Fundamental Electrical Power Concepts

2

1

1

T

T

p t v t i t

P v t i t dtT

I i t dtT

Instantaneous Power:

Average Power:

RMS (effective value):

Using RMS in Power Calculations

The total current through R is i1(t)+i2(t).

The instantaneous power is: p(t) =R(i1(t)+i2(t))2

The Average power over interval T is:

1

1

2

1 2

1t T

t

P R i t i t dtT

1 1 1

1 1 1

2 2

1 1 2 2

1 1 12

t T t T t T

t t t

R i t dt i t i t dt i t dtT T T

1

1

2 2 21 2 1 2

2t T

eff

t

RI R I RMS I RMS i t i t dtT

1

1

2 2 21 2 1 2 IFF 0

t T

eff

t

I I RMS I RMS i t i t dt

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ˆ ˆ ˆ

ˆ ˆˆ

321

321

"" are and sequences then the0 If OrthogonalnBnAban

nn

0 BA

Given two vectors:

How can we determine if the two vectors are perpendicular?

Dot Product!

BA

"" are and functions then the0 If OrthogonaltBtAdttBtA

21 range over the

"" are and functions then the0 If2

1

tt

tBtAdttBtAt

t

Orthogonal

If then they are perpendicular!

Orthogonal Functions

2coscos tBtA

2121 if coscos tBtA

Periodic functions with different periods are orthogonal.

All even functions are orthogonal to all odd functions.

Functions with non-zero average values are not orthogonal.

Any constant (DC) function is orthogonal to any function with zero average value (0-mean, or AC).

For 0-mean/AC functions:

Parseval’s Thorem

+_

+_

+_

R

+

v(t)

_

v1(t)

v2(t)

v3(t)

232

22

12 RMSVRMSVRMSVRMSV

If v1(t), v2(t) and v3(t) are orthogonal

R

RMSV

R

RMSV

R

RMSV

R

RMSV 23

22

21

2

321 PPPPT

Application #1

Application #2

0

0

21ˆˆ( )t

RMS

t

i t I i t dt

( )i t

T

t0

0 0 0

0 0 0

22 22 1 1 ˆ 0t T t t T

RMS

t t t

I i t dt i t dt dtT T

0

0

22 21 ˆˆt

RMS RMS

t

I i t dt IT T

ˆRMS RMSI I

T

80 uS100 uS

Application #2 Exampleˆ( )i t( )i t

ˆˆ ˆDC aci t I i

15 a

7 a

RMS

15 a 7 aˆ 11 amp 11 A 2DCI

4ˆ : Triangular 4 amp 0-pk A 3

ac RMSi

22

RMS

4ˆˆ( ) 11 11.24 A3

RMSi t I

RMSˆ 10.0 ARMS RMSI I

T

RMS

80ˆ 11 8.8 A100DC DCI I

T

2 2AC RMS DCi I I 2 2

RMS10 8.8 = 4.75 A

Application #3

1 1

T T

P p t dt v t i t dtT T

P = 0 if v(t) and i(t) are orthogonal waveforms in the interval T.

Application #4

1 1

T T

P p t dt v t i t dtT T

For an inductor,di

v Ldt

2

2 2

2

i T

i

L i Li T i

T T

1

i TT

i

di LP L i t dt idi

T dt T

i T i = 0 for any periodic function:

AC Power Concepts

•Source voltage waveform is assumed to be an undistorted sinusoid with zero phase angle.

•Due to reactance of the load, the current waveform may exhibit a phase shift with respect to the voltage waveform.

•Current waveforms may contain harmonic distortion components, which increases the RMS value of the current waveform, and hence the apparent power (but not real power).