<< Chapter < Page Chapter >> Page >

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
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Electrical machines. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10767/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Electrical machines' conversation and receive update notifications?