1) The document discusses the Blondel two-reaction method for analyzing alternators, which separates the armature magnetomotive force (MMF) into sine and cosine components to account for non-uniform air gaps in salient pole machines.
2) It derives an expression for the demagnetizing component of armature reaction, which is proportional to the sine of the angle between voltage and current.
3) It also derives an expression for the cross-magnetizing component of armature reaction, which is proportional to the cosine of the angle between voltage and current.
EMF EQUATION OF DC GENERATOR,DC MOTOR|DAY15|BACK EMF,TORQUE OF DC MOTOR|BASIC...Prasant Kumar
#EMF EQUATION OF DC GENERATOR
#EMF EQUATION OF DC MOTOR
#TORQUE EQUATION OF DC MOTOR
# EMF EQUATION OF DC MOTOR IN HINDI
#DERIVATION OF DC MOTOR EMF EQUATION
#FARADAY LAW OF ELECTROMAGNETIC INDUCTION
#back emf in dc motor
#back emf in dc motor in hindi
In this video you will learn about,derivation of dc machine emf equation,back emf,torque equation of dc motor,dc generated,dc motor.To understand electrical machine with trick watch all videos,
MUST UPGRADE YOUR KNOWLEDGE BY FLIPPED LEARNING
#Topic - ELECTRICAL TRANSFORMER
~ Link of all sessions are.
DAY 1 (Need/Definition)
https://youtu.be/BvaykFJ_NoE
DAY 2 (Working principle and Construction)
https://youtu.be/06rgxocihaM
DAY 3 (EMF equation and Turns Ratio)
https://youtu.be/g7e5xBPmv3Y
DAY 4 (Classification of Transformer)
https://youtu.be/6NP5L4MlvY4
DAY 5 ( Ideal and practical transformer on no load)
(Equivalent Transformer)
https://youtu.be/6LCLQC1p3lg
DAY 6 ( Losses in Transformer)
https://youtu.be/ObYNiGgd3hA
DAY 7 (O.C. and S.C. test)
https://youtu.be/8WiJRawHiTce/6LCLQC1p3lg
DAY 8 (Voltage Regulation & Efficiency)
https://youtu.be/6LCLQC1p3lg
DAY 9 (Zero Lecture)
https://youtu.be/N4xWOwgi8I4
DAY 10 (Classification of machine)
https://youtu.be/bmxnU5rC5m4
Construction of Machine
https://youtu.be/34mpphDk3gg
Working Principle of Synchronous Generator & Synchronous Motor
https://youtu.be/bkgf72M8BCY
Working Principle of Induction Motor
https://youtu.be/Lj_iQBoRiK0
Ac machines: Rotating Magnetic Field and its productionnishtha97
This is a presentation that mainly covers knowledge about the rotating magnetic field in induction motors and its production. It also covers the main conditions for the production of RMF. I hope you find it useful. :)
VTU Notes for Testing and commissioning of Electrical Equipment Department of Electrical and Electronics Faculty Name: Mrs Veena Bhat Designation: Assistant Professor Subject: Testing and Commissioning of Electrical equipment Semester: VII
EMF EQUATION OF DC GENERATOR,DC MOTOR|DAY15|BACK EMF,TORQUE OF DC MOTOR|BASIC...Prasant Kumar
#EMF EQUATION OF DC GENERATOR
#EMF EQUATION OF DC MOTOR
#TORQUE EQUATION OF DC MOTOR
# EMF EQUATION OF DC MOTOR IN HINDI
#DERIVATION OF DC MOTOR EMF EQUATION
#FARADAY LAW OF ELECTROMAGNETIC INDUCTION
#back emf in dc motor
#back emf in dc motor in hindi
In this video you will learn about,derivation of dc machine emf equation,back emf,torque equation of dc motor,dc generated,dc motor.To understand electrical machine with trick watch all videos,
MUST UPGRADE YOUR KNOWLEDGE BY FLIPPED LEARNING
#Topic - ELECTRICAL TRANSFORMER
~ Link of all sessions are.
