Theory of Electrical Machines Part I James Cale Guest Lecturer EE - PowerPoint PPT Presentation

Theory of Electrical Machines – Part I James Cale – Guest Lecturer EE 566, Fall Semester 2014 Colorado State University

Module Objective Starting from basic principles of physics, develop an understanding of the fundamental mechanics of electro-mechanical motion devices, with applications to common electrical machines such as induction and synchronous machines.

Why Electrical Machines? In the case of wind power, the prime mover and source of energy is mechanical (wind, resulting in shaft rotation). Before we can condition the power using power electronics and put it on the electrical grid, this energy must be converted first to electrical energy.

Physics Background Static Charges Static electrical charges give rise to electrostatic fields —but they don’t give rise to magnetic fields. Charges in Motion Electrons in motion (i.e., current) always give rise to magnetic fields, with an orientation specified by the “right hand rule.”

Physics Background Physical Relationship Between Current 𝑗 and Magnetic Field : 𝐼 𝐼 𝑗 𝑗 𝐼 Notation: a “ · ” denotes current coming out of the paper; an “ x ” denotes current going into the paper.

Physics Background Relationship Between Magnetic Flux 𝑁 Density , Field and Magnetization : 𝐶 𝐶 = 𝜈 0 𝐼 + 𝑁 This expression shows both the free-space field component and “bound charge” magnetization component (we’ll discuss this in a moment). Flux can also expressed as: 𝐶 = 𝜈𝐼 = 𝜈 0 𝜈 𝑠 𝐼

Physics Background 𝐶 = 𝜈𝐼 = 𝜈 0 𝜈 𝑠 𝐼 𝜈 0 : Permeability of free space 𝜈 𝑠 : Relative permeability (specific to material) 𝜈 : Permeability 𝐶 − 𝐼 What does a real characteristic look like? Refer to hand-out or obtain online (figure 7): Cale, J; Sudhoff, S; Turner, J, “An Improved Magnetic Characterization Method for Highly Permeable Materials,” Magnetics, IEEE Transactions on, vol. 42, no. 8, Aug. 2006, pg(s): 1974-1981.

Physics Background How do you obtain ? 𝜈 𝑠 Hint: the common method using IEEE 393-191 may not be accurate enough for some applications. Refer to hand-out or obtain online: Cale, J; Sudhoff, S; Turner, J, “ An Improved Magnetic Characterization Method for Highly Permeable Materials,” Magnetics, IEEE Transactions on, vol. 42, no. 8, Aug. 2006, pg(s): 1974-1981.

Physics Background What is magnetization? Ferrimagnetic Material Edge of magnetic material Magnetic “domain wall” Field from “bound charge” Random ordering of domain walls — no collective flux from magnetization outside material.

Physics Background 𝐼 Expansion of domain wall And orientation of domains in direction of externally applied field — until magnetic saturation. There is now a net magnetization outside of material, aligned with externally applied field.

Magnetic Flux Calculating Magnetic Flux Φ = 𝐶 ∙ 𝑒𝐵 Assuming the flux density is constant and everywhere orthogonal to the cross section of the material: 𝑒𝐵 𝐶 Φ = 𝐶𝐵

Physics Background Lenz’ Law: a time varying magnetic flux through a conductor loop induces a voltage which opposes the change in flux. Φ 𝑤 = − 𝑒Φ 𝑒𝑢 𝑤

Magnetic Equivalent Circuits Consider the simple, stationary device below: Φ 𝑗 + 𝑕 𝑂 𝑤 − 𝑂: number of turns of the winding 𝑕: air gap length

Magnetic Equivalent Circuits From Ampere’s Law 𝑂𝑗 = 𝐼 ∙ 𝑒𝑚 Assuming the field is constant throughout the cross section of the magnetic material and air gap: 𝑂𝑗 = 𝐼 𝑛 𝑚 𝑛 + 𝐼 𝑕 𝑕 𝐶 𝐶 = 𝜈 0 𝜈 𝑠 𝑚 𝑛 + 𝜈 0 𝑕 where 𝑚 𝑛 is the length of the magnetic material.

