1. ECE 7800: Renewable Energy Systems Topic 2: Fundamentals of Electric Power Spring 2010 © Pritpal Singh, 2010

2. 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

3. 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

4. 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

5. 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 =

6. 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

7. Good vs. Poor Power Factor Example 2.5

8. 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

9. Three-Wire Single Phase Residential Wiring

10. 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.

11. Balanced Wye-Connected 3φ To see the advantage of a 3φ system compared to a single phase system, consider the figure below.

12. 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:

13. 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

14. Balanced Wye-Connected 3φ (cont’d) Derivation of line and phase voltages

15. Delta-Connected 3φ

16. 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

17. 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:

18. 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