ECE 7800: Renewable Energy Systems

Topic 2: Fundamentals of Electric Power

Spring 2010

© Pritpal Singh, 2010

AC vs. DC PowerDC 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

voltage = = Vrms T

dttVT 0

2 )(1

Sinusoidal RMS VoltageFor a sinusoidal waveform, Vrms =

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

pkV2

Ideal LoadsResistive 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 FactorConsider a general black box as

shown below:

Consider the voltage driving this box

has rms voltage, V and phase angle = 0.

v =

tV cos2

Power Factor (cont’d)The resulting current, i =

… steps to be covered in class …

lead to power output, p is given by:

=> pave = VI cos(θ) = VI (PF)

)cos(2 tI

)cos()2cos( VItVIp

Average = 0

Good vs. Poor Power Factor

Example 2.5

Power Triangle

θ

Example 2.6

ReactivePower, Q(VAR)

Real Power, P(Watts)

P=VIcosθ

Q=VIsinθVolts-amps-reactive

Apparent powerS=VI volt-amps

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:

)cos(2 tVva 00VV a

)120cos(2 tVvb

)240cos(2 tVvc

0120VV b

0240VV c

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:

The current in the neutral is given by:

…derivation in class

)cos(2 tIia 00II a

)120cos(2 tIib

)240cos(2 tIic

0120II b

0240II c

)120cos()120cos()cos(2 tttIiiii cban

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:

where In is the rms value of the current in the nth harmonic. The rms value of current is given by:

We can show that:

...)3cos2coscos(2 321 tItItIi

...)3cos2coscos(2)( 3212 tItItIiI averms

...23

22

21 IIII rms

Total Harmonic Distortion (cont’d)

Total harmonic distortion (THDi) is a common way of expressing waveform distortion. The THDi is given by:

Example 2.12

1

24

23

22 ...

I

IIITHDi