ECE 325 –Electric Energy System Components 2‐Fundamentals ...

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Instructor: Kai SunFall 2018

ECE 325 – Electric Energy System Components2‐ Fundamentals of Electrical Circuits

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Content

• Fundamentals of electrical circuits (Ch. 2.0-2.15, 2.32-2.39)

• Active power, reactive power and apparent power (Ch. 7)

• Three-phase AC systems (Ch. 8)

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Notations: Current and Alternating Current

• Arbitrarily determine a positive direction, e.g. 12– If a current of 2A flows from 1 to 2, I=+2A– If a current of 2A flows from 2 to 1, I= -2A

G

2

1

I

I >0 I <0

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Notations: Voltage

1. Double-subscript notation:E21= +100V (the voltage between 2 and 1 is 100V, and 2 is positive w.r.t 1) E12 = -100V

2. Sign notation: Arbitrarily mark a terminal with (+); E>0 if and only if that marked terminal is positive w.r.t the other.E.g. if E21=+100V, E=E21=+100V.

100V

+

-

2

1

G

2

1

+

E

Both the double-subscript notation and sign notation apply to alternating voltage

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Notations: Alternating Voltage

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Notations: Sources and Loads

• Definition: given the instantaneous, actual polarity of voltage and actual direction of current– Actual Source: whenever current flows out of the terminal (+) – Actual Load: whenever current flows into the terminal (+)

• How about these?– Resistor, battery cell, electric motor, capacitor and inductor

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i(t)+v(t)

Resistor

Inductor Capacitor

load  source source load

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1‐Phase AC System with Sinusoidal Voltage and Current

• e , i: instantaneous voltage (V) and current (A)• Em , Im : peak values of the sinusoidal voltage (V) and current (A)• = 2f (rad/s): angular frequency, which is assumed constant here• e , i : constant phase angles (rad. or deg.) of voltage and current• Em/ 2, Im/ 2: RMS (root-mean-square, effective) values

( ) co s( )m ee t E t

( ) cos( )m ii t I t

Load

+

22 2[ ( )] =

2t m

dct T

I RTi t Rdt I RT

Equal heating effects

RMS value2m

dcII

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Phasor Representation 

• E and I are called RMS phasors of e(t) and i(t); E leads I by = e- i or in other words, I leads E by 2-

• Phasor:– mapping a time-domain sinusoidal waveform (infinitely long in

time) to a single complex number– carries the amplitude and phase angle information of a sinusoidal

signal of a common frequency () w.r.t. a chosen reference signal.

( ) cos( )( ) cos( )

m e

m i

e t ti

EIt t

Constant 2

2

me

mi

EE

II

2 cos( )| |

|2 c s( )| oe

i

E

I

t

t

| | | |

| | | |

e

i

je

ij

E E e

I I e

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Impedance

• Impedance is a complex number (in ) defined as

• Purely resistive:

• Purely inductive:

• Purely capacitive:

| | eE E

| | iI I +

def12 | | | | ( )

| |e

e ii

EE EZ Z R jXI I I

| |Z Z R

| | 90 LZ Z jX jX j L

1

2

12E

1| | 90 CZ Z jX jX jC

Z

def

e i (Impedance angle)

Impedance of branch 1-2 =Voltage drop on 1-2Current flow into 1

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Example 2‐5

• Draw the phasor diagram of the voltage and current at a frequency of 60Hz. Calculate the time interval t between the positive peaks of E and I

Solution:=2f=377 (rad/s)=21600 (deg/s)|E|=339/2=240 (V)|I|=14.1/2=10 (A)Choose an arbitrary reference to draw phasors E and It=/=30/21600=0.00139 (s)

E240V

I10A

30o

( ) 339cos(21600 90 )( ) 14.1cos(21600 120 )

e t ti t t

,t

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Kirchhoff’s Voltage Law (KVL)

• The algebraic sum of the voltages around a closed loop (CW/CCW) is zero ( voltage rises = voltage drops)– Applied to both instantaneous

voltages or voltage phasors• Loop 24312 (BCDA, CW):

E24+E43+E31+E12=0 ore24+e43+e31+e12=0

• Loop 2342 (ECF, CCW): E23+E34+E42=0 ore23+e34+e42=0

E12

E43

E31

E24

E23

4 2

13

E34

E42

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Kirchhoff’s Current Law (KCL)

• The algebraic sum of the currents arriving at a node is equal to 0.( currents in = currents out)

• Node A: I1+I3=I2+I4+I5

or i1+i3=i2+i4+i5

I1+I3+(-I2)+(-I4)+(-I5)=0 or i1+i3+(-i2)+(-i4)+(-i5)=0

A

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• KVL– Loop 24312, CW:

E24 +E43 +E31 +E12 =0I2Z2 -I3Z3 +Eb -I1Z1 =0

– Loop 2342, CCW: E23 +E34 +E42 =0 I4Z4 +I3Z3 -I2Z2 =0

– Loop 242, CW:E24 + E42 =0Ea - I2Z2 =0

+

+

Kirchhoff’s Laws and AC Circuits

• KCL– Node 2:

I5 -I2 -I4 -I1=0– Node 3:

I4 +I1 – I3=0

Ea

Eb

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Examples 2‐14 to 2‐16

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Impedance angle: >0 for inductive load and <0 for capacitive load

( ) ( ) ( ) cos( )cos( )1 cos( ) cos(2 )21 cos( ) cos[2( ) ( )]21 cos cos[2( ) ]21 cos cos cos 2( ) sin sin 2( )2

m m e i

m m e i e i

m m e i e e i

m m e

m m e e

p t e t i t E I

E I

E I

E I

E I

| | / 2

| | / 2m

m

E E

I I

e i

( ) cos [1 cos 2( )] sin sin 2( )R X

e e

p p

p E I E I

Using trigonometric identity1

cos cos cos( - ) cos( )2

A B A B A B

Instantaneous Power

( ) cos( )m ee t E t

( ) cos( )m ii t I t Load

Z

+

t