Lecture 8 - EE743Direct Current (DC)Machines - Part IIProfessor: Ali Keyhani

DC Machines Shunt-connected DC Machine

DC Machines The dynamic equations (assuming rfext=0) are:Where Lff = field self-inductance Lla= armature leakage inductance Laf = mutual inductance between the field and rotating armature coils ea = induced voltage in the armature coils (also called counter or back emf )

DC Machines

DC Machines - Shunt DC Machine Time-domain block diagram The machine equations are solved for:

DC Machines - Shunt DC Machine Time domain block diagramG1G3G2LafXXiaififia+

-+

-VaVfVf

DC Machines - Shunt DC Machine State-space equationsLetRe-writing the dynamic equations,;

DC Machines - Permanent Magnet The field flux in the Permanent Magnet machines is produced by a permanent magnet located on the stator. Therefore,

Lsfif is a constant determined by the strength of the magnet, the reluctance of the iron, and the number of turns of the armature winding.

DC Machines - Permanent Magnet Dynamic equations of a Permanent Magnet Machine

DC Machines - Permanent Magnet Dynamic equations,

DC Machines - Permanent Magnet Time domain block diagram The equations are solved by,

DC Machines - Permanent Magnet Time domain block diagram+ - +

-Va G1 G2KvKviaeaKvrrTLTe

DC Machines - Permanent Magnet State-space equationsre-writing the equations as function of states,

DC Machines - Permanent Magnet In a matrix form,

DC Machines - Permanent Magnet Transfer Function,

Let

DC Machines - Permanent Magnet The, we will have

Re-arranging the equation,

DC Machines - Permanent Magnet In a matrix representation,

DC Machines - Permanent Magnet Solving for ia

DC Machines - Permanent Magnet Let m be,

The equation is then reduced to,

DC Machines - Permanent Magnet

DC Machines - Permanent Magnet

DC Machines - Permanent Magnet The characteristic equation (or force-free equation) of the system is as shown below,

DC Machines - Permanent Magnet If < 1 , the roots are a conjugate complex pair, and the natural response consists of an exponentially decaying sinusoids.If > 1, the roots are real and the natural response consists of two exponential terms with negative real exponents.