DAY 1 (Need/Definition)
https://youtu.be/BvaykFJ_NoE
DAY 2 (Working principle and Construction)
https://youtu.be/06rgxocihaM
DAY 3 (EMF equation and Turns Ratio)
https://youtu.be/g7e5xBPmv3Y
DAY 4 (Classification of Transformer)
https://youtu.be/6NP5L4MlvY4
DAY 5 ( Ideal and practical transformer on no load)
(Equivalent Transformer)
https://youtu.be/6LCLQC1p3lg
DAY 6 ( Losses in Transformer)
https://youtu.be/ObYNiGgd3hA
DAY 7 (O.C. and S.C. test)
https://youtu.be/8WiJRawHiTce/6LCLQC1p3lg
DAY 8 (Voltage Regulation & Efficiency)
https://youtu.be/6LCLQC1p3lg
DAY 9 (Zero Lecture)
https://youtu.be/N4xWOwgi8I4
DAY 10 (Classification of machine)
https://youtu.be/bmxnU5rC5m4
Construction of Machine
https://youtu.be/34mpphDk3gg
Working Principle of Synchronous Generator & Synchronous Motor
https://youtu.be/bkgf72M8BCY
Working Principle of Induction Motor
https://youtu.be/Lj_iQBoRiK0
Ac machines: Rotating Magnetic Field and its productionnishtha97
This is a presentation that mainly covers knowledge about the rotating magnetic field in induction motors and its production. It also covers the main conditions for the production of RMF. I hope you find it useful. :)
VTU Notes for Testing and commissioning of Electrical Equipment Department of Electrical and Electronics Faculty Name: Mrs Veena Bhat Designation: Assistant Professor Subject: Testing and Commissioning of Electrical equipment Semester: VII
Elextrostatic and Electromagnetic Fields Actuators for MEMS AD/DA ConvertersCSCJournals
MEMS Analog -to-digital and digital-to- analog converters are proposed using electrostatic field and electromagnetic fields actuators. For the former, parallel deformable plates supported by springs are used with bias applied voltage which determines the amount of static displacement needed for equilibrium condition. For the latter, coil winding(s) are embedded in a rotating plate, which is exposed to a constant field of a permanent magnet, causing the plate to deflect according to the currents in the windings. In the analog-to-digital arrangement, different spring displacements are tapped off either the spring in case of electrostatic or the moving plate in case of electromagnetic actuators, corresponding to the binary decoded currents. At these off tapping points, logic high signal levels are applied at these locations so that when a certain analog voltage is applied on the moving plate of the capacitor, the spring is displaced to one of these locations, enabling different binary voltages to all switches up to that level. Similar result occurs when an analog voltage is applied on the winding. The digital binary voltages are fed to a priority encoder to obtain the digital value. In digital-to-analog arrangement, the input binary voltage is decoded to different spring locations which correspond to resistances making up a potentiometer circuit for the output analog voltage. Similarly; for the electromagnetic actuator, a number of different length coil windings are embedded within the moving plate, causing different deflections corresponding to one bit of the binary input.
Construction and components of DC Machine – Principle of operation – Lap and wave windings-EMF equations– circuit model – armature reaction –methods of excitationcommutation – interpoles compensating winding –characteristics of DC generators.
Method of Moment analysis of a printed Archimedian Spiral antenna Piyush Kashyap
A single arm Archimedean spiral printed on a grounded dielectric substrate is analyzed using the method of moments. Piecewise sinusoidal subdomain basis and test functions are used over curved segments that exactly follow the spiral curvature. Results for the input impedance obtained using the curved segmentation approach on MATLAB are compared with those obtained after simulating the model on FEKO. A comparison with published results shows that the curved segment model requires fewer segments and is therefore significantly more computationally efficient than the linear segmentation model.
A novel voltage reference without the operational amplifier and resistorsIJRES Journal
novel voltage reference has been proposed and simulated using a 0.18μm CMOS process in
this paper. A near-zero temperature coefficient voltage is achieved in virtue of the bias voltage subcirciut which
consists of two MOSFETs operating in the saturation region. The kind of bias voltage subcirciut is used to
adjust the output voltage and compensate the curvature. The output voltage is equal to the extrapolated
threshold voltage of a MOSFET at absolute zero temperature, which was about 591.5 mV for the MOSFETs we
used. The power supply rejection ratio (PSRR) is improved with three feedback loops. Although the output
voltage fluctuates with process variation, the circuit can monitor the process variation in MOSFET threshold
voltage. The simulation results show that the line regulation is 0.75 mV/V in a supply voltage range from 1.6 V
to 3.1 V and the temperature coefficient is around 10.8 ppm/℃ to 28.5 ppm/℃ at 9 different corners in a
temperature range from -20℃ to 120 ℃.