Magnetic Equivalent Circuits Now, substituting for magnetic flux and rearranging: 𝑚 𝑛 𝑕 𝑂𝑗 = 𝜈 0 𝜈 𝑠 𝐵 Φ + 𝜈 0 𝐵 Φ This is similar in form to 𝑤 = 𝑠𝑗 , where 𝑁𝑁𝐺 ≡ 𝑂𝑗 ~ 𝑤 𝑚 ℛ ≡ 𝜈𝐵 ~ 𝑠 “reluctance” to magnetic flux Note: in air, reluctance is high, in Φ ~ 𝑗 a highly permeable material, reluctance is small.

Magnetic Equivalent Circuits Our MEC analog is the following: Φ ℛ 𝑕 𝑂𝑗 ℛ 𝑛 Note: this was for 2-D. How do you obtain for 3-D fields? Refer to hand-out or obtain online: J. Cale, S. Sudhoff, and Li-Quan Tan, “ Accurately Modeling EI Core Inductors using a High-Fidelity Magnetic Equivalent Circuit Approach," Magnetics, IEEE Transactions on, vol. 42, no. 1, Jan 06, pg(s): 40-46.

Flux Linkage & Inductance Flux Linkage : 𝜇 = 𝑂Φ Inductance: = 𝑂 2 𝑀 = 𝜇 𝑗 ℛ Inductance is a measure of flux per current linking a winding — it is large when reluctance is small (e.g., in a highly permeable material).

Reluctance Machine Φ 𝑗 𝜄 + 𝑤 − Now, reluctance (and inductance) are time-varying. ℛ 𝜄 = ℛ 1 + ℛ 2 sin 𝜄 Torque is produced to minimize reluctance to flux!

Rotating MMFs Sinusoidal Winding Distributions Phase a stator winding Air 𝜚 𝑡 Iron

Rotating MMFs Now, the MMF for the a winding is (you can see this from right hand rule, e.g., flux is max at 𝜚 𝑡 = 0,−𝜌 ): 𝑏𝑡 = 𝑂 𝑡 𝑁𝑁𝐺 2 𝑗 𝑏𝑡 cos 𝜚 𝑡 If there are sinusoidal windings for the b and c phases, displaced physically by 120 degrees then: 𝑁𝑁𝐺 𝑐𝑡 = 𝑂 𝑡 2 𝑗 𝑐𝑡 cos 𝜚 𝑡 − 2𝜌 3 𝑑𝑡 = 𝑂 𝑡 2 𝑗 𝑑𝑡 cos 𝜚 𝑡 + 2𝜌 𝑁𝑁𝐺 3

Rotating MMFs For a balanced three phase set of currents: 𝑗 𝑏𝑡 = 2𝐽 𝑡 cos 𝜕 𝑓 𝑢 + 𝜄 𝑓𝑗 (0) 𝑗 𝑐𝑡 = 2𝐽 𝑡 cos 𝜕 𝑓 𝑢 − 2𝜌 3 + 𝜄 𝑓𝑗 (0) 𝑗 𝑑𝑡 = 2𝐽 𝑡 cos 𝜕 𝑓 𝑢 + 2𝜌 3 + 𝜄 𝑓𝑗 (0) 𝑁𝑁𝐺 = 𝑁𝑁𝐺 𝑏𝑡 + 𝑁𝑁𝐺 𝑐𝑡 + 𝑁𝑁𝐺 𝑑𝑡 = 𝑂 𝑡 3 2𝐽 𝑡 2 cos 𝜕 𝑓 𝑢 + 𝜄 𝑓𝑗 0 − 𝜚 𝑡 2

Rotating MMFs This means that if you hold your position fixed on the stator iron, you will see an MMF wave traveling by you! 𝑢 1 > 𝑢 0 𝑢 0

