1 Review of AC Circuits Smith College, EGR 325 March 27, 2006.

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Review of AC Circuits

Smith College, EGR 325March 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

v Vmax

waveform theoffrequency theis

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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t

v

sec/3772

sinmax

radf

tVv

2

VVV max

rms

AC Phasor Representation

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)(sin

sin

max22

max11

tVv

tVv

V1

V2

22

11 0

VV

VVt

v1v2

Reference

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)(cos

cos

max22

max11

tVv

tVv

V1

V2

22

11 0

VV

VVt

v1v2

Reference

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Phasors

tj

mj

m

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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Resistance Impedance

• Resistance in • Capacitance in F

• Inductance in H

• Z = R + jX

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Instantaneous Electric Power [p(t)]

)sin()(

)sin()(

max

max

tIti

tVtv

])2cos()[cos(2

)( maxmax tIV

t

Fixed average Zero average

V

I

)sin()sin()(*)()( maxmax ttIVtitvt

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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 Power – Time Domain

])2cos()[cos(2

)( maxmaxvivi t

IVtp

t

Q(t)

)()( tQPtp

t

p

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Complex Power

*IVS

sincos IVjIVS

VIIVIVS 0*

V

I

IMPORTANT is the power factor angle

QjPS

II

Real Power Reactive Power

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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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Operating Challenges

• Load is stochastic and is not controlled

• Power flows cannot be directed or controlled

• Electricity cannot be stored

• Everything happens in real-time

• Generation can be controlled

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Power System Variables

• Generators produce complex power– S = P + jQ– Real power, P, able to perform useful

work – Reactive power, Q, supports the system

electromagnetically

• Single system frequency, f

• Voltage profile, V

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Real Power Flow – Voltage Relation

Power (pu)

Vo

ltag

e (p

u)

• In normal system operation, frequency/real-power dynamics are decoupled from voltage/reactive-power

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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 disconnect

generators– Operators may disconnect regional tie lines

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Reactive Power Analogy

• Voltage and reactive power allow real power to flow

• Reactive power – Energy stored in capacitance and inductance– Supports the electromagnetic fields along

transmission lines– Cannot be transmitted long distances

• Analogy– Inflatable water pipes

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Voltage Collapse

• The real power demanded is above the transfer capability of a transmission line

• Return to the water pipe analogy– Load draws too much power – dips into the

stored reactive power – “collapses” the pipe

• Equations: P = V*I, I = V/Z– Load wants more power: Decrease apparent

impedance (Z), to increase current draw (I), which allows increased P

– But, if P at limit, result is to decrease V

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Power (pu)

Vo

ltag

e (p

u)

Real Power Flow – Voltage 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