1. Solar Energy Commercialization
SEC598S19
Solar Energy Commercialization
Session 03
Utility Operations - Sinusoidal Power Calculations
January 14, 2019

2. Session 03 - Value to class members
A review of electrical circuits
Sinusoidal power calculations

3. Utility Operations (References)
C.K.Alexander and M.N.O.Sadiku, Fundamentals of Electric Circuits, 3rd Ed, McGraw-Hill, 2007

4. Review of Electric Circuits
Essential Electrical Components
Voltage and Current Sources
Source of electrical energy in the circuit
Resistors
Components that represent the physical limits to current flow
Capacitors
Components that represent the electrical fields caused by charge separation
Inductors
Components that represent the magnetic fields caused by current flow

5. Review of Electric Circuits
Ref 5
Defining Relations
Resistors
Capacitors
Inductors

6. Review of Electric Circuits – The RLC Circuit
vr(t)
i (t) + – R + + –
vc(t)
vs(t) C –
vl(t) – + L
Kirchhoff Voltage Law around the loop:
vr(t) + vc(t) + vl(t) = vs(t)

7. Review of Electric Circuits – The RLC Circuit

8. Sinusoidal Analysis
After the celebrated Electricity Wars between Thomas Edison and George Westinghouse were settled, electricity for lighting and then all other uses was generated and delivered in AC (alternating current) or sinusoidal format

9. Sinusoidal Analysis

10. Sinusoidal Analysis
Period
Phase shift
Average
rms value

11. Sinusoidal Analysis - Phasors
Euler’s Identity
Sine and Cosine

12. Sinusoidal Analysis - Phasors
Moving from the time domain to the frequency domain:
Phasor Transform

13. Sinusoidal Analysis - Phasors
Resistors:
The phasor transform is:
The current and voltage for a resistor are in phase

14. Sinusoidal Analysis - Phasors
Capacitors
The phasor transform is: or
The current and voltage for a capacitor are 900 out of phase

15. Sinusoidal Analysis - Phasors
Inductors
The phasor transform is:
The current and voltage for an inductor are 900 out of phase

16. Review of Electric Circuits
Phase shifts

17. Review of Electric Circuits
Phasor Relations and Impedances
Resistors
Capacitors
Inductors

18. Review of Electric Circuits – The RLC CircuitVR I + – ZR + + – VS ZC VC – VL – + ZL
Kirchhoff Voltage Law around the loop:
VS = VR + VC + VL

19. Review of Electric Circuits – The Series RLC Circuitor

20. Review of Electric Circuits – The RLC Circuit
Z – Impedance
R – Resistance
X - Reactance

21. Sinusoidal Analysis - Instantaneous Poweror or

22. Sinusoidal Analysis - Instantaneous Power
With additional trigonometric manipulation or

23. Sinusoidal Analysis - Power
Power and Phase

24. Sinusoidal Analysis – Average Power
The instantaneous power is
The power averaged over one cycle is

25. Sinusoidal Analysis - Power
Rewritten
where

26. Sinusoidal Analysis – Average Power
Real Power (Watts):
Reactive Power (VAR):

27. Sinusoidal Analysis – Average Power
Apparent Power (VA):
In phasor format (complex power):
𝑆 = 𝑉 ∙ 𝐼 ∗

28. Sinusoidal Analysis - Power
Power Factor:

29. Sinusoidal Analysis – VARs
Amusing explanation of VARs (and other related topics)

30. Sinusoidal Analysis – VARs
A primary use of electricity is to drive electric motors
Electric motors require the creation of a magnetic field to make the shaft spin.
The electricity that magnetizes the coils (inductors) does no work and is not recorded by the electric meter – this is reactive power
The overall power the system must be designed to deliver both reactive power and real power – and the utility only gets paid for the (measured) real power
Therefore the utility measures the power factor at the meter and bills the industrial customers if it’s too low

31. Sinusoidal Analysis – Maximum Power Transfer

32. Sinusoidal Analysis – Maximum Power Transfer
KVL
Average Power delivered to load

33. Sinusoidal Analysis – Maximum Power Transfer
Results

34. Sinusoidal Analysis – Power factor correction