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In an ac circuit, if voltage V = (a+jb) and current I = (c+jd), then power is given by
ac+ad
ac+bd
bcтИТad
bc+ad
ac+bd
In an AC circuit, the complex power S is defined as the product of the voltage phasor V and the complex conjugate of the current phasor IтИЧ. For V=a+jb and I=c+jd, the conjugate is IтИЧ=cтИТjd, so S=(a+jb)(cтИТjd)=(ac+bd)+j(bcтИТad).
In an AC circuit, the complex power S is defined as the product of the voltage phasor V and the complex conjugate of the current phasor IтИЧ. For V=a+jb and I=c+jd, the conjugate is IтИЧ=cтИТjd, so S=(a+jb)(cтИТjd)=(ac+bd)+j(bcтИТad).
S=VтЛЕIтИЧ
P=Re(S)=ac+bd
Q=Im(S)=bcтИТad
The real part of the complex power, P=Re(S), represents the active power dissipated in the circuit. Calculating the product (a+jb)(cтИТjd) results in acтИТjad+jbcтИТj2bd. Since j2=тИТ1, the term тИТj2bd becomes +bd, yielding a real component of ac+bd.
Complex power S is measured in Volt-Amperes (VA).
The real part P is the active power measured in Watts (W).
The imaginary part Q is the reactive power measured in Volt-Amperes Reactive (VAR).
Taking the conjugate of the current is essential to maintain the correct phase relationship for reactive power.
Allows unified analysis of resistive and reactive components.
Simplifies power flow calculations in steady-state AC systems.
Requires understanding of complex numbers and phasor notation.
Not applicable to non-sinusoidal waveforms without Fourier transformation.
Power system load flow analysis.
Design of electrical machines and transmission lines.
AC motor power factor correction.
The active power is given by the real part of the product of voltage and the current conjugate.
Option A (ac+ad) is dimensionally incorrect for power components.
Option C (bcтИТad) represents the reactive power Q in the circuit.
B is correct тАФ The real power in an AC circuit is defined by the real part of VтЛЕIтИЧ, which evaluates to ac+bd.
Always remember the complex conjugate: when multiplying phasors for power, conjugate the current (IтИЧ) to correctly account for the phase angle difference between voltage and current.