The PSRR is -70 dB at 100Hz with a supply voltage at 1.8 V, and the
layout size is 0.012mm2. The results of simulation and post layout simulation are almost the same.
Electrical Engineering is the Branch of Engineering. Electrical Engineering field requires an understanding of core areas including Thermal and Hydraulics Prime Movers, Analog Electronic Circuits, Network Analysis and Synthesis, DC Machines and Transformers, Digital Electronic Circuits, Fundamentals of Power Electronics, Control System Engineering, Engineering Electromagnetics, Microprocessor and Microcontroller. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus https://ekeeda.com/streamdetails/stream/Electrical-Engineering
Electrical Engineering is the Branch of Engineering. Electrical Engineering field requires an understanding of core areas including Thermal and Hydraulics Prime Movers, Analog Electronic Circuits, Network Analysis and Synthesis, DC Machines and Transformers, Digital Electronic Circuits, Fundamentals of Power Electronics, Control System Engineering, Engineering Electromagnetics, Microprocessor and Microcontroller. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus Visit : https://ekeeda.com/streamdetails/stream/Electrical-Engineering
Ekeeda Provides Online Electrical and Electronics Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
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Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...
Two reaction theory
1. 03-Alternators
Prepared by Dr. M. A. Mannan Page 1 of 11
Alternators
Two Reaction Analyses [2, Ch. 16]
A multipolar machine with cylindrical rotor has a uniform air-gap, because of which
its reactance remains the same, irrespective of the spatial position of the rotor. However, a
synchronous machine with salient pole or projecting poles has non-uniform air-gap due to
which its reactance varies with the rotor position.
Blondel Two-Reaction Method [2/150/p.198]
To analyze mathematically and vector diagram of an alternator, it is inherently
assumed that the field and armature fluxes are sinusoidally distributed in the air gap. There
is a tendency for this method to give results that are somewhat higher than those of actual
test.
Sinusoidally flux distribution is nearly true for fields with distributed iron and copper
(non-salient pole m/c or smooth cylindrical m/c), but it is not true for salient-pole machine.
In order to remove the criticism of results for salient-pole machines, Professor Andre
Blondel published in 1904 the two reaction method.
Division of Armature MMF into Sine and Cosine Components
[2,150/p.198]
At zero PF lagging load, the armature-current distribution is such that the armature
MMF directly opposes the MMF of the main field.
At unity PF load, the armature MMF crowds the flux backward against the rotation.
This causes the trailing pole tips to be strengthened and the leading tips to be weakened.
When alternator is operating at normal PFs of 60 to 90 percent, lagging, a
combination of both these effects takes place.
Fig. 144 shows this in greater detail. The part of armature ATs which weakens the
main flux is shown in diagonal sectioning, and that part which strengthens it is cross-hatched.
The resulting demagnetizing ATs can be found by subtracting the latter (strengthens)
area (A2) from the former (weekens) area (A1) and by dividing by the circumferential width
ψτ of the pole shoe.
Fig. 144 shows the position of the MMF waves in space w.r.t. the field pole of an
alternator when the armature current is lagging the voltage E0, generated by the flux, by an
angle θ’ of electrical degree.
This angle θ’ of the time diagram is equal to the angle θ’ on the space diagram of Fig.
144. That is, if the armature current is in phase with E0, the maximum value of the wave A
will be shifted to a point 90o
to the right (or left, depending on direction of motion of the
pole) of the centre line of the pole.
With an armature current that lags E0 by 90o
, the maximum value of A will be
coincide with the centre line of the poles.
Derivation of the demagnetizing action of the armature reaction for the general case
will be based on the notations and symbols shown in Fig. 144.
