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
James Cale – Guest Lecturer EE 566, Fall Semester 2014 Colorado State University
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.
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.
Static electrical charges give rise to electrostatic fields—but they don’t give rise to magnetic fields.
Electrons in motion (i.e., current) always give rise to magnetic fields, with an orientation specified by the “right hand rule.”
Notation: a “·” denotes current coming out of the paper; an “x” denotes current going into the paper.
𝑗 𝐼 𝑗 𝐼
Flux can also expressed as: This expression shows both the free-space field component and “bound charge” magnetization component (we’ll discuss this in a moment).
𝜈0: Permeability of free space 𝜈𝑠: Relative permeability (specific to material) 𝜈 : Permeability
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.
𝐶 − 𝐼
𝜈𝑠
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
Edge of magnetic material Magnetic “domain wall” Field from “bound charge” Random ordering of domain walls—no collective flux from magnetization outside material.
Ferrimagnetic Material
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.
𝐼
Assuming the flux density is constant and everywhere orthogonal to the cross section of the material: Φ = 𝐶 ∙ 𝑒𝐵 Φ = 𝐶𝐵 𝐶 𝑒𝐵
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
𝑂 𝑤 𝑗
+ −
Φ
𝑂: number of turns of the winding : air gap length
From Ampere’s Law Assuming the field is constant throughout the cross section of the magnetic material and air gap: where 𝑚𝑛 is the length of the magnetic material. 𝑂𝑗 = 𝐼 ∙ 𝑒𝑚 𝑂𝑗 = 𝐼𝑛𝑚𝑛 + 𝐼 =
𝐶 𝜈0𝜈𝑠 𝑚𝑛+ 𝐶 𝜈0
Now, substituting for magnetic flux and rearranging: This is similar in form to 𝑤 = 𝑠𝑗, where “reluctance” to magnetic flux
Note: in air, reluctance is high, in a highly permeable material, reluctance is small.
𝑂𝑗 = 𝑚𝑛 𝜈0𝜈𝑠𝐵 Φ + 𝜈0𝐵 Φ 𝑁𝑁𝐺 ≡ 𝑂𝑗 ~ 𝑤 Φ ~ 𝑗 ℛ ≡ 𝑚 𝜈𝐵 ~ 𝑠
Our MEC analog is the following:
Note: this was for 2-D. How do you obtain for 3-D fields?
Refer to hand-out or obtain online:
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.
Φ 𝑂𝑗 ℛ𝑛 ℛ
Flux Linkage: Inductance: 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). 𝜇 = 𝑂Φ 𝑀 = 𝜇 𝑗 = 𝑂2 ℛ
Torque is produced to minimize reluctance to flux! 𝑤 𝑗
+ −
𝜄 Now, reluctance (and inductance) are time-varying. ℛ 𝜄 = ℛ1 + ℛ2 sin 𝜄 Φ
Sinusoidal Winding Distributions
Iron Air
𝜚𝑡
Phase a stator winding
Now, the MMF for the a winding is (you can see this from right hand rule, e.g., flux is max at 𝜚𝑡 = 0,−𝜌): If there are sinusoidal windings for the b and c phases, displaced physically by 120 degrees then: 𝑁𝑁𝐺
𝑏𝑡 = 𝑂𝑡
2 𝑗𝑏𝑡 cos 𝜚𝑡 𝑁𝑁𝐺𝑐𝑡 = 𝑂𝑡 2 𝑗𝑐𝑡 cos 𝜚𝑡 − 2𝜌 3 𝑁𝑁𝐺
𝑑𝑡 = 𝑂𝑡
2 𝑗𝑑𝑡 cos 𝜚𝑡 + 2𝜌 3
For a balanced three phase set of currents: 𝑗𝑏𝑡 = 2𝐽𝑡 cos 𝜕𝑓𝑢 + 𝜄𝑓𝑗(0) 𝑗𝑐𝑡 = 2𝐽𝑡 cos 𝜕𝑓𝑢 − 2𝜌 3 + 𝜄𝑓𝑗(0) 𝑗𝑑𝑡 = 2𝐽𝑡 cos 𝜕𝑓𝑢 + 2𝜌 3 + 𝜄𝑓𝑗(0) 𝑁𝑁𝐺 = 𝑁𝑁𝐺
𝑏𝑡+𝑁𝑁𝐺𝑐𝑡+𝑁𝑁𝐺 𝑑𝑡
= 𝑂𝑡 2 2𝐽𝑡 3 2 cos 𝜕𝑓𝑢 + 𝜄𝑓𝑗 0 − 𝜚𝑡
This means that if you hold your position fixed on the stator iron, you will see an MMF wave traveling by you!
