Lecture 31
•Sinsuoidal steady state power
• Instantaneous and average (real) power
• Reactive power
• Complex power
• Power factor
•Related educational modules:
–Section 2.9.0, 2.9.1
AC power
• Power is still the product of voltage and current:
• We are now interested in the case in which the
voltage and current are sinusoids:
v(t) and i(t) are related
• Keep in mind that power is delivered to a load
• The amplitude and phase of the voltage and current
are not independent
– They are related through the load impedance
V  I Z  V  I  Z and V   I  Z
Instantaneous AC power
• Our previous (time domain) definition of power is
called the instantaneous power
• In terms of our sinusoidal voltage & current:
• After some trigonometry and algebra:
– The power consists of a DC (constant) part and an AC
(sinusoidal part)
Graphical representation of p(t)
Alternate representation for p(t)
• Which can be decomposed into two plots:
– Average (real) power and reactive power
Average Power
• We are generally more interested in the average
power delivered to a load:
• Average power is:
– This is also called the real power (it’s the power that’s
provided to the resistive part of the load over time)
– Units are watts
RMS values
• We want to assess the power delivered by different
types of time-varying signals
– The power delivered to a resistive load:
• Find a DC (constant) value which delivers the same
average power as the time-varying signal
– Called the effective or RMS value of the signal
– Used to “compare” different time-varying signals
• Note: we want our average power to look like
an “average” current squared times resistance
or an “average” voltage squared divided by
resistance
– We want to define these “effective” values
• Note why it’s called “RMS”
RMS values – continued
• Average power:
• Effective DC value:
• Equating to time-average value:
•
,
• Annotate previous slide to show VRMS, IRMS
notation (RMS = “effective”)
Definition of RMS values
• The effective (or RMS) value of a signal is equal to
the DC value which provides the same average
power to a resistor
• For sinusoidal signal with no DC offset:
,
• Average power in terms of RMS values:
Apparent power and power factor
• Power in terms of RMS values:
• The average (real) power is the product of apparent
power and the power factor
– Apparent power:
– Power factor (pf):
(units = volt-amps = VA)
(unitless)
• Power factor is leading or lagging, to denote whether current
leads or lags voltage
Interpretation of apparent power and pf
• Power factor is a property
of the load
– For a complex load, the
power delivered to the load
is not exactly the power
supplied by the generator
– If ZL is real  pf = 1
– If ZL is imaginary  pf = 0,
and no average power is
delivered to the load
• On previous slide, mention reactive power
again.
Complex Power
• Complex power is a way to conveniently expressing
the various power parameters and their relationships

or:
S  Veff I eff v   i 
• Annotate previous slide to show real (average)
power and reactive power
Power relationships
• Complex power:
– Magnitude of S is the apparent power (units = VA)
– The real part of S is the average power (units = watts)
– The imaginary part of S is the reactive power (units = VAR)
Power Triangle
Example
• For the circuit below,
(a)
(b)
(c)
(d)
find the average power delivered by the source
find the powers absorbed by the resistor and capacitor
find the apparent and reactive powers delivered by the source
sketch a power triangle for the source
(a) find the average power delivered by the source
(b) find the powers absorbed by the resistor and capacitor
(c) find the apparent and reactive powers delivered by the source
V S  1000
I S  7.838.66
(d) sketch a power triangle for the source
• Apparent power: 391VA
• Average power: 305W
• Reactive power: -244VAR