Expressing τ, θ’, and x in radians, ψ in percent, then
∫=
+
2
'
01 sin
ψπ
θ
xdxAA ;
'
2
'
01 ]cos[
ψπ
θ +
−= xAA ; )]
2
'cos(1[1
ψπ
θ +−= AA
2. 03-Alternators
Prepared by Dr. M. A. Mannan Page 2 of 11
The area of the small part
∫∫
−−−
== 2
'
0
)'
2
(
0
2 sinsin
ψπ
θθ
ψπ
xdxAxdxAA ; 2
'
02 ]cos[
ψπ
θ −
−= xAA ;
)]
2
'cos(1[2
ψπ
θ −−= AA
Expanding
]
2
sin'sin
2
cos'[cos1
ψπ
θ
ψπ
θ −−= AAA
]
2
sin'sin
2
cos'[cos2
ψπ
θ
ψπ
θ +−= AAA
]}
2
sin'sin
2
cos'[cos{]
2
sin'sin
2
cos'[cos21
ψπ
θ
ψπ
θ
ψπ
θ
ψπ
θ +−−−−=− AAAAAA
]
2
sin'sin
2
cos'[cos]
2
sin'sin
2
cos'[cos21
ψπ
θ
ψπ
θ
ψπ
θ
ψπ
θ ++−−−=− AAAAAA
2
sin'sin
2
cos'cos
2
sin'sin
2
cos'cos21
ψπ
θ
ψπ
θ
ψπ
θ
ψπ
θ AAAAAA +++−=−
2
sin'sin221
ψπ
θAAA =−
Dividing this total area by the base ψπ to obtain the average demagnetizing ATs:
ψπ
ψπ
θ
ψπ
2
sin2
'sin21
A
AA
=
−
Demagnetizing AT.
This could be written 'sin21
θ
ψπ
AK
AA
AD =
−
= where
ψπ
ψπ
2
sin2
=K [188]
3. 03-Alternators
Prepared by Dr. M. A. Mannan Page 3 of 11
This shows that the demagnetizing ATs for a given m/c are proportional to sinθ’.
Accordingly, we may conclude that the armature reaction ATs consists of two parts.
One part is a function of Asinθ’, which is demagnetizing. The second part, which is a
function of Acosθ’, is cross-magnetizing.
A is the crest (or peak) value of the armature reaction ATs per pole. To calculate the
alternator regulation, using Fourier series analysis the value of A is obtained as follows:
pd
a
kk
P
ZI
NIA
2
9.0poleperreactionarmatureof ==
Hence the average demagnetizing ATs per pole are
'sin)2
sin2
2
9.0
('sin2
sin2
2
9.02
sin2
'sin θ
ψπ
ψπ
θ
ψπ
ψπ
ψπ
ψπ
θ pd
a
pd
a
D kk
P
ZI
kk
P
ZI
AA ===
'sinθCkk
P
ZI
A pd
a
D = (188)
where,
ψπ
ψπ
2
sin2
2
9.0
=C
The Cross-Component [2/152/p.201]
It is obvious that any sine wave can be separated into two components in quadrature,
yielding a resultant equal to the original. In fact, the analysis just followed showed a wave of
armature MMF separated into components proportional to sinθ’ and cosθ’. Such components
are shown in Fig. 145.
The cross-magnetizing component neither increases nor decreases the field excitation
as a whole, but increases the MMF acting over one half of each pole face and decrease by an
equal amount the MMF acting over the other half of each pole face. This is shown in Fig.
146(a).
The shaded portion shows approximately that part of the cross MMF that is really
effective.
4. 03-Alternators
Prepared by Dr. M. A. Mannan Page 4 of 11
In Fig. 146(b) the MMF of the cross component is shown and a curve of the flux that
it would produce. The greatly increased reluctance of the air path between the salient poles is
taken into consideration.
It is assumed that the cross field is of sinusoidal (neglecting harmonics) distribution in
space. It rotates at synchronous speed and is in a fixed position w.r.t. the poles.
It will therefore generate a voltage Ec in the armature of fundamental frequency. It
would be possible to determined component of the cross-field flux from a Fourier analysis.
This approach will not be used here. Instead, a determination will be made of the average
value of a fundamental MMF space wave which has the same area as the shaded position
between 0 and π of Fig. 146(a).
