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EE 313 Electric Machines Fall 2014
Chapter 8: DC Machinery Fundamentals
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EE 313 Electric Machines Fall 2014
Preliminary notes
 DC power systems are not very common in the contemporary
engineering practice. However, DC motors still have many practical
applications, such automobile, aircraft, and portable electronics, in
speed control applications…
 An advantage of DC motors is that it is easy to control their speed in a
wide diapason.
 DC generators are quite rare.
 Most DC machines are similar to AC machines: i.e. they have AC
voltages and current within them. DC machines have DC outputs just
because they have a mechanism converting AC voltages to DC
voltages at their terminals. This mechanism is called a commutator;
therefore, DC machines are also called commutating machines.
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EE 313 Electric Machines Fall 2014
The simplest DC machine
The simplest DC rotating machine
consists of a single loop of wire
rotating about a fixed axis. The
magnetic field is supplied by the
North and South poles of the
magnet.
Rotor is the rotating part;
Stator is the stationary part.
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EE 313 Electric Machines Fall 2014
The simplest DC machine
We notice that the rotor lies in a slot curved
in a ferromagnetic stator core, which,
together with the rotor core, provides a
constant-width air gap between the rotor and
stator.
The reluctance of air is much larger than the
reluctance of core. Therefore, the magnetic
flux must take the shortest path through the
air gap.
As a consequence, the magnetic flux is perpendicular to the rotor surface
everywhere under the pole faces.
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EE 313 Electric Machines Fall 2014
The simplest DC machine
1. Voltage induced in a rotating loop
If a rotor of a DC machine is rotated, a voltage will be induced…
The total voltage will be a sum of voltages induced on each
segment of the loop.
Voltage on each segment is:
 inde  v×B l
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EE 313 Electric Machines Fall 2014
The simplest DC machine
1) ab: velocity v is perpendicular to the magnetic field B, and the vector product v
x B points into the page. Therefore, the voltage is
 
into page under the pole face
0 beyond the pole edges
ba
vBl
e
     
   

v×B l
2) bc: In this segment, vector product v x B is perpendicular to l. Therefore, the
voltage is zero.
3) cd: In this segment, the velocity of the wire is tangential to the path of rotation.
Under the pole face, velocity v is perpendicular to the magnetic flux density B,
and the vector product v x B points out of the page. Therefore, the voltage is
 
out of page under the pole face
0 beyond the pole edges
dc
vBl
e
      
   

v×B l
4) da: In this segment, vector product v x B is perpendicular to l. Therefore, the
voltage is zero.
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EE 313 Electric Machines Fall 2014
The simplest DC machine
The total induced voltage on the loop is:
tot ba cb dc ade e e e e   
2
0
tot
vBl under the pole faces
e
beyond the pole edges
   
 
   
When the loop rotates through 1800,
segment ab is under the north pole
face instead of the south pole face.
Therefore, the direction of the voltage
on the segment reverses but its
magnitude remains constant, leading
to the total induced voltage to be
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EE 313 Electric Machines Fall 2014
The simplest DC machine
The tangential velocity of the loop’s edges is
v r
2
0
tot
r Bl under the pole faces
e
beyond the pole edges
    
 
   
where r is the radius from the axis of rotation to
the edge of the loop. The total induced voltage:
The rotor is a cylinder with surface area 2rl.
Since there are two poles, the area of the rotor
under each pole is Ap = rl. Therefore:
2
0
p
tot
A B under the pole faces
e
beyond the pole edges



   
 
    
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EE 313 Electric Machines Fall 2014
The simplest DC machine
Assuming that the flux density B is constant everywhere in the air gap under the
pole faces, the total flux under each pole is
pA B 
The total voltage is
2
0
tot
under the pole faces
e
beyond the pole edges



   
 
    
The voltage generated in any real machine depends on the following
factors:
1. The flux inside the machine;
2. The rotation speed of the machine;
3. A constant representing the construction of the machine.
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EE 313 Electric Machines Fall 2014
The simplest DC machine
2. Getting DC voltage out of a rotating
loop
A voltage out of the loop is alternatively a
constant positive value and a constant
negative value.
One possible way to convert an alternating
voltage to a constant voltage is by adding a
Commutator
segment/brush
circuitry to the
end of the loop.
Every time the
voltage of the
loop switches
direction,
contacts switch
connection.
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EE 313 Electric Machines Fall 2014
The simplest DC machine
2. Getting DC voltage out of a rotating
loop
A voltage out of the loop is alternatively a
constant positive value and a constant
negative value.
One possible way to convert an alternating
voltage to a constant voltage is by adding a
commutator
segment/brush
circuitry to the
end of the loop.
Every time the
voltage of the
loop switches
direction,
contacts switch
connection.
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EE 313 Electric Machines Fall 2014
Commutator with carbon brushes
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EE 313 Electric Machines Fall 2014
Commutator with carbon brushes
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EE 313 Electric Machines Fall 2014
The simplest DC machine
Commutator
segments
Carbon brushes
Spring
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EE 313 Electric Machines Fall 2014
The simplest DC machine
3. The induced torque in the
rotating loop
Assuming that a battery is connected
to the DC machine, the force on a
segment of a loop is:
 F i l×B
And the torque on the segment is
sinrF 
Where  is the angle between r and F.
Therefore, the torque is zero when
the loop is beyond the pole edges.
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EE 313 Electric Machines Fall 2014
The simplest DC machine
When the loop is under the pole faces:
1. Segment ab:
2. Segment bc:
3. Segment cd:
4. Segment da:
 abF i ilB l×B
 sin sin90ab rF r ilB rilB ccw     
  0bcF i l×B
sin 0bc rF  
 cdF i ilB l×B
 sin sin90cd rF r ilB rilB ccw     
  0daF i l×B
sin 0da rF  
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EE 313 Electric Machines Fall 2014
The simplest DC machine
The resulting total induced torque is
The torque in any real machine depends on the following factors:
1. The flux inside the machine;
2. The current in the machine;
3. A constant representing the construction of the machine.
2
0
ind
i under the pole faces
beyond the pole edges

 

   
 
    
ind ab bc cd da       
2
0
ind
rilB under the pole faces
beyond the pole edges

   
 
