How Complex Power S= VI*?

Derivation of Complex Power:

Here, v and i and steady state sinusoidal

signals.

( )

( )

Where,

is the Voltage phase angle.

is the Current phase angle.

Taking the instant at which the current is passing through a positive maximum as

reference. The Voltage and current can be re written as

( )

( )

By using the trigonometric identity,

( )

( )

Letting and , gives

( )

( )

Expanding the second term on the right hand side of the previous equation

By using the trigonometric identity,

( )

gives,

i

v

( )

( ) ( )

( ) ( )

Thus the Instantaneous Power can be written as,

( ) ( )

The Active Power is

( )

The Reactive Power is

( )

The Complex Power is defined as complex sum of real power and reactive power.

, Where, P = Re [S] and Q = Re[S]

( )

( )

[ ( ) ( )]

( )

( )

, Where, P = Re [S] and Q = Re[S]

Example:

Calculate the average power and the reactive power at the terminals of the

network shown in fig.

( )

( )

Determine whether the network inside the

box absorbing or delivering active power &

reactive power.

Method 1: (using Direct Formula)

( )

( ( ))

By Passive Sign Convention, Negative value of -100W indicates that the network

inside the box is delivering average power to the terminals.

( )

( ( ))

By Passive Sign Convention, Positive value of 173.21 VAR indicates that the

network inside the box is absorbing Magnetizing vars at its terminals.

i

Method 2: (using Complex Power Formula)

⌊

⌊

Therefore

( ⌊ )( ⌊ ) = 200 ⌊

S

From the above result, it is clearly shown that,

P = Re [VI*]

Q = Im [VI*]

P Q