SSS: Electric Power Systems
Sinusoidal
Steady-State:
Electric Power
• Power:
o Complex power, S = P + jQ
o The ‘power factor’
o “Real” power vs. reactive power
• Root Mean Square: RMS values
• Power transfer à maximum possible
EGR 220, Chapter 11
April 12, 2016
2
Complex Power & the Power Factor, pf
The Power Factor from Phasors
• Complex power, S = P + jQ = S<θz°
• S = P + jQ (recall the trig identity with v(t)*i(t))
1
1
= Vm I m ________+ j Vm I m _________
2
2
• pf defined as the cosine of the angle
1
P = Vm I m cos(θ v − θ i )
2
1
VI =
2
1 *
VI =
2
between V and I
o The power factor, pf = cos(θv – θi)
• Leading power factor means current leads
voltage
3
V = Vm ∠θ v
I = I m ∠θ i
4
1
Complex Power:
The Power Triangle
Discussion - Important Angles:
Ø What is the connection between
1. Phase angle of impedance, Z
Z = R + jX
2. Phase difference between V & I
3. Relative values of P & Q
Z=
V Vm∠θV Vm
=
= ∠ (θV − θ I ) = Z∠θ Z
I I m∠θ I I m
6
7
Check RMS at an Outlet
Example: Power Factor
o Find the power factor
and state if it is leading
or lagging,
X rms =
1
T
∫
T
0
x 2 dt =
Xm
2
€
8
9
2
RMS = Root Mean Square
RMS: Root Mean Square
• Most electrical equipment has two ratings
o Maximum power, energy, current, voltage...
o Average or sustained power, energy, current...
• RMS is also referred to as the effective value
• The effective value of an ac signal IS the dc
value that delivers the same average power to
a resistor
• Maximum value
o Instantaneous value è time domain
• Average value
o Phasor domain
o RMS è a new ‘average’ measurement
(the effect of the AC signal is equivalent to a DC
signal at the RMS value)
10
11
Maximum Average Power Transfer
• For circuits with complex impedance, when
does maximum power transfer occur?
Z L = RL + jX L = RTh − jX Th = Z *Th
€
12
14
3
Summary
• Power is defined with multiple methods,
depending upon the problem at hand
o Instantaneous power
o Average power
o Phasors – amplitude (peak) and phase angle
o RMS value
o Complex power and power factor
• Complex power = real power + reactive power
• S = P + jQ
• Maximum average power transfer important in
power systems
15
16
Summary: Representing Power
• Know which concepts and terms are time
domain vs. phasor domain
• Know the definition and significance of:
o
o
o
o
Instantaneous power
Average power
RMS vs. maximum value
Complex power
• Real power
• Reactive power
o Power factor
o Compensation (pf correction)
17
4