ECE 330: Power Circuits and Electromechanics

Lecture 23

2019-11-20

Last Time

• Rotating magnetic field

• Inductance matrix

• Started looking at synchronous machines

•!"#!$#!"%

='# 0 )cos(.)0 '# )sin(.)

)cos(.) )sin(.) '%

2"#2$#2"%

• Frequency condition: 34 = 3# − 3%

Today

• Why move at the synchronous speed?

• Equivalent circuit for synchronous machines

• Power consumed

• Poles

Synchronous machine

•!"#!$#!"%

='# 0 )cos(.)0 '# )sin(.)

)cos(.) )sin(.) '%

2"#2$#2"%

• Assume: 2"# = 6# cos 3#7 , 2$# = 6#sin(3#7)

2"% = 6%, . = 347 + .:

• Frequency condition: 34 = 3#

Synchronous speed

Synchronous demonstration starts at 3:30 on video.

https://www.youtube.com/watch?v=lV8iPKY-3ms

a Phase Stator Voltage

• ;"# = <#2"# +=>?@=A

• ;"# = <#6# cos 3#7 − '#6#3# sin 3#7 − )6% 3#sin 3#7 + .:• Writing as phasors

• B?@C∠.E = <#

F@C∠0° + H3#'#

F@C∠0° + I@JFK

C∠.: + 90°

Per phase circuit

• M"% =I@JFK

C: excitation voltage. Dependent on 6%

• N: torque angle

Power consumed

• OPQ = 3OSQ → OPQ = 3<U{M"% W6"∗}

• Simplifying gives: OPQ = −PZ?KB?[ \]^ _`@

• For a motor, OPQ > 0 → N < 0• For a generator, OPQ < 0 → N > 0• NOTE: to analyze a generator, flip current direction in circuit

Complex Power

• cPQ = 3d"e W6"∗ → cPQ = OPQ + HfPQ

• fPQ < 0: over-excited (supplies VARs)

• fPQ > 0: under-excited (consumes VARs)

Example

• A 3-phase 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. What is the torque angle?

• a) N = 43.2°• b) N = −43.2°• c) N = 0.754°• d) N = −0.754°

Example

• A 3-phase 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. What is the torque angle?

• a) N = 43.2°• b) N = −43.2°• c) N = 0.754°• d) N = −0.754°

Example

• A 3-phase 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. What is the power factor of the generator?

• a) Ol = 0.998 lead

• b) Ol = 0.998 lag

• c) Ol = 0.8 lag

• d) Ol = 0.8 lead

Example

• A 3-phase 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. What is the power factor of the generator?

• a) Ol = 0.998 lead

• b) Ol = 0.998 lag

• c) Ol = 0.8 lag

• d) Ol = 0.8 lead

Example on the board

• A 3-phase 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. If the power generated and the terminal voltage are the same, how much should the rotor current change to get a new

power factor of 0.8 lag?

Poles

• Everything up till now was a 2-pole system

Poles (Continued)

Equations with Multiple Poles

• Frequency condition: 34 = Cn3# where p is the number of poles

• Need because we don’t have to travel as far in θ to change current direction

• O = op34 → op = qrsI@

Example

• A 3-phase, 60 Hz, 6 pole, 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. What is the mechanical rotation rate?

• a) 34 = 377 tuv/x• b) 34 = 126 tuv/x• c) 34 = 189 tuv/x• d) 34= 1131 tuv/x

Example

• A 3-phase, 60 Hz, 6 pole, 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. What is the mechanical rotation rate?

• a) 34 = 377 tuv/x• b) 34 = 126 tuv/x• c) 34 = 189 tuv/x• d) 34= 1131 tuv/x

Example

• A 3-phase, 60 Hz, 6 pole, 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. What is the torque?

• a) op = 1.984 {|• b) op= 198.4 {|• c) op= 19.84 {|• d) op= 1984 {|

Example

• A 3-phase, 60 Hz, 6 pole, 550 V line-line generator produces 250 kW of power. The excitation voltage of the generator is 460 V/phase and the synchronous reactance is 1.2 Ω. What is the torque?

• a) op = 1.984 {|• b) op= 198.4 {|• c) op= 19.84 {|• d) op= 1984 {|