Current Linkage of the Commutator Winding and Armature Reaction

The curved function of the current linkage created by the commutator winding is com-puted in the way illustrated in Figure 2.9 for a three-phase winding. When defining the slot current Iu, the current of the short-circuited coils can be set to zero. In short-pitched coils, there may be currents flowing in opposite directions in the different coil sides of a single slot. If zb is the number of brushes, the armature current Ia is divided into brush currents I = Ia/(zb/2). Each brush current in turn is divided into two paths as conductor currents Is =I/2=Ia/2a, wherea is the total number of path pairs of the winding. In a slot, there arez_Qconductors, and thus the sum current of a slot is

I_u=z_QI_s= z Q

Ia

2a, (2.117)

wherezis the number of conductors in the complete winding. All the pole pairs of the armature are alike, and therefore it suffices to investigate only one of them, namely a two-pole winding, Figure 2.46. The curved function of the magnetic voltage therefore follows the illustration in Figure 2.53.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1

Θ Θ_ma

slots

2

ma a Θ =Θ

S N

Figure 2.53 Current linkage curve of the winding of Figure 2.46, when the commutation takes place in the coils in slots 5 and 13

The number of brushes in a commutator machine normally equals the number of poles. The number of slots between the brushes is

q= Q

2p. (2.118)

This corresponds to the number of slots per pole and phase of AC windings. The effective number of slots per pole and phase is always somewhat lower, because a part of the coils is always short-circuited. The distribution factor for an armature windingkdais obtained from Equation (2.33). For a fundamental, andm=1, it is rewritten in the form

kda1= 2p Qsin pπ

Q

. (2.119)

Armature coils are often short pitched, and the pitch factork_pais thus obtained from Equa-tion (2.32). The fundamental winding factor of a commutator winding is thus

kwa1=kda1kpa1≈ 2

π. (2.120)

When the number of slots per pole increases,kda1approaches the limit 2/π. This is the ratio of the voltage circle (polygon) diameter to half of the circle perimeter. In ordinary machines, the ratio of short pitching isW/τp>0.8, and thereforek_pal>0.95. As a result, the approxi-mate valuek_wa1=2/πis an adequate starting point in the initial manual computation. More thorough investigations have to be based on an analysis of the curved function of the current linkage. In that case, the winding has to be observed in different positions of the brushes.

Figure 2.45 shows that at the right side of the quadrature axis q, the direction of each slot current is towards the observer, and on the left, away from the observer. In other words, the rotor becomes an electromagnet with its north pole at the bottom and its south pole at the top.

The pole pair current linkage of the rotor is Θma=q Iu= Q

2p z Q

I_a 2a = z

4apIa=NaIa. (2.121) The term

Na= z

4ap (2.122)

in the equation is the number of coil turns per pole pair in a commutator armature in one parallel path, that is turns connected in series, becausez/2 is the number of all armature turns;

z/2(2a) is the number of turns in one parallel path, in other words connected in series, and finallyz/2(2a)pis the number of turns per pole pair. The current linkage calculated according to Equation (2.121) is slightly higher than in reality, because the number of slots per pole and phase includes also the slots with short-circuited coil sides. In calculation, we may employ the linear current density

Aa= Q I_u πD = 2p

πDNaIa= N_aI_a τp

. (2.123)

The current linkage of the linear current density is divided into magnetic voltages of the air gaps, the peak value of which is

Θˆδa=

τp

2

0

A_adx= 1

2A_aτp=1 2

z 2p

Ia

2a = NaIa

2 =Θma

2 , (2.124)

Θδa=Θma/2= 1

2A_aτp= 1 2

z 2p

Ia

2a = 1

2N_aI_a= Θma

2 =Θδa. (2.125) In the diagram, the peak value ˆΘδais located at the brushes (in the middle of the poles), the value varying linearly between the brushes, as illustrated with the dashed line in Figure 2.53.

Θˆδa is the armature reaction acting along the quadrature axis under one tip of a pole shoe, and it is the current linkage to be compensated. The armature current linkage also creates commutation problems, which means that the brushes have to be shifted from the q-axis by an angleεto a new position as shown in Figure 2.54.

This figure also gives the positive directions of the currentI and the respective current linkage. The current linkage can be divided into two components:

Θmd= Θma

2 sinε=Θδasinε, (2.126)

Θmq= Θma

2 cosε=Θδacosε. (2.127)

The former is called a direct component and the latter a quadrature component. The direct component magnetizes the machine either parallel or in the opposite direction to the actual field winding of the main poles of the machine. There is a demagnetizing effect if the brushes

q d

Θmd

ε I

I Θ Θ

ma

mq

Figure 2.54 Current linkage of a commutator armature and its components. To ensure better commu-tation, the brushes are not placed on the q-axis

are shifted in the direction of rotation in generator mode, or in the opposite direction of rota-tion in motoring operarota-tion; the magnetizing effect, on the contrary, is in generating operarota-tion opposite the direction of rotation, and in motoring mode in the direction of rotation. The quadrature component distorts the magnetic field of the main poles, but neither magnetizes nor demagnetizes it. This is not a phenomenon restricted to commutator machines – the reac-tion is in fact present in all rotating machines.