𝑢0 𝑢1 > 𝑢0
Suppose there are now windings on the rotor. 𝜚𝑡 𝜚𝑠 𝜄𝑠
𝜄𝑠 = 𝜕𝑠 𝜐 𝑒𝜐
𝑢
+ 𝜄𝑠(0) = 𝜚𝑠 − 𝜚𝑡
Phase a stator winding Phase a rotor winding
Self inductance of the a phase stator winding: Mutual inductance of the a phase stator winding and the a phase rotor winding (you can see from fig): We now have inductances that are dependent upon rotor position—which is typically time-varying. 𝑀𝑏𝑡𝑏𝑡 = 𝑀𝑚𝑡 + 𝑀𝑛𝑡 𝑀𝑏𝑡𝑏𝑠 = 𝑀𝑡𝑠 cos 𝜄𝑠
Stator self inductance matrix for a symmetrical three phase machine: Similarly, the rotor self inductance matrix is:
𝑴𝑡 = 𝑀𝑚𝑡 + 𝑀𝑛𝑡 − 1 2 𝑀𝑛𝑡 − 1 2 𝑀𝑛𝑡 − 1 2 𝑀𝑛𝑡 𝑀𝑚𝑡 + 𝑀𝑛𝑡 − 1 2 𝑀𝑛𝑡 − 1 2 𝑀𝑛𝑡 − 1 2 𝑀𝑛𝑡 𝑀𝑚𝑡 + 𝑀𝑛𝑡 𝑴𝒔 = 𝑀𝑚𝑠 + 𝑀𝑛𝑠 − 1 2 𝑀𝑛𝑠 − 1 2 𝑀𝑛𝑠 − 1 2 𝑀𝑛𝑠 𝑀𝑚𝑠 + 𝑀𝑛𝑠 − 1 2 𝑀𝑛𝑠 − 1 2 𝑀𝑛𝑠 − 1 2 𝑀𝑛𝑠 𝑀𝑚𝑠 + 𝑀𝑛𝑠
Mutual inductance matrix for a symmetrical three phase machine: Note the rotor position dependence of the mutual inductances.
𝑴𝑡𝑠 = 𝑀𝑡𝑠 cos 𝜄𝑠 cos 𝜄𝑠 + 2𝜌 3 cos 𝜄𝑠 − 2𝜌 3 cos 𝜄𝑠 − 2𝜌 3 cos 𝜄𝑠 cos 𝜄𝑠 + 2𝜌 3 cos 𝜄𝑠 + 2𝜌 3 cos 𝜄𝑠 − 2𝜌 3 cos 𝜄𝑠
James Cale – Guest Lecturer EE 566, Fall Semester 2014 Colorado State University
𝜚𝑡 𝜚𝑠 𝜄𝑠
as’ as bs’ bs cs cs’ ar’ ar cr cr’ br’ br
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.
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.
Fundamental Equations
Note that the flux linkage equations are still dependent on rotor angle position (from the mutual inductance matrix). After performing a coordinate transform (using Park’s Transformation), it’s possible to remove the angular dependence, but we don’t do that here.
𝒘𝑏𝑐𝑑𝑡 = 𝒔𝑡𝒋𝑏𝑐𝑑𝑡 + 𝑒 𝑒𝑢 𝝁𝑏𝑐𝑑𝑡 𝒘𝑏𝑐𝑑𝑠 = 𝒔𝑠𝒋𝑏𝑐𝑑𝑠 + 𝑒 𝑒𝑢 𝝁𝑏𝑐𝑑𝑠 𝝁𝑏𝑐𝑑𝑡 𝝁𝑏𝑐𝑑𝑠 = 𝑴𝑡 𝑴𝑡𝑠 𝑴𝑡𝑠 𝑼 𝑴𝑠 𝒋𝑏𝑐𝑑𝑡 𝒋𝑏𝑐𝑑𝑠
3 equations 3 equations 6 equations
Steady-State (only!) Equivalent T Circuit 𝑠
𝑡
𝑌𝑡 𝑌𝑠 𝑠
𝑠/s
′ ′ 𝐽
𝑡
𝐽
𝑠
′ 𝑊
𝑡
−
+
𝑌𝑁 𝑠
𝑁
s = 𝜕𝑓 − 𝜕𝑠 𝜕𝑓 = 1 − 𝜕𝑠 𝜕𝑓
Torque-Speed Curve
Where is the stable operating point?
? ?
𝑈𝑀
𝑈
𝑓
𝑈𝑀 𝑈
𝑓 = 𝑈𝑀
In steady-state, 𝜕𝑠
Where is the stable operating point?
Stable Operating Point
𝑈𝑀
Unstable
Notes:
large slope near s = 0, once the machine is in steady-state, its rotor speed will not vary much. It is like a “constant” speed machine. This is because the slightest change in speed will cause a large change in electromagnetic torque.
equations in the qd0 reference frame.
circuit.
