Power determination using complex notation is a technique commonly used in the analysis of alternating current (AC) circuits. In AC circuits, the voltage and current can be sinusoidal functions that vary with time. Complex notation, also known as phasor notation, simplifies the analysis of AC circuits by representing sinusoidal quantities as complex numbers.
In complex notation, a sinusoidal voltage or current is represented as a phasor, which is a complex number. The phasor has a magnitude (amplitude) and an angle that represents the phase shift relative to a reference. The reference angle is usually taken as the positive horizontal axis (real axis) on the complex plane.
For a sinusoidal voltage or current, the complex representation is given by:
(
)
=
⋅
cos
(
+
)
=
Re
{
⋅
(
+
)
}
V(t)=V
m
⋅cos(ωt+ϕ
v
)=Re{V
m
⋅e
j(ωt+ϕ
v
)
}
where:
(
)
V(t) is the instantaneous voltage at time
t.
V
m
is the amplitude (peak value) of the voltage.
ω is the angular frequency of the AC signal (
=
2
ω=2πf, where
f is the frequency in hertz).
ϕ
v
is the phase angle of the voltage relative to the reference.
The power in an AC circuit can be determined using complex notation by calculating the complex power, which includes both the real (active) power and the reactive power. The complex power (
S) is defined as:
=
∗
S=VI
∗
where:
V is the complex phasor representing the voltage.
∗
I
∗
is the complex conjugate of the current phasor.
The real power (
P) is the active power dissipated in the circuit and is given by the real part of the complex power:
=
Re
{
}
=
Re
{
∗
}
P=Re{S}=Re{VI
∗
}
The reactive power (
Q) represents the exchange of energy between the circuit and the energy storage elements (inductors and capacitors). It is given by the imaginary part of the complex power:
=
Im
{
}
=
Im
{
∗
}
Q=Im{S}=Im{VI
∗
}
The apparent power (
∣
∣
∣S∣) is the magnitude of the complex power and represents the total power flowing in the circuit. It is the combination of real and reactive power:
∣
∣
=
2
+
2
∣S∣=
P
2
+Q
2
In summary, complex notation simplifies the calculation of power in AC circuits by converting sinusoidal quantities into phasors represented as complex numbers. The complex power includes both active and reactive components, allowing for a comprehensive understanding of the power flow and interactions in AC circuits.