COMPLEX POWER AND POWER TRIANGLE

Complex Power and Power Triangle

Complex power S is the product of the voltage and the complex conjugate of the current.

S = VrmsIrms Cos (θv - θi) + jVrms Sin (θv - θi) 

If S is the complex power then,

S = V . I* V is the phasor representation of voltage and I* is the conjugate of current phasor.So if V is the reference phasor then V can be written as |V| ∠0.

(Usually one phasor is taken reference which makes zero degrees with real axis. It eliminates the necessity of introducing a non zero phase angle for voltage)

Let current lags voltage by an angle φ, so I = | I | ∠-φ(current phasor makes -φ degrees with real axis) I*= | I | ∠φSo,

S = |V| | I | ∠(0+φ) = |V| | I | ∠φ (For multiplication of phasors we have considered polar form to facilitate calculation)Writting the above formula for S in rectangular form we get

S = |V| | I | cos φ + j |V| | I | sin φ 

Power Triangle


POWER TRIANGLE HAS THREE SIDES:-
1-APPARENT POWER (VA)
2-REACTIVE POWER (VAR)
3-REAL POWER (W)
THE RELATIONSHIP B/W THESE CAN BE EXPRESSED BY REPRESENTING QUANTITIES AS VECTORS.
REAL POWER IS HORIZONTAL VECTOR.
REACTIVE POWER IS VERTICAL POWER.
APPARENT POWER IS HYPOTENUSE OF RT.ANGLED TRIANGLE.

(apparent power)2 = (real power)2 + (reactive power)2

RATIO OF REAL POWER TO APPARENT POWER IS CALLED POWER FACTOR.
PF=W / VA 

APPARENT POWER AND POWER FACTOR

APPARENT POWER & POWER FACTOR

• Apparent factor S is the product of RMS values of voltage & current.

• It is measured in volt-amperes or VA to distinguish it from the average or real power which is measured in WATTS.

S = VrmsIrms 

• The apparent power is so called because it seems apparent that the power should be the voltage-current product, by analogy with dc resistive circuits. 

• Power factor is the cosine of the phase difference between the voltage & current. It is also the cosine of the angle of the load impedance if V is the voltage across the load and I is the current through it. 

Pf = P / S = cos ( ɸv-ɸi )

Maximum Average Power Transfer

MAXIMUM AVERAGE POWER TRANSFER

In maximizing the power directed by a power-supplying resistive network to a load RL. Representing the circuit by its Thevenin equivalent, we proved that the maximum power would be delivered to the load if the load resistance RL = RTH. An ac circuit is connected to a load ZL and is represented by its Thevenin equivalent. The load is usually represented by an impedance. 

In rectangular form the Thevenin Impedance ZTH, and the load impedance ZL are 

ZTH = RTH + jXTH

ZL = RL + jXL

The maximum average power can be transferred to the load if 

XL = - XTH

RL = RTH

It will give us a maximum average power as;

Pmax = (VTH)^2 / 8RTH

This means that for maximum average power transfer to a purely resistive load, the load impedance is equal to the magnitude of the Thevenin impedance.