51] calculated the effective capacitance of a spherical capacitor using Proca equations. This calculation can then be generalized to any capacitor with the result that a capacitor has an additional term that is quadratic in the area of the plates of the capacitor. However, this term is not exactly the one that Dayhoff referred to. But it seems to be a very close description of it. One can add two identical capacitors C in parallel and obtain the result... 

In this model, it is assumed that the energy of the oscillator is stored in a capacitor C, its losses are represented by resistor R and the mass loading by inductor L. The static capacitance of the crystal with the electrodes is shown as capacitor C0. The transfer function of this oscillator is governed by two equations.The first is... 

Equation (1) in Scheme 37 represents the conversion of cis-A-tert-bvXy -cyclohexyl ketones of general structure 49 into the corresponding cyclobutanols of structure 50. Ephedrine proved to be the best chiral inductor, giving 25-30% enantiomeric excess [287,304,305]. The use of (+ )-ephedrine afforded the optical antipode of the photoproduct produced by the use of (— )-ephedrine, indicating that the system is well behaved. 

An enantioselective method for the synthesis of 3-functionalized 2,3-dihydrobenzofuran derivatives via an intramolecular carbolithiation reaction of allyl 2-lithioaryl ethers uses (—)-sparteine as a chiral inductor. A variety of electrophiles can be reacted with the cyclized organolithium intermediate. With certain substrates, however, /3-elimination occurs instead (Equation 140) <2005CEJ5397>. 

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