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Review of AC Circuits

Smith College, EGR 325

March 27, 2006

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Objectives

Power calculations and terminology

Expand understanding of electrical

power from simple linear circuits to

a high voltage power system

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Overview

Basic Circuits Sinusoidal waveform representation

Root mean square

Phase shift Phasors

Complex numbers

Complex impedance

Electric Power Complex: real & reactive power

Power factor and power factor correction

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ac Waveform

t

vVmax

wavefortheoffrequencytheis

2

f

f

tsinVv max

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How AC is Generated

Stator

Windings

N

S

Rotor

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Angle

v

X

N

S

f 900

1800

2700

3600

How AC is Generated

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)(sin

sin

max22

max11

tVv

tVv

V1

V2

22

11 0

VV

VVt

v1 v2

Reference

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Phasors

tjm

jm

etv

VeV

V

V

Re)(1

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Representing Power

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Power Calculations

P = VI

P = I2R

P = V2/R

S= VI

S= I2Z

S= V2/Z

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ResistanceImpedance

Resistance in

Capacitance in F

Inductance in H Z = R +jX

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Instantaneous vs. Average Power

)2cos(2

1)cos(

2

1)( ivmmivmm tIVIVtp

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Instantaneous vs. Average Power

)2cos(

2

1)cos(

2

1)( ivmmivmm tIVIVtp

Instantaneous power is written as

The average of this expression is

)cos(2

1ivmmIVP

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Real & Reactive PowerTime Domain

])2cos()[cos(2

)( maxmax vivi tIV

tp

t

Q(t)

)()( tQPtp

t

p

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Example: Current Flow

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Example: Power Flow

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Power System

Operations

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Real Power and Frequency

P and f dynamics are coupled Demand > Supply: frequency will decrease

(more energy drained from system than produced,

acts like brakes on the turbines)

Supply > Demand: frequency will increase(more energy in the power system than consumed,

acts like an accelerator so turbines spin faster)

Generation-based frequency regulation Generator inertia

Generator governors

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Frequency Problems

Imbalances in supply and demand beyond

the capabilities of these generator controls

Load may be dropped, or shed by operators

Equipment protection may disconnectgenerators

Operators may disconnect regional tie lines

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Voltage Collapse

The real power demanded is above thetransfer capability of a transmission line

Return to the water pipe analogy

Load draws too much powerdips into thestored reactive powercollapses the pipe

Equations: P = V*I, I = V/Z

Load wants more power: Decrease apparentimpedance (Z), to increase current draw (I),

which allows increased P

B u t, if P at limit, result is to decrease V

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Power (pu)

Voltage(pu)

Real Power FlowVoltage Relation

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Power System Response to Outages

Power flows on the paths of least

impedance

As elements are removed (fail), the

impedance changes and so power flows

change Instantaneously

Human and computer monitoring of and

reaction to problems is on a much slower

timescale