Power determination using complex notation involves representing voltage, current, and impedance in complex form and using complex arithmetic to calculate power. In an alternating current (AC) circuit, the power can be divided into two components: real power (active power) and reactive power.
In complex notation, voltages, currents, and impedances are represented as complex numbers with both magnitude and phase angle components. The general complex form of a quantity is:
=
β
Z=Aβ ΞΈ
Where:
Z is the complex quantity (voltage, current, or impedance).
A is the magnitude of the complex quantity.
ΞΈ is the phase angle in degrees.
The complex power (
S) is defined as:
=
β
β
S=Vβ
I
β
Where:
V is the complex voltage.
β
I
β
is the complex conjugate of the current (
I).
Real power (
P) is the average power delivered to the circuit, and it is the real part of the complex power:
=
Re
(
)
=
β
β
β
cos
β‘
(
β
)
P=Re(S)=Vβ
I
β
β
cos(ΞΈ
v
β
βΞΈ
i
β
)
Where:
Re
(
)
Re(S) represents the real part of
S.
ΞΈ
v
β
is the angle of the voltage complex quantity.
ΞΈ
i
β
is the angle of the current complex quantity.
Reactive power (
Q) is the power that flows back and forth between the source and reactive components of the circuit, and it is the imaginary part of the complex power:
=
Im
(
)
=
β
β
β
sin
β‘
(
β
)
Q=Im(S)=Vβ
I
β
β
sin(ΞΈ
v
β
βΞΈ
i
β
)
Where:
Im
(
)
Im(S) represents the imaginary part of
S.
Apparent power (
β£
β£
β£Sβ£) is the magnitude of the complex power and represents the total power supplied to the circuit:
β£
β£
=
2
+
2
β£Sβ£=
P
2
+Q
2
β
Power factor (
PF) is the ratio of real power to apparent power and is a measure of the efficiency of the circuit:
=
β£
β£
PF=
β£Sβ£
P
β
In summary, complex notation allows us to represent AC circuit quantities in a concise and mathematically elegant way, and it enables the determination of real power, reactive power, apparent power, and power factor using trigonometric functions and complex arithmetic. This notation is particularly useful in analyzing and designing AC circuits, especially those with reactive components like inductors and capacitors.