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v D = ω e λ Q size 12{v rSub { size 8{D} } = - ω rSub { size 8{e} } λ rSub { size 8{Q} } } {} (10.25)

v Q = ω e λ D size 12{v rSub { size 8{Q} } =ω rSub { size 8{e} } λ rSub { size 8{D} } } {} (10.26)

v F = R f i F size 12{v rSub { size 8{F} } =R rSub { size 8{f} } i rSub { size 8{F} } } {} (10.27)

Finally, we can write Eq. 10.21 as

T mech = 3 2 poles 2 ( λ D i Q λ Q i D ) size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] \( λ rSub { size 8{D} } i rSub { size 8{Q} } - λ rSub { size 8{Q} } i rSub { size 8{D} } \) } {} (10.28)

From this point on, we will focus our attention on machines in which the effects of saliency can be neglected. In this case, the direct- and quadrature-axis synchronous inductances are equal and we can write

L d = L q = L s size 12{L rSub { size 8{d} } =L rSub { size 8{q} } =L rSub { size 8{s} } } {} (10.29)

where L s size 12{L rSub { size 8{s} } } {} is the synchronous inductance. Substitution into Eqs. 10.22 and 10.23 and then into Eq. 10.28 gives

T mech = 3 2 poles 2 [ ( L s i D + L af i F ) i Q L s i Q i D ] size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over {2} } right ] \[ \( L rSub { size 8{s} } i rSub { size 8{D} } +L rSub { size 8{ ital "af"} } i rSub { size 8{F} } \) i rSub { size 8{Q} } - L rSub { size 8{s} } i rSub { size 8{Q} } i rSub { size 8{D} } \]} {}

= 3 2 poles 2 L af i F i Q size 12{ {}= { {3} over {2} } left [ { { ital "poles"} over {2} } right ]L rSub { size 8{ ital "af"} } i rSub { size 8{F} } i rSub { size 8{Q} } } {} (10.30)

Equation 10.30 shows that torque is produced by the interaction of the field flux (proportional to the field current) and the quadrature-axis component of the armature current, in other words the component of armature current that is orthogonal to the field flux. By analogy, we see that the direct-axis component of armature current, which is aligned with the field flux, produces no torque.

This result is fully consistent with the generalized torque expressions which are derived in Chapter 4. Consider for example the equation which expresses the torque in terms of the product of the stator and rotor mmfs ( F s size 12{F rSub { size 8{s} } } {} and F r size 12{F rSub { size 8{r} } } {} respectively) and the sine of the angle between them.

T = poles 2 μ 0 π Dl 2g F s F r sin δ sr size 12{T= - left [ { { ital "poles"} over {2} } right ] left [ { {μ rSub { size 8{0} } π ital "Dl"} over {2g} } right ]F rSub { size 8{s} } F rSub { size 8{r} } "sin"δ rSub { size 8{ ital "sr"} } } {} (10.31)

where δ r size 12{δ rSub { size 8{r} } } {} is the electrical space angle between the stator and rotor mmfs. This shows clearly that no torque will be produced by the direct-axis component of the armature mmf which, by definition, is that component of the stator mmf which is aligned with that of the field winding on the rotor.

Equation 10.31 shows the torque in a nonsalient synchronous motor is proportional to the product of the field current and the quadrature-axis component of the armature current. This is directly analogous to torque production in a dc machine for which the equations can be combined to show that the torque is proportional to the product of the field current and the armature current.

The analogy between a nonsalient synchronous machine and dc machine can be further reinforced. Consider the equation, which expresses the rms value of the line-toneutral generated voltage of a synchronous generator as

E af = ω e L af i F 2 size 12{E rSub { size 8{ ital "af"} } = { {ω rSub { size 8{e} } L rSub { size 8{ ital "af"} } i rSub { size 8{F} } } over { sqrt {2} } } } {} (10.32)

Substitution into Eq. 10.30 gives

T mech = 3 2 poles 2 E af i Q ω e size 12{T rSub { size 8{ ital "mech"} } = { {3} over {2} } left [ { { ital "poles"} over { sqrt {2} } } right ] { {E rSub { size 8{ ital "af"} } i rSub { size 8{Q} } } over {ω rSub { size 8{e} } } } } {} (10.33)

This is directly analogous to Eq. T mech = E a I a / ω m size 12{T rSub { size 8{ ital "mech"} } =E rSub { size 8{a} } I rSub { size 8{a} } /ω rSub { size 8{m} } } {} for a dc machine in which the torque is proportional to the product of the generated voltage and the armature current.

The brushes and commutator of a dc machine force the commutated armature current and armature flux along the quadrature axis such that I d size 12{I rSub { size 8{d} } } {} = 0 and it is the interaction of this quadrature-axis current with the direct-axis field flux that produces the torque. A field-oriented controller which senses the position of the rotor and controls the quadrature-axis component of armature current produces the same effect in a synchronous machine.

Although the direct-axis component of armature current does not play a role in torque production, it does play a role in determining the resultant stator flux and hence the machine terminal voltage, as can be readily shown. Specifically, from the transformation equations of Appendix C,

Questions & Answers

how do they get the third part x = (32)5/4
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ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
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Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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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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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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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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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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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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