DC & AC
Machinery
DC Generators
E = generated emf (Volt)
P = number of poles
N = speed of armature core notation
Z = total number of elements or conductors
 = flux per pole (Weber)
A = number of armature current paths
𝐙𝐏𝐍
𝐄=
𝟔𝟎𝐚
𝟖
𝟏𝐖𝐛 = 𝟏 × 𝟏𝟎 𝐌𝐚𝐱𝐰𝐞𝐥𝐥𝐬
For Lap Winding
𝐚 = 𝐦𝐏
For Wave Winding
𝐚 = 𝟐𝐦
Types of DC generators
Separately excited
𝐏𝐋
𝐈𝐋 =
𝐕𝐋
𝐈𝐚 = 𝐈𝐋
Self-excited
 Shunt generator
𝐏𝐠 = 𝐄𝐈𝐚
𝐄 = 𝐕𝐋 + 𝐈𝐚 𝐑 𝐚
𝐕𝐋
𝐈𝐬𝐡 =
𝐑 𝐬𝐡
𝐏𝐋
𝐈𝐋 =
𝐕𝐋
 Series generator
𝐈𝐚 = 𝐈𝐋 + 𝐈𝐬𝐡
𝐏𝐠 = 𝐄𝐈𝐚
𝐄 = 𝐕𝐋 + 𝐈𝐚 𝐑 𝐚
𝐈𝐚 = 𝐈𝐋 = 𝐈𝐬𝐞
𝐄 = 𝐕𝐋 + 𝐈𝐚 𝐑 𝐚
𝐏𝐠 = 𝐄𝐈𝐚
 Long-shunt compound generator
𝐈𝐬𝐞 = 𝐈𝐚
𝐏𝐋
𝐈𝐋 =
𝐕𝐋
𝐈𝐬𝐡
𝐕𝐋
=
𝐑 𝐬𝐡
𝐈𝐚 = 𝐈𝐋 + 𝐈𝐬𝐡
𝐏𝐠 = 𝐄𝐈𝐚
𝐄 = 𝐕𝐋 + 𝐈𝐚 (𝐑 𝐚 + 𝐑 𝐬𝐞 )
 Short-shunt compound generator
𝐈𝐬𝐞 = 𝐈𝐋
𝐏𝐋
𝐈𝐋 =
𝐕𝐋
𝐈𝐬𝐡
𝐕𝐋 + 𝐈𝐬𝐞 𝐑 𝐬𝐞
=
𝐑 𝐬𝐡
𝐈𝐚 = 𝐈𝐋 + 𝐈𝐬𝐡
𝐏𝐠 = 𝐄𝐈𝐚
𝐄 = 𝐕𝐋 + 𝐈𝐚 𝐑 𝐚
+ 𝐈𝐬𝐞 𝐑 𝐬𝐞
DC Motors
N = speed of armature core notation
a = number of armature current paths
P = number of poles
Z = total number of elements or conductors
 = flux per pole (Weber)
E = back emf or conductor emf
𝟔𝟎𝐚𝐄𝐛
𝐍=
𝐏𝐙
Mechanical Power Output
𝐏𝐝 = mechanical power output developed (hp or W)
N = speed of the armature rotation (RPM)
T = torque developed (lb.ft or N.m)
𝟐𝐍𝐓𝐥𝐛.𝐟𝐭
𝐏𝐝 (𝐇𝐏) =
𝟑𝟑, 𝟎𝟎𝟎
𝟐𝐍𝐓𝐍.𝐦
𝐏𝐝 =
𝟔𝟎
1 HP = 746 W
Torque Developed in the Armature
T = torque developed (N.m)
a = number of armature current paths
𝐈𝐚 = armature current P = number of poles
N = speed of armature rotation (rpm)
Z = number of conductors
𝐏𝐝 = power developed in the armature (W)
ɸ = flux per pole (Wb)
𝐄𝐛 = back emf or counter emf
𝐏𝐝
𝐏𝐙 𝐈𝐚
𝐍
𝟔.𝟐𝟖𝟑𝐚
𝐓 = 𝟗. 𝟓𝟓 ( ) or 𝐓 =
𝐏𝐝 = 𝐄𝐛 𝐈𝐚
Types of DC Motor
Series Motor
𝐈𝐚 = 𝐈𝐬𝐞 = 𝐈𝐦
𝐄𝐛 = 𝐕𝐬 − 𝐈𝐚 (𝐑 𝐚 + 𝐑 𝐬𝐞 )