   
Since Ap  rl and pA B 
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EE 313 Electric Machines Fall 2014
Example: 8.1
Figure shows a simple rotating loop between curved pole
faces connected to a battery and a resistor through a switch.
The resistor shown models the total resistance of the battery
and the wire in the machine. The physical dimensions and
characteristics of this machine are:
r = 0.5m l = 1.0m R = 0.3
B = 0.25T VB = 120V
a) What happens when the switch is closed
b) What is the machine’s max. starting current? What is its
steady state angular velocity at no load?
c) Suppose a load is attached to the loop, and the resulting
load torque is 10N.m. What would the new steady state
speed be? How much power is supplied to the shaft of the
machine? How much power is being supplied by the
battery? Is this machine a motor or a generator?
d) Suppose the machine is again unloaded, and a torque of
7.5 N.m is applied to the shaft in the direction of rotation.
What is the new steady state speed? Is this machine now
a motor or a generator?
e) Suppose the machine is running unloaded. What would
the final steady state speed of the rotor be if the flux
density were reduced to 0.20 T?
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EE 313 Electric Machines Fall 2014
Rotor (Armature Core)
Armature core is a part of
rotor on which conductors
(Armature Winding) are
wound. It consists of thin
laminations.
The main parts of an
armature Core are;
Slots :
Tooth:
Shaft:
Bearings and
commutator segments
are fixed on shaft of
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EE 313 Electric Machines Fall 2014
Rotor with Armature Winding
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EE 313 Electric Machines Fall 2014
A wound armature core
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EE 313 Electric Machines Fall 2014
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EE 313 Electric Machines Fall 2014
Commutation in a simple 4-loop
DC machine
Commutation is the process of converting the AC voltages and currents in
the rotor of a DC machine to DC voltages and currents at its terminals.
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EE 313 Electric Machines Fall 2014
A simple 4-loop DC machine has four complete loops buried in slots curved in the
laminated steel of its rotor. The pole faces are curved to make a uniform air-gap.
The four loops are laid into the slots in a special manner: the innermost wire in
each slot (end of each loop opposite to the “unprimed”) is indicated by a prime.
Loop 1 stretches
between
commutator
segments a and
b, loop 2
stretches
between
segments b and
c…
Commutation in a simple 4-loop
DC machine
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EE 313 Electric Machines Fall 2014
Commutation in a simple 4-loop
DC machine
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EE 313 Electric Machines Fall 2014
Commutation in a simple 4-loop
DC machine
At a certain time instance, when t
= 00, the 1, 2, 3’, and 4’ ends of the
loops are under the north pole face
and the 1’, 2’, 3, and 4 ends of the
loops are under the south pole face.
The voltage in each of 1, 2, 3’, and
4’ ends is given by
 
