Symmetrical Induction Machines

SYMMETRICAL INDUCTION MACHINES Introduction The ind. m/c is used in a wide variety of applications as a means of converting electric power to mech. work. It is the workhorse of the electric power industry. The classical techniques shall be used to establish the voltage and torque equations for a symmetrical ind. m/c expressed in terms of m/c variables. Next, the transformation to the arbitrary reference frame is then modified to accommodate rotating axes. Once this groundwork has been laid, the m/c voltage eqns are written in the arbitrary reference frame, entirely without a laborious exercise in trigonometry when starting from the substitution of the equations of transformations into the voltage equations expressed in m/c variables. The eqns may then be expressed in any reference frame by appropriate assignment of the reference frame speed in the arbitrary reference-frame voltage eqns rather than performing each transformation individually. Although the stationary reference frame, the reference frame fixed in the rotor, and the synchronously rotating reference frame are most frequently used, the arbitrary reference frame offers a direct means of obtaining the voltage eqns in these and all other reference frames. Figure 4.1: Two-pole, 3-phase, wye-connected symmetrical induction machine The voltage eqns in m/c variables may be expressed as: 𝑉 𝑎 𝑏 𝑐 𝑠 = 𝑅 𝑠 𝑖 𝑎 𝑏 𝑐 𝑠 + 𝑝 𝜆 𝑎 𝑏 𝑐 𝑠 (4.1) V abcs ​ =R s ​ i abcs ​ +pλ abcs ​ (4.1) 𝑉 𝑎 𝑏 𝑐 𝑟 = 𝑅 𝑟 𝑖 𝑎 𝑏 𝑐 𝑟 + 𝑝 𝜆 𝑎 𝑏 𝑐 𝑟 (4.2) V abcr ​ =R r ​ i abcr ​ +pλ abcr ​ (4.2) Where: ( 𝑓 𝑎 𝑏 𝑐 𝑠 ) 𝑇 = [ 𝑓 𝑎 𝑠 𝑓 𝑏 𝑠 𝑓 𝑐 𝑠 ] (4.3) (f abcs ​ ) T =[f as ​ f bs ​ f cs ​ ](4.3) ( 𝑓 𝑎 𝑏 𝑐 𝑟 ) 𝑇 = [ 𝑓 𝑎 𝑟 𝑓 𝑏 𝑟 𝑓 𝑐 𝑟 ] (4.4) (f abcr ​ ) T =[f ar ​ f br ​ f cr ​ ](4.4) Both 𝑅 𝑠 R s ​ and 𝑅 𝑟 R r ​ are diagonal matrices, each with equal nonzero elements. For a magnetically linear system, the flux linkages may be expressed as: [ 𝜆 𝑎 𝑏 𝑐 𝑠 𝜆 𝑎 𝑏 𝑐 𝑟 ] = [ 𝐿 𝑠 𝐿 𝑠 𝑟 ( 𝐿 𝑠 𝑟 ) 𝑇 𝐿 𝑟 ] [ 𝑖 𝑎 𝑏 𝑐 𝑠 𝑖 𝑎 𝑏 𝑐 𝑟 ] (4.5) [ λ abcs ​ λ abcr ​ ​ ]=[ L s ​ (L sr ​ ) T ​ L sr ​ L r ​ ​ ][ i abcs ​ i abcr ​ ​ ](4.5) In particular, 𝐿 𝑠 = [ 𝐿 𝑠 + 𝐿 𝑚 𝑠 − 1 2 𝐿 𝑚 𝑠 − 1 2 𝐿 𝑚 𝑠 − 1 2 𝐿 𝑚 𝑠 𝐿 𝑠 + 𝐿 𝑚 𝑠 − 1 2 𝐿 𝑚 𝑠 − 1 2 𝐿 𝑚 𝑠 − 1 2 𝐿 𝑚 𝑠 𝐿 𝑠 + 𝐿 𝑚 𝑠 ] (4.6) L s ​ = ​ L s ​ +L ms ​ − 2 1 ​ L ms ​ − 2 1 ​ L ms ​ ​ − 2 1 ​ L ms ​ L s ​ +L ms ​ − 2 1 ​ L ms ​ ​ − 2 1 ​ L ms ​ − 2 1 ​ L ms ​ L s ​ +L ms ​ ​ ​ (4.6) 𝐿 𝑟 = [ 𝐿 𝑟 + 𝐿 𝑚 𝑟 − 1 2 𝐿 𝑚 𝑟 − 1 2 𝐿 𝑚 𝑟 − 1 2 𝐿 𝑚 𝑟 𝐿 𝑟 + 𝐿 𝑚 𝑟 − 1 2 𝐿 𝑚 𝑟 − 1 2 𝐿 𝑚 𝑟 − 1 2 𝐿 𝑚 𝑟 𝐿 𝑟 + 𝐿 𝑚 𝑟 ] (4.7) L r ​ = ​ L r ​ +L mr ​ − 2 1 ​ L mr ​ − 2 1 ​ L mr ​ ​ − 2 1 ​ L mr ​ L r ​ +L mr ​ − 2 1 ​ L mr ​ ​ − 2 1 ​ L mr ​ − 2 1 ​ L mr ​ L r ​ +L mr ​ ​ ​ (4.7) Where: 𝐿 𝑠 L s ​ and 𝐿 𝑚 𝑠 L ms ​ are, respectively, the leakage and magnetizing inductances of the stator windings; 𝐿 𝑟 L r ​ and 𝐿 𝑚 𝑟 L mr ​ are for the rotor windings. The mutual inductance 𝐿 𝑠 𝑟 L sr ​ is the amplitude of the mutual inductances between stator and rotor windings: 𝐿 𝑠 𝑟 = 𝐿 𝑠 𝑟 [ cos ⁡ 𝜃 𝑟 cos ⁡ ( 𝜃 𝑟 + 2 𝜋 3 ) cos ⁡ ( 𝜃 𝑟 − 2 𝜋 3 ) cos ⁡ ( 𝜃 𝑟 − 2 𝜋 3 ) cos ⁡ 𝜃 𝑟 cos ⁡ ( 𝜃 𝑟 + 2 𝜋 3 ) cos ⁡ ( 𝜃 𝑟 + 2 𝜋 3 ) cos ⁡ ( 𝜃 𝑟 − 2 𝜋 3 ) cos ⁡ 𝜃 𝑟 ] (4.8) L sr ​ =L sr ​ ​ cosθ r ​ cos(θ r ​ − 3 2π ​ ) cos(θ r ​ + 3 2π ​ ) ​ cos(θ r ​ + 3 2π ​ ) cosθ r ​ cos(θ r ​ − 3 2π ​ ) ​ cos(θ r ​ − 3 2π ​ ) cos(θ r ​ + 3 2π ​ ) cosθ r ​ ​ ​ (4.8) In this analysis it is assumed that all effects upon the amplitude of the fundamental components of the MMF wave from due to skewing and uniformly distributed rotor windings are accounted for in the value of 𝐿 𝑠 𝑟 L sr ​ . When expressing the voltage eqns in m/c variable form, it is convenient to refer all rotor variables to the stator winding by appropriate turns ratio.