Evaluation Ac [2/153/p. 203]
xAa sin'cosθ=
∫=∫= 2
0
2
0
sin'cos22d)Area(shade
ψπψπ
θ xdxAadx
[ ] radiansMMFin
2
cos1'cos2cos'cos2d)Area(shade 2
0 ⎥⎦
⎤
⎢⎣
⎡
−=−=
ψπ
θθ
ψπ
AxA
5. 03-Alternators
Prepared by Dr. M. A. Mannan Page 5 of 11
⎥⎦
⎤
⎢⎣
⎡
−==
2
cos1
'cos2Area
MMFAverage
ψπ
π
θ
π
A
The average value of the cross MMF per pole is
'cos''cos
2
cos1
2
θθ
ψπ
π
AKAAc =⎥⎦
⎤
⎢⎣
⎡
−= where, ⎥⎦
⎤
⎢⎣
⎡
−=
2
cos1
2
'
ψπ
π
K
By substituting the value of pd
a
kk
P
ZI
A
2
9.0=
'cos)
2
cos1(
9.0
θ
ψπ
π ⎥⎦
⎤
⎢⎣
⎡
−= pd
a
c kk
P
ZI
A
'cosθCkk
P
ZI
A pd
a
c = (193)
where )
2
cos1(
9.0 ψπ
π
−=C
It should be noted that the value of Ac is based on an MMF wave having the same
area as the two shaded sections of Fig. 146(a).
The Vector Diagram [2/154/p.204]
The diagram of Fig. 147 can be built up by noting the following relationship.
When the generator delivers a current I, at a terminal voltage V and at a PF angle θ,
there will be set up an armature resistance drop IRa and a leakage reactance drop IXL in phase
and in quadrature, respectively, with the current.
These latter values add vectorially to the terminal voltage to give the EMF E, which is
generated in armature by the air-gap flux. The generated EMF can be split into two
components, in quadrature with each other, of which Ef is due to the main-pole flux and Ec is
due to the cross-flux set up in the interpole space and pole edges by the cross component of
armature reaction.
To generate Ef requires a resultant MMF along the axis of the field poles which is
designated Mf. The initial MMF of the field winding alone is M0, but the direct component of
armature reaction AD is subtracted therefrom to yield Mf. This subtraction, of course,
presumes a lagging current; for leading current the so-called demagnetizing component
would add to that of the field winding.
Fig. 147. Complete vector diagram of alternators showing voltages and MMFs.
6. 03-Alternators
Prepared by Dr. M. A. Mannan Page 6 of 11
The excitation M0 is presumed to result in a NL voltage of E0 as read from the
saturation curve, and hence the regulation would be
%100upregulation% 0
×
−
=∴
V
VE
The cross component of armature reaction sets up a flux cut by the armature
conductors and so generates the voltage Ec, which may be treated as a voltage drop.
The demagnetizing component of armature AD is an MMF subtracting arithmetically
from the field-winding M0.
Angle (φ) calculation between internal induced voltage (E) and NL voltage (E0)
The terminal voltage per phase of a generator is designated V, and the regulation is
desired at a load of I, lagging θ degrees behind V.
The effective resistance of the armature and leakage reactance are known. V, I, IRa
and IXL can be laid off on the vector diagram, and E can be calculated
αθθ +='
v= volts/AT/pole generated on the lower part of the saturation curve.
Then it is assumed that
)cos( αθ +== Ckk
P
ZI
vvAE pdcc here let IIa =
But φsinEEc = ; hence, )cos(sin αθφ +== Ckk
P
ZI
vEE pdc
Or,
E
Ckk
P
ZI
v
E
E pd
c
)cos(
sin
αθ
φ
+
==
But λφαθ +=+ ;
E
Ckk
P
ZI
v pd )cos(
sin
λφ
φ
+
=
φ
λφλφ
φ
λφ
sin
sinsincoscos
sin
)cos( −
=
+
= Ckk
P
ZI
vCkk
P
ZI
vE pdpd
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−= λ
φ
λ
φ
λφ
φ
λφ
sin
tan
cos
sin
sinsin
sin
coscos
Ckk
P
ZI
vCkk
P
ZI
vE pdpd
λ
φ
λ
sin
tan
cos
Ckk
P
ZI
vCkk
P
ZI
vE pdpd −=
φ
λ
λ
tan
cos
sin Ckk
P
ZI
vCkk
P
ZI
vE pdpd =+
λφλ costan]sin[ Ckk
P
ZI
vCkk
P
ZI
vE pdpd =+
λ
λ
φ
sin
cos
tan
P
CkvZIk
E
P
CkZIk
v
pd
pd
+
=
Since, )sin)(cos( θθ jjXRIVE La −++=
The angle (λ-θ) can be determined and likewise the angleλ. Substituting in Eq. (199),
the value of φ is determined and
7. 03-Alternators
Prepared by Dr. M. A. Mannan Page 7 of 11
λφθ +='
The determination of the angle θ′ locates the direction of the voltage Ef w.r.t. I. This
provides the necessary step in the construction of the diagram of Fig. 147.
