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 = I m cos(ωt +φ) Power into a resistive load = V 2 /R For any time-varying signal, average voltage = = V rms

Sinusoidal RMS Voltage For a sinusoidal waveform, V rms = For a wall outlet in the US, V rms = 120V and frequency = 60 Hz For a wall outlet in Europe, V rms = 240V and frequency = 50 Hz

Ideal Loads Resistive Load: I = V/RP = I 2 R or V 2 /R Capacitive Load: P = VI cos(2ωt + π/2) ; P ave = 0 Inductive Load: P = VI cos(2ωt - π/2) ; P ave = 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 =

Power Factor (cont’d) The resulting current, i = … steps to be covered in class … lead to power output, p is given by: => p ave = VI cos(θ) = VI (PF) Average = 0

Good vs. Poor Power Factor Example 2.5

Power Triangle θ Example 2.6 Reactive Power, Q (VAR) Real Power, P (Watts) P=VIcosθ Q=VIsinθ Volts-amps-reactive Apparent power S=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:

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

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 I n is the rms value of the current in the nth harmonic. The rms value of current is given by: We can show that:

Total Harmonic Distortion (cont’d) Total harmonic distortion (THD i ) is a common way of expressing waveform distortion. The THD i is given by: Example 2.12