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

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

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

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

Review of Electric Circuits Ref 5 Defining Relations Resistors Capacitors Inductors

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)

Review of Electric Circuits – The RLC Circuit

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

Sinusoidal Analysis

Sinusoidal Analysis Period Phase shift Average rms value

Sinusoidal Analysis - Phasors Euler’s Identity Sine and Cosine

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

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

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

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

Review of Electric Circuits Phase shifts

Review of Electric Circuits Phasor Relations and Impedances Resistors Capacitors Inductors

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

Review of Electric Circuits – The Series RLC Circuit or

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

Sinusoidal Analysis - Instantaneous Power or or

Sinusoidal Analysis - Instantaneous Power With additional trigonometric manipulation or

Sinusoidal Analysis - Power Power and Phase

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

Sinusoidal Analysis - Power Rewritten where

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

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

Sinusoidal Analysis - Power Power Factor:

Sinusoidal Analysis – VARs Amusing explanation of VARs (and other related topics) http://electrical-engineering-portal.com/beer-mug-and-power-factor

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

Sinusoidal Analysis – Maximum Power Transfer

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

Sinusoidal Analysis – Maximum Power Transfer Results

Sinusoidal Analysis – Power factor correction