Transcript ECE 7800: Renewable Energy Systems

ECE 7800: Renewable Energy
Systems
Topic 2: Fundamentals of Electric Power
Spring 2010
© Pritpal Singh, 2010
AC vs. DC Power
DC is a steady, constant voltage
current power source.
AC is a time varying signal (ideally
sinusoidal) => I = Im cos(ωt +φ)
Power into a resistive load = V2 /R
For any time-varying signal, average
T
voltage = 1
= Vrms
2
V (t )dt

T0
Sinusoidal RMS Voltage
For a sinusoidal waveform, Vrms = 2V pk
For a wall outlet in the US, Vrms = 120V
and frequency = 60 Hz
For a wall outlet in Europe, Vrms = 240V
and frequency = 50 Hz
Ideal Loads
Resistive Load:
I = V/R
P = I2 R or V2 /R
Capacitive Load:
P = VI cos(2ωt + π/2) ; Pave = 0
Inductive Load:
P = VI cos(2ωt - π/2) ; Pave = 0
Power Factor
Consider a general black box as
shown below:
Consider the voltage driving this box
has rms voltage, V and phase angle = 0.
v = 2V cos t

Power Factor (cont’d)
The resulting current, i =
2 I cos(t   )
… steps to be covered in class …
lead to power output, p is given by:
p  VI cos( 2t   )  VI cos(  )
Average = 0
=> pave = VI cos(θ) = VI (PF)
Good vs. Poor Power Factor
Example 2.5
Power Triangle
Reactive
Power, Q
(VAR)
Apparent power
S=VI volt-amps
Q=VIsinθ
Volts-amps-reactive
θ
P=VIcosθ
Real
Power, P
(Watts)
Example 2.6
Three-Wire Single Phase Residential Wiring
Three Phase Systems
Commercial systems in the US are
usually produced with 3 phase
synchronous generators and with 3
phase transmission lines.
3φ generators are more efficient and
offer smoother operation than single
phase generators.
3φ transmission and distribution
systems use their wires more
efficiently saving copper.
Balanced Wye-Connected 3φ
To see the advantage of a 3φ system
compared to a single phase system,
consider the figure below.
Balanced Wye-Connected 3φ (cont’d)
Suppose that each generator
produces the same voltage but 120°
shifted in phase. The phase voltages
are then given by:
va  2V cos(t )
V a V00
vb  2V cos(t  120)
V b  V1200
vc  2V cos(t  240)
V c  V2400
Balanced Wye-Connected 3φ (cont’d)
To determine the neutral current, we
need to find the current in each phase
and add them together. The current in
each phase is given by:
ia  2I cos(t )
I a  I00
ib  2I cos(t  120)
I b  I1200
ic  2I cos(t  240)
I c  I2400
The current in the neutral is given by:
in  ia  ib  ic  2I cos(t )  cos(t  120)  cos(t 120)
…derivation in class
Balanced Wye-Connected 3φ (cont’d)
Derivation of line and phase voltages
Delta-Connected 3φ
Power Quality – Harmonic Distortion
A distortion to the sinusoidal
waveform due to high frequency
components in the waveform.
Example 2.11 Harmonic Analysis
of a Square Wave
Total Harmonic Distortion
If the current of a distorted waveform
is given by:
i  2 ( I1 cos t  I 2 cos 2t  I 3 cos 3t  ...)
where In is the rms value of the
current in the nth harmonic. The rms
value of current is given by:
I rms  (i 2 ) ave 

2 ( I1 cos t  I 2 cos 2t  I 3 cos 3t  ...)
We can show that:
I rms  I12  I 22  I 32  ...

Total Harmonic Distortion (cont’d)
Total harmonic distortion (THDi) is a
common way of expressing
waveform distortion. The THDi is
given by:
THDi 
I  I  I  ...
I1
2
2
2
3
Example 2.12
2
4