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PHYSICAL STRUCTURE9
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PE 9211 Analysis of Electrical Machines
Dynamic Characteristics of Permanent Magnet DC Motor
Modes of Dynamic operation
1. Starting from stall
2. Changes in load torque
Condition: The machine supplied from a
constant – voltage source
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Mathematical Model of a PMDC Motor: 9
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This motor consists of two first order differential equation and two
algebraic equation
Armature current equation,
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Speed equation,
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Simulink Model of PMDC Motor
Motor Parameters
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Solving armature current equation
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Solving Speed equation
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Dynamic performance during starting
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Dynamic Characteristics of DC Shunt Motor
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Simulink Model of DC Shunt Motor:
Fig shows the Simulink model of DC Shunt Motor. It is constructed using
subsystems for solving each differential equations (i.e.) armature
current, field current and torque equation.
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Time domain block diagrams and state equations
Shunt connected dc machine
W.K.T
𝒗𝒂 = 𝒊𝒂𝒓𝒂 + 𝑳𝑨𝑨 𝒅𝒊𝒂𝒅𝒕
+ 𝑳𝑨𝑭𝝎𝒓𝒊𝒇 − − −− 𝟏
𝒗𝒇 = 𝒊𝒇𝑹𝒇 + 𝑳𝑭𝑭 𝒅𝒊𝒇
𝒅𝒕 − − −− 𝟐
𝑻𝒆 = 𝑻𝑳 + 𝑱 𝒅𝝎𝒓
𝒅𝒕 + 𝑩𝒎𝝎𝒓 − −− − 𝟑
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Equations (1),(2) and (3) can be written in terms of its time constants
𝒗𝒂 = 𝒓𝒂 𝟏 + 𝑳𝑨𝑨
𝒓𝒂
𝒅
𝒅𝒕 𝒊𝒂 + 𝑳𝑨𝑭𝝎𝒓𝒊𝒇
𝒗𝒂 = 𝒓𝒂 𝟏 + 𝝉𝒂 𝝆 𝒊𝒂 + 𝑳𝑨𝑭𝝎𝒓𝒊𝒇−−−−−− 𝟒
𝑯𝒆𝒓𝒆, 𝝆 ⟶𝒅
𝒅𝒕
𝒗𝒇 = 𝑹𝒇 𝟏 + 𝑳𝑭𝑭
𝑹𝒇
𝝆 𝒊𝒇
𝒗𝒇 = 𝑹𝒇 𝟏 + 𝝉𝒇 𝝆 𝒊𝒇−−−−−−−(5)
𝑻𝒆 − 𝑻𝑳 = ( 𝑩𝒎 + 𝑱 𝝆) 𝝎𝒓 −− − − 𝟔
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𝝉𝒂 ⟶ Armature time constant
𝝉𝒇 ⟶ Field time constant
𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝒕𝒉𝒆 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔 𝟒 , 𝟓 𝒂𝒏𝒅 𝟔 𝒇𝒐𝒓 𝒊𝒂 ,
𝒊𝒇, 𝒂𝒏𝒅 𝝎𝒓 𝒚𝒊𝒆𝒍𝒅𝒔
𝒊𝒂 =
𝟏𝒓𝒂
𝝉𝒂𝝆 + 𝟏 𝒗𝒂 − 𝑳𝑨𝑭𝝎𝒓𝒊𝒇 − − −− 𝟕
𝒊𝒇 =
𝟏𝑹𝒇
𝝉𝒇𝝆 + 𝟏 𝒗𝒇 − −− − 𝟖
𝝎𝒓 =𝟏
𝑱𝝆 + 𝑩𝒎 𝑻𝒆 − 𝑻𝑳 −− − − 𝟗
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Time domain block diagram of a shunt connected dc machine
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𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝒕𝒉𝒆 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔 𝟏 , 𝟐 𝒂𝒏𝒅 𝟑 𝒇𝒐𝒓 𝒅𝒊𝒂
𝒅𝒕, 𝒅𝒊𝒇
𝒅𝒕
𝒂𝒏𝒅 𝒅𝝎𝒓
𝒅𝒕 𝒚𝒊𝒆𝒍𝒅𝒔
From (1)
𝒅𝒊𝒂𝒅𝒕
= −𝒓𝒂
𝑳𝑨𝑨
𝒊𝒂 − 𝑳𝑨𝑭
𝑳𝑨𝑨
𝒊𝒇𝝎𝒓 + 𝟏
𝑳𝑨𝑨
𝒗𝒂— 𝟏𝟎
From (2)
𝒅𝒊𝒇
𝒅𝒕= −
𝑹𝒇
𝑳𝑭𝑭
𝒊𝒂 + 𝟏
𝑳𝑭𝑭
𝒗𝒇— 𝟏𝟏
From (3)
𝒅𝝎𝒓
𝒅𝒕= −
𝑩𝒎
𝑱𝝎𝒓 +
𝑳𝑨𝑭
𝑱𝒊𝒇𝒊𝒂 −
𝟏
𝑱𝑻𝑳 − −(𝟏𝟐)
State equation of shunt dc machine
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𝜌
𝒊𝒇𝒊𝒂𝝎𝒓
=
−𝑹𝒇
𝑳𝑭𝑭𝟎 𝟎
𝟎−𝒓𝒂
𝑳𝑨𝑨𝟎
𝟎 𝟎−𝑩𝒎
𝑱
𝒊𝒇𝒊𝒂𝝎𝒓
+
𝟎−𝑳𝑨𝑭𝝎𝒓
𝑳𝑨𝑨
𝑳𝑨𝑭𝒊𝒇𝒊𝒂
𝑱
+
𝟏
𝑳𝑭𝑭𝟎 𝟎
𝟎𝟏
𝑳𝑨𝑨𝟎
𝟎 𝟎−𝟏
𝑱
𝒗𝒇
𝒗𝒂
𝑻𝑳
State equations in matrix form or vector matrix form
Note: The second term on the right side contains the product of state
variables causing the system to be nonlinear.
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Permanent Magnet dc Machine
𝒗𝒇 𝒊𝒔 𝒆𝒍𝒊𝒎𝒊𝒏𝒂𝒕𝒆𝒅
𝑳𝑨𝑭𝒊𝒇 𝒊𝒔 𝒓𝒆𝒑𝒍𝒂𝒄𝒆𝒅 𝒃𝒚 𝒌𝒗
𝒌𝒗 𝒊𝒔 𝒅𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒆𝒅 𝒃𝒚
Strength of the magnetReluctance of the ironNo. of turns in the armature winding
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W.K.T
Above eqns. (1) and (2) can be written in terms of its time constants
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29Time domain block diagram of a permanent magnet DC machine
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State Equation of a permanent magnet DC machine
From (1)
From (2)
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The form in which the state equations are expressed in above eqn.
is called the fundamental form.
OR
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Advantages to using the state space representation compared with other methods.
1.The ability to easily handle systems with multiple inputs and outputs;
2.The system model includes the internal state variables as well as the output variable;
3.The model directly provides a time-domain solution, the matrix/vector modeling is very efficient from a computational standpoint for computer implementation
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404349
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