positive, out of the page
inde vBl  

v×B ×l
  positive, into the pageinde vBl   v×B ×l
The voltage in each of 1’, 2’, 3, and 4 ends is
The total voltage at the brushes of the DC machine is
4 0E e at t    
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EE 313 Electric Machines Fall 2014
Commutation in a simple 4-loop
DC machine
If the machine keeps rotating, at t = 450, loops 1 and 3 have rotated into the gap
between poles, so the voltage across each of them is zero. At the same time, the
brushes short out the commutator segments ab and cd.
This is ok since
the voltage across
loops 1 and 3 is
zero and only
loops 2 and 4 are
under the pole
faces. Therefore,
the total terminal
voltage is
2 45E e at t    
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EE 313 Electric Machines Fall 2014
Magnetic Field path due to air and
curved shape of Poles
Neutral plane of magnetic field
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EE 313 Electric Machines Fall 2014
Commutation in a simple 4-loop
DC machine
At t = 900, the loop ends 1’, 2, 3, and 4’ are under
the north pole face, and the loop ends 1, 2’, 3’, and
4 are under the south pole face. The voltages are
built up out of page for the ends under the north
pole face and into the page for the ends under the
south pole face. Four voltage-carrying ends in each
parallel path through the machine lead to the
terminal voltage of
4 90E e at t    
We notice that the voltages in loops 1 and 3 have
reversed compared to t = 00. However, the loops’
connections have also reversed, making the total
voltage being of the same polarity.
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EE 313 Electric Machines Fall 2014
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EE 313 Electric Machines Fall 2014
Commutation in a simple 4-loop
DC machine
When the voltage reverses in a loop, the connections of the loop are also switched
to keep the polarity of the terminal voltage the same.
The terminal voltage of this 4-loop DC
machine is still not constant over time,
although it is a better approximation to a
constant DC level than what is produced
by a single rotating loop.
Increasing the number of loops on the
rotor, we improve our approximation to
perfect DC voltage.
Commutator segments are usually made out of copper bars and the brushes are
made of a mixture containing graphite to minimize friction between segments and
brushes.
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EE 313 Electric Machines Fall 2014
Example of a commutator…
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EE 313 Electric Machines Fall 2014
8.6: Construction of DC Machines
All d.c. machines have five principal components viz.,
(i) Field system (Stator)
(ii) Armature core
(iii) Armature winding
(iv) Commutator
(v) Brushes
(i) Field system
 The function of the field system is to produce uniform magnetic field within
which the armature rotates. It consists of a number of poles bolted to the
inside of circular frame (generally called yoke).
 The field coils are connected in such a way that adjacent poles have
opposite polarity.
 The m.m.f. developed by the field coils produces a magnetic flux that
passes through the pole pieces, the air gap, the armature and the frame.
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EE 313 Electric Machines Fall 2014
Construction of DC Machines
(ii) Armature core
 The armature core is keyed to the machine shaft and rotates between the field
poles. It consists of slotted soft-iron laminations (about 0.4 to 0.6 mm thick) that are
stacked to form a cylindrical core as shown in Fig.
 The laminations are individually coated with a thin insulating film so that they do not
come in electrical contact with each other. The purpose of laminating the core is to
reduce the eddy current loss.
 The laminations are slotted to accommodate and provide mechanical security to the
armature winding and to give shorter air gap for the flux to cross between the pole
face and the armature “teeth”.
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EE 313 Electric Machines Fall 2014
Construction of DC Machines
(iii) Armature winding
 The slots of the armature core hold insulated conductors that are connected in
a suitable manner. This is known as armature winding. This is the winding in
which “working” e.m.f. is induced.
 The armature conductors are connected in series-parallel; the conductors being
connected in series so as to increase the voltage and in parallel paths so as to
increase the current.
 The armature winding of a d.c. machine is a closed-circuit winding; the
conductors being connected in a symmetrical manner forming a closed loop or
series of closed loops.
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EE 313 Electric Machines Fall 2014
Construction of DC Machines
(iv) Commutator
 A commutator is a mechanical rectifier which converts the alternating voltage
generated in the armature winding into direct voltage across the brushes.
 The commutator is made of copper segments insulated from each other and
mounted on the shaft of the machine.
 The armature conductors are soldered to the commutator segments in a suitable
manner.
 Great care is taken in building the
commutator because any short circuiting will
cause the brushes to bounce, producing
unacceptable sparking. The sparks may burn
the brushes and overheat.
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EE 313 Electric Machines Fall 2014
Construction of DC Machines
(v) Brushes
 The purpose of brushes is to ensure electrical
connections between the rotating commutator and
stationary external load circuit. The brushes are made of
carbon and rest on the commutator.
 The brush pressure is adjusted by means of adjustable
springs. If the brush pressure is very large, the friction
produces heating of the commutator and the brushes. On
the other hand, if it is too weak, the imperfect contact with
the commutator may produce sparking.
 Multipole machines have as many brushes as they have
poles. For example, a 4-pole machine has 4 brushes.
 The successive brushes have positive and negative
polarities. Brushes having the same polarity are
connected together so that we have two terminals viz.,
the +ve terminal and the -ve terminal.
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EE 313 Electric Machines Fall 2014
Armature winding
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EE 313 Electric Machines Fall 2014
Commutator and Armature Winding
Terminologies
Commutator Pitch (YC)
 The commutator pitch is the number of commutator segments spanned by
each coil of the winding. It is denoted by YC.
 In Fig.1 one side of the coil is connected to commutator segment 1 and
the other side connected to commutator segment 2. Therefore, the
number of commutator segments spanned by the coil is 1 i.e., YC = 1.
 In Fig. 2, one side of the coil is connected to commutator segment 1 and
the other side to commutator segment 8. Therefore, the number of
commutator segments spanned by the coil = 8 - 1 = 7 segments i.e., YC =
7.
 The commutator pitch of a winding is always a whole number. Since each
coil has two ends and as two coil connections are joined at each
commutator segment,
Number of coils = Number of commutator segments
 For example, if an armature has 30 conductors, the number of coils will be
30/2 = 15. Therefore, number of commutator segments is also 15. Note
that commutator pitch is the most important factor in determining the type
of d.c. armature winding.
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EE 313 Electric Machines Fall 2014
Commutator and Armature
Winding Terminologies
Pole-Pitch
It is the distance measured in terms of number of armature slots (or armature
conductors) per pole. Thus if a 4-pole generator has 16 coils, then number of
slots = 16.
Coil Span or Coil Pitch (YS)
 It is the distance measured in terms of the number of armature slots (or
armature conductors) spanned by a coil.
 Thus if the coil span is 9 slots, it means one side of the coil is in slot 1 and the
other side in slot 10.
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EE 313 Electric Machines Fall 2014
Commutator and Armature
Winding Terminologies
Full-Pitched Coil
 If the coil-span is equal to pole pitch, it is called full-pitched
coil.
 In this case, the e.m.f.s in the coil sides are additive and have a
phase difference of 0°. Therefore, e.m.f. induced in the coil is
maximum.
 If e.m.f. induced in one coil side is 2.5 V, then e.m.f. across the
coil terminals = 2 x 2.5 = 5 V.
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EE 313 Electric Machines Fall 2014
Commutator and Armature
Winding Terminologies
Fractional pitched coil
 If the coil span or coil pitch is less than the pole
pitch, then it is called fractional pitched coil.
 In this case, the phase difference between the e.m.f.s
in the two coil sides will not be zero so that the e.m.f. of
the coil will be less compared to full-pitched coil.
 Fractional pitch winding requires less copper but if the
pitch is too small, an appreciable reduction in the
generated e.m.f. results
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EE 313 Electric Machines Fall 2014
Commutator and Armature
Winding Terminologies
Back Pitch (YB) (start to end)
 It is the distance measured in terms of armature conductors
between the two sides of a coil at the back of the armature .
 For example, if a coil is formed by connecting conductor 1
(upper conductor in a slot) to conductor 12 (bottom
conductor in another slot) at the back of the armature, then
back pitch is YB = 12 - 1 = 11 conductors.
Front Pitch (YF) (End of first and start of second coil)
 It is the distance measured in terms of armature
conductors between the coil sides attached to any one
commutator segment .
 For example, if coil side 12 and coil side 3 are connected
to the same commutator segment, then front pitch is:
YF = 12 - 3 = 9 conductors.
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EE 313 Electric Machines Fall 2014
Front and Back Pitch
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EE 313 Electric Machines Fall 2014
Commutator and Armature
Winding Terminologies
Resultant Pitch (YR)
 It is the distance (measured in terms of armature
conductors) between the beginning of one coil and the
beginning of the next coil to which it is connected
Therefore,
YR = YF + YB
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EE 313 Electric Machines Fall 2014
Commutator and Armature
Winding Terminologies
Progressive Winding
A progressive winding is one in which, as one traces
through the winding, the connections to the commutator will
progress around the machine in the same direction as is
being traced along the path of each individual coil.
Note that YB > YF and YC = + 1.
Retrogressive Winding
A retrogressive winding is one in which, as one traces
through the winding, the connections to the commutator will
progress around the machine in the opposite direction to
that which is being traced along the path of each individual
coil.
Note that YF > YB and YC = - 1.
A retrogressive winding is seldom used because it requires
more copper.
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EE 313 Electric Machines Fall 2014
General Rules For D.C. Armature
Windings
1) The back pitch (YB) as well as front pitch (YF) should be
nearly equal to pole pitch YP. This will result in
increased e.m.f. in the coils.
2) Both pitches (YB and YF) should be odd. This will
permit all end connections (back as well as front
connection) between a conductor at the top of a slot
and one at the bottom of a slot.
3) The number of commutator segments is equal to the
number of slots or coils (or half the number of
conductors).
No. of commutator segments = No. of slots = No. of
coils
It is because each coil has two ends and two coil
connections are joined at each commutator segment
4) The winding must close upon itself i.e. it should be a
closed circuit winding.
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EE 313 Electric Machines Fall 2014
Simplex Lap Winding
Relations between Pitches
2) Both YB and YF should be >= to pole pitch (YP).
3) Average pitch =(YB + YF) / 2. It nearly equals pole
pitch (YP= Z/P).
4) Commutator pitch, YC = ± 1
YC = + 1 for progressive winding
YC = - 1 for retrogressive winding
5) The resultant pitch (YR) is even, being the arithmetical sum of two odd numbers
viz., YB and YF.
6) If Z = number of armature conductors and P = number of poles, then, It is clear
that Z/P must be an even number to make the winding possible.
1) The back and front pitches are odd. They differ numerically by
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EE 313 Electric Machines Fall 2014
Lap and Wave Winding
Comparison
Lap Winding Wave Winding
Ys= S/P Ys= S/P
Yc= +1 or -1 (Prog. Or Retrogr.) Yc= 2(C+1)/P or Yc= 2(C+1)/P
Parallel paths a=P Parallel paths a=2
Coils=slots=comm. Segments Coils=slots=comm. Segments
In general, one coil has tow conductors (active) .
If you number of coils are 10 , there will be 20 conductors (Z)
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EE 313 Electric Machines Fall 2014
Armature View and Developed
Diagram
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EE 313 Electric Machines Fall 2014
Coil Ends (front, back) Slot Number (4) Commt.Segments (+1
1,1’ 1,5 a, b
2,2’ 2,6 b, c
3,3’ 3,7 C, d
4,4’ 4,8 D,e
5,5’ 5,9 E,f
6,6’ 6,10 F,g
7,7’ 7,11 G,h
8,8’ 8,12 H,i
9,9’ 9,13 I,j
10,10’ 10,14 J,k
11,11’ 11,15 K,l
12,12’ 12,16 L,m
13,13’ 13,17(1) M,n
14,14’ 14,18(2) N,o
15,15’ 15,19(3) O,p
16,16’ 16,20(4) P,a
53
EE 313 Electric Machines Fall 2014
16 Coils, 4 Poles, Lap Winding
54
EE 313 Electric Machines Fall 2014
55
EE 313 Electric Machines Fall 2014
Lap Winding: Conclusions
1) The total number of brushes is equal to the number of poles.
2) The armature winding is divided into as many parallel paths as the number of poles.
If the total number of armature conductors is Z and P is the number of poles, then,
Number of parallel paths = P
Number of series Conductors = Z / P
In the present case, there are 32 armature conductors and 4 poles. Therefore, the
armature winding has 4 parallel paths, each consisting of 8 conductors in series.
3)
4) Total armature current,
The armature resistance can be found as under:
E.M.F.generated = E.M.F.per parallel path = average e.m.f.per conductor × Z/P
1Resistance per parallel path
No. of Parallel paths
a
l Z
R
P a P
 
