Complex Power Derivation

Complex Power Derivation:

Complex Power Derivation – Consider a single-phase load fed from a source as in Fig. 4.9. Let

Complex Power Derivation

When θ  is positive, the current lags behind voltage. This is a convenient choice of sign of θ in power systems where loads have mostly lagging power factors.

Complex power flow in the direction of current indicated is given by S =VI*

Complex Power Derivation

Here

S = complex power (VA, kVA, MVA)

|S| = apparent power (VA, kVA, MVA); it signifies rating of equipments (generators, transformers)

P = |V| |I| cos θ = real (active) power (watts, kW, MW) Q= |V| |I| sin θ = reactive power

= voltamperes reactive (VAR)

= kilovoltamperes reactive (kVAR)

= megavoltamperes reactive (MVAR)

It immediately follows from Eq. (4.17) that Q, the reactive power, is positive for lagging current (lagging power factor load) and negative for leading current (leading power factor load). With the direction of current indicated in Fig. 4.9, S = P + jQ is supplied by the source and is absorbed by the load.

Equation (4.17) can be represented by the phasor diagram of Fig. 4.10 where

Complex Power

Complex Power Derivation

If two (or more) loads are in parallel as in Fig. 4.11

Complex Power Derivation

As per Eq. (4.19), Kirchhoff’s current law applies to complex power (also applies separately to real and reactive powers).

In a series RL load carrying current I,

Complex Power Derivation

In case of a series RC load carrying current I,

Complex Power

Consider now a balanced three-phase load represented in the form of an equivalent star as shown in Fig. 4.12. The three-phase complex power fed into load is given by

Complex Power Derivation

Complex Power Derivation

Here

Complex Power Derivation

If VL, the line voltage, is expressed in kV; and IL, the line current in amperes, S is in kVA; and if the line current is in kiloamperes, S is in MVA.

In terms of load impedance Z,

Complex Power

Substituting for IL in Eq. (4.20)

Complex Power

If VL, is in kV, S is now given in MVA. Load impedance Z if required can be calculated from

Complex Power

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