Self and Mutual Inductances Suppose there are now windings on the rotor. Phase a stator winding 𝜚 𝑠 𝜄 𝑠 𝜚 𝑡 Phase a rotor winding 𝑢 𝜄 𝑠 = 𝜕 𝑠 𝜐 𝑒𝜐 + 𝜄 𝑠 (0) 0 = 𝜚 𝑠 − 𝜚 𝑡

Self and Mutual Inductances Self inductance of the a phase stator winding: 𝑀 𝑏𝑡𝑏𝑡 = 𝑀 𝑚𝑡 + 𝑀 𝑛𝑡 Mutual inductance of the a phase stator winding and the a phase rotor winding (you can see from fig): 𝑀 𝑏𝑡𝑏𝑠 = 𝑀 𝑡𝑠 cos 𝜄 𝑠 We now have inductances that are dependent upon rotor position — which is typically time-varying .

Self Inductances Stator self inductance matrix for a symmetrical three phase machine: − 1 − 1 𝑀 𝑚𝑡 + 𝑀 𝑛𝑡 2 𝑀 𝑛𝑡 2 𝑀 𝑛𝑡 − 1 − 1 𝑴 𝑡 = 2 𝑀 𝑛𝑡 𝑀 𝑚𝑡 + 𝑀 𝑛𝑡 2 𝑀 𝑛𝑡 − 1 − 1 2 𝑀 𝑛𝑡 2 𝑀 𝑛𝑡 𝑀 𝑚𝑡 + 𝑀 𝑛𝑡 Similarly, the rotor self inductance matrix is: − 1 − 1 𝑀 𝑚𝑠 + 𝑀 𝑛𝑠 2 𝑀 𝑛𝑠 2 𝑀 𝑛𝑠 − 1 − 1 𝑴 𝒔 = 2 𝑀 𝑛𝑠 𝑀 𝑚𝑠 + 𝑀 𝑛𝑠 2 𝑀 𝑛𝑠 − 1 − 1 2 𝑀 𝑛𝑠 2 𝑀 𝑛𝑠 𝑀 𝑚𝑠 + 𝑀 𝑛𝑠

Mutual Inductances Mutual inductance matrix for a symmetrical three phase machine: cos 𝜄 𝑠 + 2𝜌 cos 𝜄 𝑠 − 2𝜌 cos 𝜄 𝑠 3 3 cos 𝜄 𝑠 − 2𝜌 cos 𝜄 𝑠 + 2𝜌 𝑴 𝑡𝑠 = 𝑀 𝑡𝑠 cos 𝜄 𝑠 3 3 cos 𝜄 𝑠 + 2𝜌 cos 𝜄 𝑠 − 2𝜌 cos 𝜄 𝑠 3 3 Note the rotor position dependence of the mutual inductances.

Theory of Electrical Machines – Part II James Cale – Guest Lecturer EE 566, Fall Semester 2014 Colorado State University

Induction Machines as’ 𝜚 𝑠 bs cs 𝜄 𝑠 ar ’ br 𝜚 𝑡 cr cr ’ br ’ ar bs’ cs’ as

Induction Machines The derivation of the equations for analyzing the symmetrical induction machines is laborious and requires the use of reference frame theory — students should refer to an appropriate text*. Rather than derive all of these equations, we will focus on the intuitive reason why the induction machine works and state some fundamental results. *Krause, Wasynczuk , Sudhoff, “ Analysis of Electric Machinery and Drive Systems ,” Second edition, 2002, Wiley.

Induction Machines Recall that in the case of a three-phase machine driven by balanced, three phase stator currents a rotating MMF wave was established in the air gap. When the rotor circuits are short-circuited (as in a squirrel-cage induction machine), there is an associated MMF wave induced on the rotor. There is an associated torque whenever the rotor MMF and stator MMF waves are not synchronized. That is, it is necessary to have some “slip” between the stator and rotor MMF waves.