0. ECE 333 Renewable Energy Systems
Lecture 3:Basic Circuits, Complex Power
Prof. Tom Overbye
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign

1. Announcements Be reading Chapters 1 and 2 from the book
Be reading Chapter 3 from the book
Homework 1 is 1.1, 1.11, 2.6, 2.8, It will be covered by the first in-class quiz on Thursday Jan 29
As mentioned in lecture 2, your two lowest quiz/homework scores will be dropped

2. Engineering Insight: Modeling
Engineers use models to represent the systems we study
Guiding motto: “All models are wrong but some are useful” George Box, 1979
The engineering challenge, which can be quite difficult sometimes, is to know the limits of the underlying models.

3. Basic Electric Circuits
Ideal Voltage Source
Ideal Current Source +
Load - +
Load -

4. Example – Power to Incandescent Lamp
Find R if the lamp draws 60W at 12 V
Find the current, I
What is P if vs doubles and R stays the same? 240W +
Load -

5. Equivalent Resistance for Resistors in Series and Parallel
Resistors in series – voltage divides, current is the same + - + -
node
voltages

6. Equivalent Resistance for Resistors in Series and Parallel
Resistors in parallel – current divides, voltage is the same
branch currents
Simplification for 2 resistors + -

7. Voltage and Current Dividers
Voltage Divider + -
Current Divider + -

8. Wire Resistance
For dc systems wire resistance is key; for high voltage ac often the inductance (reactance) or capacitance (susceptance) are limiting
Resistance causes 1) losses (i2R) and 2) voltage drop (vi)
Need to consider wire resistance in both directions

9. AC: Phase Angles
Angles need to be measured with respect to a reference - depends on where we define t=0
When comparing signals, we define t=0 once and measure every other signal with respect to that reference
Choice of reference is arbitrary – the relative phase shift is what matters
Relative phase shift between signals is independent of where we define t=0

10. Example: Phase Angle Reference
Pick the bottom wave as the reference
Or pick the top as the reference- it does not matter!

11. Important Properties: RMS
RMS = root of the mean of the square
RMS for a periodic waveform
RMS for a sinusoid (derive this for homework)
In 333 we are mostly only concerned with sinusoidals

12. Important Properties: Instantaneous Power
Instantaneous power into a load
“Load sign convention” with current and power into load
positive + -
Identity

13. Important Properties: Average Power
Average power is found from
Find the average power into the load (derive this for homework)

14. Important Properties: Real Power
P is called the Real Power
cos(θV-θI) is called the Power Factor (pf)
We’ll review phasors and then come back to these definitions…

15. Review of Phasors
Phasors are used in electrical engineering (power systems) to represent sinusoids of the same frequency
A quick derivation…
Ap denotes the peak value of A(t)
Identity

16. Review of Phasors Use Euler’s Identity Identity
Written in phasor notation as
Identity
Tilde denotes a phasor
Note, a convention- the amplitude used here is the RMS value, not the peak value as used in some other classes!
Other, simplified notation
Regardless of what notation you use, it helps to be consistent.

17. Why Phasors? Simplifies calculations
Turns derivatives and integrals into algebraic equations
Makes it easier to solve AC circuits

18. Why Phasors: RLC Circuit
Solve for the current- which circuit do you prefer? + + - -

19. RLC Circuit Example

20. Complex Power S Q (θV-θI) P Q Reactive Power S Apparent power P
Power triangle S Q
(θV-θI)
Asterisk denotes complex conjugate P Q
Reactive Power S
Apparent
power P
Real Power
S = P+jQ

21. Apparent, Real, Reactive Power
P = real power (W, kW, MW)
Q = reactive power (VAr, kVAr, MVAr)
S = apparent power (VA, kVA, MVA)
Power factor angle
Power factor

22. Apparent, Real, Reactive Power
Remember ELI the ICE man
“Load sign convention” – current and power into load are assumed positive S Q
(θV-θI) P P
(θV-θI) Q S
Q and θ negative
(producing Q)
Q and θ positive
ELI
ICE
Inductive loads
I lags V (or E)
Capacitive loads
I leads V (or E)

23. Apparent, Real, Reactive Power
Relationships between P, Q, and S can be derived from the power triangle just introduced
Example: A load draws 100 kW with leading pf of What are the power factor angle, Q, and S?

24. Conservation of Power
Kirchhoff’s voltage and current laws (KVL and KCL)
Sum of voltage drops around a loop must be zero
Sum of currents into a node must be zero
Conservation of power follows
Sum of real power into every node must equal zero
Sum of reactive power into every node must equal zero

25. Conservation of Power Example
Resistor, consumed power
Inductor, consumed power

26. Power Consumption in Devices
Resistors only consume real power
Inductors only consume reactive power
Capacitors only produce reactive power

27. Example
Solve for the total power delivered by the source

28. Reactive Power Compensation
Reactive compensation is used extensively by utilities
Capacitors are used to correct the power factor
This allows reactive power to be supplied locally
Supplying reactive power locally leads to decreased line current, which results in
Decrease line losses
Ability to use smaller wires
Less voltage drop across the line

29. Power Factor Correction Example
Assume we have a 100 kVA load with pf = 0.8 lagging, and would like to correct the pf to 0.95 lagging
We have:
We want: S
Qdes.=?
P=80
This requires a capacitance of: P
Q=60
Q=-33.7
Qdes=26.3 P 29

30. Distribution System Capacitors for Power Factor Correction30