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V a = v D 2 + v Q 2 2 = ( R a i D ω e λ Q ) 2 + ( R a i Q + ω e λ D ) 2 2 size 12{V rSub { size 8{a} } = sqrt { { {v rSub { size 8{D} } rSup { size 8{2} } +v rSub { size 8{Q} } rSup { size 8{2} } } over {2} } } = sqrt { { { \( R rSub { size 8{a} } i rSub { size 8{D} } - ω rSub { size 8{e} } λ rSub { size 8{Q} } \) rSup { size 8{2} } + \( R rSub { size 8{a} } i rSub { size 8{Q} } +ω rSub { size 8{e} } λ rSub { size 8{D} } \) rSup { size 8{2} } } over {2} } } } {}

= R a i D ω e L S L m 2 L R i Q 2 + ( R a i Q + ω e L S i D ) 2 2 size 12{ {}= sqrt { { { left [R rSub { size 8{a} } i rSub { size 8{D} } - ω rSub { size 8{e} } left [L rSub { size 8{S} } - { {L rSub { size 8{m} } rSup { size 8{2} } } over {L rSub { size 8{R} } } } right ]i rSub { size 8{Q} } right ]rSup { size 8{2} } + \( R rSub { size 8{a} } i rSub { size 8{Q} } +ω rSub { size 8{e} } L rSub { size 8{S} } i rSub { size 8{D} } \) rSup { size 8{2} } } over {2} } } } {} (10.87)

These equations show that the armature flux linkages and terminal voltage are determined by both the direct- and quadrature-axis components of the armature current.

Thus, the block marked "Auxiliary Controller" in Fig. 10.20 a, which calculates the reference values for the direct- and quadrature-axis currents, must calculate the reference currents ( i D ) ref size 12{ \( i rSub { size 8{D} } \) rSub { size 8{ ital "ref"} } } {} and ( i Q ) ref size 12{ \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } } {} which achieve the desired torque subject to constraints on armature flux linkages (to avoid saturation in the motor), armature current, ( I a ) rms = ( i D 2 + i Q 2 ) / 2 size 12{ \( I rSub { size 8{a} } \) rSub { size 8{ ital "rms"} } = sqrt { \( i rSub { size 8{D} } rSup { size 8{2} } +i rSub { size 8{Q} } rSup { size 8{2} } \) /2} } {} (to avoid excessive armature heating) and armature voltage (to avoid potential insulation damage).

Note that, as we discussed with regard to synchronous machines in Section 10.2.2, the torque-control system of Fig. 10.20 a is typically imbedded within a larger control loop. One such example is the speed-control loop of Fig. 10.20 b.

The ability to independently control the rotor flux and the torque has important control implications. Consider, for example, the response of the direct-axis rotor flux to a change in direct-axis current. Equation C.66, with λ qR = 0 size 12{λ rSub { size 8{ ital "qR"} } =0} {} , becomes

0 = R aR i dR + dR dt size 12{0=R rSub { size 8{ ital "aR"} } i rSub { size 8{ ital "dR"} } + { {dλ rSub { size 8{ ital "dR"} } } over { ital "dt"} } } {} (10.88)

Substituting for i dR size 12{i rSub { size 8{ ital "dR"} } } {} in terms of λ dR size 12{λ rSub { size 8{ ital "dR"} } } {}

i dR = λ dR L m i d L R size 12{i rSub { size 8{ ital "dR"} } = { {λ rSub { size 8{ ital "dR"} } - L rSub { size 8{m} } i rSub { size 8{d} } } over {L rSub { size 8{R} } } } } {} (10.89)

gives a differential equation for the rotor flux linkages λ DR size 12{λ rSub { size 8{ ital "DR"} } } {}

dR dt + R aR L R λ dR = L m L R i d size 12{ { {dλ rSub { size 8{ ital "dR"} } } over { ital "dt"} } + left [ { {R rSub { size 8{ ital "aR"} } } over {L rSub { size 8{R} } } } right ]λ rSub { size 8{ ital "dR"} } = left [ { {L rSub { size 8{m} } } over {L rSub { size 8{R} } } } right ]i rSub { size 8{d} } } {} (10.90)

From Eq. 10.90 we see that the response of the rotor flux to a step change in direct axis current id is relatively slow; λ dR size 12{λ rSub { size 8{ ital "dR"} } } {} will change exponentially with the rotor time constant of τ R = L R / R aR size 12{τ rSub { size 8{R} } =L rSub { size 8{R} } /R rSub { size 8{ ital "aR"} } } {} . Since the torque is proportional to the product λ dR i q size 12{λ rSub { size 8{ ital "dR"} } i rSub { size 8{q} } } {} we see that fast torque response will be obtained from changes in i q size 12{i rSub { size 8{q} } } {} . Thus, for example, to implement a step change in torque, a practical control algorithm might start with a step change in ( i Q ) ref size 12{ \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } } {} to achieve the desired torque change, followed by an adjustment in ( i D ) ref size 12{ \( i rSub { size 8{D} } \) rSub { size 8{ ital "ref"} } } {} (and hence λ dR size 12{λ rSub { size 8{ ital "dR"} } } {} ) to readjust the armature current or terminal voltage as desired. This adjustment in ( i D ) ref size 12{ \( i rSub { size 8{D} } \) rSub { size 8{ ital "ref"} } } {} would be coupled with a compensating adjustment in ( i Q ) ref size 12{ \( i rSub { size 8{Q} } \) rSub { size 8{ ital "ref"} } } {} to maintain the torque at its desired value.

Control of variable-reluctance

Motors

Unlike dc and ac (synchronous or induction) machines, VRMs cannot be simply "plugged in" to a dc or ac source and then be expected to run. As is dicussed in Chapter 8, the phases must be excited with (typically unipolar) currents, and the timing of these currents must be carefully correlated with the position of the rotor poles in order to produce a useful, time-averaged torque. The result is that although the VRM itself is perhaps the simplest of rotating machines, a practical VRM drive system is relatively complex.

VRM drive systems are competitive only because this complexity can be realized easily and inexpensively through power and microelectronic circuitry. These drive systems require a fairly sophisticated level of controllability for even the simplest modes of VRM operation. Once the capability to implement this control is available, fairly sophisticated control features can be added (typically in the form of additional software) at little additional cost, further increasing the competitive position of VRM drives.

Questions & Answers

how do they get the third part x = (32)5/4
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ninjadapaul
20/(×-6^2)
Salomon
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ninjadapaul
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ninjadapaul
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ninjadapaul
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ninjadapaul
Commplementary angles
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Electrical machines. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10767/1.1
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