Sinusoidal Steady
State Power
Ch. 10 – Sinusoidal Steady State Power
2
Instantaneous Power
p = vi
v  Vm cos t   v 
i  I m cos t  i 
p
p  vi
p  vi  Vm I m cos t   v  i  cos t
Vm I m
V I
V I
cos  v  i   m m cos  v  i  cos 2t  m m sin  v  i  sin 2t
2
2
2
ELEC 250 – Summer 2015
Ch. 10 – Sinusoidal Steady State Power
Instantaneous current, voltage and power vs. ωt
ELEC 250 – Summer 2015
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Ch. 10 – Sinusoidal Steady State Power
4
Average and Reactive Power
Vm I m
cos v  i 
2
Vm I m
Q
sin v  i 
2
P
p  P  P cos 2t  Q sin 2t
 P is the average power while Q is the reactive power

Note that P is indeed the average of the instantaneous power
over one period
1 t0  T
1 t0  T
p 
p  t  dt    P  P cos 2t  Q sin 2t  dt 
t
T 0
T t0
1 t0  T
1 t0  T
1 t0  T
Pdt   P cos 2tdt   Q sin 2tdt 

t
0
T
T t0
T t0
P00  P
ELEC 250 – Summer 2015
Ch. 10 – Sinusoidal Steady State Power
Average and Reactive Power

Q arises in the presence of an inductance or a capacitance
while P arises in the presence of a resistance

P corresponds to the portion of the power that is
“consumed” by the resistive components of the circuit
(converted to heat).

Q corresponds to the power that is stored and
subsequently released by the inductive or capacitive
components of the circuit.
ELEC 250 – Summer 2015
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Ch. 10 – Sinusoidal Steady State Power
6
Power for purely resistive circuits

Assuming that
i  I m cos t t  i 
v  RI m cos t  i 
v  Vm cos t
 Then
Vm  RI m
 v  i  0
+V
-
R
Vm I m
V I
cos  v  i   m m
2
2
V I
Q  m m sin  v  i   0
2
V I
p  P  P cos 2t  m m 1  cos 2t 
2
P

i
ELEC 250 – Summer 2015
Ch. 10 – Sinusoidal Steady State Power
7
Power for purely inductive circuits

Assuming that
vL
i  I m cos t t  i 
v  Vm cos t  v 
L
+ V -
di
  LI m sin(t  i )  LI m cos(t  i  90 )  Vm cos(t  v )
dt
Vm I m
V I
cos  v  i   m m cos  90   0
2
2
V I
V I
V I
Q  m m sin  v  i   m m sin  90   m m
2
2
2
Vm I m
 LI 2 m
p  Q sin  2t   
sin  2t  
sin  2t 
2
2
P
 Then
i
Vm  L I m 
v  i  90
ELEC 250 – Summer 2015
Ch. 10 – Sinusoidal Steady State Power
8
Power for purely capacitive circuits
i

Assuming that
i C
i  I m cos t t  i 
v  Vm cos t  v 
C
+V-
dv
 CVm sin t  v   CVm cos t  v  90   I m cos t  i 
dt
 Then
Im

C
 v  i  90
Vm 
Vm I m
V I
cos  v  i   m m cos  90   0
2
2
V I
V I
V I
Q  m m sin  v  i   m m sin  90    m m
2
2
2
Vm I m
I 2m
p  Q sin  2t  
sin  2t  
sin  2t 
2
2C
P
ELEC 250 – Summer 2015
Ch. 10 – Sinusoidal Steady State Power
VAR


VAR stands for Volt Ampere Reactive
It is used to distinguish Q (the reactive power) from P (the real power)
Power Factor Concept
p
Vm I m
1 t0 T
1 t0 T
p
t
dt

P

P
cos
2

t

Q
sin
2

t
dt

cos v  i   P




T t0
T t0
2
• The average power delivered to a load for a given magnitude of a current
and a voltage, depends on the phase between these two quantities.
o For capacitive or inductive loads the phase v  i  90 and pavg= 0
o For resistive loads (θv-θi=0°) and pavg = VmIm/2
ELEC 250 – Summer 2015
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Ch. 10 – Sinusoidal Steady State Power
Power Factor


The quantity pf  cos  v  i  is the power factor
 It quantifies how effectively the provided voltage and current are
converted to power.
The quantity rf  sin  v  i  is the reactive power factor
 It quantifies the portion of power that is used (delivered or absorbed) in
the reactive elements

