Lesson 24AC 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.

AC Apparent Power is a complex quantity made up of real active power and imaginary reactive power:R jX Z

�������������� ( )

jQ AP V S

( )

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):

22 [watts, W]P I R

V

R

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.

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) 2

2 [VAR]Q I XX

V

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!

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:

Units: Volts-Amps (VA) V & I are the magnitudes, not the phasors

22 [VA]

VS VI I Z

Z

AC Power Notice the relationship between Z and S:

R j X Z��������������

( )

P j Q VA S

( )

Re

al p

ower

cal

cula

ted

with

R

Re

act

ive

po

we

r ca

lcu

late

d w

ith X

Ap

pare

nt p

ower

cal

cula

ted

with

Z

Power Triangle The power triangle graphically shows the

relationship between real (P), reactive (Q) and apparent power (S).

2 2

LP jQ

QPS

S

S

S

Example Problem 1Determine 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.

cos

sin

S IV

P S

Q S

(VA)

(W)

(VAR)

NOTE: The impedance angle and the “power factor angle” are the

same value!

Example Problem 2Determine the apparent power, total real and reactive power using the following equations:

cos

sin

S VI

P S

Q S

(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.

P1Q1

P4Q4

P3Q3

P2Q2

PTQT

1 2 3 4

1 2 3 4

T

T

P P P P P

Q Q Q Q Q

Total Power in AC Circuits Sometimes it is useful to redraw the circuit to

symbolically express the real and reactive power loads

Example Problem 3a. Determine the unknown real (P2) and reactive powers (Q3) in the circuit below.

b. Determine total apparent power

c. Draw the power triangle

d. Is the unknown element in Load #3 an inductor or capacitor?

Example Problem 4

a. Determine the value of R, PT and QT

b. Draw the power triangle and determine S.

WARNING…

Proofs for Real and reactive Power calculations follow…

AC Power to a Resistive Load In ac circuits, voltage and current are functions of time. Power at a particular instant in time is given

This is called instantaneous power.

2( sin )( sin ) sin 1 cos 22

m mm m m m

V Ip vi V t I t V I t t

Average Power to a Resistive Load

p is always positive All of the power delivered by the source is

absorbed by the load. Average power P = VmIm / 2

Average Power to a Resistive Load Using RMS values V and I

Active power is the average value of instantaneous power.

rms value of voltage2

rms value of current2

(watts)2 2 2

mRMS

mRMS

m m m mRMS RMS

VV

II

V I V IP V I

Power to an Inductive Load Consider the following circuit where

i = Im sin t .

Can we write an expression instantaneous power or pL(t) ?

Power to an Inductive Load

sin

sin( 90)

( sin 90 )( sin ) cos sin

sin 2 sin 2 sin 22 2 2

m

m

m m m m

m m m mRMS RMS

i I t

v V t

p vi V t I t V I t t

V I V It t V I t

Power to an Inductive Load p is equally positive and negative. All of the power delivered by the source is

returned. Average power PL = 0 W

Reactive Power Reactive power is the portion of power

that flows into load and then back out. It contributes nothing to average power. The power that flows into and out of a pure

inductor is reactive power only.

Power to a Capacitive Load Consider the following circuit where

i = Im sin t .

Can we write an expression instantaneous power or pC(t) ?

Power to a Capacitive Load

sin

sin( 90)

( sin 90 )( sin ) cos sin

sin 2 sin 2 sin 22 2 2

m

m

m m m m

m m m mRMS RMS

i I t

v V t

p vi V t I t V I t t

V I V It t V I t

Power to a Capacitive Load p is equally positive and negative All of the power delivered by the source is returned

(no power losses with a pure reactive load). Average power PC = 0 W

AC Power to a Resistive Load

AC Power to a Inductive Load

AC Power to a Capacitive Load