Modified Vector Diagram [2/156/p.209]
The Blondel two-reaction theory, as just presented, is closely allied to the original
form of this development.
An extension of this theory by Doherty and Nickle provides a more complete and
accurate analysis.
In deriving the separate components of armature reaction for the Blondel theory, it
was found that the directly demagnetizing portion was a function of Asinθ’ and the cross-
magnetizing component depended upon Acosθ’.
Since armature reaction ATs A naturally involve armature current, the first step in this
process is the association of the current with the terms Asinθ’ and Acosθ’. The vector
diagram of Fig. 147 is then redrawn as shown in Fig. 148.
The directly demagnetizing effect resulted in a change in voltage represented by a
difference in the length of Ef and E0. Since it is a function of Asinθ’ and is in quadrature with
the component of armature current represented by Isinθ’ it follows that the change in voltage
form E0 to Ef might be represented by a fictitious reactance drop.
'
0 df IXEE =−
Subscript “d” designated for representing direct components. Suppose Id indicated the
direct component of current I, which is in quadrature with the position of the NL voltage.
8. 03-Alternators
Prepared by Dr. M. A. Mannan Page 8 of 11
In similar manner we find that the cross-magnetizing component Ec will be
considered a fictitious reactance drop since it is proportional to cosθ′ and in quadrature with
the current component Iq=Icosθ′.
'
qqc XIE =
Xq
’
is cross reactance.
The idea of associating components of a voltage drop with suitable components of
current will be carried a step further as shown in Fig. 149.
Here the leakage reactance drop will be separated into components associated with
suitable current components so that vectorially
LqLdL XIXIIX +=
As shown in Fig. 149, the effect of the reactance Xq
’
merely replaces the effect of the
voltage Ec, but by grouping the reactances associated with the current component Iq we obtain
a new reactance
Lqq XXX += '
Similarly, associating the reactive drops, using the direct component of current, we
obtain a new reactance
Ldd XXX += '
By making of these combined terms we have a comparatively simple diagram as
shown in Fig. 150.
Fig. 150. Final step in the transition, resulting in the two-reaction diagram built up with direct
axis synchronous reactance ( Ldd XXX += '
) and quadrature axis synchronous reactance
( Lqq XXX += '
). The resistance drop can also be divided into direct and quadrature
components, or in many cases this can be neglected entirely. Its relative size is exaggerated
here.
Now, if Xd and Xq are known the vector diagram can be constructed for the
determination of regulation.
An expression for the OC voltage E0 can be obtained from the following relationship.
By trigonometric considerations
(motoring)
cos
sin
'tan
g)(generatin
cos
sin
'tan
a
q
a
q
IRV
IXV
IRV
IXV
−
−
=
+
+
=
θ
θ
θ
θ
θ
θ
Angle (α) calculation between terminal voltage (V) and NL voltage (E0)
9. 03-Alternators
Prepared by Dr. M. A. Mannan Page 9 of 11
Also 'sin;'cos θθ IIII dq == then )sin();cos( θαθα +=+= IIII dq
From the Fig. 150,
V
IRIX
V
RIXI aqadqq 'sin'cos
sin
θθ
α
−
=
−
=
V
RXI aq )'sin'cos(
sin
θθ
α
−
=
Substituting
(motoring)'
g)(generatin'
αθθ
αθθ
−=
+=
in this equation for generating and expanding to sine and cosine functions,
)]sin()cos([sin αθαθα +−+= aq RX
V
I
}]cossincos{sin}sinsincos{cos[sin θααθαθαθα +−−= aq RXIV
]cossincossinsinsincoscos[sin θααθαθαθα aaqq RRXXIV −−−=
αθθαθθα sin]cossin[cos]sincos[sin aqaq RXIRXIV +−−=
αθθαθθα cos]sincos[sin]cossin[sin aqaq RXIRXIV −=++
αθθαθθ cos]sincos[sin}]cossin{[ aqaq RXIRXIV −=++
(motoring)
]cossin[
]sincos[
cos
sin
tan
g)(generatin
]cossin[
]sincos[
cos
sin
tan
θθ
θθ
α
α
α
θθ
θθ
α
α
α
aq
aq
aq
aq
RXIV
RXI
RXIV
RXI
+−
−
==
++
−
==
This is fundamental equation and it will be used when constructing the two-reaction
diagram of the alternator.
