Download Lecture Note (Chapter 2

ECET 211 Electrical Machines and Controls
Spring, 2013
Paul I. Lin, Professor of ECET
Single- and Three-Phase Power
AC EMF Generation
 Instantaneous voltage e(t) = Em sin θ = Em sin(2ft + ) volts
  = 2ft +  radians
Electrical and Mechanical Degrees
 2-pole machine (1 pair NS pole)
o 1 sine wave - moving through a complete circle (360 )
o Mechanical degree 360 = Electrical degree 360
 4-pole machine (2 pairs NS poles)
o 2 sine waves - moving through a complete circle (360)
o Mechanical degree (360)
o Electrical degree (720)
Alternating AC Voltage and Current
v(t) = Vm sin(ωt) = Vm sin(2πft)
Let f = 60Hz, ω = 2πf = 377.7 rad/sec
Vm = 170 volts (peak or maximum voltage)
T = 1/f = 16.67 ms
Vrms (root-mean-square) can be found:
1
√2
0.707 ∙
v(t) = Vm sin(2πft) = 170 sin(ωt) = Vrms √ sin(ωt) = 120·√ sin(ωt) volts

Effective value (rms)
o Heating effect due to the power from a dc source
 Pdc = I2 R or = V2/R
o Heating effect due to the power from ac source
 Pac = i2 R = v2/R
o Pdc = Pac (equivalent)
o I rms or I (root-mean-square)
o I rms = 0.707 I m
Vrms = 0.707 Vm
i(t) = Im sin (ωt - ) = Im sin (2ft - )
 = π/6 = 30˚, Im = 30 amperes
i(t) = 30 sin (ωt - π/6) = 21.21·√ sin (2ft - π/6) amps

Frequency (f, Hz)
o Power system: 25 Hz, 50 Hz, 60 Hz

Phase relationships between v(t) and i(t):
o In phase
 v(t) = √2 ·120 sin(2 60 t)
 i(t) = 14.14 sin (2 60 t)
o Current lags the voltage by 30 degrees
 v(t) = 2 110 sin(2 60 t)
 i(t) = 14.14 sin (2 60 t - 30)
o Current leads the voltage by 30 degrees
 v(t) = 2 110 sin(2 60 t - 30)
 i(t) = 14.14 sin (2 60 t)
Phasor Diagrams
 Methods for representing AC voltages, currents, powers, resistance, reactance, and
impedance
 A Phasor: Magnitude and Direction
 Phasor Manipulations:
o Addition
o Subtraction
o Multiplication
o Division
Phasor Manipulation Examples
Example 1
Find the phasor sum of a current 3A and a current of 4 A, the two currents being in phase.
Example 2
A voltage V1 of 125-V leads a voltage V2 of 100-V by 180º. Find their phasor sum.
Example 3
A current I2 of 8A leads a current of I1 of 6A by 90º. Find their phasor sum.
Example 4
Add two voltages of 100V each that are out of phase by 30º.
Example 5
A voltage of 120V leads a voltage of 240V by 120º. What is their phasor sum?
Example 6
Subtract a current of 10A from a current of 17.3A when the 17.3A current leads the 10-A
current by 30º.
Properties of an Electrical Circuit
 Resistance: R
o IR = V/R
 Inductance (L, henry, mH)
o Inductive reactance (XL = 2fL ohms)
o IL = V/XL
 Capacitor, and Capacitance (C, farad, F)
o C = Q/V
o Capacitive Reactance (Xc = 1/(2fC) ohms)
o IC = V/XC
o Capacitors in series
 1/C = 1/C1 + 1/C2 + .. + 1/Cn
o Capacitors in parallel
 C = C1 + C2 + .. + Cn
 Reactance
o X = XL - XC ohms
 Impedance Z = R – jX = sqrt(R2 + X2) ohms
o Z = V/I ohms
Example 7
What is the effective current flowing through a resistance of 30Ω when a 120V 60 Hz voltage is
applied.
Example 8
A 0.2 H choke coil with negligible resistance is connected to 1 120V 60Hz supply. Find
(A) The inductive reactance of the coil, and
(B) The current flowing;
(C) Draw the phasor diagram showing the phase relations between the current and the applied
voltage.
Example 9
An 8-µF capacitor is connected to a 240V 60Hz circuit.
(A) Find the capacitive reactance.
(B) Find the current flowing, and
(C) Draw the phasor diagram
Example 10
A resistance of 30Ω is connected in series with an inductive reactance of 60Ω and a capacitive
reactance of 20Ω. What is the impedance of the circuit?
Demonstration
 Function generator
 Oscilloscope
 R, L, C components
Web Site References
o Educational Information
 Phase relationship: http://hyperphysics.phyastr.gsu.edu/hbase/electric/phase.html
 Inductance applet:
http://www.cco.caltech.edu/~phys1/java/phys1/Inductance/Inductance.html
 Charging capacitor: http://lectureonline.cl.msu.edu/~mmp/kap23/RC/app.htm
 RC Circuits: http://www.phy.ntnu.edu.tw/java/rc/rc.html
 Electrical and Mechanical Systems,
http://www.eng.abdn.ac.uk/~eng489/eg2558/topic2a/, and
http://www.eng.abdn.ac.uk/~eng489/eg2558/topic4d/
 MARTINDALES’s Calculators On-Line Center - Electronics/Electrical
Calculators: http://www.martindalecenter.com/Calculators4A.html
 Electromagnetic Links: http://www.oz.net/~coilgun/links.htm
o Industrial Product information
 GE Industrial Systems: http://www.geindustrial.com/cwc/products?id=lacgen
 Electric Power Systems (turbine): http://www.belyeapower.com/
 AC Power Systems: http://www.gensetcentral.com/
 AC PowerPac: http://www.acpowerpac.com/
 Savoia Generators: http://www.savoiapower.com/
 Wind Turbine Generator: http://www.windpower.dk/tour/wtrb/electric.htm