    
Current per parallel pathaI P 
56
EE 313 Electric Machines Fall 2014
Simplex Wave Winding
57
EE 313 Electric Machines Fall 2014
Design of Simplex Wave Winding
1) Both pitches YB and YF are odd and are of the same sign.
2)
3) Both YB and YF are nearly equal to pole pitch(YP) and may be equal or differ by 2.
If they differ by 2, they are one more and one less than YA.
4) Commutator pitch is given by;
5) Since YA must be a whole number, there is a restriction on the value of Z. With Z
= 180, this winding is impossible for a 4-pole machine because YA is not a whole
number.
6)
2( 1)
c
C
y
P


58
EE 313 Electric Machines Fall 2014
Simple 4-Pole Wave wound DC Macnine
59
EE 313 Electric Machines Fall 2014
Slots 9, Poles 4, Wave Wingding
(progressive)
Coil Ends (front, back) Slot Number (2) Commt.Segments (+5
1,1’ 1,3 a, f
6,6’ 6,8 f, b
2,2’ 2,4 B,g
7,7’ 7,9 G,c
3,3’ 3,5 C,h
8,8’ 8,1 H,d
4,4’ 4,6 D,i
9,9’ 9,2 I,e
5,5’ 5,7 E,a
60
EE 313 Electric Machines Fall 2014
Slots 9, Poles 4, Wave Wingding
(Developed Diagram)
61
EE 313 Electric Machines Fall 2014
Wave Winding: Conclusions
1) Only two brushes are necessary but as many brushes as there are poles may
be used.
2) The armature winding is divided into two parallel paths irrespective of the
number of poles. If the total number of armature conductors is Z and P is the
number of poles, then,
3)
5) Total armature current,
6) The armature can be wave-wound if YA or YC is a whole number.
2 Current per parallel pathaI  
62
EE 313 Electric Machines Fall 2014
Applications of Lap and Wave Windings
 In multipolar machines, for a given number of poles (P) and armature
conductors (Z), a wave winding has a higher terminal voltage than a lap
winding because it has more conductors in series.
 On the other hand, the lap winding carries more current than a wave
winding because it has more parallel paths.
 In small machines, the current-carrying capacity of the armature
conductors is not critical and in order to achieve suitable voltages, wave
windings are used.
 On the other hand, in large machines suitable voltages are easily obtained
because of the availability of large number of armature conductors and the
current carrying capacity is more critical. Hence in large machines, lap
windings are used.
63
EE 313 Electric Machines Fall 2014
The Frog-Leg Winding or Self
Equalizing Winding
 It consists of lap and wave winding combined.
 The wave winding functions as equalizer for
lap winding.
 The number of current paths in frog leg winding
is:
a = 2Pmlap
Where
P = No. Of poles on the machine
Mlap = Plex of the lap winding
64
EE 313 Electric Machines Fall 2014
The Internal Generated Voltage and Induced
Torque Equations of Real DC Machine
A: Voltage in the rotor windings of a real dc machine
The voltage out of the armature of a real dc machine is equal to the number of
conductors per current path times the voltage on each conductor.
The voltage in any single conductor: eind = e = vBl
The voltage out of the armature of a real machine is thus:
The velocity of each conductor in the rotor is: v=rw
And with ɸ=BAP and A=2rl, and if there are P poles on the machine, then the
total area A is
A
ZvBl
E
a

A
Zr Bl
E
a


2
P
A rl
A
P P

 
The total flux per pole in the machine:
(2 ) 2
P
B rl rlB
BA
P P
 
   
65
EE 313 Electric Machines Fall 2014
The Internal Generated Voltage and Induced
Torque Equations of Real DC Machine
The total flux per pole in the machine:
(2 ) 2
P
B rl rlB
BA
P P
 
   
The internal generated voltage in the machine:
2
2
2
A
A
A
Zr Bl
E
a
ZP rlB
a P
ZP
E
a
E K








  
   
  


B: Torque induced in the armature of a real dc machine
The torque in any single conductor
cond = r Icond lB
If there are ‘a’ current paths in the machine, so the current in a single conductor is given by
A
cond
I
I
a

66
EE 313 Electric Machines Fall 2014
The Internal Generated Voltage and Induced
Torque Equations of Real DC Machine
And the torque in a single conductor on the motor is: A
cond
rI lB
a
 
For Z conductors, the total induced torque in a dc machine rotor is:
The flux per pole in this machine:
A
ind
ZrlBI
a
 
(2 ) 2
P
B rl rlB
BA
P P
 
   
So the total induced torque is:
2
ind A
ZP
I
a
 


Finally,
ind AK I 
67
EE 313 Electric Machines Fall 2014
Power flow and losses in DC machines
Unfortunately, not all electrical power is converted to mechanical power by a motor
and not all mechanical power is converted to electrical power by a generator…
The efficiency of a DC machine is:
100%out
in
P
P
  
100%in loss
in
P P
P


 
or
68
EE 313 Electric Machines Fall 2014
The losses in DC machines
There are five categories of losses occurring in DC machines.
1. Electrical or copper losses – the resistive losses in the armature and field
windings of the machine.
Armature loss:
2
A A AP I R
Field loss:
2
F F FP I R
Where IA and IF are armature and field currents and RA and RF are armature and
field (winding) resistances usually measured at normal operating temperature.
69
EE 313 Electric Machines Fall 2014
The losses in DC machines
2. Brush (drop) losses – the power lost across the contact potential at the
brushes of the machine.
BD BD AP V I
Where IA is the armature current and VBD is the brush voltage drop. The voltage
drop across the set of brushes is approximately constant over a large range of
armature currents and it is usually assumed to be about 2 V.
70
EE 313 Electric Machines Fall 2014
The losses in DC machines
4. Mechanical losses – losses associated with mechanical effects: friction
(friction of the bearings) and windage (friction between the moving parts of the
machine and the air inside the casing). These losses vary as the cube of rotation
speed n3.
3. Core losses – hysteresis losses and eddy current losses.
5. Stray (Miscellaneous) losses – losses that cannot be classified in any of the
previous categories. They are usually due to inaccuracies in modeling. For many
machines, stray losses are assumed as 1% of full load.
71
EE 313 Electric Machines Fall 2014
The power-flow diagram
On of the most convenient technique to account for power losses in a
machine is the power-flow diagram.
For a DC
motor:
Electrical power is input to the machine, and the electrical and brush losses must be
subtracted. The remaining power is ideally converted from electrical to mechanical
form at the point labeled as Pconv.
72
EE 313 Electric Machines Fall 2014
The power-flow diagram
The electrical power that is converted is
conv A AP E I
And the resulting mechanical power is
conv ind mP  
After the power is converted to mechanical form, the stray losses, mechanical
losses, and core losses are subtracted, and the remaining mechanical power is
output to the load.
73
EE 313 Electric Machines Fall 2014
Simple DC Motor

