Environmental Engineering Reference
In-Depth Information
subtraction, just take the difference of these numbers. When multiplying complex numbers
in polar form, simply multiply the polar magnitudes of the complex numbers to determine
the polar magnitude of the product and add the angles of the complex numbers to determine
the angle of the product. If dividing, just divide the magnitudes and subtract the angles. Meter
measurements in an AC circuit correspond to the polar magnitudes of calculated values.
Rectangular expressions of complex quantities in an AC circuit have no direct, empirical
equivalent, although they are convenient for calculations.
All the rules and laws of DC circuits, i.e Ohm's law, Kirchhoff's laws and network analysis
methods, apply to AC circuits as well, with the exception of power calculations. The only
qualifi cation is that all variables must be expressed in complex form, taking into account
phase as well as magnitude, and all voltages and currents must be of the same frequency (in
order that their phase relationships remain constant).
A.9 Reactance and Impedance
Now expressions will be developed that describe the 'resistance' or 'opposition' offered to
the fl ow of AC currents by the three types of circuit elements. This can be found by dividing
the phasor of voltage by the resulting current phasor. The voltage will be taken as the refer-
ence phasor and polar notation will be used in these derivations.
A.9.1 Resistance
Dividing Equations (A.4) by (A.5) gives
V
I
0 °
0
Opposition offered by resistance
=
=
R
ohm
(A.18)
°
The relationship here is very simple. To derive the current through a resistor just divide
the voltage phasor by the resistance. The current and the voltage are in phase.
A.9.2 Inductance
Dividing Equation (A.10) by (A.11),
V
∠−
0
90
°
V
VL
0
°
Opposition offered by inductance
=
=
(
) ∠−
I
°
ω
90
°
(
)
=
ω
L
0
°− −90 °=
X
90
°=
j
X
ohm
(A.19)
L
L
The quantity j X L = j
fL is known as inductive reactance and represents the 'opposi-
tion' to the fl ow of current when a voltage of frequency f Hz is applied across an inductance
of L henry. The 90 ° phase lag that occurs to the current with respect to the voltage in an
inductive element is conveniently incorporated in the description of the reactance through
the operator j. There is nothing 'imaginary' of course about a reactor that consists typically
of a coil of wire. However, as will be shown later, by assigning the j operator to the reactance
the calculations are enormously assisted.
ω
L = j2
π
 
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