Shunt Motor
𝐈𝐬𝐡
𝐕𝐬
=
𝐑 𝐬𝐡
Long-shunt Compound Motor
𝐈𝐦 = 𝐈𝐚 + 𝐈𝐬𝐡
𝐄𝐛 = 𝐕𝐬 − 𝐈𝐚 𝐑 𝐚
𝐈𝐬𝐞 = 𝐈𝐚
𝐈𝐬𝐡
𝐈𝐦 = 𝐈𝐚 + 𝐈𝐬𝐡
𝐕𝐬
=
𝐑 𝐬𝐡
𝐄𝐛 = 𝐕𝐚 − 𝐈𝐚 (𝐑 𝐚 + 𝐑 𝐬𝐞 )
Short-shunt Compound Motor
𝐈𝐬𝐞 = 𝐈𝐦
𝐈𝐦 = 𝐈𝐚 + 𝐈𝐬𝐡
𝐕𝐬 − 𝐈𝐬𝐞 𝐑 𝐬𝐞
𝐈𝐬𝐡 =
𝐑 𝐬𝐡
𝐄𝐛 = 𝐕𝐚 − 𝐈𝐚 𝐑 𝐚
− 𝐈𝐬𝐞 𝐑 𝐬𝐞 )
AC Generators
Generated EMF per phase
𝐄𝐩 = generated emf per phase (Volt)
𝐤 𝐩 = pitch factor or coil span factor
𝐤 𝐝 = distribution factor or belt factor
f = frequency (Hertz)
ɸ = flux per pole (Weber)
Z = number of conductors in series per phase
n = number of slots per pole per phase
𝐄𝐩 = 𝟐. 𝟐𝟐𝐤 𝐩 𝐤 𝐝 𝐟 𝐙
𝒌𝒑 = 𝒔𝒊𝒏 [𝟗𝟎° × 𝒑𝒊𝒕𝒄𝒉
𝒑𝒊𝒕𝒄𝒉
𝒄𝒐𝒊𝒍 𝒔𝒑𝒂𝒏 𝒊𝒏 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒔𝒍𝒐𝒕𝒔
=
𝒔𝒍𝒐𝒕𝒔 𝒑𝒆𝒓 𝒑𝒐𝒍𝒆
𝒏𝜹
𝒔𝒊𝒏 ( )
𝟐
𝒌𝒅 =
𝜹
𝒏 𝒔𝒊𝒏 ( )
𝟐
𝑷𝑵
𝒇=
𝟏𝟐𝟎
Generated EMF per phase
(Electric Circuit Relationship)
𝐄𝐩 = generated emf per phase (Volt)
𝐕𝐩 = load voltage per phase
𝐑 𝐚 = armature winding resistance per phase
𝐗 𝐋𝐚 = inductive reactance of the armature
winding per phase
𝐈𝐩 = current per phase
α = angle between 𝐄𝐩 and 𝐕𝐩
𝛉 = angle between 𝐈𝐩 and 𝐕𝐩
±
±
𝐄𝐩  𝐚 = 𝐕𝐩 𝟎° + 𝐈𝐩  𝛉(𝐑 𝐚 + 𝐣𝐗 𝐋𝐚 )
𝐄𝐩
𝟐
𝟐
= √[(𝐕𝐩 𝐜𝐨𝐬 𝛉 + 𝐈𝐩 𝐑 𝐚 ) + (𝐕𝐩 𝐬𝐢𝐧 𝛉 + 𝐈𝐩 𝐗 𝐋𝐚 ) ]
Induction Motors
Slip in an Induction Motor
s = slip
𝐍𝐬𝐥𝐢𝐩 = slip speed
𝐍𝐒 = synchronous speed
𝐍𝐑 = rotor speed
f = frequency (electrical)
𝐟𝐫 = rotor frequency
P = number of poles
𝐍𝐬𝐥𝐢𝐩 𝐍𝐒 − 𝐍𝐑
𝐬=
=
𝐍𝐬
𝐍𝐒
𝐍𝐬𝐥𝐢𝐩 = 𝐍𝐒 − 𝐍𝐑
𝟏𝟐𝟎𝐟
𝐍𝐒 =
𝐏
𝐟𝐫 = 𝐬𝐟
Power Stages in an Induction Motor
𝐏𝐨𝐮𝐭 = output power (hp or W)
𝐍𝐑 = rotor speed (rpm)
T = torque developed (lb.ft or N.m)
𝐏𝐨𝐮𝐭 (𝐇𝐏) =
𝟐𝛑𝐍𝐑 𝐓𝐥𝐛.𝐟𝐭
𝟑𝟑,𝟎𝟎𝟎
or 𝐏𝐨𝐮𝐭 (𝐖) =
𝟐𝛑𝐍𝐑 𝐓𝐍.𝐦
𝟔𝟎
𝐑𝐂𝐋 = 𝐬(𝐑𝐏𝐈)
Ideal Transformer
Equivalent Circuit of an Ideal Transformer