The power factor for inductive loads is lagging (i.e. the current lags the voltage)

The power factor for capacitive loads is leading (i.e. the current leads the voltage)
ELEC 250 – Summer 2015
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Ch. 10 – Sinusoidal Steady State Power
Power Factor






Electrical industry strives for a pf = 1.
A small pf signifies a larger current is needed to the same power.
A small pf results in (proportionally) large transmission losses
Utilities charge extra for small power factors
Industry often corrects its power factor by adding banks of
capacitors
Most of the power factor issues arise because of inductive loads
(i.e. motors)
ELEC 250 – Summer 2015
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Ch. 10 – Sinusoidal Steady State Power
RMS and Power calculations

From the definitions of the average and reactive powers P and Q
Vm I m
Vm I m
P
cos  v  i  
cos  v  i   VRMS I RMS cos  v  i 
2
2 2
Q
Vm I m
V I
sin  v  i   m m sin  v  i   VRMS I RMS sin  v  i 
2
2 2
Special Case: For a resistive load v  
or
v2 t 
P t  

R
P t   i2 t  R 
P
2
V 2 RMS
1 v t 
1 1 T t0 2
2
dt 
V m cos t    dt 
T R
R T t0
R
t0 T
t0
P
t0 T
t0
1 2
1 T t0
i  t  Rdt  R  I 2 m cos 2 t    dt  RI 2 RMS
T
T t0
ELEC 250 – Summer 2015
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Ch. 10 – Sinusoidal Steady State Power
13
Complex Power

Define as Complex Power
•
Given the definition of P and Q
 Then
S  P  jQ 
S = P + jQ
Vm I m
cos  v  i 
2
V I
Q  m m sin  v  i 
2
P
Vm I m
V I
V I j  
V
I
cos v  i   j m m sin v  i   m m e  v i   m e jv m e ji
2
2
2
2
2
S
Apparent Power
RMS
*
RMS
1
 V
2
S = P 2 + Q2
ELEC 250 – Summer 2015
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Ch. 10 – Sinusoidal Steady State Power
14
Complex Power
S
eff
2
eff
*
eff
R j
Z
eff
2
eff
*
eff

2
eff
 R  jX  
X  P  jQ 
X is the reactance part of the impedance
R is the resistance part
P
Q
2
eff
2
eff
R
X
Note that an alternate reference to the RMS value is that of the effective value. Hence
VRMS
Vm
= Veff =
2
I RMS
Im
= I eff =
2
ELEC 250 – Summer 2015
Ch. 10 – Sinusoidal Steady State Power
15
Maximum Power Transfer


A similar law applies as the one we encountered for
purely resistive networks.
For maximum power transfer to load
ZL = Z
*
Th
ZTh = RTh + jXTh
Z L = RL + jX L
 The average power absorbed by the load
2
Pmax 
2
RL 
Th
2
2
RL
 RL  RTh    X L  X Th 
2

Th
2
RL
 RL  RL 
2

Th
RL
4 RL 2
2
2

Th
4 RL
Vm
2
2
VTh
Vm
2
2
P = I RL =
=
=
2
4R
4RL
8RL
ELEC 250 – Summer 2015
Ch. 10 – Sinusoidal Steady State Power
16
Proof:

If
ZTh = RTh + jXTh
 Then the current (RMS) phasor is
I=
Z L = RL + jX L
VTh
( RL + RTh ) + j ( XL + XTh )
2
 The average power at the load is
 P is maximized when the derivatives
w.r.t. RL and XL are zero
P = I RL =
2
VTh RL
( RL + RTh ) + ( XL + XTh )
2
2
 VTh 2 RL  X L  X Th 
P

 0  X L   X Th
X L  R  R 2   X  X 2  2
Th
L
Th
 L

2
 RL  RTh 2   X L  X Th 2  2 RL  RL  RTh  

V
Th
P



2
RL
 RL  RTh 2   X L  X Th 2 


2
2
 VTh  RTh 2   X L  X Th   R 2 L 

 0 R  R 2  X  X 2

 L Th 
L
Th
2
 RL  RTh 2   X L  X Th 2 


ELEC 250 – Summer 2015
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Z L  RL  jX L  RTh  jX Th  Z *Th