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Instructor: Kai SunFall 2018
ECE 325 – Electric Energy System Components2‐ Fundamentals of Electrical Circuits
2
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)
3
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
4
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
5
Notations: Alternating Voltage
6
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
7
i(t)+v(t)
Resistor
Inductor Capacitor
load source source load
8
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
9
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
10
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
11
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
12
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
13
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
14
• 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
15
Examples 2‐14 to 2‐16
16
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