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Ee 313-dc machinery fundamentals (part1)

  • 1. 1 EE 313 Electric Machines Fall 2014 Chapter 8: DC Machinery Fundamentals
  • 2. 2 EE 313 Electric Machines Fall 2014 Preliminary notes  DC power systems are not very common in the contemporary engineering practice. However, DC motors still have many practical applications, such automobile, aircraft, and portable electronics, in speed control applications…  An advantage of DC motors is that it is easy to control their speed in a wide diapason.  DC generators are quite rare.  Most DC machines are similar to AC machines: i.e. they have AC voltages and current within them. DC machines have DC outputs just because they have a mechanism converting AC voltages to DC voltages at their terminals. This mechanism is called a commutator; therefore, DC machines are also called commutating machines.
  • 3. 3 EE 313 Electric Machines Fall 2014 The simplest DC machine The simplest DC rotating machine consists of a single loop of wire rotating about a fixed axis. The magnetic field is supplied by the North and South poles of the magnet. Rotor is the rotating part; Stator is the stationary part.
  • 4. 4 EE 313 Electric Machines Fall 2014 The simplest DC machine We notice that the rotor lies in a slot curved in a ferromagnetic stator core, which, together with the rotor core, provides a constant-width air gap between the rotor and stator. The reluctance of air is much larger than the reluctance of core. Therefore, the magnetic flux must take the shortest path through the air gap. As a consequence, the magnetic flux is perpendicular to the rotor surface everywhere under the pole faces.
  • 5. 5 EE 313 Electric Machines Fall 2014 The simplest DC machine 1. Voltage induced in a rotating loop If a rotor of a DC machine is rotated, a voltage will be induced… The total voltage will be a sum of voltages induced on each segment of the loop. Voltage on each segment is:  inde  v×B l
  • 6. 6 EE 313 Electric Machines Fall 2014 The simplest DC machine 1) ab: velocity v is perpendicular to the magnetic field B, and the vector product v x B points into the page. Therefore, the voltage is   into page under the pole face 0 beyond the pole edges ba vBl e            v×B l 2) bc: In this segment, vector product v x B is perpendicular to l. Therefore, the voltage is zero. 3) cd: In this segment, the velocity of the wire is tangential to the path of rotation. Under the pole face, velocity v is perpendicular to the magnetic flux density B, and the vector product v x B points out of the page. Therefore, the voltage is   out of page under the pole face 0 beyond the pole edges dc vBl e             v×B l 4) da: In this segment, vector product v x B is perpendicular to l. Therefore, the voltage is zero.
  • 7. 7 EE 313 Electric Machines Fall 2014 The simplest DC machine The total induced voltage on the loop is: tot ba cb dc ade e e e e    2 0 tot vBl under the pole faces e beyond the pole edges           When the loop rotates through 1800, segment ab is under the north pole face instead of the south pole face. Therefore, the direction of the voltage on the segment reverses but its magnitude remains constant, leading to the total induced voltage to be
  • 8. 8 EE 313 Electric Machines Fall 2014 The simplest DC machine The tangential velocity of the loop’s edges is v r 2 0 tot r Bl under the pole faces e beyond the pole edges            where r is the radius from the axis of rotation to the edge of the loop. The total induced voltage: The rotor is a cylinder with surface area 2rl. Since there are two poles, the area of the rotor under each pole is Ap = rl. Therefore: 2 0 p tot A B under the pole faces e beyond the pole edges              
  • 9. 9 EE 313 Electric Machines Fall 2014 The simplest DC machine Assuming that the flux density B is constant everywhere in the air gap under the pole faces, the total flux under each pole is pA B  The total voltage is 2 0 tot under the pole faces e beyond the pole edges               The voltage generated in any real machine depends on the following factors: 1. The flux inside the machine; 2. The rotation speed of the machine; 3. A constant representing the construction of the machine.
  • 10. 10 EE 313 Electric Machines Fall 2014 The simplest DC machine 2. Getting DC voltage out of a rotating loop A voltage out of the loop is alternatively a constant positive value and a constant negative value. One possible way to convert an alternating voltage to a constant voltage is by adding a Commutator segment/brush circuitry to the end of the loop. Every time the voltage of the loop switches direction, contacts switch connection.
  • 11. 11 EE 313 Electric Machines Fall 2014 The simplest DC machine 2. Getting DC voltage out of a rotating loop A voltage out of the loop is alternatively a constant positive value and a constant negative value. One possible way to convert an alternating voltage to a constant voltage is by adding a commutator segment/brush circuitry to the end of the loop. Every time the voltage of the loop switches direction, contacts switch connection.
  • 12. 12 EE 313 Electric Machines Fall 2014 Commutator with carbon brushes
  • 13. 13 EE 313 Electric Machines Fall 2014 Commutator with carbon brushes
  • 14. 14 EE 313 Electric Machines Fall 2014 The simplest DC machine Commutator segments Carbon brushes Spring
  • 15. 15 EE 313 Electric Machines Fall 2014 The simplest DC machine 3. The induced torque in the rotating loop Assuming that a battery is connected to the DC machine, the force on a segment of a loop is:  F i l×B And the torque on the segment is sinrF  Where  is the angle between r and F. Therefore, the torque is zero when the loop is beyond the pole edges.
  • 16. 16 EE 313 Electric Machines Fall 2014 The simplest DC machine When the loop is under the pole faces: 1. Segment ab: 2. Segment bc: 3. Segment cd: 4. Segment da:  abF i ilB l×B  sin sin90ab rF r ilB rilB ccw        0bcF i l×B sin 0bc rF    cdF i ilB l×B  sin sin90cd rF r ilB rilB ccw        0daF i l×B sin 0da rF  
  • 17. 17 EE 313 Electric Machines Fall 2014 The simplest DC machine The resulting total induced torque is The torque in any real machine depends on the following factors: 1. The flux inside the machine; 2. The current in the machine; 3. A constant representing the construction of the machine. 2 0 ind i under the pole faces beyond the pole edges                ind ab bc cd da        2 0 ind rilB under the pole faces beyond the pole edges            Since Ap  rl and pA B 
  • 18. 18 EE 313 Electric Machines Fall 2014 Example: 8.1 Figure shows a simple rotating loop between curved pole faces connected to a battery and a resistor through a switch. The resistor shown models the total resistance of the battery and the wire in the machine. The physical dimensions and characteristics of this machine are: r = 0.5m l = 1.0m R = 0.3 B = 0.25T VB = 120V a) What happens when the switch is closed b) What is the machine’s max. starting current? What is its steady state angular velocity at no load? c) Suppose a load is attached to the loop, and the resulting load torque is 10N.m. What would the new steady state speed be? How much power is supplied to the shaft of the machine? How much power is being supplied by the battery? Is this machine a motor or a generator? d) Suppose the machine is again unloaded, and a torque of 7.5 N.m is applied to the shaft in the direction of rotation. What is the new steady state speed? Is this machine now a motor or a generator? e) Suppose the machine is running unloaded. What would the final steady state speed of the rotor be if the flux density were reduced to 0.20 T?