Steady-State Equivalent T Circuit – Doubly Fed 𝑠
𝑡
𝑌𝑡 𝑌𝑠 𝑠
𝑠 + 𝑆𝑓 /s
′ ′ 𝐽
𝑡
𝐽
𝑠
′ 𝑊
𝑡
−
+
𝑌𝑁 𝑠
𝑁
s = 𝜕𝑓 − 𝜕𝑠 𝜕𝑓 = 1 − 𝜕𝑠 𝜕𝑓 ′
Torque-Speed Curves-Various External Resistances
𝜚𝑡 𝜚𝑠 𝜄𝑠
as’ as bs’ bs cs cs’
Fundamental Equations
𝒘𝑏𝑐𝑑𝑡 = 𝒔𝑡𝒋𝑏𝑐𝑑𝑡 + 𝑒 𝑒𝑢 𝝁𝑏𝑐𝑑𝑡 𝝁𝑏𝑐𝑑𝑡 = 𝑴𝑡𝒋𝑏𝑐𝑑𝑡 + 𝝁𝑛
3 equations 3 equations
𝝁𝑛 = 𝜇𝑛 sin 𝜄𝑠 sin 𝜄𝑠 − 2𝜌 3 sin 𝜄𝑠 + 2𝜌 3
From permanent magnet, can obtain by inspection
Transforming Variables to Rotor Reference Frame
where 𝑔 can represent any variable (voltage, current, flux linkage, etc.)
𝑳𝑡 = 2 3 cos 𝜄𝑠 cos 𝜄𝑠 − 2𝜌 3 cos 𝜄𝑠 + 2𝜌 3 sin 𝜄𝑠 sin 𝜄𝑠 − 2𝜌 3 sin 𝜄𝑠 + 2𝜌 3 1 2 1 2 1 2 𝒈𝑟𝑒0𝑡 = 𝑳𝑡𝒈𝑏𝑐𝑑𝑡
𝑠 𝑠 𝑠
After transforming variables to rotor reference frame
Note that the rotor dependence is eliminated since we’re in the rotor reference frame.
𝑤𝑟𝑡 = 𝑠
𝑡𝑗𝑟𝑡 + 𝜕𝑠𝜇𝑒𝑡 + 𝑒
𝑒𝑢 𝜇𝑟𝑡 𝑤𝑒𝑡 = 𝑠
𝑡𝑗𝑒𝑡 − 𝜕𝑠𝜇𝑟𝑡 + 𝑒
𝑒𝑢 𝜇𝑒𝑡 𝑤0𝑡 = 𝑠
𝑡𝑗0𝑡 + 𝑒
𝑒𝑢 𝜇0𝑡 𝜇𝑟𝑡 = 𝑀𝑟𝑗𝑟𝑡 𝜇𝑒𝑡 = 𝑀𝑒𝑗𝑟𝑡 + 𝜇𝑛 𝜇0𝑡 = 𝑀𝑚𝑡𝑗0𝑡
𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠
Steady State Analysis
If we are supplying the phase voltages, we can generally force (through power electronics) where
𝑊
𝑟𝑡 = 𝑠 𝑡𝐽𝑟𝑡 + 𝜕𝑠𝑀𝑒𝐽𝑒𝑡 + 𝜕𝑠𝜇𝑛
𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠
𝑊
𝑒𝑡 = 𝑠 𝑡𝐽𝑒𝑡 − 𝜕𝑠𝑀𝑟𝐽𝑟𝑡
𝑈
𝑓 =
3 2 𝑄 2 𝜇𝑛𝐽𝑟𝑡 + 𝑀𝑒 − 𝑀𝑟 𝐽𝑟𝑡𝐽𝑒𝑡 𝑊
𝑟𝑡 =
2𝑤𝑡 cos 𝜚𝑤 𝑊
𝑒𝑡 = − 2𝑤𝑡 sin 𝜚𝑤
𝜚𝑤 = 𝜄𝑓𝑤 − 𝜄𝑠
This means we have the relation: Substituting we have: where This is the origin of the phasor diagram in Aliprantis’ notes, (with 𝜀 = 0).
2𝑊
𝑏𝑡 = 𝑊 𝑟𝑡 − 𝑘𝑊 𝑒𝑡
𝑘 2𝐽
𝑏𝑡 = 𝐽𝑒𝑡 + 𝑘𝐽𝑟𝑡
𝑊
𝑏𝑡 = 𝑠 𝑡 + 𝑘𝜕𝑠𝑀𝑟 𝐽 𝑏𝑡 + 𝐹
𝑏 𝐹 𝑏 = 1 2 𝜕𝑠 𝑀𝑒 − 𝑀𝑟 𝐽𝑟𝑡 + 𝜕𝑠𝜇𝑛 𝑓𝑘0
𝑠 𝑠 𝑠 𝑠 𝑠
Torque-Speed Curve 𝑈𝑀
Notes:
more accurate PMSM, since these machines can also motor (not just generate power)
currents is the same as the frequency of the stator currents (not true in IM machine).
the stator currents (e.g., through power electronics, we can control the rotor speed).
angle, can generate unique torque-speed curves.