Also
(motoring)cos
g)(generatincos
0
0
ddaq
ddaq
XIRIVE
XIRIVE
−−=
++=
α
α
Direct- and Quadrature-axis Synchronous reactance [2/157/p.212]
A direct-axis quantity is one whose magnetic effect is centered on the axis of the main
poles. Direct-axis mmf’s act on the main magnetic circuit.
A quadrature-axis quantity is one whose magnetic effect is centered on the interpoler
space.
The replacing of the MMF effects of the direct component of the armature reaction by
a fictitious reactance Xd, which contains a leakage reactance component, is, of course,
consistent with old concept of synchronous reactance.
Because of this similarity of the two reactances with which we will deal, they are
called direct-axis synchronous reactance and quadrature-axis synchronous reactance,
respectively. As shown in vector diagram, they must always be used with their respective
components of current.
Neglecting the leakage fluxes, the direct component of MMF acts on the main
magnetic circuit o the machine. The quadrature component has a magnetic circuit largely
through the air gaps and interpolar space. For this reason the quadrature-axis synchronous
reactance is smaller than the direct-axis reactance and is less affected by saturation.
In non-salient pole machine, Xd is nearly equal to Xq and would be exactly so were it
not for the slight differences in the two magnetic circuits on which they operated.
10. 03-Alternators
Prepared by Dr. M. A. Mannan Page 10 of 11
Power Developed by a Synchronous Generator [1/37.29/p.1454]
If we neglect Ra and hence Cu loss, then the power developed (Pd) by an alternator is
equal to the power output (Pout). Hence, the per phase power output of an alternator is
dout PVIP == θcos (150.0)
From Fig. 150.1, qq XIV =αsin (150.1a)
αcos0 VEXI dd −= (150.1b)
)cos( θα += IIq (150.2a)
)sin( θα += IId (150.2b)
Substituting Eq. (150.2a) in Eq. (150.1a), then
θαθαθαα sinsincoscos)cos(sin IXIXIXV qqq −=+=
αθαθα sinsinsincoscos VIXIX qq =− (150.3)
Substituting Eq. (150.2b) in Eq. (150.1b), then
θαθαθαα sinsincoscos)cos(sin IXIXIXV qqq −=+=
θαθαθαα sincoscossin)sin(cos0 IXIXXIVE ddd +=+=−
αθαθα cossincoscossin 0 VEIXIX dd −=+ (150.4)
From Eqs. (150.3) and (150.4)
αθαθα
αθαθα
cossincoscossin
sinsinsincoscos
0 VEIXIX
VIXIX
dd
qq
−=+
=−
αααθααθαα
ααθααθαα
cossinsinsincossincossinsin
sincossinsincoscoscoscos
0 VXXEIXXIXX
VXIXXIXX
qqdqdq
dqdqd
−=+
=−
αααθααθαα
ααθααθαα
cossinsinsincossincossinsin
cossinsinsincoscoscoscos
0 VXEXIXXIXX
VXIXXIXX
qqqdqd
dqdqd
−=+
=−
ααααα
θααθαα
cossinsincossin
cossinsincoscoscos
0 VXEXVX
IXXIXX
qqd
qdqd
−+=
+
αααθ sincossin)(cos 0EXVXXIXX qqdqd +−=
αααθ 2sin
2
2sin
2
sincos 0 qdqqd X
V
X
V
EXIXX −+=
αααθ 2sin
2
2sin
2
sincos 0
dqd X
V
X
V
X
E
I −+= (150.5)
Substitute (150.5) in (150.0) then
ααααα 2sin
2
)(
sin2sin
2
2sin
2
sin
2
0
22
0
qd
qd
ddqd
dout
XX
XXV
X
VE
X
V
X
V
X
VE
PP
−
+=−+==