  • 19. 19 EE 313 Electric Machines Fall 2014 Rotor (Armature Core) Armature core is a part of rotor on which conductors (Armature Winding) are wound. It consists of thin laminations. The main parts of an armature Core are; Slots : Tooth: Shaft: Bearings and commutator segments are fixed on shaft of
  • 20. 20 EE 313 Electric Machines Fall 2014 Rotor with Armature Winding
  • 21. 21 EE 313 Electric Machines Fall 2014 A wound armature core
  • 22. 22 EE 313 Electric Machines Fall 2014
  • 23. 23 EE 313 Electric Machines Fall 2014 Commutation in a simple 4-loop DC machine Commutation is the process of converting the AC voltages and currents in the rotor of a DC machine to DC voltages and currents at its terminals.
  • 24. 24 EE 313 Electric Machines Fall 2014 A simple 4-loop DC machine has four complete loops buried in slots curved in the laminated steel of its rotor. The pole faces are curved to make a uniform air-gap. The four loops are laid into the slots in a special manner: the innermost wire in each slot (end of each loop opposite to the “unprimed”) is indicated by a prime. Loop 1 stretches between commutator segments a and b, loop 2 stretches between segments b and c… Commutation in a simple 4-loop DC machine
  • 25. 25 EE 313 Electric Machines Fall 2014 Commutation in a simple 4-loop DC machine
  • 26. 26 EE 313 Electric Machines Fall 2014 Commutation in a simple 4-loop DC machine At a certain time instance, when t = 00, the 1, 2, 3’, and 4’ ends of the loops are under the north pole face and the 1’, 2’, 3, and 4 ends of the loops are under the south pole face. The voltage in each of 1, 2, 3’, and 4’ ends is given by   positive, out of the page inde vBl    v×B ×l   positive, into the pageinde vBl   v×B ×l The voltage in each of 1’, 2’, 3, and 4 ends is The total voltage at the brushes of the DC machine is 4 0E e at t    
  • 27. 28 EE 313 Electric Machines Fall 2014 Commutation in a simple 4-loop DC machine If the machine keeps rotating, at t = 450, loops 1 and 3 have rotated into the gap between poles, so the voltage across each of them is zero. At the same time, the brushes short out the commutator segments ab and cd. This is ok since the voltage across loops 1 and 3 is zero and only loops 2 and 4 are under the pole faces. Therefore, the total terminal voltage is 2 45E e at t    
  • 28. 29 EE 313 Electric Machines Fall 2014 Magnetic Field path due to air and curved shape of Poles Neutral plane of magnetic field
  • 29. 30 EE 313 Electric Machines Fall 2014 Commutation in a simple 4-loop DC machine At t = 900, the loop ends 1’, 2, 3, and 4’ are under the north pole face, and the loop ends 1, 2’, 3’, and 4 are under the south pole face. The voltages are built up out of page for the ends under the north pole face and into the page for the ends under the south pole face. Four voltage-carrying ends in each parallel path through the machine lead to the terminal voltage of 4 90E e at t     We notice that the voltages in loops 1 and 3 have reversed compared to t = 00. However, the loops’ connections have also reversed, making the total voltage being of the same polarity.
  • 30. 31 EE 313 Electric Machines Fall 2014
  • 31. 32 EE 313 Electric Machines Fall 2014 Commutation in a simple 4-loop DC machine When the voltage reverses in a loop, the connections of the loop are also switched to keep the polarity of the terminal voltage the same. The terminal voltage of this 4-loop DC machine is still not constant over time, although it is a better approximation to a constant DC level than what is produced by a single rotating loop. Increasing the number of loops on the rotor, we improve our approximation to perfect DC voltage. Commutator segments are usually made out of copper bars and the brushes are made of a mixture containing graphite to minimize friction between segments and brushes.
  • 32. 33 EE 313 Electric Machines Fall 2014 Example of a commutator…
  • 33. 34 EE 313 Electric Machines Fall 2014 8.6: Construction of DC Machines All d.c. machines have five principal components viz., (i) Field system (Stator) (ii) Armature core (iii) Armature winding (iv) Commutator (v) Brushes (i) Field system  The function of the field system is to produce uniform magnetic field within which the armature rotates. It consists of a number of poles bolted to the inside of circular frame (generally called yoke).  The field coils are connected in such a way that adjacent poles have opposite polarity.  The m.m.f. developed by the field coils produces a magnetic flux that passes through the pole pieces, the air gap, the armature and the frame.
  • 34. 35 EE 313 Electric Machines Fall 2014 Construction of DC Machines (ii) Armature core  The armature core is keyed to the machine shaft and rotates between the field poles. It consists of slotted soft-iron laminations (about 0.4 to 0.6 mm thick) that are stacked to form a cylindrical core as shown in Fig.  The laminations are individually coated with a thin insulating film so that they do not come in electrical contact with each other. The purpose of laminating the core is to reduce the eddy current loss.  The laminations are slotted to accommodate and provide mechanical security to the armature winding and to give shorter air gap for the flux to cross between the pole face and the armature “teeth”.
  • 35. 36 EE 313 Electric Machines Fall 2014 Construction of DC Machines (iii) Armature winding  The slots of the armature core hold insulated conductors that are connected in a suitable manner. This is known as armature winding. This is the winding in which “working” e.m.f. is induced.  The armature conductors are connected in series-parallel; the conductors being connected in series so as to increase the voltage and in parallel paths so as to increase the current.  The armature winding of a d.c. machine is a closed-circuit winding; the conductors being connected in a symmetrical manner forming a closed loop or series of closed loops.
  • 36. 37 EE 313 Electric Machines Fall 2014 Construction of DC Machines (iv) Commutator  A commutator is a mechanical rectifier which converts the alternating voltage generated in the armature winding into direct voltage across the brushes.  The commutator is made of copper segments insulated from each other and mounted on the shaft of the machine.  The armature conductors are soldered to the commutator segments in a suitable manner.  Great care is taken in building the commutator because any short circuiting will cause the brushes to bounce, producing unacceptable sparking. The sparks may burn the brushes and overheat.
  • 37. 38 EE 313 Electric Machines Fall 2014 Construction of DC Machines (v) Brushes  The purpose of brushes is to ensure electrical connections between the rotating commutator and stationary external load circuit. The brushes are made of carbon and rest on the commutator.  The brush pressure is adjusted by means of adjustable springs. If the brush pressure is very large, the friction produces heating of the commutator and the brushes. On the other hand, if it is too weak, the imperfect contact with the commutator may produce sparking.  Multipole machines have as many brushes as they have poles. For example, a 4-pole machine has 4 brushes.  The successive brushes have positive and negative polarities. Brushes having the same polarity are connected together so that we have two terminals viz., the +ve terminal and the -ve terminal.
