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E E 2415
Lecture 9Phasor Circuit Analysis,
Effective Value and Complex Power: Watts, VAR’s and Volt-
Amperes
Effective Value of a Sinusoid (1/3)
Average Power:
i(t) 10
Effective Value of a Sinusoid (2/3)
In our example:
Also:
The effective value is also called the Root Mean Square value or rms value.
Effective Value of a Sinusoid (3/3)
R-C Circuit Example (1/6)
Capacitive Reactance
vs(t)
40
88.42 F
+ vR - +vc-
i
Using rms phasor for voltage source.
R-C Circuit Example (2/6)
120
40
-j30 + VR -
~+VC-
~I~
Calculate Real Power:
And Reactive Power:
Apparent power is the product of voltage and current of the source.
Also:
R-C Circuit Example (3/6)
Power Factor is the ratio of real power to apparent power:
Power Factor is also the Cosine of the anglebetween the load voltage and the load current:
If the load current leads the load voltage, the power factor is leading; if it lags the load voltage,the power factor is lagging.
R-C Circuit Example (4/6)
Phasor Diagram of Voltage and Current
Current leads voltage.
R-C Circuit Example (5/6)
36.87o
I~
~V120V
2.4A
The Power Triangle showing leadingpower factor.
R-C Circuit Example (6/6)
230.4 W
-172.8 VAR288.0 VA
real
imag
Calculating Complex Power (1/2)
2S P jQ I R jX 2 *I I I
*
2* 2 2
,I c jd I c jd
then I I c d I
*S P jQ I I R jX
R jXI~
+ V -~
c
jdI~
Calculating Complex Power (2/2)
V R jX I
*S P jQ V I
From now on, we use the above method to calculate complex power.
Lagging Power Factor Example (1/4)
v(t)
5
22.97 mHi
Calculate complex power directly:
Lagging Power Factor Example (2/4)
1200°
5
j8.66 I~
Power Factor:
Power Factor is Lagging
Phasor Diagram of Voltage and Current
Lagging Power Factor Example (3/4)
60o
1200° V
12-60° A
imag
real
Power Triangle for lagging Power Factor
Lagging Power Factor Example (4/4)
1440
VA
720 W
1247 VAR's
imag
real
Phasor Power Example (1/4)
480V(rms)
0.5 j2
40
j30
-j150
+
V
-
~
I~
480 07.94 20.08
(56.75 20.75)
VI A
j
(40 30)( 150)(56.25 18.75)
40 30 150p
j jZ j
j j
Phasor Power Example (2/4)
3.550.949 94.9%
3.74PF or Lagging
*
(471 1.65 )(7.94 20.08 )
3.55 1.18
RLCS V I
V A
kW j kVAR
(56.25 18.75) 471 1.65V I j
Phasor Power Example (3/4)
*
(471 1.65 )(9.42 38.52 )
3.55 2.66
RL RLS V I
V A
kW j kVAR
471 1.659.42 38.52
(40 30)RL
VI A
j
Capacitor VAR’s:
2 24711.48
150c
VQ kVAR
Phasor Power Example (4/4)
3.55 kW
2.66 kVAR
1.18 kVAR 4.44 k
VA
3.74 kVA
1.48 kVAR}
P
Q
Impedance and Admittance
jXRZ
jBGZ
Y 1
22 XRR
G
22 XRX
B
Impedance
Admittance
Conductance
Susceptance