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 overbye@illinois.edu

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, 2.14. 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

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.

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

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 -

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

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

Voltage and Current Dividers Voltage Divider + - Current Divider + -

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

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

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

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

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

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

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…

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

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.

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

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

RLC Circuit Example

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

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

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)

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 0.85. What are the power factor angle, Q, and S?

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

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

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

Example Solve for the total power delivered by the source

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

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

Distribution System Capacitors for Power Factor Correction 30