  • 38. 39 EE 313 Electric Machines Fall 2014 Armature winding
  • 39. 40 EE 313 Electric Machines Fall 2014 Commutator and Armature Winding Terminologies Commutator Pitch (YC)  The commutator pitch is the number of commutator segments spanned by each coil of the winding. It is denoted by YC.  In Fig.1 one side of the coil is connected to commutator segment 1 and the other side connected to commutator segment 2. Therefore, the number of commutator segments spanned by the coil is 1 i.e., YC = 1.  In Fig. 2, one side of the coil is connected to commutator segment 1 and the other side to commutator segment 8. Therefore, the number of commutator segments spanned by the coil = 8 - 1 = 7 segments i.e., YC = 7.  The commutator pitch of a winding is always a whole number. Since each coil has two ends and as two coil connections are joined at each commutator segment, Number of coils = Number of commutator segments  For example, if an armature has 30 conductors, the number of coils will be 30/2 = 15. Therefore, number of commutator segments is also 15. Note that commutator pitch is the most important factor in determining the type of d.c. armature winding.
  • 40. 41 EE 313 Electric Machines Fall 2014 Commutator and Armature Winding Terminologies Pole-Pitch It is the distance measured in terms of number of armature slots (or armature conductors) per pole. Thus if a 4-pole generator has 16 coils, then number of slots = 16. Coil Span or Coil Pitch (YS)  It is the distance measured in terms of the number of armature slots (or armature conductors) spanned by a coil.  Thus if the coil span is 9 slots, it means one side of the coil is in slot 1 and the other side in slot 10.
  • 41. 42 EE 313 Electric Machines Fall 2014 Commutator and Armature Winding Terminologies Full-Pitched Coil  If the coil-span is equal to pole pitch, it is called full-pitched coil.  In this case, the e.m.f.s in the coil sides are additive and have a phase difference of 0°. Therefore, e.m.f. induced in the coil is maximum.  If e.m.f. induced in one coil side is 2.5 V, then e.m.f. across the coil terminals = 2 x 2.5 = 5 V.
  • 42. 43 EE 313 Electric Machines Fall 2014 Commutator and Armature Winding Terminologies Fractional pitched coil  If the coil span or coil pitch is less than the pole pitch, then it is called fractional pitched coil.  In this case, the phase difference between the e.m.f.s in the two coil sides will not be zero so that the e.m.f. of the coil will be less compared to full-pitched coil.  Fractional pitch winding requires less copper but if the pitch is too small, an appreciable reduction in the generated e.m.f. results
  • 43. 44 EE 313 Electric Machines Fall 2014 Commutator and Armature Winding Terminologies Back Pitch (YB) (start to end)  It is the distance measured in terms of armature conductors between the two sides of a coil at the back of the armature .  For example, if a coil is formed by connecting conductor 1 (upper conductor in a slot) to conductor 12 (bottom conductor in another slot) at the back of the armature, then back pitch is YB = 12 - 1 = 11 conductors. Front Pitch (YF) (End of first and start of second coil)  It is the distance measured in terms of armature conductors between the coil sides attached to any one commutator segment .  For example, if coil side 12 and coil side 3 are connected to the same commutator segment, then front pitch is: YF = 12 - 3 = 9 conductors.
  • 44. 45 EE 313 Electric Machines Fall 2014 Front and Back Pitch
  • 45. 46 EE 313 Electric Machines Fall 2014 Commutator and Armature Winding Terminologies Resultant Pitch (YR)  It is the distance (measured in terms of armature conductors) between the beginning of one coil and the beginning of the next coil to which it is connected Therefore, YR = YF + YB
  • 46. 47 EE 313 Electric Machines Fall 2014 Commutator and Armature Winding Terminologies Progressive Winding A progressive winding is one in which, as one traces through the winding, the connections to the commutator will progress around the machine in the same direction as is being traced along the path of each individual coil. Note that YB > YF and YC = + 1. Retrogressive Winding A retrogressive winding is one in which, as one traces through the winding, the connections to the commutator will progress around the machine in the opposite direction to that which is being traced along the path of each individual coil. Note that YF > YB and YC = - 1. A retrogressive winding is seldom used because it requires more copper.
  • 47. 48 EE 313 Electric Machines Fall 2014 General Rules For D.C. Armature Windings 1) The back pitch (YB) as well as front pitch (YF) should be nearly equal to pole pitch YP. This will result in increased e.m.f. in the coils. 2) Both pitches (YB and YF) should be odd. This will permit all end connections (back as well as front connection) between a conductor at the top of a slot and one at the bottom of a slot. 3) The number of commutator segments is equal to the number of slots or coils (or half the number of conductors). No. of commutator segments = No. of slots = No. of coils It is because each coil has two ends and two coil connections are joined at each commutator segment 4) The winding must close upon itself i.e. it should be a closed circuit winding.
  • 48. 49 EE 313 Electric Machines Fall 2014 Simplex Lap Winding Relations between Pitches 2) Both YB and YF should be >= to pole pitch (YP). 3) Average pitch =(YB + YF) / 2. It nearly equals pole pitch (YP= Z/P). 4) Commutator pitch, YC = ± 1 YC = + 1 for progressive winding YC = - 1 for retrogressive winding 5) The resultant pitch (YR) is even, being the arithmetical sum of two odd numbers viz., YB and YF. 6) If Z = number of armature conductors and P = number of poles, then, It is clear that Z/P must be an even number to make the winding possible. 1) The back and front pitches are odd. They differ numerically by
  • 49. 50 EE 313 Electric Machines Fall 2014 Lap and Wave Winding Comparison Lap Winding Wave Winding Ys= S/P Ys= S/P Yc= +1 or -1 (Prog. Or Retrogr.) Yc= 2(C+1)/P or Yc= 2(C+1)/P Parallel paths a=P Parallel paths a=2 Coils=slots=comm. Segments Coils=slots=comm. Segments In general, one coil has tow conductors (active) . If you number of coils are 10 , there will be 20 conductors (Z)
  • 50. 51 EE 313 Electric Machines Fall 2014 Armature View and Developed Diagram
  • 51. 52 EE 313 Electric Machines Fall 2014 Coil Ends (front, back) Slot Number (4) Commt.Segments (+1 1,1’ 1,5 a, b 2,2’ 2,6 b, c 3,3’ 3,7 C, d 4,4’ 4,8 D,e 5,5’ 5,9 E,f 6,6’ 6,10 F,g 7,7’ 7,11 G,h 8,8’ 8,12 H,i 9,9’ 9,13 I,j 10,10’ 10,14 J,k 11,11’ 11,15 K,l 12,12’ 12,16 L,m 13,13’ 13,17(1) M,n 14,14’ 14,18(2) N,o 15,15’ 15,19(3) O,p 16,16’ 16,20(4) P,a
  • 52. 53 EE 313 Electric Machines Fall 2014 16 Coils, 4 Poles, Lap Winding
  • 53. 54 EE 313 Electric Machines Fall 2014
