Transcript 15-16 ac power and power factor

AC Power: average power
v(t )  Vp cost 
i (t )  I p cost   
P (t )  i (t )v (t )
 I pV p cost  cost   

I p Vp
cos   I rmsVrms cos 
2 2
~ ~
 I V cos 
~ ~
~ ~
V  I Z (notice : V  I Z , phasor)
Z  R  jX
R
X
 R 2  X 2   j 
Z
Z
 Z 
cos  
R
Z
tan  
X
; use this to determine the sign
R
Example 7.2, P333
The power factor: cos(f
•The average ac power (Pav) is the power dissipated on
the load resistor.
• 0cos1, dependent on the complex load.
• ideal power factor: cos =1, Z=R, pure resistive load
v(t )  Vp cost 
i (t )  I p cost   
~~
P(t )  I V cos 
~
 I 2 Z cos 
~
 I 2R
Concept Check: thinking with phasors
The power factor in this series circuit is less than ideal.
Increasing the capacitance is found to improve the power factor.
Is the impedance in the box inductive or capacitive?
Answer: inductive
Complex Power
~ ~
V  V V
~ ~
I  I  I
~~
~
~
S  VI *  I 2 Z  V 2 / Z *
 Pav  jQ
• real power Pav: power absorbed by the load resistance.
• Q (volt-amperes reactive, VAR): exchange of energy
between the source and the reactive part of the load. No net
power is gained or lost during the process.
• S : compute by measuring the rms load voltage and
currents without regard for the phase angle.
• if Q<0, the load is capacitive, Q>0, the load is inductive
Power factor correction
• It is very common that load (motors,
power lines, etc) is inductive, i.e. Q>0.
• Common practice to bring the power
factor to unity is to connect a parallel
capacitor across the load, such that QC=Q. (the method is easy and need not
break the circuit).
Example 7.8, P345: power factor correction
~
VS  1170V ;   2 60( Hz )  377rad / sec
Z L  50  j86.7  1001.047
~
1170
~ VL
IL 

 1.17( 1.047)( A)
Z 1001.047
~~
S  VL I L *  1170  1.171.047
 68.4  j118.5(W )
~ 2
VL
117 2
XC 

 115
QC  118.5
C
1
1

 23.1F
X C 377  115
In theory, the power factor can be corrected to unity
by putting a capacitor in series with the load.
However, it is found (ex. 7.9) that this configuration
will result in significant increase in the current, and
thus increase the power required by the generator.