Lesson 24
AC Power and
Power Triangle
Learning Objectives

Define real (active) power, reactive power, average, and
apparent power.

Calculate the real, reactive, and apparent power in AC
series parallel networks.

Graph the real and reactive power of purely resistive,
inductive, or capacitive loads in AC series parallel networks
as a function of time.

Determine when power is dissipated, stored, or released in
purely resistive, inductive, or capacitive loads in AC series
parallel networks.

Use the power triangle determine relationships between
real, reactive and apparent power.
AC Power

AC Impedance is a complex quantity made up
of real resistance and imaginary reactance.
Z  R  jX

( )
AC Apparent Power is a complex quantity made
up of real active power and imaginary reactive
power:
S  P  jQ
(VA)
AC Real (Active) Power (P)


The Active power is the power that is dissipated
in the resistance of the load.
It uses the same formula used for DC (V & I are
the magnitudes, not the phasors):
2
V
2
PI R
R
[watts, W]
WARNING! #1 mistake with AC power calculations!
The Voltage in the above equation is the Voltage drop across the resistor, not
across the entire circuit!
CAUTION!
REAL value of resistance (R) is used in REAL power calculations, not
IMPEDANCE (Z)!
AC Imaginary (Reactive) Power (Q)


The reactive power is the power that is exchanged
between reactive components (inductors and capacitors)
The formulas look similar to those used by the active
power, but use reactance instead of resistances.
2
V
QI X 
X
2
[VAR]
WARNING! #1 mistake with AC power calculations!
The Voltage in the above equation is the Voltage drop across the reactance, not
across the entire circuit!


Units: Volts-Amps-Reactive (VAR)
Q is negative for a capacitor by convention and positive
for inductor.

Just like X is negative for a capacitor! (-Xcj)
AC Apparent Power (S)


The apparent power is the power that is
“appears” to flow to the load.
The magnitude of apparent power can be
calculated using similar formulas to those for
active or reactive power:
2
V
2
S  VI  I Z 
Z


[VA]
Units: Volts-Amps (VA)
V & I are the magnitudes, not the phasors
Reactive power calculated with X
Real power calculated with R

Apparent power calculated with Z
AC Power
Notice the relationship between Z and S:
ZR  j X
SP  j Q
( )
(VA)
Power Triangle

The power triangle graphically shows the
relationship between real (P), reactive (Q) and
apparent power (S).
S  P2  Q2
S  P  jQL
S  S 
Example Problem 1
Determine the real and reactive power of each
component.
Determine the apparent power delivered by the
source.
Real and Reactive Power

The power triangle also shows that we can find
real (P) and reactive (Q) power.
S  IV
P  S cos
Q  S sin 
(VA)
(W)
(VAR)
NOTE: The impedance angle and
the “power factor angle” are the
same value!
Example Problem 2
Determine the apparent power, total real and
reactive power using the following equations:
S  VI
P  S cos
Q  S sin 
(VA)
(W)
(VAR)
Total Power in AC Circuits


The total power real (PT) and reactive power
(QT) is simply the sum of the real and reactive
power for each individual circuit elements.
How elements are connected does not matter
for computation of total power.
P1
Q1
PT  P1  P2  P3 PP4
T
QT  Q1  Q2  Q3 QQT4
P2
Q2
P3
Q3
P4
Q4
Total Power in AC Circuits

Sometimes it is useful to redraw the circuit to
symbolically express the real and reactive power loads
Example Problem 3
a.
b.
c.
d.
Determine the unknown real (P2) and reactive powers
(Q3) in the circuit below.
Determine total apparent power
Draw the power triangle
Is the unknown element in Load #3 an inductor or
capacitor?
Example Problem 4
a.
b.
Determine the value of R, PT and QT
Draw the power triangle and determine S.
Use of complex numbers in Power calculations



AC power can be calculated using complex equations.
Apparent Power can be represented as a complex number
The resultant can be used to determine real and reactive power by
changing it to rectangular form.
I*is complex conjugate of I

S  VI  P  jQ
S
V
Z
P

2
QC
2

I Z
S
NOTE!
The complex conjugate of Current is used to make the power angle the same as
the impedance angle!
Power Factor


Power factor (FP) tells us what portion of the
apparent power (S) is actually real power (P).
Power factor is a ratio given by
FP = P / S

Power factor is expressed as a number
between 0 to 1.0 (or as a percent from 0% to
100%)
Power Factor

From the power triangle it can be seen that
FP = P / S = cos 

Power factor angle is thus given
 = cos-1(P / S)



For a pure resistance,  = 0º
For a pure inductance,  = 90º
For a pure capacitance,  = -90º
S
Q
NOTE: Ө is the phase angle of ZT, not the
current or voltage.

P
Unity power factor (FP = 1)




Implies that all of a load’s apparent power is
real power (S = P).
If FP = 1, then  = 0º.
It could also be said that the load looks purely
resistive.
Load current and voltage are in phase.

P,S
Q=0
Lagging power factor ( > 0º)

The load current lags load voltage

Implies that the load looks inductive.
S
Q

P
VARind
ELI
Leading power factor ( < 0º)

The load current leads load voltage ICE

Implies that the load looks capacitive.
P

Q
S
VARcap
Example Problem 5
a. Determine P,Q,S and the power factor for this circuit.
Draw the power triangle.
b. Is it a leading or lagging power factor?
c. Is the circuit inductive or capacitive?
Example Problem 6
a. Determine total current, apparent power, and the power
factor for this circuit. Is it a leading or lagging power
factor?
b. Determine total current, apparent power, and the power
factor if the capacitor reactance is decreased to 40
ohms. What kind of power factor does it have?