Lecture 31•Sinsuoidal steady state power

• Instantaneous and average (real) power• Reactive power• Complex power• Power factor

•Related educational materials: – Chapter 12.1 - 12.5

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

ZIV ZIV ZIV

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: (units = volt-amps = VA)

– Power factor (pf): (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:

iveffeff IVS

• 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) 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(d) 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

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663887 ..I S

(d) sketch a power triangle for the source

• Apparent power: 391VA• Average power: 305W• Reactive power: -244VAR