  • 54. 55 EE 313 Electric Machines Fall 2014 Lap Winding: Conclusions 1) The total number of brushes is equal to the number of poles. 2) The armature winding is divided into as many parallel paths as the number of poles. If the total number of armature conductors is Z and P is the number of poles, then, Number of parallel paths = P Number of series Conductors = Z / P In the present case, there are 32 armature conductors and 4 poles. Therefore, the armature winding has 4 parallel paths, each consisting of 8 conductors in series. 3) 4) Total armature current, The armature resistance can be found as under: E.M.F.generated = E.M.F.per parallel path = average e.m.f.per conductor × Z/P 1Resistance per parallel path No. of Parallel paths a l Z R P a P        Current per parallel pathaI P 
  • 55. 56 EE 313 Electric Machines Fall 2014 Simplex Wave Winding
  • 56. 57 EE 313 Electric Machines Fall 2014 Design of Simplex Wave Winding 1) Both pitches YB and YF are odd and are of the same sign. 2) 3) Both YB and YF are nearly equal to pole pitch(YP) and may be equal or differ by 2. If they differ by 2, they are one more and one less than YA. 4) Commutator pitch is given by; 5) Since YA must be a whole number, there is a restriction on the value of Z. With Z = 180, this winding is impossible for a 4-pole machine because YA is not a whole number. 6) 2( 1) c C y P  
  • 57. 58 EE 313 Electric Machines Fall 2014 Simple 4-Pole Wave wound DC Macnine
  • 58. 59 EE 313 Electric Machines Fall 2014 Slots 9, Poles 4, Wave Wingding (progressive) Coil Ends (front, back) Slot Number (2) Commt.Segments (+5 1,1’ 1,3 a, f 6,6’ 6,8 f, b 2,2’ 2,4 B,g 7,7’ 7,9 G,c 3,3’ 3,5 C,h 8,8’ 8,1 H,d 4,4’ 4,6 D,i 9,9’ 9,2 I,e 5,5’ 5,7 E,a
  • 59. 60 EE 313 Electric Machines Fall 2014 Slots 9, Poles 4, Wave Wingding (Developed Diagram)
  • 60. 61 EE 313 Electric Machines Fall 2014 Wave Winding: Conclusions 1) Only two brushes are necessary but as many brushes as there are poles may be used. 2) The armature winding is divided into two parallel paths irrespective of the number of poles. If the total number of armature conductors is Z and P is the number of poles, then, 3) 5) Total armature current, 6) The armature can be wave-wound if YA or YC is a whole number. 2 Current per parallel pathaI  
  • 61. 62 EE 313 Electric Machines Fall 2014 Applications of Lap and Wave Windings  In multipolar machines, for a given number of poles (P) and armature conductors (Z), a wave winding has a higher terminal voltage than a lap winding because it has more conductors in series.  On the other hand, the lap winding carries more current than a wave winding because it has more parallel paths.  In small machines, the current-carrying capacity of the armature conductors is not critical and in order to achieve suitable voltages, wave windings are used.  On the other hand, in large machines suitable voltages are easily obtained because of the availability of large number of armature conductors and the current carrying capacity is more critical. Hence in large machines, lap windings are used.
  • 62. 63 EE 313 Electric Machines Fall 2014 The Frog-Leg Winding or Self Equalizing Winding  It consists of lap and wave winding combined.  The wave winding functions as equalizer for lap winding.  The number of current paths in frog leg winding is: a = 2Pmlap Where P = No. Of poles on the machine Mlap = Plex of the lap winding
  • 63. 64 EE 313 Electric Machines Fall 2014 The Internal Generated Voltage and Induced Torque Equations of Real DC Machine A: Voltage in the rotor windings of a real dc machine The voltage out of the armature of a real dc machine is equal to the number of conductors per current path times the voltage on each conductor. The voltage in any single conductor: eind = e = vBl The voltage out of the armature of a real machine is thus: The velocity of each conductor in the rotor is: v=rw And with ɸ=BAP and A=2rl, and if there are P poles on the machine, then the total area A is A ZvBl E a  A Zr Bl E a   2 P A rl A P P    The total flux per pole in the machine: (2 ) 2 P B rl rlB BA P P      
  • 64. 65 EE 313 Electric Machines Fall 2014 The Internal Generated Voltage and Induced Torque Equations of Real DC Machine The total flux per pole in the machine: (2 ) 2 P B rl rlB BA P P       The internal generated voltage in the machine: 2 2 2 A A A Zr Bl E a ZP rlB a P ZP E a E K                     B: Torque induced in the armature of a real dc machine The torque in any single conductor cond = r Icond lB If there are ‘a’ current paths in the machine, so the current in a single conductor is given by A cond I I a 
  • 65. 66 EE 313 Electric Machines Fall 2014 The Internal Generated Voltage and Induced Torque Equations of Real DC Machine And the torque in a single conductor on the motor is: A cond rI lB a   For Z conductors, the total induced torque in a dc machine rotor is: The flux per pole in this machine: A ind ZrlBI a   (2 ) 2 P B rl rlB BA P P       So the total induced torque is: 2 ind A ZP I a     Finally, ind AK I 
  • 66. 67 EE 313 Electric Machines Fall 2014 Power flow and losses in DC machines Unfortunately, not all electrical power is converted to mechanical power by a motor and not all mechanical power is converted to electrical power by a generator… The efficiency of a DC machine is: 100%out in P P    100%in loss in P P P     or
  • 67. 68 EE 313 Electric Machines Fall 2014 The losses in DC machines There are five categories of losses occurring in DC machines. 1. Electrical or copper losses – the resistive losses in the armature and field windings of the machine. Armature loss: 2 A A AP I R Field loss: 2 F F FP I R Where IA and IF are armature and field currents and RA and RF are armature and field (winding) resistances usually measured at normal operating temperature.
  • 68. 69 EE 313 Electric Machines Fall 2014 The losses in DC machines 2. Brush (drop) losses – the power lost across the contact potential at the brushes of the machine. BD BD AP V I Where IA is the armature current and VBD is the brush voltage drop. The voltage drop across the set of brushes is approximately constant over a large range of armature currents and it is usually assumed to be about 2 V.
  • 69. 70 EE 313 Electric Machines Fall 2014 The losses in DC machines 4. Mechanical losses – losses associated with mechanical effects: friction (friction of the bearings) and windage (friction between the moving parts of the machine and the air inside the casing). These losses vary as the cube of rotation speed n3. 3. Core losses – hysteresis losses and eddy current losses. 5. Stray (Miscellaneous) losses – losses that cannot be classified in any of the previous categories. They are usually due to inaccuracies in modeling. For many machines, stray losses are assumed as 1% of full load.
  • 70. 71 EE 313 Electric Machines Fall 2014 The power-flow diagram On of the most convenient technique to account for power losses in a machine is the power-flow diagram. For a DC motor: Electrical power is input to the machine, and the electrical and brush losses must be subtracted. The remaining power is ideally converted from electrical to mechanical form at the point labeled as Pconv.
  • 71. 72 EE 313 Electric Machines Fall 2014 The power-flow diagram The electrical power that is converted is conv A AP E I And the resulting mechanical power is conv ind mP   After the power is converted to mechanical form, the stray losses, mechanical losses, and core losses are subtracted, and the remaining mechanical power is output to the load.
  • 72. 73 EE 313 Electric Machines Fall 2014 Simple DC Motor