Circuits IIEE221

Unit 3Instructor: Kevin D. Donohue

Instantaneous, Average, RMS, and Apparent Power, and, Maximum Power

Transfer, and Power Factors

Power Definitions/Units:

Work is in units of newton-meters or joules same as energy.

1 joule is work done for 1 ampere passed through 1 ohm for 1 second, or work done by a force of 1 Newton applied over 1 meter.

Power is a measure of the rate at which work is done and is in units ofjoules per seconds, 746 Watts in one horsepower.

Power Conversion

Electric motors and generators convert electrical power to mechanical power and vise versa.

Other devices exist that convert electric power to light, heat, sound, …. and vise versa

Instantaneous and Average Power

The instantaneous power is the power absorbed by an element at an instance of time. In an electric circuit this is given by:

where i(t) is the current through the element and v(t) is the voltage drop over the element.

Average Power over some time interval T is given by:

If the i and v product is periodic, then PAV can be reduced to the integration over a single period. For random/non-periodic signals T goes to infinity.

)()()( titvtP

T

AV dttitvT

P0

)()(1

Instantaneous Power

Instantaneous power is simply computed by the product of 2 functions. For sinusoidal problems, trig identities or phasors can be used to simplify the products and result in easier formulae for average power.

Example: Write a Matlab script that plots the instantaneous power of a cosine voltage across an impedance load. Write the script so the voltage waveform and impedance can be easily changed and instantaneous power replotted.

Matlab Scripts

A script is a series of Matlab command line instructions typed in a text file and stored with an *.m extension. This is referred to as an mfile.

To run these commands change your current directory associated with the Matlab work space to the one containing the mfile. Then type the base name of the file in the Matlab command line.

Matlab Scripts

For this example comments are included in the script, and are identified by text following a “%” character.

The functions plot and cos are used in this script. Type help plot or help cos to get a full description of these functions in the Matlab workspace.

Matlab Scripts

% This script will plot the instantaneous power absorbed% by an impedance load with a sinusoidal voltage over it.

% Set the parameters of the Voltage signalf = 1000; % Frequency of signal in Hzph = 30; % Phase of signal in degreesA=2; % amplitude of signal

%Set impedance valuezm = 8; % Impedance magnitudezp = -40; % Impedance phase in degrees

% Create a time axis of 2 periods over which to plot the powertp = 1/f; % Determine periodt = 2*tp*[0:5000]/5000; % Make a 5001 point time axis (row vector) over 2 periods

Matlab Scripts

% Create a vector of points for voltage v = A*cos(2*pi*f*t + pi*ph/180);% Create a vector of points for the current (adjust magnitude and phase% based on impedancei = (A/zm)*cos(2*pi*f*t + pi*(ph-zp)/180);% Take an element by element product to get powerp = v.*i;% Plot itplot(t,p)title(['Instantaneous Power '] )xlabel(['Seconds'])ylabel(['Watts'])

Result

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Instantaneous Power

Seconds

Wat

ts

What is the meaning of negative instantaneous power?

Average Power in Periodic Signals

Given a sinusoidal voltage and current in a device:

Show:

)cos()(

)cos()(

ii

vv

tAti

tAtv

)cos(2

1ivivAV AAP

Average Power in Periodic Signals

Given a phasor representation of a voltage and current in a device:

Show:

ii

vv

AI

AV

ˆ

ˆ

)cos(2

1ˆˆRe2

1 *ivivAV AAIVP

Instantaneous and Average Power

Conservation of Power

In a given circuit the average power absorb (denoted by positive values) equals the power delivered (denoted by negative values).

For a circuit with N elements the sum of all power is zero:

N

iiP

1

0

Note: These are all real or average power values.

Passive Sign ConventionIt is assumed that positive

charge entering the positive terminal of an element implies power absorbed by the element.

Therefore, charge leaving the positive terminal of an element implies power supplied or delivered by the element.

If the words absorbed or supplied are not given with a power value, power absorbed will be assumed. (A negative sign will imply power supplied).

I

V

I

V

Example Average PowerFind the average power each of the elements given is(t) = 3cos(1000t)A

is

ia

25Ω

80µF

10mH

5Ω

10ia

P25Ω=34.61WPL=PC=0P5Ω=21.13WPCCVS=-5.54WPis=-50.19W

Maximum Power TransferGiven a Thévenin circuit with load ZL:

Show that for a maximum power transfer to the load:

And the maximum power transfer is:

*ˆˆthL ZZ

Zth

ZLVth

]ˆRe[8

ˆ

*

2

max

th

th

Z

VP

Example Maximum PowerFind the impedance of the load Z to result in the maximum power transfer, and find the resulting power. Assume vs(t) = 110cos(377t) V

Z=2.87-75.88Ith=1.6583.12AVth = 4.71159VPz=3.98W

vs Z8Ω40µF

7.5mH12Ω

Root Mean Square (RMS) Values

The RMS value of a periodic current or voltage is its DC equivalent value for delivering average power to a resistor.

T T

rmsAV RIdtT

tiRdttRi

TP 2

22 )(

)(1

T

rms dttiT

I )(1 2

T T

rmsAV R

Vdt

T

tv

Rdt

R

tv

TP

222 )(1)(1

T

rms dttvT

V )(1 2

RMS Formula for Sinusoids

The RMS value for a sinusoid of any frequency or phase is its amplitude divided by square root of 2.

TT

rms

Trms

Trms

Trms

dttjtjT

Adt

T

AI

dttjtjT

AI

dttjtj

T

AI

dttAT

I

))(2exp())(2exp(4

24

))(2exp()0exp(2))(2exp(4

2

))(exp())(exp(

)(cos1

22

2

22

22

202

4

2 AT

T

AI rms

Apparent Power and Power FactorApparent power (S) for sinusoidal waveforms is the product of the RMS voltage and current magnitudes without regard to their phase offsets. It is in units of Volt-Amps (VA). The cosine of their phase difference is the power factor (pf).

)cos()(

)cos()(

ii

vv

tAti

tAtv

)cos()cos(2

1ˆˆRe2

1 *ivrmsrmsivivAV IVAAIVP

ii

vv

AI

AV

ˆ

ˆ

power)(complex )(expˆ

factor)(power )cos(pf

power) (reactive )sin(

power) (real )cos(

power)(apparent

ivrmsrms

iv

iv

ivAV

rmsrms

jIVS

SQ

SP

IVS

Leading and Lagging

The sign of phase difference between the voltage and current is related to the angle of the impedance of the load. The terms leading and lagging are used to describe this property:

Load) Real(Only 0

Load) (Inductive lagging pf0

Load) e(Capacitiv leading pf0

iv

iv

iv